NATIONAL BUREAU STANDARDS REPORT · 10388 ELASTICSTABILITYANDDEFLECTIONS...
76
N9T-~f9R P USL I CAT tOtr-ei^^fOiT^ffH^-g- NATIONAL BUREAU OF STANDARDS REPORT 10 388 ELASTIC STABILITY AND DEFLECTIONS OF ECCENTRICALLY-LOADED COMPRESSION MEMBERS MADE OF MATERIAL WITH NO TENSILE STRENGTH U.S. DEPARTMENT OF COMMERCE NATIONAL BUREAU OF STANDARDS
Transcript of NATIONAL BUREAU STANDARDS REPORT · 10388 ELASTICSTABILITYANDDEFLECTIONS...
N9T-~f9R P USL ICAT tOtr-ei^^fOiT^ffH^-g-
NATIONAL BUREAU OF STANDARDS REPORT
10 388
ELASTIC STABILITY AND DEFLECTIONS
OF ECCENTRICALLY-LOADED COMPRESSION
MEMBERS MADE OF MATERIAL WITH
NO TENSILE STRENGTH
U.S. DEPARTMENT OF COMMERCE
NATIONAL BUREAU OF STANDARDS
NATIONAL BUREAU OF STANDARDS
The National Bureau of Standards ' was established by an act of Congress March 3, 1901. Today,
in addition to serving as the Nation’s central measurement laboratory, the Bureau is a principal
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NATIONAL BUREAU OF STANDARDS REPORT
NBS PROJECT
4212111 November 23, 1970
NBS REPORT
10 388
ELASTIC STABILITY AND DEFLECTIONS
OF ECCENTRICALLY-LOADED COMPRESSION
MEMBERS MADE OF MATERIAL WITH
NO TENSILE STRENGTH
By
F el ix Y. Yokel
IMPORTANT NOTICE
NATIONAL BUREAU OF STANDARfor use within the Government. Before
and review. For this reason, the public,
whole or in part, is not authorized un
Bureau of Standards, Washington, D.C.
the Report has been specifically prepare
Approved for public release by the
director of the National Institute of
Standards and Technology (NIST)
on October 9, 2015
jnting documents intended
°d to additional evaluation
of this Report, either in
of the Director, National
ivernment agency for which
r its own use.
<NBS>
U.S. DEPARTMENT OF COMMERCE
NATIONAL BUREAU OF STANDARDS
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TABLE