Analysis of Beams on Elastic Half-Space - J. Ross Publishing · Analysis of Beams on Elastic...
Transcript of Analysis of Beams on Elastic Half-Space - J. Ross Publishing · Analysis of Beams on Elastic...
179
4Analysis of Beams on
Elastic Half-Space
4.1 IntroductionAs mentioned earlier in Chapter 1, some soils under applied pressure behave as elastic material following the rules of the theory of elasticity, and soil models such as elas-tic half-space and elastic layer may produce more realistic practical results compared to results obtained by using Winkler foundation. But analysis of beams supported on elastic half-space is a difficult problem and few works are devoted to this subject. Most known publications are scientific and research papers written by academics and researchers that cannot be used by practicing engineers. Nevertheless, there are some methods that allow performing analysis of simple beams supported on elastic half-space as well as on elastic layer. These methods can be used for hand calculations and computer analysis as well. Chapter 4 presents equations and tables for analysis of free supported beams on elastic half-space developed by Borowicka (1938, 1939) and Gorbunov-Posadov (1940, 1949, 1953, 1984). Theoretical bases of the method are not discussed in this book. The reader can find a detailed explanation of the method in the original publications mentioned above and a good review of the method by Selvadurai (1979). This chapter also includes the analysis of complex beams, such as beams with various boundary conditions, various continuous beams, stepped beams developed by the author of this book (Tsudik 2006), and analysis of frames on elastic half-space including direct methods as well as iterative methods. Tables are developed for beams that meet the following two requirements: a/b ≥ 7 and the width of the beam is nar-row enough so the bending in the transverse direction can be ignored; a is half of the length of the beam and b is half of the width of the beam, as shown in Figures 4.1 and 4.2. Tables are developed for three categories of beams: 1. rigid beams, 2. short beams, and 3. long beams.
In order to find out if the beam belongs to rigid beams, the following parameter is obtained:
/t E a b E I2 10
3 21
r o= -` j: D (4.1)
All tables can be found at the end of the chapter.
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180 Analysis of Structures on Elastic Foundations
where E0 is the modulus of elasticity of the soil, o is Poisson’s ratio of the soil, E1 is the modulus of elasticity of the beam material, and I is the moment of inertia of the beam. The beam belongs to rigid beams when t ≤ 0.5. The beam belongs to rigid beams also when a = a/b < 20 and 0.5 ≤ t ≥ 1.
In order to find out if the beam belongs to long beams, parameter L of the system
beam-soil is found as follows: ( )
Lb E
E I2 1
0
12
3o
=-
l where b′ = 2b and coefficients m and
b are obtained as Lam = and
Lbb = . The beam belongs to long beams when b < 0.15
and m > 1 or when b ≤ 0.30 and m > 2 or when b ≤ 0.50 and m > 3.5. If parameters of the beam do not meet requirements for rigid and long beams, the beam belongs to the short beams and analysis of such beams is performed using tables for analysis of short beams. Tables for analysis of short and long beams are shown at the end of the chapter. Now, we can start with the analysis of rigid beams. Tables and equations presented in Chapter 4 are developed only for analysis of simple free supported beams.
The author of this book (Tsudik 2006) proposed to use the method of forces for analysis of complex beams that includes simple beams with various boundary con-ditions and continuous beams, including beams with various intermediate and end supports.
4.2 Analysis of Rigid BeamsTables 4.1–4.5 are developed for analysis of rigid beams on elastic half-space for two types of loads: concentrated vertical loads and moments applied to the center of the beam. If a concentrated vertical load P0 is applied to the center of the beam, the fol-lowing equations are used:
ppb aP
0
0
$=
l, , , ,M MQ P a Y Y
E aP
tgQ P 1 00 0 0 0 0
0
20! $ $ $ $o {== =
-= (4.2)
Figure 4.1
Figure 4.2
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Analysis of Beams on Elastic Half-Space 181
In equations 4.2, p is the soil pressure, Q is the shearing force, M is the moment, and Y is the settlement of the beam. Coefficients p
0, Q
0, M
0, and Y 0 are dimensionless and
taken from Table 4.2. Rotation of the beam for this type of load, tg{ = 0. If the beam is loaded with a concentrated moment m0 equations 4.3 are used:
1 1, ,
,
Qam
M m
Y tgE
xam
tg tgE a
m
ppb am
1 1
00
10
2
20
10
2
0
0
1 20 !
! $
!
{ o { { o
= =
=-
=-
= Q Ml
_
`
a
bbb
bb
(4.3)
In these equations, x = x′/a, where x′ is the distance of the cross section from the cen-ter of the beam and is always positive regardless of where the cross section is located, whether at the left or right half of the beam. Two signs (±) mean that the upper sign belongs to the right half and the lower sign belongs to the left half of the beam. Tables 4.2 and 4.4 do not contain information for beams with 7 ≤ a < 10. In order to obtain the soil pressure, shearing forces, and moments in this case, it is recommended to use data for a = 10. However, settlements and rotations of the beam, when a < 10, cannot be obtained from these tables and have to be found from equations 4.4 shown below:
E
KAP
tgE a
K m
1
1
p
x
0
2
00
0
2
31$
~ o
{ o
=-
=-
_
`
a
bbb
bb
(4.4)
In 4.4, coefficient K0 is dimensionless and is obtained from Table 4.1; A is the area of the foundation. Coefficient K1 is also dimensionless and obtained from Table 4.1; Kx is the moment applied to the beam in the direction of the length. Table 4.1 is built using specified values of these two coefficients. Equations 4.4 are obtained for totally rigid individual rectangular foundations and used here for beam analysis when a < 10. Analysis of the beam is performed as follows. Each half of the beam is divided into 10 sections, as shown in Figure 4.3. Since the loads are symmetrical or asymmetrical, coef-ficients are given only for the right half of the beam starting from point 0.
If the beam has a series of vertical concentrated loads, they have to be replaced with one load and one moment applied to the center of the beam. If the beam has
Figure 4.3
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182 Analysis of Structures on Elastic Foundations
a distributed load located at any part of the beam, this load is replaced with a series of concentrated vertical loads and all these loads, in turn, are also replaced with one moment and one vertical load applied to the center of the beam. The total moment applied to the center of the beam m0 is equal to the sum of all moments produced by vertical loads Pi and moments mi applied to the beam. The total vertical load P0 is equal to the sum of all vertical loads Pi including distributed loads q(x). The moment is positive if applied clockwise. When P0 and m0 are found, the soil pressure is obtained as follows:
0 a ap pp
P m0
1 20!=
b bl l (4.5)
Coefficients p0 and p
1 are obtained from Tables 4.2 and 4.4, respectively, taking into
account the actual value of parameters a = a/b and x = x′/a at each point of the beam. Vertical concentrated loads Pi, distributed loads q(x), and moments applied to the beam will produce shear forces Q(x). The shearing force at any point of the beam is obtained from equation 4.6:
Q QQ x Pam
Q0 0 1
0ext
!= + -` j (4.6)
where Q0 and Q
1 are found from Tables 4.2 and 4.4, respectively, and Qext for the
right half of the beam is equal to P Px 0-/ . For the left half of the beam Q Pxext
=/ , where Px/ includes all vertical loads located at distance x between the left end of the beam and the point the shear is obtained. Vertical concentrated loads Pi, distributed loads q(x), and moments mi will also produce moments that are obtained from equa-tion 4.7:
M M P a M m M0 0 1 0 ext!= + (4.7)
In 4.7 coefficients M0 and M
1 are obtained from Tables 4.2 and 4.4, respectively. Mext
for the right half of the beam are found from equation 4.8:
M M m P ax mx x 0 0ext= + + -// (4.8)
where Mx/ is the sum of all moments produced by all vertical concentrated (Pi) and distributed loads q(x) located at left from point x; mx/ is the sum of all concentrated moments applied to the beam and located at left from point x.
For the left half of the beam:
M M mx xext= +// (4.9)
The settlement of the beam is obtained from equation 4.10:
gY Y P t xam
aE1
0 0 10
0
2
! {o
=-
e`o
j (4.10)
The soil pressure, shear forces, moments, settlements, and rotations of the beam can be found using equations presented above and Tables 4.1–4.5. A simple numerical ex-ample that illustrates analysis of rigid beams supported on elastic half-space is shown next.
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Analysis of Beams on Elastic Half-Space 183
Example 4.1Given: A reinforced concrete beam loaded with a vertical load P = 200t is shown in Figure 4.4. The length of the beam is 2a = 10m, the cross section is equal to b′ ∙ h = 2m ∙ 1.5m(h) = 3m2 (b′ = 2b), and the moment of inertia of the beam section is equal
to h . .I12 12
2 1 5 0 5625m3 3
3$= = =
bl . The beam is loaded with a vertical concentrated
load equal to 200t. The modulus of elasticity of the soil is E = 300kg/cm2 = 3000t/m2; Poisson’s ratio m = 0.4 and the modulus of elasticity of the concrete is E1 = 3,000,000t/m2. Find the soil pressure, shearing forces, and moments. Build the soil pressure, shear, and moment diagrams.
Figure 4.4
Solution:
1. Find parameter t = rEa3b/[2(1 − m2)EI] = 3.14 ∙ 3,000 ∙ 53 ∙ 1/(1.68 ∙ 0.5625 ∙ 3,000,000) = 0.415. Since t < 0.5, the beam is rigid.
2. Find ba
15 5a = = = and coefficients p
0 from Table 4.2. As can be seen, the
closest parameter a = 10. So all coefficients are obtained for a = 10. The soil pressure diagram is shown in Figure 4.5.
3. Find the soil pressure. The soil pressure is obtained, for convenience, only in all other points of the beam.
. ./ /pp P b a 0 439 200 1 5 17 56t/m0 0 0
$ $ $= = = ; p0.1 = 0.440 ∙ 40 = 17.6t/m; p0.2 = 0.442 ∙ 40 = 17.68t/m; p0.3 = 0.446 ∙ 40 = 17.84t/m; p0.4 = 0.445 ∙ 40 = 18.2t/m; p0.5 = 0.462 ∙ 40 = 18.48t/m; p0.6 = 0.475 ∙ 40 = 19t/m; p0.7 = 0.498 ∙ 40 = 19.92t/m; p0.8 = 0.541 ∙ 40 = 21.64t/m; p0.9 = 0.632 ∙ 40 = 25.28t/m; p1.0 = 0.842 ∙ 40 = 33.68t/m
Figure 4.5 The soil pressure diagram
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184 Analysis of Structures on Elastic Foundations
The total soil pressure applied to the beam is equal to:
. 198.2p p0 5 tip
p
0.
.
0 1
1 0
+ =/ , that is close to the load applied to the beam P0 = 200t. The
difference is only 0.9%. The shear diagram is built analogously, as shown in Figure 4.6. Shear forces are found below using the second equation from 4.2:
.QQ P 0 5 200 100t0 0 0
$ $=- =- =- ; Q0.1 = −0.456 ∙ 200 = −91.2t; Q0.2 = −0.412 ∙ 200 = −52.4t; Q0.3 = −0.367 ∙ 200 = −73.4t; Q0.4 = −0.322 ∙ 200 = −64.4t; Q0.5 = −0.277 ∙ 200 = −55.4t; Q0.6 = −0.230 ∙ 200 = −46.0t; Q0.7 = −0.182 ∙ 200 = −36.4t; Q0.8 = −0.130 ∙ 200 = −26.4t; Q0.9 = −0.072 ∙ 200 = −14.4t; Q1.0 = 0.
Obtained moments and the moment are shown below. The moment diagram is shown in Figure 4.7.
. .MM P a 0 2703 200 5 270 3tm0 0 0
$ $ $ $= = = ; M0.1 = 0.2225 ∙ 200 ∙ 5 = 222.5tm; M0.2 = 0.1791 ∙ 200 ∙ 5 = 179.1tm; M0.3 = 0.1401 ∙ 200 ∙ 5 = 140.1tm; M0.4 = 0.1056 ∙ 200 ∙ 5 = 105.6tm; M0.5 = 0.0756 ∙ 200 ∙ 5 = 75.6tm; M0.6 = 0.0502 ∙ 200 ∙ 5 = 50.2tm; M0.7 = 0.0295 ∙ 200 ∙ 5 = 29.5tm; M0.8 = 0.0139 ∙ 200 ∙ 5 = 13.9tm; M0.9 = 0.0037 ∙ 200 ∙ 5 = 3.7tm; M1.0 = 0
Taking into account that a = 5 < 10, the settlement of the beam is obtained from the first equation 4.4:
. .,
.E
KAP1
3 0001 0 4 0 77
10 2200 0 964cmp
0
2
00
2
$ $$
~ o=-
=-
=
The soil pressure, shear, and moment diagrams are shown in Figures 4.5, 4.6, and 4.7, respectively.
Figure 4.6 Shear diagram
Figure 4.7 Moment diagram
SAMPLE C
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Analysis of Beams on Elastic Half-Space 185
Example 4.2The given beam is loaded with asymmetrical loads as shown in Figure 4.8, assuming P1 = 100t M1 = 10t 3x m
1=l x 2m
2=l . Find the shear forces and rotation of the beam.
Figure 4.8
Solution:
1. The total moment applied to the center of the beam is equal to:
175tm25 100m M P a x 5 30 1= - - = - - =-l` `j j
2. The total vertical concentrated load applied to the center of the beam is equal to:
P0 = P = 100t
3. The shear forces are obtained below. For the right half of the beam we have:
. ( . )( )
.Q Q P Qam
Q 0 5 100 0 7095175
25 185t0 0 0 1
0ext
$ $= + - =- + --
=-
. ( . )( )
.Q Q P Qam
Q 0 456 100 0 7045175
20 96t.0 1 0 0 1
0ext
$ $= + - =- + --
=-
. ( . )( )
.Q Q P Qam
Q 0 412 100 0 6865175
17 19t.0 2 0 0 1
0ext
$ $= + - =- + --
=-
. ( . )( )
.Q Q P Qam
Q 0 367 100 0 6585175
13 67t.0 3 0 0 1
0ext
$ $= + - =- + --
=-
. ( . )( )
.Q Q P Qam
Q 0 322 100 0 6175175
10 605t.0 4 0 0 1
0ext
$ $= + - =- + --
=-
. ( . )( )
.Q Q P Qam
Q 0 277 100 0 5635175
7 995t.0 5 0 0 1
0ext
$ $= + - =- + --
=-
. ( . )( )
.Q Q P Qam
Q 0 230 100 0 4965175
5 64t.0 6 0 0 1
0ext
$ $= + - =- + --
=-
SAMPLE C
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186 Analysis of Structures on Elastic Foundations
. ( . )( )
.Q Q P Qam
Q 0 182 100 0 4135175
3 545t.0 7 0 0 1
0ext
$ $= + - =- + --
=-
. ( . )( )
.Q Q P Qam
Q 0 130 100 0 3115175
2 115t.0 8 0 0 1
0ext
$ $= + - =- + --
=-
. ( . )( )
.Q Q P Qam
Q 0 072 100 0 1805175
0 900t.0 9 0 0 1
0ext
$ $= + - =- + --
=-
Q1 = 0
For the left half of the beam:
. ( . )( )
.Q Q P Qam
Q 0 5 100 0 7095175
25 185100 t0 0 0 1
0ext
$ $= + - = --
- =-
. ( . )( )
.Q Q P Qam
Q 0 456 100 0 7045175
100 29 76t.0 1 0 0 1
0ext
$ $= + - = --
- =--
. ( . )( )
.Q Q P Qam
Q 0 412 100 0 6865175
100 35 79t.0 2 0 0 1
0ext
$= + - = + --
- =--
.100 40 27t=-. ( . )( )
Q Q P Qam
Q 0 367 100 0 6585175
.0 3 0 0 1
0ext
$= + - = + --
--
. ( . )( )
.Q Q P Qam
Q 0 322 100 0 6175175
100 46 205t.0 4 0 0 1
0ext
$= + - = + --
- =--
. ( . )( )
.Q Q P Qam
Q 0 277 100 0 5635175
47 401t.0 5 0 0 1
0ext
$= + - = + --
=-
. ( . )( )
.Q Q P Qam
Q 0 230 100 0 4965175
40 36t.0 6 0 0 1
0ext
$= + - = + --
=-
. ( . )( )
.Q Q P Qam
Q 0 182 100 0 4135175
32 655t.0 7 0 0 1
0ext
$= + - = + --
=-
. ( . )( )
.Q Q P Qam
Q 0 130 100 0 3115175
23 885t.0 8 0 0 1
0ext
$= + - = + --
=-
. ( . )( )
.Q Q P Qam
Q 0 072 100 0 1805175
13 5t.0 9 0 0 1
0ext
$= + - = + --
=-
Q−1.0 = 0.
Taking into account that a = 5 < 10, rotation of the foundation is obtained using the second equation from 4.4.
The shear diagram is shown in Figure 4.9.
. . .,
tgE a
K m13 000
1 0 4 1 452175
0 00888x
0
2
31
2
3$ $ ${ o=-
=- -
=-` j
m0 = mx = −175tm. K1 = 1.45 is obtained from Table 4.1 for a = 5.
SAMPLE C
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Analysis of Beams on Elastic Half-Space 187
4.3 Short Beam AnalysisAnalysis of short beams is performed using Tables 4.6–4.13 that are developed for analysis of beams with concentrated vertical loads P and moments m. Table 4.14 is developed for analysis of short beams with a uniformly distributed load q along the length of the beam. The beam is divided into 20 sections of equal length. The length of each section is equal to 0.1 a where a is half of the beam’s length, as shown previ-ously in Figure 4.3. Although the tables are developed for beams with a/b = 10, they produce good practical results when 7 ≤ a/b ≥ 15.
Tables 4.6–4.9 and equations 4.11 allow obtaining moments, soil pressures, shear forces, and settlements of the beam due to applied vertical load P. These equations look as follows:
p ~,M MPa pb aP Q QP
E aP1
0
2
~o
= = = =-
l
` j
(4.11)
In order to use Tables 4.6–4.9 and equations 4.11, we have to obtain parameter t from
equation 4.1 and two other parameters add = and x x
a=l , where d is the distance
between the point the load is applied and the center of the beam, and x′ is the distance from the center of the beam and the point in which M, p, Q, and ~ are obtained. It is important to mention that coefficients M and Q have to be multiplied by 10−3; coef-ficients p and ~ have to be multiplied by 10−2.
Tables 4.10–4.13 and equations 4.12 allow calculating moments, shear forces, soil pressures, and settlements of the beam due to applied concentrated moments m:
a~p, , ,M Mm Q Q
am p
bm
E am1
20
2
2
~o
= = = =-
l
` j
(4.12)
Figure 4.9 Shear diagram
SAMPLE C
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188 Analysis of Structures on Elastic Foundations
The use of Tables 4.10–4.13 and equations 4.12 requires the following data: parameter
t that is obtained from equation 4.1, parameters add = and x x
a=l , where d is the distance
between the center of the beam and the point the moment m is applied, and x′ is the distance between the center of the beam and the point in which M, Q, p, and ~ have to be found. If the distance d is not divisible by 0.1a, the concentrated load (vertical load or moment) should be moved to the closest point the beam is divided. All tables are developed for the following numerical values of t: 1, 2, 5, and 10. If t < 0.75 analy-sis is performed using tables for rigid beams (Tables 4.1–4.5), when t > 10 analysis is performed using Tables 4.15–4.19 for analysis of long beams.
All numerical values of M and Q in the tables, for convenience, are multiplied by 1000 and all numerical values of p and Y are multiplied by 100. If, for example, in tables for M or Q the values are shown as equal to −045, the actual value of this coef-ficient is equal to −0.045, while in tables for p and Y it is equal to −0.45.
In Tables 4.6–4.9, the shearing force Q is shown in bold at all points of the beam, where the vertical load P is applied. This shear is equal to the shear QL
at left from the point the load is applied. The shear at right from this point is found as follows:
R LQ Q 1= - . When the beam is loaded at any point with a moment m as shown in tables 4.10–4.13, the moment shown bold is the moment at left from the point where the moment is applied and is equal to ML . The moment at right from that point is equal to M M 1L +=R . Two numerical examples illustrate the use of Tables 4.6–4.9 and equations 4.11 and Tables 4.10–4.13 and equations 4.12.
Example 4.3The given beam loaded with a vertical concentrated load is shown in Figure 4.10. The data are: the modulus of elasticity of the soil E0 = 3,000t/m2, the total length of the beam 2a = 12m, b′ = 2b = 1.6m, Poisson’s ratio o = 0.40, the modulus of elasticity of the beam material E1 = 2,600,000t/m2, the height of the beam h = 1m, the load P = 100t applied to the beam, and x′ = 2.4m at right from the center. Build the moment and shear diagrams.
Figure 4.10
To find out the category the beam belongs to, we find parameter t:
. . . . . .
/
, / , ,
E a b E I2 1
3 14 2 300 6 0 8 1 68 2 600 00012
1 6 1 2 14 0 5
t 03 2
1
33
$ $ $ $ $ $ 2
r o= -
= =
` j: D
SAMPLE C
HAPTER
Analysis of Beams on Elastic Half-Space 189
Since t = 2.14 > 0.5, the beam is not rigid. Since . ,
, , .
.L1 6 2 300
2 3 000 00012
1 6 1
6 013
3
3
$
$ $
= = ,
and b = b/L = 0.8/6.013 = 0.133 < 0.15, but m = a/L = 6/6.013 = 0.997 < 1, the beam belongs to short beams.
Solution:
Using Table 4.7 for t = 2 and d = d/a = 2.4/6 = 0.4, we can find the moments, shear forces, soil pressure, and settlements in all 21 points of the beam. Below, the moments are obtained using the first equation from 4.11:
.M MPa M M100 6 10 0 63$ $ $= = =- .M−1 = 0, M−0.9 = 0.6tm, M−0.8 = −1.2tm, M−0.7 = −1.8tm, M−0.6 = −2.4tm, M−0.5 = −1.8tm, M−0.4 = 0, M−0.3 = 3.0tm, M−0.2 = 8.4tm, M−0.1 = 16.2tm, M−0.0 = 27.0tm, M0.1 = 40.80tm, M0.2 = 58.20tm, M0.3 = 79.20tm, M0.4 = 105.0tm, M0.5 = 74.40tm, M0.6 = 49.20tm, M0.7 = 28.80tm, M0.8 = 13.80tm, M0.9 = 3.60tm, M1.0 = 0.
The moment diagram is shown in Figure 4.11.The shear diagram can be built analogously. Shear forces are obtained using the
third equation from 4.11:
.Q QP Q Q100 10 0 13$= = =-
Q−1 = 0, Q−0.9 = −11 ∙ 0.1 = −1.1t, Q−0.8 = −15 ∙ 0.1 = −1.5t, Q−0.7 = −11 ∙ 0.1 = −1.1t, Q−0.6 = 0 ∙ 0.1 = 0, Q−0.5 = 17 ∙ 0.1 = 1.7t, Q−0.4 = 42 ∙ 0.1 = 4.2t, Q−0.3 = 72 ∙ 0.1 = 7.2t, Q−0.2 = 109 ∙ 0.1 = 10.9t, Q−0.1 = 152 ∙ 0.1 = 15.2t, Q0 = 202 ∙ 0.1 = 20.2t, Q0.1 = 258 ∙ 0.1 = 25.8t, Q0.2 = 320 ∙ 0.1 = 32.0t, Q0.3 = 338 ∙ 0.1 = 33.8t, Q0.4 = 461 ∙ 0.1 = 46.1t, Q0.5 = −462 ∙ 0.1 = −46.2t, Q0.6 = −382 ∙ 0.1 = −38.2t, Q0.7 = −299 ∙ 0.1 = −29.9t, Q0.8 = −213 ∙ 0.1 = −21.3t, Q0.9 = −117 ∙ 0.1 = −11.7t, Q1 = 0.
The shear diagram is shown in Figure 4.12. Numerical values of the moments and shear forces are found for all 21 points of the beam, but shown, for convenience, in both diagrams only at all other points.
Figure 4.11 Moment diagram
SAMPLE C
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190 Analysis of Structures on Elastic Foundations
Example 4.4The given beam is shown in Figure 4.13. Data: t = 2,
.
. .ad
5 42 7 0 5d = = = , m = 240tm.
Find the soil pressure and settlements of the beam.
Figure 4.13
By introducing the given numerical data into equations 4.12 we find:
p p p
~~ ~
, ,.
.. .
,
.. .
.
,
QM M Q QM m Qam p
b am
E am
2405 4240 44 44
1 4 5 4240 5 88
12 100 5 40 84 240 0 0032921
2 2
02
2
2
$
$$~
o
= = = = = = = =
=-
= =
l
` j
Now, the soil pressure and settlements of the beam can be found.
p−1.0 = −5.88 ∙ 2.1 = −12.3480t/m, p−0.9 = −5.88 ∙ 1.43 = −8.4080t/m, p−0.8 = −5.88 ∙ 1.08 = −6.3504t/m, p−0.7 = −5.88 ∙ 0.87 = −5.1156t/m, p−0.6 = −5.88 ∙ 0.73 = −4.2924t/m, p−0.5 = −5.88 ∙ 0.60 = −3.5280t/m, p−0.4 = −5.88 ∙ 0.49 = −2.8800t/m, p−0.3 = −5.88 ∙ 0.38 = −2.2344t/m, p−0.2 = −5.88 ∙ 0.27 = −1.5876t/m, p−0.1 = −5.88 ∙ 0.17 = −1.0000t/m, p−0.0 = −5.88 ∙ 0.07 = −0.4116t/m, p0.1 = 5.88 ∙ 0.04 = 0.2352t/m, p0.2 = 5.88 ∙ 0.15 = 0.8820t/m, p0.3 = 5.88 ∙ 0.28 = 1.6464t/m, p0.4 = 5.88 ∙ 0.43 = 2.5284t/m, p0.5 = 5.88 ∙ 0.58 = 3.4104t/m, p0.6 = 5.88 ∙ 0.75 = 4.4100t/m, p0.7 = 5.88 ∙ 0.95 = 5.5860t/m, p0.8 = 5.88 ∙ 1.21 = 7.1148t/m, p0.9 = 5.88 ∙ 1.60 = 9.4080t/m, p1.0 = 5.88 ∙ 2.32 = 13.6416t/m.
Settlements of the beam are found analogously:
Figure 4.12 Shear diagram
SAMPLE C
HAPTER
Analysis of Beams on Elastic Half-Space 191
..
. .,
mE a
m0 329
12 100 5 40 84 240 0 0032921 cm
02
2
2$$ ~~ ~
o~ ==
-= =
` j
~−1.0 = −0.32921 ∙ 0.87 = −0.286cm, ~−0.9 = −0.32921 ∙ 0.90 = −0.296cm, ~−0.8 = −0.32921 ∙ 0.94 = −0.395cm, ~−0.7 = −0.32921 ∙ 0.96 = −0.316cm, ~−0.6 = −0.32921 ∙ 0.98 = −0.322cm, ~−0.5 = −0.32921 ∙ 0.99 = −0.326cm, ~−0.4 = −0.32921 ∙ 1.00 = −0.329cm, ~−0.3 = −0.32921 ∙ 0.98 = −0.322cm, ~−0.2 = −0.32921 ∙ 0.94 = −0.395cm, ~−0.1 = −0.32921 ∙ 0.87 = −0.286cm, ~0.00 = −0.32921 ∙ 0.76 = −0.250cm, ~0.1 = −0.32921 ∙ 0.60 = −0.198cm, ~0.2 = −0.32921 ∙ 0.39 = −0.128cm, ~0.3 = −0.32921 ∙ 0.12 = −0.0395cm, ~0.4 = 0.32921 ∙ 0.23 = 0.076cm, ~0.5 = 0.32921 ∙ 0.67 = 0.221cm, ~0.6 = 0.32921 ∙ 1.14 = 0.375cm, ~0.7 = 0.32921 ∙ 1.58 = 0.520cm, ~0.8 = 0.32921 ∙ 2.02 = 0.665cm, ~0.9 = 0.32921 ∙ 2.44 = 0.665cm, ~1.0 = 0.32921 ∙ 2.87 = 0.945cm.
Soil pressure and settlements’ diagrams are shown in Figures 4.14 and 4.15, respec-tively. Numerical values of the soil pressure and settlements, for convenience, are not shown at all points of the beam. Moment and shear diagrams can be built analogously.
Figure 4.14 Soil pressure diagram
Figure 4.15 Diagram of the settlements
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192 Analysis of Structures on Elastic Foundations
4.4 Analysis of Short Beams with Uniformly Distributed LoadsTables and equations for such type of analysis are developed only for beams loaded completely from the left to the right ends, as shown in Figure 4.16.
Figure 4.16
Equations for obtaining M, Q, p, and Y are shown below:
p, ,,M aq p q Y YE
qM a q Q Q 12
0
2o= = =
-= (4.13)
Coefficients M , Q , p , and Y are taken from Table 4.14. Since the load applied to the beam is symmetrical, all coefficients are given only for the right half of the beam. Coefficient Q for the left half of the beam should be taken with the opposite sign.
Parameter x has the same meaning as in the tables shown earlier and is equal to xax
=l ,
where x′ is the distance from the center of the beam and the point coefficients M , Q , p , and Y are found. A numerical example is shown below.
It is important to mention that rotations of short beams at all points, except the left and right ends, are found from the following formula: tg{ = (Yi+1 − Yi−1)/0.2a. Rotations of the beam at the ends are found as follows:
tg{ = (Yi − Yi−1)/0.1a
where Yi is the settlement of the beam at point i.
Example 4.5The given beam is 10m long, parameter t = 10m. The beam is loaded with a uniformly distributed load q = 100t/m, coefficient t = 10. Build the moment and soil pressure diagrams.
Table 4.14 and equations 4.13 are used to build the moment and soil pressure dia-grams shown in Figures 4.17 and 4.18, respectively. Because of the beam symmetry both diagrams are built only for the right half of the beam.
Let us check the equilibrium of vertical loads applied to the beam. The total uni-formly distributed load applied to the beam is equal to 100 ∙ 5 = 500t. The total soil pressure is equal to:
98 ∙ 0.25 + 2 ∙ 97 ∙ 0.5 + 3 ∙ 96 ∙ 0.5 + 93 ∙ 0.5 + 95 ∙ 0.5 + 100 ∙ 0.5 + 110 ∙ 0.5 + 134 ∙ 0.25 = 498t
As shown, the difference between the total applied load and total soil pressure is neg-ligible: (498 ≈ 500).
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Analysis of Beams on Elastic Half-Space 193
4.5 Analysis of Long BeamsAs mentioned earlier, the beam belongs to the category of long beams when:
b < 0.15 and m > 1 or b ≤ 0.3 and m > 2 or b ≤ 0.5 and m > 3.5, where Lbb = and
aL
m = 1
Lb E
E I2
0
12
3m
=-
l
` j. Tables are developed only for analysis of beams
with concentrated vertical loads. If the beam is loaded with distributed loads, these loads have to be replaced with a series of concentrated vertical loads. When a moment is applied to the beam, the beam should be analyzed as a short beam. Analysis is per-formed using Tables 4.17–4.21 and the following equations:
p , , ,pLP M MPL Q QP Y Y
E LP1
0
2
$m= = = =-
(4.14)
Coefficients p , M , Q , and Y are obtained from Tables 4.17–4.21 that are developed for the following numerical values of b: 0.025, 0.075, 0.15, 0.3, 0.5. Analysis is per-formed in several steps:
Step 1: Choose the right table for analysis taking into account the following:
When 0.01 ≤ b ≤ 0.04, use Table 4.17 for b = 0.025.When 0.04 < b ≤ 0.10, use Table 4.18 for b = 0.075.
Figure 4.17 Moment diagram
Figure 4.18 Soil pressure diagram
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194 Analysis of Structures on Elastic Foundations
When 0.10 < b ≤ 0.20, use Table 4.19 for b = 0.15.When 0.20 < b ≤ 0.40, use Table 4.20 for b = 0.30.When 0.40 < b ≤ 0.70, use Table 4.21 for b = 0.50.
If the beam has several vertical loads Pi, as shown in Figure 4.19, analysis is performed for each load and final results are obtained by superposition. Analysis of long beams is based on analysis of infinite and semi-infinite beams.
Figure 4.19
Step 2: After the table for analysis is chosen, the so-called conditional distances be-tween each load and the left and right ends of the beam have to be found. Conditional distances, ali and ari, are obtained as follows:
Ld
Ld
lili
riria a= =
where dli and dri are the actual distances between the load and the left and right ends, respectively. Both types of distances are shown in Figure 4.19. When performing anal-ysis of long beams two cases are possible:
1. When b ≤ 0.2 and one of two values (ali, ari) is less than 1, or when b > 0.2 and one of the same values is less than 2. In this case analysis of the beam is per-formed using the smallest of the two a. The beam is analyzed as a semi-infinite beam.
2. When b ≤ 0.2 and one of two values (ali, ari) is larger than 1, or when b > 0.2 and one of the same two values is larger than 2. In this case analysis is per-formed using a = ∞. The beam is analyzed as an infinite beam.
Example 4.6Given: A beam shown in Figure 4.20, L = 2m b = 0.075 al = 0.4 ar = 2.2 E0 = 180kg/cm2 P = 60t o0 = 0.3 l = 6.2m.
Figure 4.20
SAMPLE C
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Analysis of Beams on Elastic Half-Space 195
Find the soil pressure and settlements of the beam.
Solution:
p p ppLP
20060000 300= = = . .
,Y YY
E LP1
1801 0 3
20060 000
1 50
02 2
$m
=-
=-
=
Analysis is performed in Table 4.15.The soil pressure and settlements’ diagrams are shown in Figures 4.21 and 4.22,
respectively. The soil pressure is shown in kg/m, the settlements in cm.
Figure 4.21 Soil pressure diagram
Figure 4.22 Settlements diagram
Example 4.7The given beam is shown in Figure 4.23.
L = 3m, b = 0.3 . .33 6 1 2la = = , . .
34 2 1 4ra = = , Q QQ P 80$= =
Find the shear forces and build the shear diagram.Analysis is performed in Table 4.16. Using Table 4.20 for b = 0.3 and al = 1.2, nu-
merical values of Q are found and final shear forces are obtained. The shear diagram is shown in Figure 4.24. The soil pressure, moments, and settlements of the beam can be obtained analogously.
SAMPLE C
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196 Analysis of Structures on Elastic Foundations
4.6 Analysis of Complex BeamsAll tables presented and used in this chapter are developed for analysis of free sup-ported beams. When the ends of the beam are restrained against various deflections, analysis becomes more complex. For example, analysis of the beam shown in Figure 4.25 is performed easily using one of the tables presented in this chapter. Analysis of the beam shown in Figure 4.26, with one restraint against vertical deflection at the left end and two other restraints against vertical deflection and rotation at the right end, requires solving a system of three equations that reflect the actual boundary condi-tions. Analysis, in principle, is not different from the analysis used in Chapter 3 for finite beams on Winkler foundation. The same method is applied to analysis of beams supported on elastic half-space using tables and equations introduced above. Analysis of the beam shown in Figure 4.26 is performed as follows: After removal of all three restraints and replacing them with unknown reactions X1, X2, and X3, the beam will look as shown in Figure 4.27.
Figure 4.23
Figure 4.24 Shear diagram
Figure 4.25 Figure 4.26
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Analysis of Beams on Elastic Half-Space 197
Now, the beam is analyzed applying the given load P and all three unknown re-actions successively assuming that all unknown reactions are equal to one unit. All analyses are performed using tables and equations presented earlier. Using the results of these analyses, the following system of equations is written:
X X XX X XX X X
000
P
P
P
11 1 12 2 13 3 1
21 1 22 2 23 3 2
31 1 32 2 33 3 3
d d d d
d d d d
d d d d
+ + + =
+ + + =
+ + + =
_
`
a
bb
bb (4.15)
In this system, dik is deflection of the beam at point i due to one unit load applied at point k, diP is the deflection of the beam at point i due to the given load P. By solv-ing system 4.15, reactions X1, X2, and X3 are found and final analysis of the beam is performed by applying to the beam all given loads and reactions. The same method can be used for analysis of continuous beams, including beams with spring supports. For example, analysis of the beam shown in Figure 4.28 is performed as follows: Af-ter removing all restraints-supports and replacing them with unknown reactions, the given beam will look as shown in Figure 4.29. The total number of unknowns in this case is equal to the number of removed restraints. The system of equations will look as follows:
X X X X XX X X X X
X XC
X X X
X X X X XX X X X X
00
1 0
00
P
P
DP
P
P
3 4 5
3 4 5
33 3 4 5
3 4 5
3 4 55 5 5
11 1 12 13 14 15 1
21 22 2 23 24 25 2
31 32 2 34 35 3
41 42 2 43 44 45 4
51 52 2 53 54
2
1
1
1
1
d d d d d d
d d d d d d
d d d d d d
d d d d d d
d d d d d d
+ + + + + =
+ + + + + =
+ + + + + + =
+ + + + + =
+ + + + + =
e o
_
`
a
bbbb
bbbb
(4.16)
In this system, CD is the rigidity of an elastic support D. All other coefficients in this system have the same meaning as in system 4.15. By solving the system of equations 4.16 all unknown reactions are found. Final analysis of the beam is performed by ap-plying to the beam the given loads and obtained reactions.
As shown, analysis of these types of beams is much more time consuming compared to analysis of beams with free supported ends. Moreover, the method can be used only when the given beam is replaced with one free supported beam with the given loads and unknown reactions, as shown in Figure 4.29. When analysis requires replacement of the given complex beam with several simple beams, the method cannot be used because each simple beam supported on elastic half-space produces settlements not
Figure 4.27
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198 Analysis of Structures on Elastic Foundations
only under its loaded area, but also under other neighboring beams that are not taken into account. This method of analysis is justified for beams on Winkler foundation where all beams work independently without affecting each other, but when the beam is supported on elastic half-space, this method cannot be applied directly. However, we propose a simple method that can resolve this problem. Let us assume we have to analyze a pin-connected beam shown in Figure 4.30. Analysis is performed in the fol-lowing steps:
Step 1: The given beam is divided into two simple beams, as shown in Figure 4.31, and unknown forces of interaction X1 are applied to both beams at points 2. Taking into account that settlements of the beams 1–2 and 2–3 at point 2 are the same, equation 4.17 is written:
X 0( ) ( ) P P1 1 2 1 2 3 1 1 11 2
~ ~ ~ ~+ + + =- -
` j (4.17)
where ~1(1−2) and ~1(2−3) are settlements in direction X1 of the beams 1–2 and 2–3 due to one unit loads applied at point 2; P1 1
~ and P1 2~ are settlements of the beams 1–2
and 2–3 due to the given loads P1 and P2. Reactions X1 obtained from 4.17 are used for separate analyses of both beams.
Step 2: Both beams are analyzed separately as two simple beams.
Step 3: In order to specify analysis of one of the beams, let’s say beam 1–2, by tak-ing into account the soil pressure produced by the beam 2–3, the soil pressure under beam 2–3 is replaced with individual loads-stamps applied to the soil in 21 points. For example, the load applied at point i can be found from equation 4.18:
Pp l20i
i= (4.18)
Figure 4.28
Figure 4.29
SAMPLE C
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Analysis of Beams on Elastic Half-Space 199
where pi is the soil pressure per one unit of the beam length 2–3 and l is the length of the beam. By applying loads Pi to the half-space under beam 2–3 and using equa-tions 1.23 and 1.25 from Chapter 1, additional settlements under beam 1–2 are found. These settlements will produce additional moments and shear forces in beam 1–2. As-suming that beam 1–2 is supported at all 21 points on non-yielding supports, as shown in Figure 4.33, and applying the settlements to all supports, analysis of beam 1–2 is performed.
Now, by summing obtained results with the results of the original analysis of beam 1–2, final moments and shear forces in beam 1–2 are found. The same method is used for analysis of beam 2–3.
It is obvious that this method can be applied to analysis of stepped beams and groups of close located foundations.
Figure 4.30
Figure 4.31
Figure 4.32
Additional settlements beam 1-2 Soil pressure diagram beam 2-3
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200 Analysis of Structures on Elastic Foundations
4.7 Analysis of Frames on Elastic Half-SpaceAnalysis of continuous foundations supported on the elastic half-space discussed above can take into account the rigidity of the superstructure. A method of analysis of foundations and 2D frames using the frame shown in Figure 4.34 is explained below. Analysis is performed in the following order:
The given system frame-foundation is divided into two parts: frame and foundation, as shown in Figure 4.35. All frame supports are restrained against any deflections. By applying successively to the frame the given loads and one unit deflections at all points of support, the following system of equations is written:
...
.......................................................
...
X r r r rX r r r r
X r r r r
n n P
n n P
n n n nn n nP
1 11 1 12 2 1 1
2 21 1 22 2 2 2
1 1 2 2
~ ~ ~
~ ~ ~
~ ~ ~
= + + + +
= + + + +
= + + + +
_
`
a
bbb
bb
(4.19)
In this system of equations:
Xi is the unknown reaction at point irij is the reaction at point i due to one unit load applied at point j~i is the settlement or rotation at point iriP is the reaction at point i due to the given loads applied to the frame.
Now, by applying the given loads, and unknown reactions acting in the opposite direc-tion to the foundation, all deflections of the foundation can be expressed as follows:
...
.............................................................
...
X X XX X X
X X X
n n P
n n P
n n n nn n nP
1 11 1 12 2 1 1
2 21 1 22 2 2 2
1 1 2 2
~ ~
~ ~
~ ~
D D D
D D D
D D D
= + + + +
= + + + +
= + + + +
_
`
a
bbb
bb
(4.20)
In this system of equations, Dij is deflection of the foundation at point i due to reaction equal to one unit and applied to point j, and ~iP is the deflection of the foundation at point i due to the given loads applied to the foundation.
The number of unknowns can be reduced by introducing reactions Xi from 4.19 into the system of equations 4.20 that leads to a system of equations with only n un-known deflections. In this case, the stiffness method is used. If deflections ~i from the
Figure 4.33 Beam 1–2 supported on 21 supports
SAMPLE C
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Analysis of Beams on Elastic Half-Space 201
system of equations 4.20 are introduced into the system of equations 4.19, we again obtain a system of equations with only n unknown reactions. In this case, the method of forces is used.
By solving both systems of equations 4.19 and 4.20 as one system, we find 2n un-knowns: n deflections and n reactions. In this case we use the so-called combined method of analysis. Final analysis of the frame is performed by applying to the frame the given loads and deflections of all supports. Final analysis of the foundation is per-formed by applying to the foundation the given loads and obtained reactions acting in the opposite direction. It should be mentioned that the iterative method is another way to perform analysis. This method was discussed in Chapter 3 in the analysis of frames on Winkler foundation. The only difference between the analyses described in Chapter 3 and analysis in this chapter is the method of obtaining deflections of the foundation.
Figure 4.34 Figure 4.35
α 1 2 3 4 5 6 7 8 9 10
K0 0.88 0.86 0.82 0.79 0.77 0.74 0.73 0.71 0.69 0.67
K1 0.52 0.85 1.10 1.30 1.45 1.60 1.70 1.80 1.90 2.00
Table 4.1 Coefficients K0 and K1SAMPLE C
HAPTER
202 Analysis of Structures on Elastic Foundations
αx
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
P0
10 0.439 0.440 0.442 0.446 0.455 0.462 0.475 0.498 0.541 0.632 0.842
15 0.447 0.448 0.450 0.454 0.459 0.468 0.480 0.500 0.537 0.611 0.773
20 0.452 0.453 0.455 0.458 0.464 0.471 0.483 0.502 0.535 0.600 0.737
30 0.458 0.459 0.460 0.463 0.468 0.475 0.486 0.503 0.532 0.586 0.699
50 0.464 0.465 0.466 0.468 0.473 0.480 0.489 0.503 0.538 0.574 0.665
100 0.469 0.470 0.471 0.473 0.477 0.483 0.481 0.504 0.525 0.561 0.634
Q0
10 −0.500 −0.456 −0.412 −0.367 −0.322 −0.277 −0.230 −0.182 −0.130 −0.072 0
15 −0.500 −0.455 −0.411 −0.365 −0.320 −0.273 −0.225 −0.177 −0.125 −0.068 0
20 −0.500 −0.454 −0.410 −0.363 −0.317 −0.271 −0.223 −0.174 −0.122 −0.066 0
30 −0.500 −0.454 −0.408 −0.362 −0.315 −0.268 −0.220 −0.171 −0.119 −0.063 0
50 −0.500 −0.453 −0.407 −0.360 −0.313 −0.265 −0.217 −0.168 −0.116 −0.061 0
100 −0.500 −0.453 −0.406 −0.359 −0.311 −0.263 −0.214 −0.165 −0.113 −0.058 0
M0
10 0.2703 0.2225 0.1791 0.1401 0.1056 0.0756 0.0502 0.0295 0.0139 0.0037 0
15 0.2672 0.2194 0.1761 0.1373 0.1931 0.0735 0.0486 0.0284 0.0132 0.0035 0
20 0.2654 0.2176 0.1744 0.1357 0.1017 0.0722 0.0475 0.0277 0.0128 0.0034 0
30 0.2634 0.2156 0.1725 0.1340 0.1002 0.0710 0.0465 0.0270 0.0125 0.0033 0
50 0.2615 0.2137 0.1707 0.1323 0.0986 0.0697 0.0455 0.0263 0.0120 0.0031 0
100 0.2596 0.2120 0.1690 0.1307 0.0972 0.0685 0.0446 0.0256 0.0117 0.0030 0
Table 4.2 Dimensionless coefficients for analysis of rigid beams loaded with a verticalconcentrated load P0 applied at the center of the beam
a 10 15 20 30 50 100
Y0
1.081 1,210 1.302 1.431 1.595 1.814
Table 4.3 Dimensionless coefficients Y0 for rigid beams loaded
with a vertical concentrated load P0 applied at the center of the beamSAM
PLE CHAPTER
Analysis of Beams on Elastic Half-Space 203
αx
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
p0
10 0 0.114 0.229 0.347 0.470 0.600 0.746 0.918 1.148 1.506 2.155
15 0 0.119 0.239 0.362 0.490 0.625 0.772 0.943 1.162 1.483 2.024
20 0 0.122 0.245 0.371 0.501 0.638 0.786 0.957 1.169 1.449 1.957
30 0 0.125 0.252 0.381 0.513 0.652 0.802 0.971 1.176 1.415 1.885
50 0 0.129 0.258 0.390 0.525 0.667 0.817 0.984 1.181 1.414 1.820
100 0 0.133 0.265 0.400 0.538 0.680 0.831 0.996 1.86 1.415 1.761
Q0
10 −0.709 −0.704 −0.686 −0.658 −0.617 −0.563 −0.496 −0.413 −0.311 −0.180 0
15 −0.716 −0.710 −0.692 −0.662 −0.619 −0.564 −0.494 −0.408 −0.304 −0.173 0
20 −0.719 −0.713 −0.695 −0.664 −0.621 −0.564 −0.493 −0.406 −0.300 −0.169 0
30 −0.723 −0.717 −0.698 −0.667 −0.622 −0.564 −0.491 −0.403 −0.296 −0.165 0
50 −0.727 −0.721 −0.701 -.0669 −0.623 −0.564 −0.490 −0.400 −0.292 −0.162 0
100 −0.731 −0.724 −0.705 −0.671 −0.624 −0.564 −0.488 −0.397 −0.288 −0.158 0
M0
10 0.5 0.4293 0.3597 0.2924 0.2285 0.1694 0.1163 0.0707 0.0343 0.0095 0
15 0.5 0.4285 0.3584 0.2907 0.2265 0.1672 0.1142 0.0690 0.0332 0.0091 0
20 0.5 0.4283 0.3577 0.2897 0.2253 0.1660 0.1130 0.0679 0.0324 0.0088 0
30 0.5 0.4297 0.3570 0.2887 0.2241 0.1647 0.1119 0.0670 0.0319 0.0084 0
50 0.5 0.4275 0.3563 0.2876 0.2229 0.1635 0.1107 0.0660 −0.0313 0.0083 0
100 0.5 0.4271 0.3556 0.2867 0.2217 0.1622 01095 0.0651 −0.0306 0.0081 0
Table 4.4 Dimensionless coefficients for analysis of rigid beams loaded with a concentrated moment m0 applied at the center of the beam
a 10 15 20 30 50 100
tg1{ 2.088 2.466 3.737 3.122 3.606 4.311
Table 4.5 Coefficients tg1{ for rigid beams loaded with a concen-
trated moment m0SAMPLE C
HAPTER
204
t =
1
Co
ef.
δx
−1
−0.
9−
0.8
−0.
7−
0.6
−0.
5−
0.4
−0.
3−
0.2
−0.
10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
M
0
000
301
002
203
805
908
511
615
219
324
119
315
211
608
505
903
802
201
000
30
Q0
050
096
140
186
234
283
335
388
444
500*
−44
4−
338
−33
5−
283
−23
4−
186
−14
0−
096
−05
00
p05
504
704
504
504
604
805
005
305
505
605
605
605
505
305
004
804
604
504
504
705
5
~08
709
209
710
210
711
111
611
912
212
412
512
412
211
911
611
110
710
209
709
208
7
M
0.1
000
200
701
502
704
306
308
811
715
219
223
718
814
510
807
605
002
901
300
30
Q0
034
068
102
139
180
223
269
319
372
426
482*
−46
0−
403
−34
6−
290
−23
5−
181
−12
7−
069
0
p0.
3603
303
403
603
804
204
504
805
105
405
605
705
705
705
605
50.
540.
5405
606
207
7
~06
907
608
208
809
510
110
611
211
612
112
412
612
712
512
412
211
911
611
310
910
6
M
0.2
000
100
400
901
702
904
306
108
411
214
418
222
617
613
109
306
103
601
700
40
Q0
019
040
066
095
128
165
205
250
300
352
408
467*
−47
3−
412
−35
0−
288
−22
4−
158
−08
50
p01
902
002
302
703
103
503
904
304
705
105
505
705
906
106
106
206
306
506
907
709
6
~05
005
906
607
408
108
909
710
411
111
712
212
713
013
213
413
413
413
213
113
112
9
M
0.3
000
000
100
300
801
402
403
605
307
309
912
916
620
815
711
207
404
402
000
50
Q0
005
015
032
053
079
109
145
185
230
280
335
393
454*
−48
1−
410
−33
0−
270
−19
2−
106
0
p00
200
801
401
902
402
803
303
804
304
805
205
606
005
306
606
907
107
508
109
412
1
~03
304
305
206
006
907
808
709
510
411
212
012
613
213
714
214
414
614
815
015
115
2
M
0.4
000
0−
001
−00
1−
001
001
005
012
023
037
055
079
107
142
184
132
088
052
024
007
0
Q0
−00
7−
007
000
012
031
056
085
121
162
208
260
318
380
448*
−47
940
2−
319
−22
9−
127
0
p−
010
−00
300
301
001
602
202
703
303
804
404
905
506
006
507
007
508
008
609
511
214
5
~01
502
503
604
605
606
607
608
609
610
611
612
413
414
214
915
516
016
516
917
417
8
0.5
0−
001
−00
4−
007
−01
0−
012
−01
3−
011
−00
700
001
202
805
007
711
115
310
206
102
900
80
0−
021
−03
1−
032
−02
7−
015
003
028
058
095
138
188
244
307
376
453*
−46
2−
370
−26
7−
148
0
−02
7−
015
−00
500
200
901
502
102
703
404
004
605
305
906
607
308
008
809
710
913
117
0
001
102
203
304
405
506
607
808
910
111
212
213
414
515
516
517
418
118
919
720
4
0.6
0−
002
−00
6−
012
−01
8−
024
−03
0−
034
−03
5−
034
−02
9−
020
−00
601
404
107
511
807
003
300
90
0−
032
−05
2−
063
−06
5−
059
−04
7−
029
−00
302
906
811
516
923
230
338
447
4*−
425
−30
9−
172
0
−03
9−
026
−01
5−
006
002
009
015
022
029
036
043
051
068
067
075
085
096
108
125
151
198
−01
6−
004
007
020
031
044
056
068
081
094
107
120
134
148
161
174
186
199
210
221
233
0.7
0−
002
−00
8−
017
−02
6−
037
−04
7−
056
−06
3−
069
−07
0−
068
−06
1−
049
−02
9−
002
034
080
038
010
0
0−
044
−07
4−
092
−10
2−
103
−09
7−
085
−06
4−
036
000
044
096
158
230
313
409
519*
−35
2−
198
0
−05
2−
036
−02
4−
014
−00
500
200
901
702
403
204
004
805
706
707
708
910
311
914
017
222
9
−03
0−
018
−00
600
702
003
304
606
007
408
810
211
813
314
916
518
219
921
423
024
626
2
0.8
0−
003
−01
0−
021
−03
5−
049
−06
4−
078
−09
1−
103
−11
1−
116
−11
6−
111
−09
9−
079
−05
0−
010
043
012
0
0−
055
−09
5−
122
−13
9−
147
−14
7−
139
−12
4−
101
−06
9−
028
023
084
157
243
344
462
603*
−22
40
−06
5−
047
−03
3−
022
−01
2−
004
004
011
019
027
036
046
056
067
079
093
109
129
155
194
259
−04
7−
033
−02
0−
006
008
022
036
050
066
081
097
115
132
151
170
190
211
231
252
272
293
0.9
0−
003
−01
3−
026
−04
3−
061
−08
−10
0−
119
−13
6−
152
−16
4−
171
−17
3−
169
−15
6−
133
−10
0−
052
013
0
0−
067
−11
6−
151
−17
5−
190
−19
6−
194
−18
7−
165
−13
7−
099
−05
001
008
417
227
840
455
874
8*0
−07
9−
057
−04
1−
029
−01
9−
010
−00
200
601
402
303
304
305
406
708
109
711
513
817
021
229
3
−06
2−
047
−03
3−
018
−00
401
102
504
105
807
409
211
113
115
217
419
822
224
727
229
832
5
1.0
0−
004
−01
5−
031
−05
1−
073
−09
7−
122
−14
7−
170
−19
2−
213
−22
6−
235
−23
3−
232
−21
7−
189
147
085
0
0−
079
−13
7−
180
−21
2−
233
−24
5−
249
−24
4−
229
−20
5−
170
−12
3−
064
011
102
212
347
512
721
1000
*
−09
2−
067
−05
0−
037
−02
6−
017
−00
800
101
001
902
904
105
306
608
210
012
114
818
423
832
6
−07
6−
061
−04
6−
031
−01
6−
001
016
032
050
068
088
108
130
154
179
206
233
263
293
325
356
Tab
le 4
.6 D
imen
sion
less
coe
ffici
ents
for
ana
lysi
s of
sho
rt b
eam
s lo
aded
with
a c
once
ntra
ted
ver
tical
load
P
SAMPLE C
HAPTER
205
t =
1
Co
ef.
δx
−1
−0.
9−
0.8
−0.
7−
0.6
−0.
5−
0.4
−0.
3−
0.2
−0.
10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
000
301
002
203
805
908
511
615
219
324
119
315
211
608
505
903
802
201
000
30
005
009
614
018
623
428
333
538
844
450
0*−
444
−33
8−
335
−28
3−
234
−18
6−
140
−09
6−
050
0
055
047
045
045
046
048
050
053
055
056
056
056
055
053
050
048
046
045
045
047
055
087
092
097
102
107
111
116
119
122
124
125
124
122
119
116
111
107
102
097
092
087
0.1
000
200
701
502
704
306
308
811
715
219
223
718
814
510
807
605
002
901
300
30
003
406
810
213
918
022
326
931
937
242
648
2*−
460
−40
3−
346
−29
0−
235
−18
1−
127
−06
90
0.36
033
034
036
038
042
045
048
051
054
056
057
057
057
056
055
0.54
0.54
056
062
077
069
076
082
088
095
101
106
112
116
121
124
126
127
125
124
122
119
116
113
109
106
0.2
000
100
400
901
702
904
306
108
411
214
418
222
617
613
109
306
103
601
700
40
001
904
006
609
512
816
520
525
030
035
240
846
7*−
473
−41
2−
350
−28
8−
224
−15
8−
085
0
019
020
023
027
031
035
039
043
047
051
055
057
059
061
061
062
063
065
069
077
096
050
059
066
074
081
089
097
104
111
117
122
127
130
132
134
134
134
132
131
131
129
0.3
000
000
100
300
801
402
403
605
307
309
912
916
620
815
711
207
404
402
000
50
000
501
503
205
307
910
914
518
523
028
033
539
345
4*−
481
−41
0−
330
−27
0−
192
−10
60
002
008
014
019
024
028
033
038
043
048
052
056
060
053
066
069
071
075
081
094
121
033
043
052
060
069
078
087
095
104
112
120
126
132
137
142
144
146
148
150
151
152
0.4
000
0−
001
−00
1−
001
001
005
012
023
037
055
079
107
142
184
132
088
052
024
007
0
0−
007
−00
700
001
203
105
608
512
116
220
826
031
838
044
8*−
479
402
−31
9−
229
−12
70
−01
0−
003
003
010
016
022
027
033
038
044
049
055
060
065
070
075
080
086
095
112
145
015
025
036
046
056
066
076
086
096
106
116
124
134
142
149
155
160
165
169
174
178
M
0.5
0−
001
−00
4−
007
−01
0−
012
−01
3−
011
−00
700
001
202
805
007
711
115
310
206
102
900
80
Q0
−02
1−
031
−03
2−
027
−01
500
302
805
809
513
818
824
430
737
645
3*−
462
−37
0−
267
−14
80
p−
027
−01
5−
005
002
009
015
021
027
034
040
046
053
059
066
073
080
088
097
109
131
170
~0
011
022
033
044
055
066
078
089
101
112
122
134
145
155
165
174
181
189
197
204
M
0.6
0−
002
−00
6−
012
−01
8−
024
−03
0−
034
−03
5−
034
−02
9−
020
−00
601
404
107
511
807
003
300
90
Q0
−03
2−
052
−06
3−
065
−05
9−
047
−02
9−
003
029
068
115
169
232
303
384
474*
−42
5−
309
−17
20
p−
039
−02
6−
015
−00
600
200
901
502
202
903
604
305
106
806
707
508
509
610
812
515
119
8
~−
016
−00
400
702
003
104
405
606
808
109
410
712
013
414
816
117
418
619
921
022
123
3
M
0.7
0−
002
−00
8−
017
−02
6−
037
−04
7−
056
−06
3−
069
−07
0−
068
−06
1−
049
−02
9−
002
034
080
038
010
0
Q0
−04
4−
074
−09
2−
102
−10
3−
097
−08
5−
064
−03
600
004
409
615
823
031
340
951
9*−
352
−19
80
p−
052
−03
6−
024
−01
4−
005
002
009
017
024
032
040
048
057
067
077
089
103
119
140
172
229
~−
030
−01
8−
006
007
020
033
046
060
074
088
102
118
133
149
165
182
199
214
230
246
262
M
0.8
0−
003
−01
0−
021
−03
5−
049
−06
4−
078
−09
1−
103
−11
1−
116
−11
6−
111
−09
9−
079
−05
0−
010
043
012
0
Q0
−05
5−
095
−12
2−
139
−14
7−
147
−13
9−
124
−10
1−
069
−02
802
308
415
724
334
446
260
3*−
224
0
p−
065
−04
7−
033
−02
2−
012
−00
400
401
101
902
703
604
605
606
707
909
310
912
915
519
425
9
~−
047
−03
3−
020
−00
600
802
203
605
006
608
109
711
513
215
117
019
021
123
125
227
229
3
M
0.9
0−
003
−01
3−
026
−04
3−
061
−08
−10
0−
119
−13
6−
152
−16
4−
171
−17
3−
169
−15
6−
133
−10
0−
052
013
0
Q0
−06
7−
116
−15
1−
175
−19
0−
196
−19
4−
187
−16
5−
137
−09
9−
050
010
084
172
278
404
558
748*
0
p−
079
−05
7−
041
−02
9−
019
−01
0−
002
006
014
023
033
043
054
067
081
097
115
138
170
212
293
~−
062
−04
7−
033
−01
8−
004
011
025
041
058
074
092
111
131
152
174
198
222
247
272
298
325
M
1.0
0−
004
−01
5−
031
−05
1−
073
−09
7−
122
−14
7−
170
−19
2−
213
−22
6−
235
−23
3−
232
−21
7−
189
147
085
0
Q0
−07
9−
137
−18
0−
212
−23
3−
245
−24
9−
244
−22
9−
205
−17
0−
123
−06
401
110
221
234
751
272
110
00*
p−
092
−06
7−
050
−03
7−
026
−01
7−
008
001
010
019
029
041
053
066
082
100
121
148
184
238
326
~−
076
−06
1−
046
−03
1−
016
−00
101
603
205
006
808
810
813
015
417
920
623
326
329
332
535
6
SAMPLE C
HAPTER
206
t =
2
Co
ef.
δx
−1
−0.
9−
0.8
−0.
7−
0.6
−0.
5−
0.4
−0.
3−
0.2
−0.
10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
M
0
000
200
701
603
004
707
009
813
217
321
917
313
209
807
004
703
001
600
700
20
Q0
036
072
111
155
202
254
310
371
434
500*
−43
4−
371
−31
0−
254
−20
2−
155
−11
107
203
60
p03
903
503
704
104
504
905
405
806
206
506
606
506
205
805
404
904
504
103
703
503
9
~07
108
008
909
710
611
412
112
713
313
613
713
613
312
712
111
410
609
708
908
007
1
M
0.1
000
100
401
002
003
305
007
209
913
217
121
716
912
809
306
404
102
301
100
30
Q0
021
046
076
110
150
195
246
301
361
424
489*
−44
5−
381
−31
8−
259
−20
3−
152
−10
3−
055
0
p02
002
302
703
103
704
204
805
305
806
206
406
506
506
406
105
705
304
904
805
0−
61
~05
706
607
508
509
410
311
111
912
613
213
613
813
713
412
912
411
711
010
209
508
7
M
0.2
000
000
200
501
001
903
104
706
809
412
616
420
815
911
708
205
303
001
400
40
Q0
007
022
043
070
103
140
183
232
287
348
412
478*
−45
4−
387
−32
2−
258
−19
6−
134
−07
10
p00
501
101
802
403
003
504
004
605
205
806
306
606
706
706
606
506
306
206
206
507
8
~04
005
006
007
108
109
110
111
011
912
713
313
814
114
214
013
713
312
812
311
811
3
M
0.3
000
0−
001
000
002
007
014
025
040
059
084
114
151
194
145
102
067
039
019
005
0
Q0
005
001
014
034
059
089
126
169
219
274
335
400
468*
−46
1−
389
−31
7−
244
−17
1−
093
0
p−
010
001
010
017
022
028
034
040
046
053
058
063
067
069
071
072
072
072
074
083
106
~02
603
604
705
806
807
908
910
011
011
912
813
614
214
614
814
814
714
514
214
013
7
M
0.4
000
1−
002
−00
3−
004
−00
300
000
501
402
704
506
809
713
217
512
408
204
802
300
60
Q0
−01
1−
015
−01
100
001
704
207
210
915
220
225
832
038
846
1*−
462
−38
2−
299
−21
3−
117
0
p−
014
−00
700
000
701
402
102
703
404
004
605
305
906
507
107
507
908
108
409
010
313
4
~01
002
103
204
305
406
607
708
809
911
012
113
114
014
815
515
916
216
316
416
516
5
0.5
0−
001
−00
4−
008
−01
1−
014
−01
5−
014
−01
1−
004
007
023
044
071
106
148
098
058
027
007
0
0−
023
−03
5−
037
−03
2−
021
−00
302
005
108
813
318
424
330
838
146
2*−
451
−35
7−
256
−14
10
−03
1−
017
−00
600
100
801
402
102
703
404
104
805
506
206
907
708
409
009
710
712
516
1
−00
1−
010
021
032
043
055
066
078
090
102
114
126
138
149
159
168
175
181
186
191
196
0.6
0−
002
−00
6−
011
−01
7−
023
−02
8−
032
−03
4−
033
−02
8−
020
−00
601
404
007
411
706
903
300
90
0−
003
−04
9−
059
−06
1−
057
−04
6−
029
−00
502
606
411
116
522
830
038
247
5*−
422
−30
5−
170
0
−03
5−
025
−01
4−
006
001
008
014
020
027
035
042
050
059
067
077
087
098
109
124
149
195
−01
5−
003
008
019
031
042
054
067
079
093
106
120
134
149
163
176
180
200
211
221
231
0.7
0−
002
−00
7−
014
−02
3−
032
−04
1−
049
−05
6−
061
−06
3−
061
−05
5−
043
−02
500
103
608
103
901
00
0−
038
−06
4−
081
−08
9−
091
−08
7−
077
−05
9−
034
−00
203
908
914
821
830
139
851
0*−
358
−20
20
−04
3−
032
−02
1−
012
−00
500
100
701
402
102
903
704
505
406
407
608
910
412
114
217
523
4
−02
4−
013
−00
200
902
103
204
405
707
008
409
811
413
014
716
418
219
921
723
324
926
5
0.8
0−
002
−00
9−
018
−02
9−
041
−05
4−
066
−07
8−
088
−09
7−
102
−10
3−
099
−08
907
1−
044
−00
904
501
20
0−
045
−07
8−
102
−11
7−
126
−12
8−
193
−11
2−
094
−06
8−
032
013
069
138
221
321
440
585*
−23
40
−05
2−
039
−02
8−
019
−01
2−
005
001
007
014
022
030
040
050
062
075
091
109
131
160
203
272
−03
7−
026
−01
5−
004
008
020
032
045
059
074
089
107
125
145
166
188
211
235
258
281
304
0.9
0−
003
−01
0−
021
−03
4−
050
−06
6−
083
−10
0−
116
−13
0−
142
−15
0−
154
−15
2−
142
−12
3−
093
−04
801
40
0−
053
−09
3−
123
−14
5−
158
−16
8−
170
−16
5−
159
−13
3−
103
−06
3−
010
057
140
243
370
522
730*
0
−06
1−
046
−03
4−
025
−01
8−
011
−00
500
100
801
602
503
504
605
907
509
311
414
117
723
131
7
−04
7−
036
−02
5−
014
003
009
021
034
049
064
081
100
120
142
167
193
221
251
281
313
344
1.0
0−
003
−01
2−
024
−04
0−
058
−07
9−
100
−12
2−
164
−16
4−
182
−19
8−
210
−21
5−
217
−20
3−
180
−14
2−
084
0
0−
061
−10
8−
144
−17
2−
193
−20
8−
216
−21
8−
212
−19
8−
174
−13
8−
089
−02
505
916
530
047
169
510
00*
−07
0−
053
−04
1−
032
−02
4−
018
−01
2−
005
002
010
019
029
043
056
074
094
119
151
194
258
361
−05
7−
046
−03
6−
025
−01
4−
002
010
024
038
055
073
093
115
140
168
198
231
167
305
344
385
Tab
le 4
.7 D
imen
sion
less
coe
ffici
ents
for
ana
lysi
s of
sho
rt b
eam
s lo
aded
with
a c
once
ntra
ted
ver
tical
load
P
SAMPLE C
HAPTER
207
t =
2
Co
ef.
δx
−1
−0.
9−
0.8
−0.
7−
0.6
−0.
5−
0.4
−0.
3−
0.2
−0.
10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
000
200
701
603
004
707
009
813
217
321
917
313
209
807
004
703
001
600
700
20
003
607
211
115
520
225
431
037
143
450
0*−
434
−37
1−
310
−25
4−
202
−15
5−
111
072
036
0
039
035
037
041
045
049
054
058
062
065
066
065
062
058
054
049
045
041
037
035
039
071
080
089
097
106
114
121
127
133
136
137
136
133
127
121
114
106
097
089
080
071
0.1
000
100
401
002
003
305
007
209
913
217
121
716
912
809
306
404
102
301
100
30
002
104
607
611
015
019
524
630
136
142
448
9*−
445
−38
1−
318
−25
9−
203
−15
2−
103
−05
50
020
023
027
031
037
042
048
053
058
062
064
065
065
064
061
057
053
049
048
050
−61
057
066
075
085
094
103
111
119
126
132
136
138
137
134
129
124
117
110
102
095
087
0.2
000
000
200
501
001
903
104
706
809
412
616
420
815
911
708
205
303
001
400
40
000
702
204
307
010
314
018
323
228
734
841
247
8*−
454
−38
7−
322
−25
8−
196
−13
4−
071
0
005
011
018
024
030
035
040
046
052
058
063
066
067
067
066
065
063
062
062
065
078
040
050
060
071
081
091
101
110
119
127
133
138
141
142
140
137
133
128
123
118
113
0.3
000
0−
001
000
002
007
014
025
040
059
084
114
151
194
145
102
067
039
019
005
0
000
500
101
403
405
908
912
616
921
927
433
540
046
8*−
461
−38
9−
317
−24
4−
171
−09
30
−01
000
101
001
702
202
803
404
004
605
305
806
306
706
907
107
207
207
207
408
310
6
026
036
047
058
068
079
089
100
110
119
128
136
142
146
148
148
147
145
142
140
137
0.4
000
1−
002
−00
3−
004
−00
300
000
501
402
704
506
809
713
217
512
408
204
802
300
60
0−
011
−01
5−
011
000
017
042
072
109
152
202
258
320
388
461*
−46
2−
382
−29
9−
213
−11
70
−01
4−
007
000
007
014
021
027
034
040
046
053
059
065
071
075
079
081
084
090
103
134
010
021
032
043
054
066
077
088
099
110
121
131
140
148
155
159
162
163
164
165
165
M
0.5
0−
001
−00
4−
008
−01
1−
014
−01
5−
014
−01
1−
004
007
023
044
071
106
148
098
058
027
007
0
Q0
−02
3−
035
−03
7−
032
−02
1−
003
020
051
088
133
184
243
308
381
462*
−45
1−
357
−25
6−
141
0
p−
031
−01
7−
006
001
008
014
021
027
034
041
048
055
062
069
077
084
090
097
107
125
161
~−
001
−01
002
103
204
305
506
607
809
010
211
412
613
814
915
916
817
518
118
619
119
6
M
0.6
0−
002
−00
6−
011
−01
7−
023
−02
8−
032
−03
4−
033
−02
8−
020
−00
601
404
007
411
706
903
300
90
Q0
−00
3−
049
−05
9−
061
−05
7−
046
−02
9−
005
026
064
111
165
228
300
382
475*
−42
2−
305
−17
00
p−
035
−02
5−
014
−00
600
100
801
402
002
703
504
205
005
906
707
708
709
810
912
414
919
5
~−
015
−00
300
801
903
104
205
406
707
909
310
612
013
414
916
317
618
020
021
122
123
1
M
0.7
0−
002
−00
7−
014
−02
3−
032
−04
1−
049
−05
6−
061
−06
3−
061
−05
5−
043
−02
500
103
608
103
901
00
Q0
−03
8−
064
−08
1−
089
−09
1−
087
−07
7−
059
−03
4−
002
039
089
148
218
301
398
510*
−35
8−
202
0
p−
043
−03
2−
021
−01
2−
005
001
007
014
021
029
037
045
054
064
076
089
104
121
142
175
234
~−
024
−01
3−
002
009
021
032
044
057
070
084
098
114
130
147
164
182
199
217
233
249
265
M
0.8
0−
002
−00
9−
018
−02
9−
041
−05
4−
066
−07
8−
088
−09
7−
102
−10
3−
099
−08
907
1−
044
−00
904
501
20
Q0
−04
5−
078
−10
2−
117
−12
6−
128
−19
3−
112
−09
4−
068
−03
201
306
913
822
132
144
058
5*−
234
0
p−
052
−03
9−
028
−01
9−
012
−00
500
100
701
402
203
004
005
006
207
509
110
913
116
020
327
2
~−
037
−02
6−
015
−00
400
802
003
204
505
907
408
910
712
514
516
618
821
123
525
828
130
4
M
0.9
0−
003
−01
0−
021
−03
4−
050
−06
6−
083
−10
0−
116
−13
0−
142
−15
0−
154
−15
2−
142
−12
3−
093
−04
801
40
Q0
−05
3−
093
−12
3−
145
−15
8−
168
−17
0−
165
−15
9−
133
−10
3−
063
−01
005
714
024
337
052
273
0*0
p−
061
−04
6−
034
−02
5−
018
−01
1−
005
001
008
016
025
035
046
059
075
093
114
141
177
231
317
~−
047
−03
6−
025
−01
400
300
902
103
404
906
408
110
012
014
216
719
322
125
128
131
334
4
M
1.0
0−
003
−01
2−
024
−04
0−
058
−07
9−
100
−12
2−
164
−16
4−
182
−19
8−
210
−21
5−
217
−20
3−
180
−14
2−
084
0
Q0
−06
1−
108
−14
4−
172
−19
3−
208
−21
6−
218
−21
2−
198
−17
4−
138
−08
9−
025
059
165
300
471
695
1000
*
p−
070
−05
3−
041
−03
2−
024
−01
8−
012
−00
500
201
001
902
904
305
607
409
411
915
119
425
836
1
~−
057
−04
6−
036
−02
5−
014
−00
201
002
403
805
507
309
311
514
016
819
823
116
730
534
438
5
SAMPLE C
HAPTER
208
t =
5
Co
ef.
δx
−1
−0.
9−
0.8
−0.
7−
0.6
−0.
5−
0.4
−0.
3−
0.2
−0.
10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
M
0
000
000
200
701
402
604
306
609
613
418
013
409
606
604
302
601
400
700
200
00
Q0
011
029
058
096
143
199
263
336
416
500*
−41
6−
336
−26
3−
199
−14
3−
096
−05
8−
029
−01
10
p01
001
402
303
304
305
106
006
907
708
208
408
207
706
906
005
104
303
302
301
401
0
~04
205
807
308
810
211
713
014
215
215
916
215
915
214
213
011
710
208
807
305
804
2
M
0.1
000
000
000
200
601
402
504
206
509
613
317
913
309
606
604
302
601
400
600
20
Q0
−00
100
902
805
709
514
219
926
533
641
750
0*−
418
−33
8−
264
−19
8−
141
−09
6−
060
−03
10
p00
700
401
502
403
304
305
206
107
007
708
108
308
107
707
106
205
104
003
102
903
5
~03
605
006
407
809
210
612
013
214
315
215
916
015
714
913
912
611
209
708
106
605
0
M
0.2
000
100
200
200
100
301
002
203
906
109
112
817
412
809
006
003
702
000
900
20
Q0
−01
2−
011
003
026
056
094
139
195
260
335
415
498*
−41
9−
340
−26
6−
199
−13
9−
087
−04
20
p−
018
−00
500
801
902
703
304
105
006
107
007
808
208
308
107
707
106
405
604
804
304
4
~02
403
705
106
507
909
310
612
013
314
415
416
016
316
015
414
413
312
010
709
308
0
M
0.3
0−
001
−00
3−
006
−00
6−
005
−00
200
501
703
305
608
512
216
712
108
805
202
901
300
30
Q0
−02
1−
024
−01
600
102
305
208
013
719
325
933
241
149
3*−
424
−34
2−
264
−19
3−
129
−06
90
p−
031
−01
103
501
301
902
603
304
205
206
107
007
608
008
308
308
007
506
706
106
208
1
~01
6−
028
041
054
067
080
093
106
120
132
144
153
160
164
162
156
148
138
127
116
105
M
0.4
0−
001
−00
3−
006
−00
9−
010
−01
0−
007
000
010
026
048
076
112
156
109
070
041
019
005
0
Q0
−01
7−
027
−02
9−
022
−00
701
504
608
413
018
524
932
030
948
4*−
428
−34
1−
257
−17
8−
097
0
p−
020
−01
4−
006
002
011
019
027
034
042
050
059
068
076
082
087
088
086
081
079
086
113
~00
1−
013
025
038
050
063
076
090
104
117
131
143
154
163
168
168
165
160
153
146
139
0.5
0−
001
−00
5−
009
−01
3−
017
−01
9−
019
−01
7−
012
−00
201
203
205
909
413
809
005
202
400
60
0−
027
−03
9−
042
−03
9−
030
−01
500
603
607
311
917
323
630
838
947
8*−
428
−33
0−
230
−12
40
−03
6−
019
−00
700
000
601
101
802
503
304
105
005
806
707
608
509
209
709
910
111
114
3
000
010
021
031
042
053
065
077
090
104
118
132
145
158
169
176
180
181
180
178
176
0.6
0−
001
−00
5−
009
−01
5−
020
−02
5−
028
−03
0−
030
−02
6−
019
−00
601
203
807
111
406
703
100
80
0−
025
−04
2−
051
−05
4−
050
−04
2−
029
−00
901
805
409
915
321
829
337
947
7*−
414
−29
6−
162
0
−02
6−
022
−01
3−
005
000
005
010
016
023
032
040
049
059
070
081
092
103
113
124
144
187
010
−00
100
901
902
903
905
106
207
508
910
411
913
615
216
818
319
520
421
121
822
3
0.7
0−
001
−00
5−
010
−01
6−
023
−03
0−
037
−42
−04
7−
048
−04
7−
042
−03
2−
016
007
040
083
040
011
0
0−
026
−04
6−
059
−00
6−
069
−06
8−
063
−05
1−
032
−00
503
007
412
819
627
837
849
5*36
820
70
−02
5−
024
−01
6−
009
−00
4−
001
003
008
015
023
031
039
049
060
074
091
109
127
147
179
244
−01
0−
002
−00
601
302
203
104
005
106
207
509
010
612
314
216
218
220
222
123
825
326
8
0.8
0−
001
−00
5−
011
−01
8−
026
−03
5−
045
−05
4−
062
−06
9−
075
−07
7−
076
−06
8−
055
−03
200
204
901
30
0−
026
−04
9−
066
−07
8−
087
−09
1−
092
−08
9−
080
−06
4−
040
−00
604
110
217
927
739
955
2*−
254
0
−02
6−
024
−02
0−
015
−01
0−
006
−00
300
100
601
202
002
904
005
306
808
710
913
617
121
929
6
−01
9−
012
−00
600
100
901
702
503
504
605
907
409
211
113
315
718
421
224
127
029
932
6
0.9
0−
001
−00
5−
012
−02
0−
030
−04
0−
052
−06
5−
078
−09
0−
102
−11
1−
118
−12
0−
116
−10
3−
080
−04
1−
016
0
0−
028
−05
3−
073
−08
9−
103
−11
4−
123
−12
7−
127
−12
1−
107
−08
3−
046
007
079
176
303
470
693*
0
−03
0−
027
−02
2−
018
−01
5−
012
−01
0−
007
−00
300
301
001
903
004
406
208
411
114
519
125
836
5
−02
1−
016
−01
1−
006
000
007
014
022
033
045
060
077
098
123
151
183
218
257
298
341
384
1.0
0−
001
−00
6−
013
−02
2−
032
−04
5−
060
−07
6−
093
−11
1−
128
−14
5−
160
−17
1−
177
175
−16
1−
132
−08
10
0−
029
−05
6−
079
−10
0−
120
−13
7−
153
−16
6−
175
−17
8−
174
−16
0−
132
−08
8−
021
074
207
388
640
1000
*
−03
0−
028
−02
5−
022
−02
0−
018
−01
7−
014
−01
1−
006
000
008
020
035
055
080
112
154
212
298
433
−02
4−
020
−01
6−
013
−00
9−
004
002
010
019
030
045
063
055
112
144
183
225
273
327
384
444
Tab
le 4
.8 D
imen
sion
less
coe
ffici
ents
for
ana
lysi
s of
sho
rt b
eam
s lo
aded
with
a c
once
ntra
ted
ver
tical
load
P
SAMPLE C
HAPTER
209
t =
5
Co
ef.
δx
−1
−0.
9−
0.8
−0.
7−
0.6
−0.
5−
0.4
−0.
3−
0.2
−0.
10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
000
000
200
701
402
604
306
609
613
418
013
409
606
604
302
601
400
700
200
00
001
102
905
809
614
319
926
333
641
650
0*−
416
−33
6−
263
−19
9−
143
−09
6−
058
−02
9−
011
0
010
014
023
033
043
051
060
069
077
082
084
082
077
069
060
051
043
033
023
014
010
042
058
073
088
102
117
130
142
152
159
162
159
152
142
130
117
102
088
073
058
042
0.1
000
000
000
200
601
402
504
206
509
613
317
913
309
606
604
302
601
400
600
20
0−
001
009
028
057
095
142
199
265
336
417
500*
−41
8−
338
−26
4−
198
−14
1−
096
−06
0−
031
0
007
004
015
024
033
043
052
061
070
077
081
083
081
077
071
062
051
040
031
029
035
036
050
064
078
092
106
120
132
143
152
159
160
157
149
139
126
112
097
081
066
050
0.2
000
100
200
200
100
301
002
203
906
109
112
817
412
809
006
003
702
000
900
20
0−
012
−01
100
302
605
609
413
919
526
033
541
549
8*−
419
−34
0−
266
−19
9−
139
−08
7−
042
0
−01
8−
005
008
019
027
033
041
050
061
070
078
082
083
081
077
071
064
056
048
043
044
024
037
051
065
079
093
106
120
133
144
154
160
163
160
154
144
133
120
107
093
080
0.3
0−
001
−00
3−
006
−00
6−
005
−00
200
501
703
305
608
512
216
712
108
805
202
901
300
30
0−
021
−02
4−
016
001
023
052
080
137
193
259
332
411
493*
−42
4−
342
−26
4−
193
−12
9−
069
0
−03
1−
011
035
013
019
026
033
042
052
061
070
076
080
083
083
080
075
067
061
062
081
016
−02
804
105
406
708
009
310
612
013
214
415
316
016
416
215
614
813
812
711
610
5
0.4
0−
001
−00
3−
006
−00
9−
010
−01
0−
007
000
010
026
048
076
112
156
109
070
041
019
005
0
0−
017
−02
7−
029
−02
2−
007
015
046
084
130
185
249
320
309
484*
−42
8−
341
−25
7−
178
−09
70
−02
0−
014
−00
600
201
101
902
703
404
205
005
906
807
608
208
708
808
608
107
908
611
3
001
−01
302
503
805
006
307
609
010
411
713
114
315
416
316
816
816
516
015
314
613
9
M
0.5
0−
001
−00
5−
009
−01
3−
017
−01
9−
019
−01
7−
012
−00
201
203
205
909
413
809
005
202
400
60
Q0
−02
7−
039
−04
2−
039
−03
0−
015
006
036
073
119
173
236
308
389
478*
−42
8−
330
−23
0−
124
0
p−
036
−01
9−
007
000
006
011
018
025
033
041
050
058
067
076
085
092
097
099
101
111
143
~00
001
002
103
104
205
306
507
709
010
411
813
214
515
816
917
618
018
118
017
817
6
M
0.6
0−
001
−00
5−
009
−01
5−
020
−02
5−
028
−03
0−
030
−02
6−
019
−00
601
203
807
111
406
703
100
80
Q0
−02
5−
042
−05
1−
054
−05
0−
042
−02
9−
009
018
054
099
153
218
293
379
477*
−41
4−
296
−16
20
p−
026
−02
2−
013
−00
500
000
501
001
602
303
204
004
905
907
008
109
210
311
312
414
418
7
~01
0−
001
009
019
029
039
051
062
075
089
104
119
136
152
168
183
195
204
211
218
223
M
0.7
0−
001
−00
5−
010
−01
6−
023
−03
0−
037
−42
−04
7−
048
−04
7−
042
−03
2−
016
007
040
083
040
011
0
Q0
−02
6−
046
−05
9−
006
−06
9−
068
−06
3−
051
−03
2−
005
030
074
128
196
278
378
495*
368
207
0
p−
025
−02
4−
016
−00
9−
004
−00
100
300
801
502
303
103
904
906
007
409
110
912
714
717
924
4
~−
010
−00
2−
006
013
022
031
040
051
062
075
090
106
123
142
162
182
202
221
238
253
268
M
0.8
0−
001
−00
5−
011
−01
8−
026
−03
5−
045
−05
4−
062
−06
9−
075
−07
7−
076
−06
8−
055
−03
200
204
901
30
Q0
−02
6−
049
−06
6−
078
−08
7−
091
−09
2−
089
−08
0−
064
−04
0−
006
041
102
179
277
399
552*
−25
40
p−
026
−02
4−
020
−01
5−
010
−00
6−
003
001
006
012
020
029
040
053
068
087
109
136
171
219
296
~−
019
−01
2−
006
001
009
017
025
035
046
059
074
092
111
133
157
184
212
241
270
299
326
M
0.9
0−
001
−00
5−
012
−02
0−
030
−04
0−
052
−06
5−
078
−09
0−
102
−11
1−
118
−12
0−
116
−10
3−
080
−04
1−
016
0
Q0
−02
8−
053
−07
3−
089
−10
3−
114
−12
3−
127
−12
7−
121
−10
7−
083
−04
600
707
917
630
347
069
3*0
p−
030
−02
7−
022
−01
8−
015
−01
2−
010
−00
7−
003
003
010
019
030
044
062
084
111
145
191
258
365
~−
021
−01
6−
011
−00
600
000
701
402
203
304
506
007
709
812
315
118
321
825
729
834
138
4
M
1.0
0−
001
−00
6−
013
−02
2−
032
−04
5−
060
−07
6−
093
−11
1−
128
−14
5−
160
−17
1−
177
175
−16
1−
132
−08
10
Q0
−02
9−
056
−07
9−
100
−12
0−
137
−15
3−
166
−17
5−
178
−17
4−
160
−13
2−
088
−02
107
420
738
864
010
00*
p−
030
−02
8−
025
−02
2−
020
−01
8−
017
−01
4−
011
−00
600
000
802
003
505
508
011
215
421
229
843
3
~−
024
−02
0−
016
−01
3−
009
−00
400
201
001
903
004
506
305
511
214
418
322
527
332
738
444
4
SAMPLE C
HAPTER
210
t =
10
Co
ef.
δx
−1
−0.
9−
0.8
−0.
7−
0.6
−0.
5−
0.4
−0.
3−
0.2
−0.
10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
M
0
000
0−
001
−00
100
200
902
204
006
610
114
610
106
604
002
200
900
2−
001
−00
800
000
Q0
−00
7−
003
016
048
093
150
221
304
399
500*
−39
9−
304
−22
1−
150
−09
3−
048
−01
600
300
700
p−
008
−00
301
102
603
905
106
407
709
009
010
209
909
007
706
405
103
902
601
1−
003
−00
8
~02
003
905
907
909
811
813
715
416
917
918
417
916
915
413
711
809
807
905
903
902
0
M
0.1
0−
001
−00
3−
004
−00
4−
001
006
019
038
065
101
147
103
068
042
024
013
006
003
001
0
Q0
−01
7−
018
−00
701
505
009
615
623
031
540
850
7*−
395
−30
1−
216
−14
3−
087
−04
8−
027
−01
40
p−
026
−00
900
501
702
804
005
306
708
009
009
709
909
709
008
006
504
702
901
501
102
1
~02
504
205
807
409
110
812
514
215
717
017
918
117
516
214
512
510
408
105
903
601
3
M
0.2
0−
002
−00
5−
008
−00
9−
008
−00
400
301
603
506
209
814
510
106
604
002
201
100
400
10
Q0
−02
5−
033
−02
4−
005
021
055
090
156
230
316
412
512*
−39
1−
299
−21
8−
147
−08
9−
046
−01
80
p−
032
−01
600
201
502
302
903
805
006
508
009
209
910
009
508
707
606
405
103
602
101
7
~01
703
204
706
207
709
310
912
614
315
817
218
118
417
916
715
113
211
209
107
004
9
M
0.3
0−
002
−00
5−
009
−01
2−
013
−01
2−
008
−00
101
303
206
009
714
309
906
403
802
100
900
20
Q0
−00
3−
041
−03
4−
020
−00
202
205
610
316
423
732
241
350
9*−
394
−30
0−
215
−14
4−
090
−04
80
p−
047
−01
900
101
101
602
102
904
005
406
707
908
809
409
709
609
007
906
304
604
106
3
~01
602
804
005
206
507
909
410
912
514
115
717
017
918
217
716
515
013
111
209
207
2
M
0.4
0−
001
−00
3−
007
−01
0−
014
−01
5−
014
−01
1−
002
011
030
057
093
138
093
058
033
015
004
0
Q0
−01
6−
031
−03
8−
036
−02
5−
006
022
058
104
162
231
311
403
502*
−39
7−
300
−21
5−
143
−07
80
p−
016
−01
6−
011
003
006
015
023
032
041
052
063
075
086
096
101
100
092
078
066
067
095
~−
003
009
021
033
046
059
073
088
104
121
138
154
168
178
183
179
170
157
142
126
109
0.5
0−
001
−00
5−
009
−01
3−
017
−02
0−
022
−02
1−
017
−01
000
302
204
808
212
608
004
502
000
50
0−
029
−04
0−
042
−04
0−
034
−02
4−
006
019
054
000
155
222
300
391
491*
−40
4−
299
−20
0−
106
0
−04
1−
018
−00
600
000
400
801
402
103
004
005
006
107
308
509
610
410
610
209
509
612
3
006
014
023
031
041
051
062
074
088
103
119
136
153
168
180
187
187
181
173
163
153
0.6
0−
001
−00
3−
007
−01
2−
016
−02
0−
024
−02
6−
026
−02
4−
018
−00
701
003
406
711
006
403
000
80
0−
018
−03
4−
042
−04
4−
043
−03
8−
030
−01
600
704
008
313
720
328
337
648
2*−
402
−28
0−
151
0
−01
3−
019
−01
2−
005
000
003
006
011
018
028
038
048
060
073
086
100
112
119
124
136
176
006
008
010
018
026
035
046
057
070
085
101
119
138
157
176
192
204
210
212
212
212
0.7
000
0−
003
−00
6−
010
−01
5−
020
−02
6−
030
−03
4−
036
−03
6−
032
−02
4−
1001
204
208
404
601
10
0−
015
−03
0−
040
−04
5−
049
−05
2−
051
−04
5−
031
−00
902
005
910
917
325
036
148
6*−
372
−20
80
−00
7−
017
−01
2−
007
−00
4−
004
−00
200
301
001
802
603
404
405
607
309
411
513
415
117
925
0
003
008
013
018
024
030
037
046
056
068
082
099
118
139
161
184
206
226
241
253
265
0.8
000
000
200
601
001
502
102
703
404
104
705
205
605
605
204
102
200
805
201
40
0−
010
−02
5−
038
−04
8−
055
−06
1−
066
−06
8−
067
−06
0−
045
−02
001
807
114
424
036
452
4*−
270
0
−00
4−
014
−01
4−
011
−00
8−
007
−00
5−
004
−00
100
401
102
003
104
506
208
310
914
118
123
431
3
−00
8−
004
000
005
009
014
020
028
037
048
062
079
099
123
150
180
214
248
282
313
344
0.9
000
0−
002
−00
5−
009
−01
5−
021
−02
9−
038
−04
7−
058
−06
8−
078
−08
6−
092
−09
2−
085
−06
7−
035
018
0
0−
010
−02
4−
036
−04
7−
059
−07
0−
082
−09
3−
101
−10
6−
104
−09
3−
070
−03
2−
029
117
242
416
657*
0
−00
6−
013
−01
3−
012
−01
2−
011
−01
1−
011
−01
000
7−
002
006
016
030
049
073
105
147
203
285
412
−00
3−
002
000
001
001
066
010
015
022
031
043
059
079
104
135
171
214
261
313
267
421
1.0
000
0−
002
−00
4−
008
−01
4−
021
−03
0−
041
−05
4−
068
−08
4−
100
−11
7−
137
−43
−14
8−
142
−12
2−
078
0
0−
008
−02
0−
033
−04
8−
064
−08
1−
099
−11
8−
136
−15
2−
163
−16
7−
159
−13
5−
086
−00
512
030
858
510
00*
−00
4−
011
013
−01
4−
015
−01
6−
018
−01
9−
018
−01
701
400
800
101
503
506
810
115
322
733
650
7
−00
1−
001
−00
2−
002
−00
2−
002
−00
100
200
601
402
403
905
908
612
016
221
427
534
542
250
4
Tab
le 4
.9 D
imen
sion
less
coe
ffici
ents
for
ana
lysi
s of
sho
rt b
eam
s lo
aded
with
a c
once
ntra
ted
ver
tical
load
P
SAMPLE C
HAPTER
211
t =
10
Co
ef.
δx
−1
−0.
9−
0.8
−0.
7−
0.6
−0.
5−
0.4
−0.
3−
0.2
−0.
10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
000
0−
001
−00
100
200
902
204
006
610
114
610
106
604
002
200
900
2−
001
−00
800
000
0−
007
−00
301
604
809
315
022
130
439
950
0*−
399
−30
4−
221
−15
0−
093
−04
8−
016
003
007
00
−00
8−
003
011
026
039
051
064
077
090
090
102
099
090
077
064
051
039
026
011
−00
3−
008
020
039
059
079
098
118
137
154
169
179
184
179
169
154
137
118
098
079
059
039
020
0.1
0−
001
−00
3−
004
−00
4−
001
006
019
038
065
101
147
103
068
042
024
013
006
003
001
0
0−
017
−01
8−
007
015
050
096
156
230
315
408
507*
−39
5−
301
−21
6−
143
−08
7−
048
−02
7−
014
0
−02
6−
009
005
017
028
040
053
067
080
090
097
099
097
090
080
065
047
029
015
011
021
025
042
058
074
091
108
125
142
157
170
179
181
175
162
145
125
104
081
059
036
013
0.2
0−
002
−00
5−
008
−00
9−
008
−00
400
301
603
506
209
814
510
106
604
002
201
100
400
10
0−
025
−03
3−
024
−00
502
105
509
015
623
031
641
251
2*−
391
−29
9−
218
−14
7−
089
−04
6−
018
0
−03
2−
016
002
015
023
029
038
050
065
080
092
099
100
095
087
076
064
051
036
021
017
017
032
047
062
077
093
109
126
143
158
172
181
184
179
167
151
132
112
091
070
049
0.3
0−
002
−00
5−
009
−01
2−
013
−01
2−
008
−00
101
303
206
009
714
309
906
403
802
100
900
20
0−
003
−04
1−
034
−02
0−
002
022
056
103
164
237
322
413
509*
−39
4−
300
−21
5−
144
−09
0−
048
0
−04
7−
019
001
011
016
021
029
040
054
067
079
088
094
097
096
090
079
063
046
041
063
016
028
040
052
065
079
094
109
125
141
157
170
179
182
177
165
150
131
112
092
072
0.4
0−
001
−00
3−
007
−01
0−
014
−01
5−
014
−01
1−
002
011
030
057
093
138
093
058
033
015
004
0
0−
016
−03
1−
038
−03
6−
025
−00
602
205
810
416
223
131
140
350
2*−
397
−30
0−
215
−14
3−
078
0
−01
6−
016
−01
100
300
601
502
303
204
105
206
307
508
609
610
110
009
207
806
606
709
5
−00
300
902
103
304
605
907
308
810
412
113
815
416
817
818
317
917
015
714
212
610
9
M
0.5
0−
001
−00
5−
009
−01
3−
017
−02
0−
022
−02
1−
017
−01
000
302
204
808
212
608
004
502
000
50
Q0
−02
9−
040
−04
2−
040
−03
4−
024
−00
601
905
400
015
522
230
039
149
1*−
404
−29
9−
200
−10
60
p−
041
−01
8−
006
000
004
008
014
021
030
040
050
061
073
085
096
104
106
102
095
096
123
~00
601
402
303
104
105
106
207
408
810
311
913
615
316
818
018
718
718
117
316
315
3
M
0.6
0−
001
−00
3−
007
−01
2−
016
−02
0−
024
−02
6−
026
−02
4−
018
−00
701
003
406
711
006
403
000
80
Q0
−01
8−
034
−04
2−
044
−04
3−
038
−03
0−
016
007
040
083
137
203
283
376
482*
−40
2−
280
−15
10
p−
013
−01
9−
012
−00
500
000
300
601
101
802
803
804
806
007
308
610
011
211
912
413
617
6
~00
600
801
001
802
603
504
605
707
008
510
111
913
815
717
619
220
421
021
221
221
2
M
0.7
000
0−
003
−00
6−
010
−01
5−
020
−02
6−
030
−03
4−
036
−03
6−
032
−02
4−
1001
204
208
404
601
10
Q0
−01
5−
030
−04
0−
045
−04
9−
052
−05
1−
045
−03
1−
009
020
059
109
173
250
361
486*
−37
2−
208
0
p−
007
−01
7−
012
−00
7−
004
−00
4−
002
003
010
018
026
034
044
056
073
094
115
134
151
179
250
~00
300
801
301
802
403
003
704
605
606
808
209
911
813
916
118
420
622
624
125
326
5
M
0.8
000
000
200
601
001
502
102
703
404
104
705
205
605
605
204
102
200
805
201
40
Q0
−01
0−
025
−03
8−
048
−05
5−
061
−06
6−
068
−06
7−
060
−04
5−
020
018
071
144
240
364
524*
−27
00
p−
004
−01
4−
014
−01
1−
008
−00
7−
005
−00
4−
001
004
011
020
031
045
062
083
109
141
181
234
313
~−
008
−00
400
000
500
901
402
002
803
704
806
207
909
912
315
018
021
424
828
231
334
4
M
0.9
000
0−
002
−00
5−
009
−01
5−
021
−02
9−
038
−04
7−
058
−06
8−
078
−08
6−
092
−09
2−
085
−06
7−
035
018
0
Q0
−01
0−
024
−03
6−
047
−05
9−
070
−08
2−
093
−10
1−
106
−10
4−
093
−07
0−
032
−02
911
724
241
665
7*0
p−
006
−01
3−
013
−01
2−
012
−01
1−
011
−01
1−
010
007
−00
200
601
603
004
907
310
514
720
328
541
2
~−
003
−00
200
000
100
106
601
001
502
203
104
305
907
910
413
517
121
426
131
326
742
1
M
1.0
000
0−
002
−00
4−
008
−01
4−
021
−03
0−
041
−05
4−
068
−08
4−
100
−11
7−
137
−43
−14
8−
142
−12
2−
078
0
Q0
−00
8−
020
−03
3−
048
−06
4−
081
−09
9−
118
−13
6−
152
−16
3−
167
−15
9−
135
−08
6−
005
120
308
585
1000
*
p−
004
−01
101
3−
014
−01
5−
016
−01
8−
019
−01
8−
017
014
008
001
015
035
068
101
153
227
336
507
~−
001
−00
1−
002
−00
2−
002
−00
2−
001
002
006
014
024
039
059
086
120
162
214
275
345
422
504
SAMPLE C
HAPTER
212
t =
1
Co
ef.
δx
−1
−0.
9−
0.8
−0.
7−
0.6
−0.
5−
0.4
−0.
3−
0.2
−0.
10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
M
0
0−
009
−03
2−
066
−11
0−
162
−22
0−
285
−35
3−
426
500*
426
353
285
220
162
110
066
032
009
0
Q0
−16
2−
292
−39
6−
581
−55
2−
614
−66
7−
709
−73
6−
745
−73
6−
709
−66
7−
614
−55
2−
481
−39
6−
292
−16
20
p−
180
−14
5−
115
−09
2−
077
−06
6−
058
−04
8−
035
−01
80
018
035
048
058
066
077
093
115
145
180
~−
190
−17
5−
159
−14
3−
127
−10
9−
091
−07
1−
050
−02
60
026
050
071
091
109
127
143
159
175
190
M
0.1
0−
008
−02
9−
061
−10
2−
151
−20
7−
268
−33
5−
406
−47
9−
553
373
302
235
173
119
072
035
010
0
Q0
−14
6−
270
−37
1−
454
−52
4−
586
−64
1−
685
−72
3−
742
−74
3−
725
−69
3−
644
−58
3−
510
−42
2−
316
−18
00
p−
155
−13
6−
112
−09
1−
075
−06
6−
059
−05
2−
041
−02
7−
010
008
026
041
055
067
080
096
119
154
210
~−
199
−18
4−
168
−15
3−
136
−12
0−
102
−08
3−
062
−03
9−
013
015
045
071
095
119
141
162
183
204
224
M
0.2
0−
007
−02
6−
056
−09
6−
142
−19
6−
255
−31
9−
388
−46
0−
534
−60
831
924
918
412
607
703
701
00
Q0
−13
5−
250
−34
9−
431
−50
2−
564
−61
9−
667
−70
6−
732
−74
4−
739
−71
6−
675
−61
5−
540
−44
8−
338
−19
60
p−
145
−12
5−
107
−09
0−
076
−06
6−
058
−05
1−
043
−03
3−
020
−00
401
403
205
106
808
310
012
316
323
6
~−
189
−17
6−
162
−14
8−
134
−12
0−
104
−08
7−
069
−04
8−
025
001
030
060
088
115
140
164
188
212
236
M
0.3
0−
006
−02
5−
053
−09
0−
135
−18
6−
243
−30
6−
373
−44
3−
516
−59
0−
665
263
196
135
082
040
011
0
Q0
−12
9−
236
−32
8−
409
−48
1−
544
−60
0−
648
−68
8−
719
−73
9−
745
−77
3−
701
−54
8−
573
−47
7−
360
−21
10
p−
143
−11
6−
099
−08
6−
076
−06
7−
060
−05
2−
044
−03
6−
025
−01
300
202
104
306
508
510
513
017
325
9
~−
170
−15
9−
148
−13
7−
125
−11
3−
101
−08
7−
071
−05
3−
033
−01
001
604
607
610
613
416
418
821
424
1
M
0.4
0−
006
−02
4−
052
−08
7−
130
−18
0−
236
−29
7−
362
−43
1−
503
−57
7−
652
725
205
141
086
041
011
1
Q0
−12
7−
229
−31
5−
394
−46
6−
530
−58
6−
633
−67
3−
706
−73
1−
745
−74
372
8−
673
−60
0−
502
−37
8−
219
0
p−
152
−11
1−
092
−08
2−
076
−06
9−
060
−05
1−
044
−03
7−
025
−02
0−
006
012
011
060
086
111
139
182
265
~−
152
−14
3−
134
−12
5−
116
−10
7−
096
−08
5−
071
−05
7−
039
−01
900
403
106
209
512
715
718
621
624
6
0.5
0−
009
−03
2−
063
−09
914
2−
198
−24
4−
302
−36
4−
429
−49
7−
568
−54
0−
711
−78
115
409
604
801
40
0−
175
−27
0−
338
−39
9−
457
−51
0−
557
−59
9−
636
−66
8−
696
−71
5−
721
−71
0−
677
−61
8−
534
−42
0−
258
0
−24
8−
122
−07
6−
063
−05
9−
056
−05
0−
044
−03
9−
035
−03
0−
024
−01
400
102
204
607
109
813
319
134
0
−14
2−
134
−12
7−
120
−11
2−
104
−09
5−
085
−07
3−
059
−04
3−
025
−00
302
205
108
512
015
519
022
325
7
0.6
0−
007
−02
6−
054
−08
8−
130
−17
8−
231
−28
9−
352
−41
8−
488
−56
0−
533
−70
7−
778
−84
409
604
701
30
0−
138
−23
4−
313
−38
3−
417
−50
6−
559
−60
6−
647
−68
2−
710
−72
9−
735
−72
4−
693
−63
6−
548
−42
3−
249
0
−17
3−
112
−08
6−
073
−06
7−
062
−05
6−
050
−01
4−
038
−03
2−
024
−01
300
102
004
307
110
514
720
630
0
−14
6−
139
−13
1−
123
−11
6−
107
−09
8−
087
−07
6−
062
−04
6−
028
−00
601
904
808
112
016
120
124
128
1
0.7
0−
005
−01
9−
041
−07
2−
110
−15
5−
201
−26
4−
326
−39
2−
462
−53
5−
610
−68
5−
758
−82
8−
892
054
015
0
0−
094
−18
4−
268
−34
6−
418
−48
3−
543
−59
6−
644
−68
4−
717
−73
9−
749
−74
5−
722
−67
5−
598
−47
4−
286
0
−09
3−
093
−08
7−
081
−07
5−
068
−06
2−
057
−05
1−
044
−03
6−
027
−01
7−
003
013
034
061
099
151
220
353
−15
1−
143
−13
5−
127
−11
9−
110
−10
1−
190
−07
8−
064
−04
8−
030
−00
801
704
608
011
816
220
825
430
0
0.8
0−
006
−02
3−
048
−08
1−
121
−16
7−
219
−27
6−
339
−40
5−
475
−54
7−
621
−69
4−
766
−83
4−
896
−94
901
40
0−
116
−21
2−
294
−36
6−
430
−49
1−
547
−59
9−
645
−68
3−
712
−73
1−
737
−73
9−
705
−65
7−
578
−45
5−
271
0
−12
8−
105
−08
8−
076
−06
8−
062
−05
8−
054
−04
9−
042
−03
4−
024
−01
300
0−
015
−03
506
209
915
022
232
6
−15
7−
148
−14
0−
132
−12
3−
113
−10
4−
092
−08
0−
066
−04
9−
030
−00
801
804
708
112
016
421
426
732
0
0.9
0−
006
−02
2−
048
−08
1−
120
−16
6−
218
−27
6−
338
−40
4−
474
−54
6−
620
−69
4−
766
−83
4−
896
−94
8−
985
0
0−
115
−21
0−
293
−36
5−
431
−49
1−
547
−59
8−
644
−68
2−
712
−73
1−
738
−73
1−
706
−65
7−
578
−45
6−
274
0
−12
6−
104
−08
8−
077
−06
9−
063
−05
8−
054
−04
8−
042
−03
4−
025
−01
400
001
603
606
209
814
822
133
5
−15
3−
146
−13
7−
129
−12
1−
112
−10
2−
092
−08
0−
066
049
−03
1−
009
016
045
079
117
161
210
265
324
1.0
0−
006
−02
2−
048
−08
1−
120
−16
7−
219
−27
6−
338
−40
4−
474
−54
7−
620
−69
4−
766
−83
4−
896
−94
8−
993
−10
3
0−
116
−21
1−
292
−36
5−
432
−49
2−
548
−59
9−
644
−68
2−
711
−73
1−
739
−73
2−
706
−65
6−
576
−45
5−
274
0
−12
9−
104
−08
7−
077
−06
9−
063
−05
8−
053
−04
8−
041
−03
4−
025
−01
4−
001
016
036
063
098
147
220
336
150
−14
2−
134
−12
7−
118
−11
0−
101
−09
0−
079
−06
5−
049
−03
1−
010
005
044
077
115
158
207
263
324
Tab
le 4
.10
Dim
ensi
onle
ss c
oeffi
cien
ts f
or a
naly
sis
of s
hort
bea
ms
load
ed w
ith a
con
cent
rate
d m
omen
t m
SAMPLE C
HAPTER
213
t =
1
Co
ef.
δx
−1
−0.
9−
0.8
−0.
7−
0.6
−0.
5−
0.4
−0.
3−
0.2
−0.
10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
0−
009
−03
2−
066
−11
0−
162
−22
0−
285
−35
3−
426
500*
426
353
285
220
162
110
066
032
009
0
0−
162
−29
2−
396
−58
1−
552
−61
4−
667
−70
9−
736
−74
5−
736
−70
9−
667
−61
4−
552
−48
1−
396
−29
2−
162
0
−18
0−
145
−11
5−
092
−07
7−
066
−05
8−
048
−03
5−
018
001
803
504
805
806
607
709
311
514
518
0
−19
0−
175
−15
9−
143
−12
7−
109
−09
1−
071
−05
0−
026
002
605
007
109
110
912
714
315
917
519
0
0.1
0−
008
−02
9−
061
−10
2−
151
−20
7−
268
−33
5−
406
−47
9−
553
373
302
235
173
119
072
035
010
0
0−
146
−27
0−
371
−45
4−
524
−58
6−
641
−68
5−
723
−74
2−
743
−72
5−
693
−64
4−
583
−51
0−
422
−31
6−
180
0
−15
5−
136
−11
2−
091
−07
5−
066
−05
9−
052
−04
1−
027
−01
000
802
604
105
506
708
009
611
915
421
0
−19
9−
184
−16
8−
153
−13
6−
120
−10
2−
083
−06
2−
039
−01
301
504
507
109
511
914
116
218
320
422
4
0.2
0−
007
−02
6−
056
−09
6−
142
−19
6−
255
−31
9−
388
−46
0−
534
−60
831
924
918
412
607
703
701
00
0−
135
−25
0−
349
−43
1−
502
−56
4−
619
−66
7−
706
−73
2−
744
−73
9−
716
−67
5−
615
−54
0−
448
−33
8−
196
0
−14
5−
125
−10
7−
090
−07
6−
066
−05
8−
051
−04
3−
033
−02
0−
004
014
032
051
068
083
100
123
163
236
−18
9−
176
−16
2−
148
−13
4−
120
−10
4−
087
−06
9−
048
−02
500
103
006
008
811
514
016
418
821
223
6
0.3
0−
006
−02
5−
053
−09
0−
135
−18
6−
243
−30
6−
373
−44
3−
516
−59
0−
665
263
196
135
082
040
011
0
0−
129
−23
6−
328
−40
9−
481
−54
4−
600
−64
8−
688
−71
9−
739
−74
5−
773
−70
1−
548
−57
3−
477
−36
0−
211
0
−14
3−
116
−09
9−
086
−07
6−
067
−06
0−
052
−04
4−
036
−02
5−
013
002
021
043
065
085
105
130
173
259
−17
0−
159
−14
8−
137
−12
5−
113
−10
1−
087
−07
1−
053
−03
3−
010
016
046
076
106
134
164
188
214
241
0.4
0−
006
−02
4−
052
−08
7−
130
−18
0−
236
−29
7−
362
−43
1−
503
−57
7−
652
725
205
141
086
041
011
1
0−
127
−22
9−
315
−39
4−
466
−53
0−
586
−63
3−
673
−70
6−
731
−74
5−
743
728
−67
3−
600
−50
2−
378
−21
90
−15
2−
111
−09
2−
082
−07
6−
069
−06
0−
051
−04
4−
037
−02
5−
020
−00
601
201
106
008
611
113
918
226
5
−15
2−
143
−13
4−
125
−11
6−
107
−09
6−
085
−07
1−
057
−03
9−
019
004
031
062
095
127
157
186
216
246
M
0.5
0−
009
−03
2−
063
−09
914
2−
198
−24
4−
302
−36
4−
429
−49
7−
568
−54
0−
711
−78
115
409
604
801
40
Q0
−17
5−
270
−33
8−
399
−45
7−
510
−55
7−
599
−63
6−
668
−69
6−
715
−72
1−
710
−67
7−
618
−53
4−
420
−25
80
p−
248
−12
2−
076
−06
3−
059
−05
6−
050
−04
4−
039
−03
5−
030
−02
4−
014
001
022
046
071
098
133
191
340
~−
142
−13
4−
127
−12
0−
112
−10
4−
095
−08
5−
073
−05
9−
043
−02
5−
003
022
051
085
120
155
190
223
257
M
0.6
0−
007
−02
6−
054
−08
8−
130
−17
8−
231
−28
9−
352
−41
8−
488
−56
0−
533
−70
7−
778
−84
409
604
701
30
Q0
−13
8−
234
−31
3−
383
−41
7−
506
−55
9−
606
−64
7−
682
−71
0−
729
−73
5−
724
−69
3−
636
−54
8−
423
−24
90
p−
173
−11
2−
086
−07
3−
067
−06
2−
056
−05
0−
014
−03
8−
032
−02
4−
013
001
020
043
071
105
147
206
300
~−
146
−13
9−
131
−12
3−
116
−10
7−
098
−08
7−
076
−06
2−
046
−02
8−
006
019
048
081
120
161
201
241
281
M
0.7
0−
005
−01
9−
041
−07
2−
110
−15
5−
201
−26
4−
326
−39
2−
462
−53
5−
610
−68
5−
758
−82
8−
892
054
015
0
Q0
−09
4−
184
−26
8−
346
−41
8−
483
−54
3−
596
−64
4−
684
−71
7−
739
−74
9−
745
−72
2−
675
−59
8−
474
−28
60
p−
093
−09
3−
087
−08
1−
075
−06
8−
062
−05
7−
051
−04
4−
036
−02
7−
017
−00
301
303
406
109
915
122
035
3
~−
151
−14
3−
135
−12
7−
119
−11
0−
101
−19
0−
078
−06
4−
048
−03
0−
008
017
046
080
118
162
208
254
300
M
0.8
0−
006
−02
3−
048
−08
1−
121
−16
7−
219
−27
6−
339
−40
5−
475
−54
7−
621
−69
4−
766
−83
4−
896
−94
901
40
Q0
−11
6−
212
−29
4−
366
−43
0−
491
−54
7−
599
−64
5−
683
−71
2−
731
−73
7−
739
−70
5−
657
−57
8−
455
−27
10
p−
128
−10
5−
088
−07
6−
068
−06
2−
058
−05
4−
049
−04
2−
034
−02
4−
013
000
−01
5−
035
062
099
150
222
326
~−
157
−14
8−
140
−13
2−
123
−11
3−
104
−09
2−
080
−06
6−
049
−03
0−
008
018
047
081
120
164
214
267
320
M
0.9
0−
006
−02
2−
048
−08
1−
120
−16
6−
218
−27
6−
338
−40
4−
474
−54
6−
620
−69
4−
766
−83
4−
896
−94
8−
985
0
Q0
−11
5−
210
−29
3−
365
−43
1−
491
−54
7−
598
−64
4−
682
−71
2−
731
−73
8−
731
−70
6−
657
−57
8−
456
−27
40
p−
126
−10
4−
088
−07
7−
069
−06
3−
058
−05
4−
048
−04
2−
034
−02
5−
014
000
016
036
062
098
148
221
335
~−
153
−14
6−
137
−12
9−
121
−11
2−
102
−09
2−
080
−06
604
9−
031
−00
901
604
507
911
716
121
026
532
4
M
1.0
0−
006
−02
2−
048
−08
1−
120
−16
7−
219
−27
6−
338
−40
4−
474
−54
7−
620
−69
4−
766
−83
4−
896
−94
8−
993
−10
3
Q0
−11
6−
211
−29
2−
365
−43
2−
492
−54
8−
599
−64
4−
682
−71
1−
731
−73
9−
732
−70
6−
656
−57
6−
455
−27
40
p−
129
−10
4−
087
−07
7−
069
−06
3−
058
−05
3−
048
−04
1−
034
−02
5−
014
−00
101
603
606
309
814
722
033
6
~15
0−
142
−13
4−
127
−11
8−
110
−10
1−
090
−07
9−
065
−04
9−
031
−01
000
504
407
711
515
820
726
332
4
SAMPLE C
HAPTER
214
t =
2
Co
ef.
δx
−1
−0.
9−
0.8
−0.
7−
0.6
−0.
5−
0.4
−0.
3−
0.2
−0.
10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
M
0
0−
008
−03
0−
063
−10
6−
156
−21
4−
278
−34
8−
423
−50
042
334
827
821
415
610
606
303
000
80
Q0
−15
1−
279
−38
2−
466
−54
0−
609
−67
3−
731
−76
4−
777
−76
4−
731
−67
3−
609
−54
0−
466
−38
2−
279
−15
10
p−
157
−14
1−
115
−09
2−
078
−07
1−
067
−06
0−
046
−02
50
025
046
000
067
071
078
092
115
141
157
~−
171
−16
0−
149
−13
7−
126
−11
3−
097
−07
9−
057
−03
10
031
057
079
097
113
126
137
149
160
171
M
0.1
0−
009
−03
2−
068
−11
2−
164
−22
3−
286
−35
4−
424
−49
5−
567
362
294
229
169
116
070
034
009
0
Q0
−16
8−
298
−40
2−
486
−55
5−
611
−65
6−
689
−71
0−
719
−71
4−
697
−66
7−
624
−56
9−
499
−41
4−
309
−17
60
p−
192
−14
7−
115
−09
3−
076
−06
3−
050
−03
9−
027
−01
5−
002
011
024
036
049
062
077
094
117
151
205
~−
192
−18
1−
169
−15
9−
146
−13
2−
116
−09
9−
078
−05
3−
023
013
049
080
108
132
154
175
195
214
233
M
0.2
0−
009
−03
2−
067
−11
1−
162
−22
0−
283
−35
0−
420
−49
1−
563
−63
329
823
217
211
807
103
400
90
Q0
−16
6−
294
−39
7−
481
−55
0−
606
−65
1−
684
−70
6−
718
−71
5−
700
−67
3−
631
−57
6−
506
−42
0−
314
−17
90
p−
190
−14
4−
114
−09
2−
076
−06
2−
050
−03
9−
028
−01
6−
005
008
021
034
048
062
078
095
118
153
210
~−
175
−16
7−
159
−15
1−
142
−13
2−
120
−10
6−
090
−06
9−
045
−01
402
306
109
412
515
317
920
422
825
3
M
0.3
0−
008
−03
1−
066
−10
9−
161
−21
8−
280
−34
7−
416
−48
7−
559
−62
9−
698
236
174
100
072
035
010
0
Q0
−16
4−
291
−39
2−
476
−54
6−
602
−64
7−
680
−70
3−
714
−71
4−
702
−67
6−
637
−58
3−
513
−42
6−
319
−18
30
p−
190
−14
2−
113
−09
2−
076
−06
3−
050
−03
9−
028
−01
7−
006
006
019
032
046
062
078
096
112
155
216
~−
140
−13
6−
133
−12
9−
125
−12
0−
113
−10
4−
093
−07
8−
059
−03
4−
004
033
073
108
141
172
201
230
260
M
0.4
0−
008
−03
1−
065
−10
9−
159
−21
7−
279
−34
5−
414
−48
4−
557
−62
7−
696
−76
217
612
107
303
501
00
Q0
−16
4−
289
−38
9−
473
−54
2−
598
−64
3−
677
−69
9−
711
−71
2−
702
−67
9−
641
−58
9−
519
−43
2−
323
−18
40
p−
191
−14
1−
111
−09
1−
076
−06
3−
051
−03
9−
028
−01
7−
007
009
017
030
045
061
078
097
122
158
217
~−
104
−10
5−
106
−10
6−
106
−10
6−
104
−09
9−
093
−08
3−
069
−05
0−
025
006
045
087
125
162
197
232
267
0.5
0−
009
−03
3−
068
−11
1−
162
−21
9−
280
−34
6−
414
−48
4−
554
−62
5−
693
−75
9−
821
124
075
037
010
0
0−
173
−29
7−
394
−47
4−
540
−59
4−
637
−67
0−
692
−70
4−
705
−69
6−
674
−63
9−
590
−52
3−
438
−33
1−
193
0
−21
0−
143
−10
8−
087
−07
3−
060
−04
9−
038
−02
7−
017
−00
700
401
502
804
305
807
509
512
116
023
2
−08
7−
090
−09
4−
096
−09
8−
099
−10
0−
098
−09
4−
087
−07
6−
060
−03
9−
012
023
067
114
158
202
244
287
0.6
0−
009
−03
2−
067
−10
9−
159
−21
6−
217
−34
3−
411
−48
1−
552
−62
3−
692
−75
8−
820
−87
607
503
601
00
0−
165
−29
0−
388
−47
0−
538
−59
3−
637
−67
1−
694
−70
6−
708
−69
9−
677
−64
3−
593
−52
7−
442
−33
3−
191
0
−19
5−
141
−11
0−
089
−07
4−
062
−05
0−
039
−02
6−
018
−00
700
401
502
704
205
707
509
612
416
222
5
−09
7−
100
−10
2−
104
−10
5−
106
−10
5−
103
−09
9−
091
−08
0−
064
−04
3−
017
018
061
113
170
225
278
332
0.7
0−
008
−03
0−
063
−10
5−
155
−21
1−
272
−33
7−
405
−47
5−
546
−61
7−
686
−75
3−
815
−87
2−
922
038
010
0
0−
156
−27
8−
378
−46
1−
530
−58
8−
633
−66
8−
693
−70
7−
710
−70
1−
681
−64
8−
600
−53
6−
453
−34
4−
199
0
178
−13
7−
110
−09
1−
076
−06
3−
051
−04
0−
030
−01
9−
008
003
014
027
040
055
073
095
125
168
236
−10
8−
109
−11
1−
111
−11
2−
112
−11
1−
108
−10
2−
095
−08
3−
067
−04
6−
019
015
057
109
171
237
303
369
0.8
0−
008
−03
1−
065
−10
7−
157
−21
3−
275
−34
0−
408
−47
8−
549
−62
0−
689
−75
5−
817
−87
4−
923
−96
301
00
0−
161
−28
5−
384
−46
5−
534
−59
0−
635
−66
9−
693
−70
7−
709
−70
0−
678
−64
4−
596
−53
2−
448
−34
0−
196
0
−18
6−
140
−11
0−
090
−07
4−
062
−05
0−
040
−02
9−
019
−00
900
301
502
704
105
607
309
512
416
623
0
−12
0−
120
−12
0−
120
−12
0−
118
−11
6−
112
−10
6−
097
−08
4−
067
−04
5−
017
018
061
113
176
246
328
416
0.9
0−
008
−03
1−
054
−10
7−
157
−21
3−
275
−34
0−
408
−47
8−
549
−62
0−
689
−75
5−
817
−87
4−
923
−96
2−
990
0
0−
160
−28
4−
384
−46
6−
534
−59
0−
635
−66
9−
693
−70
6−
709
−70
0−
679
−64
5−
596
−53
2−
448
−34
0−
196
0
−18
5−
139
−11
0−
090
−07
4−
062
−05
0−
040
−02
9−
019
−00
800
301
502
704
105
607
309
512
416
623
2
−11
4−
115
−11
6−
116
−11
6−
115
−11
3−
110
−10
4−
096
−08
5−
069
−04
8−
020
014
056
108
169
241
326
416
1.0
0−
008
−03
1−
064
−10
7−
157
−21
3−
275
−34
0−
408
−47
8−
549
−62
0−
689
−75
5−
817
−87
4−
923
−96
2−
990
−10
3
0−
168
−28
4−
384
−46
6−
534
−59
0−
635
−66
9−
693
−70
6−
709
−70
0−
679
−64
5−
596
−53
2−
448
−33
9−
196
0
−18
6−
139
−11
0−
090
−07
5−
062
−05
0−
040
−02
9−
019
−00
800
301
502
704
105
607
409
512
416
623
2
−10
6−
108
−10
9−
111
−11
1−
111
−11
1−
108
−10
3−
095
−08
5−
069
−04
9−
022
011
053
103
164
236
320
416
Tab
le 4
.11
Dim
ensi
onle
ss c
oeffi
cien
ts f
or a
naly
sis
of s
hort
bea
ms
load
ed w
ith a
mom
ent m
SAMPLE C
HAPTER
215
t =
2
Co
ef.
δx
−1
−0.
9−
0.8
−0.
7−
0.6
−0.
5−
0.4
−0.
3−
0.2
−0.
10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
0−
008
−03
0−
063
−10
6−
156
−21
4−
278
−34
8−
423
−50
042
334
827
821
415
610
606
303
000
80
0−
151
−27
9−
382
−46
6−
540
−60
9−
673
−73
1−
764
−77
7−
764
−73
1−
673
−60
9−
540
−46
6−
382
−27
9−
151
0
−15
7−
141
−11
5−
092
−07
8−
071
−06
7−
060
−04
6−
025
002
504
600
006
707
107
809
211
514
115
7
−17
1−
160
−14
9−
137
−12
6−
113
−09
7−
079
−05
7−
031
003
105
707
909
711
312
613
714
916
017
1
0.1
0−
009
−03
2−
068
−11
2−
164
−22
3−
286
−35
4−
424
−49
5−
567
362
294
229
169
116
070
034
009
0
0−
168
−29
8−
402
−48
6−
555
−61
1−
656
−68
9−
710
−71
9−
714
−69
7−
667
−62
4−
569
−49
9−
414
−30
9−
176
0
−19
2−
147
−11
5−
093
−07
6−
063
−05
0−
039
−02
7−
015
−00
201
102
403
604
906
207
709
411
715
120
5
−19
2−
181
−16
9−
159
−14
6−
132
−11
6−
099
−07
8−
053
−02
301
304
908
010
813
215
417
519
521
423
3
0.2
0−
009
−03
2−
067
−11
1−
162
−22
0−
283
−35
0−
420
−49
1−
563
−63
329
823
217
211
807
103
400
90
0−
166
−29
4−
397
−48
1−
550
−60
6−
651
−68
4−
706
−71
8−
715
−70
0−
673
−63
1−
576
−50
6−
420
−31
4−
179
0
−19
0−
144
−11
4−
092
−07
6−
062
−05
0−
039
−02
8−
016
−00
500
802
103
404
806
207
809
511
815
321
0
−17
5−
167
−15
9−
151
−14
2−
132
−12
0−
106
−09
0−
069
−04
5−
014
023
061
094
125
153
179
204
228
253
0.3
0−
008
−03
1−
066
−10
9−
161
−21
8−
280
−34
7−
416
−48
7−
559
−62
9−
698
236
174
100
072
035
010
0
0−
164
−29
1−
392
−47
6−
546
−60
2−
647
−68
0−
703
−71
4−
714
−70
2−
676
−63
7−
583
−51
3−
426
−31
9−
183
0
−19
0−
142
−11
3−
092
−07
6−
063
−05
0−
039
−02
8−
017
−00
600
601
903
204
606
207
809
611
215
521
6
−14
0−
136
−13
3−
129
−12
5−
120
−11
3−
104
−09
3−
078
−05
9−
034
−00
403
307
310
814
117
220
123
026
0
0.4
0−
008
−03
1−
065
−10
9−
159
−21
7−
279
−34
5−
414
−48
4−
557
−62
7−
696
−76
217
612
107
303
501
00
0−
164
−28
9−
389
−47
3−
542
−59
8−
643
−67
7−
699
−71
1−
712
−70
2−
679
−64
1−
589
−51
9−
432
−32
3−
184
0
−19
1−
141
−11
1−
091
−07
6−
063
−05
1−
039
−02
8−
017
−00
700
901
703
004
506
107
809
712
215
821
7
−10
4−
105
−10
6−
106
−10
6−
106
−10
4−
099
−09
3−
083
−06
9−
050
−02
500
604
508
712
516
219
723
226
7
M
0.5
0−
009
−03
3−
068
−11
1−
162
−21
9−
280
−34
6−
414
−48
4−
554
−62
5−
693
−75
9−
821
124
075
037
010
0
Q0
−17
3−
297
−39
4−
474
−54
0−
594
−63
7−
670
−69
2−
704
−70
5−
696
−67
4−
639
−59
0−
523
−43
8−
331
−19
30
p−
210
−14
3−
108
−08
7−
073
−06
0−
049
−03
8−
027
−01
7−
007
004
015
028
043
058
075
095
121
160
232
~−
087
−09
0−
094
−09
6−
098
−09
9−
100
−09
8−
094
−08
7−
076
−06
0−
039
−01
202
306
711
415
820
224
428
7
M
0.6
0−
009
−03
2−
067
−10
9−
159
−21
6−
217
−34
3−
411
−48
1−
552
−62
3−
692
−75
8−
820
−87
607
503
601
00
Q0
−16
5−
290
−38
8−
470
−53
8−
593
−63
7−
671
−69
4−
706
−70
8−
699
−67
7−
643
−59
3−
527
−44
2−
333
−19
10
p−
195
−14
1−
110
−08
9−
074
−06
2−
050
−03
9−
026
−01
8−
007
004
015
027
042
057
075
096
124
162
225
~−
097
−10
0−
102
−10
4−
105
−10
6−
105
−10
3−
099
−09
1−
080
−06
4−
043
−01
701
806
111
317
022
527
833
2
M
0.7
0−
008
−03
0−
063
−10
5−
155
−21
1−
272
−33
7−
405
−47
5−
546
−61
7−
686
−75
3−
815
−87
2−
922
038
010
0
Q0
−15
6−
278
−37
8−
461
−53
0−
588
−63
3−
668
−69
3−
707
−71
0−
701
−68
1−
648
−60
0−
536
−45
3−
344
−19
90
p17
8−
137
−11
0−
091
−07
6−
063
−05
1−
040
−03
0−
019
−00
800
301
402
704
005
507
309
512
516
823
6
~−
108
−10
9−
111
−11
1−
112
−11
2−
111
−10
8−
102
−09
5−
083
−06
7−
046
−01
901
505
710
917
123
730
336
9
M
0.8
0−
008
−03
1−
065
−10
7−
157
−21
3−
275
−34
0−
408
−47
8−
549
−62
0−
689
−75
5−
817
−87
4−
923
−96
301
00
Q0
−16
1−
285
−38
4−
465
−53
4−
590
−63
5−
669
−69
3−
707
−70
9−
700
−67
8−
644
−59
6−
532
−44
8−
340
−19
60
p−
186
−14
0−
110
−09
0−
074
−06
2−
050
−04
0−
029
−01
9−
009
003
015
027
041
056
073
095
124
166
230
~−
120
−12
0−
120
−12
0−
120
−11
8−
116
−11
2−
106
−09
7−
084
−06
7−
045
−01
701
806
111
317
624
632
841
6
M
0.9
0−
008
−03
1−
054
−10
7−
157
−21
3−
275
−34
0−
408
−47
8−
549
−62
0−
689
−75
5−
817
−87
4−
923
−96
2−
990
0
Q0
−16
0−
284
−38
4−
466
−53
4−
590
−63
5−
669
−69
3−
706
−70
9−
700
−67
9−
645
−59
6−
532
−44
8−
340
−19
60
p−
185
−13
9−
110
−09
0−
074
−06
2−
050
−04
0−
029
−01
9−
008
003
015
027
041
056
073
095
124
166
232
~−
114
−11
5−
116
−11
6−
116
−11
5−
113
−11
0−
104
−09
6−
085
−06
9−
048
−02
001
405
610
816
924
132
641
6
M
1.0
0−
008
−03
1−
064
−10
7−
157
−21
3−
275
−34
0−
408
−47
8−
549
−62
0−
689
−75
5−
817
−87
4−
923
−96
2−
990
−10
3
Q0
−16
8−
284
−38
4−
466
−53
4−
590
−63
5−
669
−69
3−
706
−70
9−
700
−67
9−
645
−59
6−
532
−44
8−
339
−19
60
p−
186
−13
9−
110
−09
0−
075
−06
2−
050
−04
0−
029
−01
9−
008
003
015
027
041
056
074
095
124
166
232
~−
106
−10
8−
109
−11
1−
111
−11
1−
111
−10
8−
103
−09
5−
085
−06
9−
049
−02
201
105
310
316
423
632
041
6
SAMPLE C
HAPTER
216
t =
5
Co
ef.
δx
−1
−0.
9−
0.8
−0.
7−
0.6
−0.
5−
0.4
−0.
3−
0.2
−0.
10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
M
0
0−
008
−02
6−
055
−09
4−
141
−19
6−
260
−33
3−
414
−50
0−
414
333
260
196
141
094
055
026
007
0
Q0
−11
8−
243
−34
4−
427
−50
6−
594
−68
9−
777
−84
0−
864
−84
0−
777
−68
9−
594
−50
6−
427
−34
4−
243
−11
80
p−
092
−13
0−
114
−08
9−
078
−08
3−
093
−09
4−
077
−04
500
004
507
709
409
308
307
808
911
413
009
2
~−
113
−11
7−
120
−12
2−
122
−12
0−
113
−09
9−
078
0045
000
045
078
099
113
120
122
122
120
117
113
M
0.1
0−
004
−01
6−
039
−07
0−
109
−15
5−
210
−27
6−
351
−43
3−
519
395
314
241
176
120
073
036
011
0
Q0
−06
4−
175
−27
4−
351
−42
4−
507
−60
4−
705
−79
2−
848
−86
3−
837
−77
7−
096
−60
6−
514
−42
0−
314
−18
10
p−
002
−10
3−
109
−08
6−
077
−07
6−
091
−10
1−
097
−07
4−
037
006
045
073
087
092
092
097
116
154
210
~−
187
−18
4−
181
−17
8−
173
−16
6−
155
−14
0−
118
−08
7−
045
011
068
112
146
172
193
210
224
237
250
M
0.2
0−
001
−00
9−
025
−05
0−
083
−12
3−
172
−22
9−
297
−37
3−
456
−54
337
028
621
014
408
904
401
30
Q0
−03
3−
118
−20
8−
289
−36
4−
443
−53
2−
627
−72
1−
802
−85
6−
876
−85
7−
800
714
610
500
383
236
0
p01
5−
069
−09
2−
087
−07
7−
076
−08
3−
093
097
−08
9−
070
−03
900
003
907
309
710
811
212
617
930
5
~−
161
−16
1−
163
−16
4−
164
−16
2−
158
−15
0−
136
−11
5−
084
−04
2−
015
073
120
157
188
214
237
259
280
M
0.3
0−
001
−00
7−
020
−04
0−
067
−10
1−
144
−19
6−
257
−32
6−
405
−49
0−
580
330
244
167
102
050
014
0
Q0
−03
9−
095
−16
1−
232
−30
8−
389
−47
3−
561
−65
3−
741
−82
2−
882
−90
8−
880
−82
2−
715
−58
4−
442
−27
30
p−
026
−04
9−
062
−06
9−
074
−07
8−
082
−08
6−
090
−09
1−
086
−07
2−
045
−00
504
308
912
213
714
920
237
2
~−
084
−09
4−
104
−11
4−
123
−13
0−
136
−13
8−
136
−12
7−
111
−08
3−
043
012
071
118
158
193
224
254
283
M
0.4
0−
001
−00
7−
017
−03
3−
055
−08
6−
125
−17
1−
226
−28
0−
363
−44
5−
535
−63
027
618
911
405
501
40
Q0
−04
8−
078
−12
3−
189
−26
7−
348
−42
7−
505
−58
9−
681
−77
8−
868
−93
3−
952
−91
4−
817
−67
6−
504
−29
80
p−
077
−03
0−
034
−05
6−
074
−08
0−
083
−08
5−
086
−08
8−
096
−09
6−
080
−04
500
806
812
215
818
523
638
8
~−
005
−02
4−
062
−06
0−
078
−09
5−
110
−12
2−
130
−13
3−
128
−11
4−
089
−04
900
706
812
016
620
824
828
8
0.5
0−
001
−00
5−
013
−02
5−
044
−07
1−
106
−14
8−
198
−25
7−
325
−40
5−
493
−58
9−
688
218
133
064
017
0
0−
038
−05
7−
094
−15
6−
232
−31
0−
385
−45
9−
540
−63
3−
737
−84
2−
931
−98
1−
976
−90
6−
775
−59
0−
347
0
−06
8−
019
−02
5−
050
−07
1−
073
−07
7−
079
−07
9−
086
−09
9−
107
−10
0−
073
−02
503
710
215
921
128
243
5
028
−00
6−
016
−03
7−
058
−07
9−
098
−11
4−
128
−13
6−
138
−13
2−
116
−08
6−
041
022
092
155
213
270
325
0.6
0−
001
−00
6−
013
−02
3−
040
−06
3−
−93
−13
2−
178
−23
4−
300
−37
6−
461
−55
1−
654
−75
315
407
602
10
0−
034
−05
4−
084
−13
3−
196
−26
7−
343
−42
3−
510
−60
6−
708
−81
0−
902
−97
1−
999
−97
1−
870
−68
3−
389
0
−05
6−
021
−02
2−
039
−05
7−
068
−07
4−
078
−08
3−
091
−09
9−
103
−09
9−
083
−05
1−
003
062
142
233
337
469
−00
4−
023
−04
1−
059
−07
7−
094
−11
0−
124
−13
5−
142
−14
3−
136
−12
0−
092
−04
901
109
218
126
634
742
8
0.7
0−
001
−00
4−
009
−01
9−
034
−05
4−
081
−11
7−
161
−21
5−
278
−35
2−
435
−52
7−
625
−72
6−
824
089
026
0
0−
016
−04
1−
075
−11
9−
173
−23
8−
312
−39
6−
488
−58
6−
687
−78
6−
878
−95
4−
998
−96
5−
939
−77
4−
475
0
−01
0−
020
−02
9−
039
−04
9−
059
−07
0−
079
−08
8−
095
−10
0−
100
−09
7−
085
−06
4−
029
027
110
226
379
580
−03
7−
052
−06
6−
081
−09
6−
110
−12
2−
133
−14
2−
146
−14
6−
139
−12
2−
095
−05
400
408
218
329
440
351
0
0.8
0−
001
−00
3−
009
−01
8−
031
−05
0−
076
−11
0−
153
−20
7−
270
−34
2−
424
−51
4−
611
−71
2−
813
−90
302
80
0−
008
−03
9−
074
−11
2−
159
−22
0−
297
−38
6−
483
−58
2−
679
−77
2−
860
−93
8−
997
−01
8−
972
−82
0−
514
0
017
−02
5−
034
−03
5−
041
−05
3−
069
−08
4−
094
−09
9−
098
−09
5−
091
−08
4−
071
−04
400
709
222
040
064
0
−07
2−
083
−09
4−
105
−11
6−
126
−13
6−
143
−14
8−
149
−14
6−
136
−11
7−
088
−04
501
409
319
432
246
260
1
0.9
000
0−
002
−00
7−
016
−02
9−
049
−07
5−
108
−15
2−
204
−26
7−
340
−42
2−
512
−61
0−
711
−81
0−
902
−97
10
0−
002
−03
1−
068
−11
2−
163
−22
4−
298
−38
4−
478
−57
7−
677
−77
4−
864
−94
3−
998
−99
6−
970
−82
6−
528
0
021
−02
0−
035
−04
0−
046
−05
6−
068
−08
0−
090
−09
7−
099
−09
9−
094
−08
5−
069
−04
000
908
821
039
668
1
−05
6−
068
−08
1−
094
−10
6−
118
−12
9−
138
−14
5−
148
−14
7−
139
−12
3−
095
−05
500
207
917
830
245
662
4
1.0
0−
000
−00
2−
007
−01
6−
029
−04
9−
076
−11
0−
153
−20
5−
258
−34
0−
422
−51
3−
611
−71
2−
812
−90
2−
971
−10
3
0−
007
−03
1−
066
−11
1−
165
−22
9−
303
−38
6−
436
−57
3−
674
−77
4−
868
−94
7−
103
−11
14−
965
−82
0−
528
0
005
−01
7−
030
−04
0−
049
−05
9−
069
−07
8−
087
−09
4−
099
−10
1−
098
−08
8−
069
−03
601
409
020
739
169
1
036
−05
0−
065
−98
0−
094
−10
8−
121
−13
2−
141
−14
6−
147
−14
1−
126
−10
1−
062
−00
606
816
628
944
062
3
Tab
le 4
.12
Dim
ensi
onle
ss c
oeffi
cien
ts f
or a
naly
sis
of s
hort
bea
ms
load
ed w
ith a
con
cent
rate
d m
omen
t m
SAMPLE C
HAPTER
217
t =
5
Co
ef.
δx
−1
−0.
9−
0.8
−0.
7−
0.6
−0.
5−
0.4
−0.
3−
0.2
−0.
10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
0−
008
−02
6−
055
−09
4−
141
−19
6−
260
−33
3−
414
−50
0−
414
333
260
196
141
094
055
026
007
0
0−
118
−24
3−
344
−42
7−
506
−59
4−
689
−77
7−
840
−86
4−
840
−77
7−
689
−59
4−
506
−42
7−
344
−24
3−
118
0
−09
2−
130
−11
4−
089
−07
8−
083
−09
3−
094
−07
7−
045
000
045
077
094
093
083
078
089
114
130
092
−11
3−
117
−12
0−
122
−12
2−
120
−11
3−
099
−07
800
4500
004
507
809
911
312
012
212
212
011
711
3
0.1
0−
004
−01
6−
039
−07
0−
109
−15
5−
210
−27
6−
351
−43
3−
519
395
314
241
176
120
073
036
011
0
0−
064
−17
5−
274
−35
1−
424
−50
7−
604
−70
5−
792
−84
8−
863
−83
7−
777
−09
6−
606
−51
4−
420
−31
4−
181
0
−00
2−
103
−10
9−
086
−07
7−
076
−09
1−
101
−09
7−
074
−03
700
604
507
308
709
209
209
711
615
421
0
−18
7−
184
−18
1−
178
−17
3−
166
−15
5−
140
−11
8−
087
−04
501
106
811
214
617
219
321
022
423
725
0
0.2
0−
001
−00
9−
025
−05
0−
083
−12
3−
172
−22
9−
297
−37
3−
456
−54
337
028
621
014
408
904
401
30
0−
033
−11
8−
208
−28
9−
364
−44
3−
532
−62
7−
721
−80
2−
856
−87
6−
857
−80
071
461
050
038
323
60
015
−06
9−
092
−08
7−
077
−07
6−
083
−09
309
7−
089
−07
0−
039
000
039
073
097
108
112
126
179
305
−16
1−
161
−16
3−
164
−16
4−
162
−15
8−
150
−13
6−
115
−08
4−
042
−01
507
312
015
718
821
423
725
928
0
0.3
0−
001
−00
7−
020
−04
0−
067
−10
1−
144
−19
6−
257
−32
6−
405
−49
0−
580
330
244
167
102
050
014
0
0−
039
−09
5−
161
−23
2−
308
−38
9−
473
−56
1−
653
−74
1−
822
−88
2−
908
−88
0−
822
−71
5−
584
−44
2−
273
0
−02
6−
049
−06
2−
069
−07
4−
078
−08
2−
086
−09
0−
091
−08
6−
072
−04
5−
005
043
089
122
137
149
202
372
−08
4−
094
−10
4−
114
−12
3−
130
−13
6−
138
−13
6−
127
−11
1−
083
−04
301
207
111
815
819
322
425
428
3
0.4
0−
001
−00
7−
017
−03
3−
055
−08
6−
125
−17
1−
226
−28
0−
363
−44
5−
535
−63
027
618
911
405
501
40
0−
048
−07
8−
123
−18
9−
267
−34
8−
427
−50
5−
589
−68
1−
778
−86
8−
933
−95
2−
914
−81
7−
676
−50
4−
298
0
−07
7−
030
−03
4−
056
−07
4−
080
−08
3−
085
−08
6−
088
−09
6−
096
−08
0−
045
008
068
122
158
185
236
388
−00
5−
024
−06
2−
060
−07
8−
095
−11
0−
122
−13
0−
133
−12
8−
114
−08
9−
049
007
068
120
166
208
248
288
M
0.5
0−
001
−00
5−
013
−02
5−
044
−07
1−
106
−14
8−
198
−25
7−
325
−40
5−
493
−58
9−
688
218
133
064
017
0
Q0
−03
8−
057
−09
4−
156
−23
2−
310
−38
5−
459
−54
0−
633
−73
7−
842
−93
1−
981
−97
6−
906
−77
5−
590
−34
70
p−
068
−01
9−
025
−05
0−
071
−07
3−
077
−07
9−
079
−08
6−
099
−10
7−
100
−07
3−
025
037
102
159
211
282
435
~02
8−
006
−01
6−
037
−05
8−
079
−09
8−
114
−12
8−
136
−13
8−
132
−11
6−
086
−04
102
209
215
521
327
032
5
M
0.6
0−
001
−00
6−
013
−02
3−
040
−06
3−
−93
−13
2−
178
−23
4−
300
−37
6−
461
−55
1−
654
−75
315
407
602
10
Q0
−03
4−
054
−08
4−
133
−19
6−
267
−34
3−
423
−51
0−
606
−70
8−
810
−90
2−
971
−99
9−
971
−87
0−
683
−38
90
p−
056
−02
1−
022
−03
9−
057
−06
8−
074
−07
8−
083
−09
1−
099
−10
3−
099
−08
3−
051
−00
306
214
223
333
746
9
~−
004
−02
3−
041
−05
9−
077
−09
4−
110
−12
4−
135
−14
2−
143
−13
6−
120
−09
2−
049
011
092
181
266
347
428
M
0.7
0−
001
−00
4−
009
−01
9−
034
−05
4−
081
−11
7−
161
−21
5−
278
−35
2−
435
−52
7−
625
−72
6−
824
089
026
0
Q0
−01
6−
041
−07
5−
119
−17
3−
238
−31
2−
396
−48
8−
586
−68
7−
786
−87
8−
954
−99
8−
965
−93
9−
774
−47
50
p−
010
−02
0−
029
−03
9−
049
−05
9−
070
−07
9−
088
−09
5−
100
−10
0−
097
−08
5−
064
−02
902
711
022
637
958
0
~−
037
−05
2−
066
−08
1−
096
−11
0−
122
−13
3−
142
−14
6−
146
−13
9−
122
−09
5−
054
004
082
183
294
403
510
M
0.8
0−
001
−00
3−
009
−01
8−
031
−05
0−
076
−11
0−
153
−20
7−
270
−34
2−
424
−51
4−
611
−71
2−
813
−90
302
80
Q0
−00
8−
039
−07
4−
112
−15
9−
220
−29
7−
386
−48
3−
582
−67
9−
772
−86
0−
938
−99
7−
018
−97
2−
820
−51
40
p01
7−
025
−03
4−
035
−04
1−
053
−06
9−
084
−09
4−
099
−09
8−
095
−09
1−
084
−07
1−
044
007
092
220
400
640
~−
072
−08
3−
094
−10
5−
116
−12
6−
136
−14
3−
148
−14
9−
146
−13
6−
117
−08
8−
045
014
093
194
322
462
601
M
0.9
000
0−
002
−00
7−
016
−02
9−
049
−07
5−
108
−15
2−
204
−26
7−
340
−42
2−
512
−61
0−
711
−81
0−
902
−97
10
Q0
−00
2−
031
−06
8−
112
−16
3−
224
−29
8−
384
−47
8−
577
−67
7−
774
−86
4−
943
−99
8−
996
−97
0−
826
−52
80
p02
1−
020
−03
5−
040
−04
6−
056
−06
8−
080
−09
0−
097
−09
9−
099
−09
4−
085
−06
9−
040
009
088
210
396
681
~−
056
−06
8−
081
−09
4−
106
−11
8−
129
−13
8−
145
−14
8−
147
−13
9−
123
−09
5−
055
002
079
178
302
456
624
M
1.0
0−
000
−00
2−
007
−01
6−
029
−04
9−
076
−11
0−
153
−20
5−
258
−34
0−
422
−51
3−
611
−71
2−
812
−90
2−
971
−10
3
Q0
−00
7−
031
−06
6−
111
−16
5−
229
−30
3−
386
−43
6−
573
−67
4−
774
−86
8−
947
−10
3−
1114
−96
5−
820
−52
80
p00
5−
017
−03
0−
040
−04
9−
059
−06
9−
078
−08
7−
094
−09
9−
101
−09
8−
088
−06
9−
036
014
090
207
391
691
~03
6−
050
−06
5−
980
−09
4−
108
−12
1−
132
−14
1−
146
−14
7−
141
−12
6−
101
−06
2−
006
068
166
289
440
623
SAMPLE C
HAPTER
218
t =
10
Co
ef.
δx
−1
−0.
9−
0.8
−0.
7−
0.6
−0.
5−
0.4
−0.
3−
0.2
−0.
10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
M
0
0−
006
−01
9−
044
−07
8−
119
−17
0−
234
−31
2−
402
−50
0−
402
−31
2−
234
−17
0−
119
−07
8−
044
−01
9−
006
0
Q0
−07
0−
193
−29
3−
370
−45
6−
570
−70
9−
849
−95
3−
991
−95
3−
849
−70
9−
570
−45
6−
370
−29
3−
193
−07
00
p01
1−
119
−11
6−
085
−07
6−
098
−12
9−
145
−12
8−
075
007
512
814
512
909
807
608
511
611
901
1
~−
029
−05
2−
074
−09
5−
113
−12
6−
132
−12
6−
106
−06
60
066
106
126
132
126
113
095
074
052
029
M
0.1
000
0−
005
−02
1−
015
−07
7−
116
−16
8−
234
−31
5−
407
−50
539
930
923
116
611
206
903
501
20
Q0
010
−10
4−
208
−25
0−
350
−44
9−
584
−73
6−
873
−96
1−
983
−03
7−
841
−71
8−
594
−48
3−
386
−28
9−
167
0
p15
6−
086
−12
0−
085
−06
5−
081
−11
8−
148
−15
0−
117
057
013
075
114
127
119
103
093
105
143
191
~−
208
−21
0−
213
−21
5−
216
−21
4−
207
−19
2−
167
−12
606
602
010
716
821
023
625
426
226
626
827
0
M
0.2
000
300
4−
003
−02
0−
044
−07
7−
119
−17
4−
242
−32
4−
416
−51
438
829
621
4−
145
090
016
014
0
Q0
045
−03
0−
126
−20
8−
283
−37
0−
481
−61
2−
751
−87
3−
959
−99
1−
962
−87
7−
754
−61
8−
492
−38
1−
248
0
p15
4−
037
−09
6−
090
−07
6−
078
−09
8−
123
−13
8−
134
−10
7−
061
−00
205
810
713
413
411
611
117
035
0
~−
151
−19
5−
195
−19
5−
200
−20
3−
204
−19
9−
187
−16
4−
125
−06
6−
020
107
169
212
242
252
277
288
298
M
0.3
000
300
300
0−
010
−02
8−
053
−05
8−
133
−19
0−
260
−34
2−
436
−53
736
026
217
710
705
401
50
Q0
014
−01
4−
067
−13
4−
212
−30
0−
398
−50
8−
631
−76
0−
884
−98
3−
1033
−10
15−
924
−77
8−
614
−46
3−
302
0
p01
7−
011
−04
3−
061
−07
2−
083
−09
3−
104
−11
7−
127
−12
9−
116
−07
8−
018
056
123
161
160
145
197
459
~−
044
−06
2−
081
−10
1−
120
−13
8−
155
−16
8−
175
−17
4−
161
−13
1−
080
000
081
140
182
213
237
257
276
M
0.4
000
200
000
0−
005
−01
7−
037
−06
7−
104
−15
1−
210
282
−36
9−
470
−57
931
121
012
505
901
50
Q0
−01
9−
004
−02
0−
079
−16
2−
250
−33
4−
421
−52
3−
649
−79
7−
945
−10
61−
1109
−10
68−
941
−75
7−
553
−33
30
p07
601
500
5−
039
−07
5−
081
−08
5−
089
−09
2−
113
−13
6−
153
−13
8−
087
−00
508
716
219
920
724
846
4
~10
206
803
400
1−
033
−06
7−
099
−12
9−
154
−17
3−
185
−17
8−
156
−11
0−
035
046
107
154
194
229
263
0.5
000
200
200
400
3−
005
−02
2−
046
−07
7−
117
−16
7−
231
−31
1−
407
−51
7−
632
255
155
074
019
0
001
001
801
0−
013
−12
3−
204
−27
0−
353
−44
3−
564
−71
6−
884
−10
38−
1139
−11
57−
1078
−91
5−
689
−40
70
073
028
016
−01
3−
071
−07
3−
078
−07
9−
079
−10
4−
137
−16
4−
167
−13
3−
063
030
124
198
251
323
528
154
117
079
042
001
−03
5−
072
−10
9−
142
−17
0−
181
−20
1−
197
−17
2−
122
−03
805
412
919
525
531
6
0.6
000
000
0−
001
−00
1−
003
−01
4−
032
−05
8−
092
−13
8−
197
−27
1−
360
−46
4−
579
−69
719
009
302
50
0−
011
−01
1−
013
−01
9−
074
−14
2−
217
−30
0−
399
−51
9−
661
−81
6−
970
−11
00−
1178
−11
76−
1071
−84
5−
491
0
056
018
017
−01
5−
046
−06
4−
071
−07
8−
090
−10
9−
131
−15
1−
158
−14
6−
108
−04
305
016
328
942
056
7
082
052
022
−00
7−
037
−06
7−
097
−12
5−
152
−17
5−
192
−20
0−
195
−17
3−
128
−05
405
718
129
339
950
2
0.7
000
100
300
500
500
3−
005
−01
8−
040
−07
1−
114
−16
9−
239
−32
2−
421
−53
2−
651
−77
211
703
40
001
902
401
5−
009
−04
8−
103
−17
4−
262
−36
7−
488
−62
1−
765
−91
2−
1050
−11
62−
1219
−11
82−
1002
−62
50
028
011
−00
2−
016
−03
1−
047
−06
3−
080
−09
6−
113
−12
7−
139
−14
6−
145
−12
9−
090
−01
710
026
949
276
5
007
−01
4−
034
−05
5−
076
−09
7−
119
−14
0−
160
−17
8−
191
−19
6−
191
−17
0−
129
−06
104
118
434
549
864
8
0.8
000
100
400
506
600
4−
001
−01
2−
032
−06
2−
104
−15
9−
226
−30
7−
401
−50
8−
628
−75
2−
870
039
0
000
002
201
0−
003
−02
7−
076
−15
1−
249
−36
3−
484
−60
8−
737
−87
2−
1011
−11
42−
1233
−12
35−
1082
−69
50
082
000
−01
4−
021
−01
6−
035
−06
2−
088
−10
7−
118
−12
3−
126
−13
2−
138
−13
8−
117
−05
706
325
752
987
4
−06
7−
080
−09
2−
105
−11
8−
131
−14
4−
158
−17
0−
181
−18
8−
188
−17
7−
152
−10
8−
038
064
206
396
610
821
0.9
000
200
600
901
000
800
2−
010
−03
0−
060
−10
2−
155
−22
2−
303
−39
8−
506
−62
5−
749
−86
6−
959
0
004
203
701
9−
005
−03
8−
087
−15
7−
247
−35
4−
474
−60
3−
739
−87
9−
1018
−11
42−
1226
−12
29−
1091
−72
30
087
−01
0−
016
−02
1−
027
−04
0−
059
−08
0−
099
−11
4−
125
−13
3−
139
−14
1−
135
−10
9−
052
059
236
520
956
−03
6−
051
−06
6−
081
−09
7−
114
−13
1−
148
−16
4−
178
−18
9−
193
−18
7−
167
−12
7−
062
035
172
357
597
865
1.0
000
200
500
901
000
800
1−
012
−03
2−
062
−10
3−
156
−22
2−
304
−40
0−
509
−62
8−
751
−86
7−
960
−10
3
003
303
702
4−
003
−04
2−
096
−16
4−
248
−34
9−
466
−59
7−
740
−88
7−
1028
−11
48−
1224
−12
19−
1081
−72
30
−05
5−
015
−00
5−
020
−03
3−
046
−06
1−
076
−09
2−
109
−12
4−
138
−14
6−
146
−13
4−
102
−04
206
023
051
097
7
004
−01
5−
034
−05
3−
073
−09
4−
115
−13
6−
156
−17
5−
189
−19
6−
194
−17
7−
141
−08
001
414
833
056
786
4
Tab
le 4
.13
Dim
ensi
onle
ss c
oeffi
cien
ts f
or a
naly
sis
of s
hort
bea
ms
load
ed w
ith a
mom
ent m
SAMPLE C
HAPTER
219
t =
10
Co
ef.
δx
−1
−0.
9−
0.8
−0.
7−
0.6
−0.
5−
0.4
−0.
3−
0.2
−0.
10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
0−
006
−01
9−
044
−07
8−
119
−17
0−
234
−31
2−
402
−50
0−
402
−31
2−
234
−17
0−
119
−07
8−
044
−01
9−
006
0
0−
070
−19
3−
293
−37
0−
456
−57
0−
709
−84
9−
953
−99
1−
953
−84
9−
709
−57
0−
456
−37
0−
293
−19
3−
070
0
011
−11
9−
116
−08
5−
076
−09
8−
129
−14
5−
128
−07
50
075
128
145
129
098
076
085
116
119
011
−02
9−
052
−07
4−
095
−11
3−
126
−13
2−
126
−10
6−
066
006
610
612
613
212
611
309
507
405
202
9
0.1
000
0−
005
−02
1−
015
−07
7−
116
−16
8−
234
−31
5−
407
−50
539
930
923
116
611
206
903
501
20
001
0−
104
−20
8−
250
−35
0−
449
−58
4−
736
−87
3−
961
−98
3−
037
−84
1−
718
−59
4−
483
−38
6−
289
−16
70
156
−08
6−
120
−08
5−
065
−08
1−
118
−14
8−
150
−11
705
701
307
511
412
711
910
309
310
514
319
1
−20
8−
210
−21
3−
215
−21
6−
214
−20
7−
192
−16
7−
126
066
020
107
168
210
236
254
262
266
268
270
0.2
000
300
4−
003
−02
0−
044
−07
7−
119
−17
4−
242
−32
4−
416
−51
438
829
621
4−
145
090
016
014
0
004
5−
030
−12
6−
208
−28
3−
370
−48
1−
612
−75
1−
873
−95
9−
991
−96
2−
877
−75
4−
618
−49
2−
381
−24
80
154
−03
7−
096
−09
0−
076
−07
8−
098
−12
3−
138
−13
4−
107
−06
1−
002
058
107
134
134
116
111
170
350
−15
1−
195
−19
5−
195
−20
0−
203
−20
4−
199
−18
7−
164
−12
5−
066
−02
010
716
921
224
225
227
728
829
8
0.3
000
300
300
0−
010
−02
8−
053
−05
8−
133
−19
0−
260
−34
2−
436
−53
736
026
217
710
705
401
50
001
4−
014
−06
7−
134
−21
2−
300
−39
8−
508
−63
1−
760
−88
4−
983
−10
33−
1015
−92
4−
778
−61
4−
463
−30
20
017
−01
1−
043
−06
1−
072
−08
3−
093
−10
4−
117
−12
7−
129
−11
6−
078
−01
805
612
316
116
014
519
745
9
−04
4−
062
−08
1−
101
−12
0−
138
−15
5−
168
−17
5−
174
−16
1−
131
−08
000
008
114
018
221
323
725
727
6
0.4
000
200
000
0−
005
−01
7−
037
−06
7−
104
−15
1−
210
282
−36
9−
470
−57
931
121
012
505
901
50
0−
019
−00
4−
020
−07
9−
162
−25
0−
334
−42
1−
523
−64
9−
797
−94
5−
1061
−11
09−
1068
−94
1−
757
−55
3−
333
0
076
015
005
−03
9−
075
−08
1−
085
−08
9−
092
−11
3−
136
−15
3−
138
−08
7−
005
087
162
199
207
248
464
102
068
034
001
−03
3−
067
−09
9−
129
−15
4−
173
−18
5−
178
−15
6−
110
−03
504
610
715
419
422
926
3
M
0.5
000
200
200
400
3−
005
−02
2−
046
−07
7−
117
−16
7−
231
−31
1−
407
−51
7−
632
255
155
074
019
0
Q0
010
018
010
−01
3−
123
−20
4−
270
−35
3−
443
−56
4−
716
−88
4−
1038
−11
39−
1157
−10
78−
915
−68
9−
407
0
p07
302
801
6−
013
−07
1−
073
−07
8−
079
−07
9−
104
−13
7−
164
−16
7−
133
−06
303
012
419
825
132
352
8
~15
411
707
904
200
1−
035
−07
2−
109
−14
2−
170
−18
1−
201
−19
7−
172
−12
2−
038
054
129
195
255
316
M
0.6
000
000
0−
001
−00
1−
003
−01
4−
032
−05
8−
092
−13
8−
197
−27
1−
360
−46
4−
579
−69
719
009
302
50
Q0
−01
1−
011
−01
3−
019
−07
4−
142
−21
7−
300
−39
9−
519
−66
1−
816
−97
0−
1100
−11
78−
1176
−10
71−
845
−49
10
p05
601
801
7−
015
−04
6−
064
−07
1−
078
−09
0−
109
−13
1−
151
−15
8−
146
−10
8−
043
050
163
289
420
567
~08
205
202
2−
007
−03
7−
067
−09
7−
125
−15
2−
175
−19
2−
200
−19
5−
173
−12
8−
054
057
181
293
399
502
M
0.7
000
100
300
500
500
3−
005
−01
8−
040
−07
1−
114
−16
9−
239
−32
2−
421
−53
2−
651
−77
211
703
40
Q0
019
024
015
−00
9−
048
−10
3−
174
−26
2−
367
−48
8−
621
−76
5−
912
−10
50−
1162
−12
19−
1182
−10
02−
625
0
p02
801
1−
002
−01
6−
031
−04
7−
063
−08
0−
096
−11
3−
127
−13
9−
146
−14
5−
129
−09
0−
017
100
269
492
765
~00
7−
014
−03
4−
055
−07
6−
097
−11
9−
140
−16
0−
178
−19
1−
196
−19
1−
170
−12
9−
061
041
184
345
498
648
M
0.8
000
100
400
506
600
4−
001
−01
2−
032
−06
2−
104
−15
9−
226
−30
7−
401
−50
8−
628
−75
2−
870
039
0
Q0
000
022
010
−00
3−
027
−07
6−
151
−24
9−
363
−48
4−
608
−73
7−
872
−10
11−
1142
−12
33−
1235
−10
82−
695
0
p08
200
0−
014
−02
1−
016
−03
5−
062
−08
8−
107
−11
8−
123
−12
6−
132
−13
8−
138
−11
7−
057
063
257
529
874
~−
067
−08
0−
092
−10
5−
118
−13
1−
144
−15
8−
170
−18
1−
188
−18
8−
177
−15
2−
108
−03
806
420
639
661
082
1
M
0.9
000
200
600
901
000
800
2−
010
−03
0−
060
−10
2−
155
−22
2−
303
−39
8−
506
−62
5−
749
−86
6−
959
0
Q0
042
037
019
−00
5−
038
−08
7−
157
−24
7−
354
−47
4−
603
−73
9−
879
−10
18−
1142
−12
26−
1229
−10
91−
723
0
p08
7−
010
−01
6−
021
−02
7−
040
−05
9−
080
−09
9−
114
−12
5−
133
−13
9−
141
−13
5−
109
−05
205
923
652
095
6
~−
036
−05
1−
066
−08
1−
097
−11
4−
131
−14
8−
164
−17
8−
189
−19
3−
187
−16
7−
127
−06
203
517
235
759
786
5
M
1.0
000
200
500
901
000
800
1−
012
−03
2−
062
−10
3−
156
−22
2−
304
−40
0−
509
−62
8−
751
−86
7−
960
−10
3
Q0
033
037
024
−00
3−
042
−09
6−
164
−24
8−
349
−46
6−
597
−74
0−
887
−10
28−
1148
−12
24−
1219
−10
81−
723
0
p−
055
−01
5−
005
−02
0−
033
−04
6−
061
−07
6−
092
−10
9−
124
−13
8−
146
−14
6−
134
−10
2−
042
060
230
510
977
~00
4−
015
−03
4−
053
−07
3−
094
−11
5−
136
−15
6−
175
−18
9−
196
−19
4−
177
−14
1−
080
014
148
330
567
864
SAMPLE C
HAPTER
220 Analysis of Structures on Elastic Foundations
tx
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
M
1 032 032 031 028 0.25 021 016 012 007 002 0
2 0.27 0.27 0.27 0.24 0.21 0.19 0.15 0.11 0.06 0.02 0
5 020 020 019 017 0.16 014 011 008 004 002 0
10 012 012 012 012 011 010 009 006 004 001 0
Q
1 0 −009 −019 −028 −035 −043 −049 −052 −049 −036 0
2 0 −008 −016 −022 −030 −036 −042 −045 −046 −032 0
5 0 −004 −010 −015 −020 −024 −030 −033 −033 −023 0
10 0 −0.02 −005 −008 −010 −015 −019 −023 −026 −020 0
p
1 090 090 091 091 092 093 095 100 107 122 153
2 092 092 092 092 093 094 096 098 116 147 198
5 096 095 095 095 095 094 094 093 102 114 141
10 098 097 097 096 093 095 096 096 100 110 134
xY
1 221 220 220 219 218 217 217 216 215 213 212
2 222 222 221 220 220 218 217 214 212 211 208
5 226 226 225 224 223 218 217 213 210 205 201
10 230 230 229 227 224 221 216 212 207 202 196
Table 4.14 Dimensionless coefficients for analysis of beams loaded with a uniformly distributed load
p 0 0.2 0.4 06 0.8 1.0 1.2 1.4 1.6 2.0 2.4 2.8 3.0
p 1.487 1.322 0.981 0.772 0.582 0.418 0.284 0.179 0.100 0.004 −0.034 −0.042 −0.042
p 446 400 294 230 175 125 85 54 30 1.2 −10 −12 −12
Y 2.57 2.32 2.07 1.76 1.44 1.16 0.89 0.67 0.49 0.24 0.12 0.05 0.04
Y 3.86 3.48 3.11 2.64 2.16 1.74 1.34 1.00 0.74 0.36 0.18 0.08 0.06
Table 4.15 Soil pressure and settlements
p 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6
Q 0.00 0.051 0.118 0.200 0.296 0.397 0.500 −0.400 −0.310 −0.231 −0.164 −0.11 −0.067 −0.036
Q 0.0 4.1 9.4 16 24 32 40/−40 −32 −24.8 −18.5 −13.1 −8.8 −5.4 −2.9
Table 4.16 Shear forces
SAMPLE C
HAPTER
Analysis of Beams on Elastic Half-Space 221
β =
0.0
25
ζα
00.
10.
20.
30.
40.
50.
60.
70.
80.
91.
0∞
p
0.0
3.22
72.
747
2.25
91.
828
1.40
21.
101
0.86
70.
723
0.57
00.
434
0.27
30.
761
0.1
2.62
62.
296
1.96
31.
644
1.33
51.
057
0.82
50.
643
0.48
20.
340
0.21
90.
733
0.2
2.10
51.
805
1.68
31.
465
1.24
81.
034
0.82
90.
643
0.48
00.
342
0.22
60.
668
0.3
1.65
81.
538
1.42
11.
291
1.15
91.
106
0.84
60.
680
0.53
70.
395
0.28
40.
602
0.4
1.28
21.
229
1.18
21.
121
1.05
90.
967
0.85
70.
727
0.58
80.
480
0.37
10.
552
0.5
0.96
30.
959
0.96
20.
958
0.94
70.
911
0.85
10.
765
0.66
90.
569
0.47
10.
509
0.6
0.69
80.
730
0.76
80.
802
0.83
10.
839
0.82
50.
775
0.72
30.
649
0.57
30.
456
0.7
0.48
10.
538
0.60
30.
660
0.71
50.
755
0.77
50.
772
o.75
00.
711
0.65
90.
388
0.8
0.30
60.
376
0.45
40.
528
0.60
10.
665
0.71
40.
745
0.75
70.
751
0.72
40.
310
0.9
0.16
50.
246
0.32
80.
413
0.49
40.
575
0.64
50.
703
0.74
20.
759
0.73
60.
233
1.0
0.05
70.
141
0.23
00.
314
0.40
10.
484
0.56
30.
634
0.69
30.
737
0.76
20.
164
1.2
−0.
085
−0.
003
0.07
90.
160
0.24
30.
325
0.40
90.
494
0.57
60.
650
0.71
20.
065
1.4
−0.
152
−0.
083
−0.
012
0.05
70.
127
0.19
90.
275
0.35
30.
432
0.50
70.
583
0.01
3
1.6
−0.
170
−0.
115
−0.
062
−0.
006
0.04
70.
106
0.16
70.
233
0.30
00.
367
0.43
8−
0.00
9
1.8
−0.
158
−0.
121
−0.
083
−0.
040
−0.
003
0.04
30.
089
0.14
10.
192
0.24
30.
301
−0.
016
2.0
−0.
134
−0.
109
−0.
084
−0.
055
−0.
030
0.00
20.
035
0.07
30.
110
0.14
901
92−
0.01
6
2.2
−0.
106
−0.
091
−0.
074
−0.
058
−0.
042
−0.
023
−0.
002
0.02
40.
051
0.07
90.
111
−0.
014
2.4
−0.
076
−0.
068
−0.
059
−0.
051
−0.
043
−0.
034
−0.
019
−0.
008
0.01
30.
028
0.05
0−
0.01
2
2.6
−0.
052
−0.
050
−0.
045
−0.
042
−0.
039
−0.
035
−0.
031
−0.
017
−0.
013
−0.
009
−0.
003
−0.
009
2.8
−0.
032
−0.
033
−0.
033
−0.
033
−0.
035
−0.
035
−0.
034
−0.
032
−0.
031
−0.
031
−0.
031
−0.
007
3.0
−0.
021
−0.
020
−0.
020
−0.
024
−0.
028
−0.
034
−0.
040
−0.
044
−0.
039
−0.
039
−0.
030
−0.
005
Tab
le 4
.17
Dim
ensi
onle
ss c
oeffi
cien
ts f
or a
naly
sis
of lo
ng b
eam
s lo
aded
with
a c
once
ntra
ted
ver
tical
load
P
SAMPLE C
HAPTER
222 Analysis of Structures on Elastic Foundations
β =
0.0
25
ζα
00.
10.
20.
30.
40.
50.
60.
70.
80.
91.
0∞
M
0.0
00
00
00
00
00
00.
176
0.1
0.08
50.
013
0.01
10.
007
0.00
70.
006
0.00
60.
005
0.00
40.
003
0.00
20.
130
0.2
−0.
142
−0.
050
0.04
10.
033
0.02
80.
022
0.01
80.
015
0.01
10.
007
0.00
50.
091
0.3
−0.
181
−0.
094
−0.
011
0.07
20.
059
0.04
80.
039
0.03
10.
022
0.01
70.
011
0.05
8
0.4
−0.
202
−0.
124
−0.
050
0.02
60.
104
0.08
50.
068
0.05
40.
041
0.03
00.
020
0.03
2
0.5
−0.
209
−0.
142
−0.
076
−0.
007
0.05
90.
131
0.10
50.
085
0.06
50.
046
0.03
10.
011
0.6
−0.
207
−0.
150
−0.
092
−0.
035
0.02
20.
085
0.15
20.
122
0.09
20.
070
0.04
8−
0.00
6
0.7
−0.
198
−0.
150
−0.
102
−0.
054
−0.
004
0.05
00.
107
0.16
80.
131
0.10
00.
070
−0.
016
0.8
−0.
185
−0.
144
−0.
105
−0.
067
−0.
024
0.02
00.
068
0.12
00.
176
0.13
70.
100
−0.
023
0.9
−0.
168
−0.
135
−0.
105
−0.
072
−0.
039
−0.
002
0.03
90.
081
0.13
00.
179
0.13
7−
0.02
7
1.0
−0.
150
−0.
124
−0.
100
−0.
074
−0.
048
−0.
018
0.01
50.
050
0.08
90.
135
0.18
1−
0.02
9
1.2
−0.
113
−0.
098
−0.
085
−0.
070
−0.
056
−0.
039
−0.
018
0.00
40.
030
0.05
90.
092
−0.
028
1.4
−0.
080
−0.
072
−0.
067
−0.
061
−0.
054
−0.
044
−0.
033
−0.
020
−0.
007
0.01
10.
031
−0.
024
1.6
−0.
050
−0.
054
−0.
048
−0.
048
−0.
046
−0.
043
−0.
039
−0.
033
−0.
028
−0.
018
−0.
007
−0.
020
1.8
−0.
028
−0.
030
−0.
033
−0.
035
−0.
035
−0.
037
−0.
037
−0.
035
−0.
034
−0.
031
−0.
026
−0.
016
2.0
−0.
013
−0.
017
−0.
020
−0.
024
−0.
026
−0.
031
−0.
031
−0.
032
−0.
035
−0.
035
−0.
035
−0.
012
2.2
−0.
004
−0.
007
−0.
011
−0.
015
−0.
018
−0.
024
−0.
024
−0.
026
−0.
030
−0.
033
−0.
035
−0.
010
2.4
0.00
20.
000
−0.
006
−0.
007
−0.
011
−0.
017
−0.
017
−0.
018
−0.
024
−0.
028
−0.
031
−0.
006
2.6
0.00
50.
002
−0.
002
−0.
003
−0.
005
−0.
009
−0.
009
−0.
013
−0.
017
−0.
020
−0.
024
−0.
006
2.8
0.00
60.
004
0.00
00.
000
−0.
002
−0.
004
−0.
004
−0.
006
−0.
011
−0.
015
−0.
018
−0.
005
3.0
0.00
60.
004
0.00
20.
001
0.00
0−
0.00
1−
0.00
2−
0.00
2−
0.00
5−
0.00
9−
0.01
3−
0.00
4
0.0
0.00
00.
000
0.00
00.
000
0.00
00.
000
0.00
00.
000
0.00
00.
000
0.00
−0.
500
0.1
−0.
708
0.25
20.
220
0.17
40.
137
0.10
60.
083
0.96
60.
050
0.03
80.
026
−0.
425
0.2
−0.
472
−0.
539
0.39
30.
330
0.26
60.
211
0.16
60.
128
0.09
10.
071
0.04
7−
0.35
5
0.3
−0.
284
−0.
367
−0.
452
0.46
70.
387
0.31
30.
250
0.19
40.
146
0.10
70.
072
−0.
292
0.4
−0.
138
−0.
229
−0.
322
−0.
413
0.49
70.
412
0.33
50.
264
0.20
20.
150
0.10
3−
0.23
4
0.5
−0.
026
−0.
120
−0.
215
−0.
309
−0.
402
0.50
60.
420
0.34
00.
266
0.20
20.
144
−0.
181
0.6
0.05
6−
0.03
6−
0.12
9−
0.22
1−
0.31
4−
0.40
70.
504
0.41
70.
386
0.26
30.
197
−0.
132
0.7
0.11
30.
027
−0.
061
−0.
149
−0.
237
−0.
328
−0.
416
0.49
50.
413
0.33
30.
261
−0.
090
0.8
0.15
30.
073
−0.
009
−0.
089
−0.
171
−0.
256
−0.
340
−0.
427
0.48
80.
407
0.33
0−
0.05
5
0.9
0.17
60.
103
0.03
0−
0.04
2−
0.11
6−
0.19
4−
0.27
2−
0.35
4−
0.43
80.
482
0.40
2−
0.02
8
1.0
0.18
70.
122
0.05
8−
0.00
6−
0.07
2−
0.14
1−
0.21
2−
0.28
8−
0.36
5−
0.44
20.
482
−0.
008
1.2
0.18
30.
135
0.08
80.
040
−0.
008
−0.
060
−0.
114
−0.
174
−0.
237
−0.
303
−0.
373
0.01
4
1.4
0.15
90.
126
0.09
30.
061
0.02
8−
0.00
9−
0.04
7−
0.09
0−
0.13
7−
0.18
8−
0.24
30.
021
1.6
0.14
60.
106
0.08
60.
065
0.04
80.
021
−0.
004
−0.
033
−0.
064
−0.
100
−0.
141
0.02
1
1.8
0.09
20.
091
0.07
00.
060
0.04
90.
036
0.02
2−
0.00
4−
0.01
6−
0.04
0−
0.06
70.
018
2.0
0.06
20.
058
0.05
30.
050
0.04
60.
040
0.03
40.
025
0.01
9−
0.00
1−
0.01
80.
014
2.2
0.03
90.
038
0.03
80.
039
0.03
90.
038
0.03
70.
035
0.02
90.
022
0.01
30.
011
2.4
0.02
00.
022
0.02
40.
027
0.03
00.
032
0.03
40.
036
0.03
50.
032
0.02
80.
009
2.6
0.00
80.
011
0.01
40.
018
0.02
20.
025
0.02
90.
032
0.03
30.
033
0.03
20.
007
2.8
−0.
001
0.00
20.
006
0.01
00.
014
0.01
80.
022
0.02
70.
029
0.02
90.
029
0.00
5
3.0
−0.
005
−0.
003
0.00
00.
005
0.00
80.
012
0.01
60.
020
0.02
20.
022
0.02
20.
004
Tab
le 4
.17
Con
’t
SAMPLE C
HAPTER
Analysis of Beams on Elastic Half-Space 223
β =
0.0
25
ζα
00.
10.
20.
30.
40.
50.
60.
70.
80.
91.
0∞
0.0
00
00
00
00
00
00.
176
0.1
0.08
50.
013
0.01
10.
007
0.00
70.
006
0.00
60.
005
0.00
40.
003
0.00
20.
130
0.2
−0.
142
−0.
050
0.04
10.
033
0.02
80.
022
0.01
80.
015
0.01
10.
007
0.00
50.
091
0.3
−0.
181
−0.
094
−0.
011
0.07
20.
059
0.04
80.
039
0.03
10.
022
0.01
70.
011
0.05
8
0.4
−0.
202
−0.
124
−0.
050
0.02
60.
104
0.08
50.
068
0.05
40.
041
0.03
00.
020
0.03
2
0.5
−0.
209
−0.
142
−0.
076
−0.
007
0.05
90.
131
0.10
50.
085
0.06
50.
046
0.03
10.
011
0.6
−0.
207
−0.
150
−0.
092
−0.
035
0.02
20.
085
0.15
20.
122
0.09
20.
070
0.04
8−
0.00
6
0.7
−0.
198
−0.
150
−0.
102
−0.
054
−0.
004
0.05
00.
107
0.16
80.
131
0.10
00.
070
−0.
016
0.8
−0.
185
−0.
144
−0.
105
−0.
067
−0.
024
0.02
00.
068
0.12
00.
176
0.13
70.
100
−0.
023
0.9
−0.
168
−0.
135
−0.
105
−0.
072
−0.
039
−0.
002
0.03
90.
081
0.13
00.
179
0.13
7−
0.02
7
1.0
−0.
150
−0.
124
−0.
100
−0.
074
−0.
048
−0.
018
0.01
50.
050
0.08
90.
135
0.18
1−
0.02
9
1.2
−0.
113
−0.
098
−0.
085
−0.
070
−0.
056
−0.
039
−0.
018
0.00
40.
030
0.05
90.
092
−0.
028
1.4
−0.
080
−0.
072
−0.
067
−0.
061
−0.
054
−0.
044
−0.
033
−0.
020
−0.
007
0.01
10.
031
−0.
024
1.6
−0.
050
−0.
054
−0.
048
−0.
048
−0.
046
−0.
043
−0.
039
−0.
033
−0.
028
−0.
018
−0.
007
−0.
020
1.8
−0.
028
−0.
030
−0.
033
−0.
035
−0.
035
−0.
037
−0.
037
−0.
035
−0.
034
−0.
031
−0.
026
−0.
016
2.0
−0.
013
−0.
017
−0.
020
−0.
024
−0.
026
−0.
031
−0.
031
−0.
032
−0.
035
−0.
035
−0.
035
−0.
012
2.2
−0.
004
−0.
007
−0.
011
−0.
015
−0.
018
−0.
024
−0.
024
−0.
026
−0.
030
−0.
033
−0.
035
−0.
010
2.4
0.00
20.
000
−0.
006
−0.
007
−0.
011
−0.
017
−0.
017
−0.
018
−0.
024
−0.
028
−0.
031
−0.
006
2.6
0.00
50.
002
−0.
002
−0.
003
−0.
005
−0.
009
−0.
009
−0.
013
−0.
017
−0.
020
−0.
024
−0.
006
2.8
0.00
60.
004
0.00
00.
000
−0.
002
−0.
004
−0.
004
−0.
006
−0.
011
−0.
015
−0.
018
−0.
005
3.0
0.00
60.
004
0.00
20.
001
0.00
0−
0.00
1−
0.00
2−
0.00
2−
0.00
5−
0.00
9−
0.01
3−
0.00
4
Q
0.0
0.00
00.
000
0.00
00.
000
0.00
00.
000
0.00
00.
000
0.00
00.
000
0.00
−0.
500
0.1
−0.
708
0.25
20.
220
0.17
40.
137
0.10
60.
083
0.96
60.
050
0.03
80.
026
−0.
425
0.2
−0.
472
−0.
539
0.39
30.
330
0.26
60.
211
0.16
60.
128
0.09
10.
071
0.04
7−
0.35
5
0.3
−0.
284
−0.
367
−0.
452
0.46
70.
387
0.31
30.
250
0.19
40.
146
0.10
70.
072
−0.
292
0.4
−0.
138
−0.
229
−0.
322
−0.
413
0.49
70.
412
0.33
50.
264
0.20
20.
150
0.10
3−
0.23
4
0.5
−0.
026
−0.
120
−0.
215
−0.
309
−0.
402
0.50
60.
420
0.34
00.
266
0.20
20.
144
−0.
181
0.6
0.05
6−
0.03
6−
0.12
9−
0.22
1−
0.31
4−
0.40
70.
504
0.41
70.
386
0.26
30.
197
−0.
132
0.7
0.11
30.
027
−0.
061
−0.
149
−0.
237
−0.
328
−0.
416
0.49
50.
413
0.33
30.
261
−0.
090
0.8
0.15
30.
073
−0.
009
−0.
089
−0.
171
−0.
256
−0.
340
−0.
427
0.48
80.
407
0.33
0−
0.05
5
0.9
0.17
60.
103
0.03
0−
0.04
2−
0.11
6−
0.19
4−
0.27
2−
0.35
4−
0.43
80.
482
0.40
2−
0.02
8
1.0
0.18
70.
122
0.05
8−
0.00
6−
0.07
2−
0.14
1−
0.21
2−
0.28
8−
0.36
5−
0.44
20.
482
−0.
008
1.2
0.18
30.
135
0.08
80.
040
−0.
008
−0.
060
−0.
114
−0.
174
−0.
237
−0.
303
−0.
373
0.01
4
1.4
0.15
90.
126
0.09
30.
061
0.02
8−
0.00
9−
0.04
7−
0.09
0−
0.13
7−
0.18
8−
0.24
30.
021
1.6
0.14
60.
106
0.08
60.
065
0.04
80.
021
−0.
004
−0.
033
−0.
064
−0.
100
−0.
141
0.02
1
1.8
0.09
20.
091
0.07
00.
060
0.04
90.
036
0.02
2−
0.00
4−
0.01
6−
0.04
0−
0.06
70.
018
2.0
0.06
20.
058
0.05
30.
050
0.04
60.
040
0.03
40.
025
0.01
9−
0.00
1−
0.01
80.
014
2.2
0.03
90.
038
0.03
80.
039
0.03
90.
038
0.03
70.
035
0.02
90.
022
0.01
30.
011
2.4
0.02
00.
022
0.02
40.
027
0.03
00.
032
0.03
40.
036
0.03
50.
032
0.02
80.
009
2.6
0.00
80.
011
0.01
40.
018
0.02
20.
025
0.02
90.
032
0.03
30.
033
0.03
20.
007
2.8
−0.
001
0.00
20.
006
0.01
00.
014
0.01
80.
022
0.02
70.
029
0.02
90.
029
0.00
5
3.0
−0.
005
−0.
003
0.00
00.
005
0.00
80.
012
0.01
60.
020
0.02
20.
022
0.02
20.
004
SAMPLE C
HAPTER
224 Analysis of Structures on Elastic Foundations
β =
0.0
25
ζα
00.
10.
20.
30.
40.
50.
60.
70.
80.
91.
0∞
Y
06.
926.
005.
124.
283.
522.
842.
201.
721.
240.
880.
602.
14
0.1
6.00
5.32
4.64
3.96
3.32
2.76
2.28
1.80
1.40
1.08
0.80
2.11
0.2
5.12
4.64
4.16
3.68
3.20
2.88
2.34
1.92
1.56
1.24
1.00
2.02
0.3
4.28
3.96
3.68
3.44
3.08
2.72
2.40
2.08
1.76
1.48
1.2
1.90
0.4
3.52
3.32
3.20
3.08
2.88
2.64
2.43
2.16
1.88
1.64
1.40
1.76
0.5
2.84
2.76
2.88
2.72
2.64
2.56
2.44
2.24
2.04
1.80
1.60
1.60
0.6
2.20
2.28
2.34
2.40
2.43
2.44
2.36
2.24
2.12
1.96
1.78
1.44
0.7
1.72
1.80
1.92
2.08
2.16
2.24
2.24
2.24
2.16
2.04
1.92
1.28
0.8
1.24
1.40
1.56
1.76
1.88
2.04
2.12
2.16
2.20
2.12
2.04
1.13
0.9
0.88
1.08
1.24
1.48
1.64
1.80
1.96
2.04
2.12
2.16
2.12
0.99
1.0
0.60
0.80
1.00
1.20
1.40
1.60
1.76
1.92
2.04
2.12
2.16
0.85
1.2
0.16
0.36
0.56
0.80
1.00
1.20
1.40
1.56
1.76
1.52
2.04
0.62
1.4
−0.
080.
080.
280.
480.
680.
881.
041.
241.
401.
601.
760.
43
1.6
−0.
16−
0.04
0.12
0.24
0.44
0.60
0.76
0.92
1.08
1.24
1.40
0.28
1.8
−0.
20−
0.12
0.00
0.16
0.28
0.40
0.52
0.64
0.80
0.96
1.08
0.16
2.0
−0.
20−
0.12
−0.
040.
080.
160.
280.
360.
440.
560.
640.
800.
07
2.2
−0.
12−
0.12
−0.
040.
040.
120.
160.
240.
320.
400.
480.
560.
00
2.4
−0.
08−
0.08
−0.
040.
040.
080.
120.
160.
200.
280.
320.
36−
0.06
2.6
−0.
04−
0.04
0.00
0.04
0.04
0.08
0.12
0.16
0.16
0.20
0.24
−0.
11
2.8
0.00
0.00
0.04
0.04
0.04
0.08
0.08
0.12
0.12
0.12
0.16
−0.
15
3.0
0.04
0.04
0.04
0.04
0.04
0.04
0.04
0.08
0.08
0.08
0.08
−0.
18
Tab
le 4
.17
Con
’t
SAMPLE C
HAPTER
Analysis of Beams on Elastic Half-Space 225
β =
0.0
75
ζα
00.
10.
20.
30.
40.
50.
60.
70.
80.
91.
0∞
p
0.0
2.80
02.
481
2.13
11.
803
1.48
71.
238
1.01
50.
851
0.69
70.
547
0.40
50.
620
0.1
2.29
52.
064
1.81
41.
575
1.33
21.
125
0.92
70.
762
0.61
50.
469
0.34
10.
605
0.2
1.87
61.
716
1.54
91.
387
1.20
81.
041
0.87
30.
726
0.58
90.
641
0.36
10.
572
0.3
1.52
51.
421
1.31
51.
203
1.09
20.
970
0.84
40.
714
0.59
40.
485
0.38
80.
536
0.4
0.22
91.
170
1.11
11.
046
0.98
10.
902
0.81
30.
714
0.61
60.
521
0.43
20.
494
0.5
0.97
90.
945
0.93
00.
903
0.87
50.
832
0.77
90.
712
0.63
80.
561
0.48
40.
460
0.6
0.76
80.
769
0.77
10.
772
0.77
20.
761
0.73
90.
702
0.66
40.
596
0.53
40.
417
0.7
0.58
90.
610
0.63
20.
654
0.67
40.
687
0.68
90.
678
0.65
50.
623
0.58
10.
369
0.8
0.43
90.
474
0.51
00.
546
0.58
20.
612
0.63
50.
646
0.64
80.
637
0.61
40.
318
0.9
0.31
40.
359
0.40
40.
450
0.49
60.
539
0.57
70.
608
0.62
70.
633
0.62
70.
266
1.0
0.20
90.
261
0.31
30.
365
0.41
80.
469
0.51
60.
555
0.59
10.
616
0.62
90.
216
1.2
0.05
50.
112
0.17
00.
233
0.28
40.
341
0.39
70.
452
0.50
50.
553
0.59
20.
132
1.4
−0.
043
0.01
20.
368
0.12
30.
179
0.23
40.
292
0.34
80.
401
0.45
80.
510
0.07
1
1.6
−0.
099
−0.
050
0.00
00.
050
0.10
00.
151
0.20
40.
256
0.30
90.
362
0.41
50.
131
1.8
−0.
126
−0.
084
−0.
042
0.00
10.
043
0.08
80.
132
0.17
90.
225
0.27
20.
320
0.00
8
2.0
−0.
132
−0.
098
−0.
064
−0.
029
0.00
40.
040
0.07
70.
117
0.15
50.
195
0.23
70.
002
2.2
−0.
125
−0.
099
−0.
073
−0.
046
−0.
020
0.00
70.
036
0.06
70.
099
0.13
30.
167
−0.
009
2.4
−0.
111
−0.
092
−0.
072
−0.
061
−0.
034
−0.
013
0.00
60.
025
0.05
40.
081
0.10
9−
0.01
1
2.6
−0.
094
−0.
080
−0.
067
−0.
054
−0.
041
−0.
026
−0.
011
0.00
70.
021
0.03
80.
055
−0.
011
2.8
−0.
076
−0.
068
−0.
059
−0.
051
−0.
042
−0.
034
−0.
025
−0.
016
−0.
006
0.00
30.
014
−0.
010
3.0
−0.
058
−0.
054
−0.
049
−0.
046
−0.
042
−0.
040
−0.
038
−0.
035
−0.
029
−0.
021
−0.
011
−0.
008
Tab
le 4
.18
Dim
ensi
onle
ss c
oeffi
cien
ts f
or a
naly
sis
of lo
ng b
eam
s lo
aded
with
a c
once
ntra
ted
ver
tical
load
P
SAMPLE C
HAPTER
226 Analysis of Structures on Elastic Foundations
β =
0.0
75
ζα
00.
10.
20.
30.
40.
50.
60.
70.
80.
91.
0∞
M
0.0
0.00
00.
000
0.00
00.
000
0.00
00.
000
0.00
00.
000
0.00
00.
000
0.00
00.
214
0.1
−0.
087
−0.
011
0.00
90.
007
0.00
60.
006
0.00
60.
005
0.00
40.
003
0.00
20.
167
0.2
−0.
150
−0.
057
0.03
90.
033
0.02
80.
022
0.02
00.
018
0.01
50.
011
0.00
70.
127
0.3
−0.
195
−0.
106
−0.
019
0.07
00.
061
0.05
00.
043
0.03
70.
030
0.02
40.
017
0.09
1
0.4
−0.
224
−0.
152
−0.
061
0.02
20.
104
0.08
70.
074
0.06
30.
052
0.03
90.
030
0.06
0
0.5
−0.
242
−0.
169
−0.
093
0.01
70.
057
0.13
80.
113
0.09
60.
078
0.05
40.
046
0.03
6
0.6
−0.
248
−0.
182
−0.
115
−0.
046
0.02
00.
087
0.15
10.
137
0.11
30.
089
0.06
90.
016
0.7
−0.
246
−0.
189
−0.
130
−0.
070
−0.
009
0.05
00.
115
0.18
50.
154
0.12
20.
096
−0.
001
0.8
−0.
241
−0.
191
−0.
139
−0.
085
−0.
033
0.01
90.
078
0.13
90.
202
0.16
30.
130
−0.
013
0.9
−0.
230
−0.
185
−0.
141
−0.
096
−0.
050
−0.
004
0.04
40.
098
0.15
40.
209
0.17
1−
0.02
3
1.0
−0.
217
−0.
180
−0.
141
−0.
119
−0.
063
−0.
024
0.01
90.
065
0.11
30.
163
0.21
5−
0.02
9
1.2
−0.
183
−0.
158
−0.
132
−0.
138
−0.
076
−0.
050
−0.
019
0.01
50.
050
0.08
50.
128
−0.
036
1.4
−0.
148
−0.
135
−0.
109
−0.
096
−0.
080
−0.
061
−0.
041
−0.
017
0.00
70.
031
0.06
1−
0.03
8
1.6
−0.
113
−0.
104
−0.
094
−0.
083
−0.
074
−0.
063
−0.
050
−0.
037
−0.
020
−0.
004
0.01
3−
0.03
6
1.8
−0.
083
−0.
076
−0.
074
−0.
068
−0.
065
−0.
059
−0.
052
−0.
044
−0.
035
−0.
026
−0.
015
−0.
034
2.0
−0.
057
−0.
056
−0.
056
−0.
054
−0.
052
−0.
052
−0.
048
−0.
044
−0.
041
−0.
037
−0.
032
−0.
031
2.2
−0.
037
−0.
037
−0.
039
−0.
041
−0.
041
−0.
043
−0.
043
−0.
041
−0.
039
−0.
039
−0.
039
−0.
027
2.4
−0.
022
−0.
024
−0.
026
−0.
028
−0.
032
−0.
033
−0.
035
−0.
035
−0.
035
−0.
037
−0.
039
−0.
022
2.6
−0.
011
−0.
013
−0.
017
−0.
014
−0.
022
−0.
022
−0.
026
−0.
026
−0.
028
−0.
032
−0.
035
−0.
018
2.8
−0.
004
−0.
006
−0.
009
−0.
011
−0.
013
−0.
017
−0.
017
−0.
019
−0.
020
−0.
024
−0.
028
−0.
015
3.0
0.00
0−
0.00
2−
0.00
2−
0.00
6−
0.00
6−
0.00
7−
0.00
9−
0.01
1−
0.01
1−
0.01
5−
0.02
0−
0.01
1
0.0
0.00
00.
000
0.00
00.
000
0.00
00.
000
0.00
00.
000
0.00
00.
000
0.00
0−
0.50
0
0.1
−0.
744
0.22
80.
196
0.16
80.
141
0.11
70.
097
0.07
90.
065
0.05
20.
039
−0.
439
0.2
−0.
535
−0.
584
0.36
40.
316
0.27
40.
288
0.18
70.
156
0.12
40.
103
0.07
5−
0.39
1
0.3
−0.
365
−0.
426
−0.
492
0.44
50.
383
0.32
60.
274
0.22
60.
184
0.14
60.
112
−0.
325
0.4
−0.
228
−0.
298
−0.
372
−0.
443
0.48
00.
419
0.35
60.
297
0.24
40.
196
0.14
7−
0.27
4
0.5
−0.
117
−0.
192
−0.
269
−0.
345
−0.
421
0.50
60.
435
0.38
70.
307
0.25
00.
197
−0.
225
0.6
0.03
1−
0.10
7−
0.18
4−
0.26
1−
0.33
8−
0.41
40.
511
0.43
90.
371
0.30
80.
248
−0.
181
0.7
0.03
7−
0.03
5−
0.11
5−
0.19
0−
0.26
6−
0.34
2−
0.41
80.
508
0.43
60.
369
0.30
5−
0.14
2
0.8
0.08
90.
017
−0.
057
−0.
130
−0.
203
−0.
277
−0.
351
−0.
425
0.50
30.
432
0.36
5−
0.10
6
0.9
0.12
50.
058
−0.
012
−0.
080
−0.
150
−0.
219
−0.
289
−0.
362
−0.
433
0.49
60.
426
−0.
079
1.0
0.15
20.
008
0.02
4−
0.04
0−
0.10
4−
0.16
9−
0.23
6−
0.30
4−
0.37
2−
0.44
10.
491
−0.
054
1.2
0.17
70.
125
0.07
10.
019
−0.
034
−0.
088
−0.
145
−0.
203
−0.
262
−0.
317
−0.
388
−0.
020
1.4
0.17
70.
131
0.09
50.
054
0.01
2−
0.03
2−
0.07
6−
0.12
4−
0.17
2−
0.22
4−
0.27
70.
004
1.6
0.16
30.
132
0.10
10.
071
0.03
90.
007
−0.
027
−0.
063
−0.
101
−0.
141
−0.
185
0.01
0
1.8
0.14
00.
118
0.09
60.
075
0.05
50.
031
0.00
6−
0.02
0−
0.04
7−
0.07
8−
0.11
10.
013
2.0
0.11
30.
100
0.08
50.
074
0.05
70.
043
0.02
70.
009
−0.
009
−0.
031
−0.
055
0.01
5
2.2
0.08
80.
079
0.07
20.
065
0.05
60.
048
0.03
90.
028
0.01
6−
0.00
2−
0.01
50.
013
2.4
0.06
40.
061
0.05
70.
054
0.05
00.
047
0.04
20.
037
0.03
10.
027
0.01
20.
011
2.6
0.04
40.
044
0.04
30.
044
0.04
30.
043
0.04
10.
040
0.03
80.
034
0.02
80.
009
2.8
0.02
70.
029
0.03
00.
033
0.03
50.
036
0.03
80.
038
0.03
90.
041
0.03
60.
007
3.0
0.01
30.
017
0.02
00.
023
0.02
60.
030
0.03
20.
033
0.03
60.
037
0.03
60.
004
Tab
le 4
.18
Con
’t
SAMPLE C
HAPTER
Analysis of Beams on Elastic Half-Space 227
β =
0.0
75
ζα
00.
10.
20.
30.
40.
50.
60.
70.
80.
91.
0∞
0.0
0.00
00.
000
0.00
00.
000
0.00
00.
000
0.00
00.
000
0.00
00.
000
0.00
00.
214
0.1
−0.
087
−0.
011
0.00
90.
007
0.00
60.
006
0.00
60.
005
0.00
40.
003
0.00
20.
167
0.2
−0.
150
−0.
057
0.03
90.
033
0.02
80.
022
0.02
00.
018
0.01
50.
011
0.00
70.
127
0.3
−0.
195
−0.
106
−0.
019
0.07
00.
061
0.05
00.
043
0.03
70.
030
0.02
40.
017
0.09
1
0.4
−0.
224
−0.
152
−0.
061
0.02
20.
104
0.08
70.
074
0.06
30.
052
0.03
90.
030
0.06
0
0.5
−0.
242
−0.
169
−0.
093
0.01
70.
057
0.13
80.
113
0.09
60.
078
0.05
40.
046
0.03
6
0.6
−0.
248
−0.
182
−0.
115
−0.
046
0.02
00.
087
0.15
10.
137
0.11
30.
089
0.06
90.
016
0.7
−0.
246
−0.
189
−0.
130
−0.
070
−0.
009
0.05
00.
115
0.18
50.
154
0.12
20.
096
−0.
001
0.8
−0.
241
−0.
191
−0.
139
−0.
085
−0.
033
0.01
90.
078
0.13
90.
202
0.16
30.
130
−0.
013
0.9
−0.
230
−0.
185
−0.
141
−0.
096
−0.
050
−0.
004
0.04
40.
098
0.15
40.
209
0.17
1−
0.02
3
1.0
−0.
217
−0.
180
−0.
141
−0.
119
−0.
063
−0.
024
0.01
90.
065
0.11
30.
163
0.21
5−
0.02
9
1.2
−0.
183
−0.
158
−0.
132
−0.
138
−0.
076
−0.
050
−0.
019
0.01
50.
050
0.08
50.
128
−0.
036
1.4
−0.
148
−0.
135
−0.
109
−0.
096
−0.
080
−0.
061
−0.
041
−0.
017
0.00
70.
031
0.06
1−
0.03
8
1.6
−0.
113
−0.
104
−0.
094
−0.
083
−0.
074
−0.
063
−0.
050
−0.
037
−0.
020
−0.
004
0.01
3−
0.03
6
1.8
−0.
083
−0.
076
−0.
074
−0.
068
−0.
065
−0.
059
−0.
052
−0.
044
−0.
035
−0.
026
−0.
015
−0.
034
2.0
−0.
057
−0.
056
−0.
056
−0.
054
−0.
052
−0.
052
−0.
048
−0.
044
−0.
041
−0.
037
−0.
032
−0.
031
2.2
−0.
037
−0.
037
−0.
039
−0.
041
−0.
041
−0.
043
−0.
043
−0.
041
−0.
039
−0.
039
−0.
039
−0.
027
2.4
−0.
022
−0.
024
−0.
026
−0.
028
−0.
032
−0.
033
−0.
035
−0.
035
−0.
035
−0.
037
−0.
039
−0.
022
2.6
−0.
011
−0.
013
−0.
017
−0.
014
−0.
022
−0.
022
−0.
026
−0.
026
−0.
028
−0.
032
−0.
035
−0.
018
2.8
−0.
004
−0.
006
−0.
009
−0.
011
−0.
013
−0.
017
−0.
017
−0.
019
−0.
020
−0.
024
−0.
028
−0.
015
3.0
0.00
0−
0.00
2−
0.00
2−
0.00
6−
0.00
6−
0.00
7−
0.00
9−
0.01
1−
0.01
1−
0.01
5−
0.02
0−
0.01
1
Q
0.0
0.00
00.
000
0.00
00.
000
0.00
00.
000
0.00
00.
000
0.00
00.
000
0.00
0−
0.50
0
0.1
−0.
744
0.22
80.
196
0.16
80.
141
0.11
70.
097
0.07
90.
065
0.05
20.
039
−0.
439
0.2
−0.
535
−0.
584
0.36
40.
316
0.27
40.
288
0.18
70.
156
0.12
40.
103
0.07
5−
0.39
1
0.3
−0.
365
−0.
426
−0.
492
0.44
50.
383
0.32
60.
274
0.22
60.
184
0.14
60.
112
−0.
325
0.4
−0.
228
−0.
298
−0.
372
−0.
443
0.48
00.
419
0.35
60.
297
0.24
40.
196
0.14
7−
0.27
4
0.5
−0.
117
−0.
192
−0.
269
−0.
345
−0.
421
0.50
60.
435
0.38
70.
307
0.25
00.
197
−0.
225
0.6
0.03
1−
0.10
7−
0.18
4−
0.26
1−
0.33
8−
0.41
40.
511
0.43
90.
371
0.30
80.
248
−0.
181
0.7
0.03
7−
0.03
5−
0.11
5−
0.19
0−
0.26
6−
0.34
2−
0.41
80.
508
0.43
60.
369
0.30
5−
0.14
2
0.8
0.08
90.
017
−0.
057
−0.
130
−0.
203
−0.
277
−0.
351
−0.
425
0.50
30.
432
0.36
5−
0.10
6
0.9
0.12
50.
058
−0.
012
−0.
080
−0.
150
−0.
219
−0.
289
−0.
362
−0.
433
0.49
60.
426
−0.
079
1.0
0.15
20.
008
0.02
4−
0.04
0−
0.10
4−
0.16
9−
0.23
6−
0.30
4−
0.37
2−
0.44
10.
491
−0.
054
1.2
0.17
70.
125
0.07
10.
019
−0.
034
−0.
088
−0.
145
−0.
203
−0.
262
−0.
317
−0.
388
−0.
020
1.4
0.17
70.
131
0.09
50.
054
0.01
2−
0.03
2−
0.07
6−
0.12
4−
0.17
2−
0.22
4−
0.27
70.
004
1.6
0.16
30.
132
0.10
10.
071
0.03
90.
007
−0.
027
−0.
063
−0.
101
−0.
141
−0.
185
0.01
0
1.8
0.14
00.
118
0.09
60.
075
0.05
50.
031
0.00
6−
0.02
0−
0.04
7−
0.07
8−
0.11
10.
013
2.0
0.11
30.
100
0.08
50.
074
0.05
70.
043
0.02
70.
009
−0.
009
−0.
031
−0.
055
0.01
5
2.2
0.08
80.
079
0.07
20.
065
0.05
60.
048
0.03
90.
028
0.01
6−
0.00
2−
0.01
50.
013
2.4
0.06
40.
061
0.05
70.
054
0.05
00.
047
0.04
20.
037
0.03
10.
027
0.01
20.
011
2.6
0.04
40.
044
0.04
30.
044
0.04
30.
043
0.04
10.
040
0.03
80.
034
0.02
80.
009
2.8
0.02
70.
029
0.03
00.
033
0.03
50.
036
0.03
80.
038
0.03
90.
041
0.03
60.
007
3.0
0.01
30.
017
0.02
00.
023
0.02
60.
030
0.03
20.
033
0.03
60.
037
0.03
60.
004
SAMPLE C
HAPTER
228 Analysis of Structures on Elastic Foundations
β =
0.0
75
ζα
00.
10.
20.
30.
40.
50.
60.
70.
80.
91.
0∞
Y
04.
313.
843.
402.
972.
572.
21.
851.
551.
271.
040.
851.
37
0.1
3.84
3.47
3.11
2.79
2.44
2.12
1.83
1.54
1.29
1.09
0.93
1.35
0.2
3.40
3.11
2.85
2.59
2.32
2.05
1.80
1.56
1.35
1.16
1.00
1.32
0.3
2.97
2.79
2.59
2.40
2.20
1.99
1.78
1.57
1.40
1.23
1.09
1.27
0.4
2.57
2.44
2.32
2.20
2.07
1.92
1.76
1.59
1.44
1.29
1.16
1.21
0.5
2.20
2.12
2.05
1.99
1.92
1.81
1.71
1.59
1.47
1.35
1.23
1.13
0.6
1.85
1.83
1.80
1.78
1.76
1.71
1.65
1.56
1.49
1.39
1.29
1.05
0.7
1.55
1.54
1.56
1.57
1.59
1.59
1.56
1.52
1.48
1.43
1.35
0.97
0.8
1.27
1.29
1.35
1.40
1.44
1.47
1.49
1.48
1.48
1.45
1.41
0.89
0.9
1.04
1.09
1.16
1.23
1.29
1.35
1.39
1.43
1.45
1.44
1.42
0.80
1.0
0.85
0.93
1.00
1.09
1.16
1.23
1.29
1.35
1.41
1.42
1.44
0.73
1.2
0.49
0.59
0.68
0.80
0.89
0.97
1.08
1.15
1.25
1.31
1.32
0.60
1.4
0.27
0.36
0.45
0.57
0.67
0.76
0.87
0.96
1.07
1.15
1.24
0.48
1.6
0.09
0.20
0.29
0.40
0.49
0.57
0.69
0.77
0.88
0.96
1.08
0.35
1.8
0.00
0.09
0.17
0.27
0.35
0.44
0.53
0.61
0.72
0.80
0.89
0.26
2.0
−0.
050.
030.
090.
170.
240.
320.
400.
470.
570.
640.
730.
17
2.2
−0.
07−
0.01
0.04
0.11
0.17
0.23
0.29
0.36
0.44
0.51
0.57
0.10
2.4
−0.
07−
0.03
0.01
0.07
0.12
0.16
0.21
0.27
0.33
0.39
0.44
0.04
2.6
−0.
06−
0.03
0.00
0.05
0.08
0.11
0.15
0.19
0.24
0.38
0.38
0.00
2.8
−0.
05−
0.01
0.00
0.04
0.05
0.08
0.10
0.13
0.17
0.20
0.23
−0.
03
3.0
−0.
030.
000.
000.
030.
040.
050.
070.
080.
110.
130.
15−
0.07
Tab
le 4
.18
Con
’t
SAMPLE C
HAPTER
Analysis of Beams on Elastic Half-Space 229
β =
0.1
5
ζα
00.
10.
20.
30.
40.
50.
60.
70.
80.
91.
0∞
p
0.0
2.73
22.
435
2.12
81.
841
1.55
21.
316
1.10
70.
946
0.79
40.
640
0.50
70.
555
0.1
2.20
01.
981
1.77
31.
562
1.35
11.
159
0.97
50.
827
0.68
70.
557
0.44
60.
548
0.2
1.78
11.
635
1.48
81.
339
1.19
01.
043
0.89
80.
763
0.63
90.
523
0.42
50.
529
0.3
1.44
41.
349
1.25
81.
154
1.05
50.
949
0.83
80.
727
0.61
90.
520
0.43
20.
503
0.4
1.17
31.
115
1.05
70.
998
0.93
80.
868
0.79
00.
706
0.61
90.
535
0.45
60.
476
0.5
0.94
80.
948
0.89
00.
861
0.83
10.
793
0.74
50.
687
0.62
40.
557
0.49
00.
452
0.6
0.76
10.
753
0.74
60.
741
0.73
30.
720
0.69
80.
666
0.62
40.
575
0.52
20.
426
0.7
0.60
40.
613
0.62
30.
634
0.64
40.
649
0.64
70.
636
0.61
50.
586
0.55
00.
393
0.8
0.47
10.
493
0.51
50.
538
0.56
10.
581
0.59
40.
603
0.59
70.
590
0.57
00.
352
0.9
0.35
70.
389
0.41
90.
453
0.48
40.
516
0.54
30.
565
0.57
80.
582
0.57
60.
302
1.0
0.26
40.
302
0.33
90.
380
0.41
60.
454
0.48
60.
520
0.54
50.
564
0.57
30.
252
1.2
0.11
80.
163
0.20
80.
252
0.29
70.
341
0.38
50.
428
0.46
90.
508
0.53
90.
155
1.4
0.01
90.
065
0.11
10.
157
0.20
30.
249
0.29
90.
340
0.38
50.
433
0.47
10.
081
1.6
−0.
046
−0.
003
0.04
10.
085
0.12
90.
173
0.21
70.
261
0.30
50.
330
0.39
40.
035
1.8
−0.
083
−0.
045
−0.
007
0.03
40.
072
0112
0.15
30.
194
0.23
40.
278
0.31
60.
009
2.0
−0.
102
−0.
069
−0.
037
−0.
002
0.03
10.
065
0.10
10.
137
0.17
30.
210
0.24
6−
0.00
3
2.2
−0.
109
−0.
081
−0.
054
−0.
026
0.00
20.
030
0.05
90.
090
0.12
10.
153
0.18
6−
0.00
9
2.4
−0.
105
−0.
083
−0.
061
−0.
039
−0.
016
0.00
50.
028
0.05
10.
077
0.10
50.
132
−0.
011
2.6
−0.
097
−0.
083
−0.
064
−0.
047
−0.
030
−0.
013
0.00
40.
024
0.04
20.
062
0.08
0−
0.01
1
2.8
−0.
087
−0.
081
−0.
063
−0.
050
−0.
039
−0.
027
−0.
014
−0.
001
0.01
20.
024
0.03
6−
0.01
0
3.0
−0.
076
−0.
075
−0.
061
−0.
053
−0.
046
−0.
040
−0.
032
−0.
025
−0.
017
−0.
007
−0.
003
−0.
008
Tab
le 4
.19
Dim
ensi
onle
ss c
oeffi
cien
ts f
or a
naly
sis
of lo
ng b
eam
s lo
aded
with
a c
once
ntra
ted
ver
tical
load
P
SAMPLE C
HAPTER
230 Analysis of Structures on Elastic Foundations
β =
0.1
5
ζα
00.
10.
20.
30.
40.
50.
60.
70.
80.
91.
0∞
M
0.0
0.00
00.
000
0.00
00.
000
0.00
00.
000
0.00
00.
000
0.00
00.
000
0.00
00.
230
0.1
−0.
087
0.01
20.
009
0.00
90.
008
0.00
50.
005
0.00
40.
003
0.00
30.
003
0.18
3
0.2
−0.
152
−0.
056
0.03
70.
033
0.02
90.
024
0.02
00.
017
0.01
40.
012
0.01
00.
141
0.3
−0.
200
−0.
110
−0.
021
0.07
00.
061
0.05
20.
044
0.03
70.
031
0.02
60.
020
0.10
5
0.4
−0.
232
−0.
148
−0.
065
0.02
00.
105
0.09
00.
077
0.06
60.
055
0.04
60.
036
0.07
2
0.5
−0.
253
−0.
176
−0.
099
−0.
021
0.05
80.
137
0.11
60.
100
0.08
50.
070
0.05
60.
046
0.6
−0.
265
−0.
194
−0.
124
−0.
053
0.01
90.
091
0.16
50.
139
0.12
00.
100
0.08
10.
024
0.7
−0.
269
−0.
205
−0.
142
−0.
077
−0.
012
−0.
053
0.12
00.
191
0.16
30.
137
0.11
00.
006
0.8
−0.
267
−0.
210
−0.
153
−0.
096
−0.
037
0.02
10.
081
0.14
50.
210
0.17
80.
147
−0.
006
0.9
−0.
259
−0.
209
−0.
158
−0.
108
−0.
057
−0.
005
0.04
90.
106
0.16
40.
225
0.19
0−
0.01
8
1.0
−0.
249
−0.
205
−0.
161
−0.
116
−0.
071
−0.
026
0.02
20.
124
0.12
40.
179
0.23
8−
0.02
5
1.2
−0.
221
−0.
188
−0.
156
−0.
122
−0.
088
−0.
054
−0.
019
0.02
00.
060
0.10
30.
147
−0.
033
1.4
−0.
188
−0.
165
−0.
142
−0.
118
−0.
093
−0.
069
−0.
044
−0.
015
0.01
40.
047
0.08
0−
0.03
5
1.6
−0.
154
−0.
139
−0.
124
−0.
107
−0.
091
−0.
074
−0.
056
−0.
036
−0.
016
0.00
70.
031
−0.
033
1.8
−0.
122
−0.
133
−0.
103
−0.
093
−0.
083
−0.
072
−0.
061
−0.
047
−0.
034
−0.
018
−0.
002
−0.
030
2.0
−0.
093
−0.
088
−0.
083
−0.
077
−0.
072
−0.
066
−0.
059
−0.
051
−0.
043
−0.
033
−0.
022
−0.
026
2.2
−0.
068
−0.
066
−0.
065
−0.
062
−0.
059
−0.
057
−0.
053
−0.
049
−0.
044
−0.
039
−0.
032
−0.
022
2.4
−0.
047
−0.
048
−0.
048
−0.
047
−0.
047
−0.
046
−0.
045
−0.
042
−0.
041
−0.
038
−0.
035
−0.
018
2.6
−0.
031
−0.
033
−0.
034
−0.
034
−0.
035
−0.
036
−0.
035
−0.
031
−0.
034
−0.
033
−0.
032
−0.
015
2.8
−0.
013
−0.
021
−0.
023
−0.
023
−0.
024
−0.
025
−0.
026
−0.
026
−0.
026
−0.
026
−0.
026
−0.
013
3.0
−0.
010
−0.
011
−0.
014
−0.
014
−0.
015
−0.
016
−0.
017
−0.
017
−0.
017
−0.
019
−0.
019
−0.
010
0.0
0.00
00.
000
0.00
00.
000
0.00
00.
000
0.00
00.
000
0.00
00.
000
0.00
0−
0.50
0
0.1
−0.
755
0.21
90.
196
0.16
80.
144
0.12
40.
104
0.08
90.
074
0.06
20.
046
−0.
445
0.2
−0.
556
−0.
600
0.35
80.
313
0.27
10.
234
0.19
80.
168
0.14
00.
116
0.09
0−
0.39
1
0.3
−0.
395
−0.
454
−0.
505
0.43
90.
384
0.33
40.
285
0.24
20.
203
0.16
70.
133
−0.
339
0.4
−0.
265
−0.
328
−0.
390
−0.
454
0.48
30.
424
0.36
60.
313
0.26
40.
219
0.17
7−
0.29
0
0.5
−0.
159
−0.
226
−0.
293
−0.
293
−0.
428
0.50
70.
443
0.38
20.
326
0.27
30.
224
−0.
244
0.6
−0.
075
−0.
143
−0.
211
−0.
281
−0.
350
−0.
417
0.51
50.
450
0.38
80.
329
0.27
5−
0.20
0
0.7
−0.
007
−0.
075
−0.
143
−0.
213
−0.
280
−0.
349
−0.
418
0.51
50.
451
0.38
80.
329
−0.
159
0.8
0.04
7−
0.02
0−
0.08
6−
0.15
4−
0.22
1−
0.28
7−
0.35
6−
0.42
30.
512
0.47
70.
385
−0.
123
0.9
0.08
80.
024
−0.
040
−0.
104
−0.
169
−0.
232
−0.
299
−0.
364
−0.
429
0.50
60.
442
−0.
090
1.0
0.11
90.
059
−0.
002
−0.
063
−0.
123
−0.
184
−0.
248
−0.
311
−0.
373
−0.
436
0.50
1−
0.06
0
1.2
0.15
70.
105
0.05
20.
000
−0.
053
−0.
105
−0.
160
−0.
215
−0.
272
−0.
330
−0.
389
−0.
021
1.4
0.17
00.
126
0.08
30.
040
−0.
003
−0.
047
−0.
093
−0.
139
−0.
187
−0.
236
−0.
287
0.00
3
1.6
0.16
70.
132
0.09
90.
064
0.03
0−
0.00
5−
0.04
2−
0.07
9−
0.11
8−
0.15
8−
0.20
10.
017
1.8
0.15
30.
128
0.10
20.
075
0.05
00.
023
−0.
005
−0.
034
−0.
064
−0.
096
−0.
130
0.02
0
2.0
0.13
40.
116
0.09
70.
078
0.06
00.
041
0.02
1−
0.00
1−
0.02
3−
0.04
7−
0.07
40.
019
2.2
0.11
40.
102
0.08
80.
076
0.06
30.
061
0.03
70.
022
0.01
6−
0.01
1−
0.03
00.
018
2.4
0.09
20.
084
0.07
60.
069
0.06
10.
054
0.04
50.
036
0.02
60.
014
0.00
10.
015
2.6
0.07
20.
068
0.06
40.
060
0.05
60.
053
0.04
80.
043
0.03
70.
030
0.02
20.
012
2.8
0.05
30.
052
0.05
20.
050
0.04
90.
049
0.04
70.
045
0.04
30.
039
0.03
40.
910
3.0
0.03
70.
037
0.03
90.
040
0.04
10.
042
0.04
30.
043
0.04
20.
040
0.03
80.
009
Tab
le 4
.19
Con
’t
SAMPLE C
HAPTER
Analysis of Beams on Elastic Half-Space 231
β =
0.1
5
ζα
00.
10.
20.
30.
40.
50.
60.
70.
80.
91.
0∞
0.0
0.00
00.
000
0.00
00.
000
0.00
00.
000
0.00
00.
000
0.00
00.
000
0.00
00.
230
0.1
−0.
087
0.01
20.
009
0.00
90.
008
0.00
50.
005
0.00
40.
003
0.00
30.
003
0.18
3
0.2
−0.
152
−0.
056
0.03
70.
033
0.02
90.
024
0.02
00.
017
0.01
40.
012
0.01
00.
141
0.3
−0.
200
−0.
110
−0.
021
0.07
00.
061
0.05
20.
044
0.03
70.
031
0.02
60.
020
0.10
5
0.4
−0.
232
−0.
148
−0.
065
0.02
00.
105
0.09
00.
077
0.06
60.
055
0.04
60.
036
0.07
2
0.5
−0.
253
−0.
176
−0.
099
−0.
021
0.05
80.
137
0.11
60.
100
0.08
50.
070
0.05
60.
046
0.6
−0.
265
−0.
194
−0.
124
−0.
053
0.01
90.
091
0.16
50.
139
0.12
00.
100
0.08
10.
024
0.7
−0.
269
−0.
205
−0.
142
−0.
077
−0.
012
−0.
053
0.12
00.
191
0.16
30.
137
0.11
00.
006
0.8
−0.
267
−0.
210
−0.
153
−0.
096
−0.
037
0.02
10.
081
0.14
50.
210
0.17
80.
147
−0.
006
0.9
−0.
259
−0.
209
−0.
158
−0.
108
−0.
057
−0.
005
0.04
90.
106
0.16
40.
225
0.19
0−
0.01
8
1.0
−0.
249
−0.
205
−0.
161
−0.
116
−0.
071
−0.
026
0.02
20.
124
0.12
40.
179
0.23
8−
0.02
5
1.2
−0.
221
−0.
188
−0.
156
−0.
122
−0.
088
−0.
054
−0.
019
0.02
00.
060
0.10
30.
147
−0.
033
1.4
−0.
188
−0.
165
−0.
142
−0.
118
−0.
093
−0.
069
−0.
044
−0.
015
0.01
40.
047
0.08
0−
0.03
5
1.6
−0.
154
−0.
139
−0.
124
−0.
107
−0.
091
−0.
074
−0.
056
−0.
036
−0.
016
0.00
70.
031
−0.
033
1.8
−0.
122
−0.
133
−0.
103
−0.
093
−0.
083
−0.
072
−0.
061
−0.
047
−0.
034
−0.
018
−0.
002
−0.
030
2.0
−0.
093
−0.
088
−0.
083
−0.
077
−0.
072
−0.
066
−0.
059
−0.
051
−0.
043
−0.
033
−0.
022
−0.
026
2.2
−0.
068
−0.
066
−0.
065
−0.
062
−0.
059
−0.
057
−0.
053
−0.
049
−0.
044
−0.
039
−0.
032
−0.
022
2.4
−0.
047
−0.
048
−0.
048
−0.
047
−0.
047
−0.
046
−0.
045
−0.
042
−0.
041
−0.
038
−0.
035
−0.
018
2.6
−0.
031
−0.
033
−0.
034
−0.
034
−0.
035
−0.
036
−0.
035
−0.
031
−0.
034
−0.
033
−0.
032
−0.
015
2.8
−0.
013
−0.
021
−0.
023
−0.
023
−0.
024
−0.
025
−0.
026
−0.
026
−0.
026
−0.
026
−0.
026
−0.
013
3.0
−0.
010
−0.
011
−0.
014
−0.
014
−0.
015
−0.
016
−0.
017
−0.
017
−0.
017
−0.
019
−0.
019
−0.
010
Q
0.0
0.00
00.
000
0.00
00.
000
0.00
00.
000
0.00
00.
000
0.00
00.
000
0.00
0−
0.50
0
0.1
−0.
755
0.21
90.
196
0.16
80.
144
0.12
40.
104
0.08
90.
074
0.06
20.
046
−0.
445
0.2
−0.
556
−0.
600
0.35
80.
313
0.27
10.
234
0.19
80.
168
0.14
00.
116
0.09
0−
0.39
1
0.3
−0.
395
−0.
454
−0.
505
0.43
90.
384
0.33
40.
285
0.24
20.
203
0.16
70.
133
−0.
339
0.4
−0.
265
−0.
328
−0.
390
−0.
454
0.48
30.
424
0.36
60.
313
0.26
40.
219
0.17
7−
0.29
0
0.5
−0.
159
−0.
226
−0.
293
−0.
293
−0.
428
0.50
70.
443
0.38
20.
326
0.27
30.
224
−0.
244
0.6
−0.
075
−0.
143
−0.
211
−0.
281
−0.
350
−0.
417
0.51
50.
450
0.38
80.
329
0.27
5−
0.20
0
0.7
−0.
007
−0.
075
−0.
143
−0.
213
−0.
280
−0.
349
−0.
418
0.51
50.
451
0.38
80.
329
−0.
159
0.8
0.04
7−
0.02
0−
0.08
6−
0.15
4−
0.22
1−
0.28
7−
0.35
6−
0.42
30.
512
0.47
70.
385
−0.
123
0.9
0.08
80.
024
−0.
040
−0.
104
−0.
169
−0.
232
−0.
299
−0.
364
−0.
429
0.50
60.
442
−0.
090
1.0
0.11
90.
059
−0.
002
−0.
063
−0.
123
−0.
184
−0.
248
−0.
311
−0.
373
−0.
436
0.50
1−
0.06
0
1.2
0.15
70.
105
0.05
20.
000
−0.
053
−0.
105
−0.
160
−0.
215
−0.
272
−0.
330
−0.
389
−0.
021
1.4
0.17
00.
126
0.08
30.
040
−0.
003
−0.
047
−0.
093
−0.
139
−0.
187
−0.
236
−0.
287
0.00
3
1.6
0.16
70.
132
0.09
90.
064
0.03
0−
0.00
5−
0.04
2−
0.07
9−
0.11
8−
0.15
8−
0.20
10.
017
1.8
0.15
30.
128
0.10
20.
075
0.05
00.
023
−0.
005
−0.
034
−0.
064
−0.
096
−0.
130
0.02
0
2.0
0.13
40.
116
0.09
70.
078
0.06
00.
041
0.02
1−
0.00
1−
0.02
3−
0.04
7−
0.07
40.
019
2.2
0.11
40.
102
0.08
80.
076
0.06
30.
061
0.03
70.
022
0.01
6−
0.01
1−
0.03
00.
018
2.4
0.09
20.
084
0.07
60.
069
0.06
10.
054
0.04
50.
036
0.02
60.
014
0.00
10.
015
2.6
0.07
20.
068
0.06
40.
060
0.05
60.
053
0.04
80.
043
0.03
70.
030
0.02
20.
012
2.8
0.05
30.
052
0.05
20.
050
0.04
90.
049
0.04
70.
045
0.04
30.
039
0.03
40.
910
3.0
0.03
70.
037
0.03
90.
040
0.04
10.
042
0.04
30.
043
0.04
20.
040
0.03
80.
009
Y
SAMPLE C
HAPTER
232 Analysis of Structures on Elastic Foundations
β =
0.1
5
ζα
00.
10.
20.
30.
40.
50.
60.
70.
80.
91.
0∞
03.
042.
752.
482.
211.
961.
721.
501.
301.
110.
950.
791.
07
0.1
2.75
2.53
2.31
2.09
1.87
1.67
1.47
1.3
1.13
0.98
0.84
1.06
0.2
2.48
2.31
2.12
1.95
1.77
1.60
1.44
1.29
1.14
1.00
0.88
1.04
0.3
2.21
2.09
1.95
1.81
1.67
1.51
1.41
1.27
1.15
1.03
0.92
1.01
0.4
1.96
1.87
1.77
1.67
1.59
1.49
1.38
1.27
1.17
1.06
0.97
0.98
0.5
1.72
1.67
1.60
1.51
1.49
1.41
1.33
1.25
1.17
1.08
1.00
0.94
0.6
1.50
1.47
1.44
1.41
1.38
1.33
1.29
1.24
1.17
1.11
1.06
0.85
0.7
1.30
1.30
1.29
1.27
1.27
1.25
1.24
1.21
1.17
1.11
1.07
0.80
0.8
1.11
1.13
1.14
1.15
1.17
1.17
1.17
1.17
1.14
1.11
1.07
0.80
0.9
0.95
0.98
1.00
1.03
1.06
1.08
1.11
1.11
1.11
1.11
1.09
0.76
1.0
0.79
0.84
0.88
0.92
0.97
1.00
1.03
1.06
1.07
1.09
1.09
0.65
1.2
0.55
0.61
0.66
0.72
0.78
0.83
0.89
0.93
0.97
1.01
1.05
0.55
1.4
0.36
0.43
0.49
0.55
0.62
0.68
0.75
0.81
0.86
0.91
0.97
0.46
1.6
0.22
0.29
0.35
0.41
0.49
0.55
0.62
0.68
0.74
0.80
0.86
0.36
1.8
0.12
0.18
0.24
0.31
0.37
0.43
0.50
0.56
0.62
0.68
0.75
0.28
2.0
0.05
0.11
0.17
0.22
0.28
0.34
0.40
0.46
0.51
0.57
0.63
0.15
2.2
0.01
0.06
0.11
0.16
0.21
0.26
0.32
0.37
0.42
0.47
0.53
0.15
2.4
−0.
010.
030.
070.
110.
160.
200.
250.
290.
330.
370.
430.
10
2.6
−0.
020.
010.
050.
080.
110.
140.
190.
230.
260.
290.
340.
05
2.8
−0.
020.
000.
030.
050.
090.
110.
140.
170.
190.
220.
260.
02
3.0
−0.
01−
0.01
0.02
0.04
0.06
0.08
0.11
0.12
0.13
0.15
0.18
−0.
01
Tab
le 4
.19
Con
’t
SAMPLE C
HAPTER
Analysis of Beams on Elastic Half-Space 233
β =
0.3
ζα
00.
10.
20.
30.
40.
50.
60.
70.
80.
91.
0∞
p
0.0
2.25
1.81
1.34
0.95
0.54
0.34
0.24
0.28
0.31
0.28
0.13
0.49
0.2
1.65
1.55
1.14
0.89
0.64
0.44
0.29
0.20
0.13
0.07
0.03
0.47
0.4
1.18
1.06
0.94
0.80
0.67
0.52
0.37
0.23
0.11
0.03
0.02
0.43
0.6
0.82
0.78
0.75
0.70
0.65
0.56
0.45
0.30
0.18
0.09
0.02
0.36
0.8
0.54
0.56
0.58
0.59
0.60
0.57
0.50
0.39
0.28
0.19
0.11
0.30
1.0
0.34
0.37
0.44
0.49
0.53
0.54
0.52
0.45
0.38
0.30
0.22
0.23
1.2
0.19
0.25
0.32
0.39
0.47
0.49
0.51
0.49
0.45
0.39
0.33
0.17
1.4
0.08
0.15
0.22
0.30
0.37
0.43
0.48
0.50
0.49
0.46
0.42
0.13
1.6
0.00
0.07
0.14
0.22
0.29
0.36
0.43
0.47
0.50
0.50
0.48
0.09
1.8
−0.
040.
020.
080.
150.
220.
300.
370.
430.
490.
500.
510.
06
2.0
−0.
08−
0.02
0.04
0.10
0.16
0.23
0.30
0.37
0.43
0.48
0.51
0.04
2.2
−0.
09−
0.03
0.01
0.06
0.12
0.18
0.24
0.31
0.38
0.44
0.48
0.02
2.4
−0.
09−
0.05
−0.
020.
030.
080.
130.
180.
250.
310.
380.
440.
01
2.6
−0.
09−
0.06
−0.
030.
010.
050.
090.
140.
190.
250.
320.
380.
00
2.8
−0.
09−
0.06
−0.
030.
000.
030.
060.
100.
140.
200.
250.
320.
00
3.0
−0.
08−
.005
−0.
04−
0.01
0.01
0.04
0.07
0.10
0.15
0.20
0.25
0.00
3.2
−0.
07−
0.05
−0.
04−
0.02
0.00
0.02
0.04
0.07
0.11
0.15
0.19
−0.
01
3.4
−0.
06−
0.05
−0.
03−
0.02
−0.
010.
010.
030.
050.
080.
100.
14−
0.01
3.6
−0.
05−
0.04
−0.
03−
0.02
−0.
010.
000.
010.
030.
050.
070.
10−
0.01
3.8
−0.
04−
0.03
−0.
03−
0.02
−0.
02−
0.01
0.00
0.02
0.03
0.05
0.07
−0.
01
4.0
−0.
03−
0.03
−0.
03−
0.02
−0.
02−
0.01
0.00
0.01
0.02
0.03
0.05
−0.
01
Tab
le 4
.20
Dim
ensi
onle
ss c
oeffi
cien
ts f
or a
naly
sis
of lo
ng b
eam
s lo
aded
with
a c
once
ntra
ted
ver
tical
load
P
SAMPLE C
HAPTER
234 Analysis of Structures on Elastic Foundations
β =
0.3
ζα
00.
10.
20.
30.
40.
50.
60.
70.
80.
91.
0∞
M
0.0
0.00
00.
000
0.00
00.
000
0.00
00.
000
0.00
00.
000
0.00
00.
000
0.00
00.
282
0.2
−0.
159
0.03
40.
026
0.01
90.
012
0.00
80.
005
0.00
50.
005
0.00
40.
003
0.19
2
0.4
−0.
252
−0.
077
0.09
60.
073
0.04
90.
028
0.02
20.
019
0.01
60.
012
0.00
70.
120
0.6
−0.
296
−0.
145
0.00
50.
159
0.11
20.
079
0.05
30.
041
0.03
10.
021
0.01
20.
066
0.8
−0.
310
−0.
182
−0.
057
0.07
30.
202
0.14
60.
103
0.07
50.
053
0.03
30.
017
0.02
6
1.0
−0.
300
−0.
196
−0.
095
0.01
10.
115
0.23
70.
172
0.12
50.
087
0.05
40.
026
−0.
002
1.2
−0.
280
−0.
194
−0.
115
−0.
032
0.05
00.
149
0.26
20.
193
0.13
60.
086
0.04
4−
0.02
2
1.4
−0.
246
−0.
183
−0.
123
−0.
067
0.00
30.
081
0.17
20.
281
0.20
20.
133
0.07
7−
0.03
3
1.6
−0.
212
−0.
166
−0.
120
−0.
075
−0.
036
0.03
00.
101
0.18
90.
289
0.20
00.
124
−0.
039
1.8
−0.
178
−0.
145
−0.
114
−0.
081
−0.
049
−0.
006
0.04
70.
115
0.19
50.
288
0.19
1−
0.04
1
2.0
−0.
146
−0.
124
−0.
110
−0.
82−
0.06
0−
0.03
10.
010
0.05
80.
120
0.19
10.
277
−0.
041
2.2
−0.
117
−0.
104
−0.
091
−0.
078
−0.
065
−0.
045
−0.
020
0.01
80.
062
0.11
60.
189
−0.
039
2.4
−0.
090
−0.
084
−0.
078
−0.
072
−0.
065
−0.
054
−0.
037
−0.
012
0.02
00.
059
0.11
0−
0.03
7
2.6
−0.
068
−0.
067
−0.
065
−0.
064
−0.
062
−0.
058
−0.
047
−0.
031
−0.
010
0.01
70.
055
−0.
033
2.8
−0.
050
−0.
051
−0.
054
−0.
055
−0.
057
−0.
056
−0.
052
−0.
043
−0.
031
−0.
013
0.01
3−
0.03
0
3.0
−0.
035
−0.
039
−0.
043
−0.
047
−0.
051
−0.
053
−0.
053
−0.
049
−0.
043
−0.
032
−0.
015
−0.
027
3.2
−0.
023
−0.
027
−0.
033
−0.
039
−0.
044
−0.
048
−0.
050
−0.
051
−0.
049
−0.
044
−0.
034
−0.
023
3.4
−0.
013
−0.
019
−0.
025
−0.
031
−0.
037
−0.
042
−0.
047
−0.
050
−0.
050
−0.
049
−0.
044
−0.
020
3.6
−0.
007
−0.
013
−0.
019
−0.
025
−0.
031
−0.
036
−0.
041
−0.
046
−0.
049
−0.
050
−0.
049
−0.
018
3.8
−0.
002
−0.
008
−0.
013
−0.
019
−0.
025
−0.
030
−0.
036
−0.
041
−0.
045
−0.
049
−0.
050
−0.
015
4.0
0.00
1−
0.00
4−
0.00
9−
0.01
5−
0.02
0−
0.02
5−
0.03
0−
0.03
5−
0.04
0−
0.04
5−
0.04
8−
0.01
3
0.0
0.00
00.
000
0.00
00.
000
0.00
00.
000
0.00
00.
000
0.00
00.
000
0.00
0−
0.50
0
0.2
−0.
612
0.32
10.
246
0.18
40.
119
0.07
80.
051
0.04
60.
040
0.03
20.
020
−0.
403
0.4
−0.
331
−0.
435
0.45
40.
353
0.22
40.
174
0.11
80.
099
0.06
20.
041
0.02
1−
0.31
3
0.6
−0.
133
−0.
253
−0.
377
0.50
30.
384
0.25
60.
200
0.14
80.
091
0.05
20.
020
−0.
234
0.8
−0.
001
−0.
120
−0.
244
−0.
368
0.51
00.
397
0.29
60.
210
0.13
70.
079
0.03
2−
0.16
8
1.0
0.08
8−
0.02
6−
0.14
2−
0.26
0−
0.37
60.
508
0.39
70.
294
0.20
30.
128
0.06
5−
0.11
5
1.2
0.13
90.
036
−0.
066
−0.
173
−0.
277
−0.
388
0.50
00.
390
0.28
30.
197
0.12
1−
0.07
4
1.4
0.16
50.
076
−0.
013
−0.
105
−0.
195
−0.
295
−0.
400
0.48
90.
381
0.28
40.
195
−0.
043
1.6
0.17
30.
097
0.02
3−
0.05
3−
0.12
9−
0.21
5−
0.31
0−
0.41
10.
481
0.38
00.
285
−0.
025
1.8
0.16
20.
105
0.04
6−
0.01
6−
0.07
8−
0.14
9−
0.23
1−
0.32
3−
0.42
10.
481
0.38
40.
006
2.0
0.15
60.
106
0.05
70.
009
−0.
039
−0.
096
−0.
164
−0.
243
−0.
330
−0.
421
0.38
60.
006
2.2
0.14
00.
101
0.06
40.
026
−0.
011
−0.
056
−0.
110
−0.
174
−0.
248
−0.
329
−0.
417
0.01
2
2.4
0.12
00.
092
0.06
40.
037
0.00
9−
0.02
5−
0.06
7−
0.11
8−
0.18
1−
0.24
6−
0.32
30.
016
2.6
0.10
10.
082
0.06
20.
040
0.02
2−
0.00
4−
0.03
6−
0.07
6−
0.12
5−
0.17
7−
0.24
10.
017
2.8
0.08
40.
070
0.05
70.
042
0.02
90.
012
−0.
042
−0.
042
−0.
078
−0.
121
−0.
176
0.01
7
3.0
0.06
60.
058
0.05
10.
040
0.03
30.
022
0.00
5−
0.01
6−
0.04
3−
0.07
6−
0.11
60.
016
3.2
0.05
20.
047
0.04
40.
037
0.03
40.
027
0.01
60.
001
−0.
014
−0.
041
−0.
071
0.01
5
3.4
0.03
90.
038
0.03
60.
033
0.03
30.
030
0.02
30.
014
−0.
001
0.01
5−
0.03
70.
014
3.6
0.02
90.
029
0.02
90.
029
0.03
10.
030
0.02
70.
022
0.01
30.
002
−0.
014
0.01
3
3.8
0.02
00.
022
0.02
30.
025
0.02
70.
030
0.02
90.
028
0.02
20.
014
0.00
30.
011
4.0
0.01
50.
016
0.01
70.
021
0.02
40.
028
0.02
90.
030
0.02
60.
022
0.01
40.
010
Tab
le 4
.20
Con
’t
SAMPLE C
HAPTER
Analysis of Beams on Elastic Half-Space 235
β =
0.3
ζα
00.
10.
20.
30.
40.
50.
60.
70.
80.
91.
0∞
0.0
0.00
00.
000
0.00
00.
000
0.00
00.
000
0.00
00.
000
0.00
00.
000
0.00
00.
282
0.2
−0.
159
0.03
40.
026
0.01
90.
012
0.00
80.
005
0.00
50.
005
0.00
40.
003
0.19
2
0.4
−0.
252
−0.
077
0.09
60.
073
0.04
90.
028
0.02
20.
019
0.01
60.
012
0.00
70.
120
0.6
−0.
296
−0.
145
0.00
50.
159
0.11
20.
079
0.05
30.
041
0.03
10.
021
0.01
20.
066
0.8
−0.
310
−0.
182
−0.
057
0.07
30.
202
0.14
60.
103
0.07
50.
053
0.03
30.
017
0.02
6
1.0
−0.
300
−0.
196
−0.
095
0.01
10.
115
0.23
70.
172
0.12
50.
087
0.05
40.
026
−0.
002
1.2
−0.
280
−0.
194
−0.
115
−0.
032
0.05
00.
149
0.26
20.
193
0.13
60.
086
0.04
4−
0.02
2
1.4
−0.
246
−0.
183
−0.
123
−0.
067
0.00
30.
081
0.17
20.
281
0.20
20.
133
0.07
7−
0.03
3
1.6
−0.
212
−0.
166
−0.
120
−0.
075
−0.
036
0.03
00.
101
0.18
90.
289
0.20
00.
124
−0.
039
1.8
−0.
178
−0.
145
−0.
114
−0.
081
−0.
049
−0.
006
0.04
70.
115
0.19
50.
288
0.19
1−
0.04
1
2.0
−0.
146
−0.
124
−0.
110
−0.
82−
0.06
0−
0.03
10.
010
0.05
80.
120
0.19
10.
277
−0.
041
2.2
−0.
117
−0.
104
−0.
091
−0.
078
−0.
065
−0.
045
−0.
020
0.01
80.
062
0.11
60.
189
−0.
039
2.4
−0.
090
−0.
084
−0.
078
−0.
072
−0.
065
−0.
054
−0.
037
−0.
012
0.02
00.
059
0.11
0−
0.03
7
2.6
−0.
068
−0.
067
−0.
065
−0.
064
−0.
062
−0.
058
−0.
047
−0.
031
−0.
010
0.01
70.
055
−0.
033
2.8
−0.
050
−0.
051
−0.
054
−0.
055
−0.
057
−0.
056
−0.
052
−0.
043
−0.
031
−0.
013
0.01
3−
0.03
0
3.0
−0.
035
−0.
039
−0.
043
−0.
047
−0.
051
−0.
053
−0.
053
−0.
049
−0.
043
−0.
032
−0.
015
−0.
027
3.2
−0.
023
−0.
027
−0.
033
−0.
039
−0.
044
−0.
048
−0.
050
−0.
051
−0.
049
−0.
044
−0.
034
−0.
023
3.4
−0.
013
−0.
019
−0.
025
−0.
031
−0.
037
−0.
042
−0.
047
−0.
050
−0.
050
−0.
049
−0.
044
−0.
020
3.6
−0.
007
−0.
013
−0.
019
−0.
025
−0.
031
−0.
036
−0.
041
−0.
046
−0.
049
−0.
050
−0.
049
−0.
018
3.8
−0.
002
−0.
008
−0.
013
−0.
019
−0.
025
−0.
030
−0.
036
−0.
041
−0.
045
−0.
049
−0.
050
−0.
015
4.0
0.00
1−
0.00
4−
0.00
9−
0.01
5−
0.02
0−
0.02
5−
0.03
0−
0.03
5−
0.04
0−
0.04
5−
0.04
8−
0.01
3
Q
0.0
0.00
00.
000
0.00
00.
000
0.00
00.
000
0.00
00.
000
0.00
00.
000
0.00
0−
0.50
0
0.2
−0.
612
0.32
10.
246
0.18
40.
119
0.07
80.
051
0.04
60.
040
0.03
20.
020
−0.
403
0.4
−0.
331
−0.
435
0.45
40.
353
0.22
40.
174
0.11
80.
099
0.06
20.
041
0.02
1−
0.31
3
0.6
−0.
133
−0.
253
−0.
377
0.50
30.
384
0.25
60.
200
0.14
80.
091
0.05
20.
020
−0.
234
0.8
−0.
001
−0.
120
−0.
244
−0.
368
0.51
00.
397
0.29
60.
210
0.13
70.
079
0.03
2−
0.16
8
1.0
0.08
8−
0.02
6−
0.14
2−
0.26
0−
0.37
60.
508
0.39
70.
294
0.20
30.
128
0.06
5−
0.11
5
1.2
0.13
90.
036
−0.
066
−0.
173
−0.
277
−0.
388
0.50
00.
390
0.28
30.
197
0.12
1−
0.07
4
1.4
0.16
50.
076
−0.
013
−0.
105
−0.
195
−0.
295
−0.
400
0.48
90.
381
0.28
40.
195
−0.
043
1.6
0.17
30.
097
0.02
3−
0.05
3−
0.12
9−
0.21
5−
0.31
0−
0.41
10.
481
0.38
00.
285
−0.
025
1.8
0.16
20.
105
0.04
6−
0.01
6−
0.07
8−
0.14
9−
0.23
1−
0.32
3−
0.42
10.
481
0.38
40.
006
2.0
0.15
60.
106
0.05
70.
009
−0.
039
−0.
096
−0.
164
−0.
243
−0.
330
−0.
421
0.38
60.
006
2.2
0.14
00.
101
0.06
40.
026
−0.
011
−0.
056
−0.
110
−0.
174
−0.
248
−0.
329
−0.
417
0.01
2
2.4
0.12
00.
092
0.06
40.
037
0.00
9−
0.02
5−
0.06
7−
0.11
8−
0.18
1−
0.24
6−
0.32
30.
016
2.6
0.10
10.
082
0.06
20.
040
0.02
2−
0.00
4−
0.03
6−
0.07
6−
0.12
5−
0.17
7−
0.24
10.
017
2.8
0.08
40.
070
0.05
70.
042
0.02
90.
012
−0.
042
−0.
042
−0.
078
−0.
121
−0.
176
0.01
7
3.0
0.06
60.
058
0.05
10.
040
0.03
30.
022
0.00
5−
0.01
6−
0.04
3−
0.07
6−
0.11
60.
016
3.2
0.05
20.
047
0.04
40.
037
0.03
40.
027
0.01
60.
001
−0.
014
−0.
041
−0.
071
0.01
5
3.4
0.03
90.
038
0.03
60.
033
0.03
30.
030
0.02
30.
014
−0.
001
0.01
5−
0.03
70.
014
3.6
0.02
90.
029
0.02
90.
029
0.03
10.
030
0.02
70.
022
0.01
30.
002
−0.
014
0.01
3
3.8
0.02
00.
022
0.02
30.
025
0.02
70.
030
0.02
90.
028
0.02
20.
014
0.00
30.
011
4.0
0.01
50.
016
0.01
70.
021
0.02
40.
028
0.02
90.
030
0.02
60.
022
0.01
40.
010
SAMPLE C
HAPTER
236 Analysis of Structures on Elastic Foundations
β =
0.3
ζα
00.
10.
20.
30.
40.
50.
60.
70.
80.
91.
0∞
Y
0.0
2.29
1.92
1.56
1.25
1.56
1.25
1.01
0.78
0.57
0.40
0.12
0.78
0.2
1.92
1.66
1.40
1.14
0.99
0.80
0.62
0.46
0.35
0.27
0.22
0.77
0.4
1.56
1.40
1.24
1.09
0.97
0.82
0.66
0.52
0.40
0.33
0.29
0.72
0.6
1.25
1.14
1.09
1.03
0.95
0.84
0.72
0.61
0.50
0.43
0.38
0.66
0.8
1.01
0.99
0.97
0.95
0.92
0.86
0.77
0.67
0.58
0.43
0.38
0.66
1.0
0.78
0.80
0.82
0.84
0.86
0.84
0.80
0.73
0.65
0.59
0.54
0.53
1.2
0.57
0.62
0.66
0.72
0.77
0.80
0.79
0.76
0.71
0.65
0.61
0.45
1.4
0.40
0.46
0.52
0.61
0.67
0.73
0.76
0.77
0.75
0.71
0.67
0.39
1.6
0.28
0.35
0.40
0.50
0.58
0.65
0.71
0.75
0.76
0.75
0.72
0.32
1.8
0.19
0.27
0.33
0.43
0.51
0.59
0.65
0.71
0.75
0.78
0.77
0.26
2.0
0.12
0.22
0.29
0.38
0.46
0.54
0.61
0.67
0.72
0.77
0.79
0.21
2.2
0.07
0.16
0.22
0.31
0.38
0.45
0.52
0.59
0.66
0.72
0.77
0.16
2.4
0.03
0.12
0.17
0.25
0.31
0.38
0.45
0.52
0.59
0.66
0.72
0.12
2.6
0.00
0.08
0.14
0.21
0.26
0.33
0.39
0.46
0.53
0.60
0.66
0.08
2.8
−0.
010.
060.
110.
170.
220.
270.
330.
400.
460.
530.
600.
05
3.0
−0.
010.
040.
090.
140.
180.
230.
280.
340.
400.
460.
530.
02
3.2
−0.
010.
020.
070.
120.
150.
200.
240.
290.
340.
400.
460.
00
3.4
−0.
010.
010.
050.
100.
130.
170.
200.
250.
290.
340.
39−
0.02
3.6
−0.
010.
000.
040.
090.
110.
140.
170.
210.
250.
290.
34−
0.04
3.8
0.00
0.00
0.03
0.07
0.10
0.12
0.15
0.18
0.21
0.25
0.28
−0.
05
4.0
0.00
0.00
0.02
0.06
0.09
0.11
0.13
0.16
0.18
0.21
0.24
−0.
07
Tab
le 4
.20
Con
’t
SAMPLE C
HAPTER
Analysis of Beams on Elastic Half-Space 237
β =
0.5
ζα
00.
10.
20.
30.
40.
50.
60.
70.
80.
91.
0∞
p
0.0
2.27
1.86
1.43
1.04
0.64
0.39
0.24
0.22
0.21
0.18
0.14
0.44
0.2
1.63
1.39
1.15
0.92
0.68
0.48
0.31
0.19
0.11
0.05
0.01
0.43
0.4
1.15
0.96
0.92
0.80
0.67
0.53
0.39
0.25
0.14
0.05
0.00
0.40
0.6
0.79
0.75
0.72
0.68
0.64
0.56
0.46
0.33
0.22
0.12
0.05
0.36
0.8
0.53
0.54
0.55
0.56
0.57
0.55
0.50
0.41
0.32
0.23
0.14
0.31
1.0
0.32
0.37
0.42
0.46
0.50
0.52
0.51
0.47
0.40
0.33
0.25
0.25
1.2
0.20
0.25
0.30
0.37
0.42
0.47
0.49
0.49
0.46
0.41
0.34
0.20
1.4
0.09
0.15
0.22
0.28
0.35
0.41
0.46
0.48
0.49
0.46
0.42
0.14
1.6
0.02
0.09
0.15
0.21
0.28
0.34
0.41
0.46
0.48
0.49
0.47
0.10
1.8
−0.
020.
040.
090.
150.
220.
280.
350.
410.
460.
480.
490.
07
2.0
−0.
050.
000.
060.
110.
160.
220.
290.
350.
400.
460.
490.
04
2.2
−0.
07−
0.03
0.03
0.07
0.12
0.17
0.23
0.29
0.35
0.41
0.46
0.02
2.4
−0.
08−
0.04
0.01
0.04
0.09
0.14
0.18
0.23
0.29
0.36
0.41
0.00
2.6
−0.
08−
0.05
−0.
010.
020.
060.
100.
140.
190.
250.
300.
350.
00
2.8
−0.
07−
0.05
−0.
020.
010.
040.
060.
100.
140.
190.
240.
300.
00
3.0
−0.
07−
0.05
−0.
020.
000.
020.
050.
070.
110.
150.
190.
24−
0.01
3.2
−0.
07−
0.05
−0.
03−
0.01
0.01
0.03
0.05
0.08
0.11
0.15
0.19
−0.
01
3.4
−0.
06−
0.04
−0.
03−
0.01
0.00
0.02
0.04
0.06
0.08
0.10
0.14
−0.
01
3.6
−0.
05−
0.04
−0.
03−
0.02
−0.
010.
010.
020.
040.
060.
080.
10−
0.01
3.8
−0.
04−
0.03
−0.
03−
0.02
−0.
010.
000.
010.
030.
040.
060.
08−
0.01
4.0
−0.
03−
0.03
−0.
03−
0.02
−0.
010.
000.
010.
020.
030.
040.
060.
00
Tab
le 4
.21
Dim
ensi
onle
ss c
oeffi
cien
ts f
or a
naly
sis
of lo
ng b
eam
s lo
aded
with
a c
once
ntra
ted
ver
tical
load
P
SAMPLE C
HAPTER
238 Analysis of Structures on Elastic Foundations
β =
0.5
ζα
00.
10.
20.
30.
40.
50.
60.
70.
80.
91.
0∞
M
0.0
0.00
00.
000
0.00
00.
000
0.00
00.
000
0.00
00.
000
0.00
00.
000
0.00
00.
300
0.2
−0.
159
0.03
40.
026
0.02
00.
013
0.00
90.
005
0.00
40.
003
0.00
20.
001
0.20
7
0.4
−0.
253
−0.
076
0.09
90.
077
0.05
30.
036
0.02
30.
017
0.01
10.
008
0.00
40.
135
0.6
−0.
300
−0.
144
0.00
90.
165
0.12
10.
085
0.05
70.
039
0.02
50.
019
0.00
60.
075
0.8
−0.
316
−0.
183
−0.
052
0.08
10.
217
0.15
60.
108
0.07
40.
048
0.02
70.
012
0.03
3
1.0
−0.
310
−0.
199
−0.
091
0.01
90.
128
0.25
00.
180
0.12
60.
083
0.04
90.
023
0.00
1
1.2
−0.
291
−0.
201
−0.
113
−0.
025
0.06
40.
162
0.27
20.
197
0.13
40.
084
0.04
3−
0.02
0
1.4
−0.
264
−0.
192
−0.
123
−0.
053
0.01
60.
094
0.18
30.
287
0.20
40.
135
0.07
8−
0.03
3
1.6
−0.
232
−0.
177
−0.
124
−0.
071
−0.
018
0.04
20.
113
0.19
70.
293
0.20
40.
129
−0.
041
1.8
−0.
200
−0.
159
−0.
119
−0.
080
−0.
041
0.00
50.
058
0.12
40.
201
0.29
30.
198
−0.
044
2.0
−0.
169
−0.
139
−0.
111
−0.
082
−0.
055
−0.
022
0.01
80.
068
0.12
80.
201
0.28
8−
0.04
5
2.2
−0.
140
−0.
120
−0.
101
−0.
081
−0.
062
−0.
039
−0.
011
0.02
60.
071
0.12
70.
196
−0.
044
2.4
−0.
114
−0.
101
−0.
089
−0.
077
−0.
065
−0.
050
−0.
030
−0.
006
0.02
80.
070
0.12
6−
0.04
3
2.6
−0.
091
−0.
083
−0.
077
−0.
070
−0.
064
−0.
055
−0.
043
−0.
025
−0.
003
0.02
80.
067
−0.
029
2.8
−0.
070
−0.
068
−0.
066
−0.
063
−0.
061
−0.
057
−0.
051
−0.
040
−0.
024
−0.
004
0.02
4−
0.01
7
3.0
−0.
053
−0.
054
−0.
055
−0.
055
−0.
056
−0.
056
−0.
053
−0.
047
−0.
039
−0.
025
−0.
006
−0.
012
3.2
−0.
039
−0.
042
−0.
045
−0.
048
−0.
051
−0.
053
−0.
054
−0.
051
−0.
046
−0.
039
−0.
027
−0.
009
3.4
−0.
028
−0.
032
−0.
036
−0.
041
−0.
045
−0.
048
−0.
051
−0.
051
−0.
050
−0.
046
−0.
039
−0.
009
3.6
−0.
019
−0.
024
−0.
029
−0.
034
−0.
038
−0.
043
−0.
047
−0.
019
−0.
050
−0.
046
−0.
008
−0.
008
3.8
−0.
012
−0.
017
−0.
022
−0.
028
−0.
033
−0.
038
−0.
042
−0.
046
−0.
049
−0.
050
−0.
049
−0.
008
4.0
−0.
007
−0.
012
−0.
017
−0.
022
−0.
027
−0.
032
−0.
037
−0.
041
−0.
045
−0.
048
−0.
049
−0.
008
0.0
0.00
00.
000
0.00
00.
000
0.00
00.
000
0.00
00.
000
0.00
00.
000
0.00
0−
0.50
0
0.2
−0.
614
0.32
30.
258
0.19
60.
133
0.08
80.
055
0.03
90.
029
0.02
00.
014
−0.
411
0.4
−0.
339
−0.
436
0.48
40.
367
0.26
90.
189
0.12
60.
083
0.05
30.
030
0.01
3−
0.32
9
0.6
−0.
147
−0.
258
−0.
372
0.51
40.
400
0.29
90.
211
0.14
20.
088
0.08
20.
036
−0.
186
0.8
−0.
017
−0.
127
−0.
245
−0.
362
0.52
20.
410
0.30
70.
218
0.14
20.
082
0.03
6−
0.18
6
1.0
0.06
8−
0.04
0−
0.14
9−
0.26
0−
0.37
10.
516
0.40
80.
305
0.21
40.
137
0.07
5−
0.13
0
1.2
0.12
00.
021
−0.
075
−0.
178
−0.
278
−0.
384
0.50
90.
401
0.30
20.
211
0.13
4−
0.08
5
1.4
0.14
90.
061
−0.
026
−0.
113
−0.
202
−0.
296
−0.
396
0.49
90.
397
0.29
90.
211
−0.
050
1.6
0.16
00.
084
0.01
0−
0.06
4−
0.14
0−
0.22
1−
0.31
0−
0.40
60.
494
0.39
50.
300
−0.
027
1.8
0.16
00.
096
0.03
4−
0.02
8−
0.09
0−
0.15
9−
0.23
5−
0.32
0−
0.41
20.
493
0.39
7−
0.01
1
2.0
0.15
20.
100
0.04
8−
0.00
1−
0.05
3−
0.10
8−
0.17
1−
0,24
3−
0.32
5−
0.41
30.
495
0.00
0
2.2
0.13
90.
097
0.05
60.
016
−0.
025
−0.
069
−0.
119
−0.
179
−0.
247
−0.
325
−0.
410
0.00
6
2.4
0.12
40.
092
0.05
90.
028
−0.
004
−0.
039
−0.
080
−0.
126
−0.
182
−0.
248
−0.
322
0.00
9
2.6
0.10
90.
083
0.06
00.
035
0.01
1−
0.01
6−
0.04
7−
0.08
6−
0.13
0−
0.18
3−
0.24
40.
010
2.8
0.09
20.
074
0.05
60.
038
0.02
10.
001
−0.
023
−0.
051
−0.
087
−0.
129
−0.
178
0.00
9
3.0
0.07
70.
064
0.05
20.
039
0.02
70.
012
−0.
006
−0.
028
0.05
3−
0.08
6−
0.12
50.
008
3.2
0.06
30.
055
0.04
70.
038
0.03
10.
019
0.00
7−
0.00
9−
0.02
8−
0.05
2−
0.08
20.
007
3.4
0.05
00.
045
0.04
10.
035
0.03
10.
024
0.01
60.
004
−0.
009
−0.
027
−0.
049
0.00
5
3.6
0.04
00.
037
0.03
50.
032
0.03
00.
026
0.02
10.
014
0.00
4−
0.00
8−
0.02
50.
004
3.8
0.03
10.
030
0.02
90.
029
0.02
80.
027
0.02
50.
021
0.01
40.
005
−0.
007
0.00
3
4.0
0.02
30.
023
0.02
30.
025
0.02
50.
027
0.02
70.
025
0.02
10.
014
0.00
60.
002
Tab
le 4
.21
Con
’t
SAMPLE C
HAPTER
Analysis of Beams on Elastic Half-Space 239
β =
0.5
ζα
00.
10.
20.
30.
40.
50.
60.
70.
80.
91.
0∞
0.0
0.00
00.
000
0.00
00.
000
0.00
00.
000
0.00
00.
000
0.00
00.
000
0.00
00.
300
0.2
−0.
159
0.03
40.
026
0.02
00.
013
0.00
90.
005
0.00
40.
003
0.00
20.
001
0.20
7
0.4
−0.
253
−0.
076
0.09
90.
077
0.05
30.
036
0.02
30.
017
0.01
10.
008
0.00
40.
135
0.6
−0.
300
−0.
144
0.00
90.
165
0.12
10.
085
0.05
70.
039
0.02
50.
019
0.00
60.
075
0.8
−0.
316
−0.
183
−0.
052
0.08
10.
217
0.15
60.
108
0.07
40.
048
0.02
70.
012
0.03
3
1.0
−0.
310
−0.
199
−0.
091
0.01
90.
128
0.25
00.
180
0.12
60.
083
0.04
90.
023
0.00
1
1.2
−0.
291
−0.
201
−0.
113
−0.
025
0.06
40.
162
0.27
20.
197
0.13
40.
084
0.04
3−
0.02
0
1.4
−0.
264
−0.
192
−0.
123
−0.
053
0.01
60.
094
0.18
30.
287
0.20
40.
135
0.07
8−
0.03
3
1.6
−0.
232
−0.
177
−0.
124
−0.
071
−0.
018
0.04
20.
113
0.19
70.
293
0.20
40.
129
−0.
041
1.8
−0.
200
−0.
159
−0.
119
−0.
080
−0.
041
0.00
50.
058
0.12
40.
201
0.29
30.
198
−0.
044
2.0
−0.
169
−0.
139
−0.
111
−0.
082
−0.
055
−0.
022
0.01
80.
068
0.12
80.
201
0.28
8−
0.04
5
2.2
−0.
140
−0.
120
−0.
101
−0.
081
−0.
062
−0.
039
−0.
011
0.02
60.
071
0.12
70.
196
−0.
044
2.4
−0.
114
−0.
101
−0.
089
−0.
077
−0.
065
−0.
050
−0.
030
−0.
006
0.02
80.
070
0.12
6−
0.04
3
2.6
−0.
091
−0.
083
−0.
077
−0.
070
−0.
064
−0.
055
−0.
043
−0.
025
−0.
003
0.02
80.
067
−0.
029
2.8
−0.
070
−0.
068
−0.
066
−0.
063
−0.
061
−0.
057
−0.
051
−0.
040
−0.
024
−0.
004
0.02
4−
0.01
7
3.0
−0.
053
−0.
054
−0.
055
−0.
055
−0.
056
−0.
056
−0.
053
−0.
047
−0.
039
−0.
025
−0.
006
−0.
012
3.2
−0.
039
−0.
042
−0.
045
−0.
048
−0.
051
−0.
053
−0.
054
−0.
051
−0.
046
−0.
039
−0.
027
−0.
009
3.4
−0.
028
−0.
032
−0.
036
−0.
041
−0.
045
−0.
048
−0.
051
−0.
051
−0.
050
−0.
046
−0.
039
−0.
009
3.6
−0.
019
−0.
024
−0.
029
−0.
034
−0.
038
−0.
043
−0.
047
−0.
019
−0.
050
−0.
046
−0.
008
−0.
008
3.8
−0.
012
−0.
017
−0.
022
−0.
028
−0.
033
−0.
038
−0.
042
−0.
046
−0.
049
−0.
050
−0.
049
−0.
008
4.0
−0.
007
−0.
012
−0.
017
−0.
022
−0.
027
−0.
032
−0.
037
−0.
041
−0.
045
−0.
048
−0.
049
−0.
008
Q
0.0
0.00
00.
000
0.00
00.
000
0.00
00.
000
0.00
00.
000
0.00
00.
000
0.00
0−
0.50
0
0.2
−0.
614
0.32
30.
258
0.19
60.
133
0.08
80.
055
0.03
90.
029
0.02
00.
014
−0.
411
0.4
−0.
339
−0.
436
0.48
40.
367
0.26
90.
189
0.12
60.
083
0.05
30.
030
0.01
3−
0.32
9
0.6
−0.
147
−0.
258
−0.
372
0.51
40.
400
0.29
90.
211
0.14
20.
088
0.08
20.
036
−0.
186
0.8
−0.
017
−0.
127
−0.
245
−0.
362
0.52
20.
410
0.30
70.
218
0.14
20.
082
0.03
6−
0.18
6
1.0
0.06
8−
0.04
0−
0.14
9−
0.26
0−
0.37
10.
516
0.40
80.
305
0.21
40.
137
0.07
5−
0.13
0
1.2
0.12
00.
021
−0.
075
−0.
178
−0.
278
−0.
384
0.50
90.
401
0.30
20.
211
0.13
4−
0.08
5
1.4
0.14
90.
061
−0.
026
−0.
113
−0.
202
−0.
296
−0.
396
0.49
90.
397
0.29
90.
211
−0.
050
1.6
0.16
00.
084
0.01
0−
0.06
4−
0.14
0−
0.22
1−
0.31
0−
0.40
60.
494
0.39
50.
300
−0.
027
1.8
0.16
00.
096
0.03
4−
0.02
8−
0.09
0−
0.15
9−
0.23
5−
0.32
0−
0.41
20.
493
0.39
7−
0.01
1
2.0
0.15
20.
100
0.04
8−
0.00
1−
0.05
3−
0.10
8−
0.17
1−
0,24
3−
0.32
5−
0.41
30.
495
0.00
0
2.2
0.13
90.
097
0.05
60.
016
−0.
025
−0.
069
−0.
119
−0.
179
−0.
247
−0.
325
−0.
410
0.00
6
2.4
0.12
40.
092
0.05
90.
028
−0.
004
−0.
039
−0.
080
−0.
126
−0.
182
−0.
248
−0.
322
0.00
9
2.6
0.10
90.
083
0.06
00.
035
0.01
1−
0.01
6−
0.04
7−
0.08
6−
0.13
0−
0.18
3−
0.24
40.
010
2.8
0.09
20.
074
0.05
60.
038
0.02
10.
001
−0.
023
−0.
051
−0.
087
−0.
129
−0.
178
0.00
9
3.0
0.07
70.
064
0.05
20.
039
0.02
70.
012
−0.
006
−0.
028
0.05
3−
0.08
6−
0.12
50.
008
3.2
0.06
30.
055
0.04
70.
038
0.03
10.
019
0.00
7−
0.00
9−
0.02
8−
0.05
2−
0.08
20.
007
3.4
0.05
00.
045
0.04
10.
035
0.03
10.
024
0.01
60.
004
−0.
009
−0.
027
−0.
049
0.00
5
3.6
0.04
00.
037
0.03
50.
032
0.03
00.
026
0.02
10.
014
0.00
4−
0.00
8−
0.02
50.
004
3.8
0.03
10.
030
0.02
90.
029
0.02
80.
027
0.02
50.
021
0.01
40.
005
−0.
007
0.00
3
4.0
0.02
30.
023
0.02
30.
025
0.02
50.
027
0.02
70.
025
0.02
10.
014
0.00
60.
002
SAMPLE C
HAPTER
240 Analysis of Structures on Elastic Foundations
β =
0.5
ζα
00.
10.
20.
30.
40.
50.
60.
70.
80.
91.
0∞
Y
0.0
1.69
1.43
1.21
1.01
0.82
0.66
0.52
0.40
0.31
0.24
0.20
0.60
0.2
1.43
1.25
1.08
0.94
0.78
0.66
0.54
0.44
0.36
0.29
0.24
0.59
0.4
1.21
1.08
0.97
0.87
0.76
0.66
0.56
0.48
0.40
0.34
0.28
0.56
0.6
1.01
0.94
0.87
0.80
0.73
0.66
0.59
0.51
0.45
0.39
0.32
0.52
0.8
0.82
0.78
0.76
0.73
0.71
0.66
0.61
0.54
0.50
0.44
0.39
0.47
1.0
0.66
0.66
0.66
0.66
0.66
0.65
0.62
0.57
0.54
0.49
0.44
0.43
1.2
0.52
0.54
0.56
0.59
0.61
0.62
0.62
0.60
0.57
0.53
0.48
0.38
1.4
0.40
0.44
0.48
0.51
0.54
0.57
0.60
0.60
0.59
0.56
0.53
0.33
1.6
0.31
0.36
0.40
0.45
0.50
0.54
0.57
0.59
0.60
0.59
0.57
0.28
1.8
0.24
0.29
0.34
0.39
0.44
0.49
0.53
0.56
0.59
0.61
0.60
0.24
2.0
0.20
0.24
0.28
0.32
0.39
0.44
0.48
0.53
0.57
0.60
0.61
0.20
2.2
0.16
0.19
0.24
0.29
0.34
0.39
0.43
0.48
0.52
0.57
0.60
0.16
2.4
0.13
0.15
0.20
0.24
0.29
0.34
0.39
0.43
0.48
0.52
0.57
0.12
2.6
0.10
0.13
0.16
0.21
0.25
0.30
0.34
0.39
0.44
0.48
0.53
0.09
2.8
0.08
0.11
0.14
0.18
0.22
0.26
0.30
0.34
0.39
0.44
0.49
0.05
3.0
0.07
0.09
0.12
0.15
0.19
0.23
0.26
0.30
0.34
0.39
0.44
0.03
3.2
0.06
0.08
0.10
0.13
0.16
0.20
0.23
0.27
0.30
0.35
0.39
0.00
3.4
0.05
0.07
0.09
0.12
0.14
0.17
0.20
0.23
0.27
0.31
0.35
−0.
03
3.6
0.04
0.06
0.08
0.10
0.12
0.15
0.18
0.20
0.23
0.27
0.30
−0.
05
3.8
0.03
0.05
0.07
0.09
0.11
0.13
0.15
0.18
0.20
0.23
0.27
−0.
07
4.0
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.16
0.18
0.20
0.23
−0.
09
Tab
le 4
.21
Con
’t
SAMPLE C
HAPTER
Analysis of Beams on Elastic Half-Space 241
References1. Borowicka, H. 1938. The distribution of pressure under a uniformly loaded strip resting on
elastic isotropic ground. 2nd Congress International Association for Bridge and Structural En-gineering (Berlin), Final Report, VIII, 3.
2. Borowicka, H. 1939. Druckverteilung unterelastischen Platten. ING. Arch. (Berlin) 10(2): 113–125.
3. Gorbunov-Posadov, M. I. 1940. Analysis of beams and plates on elastic half-space. Applied Mechanics and Mathematics 4(3): 60–80.
4. Gorbunov-Posadov, M. I. 1949. Beams and slabs on elastic foundation. Moscow: Mashstroiizdat.5. Gorbunov-Posadov, M. I. 1953. Analysis of structures on elastic foundation. Moscow:
Gosstroiizdat.6. Gorbunov-Posadov, M. I. 1984. Analysis of structures on elastic foundation. 3rd ed. Moscow:
Stroiizdat, pp. 252–298.7. Selvadurai, A.P.S. 1979. Elastic analysis of soil-foundation interaction. Amsterdam/Oxford/
New York: Elsevier Scientific, pp. 11–121.8. Tsudik E. A. 2006. Analysis of beams and frames on elastic foundation. Canada/UK: Trafford,
pp. 29–47.
SAMPLE C
HAPTER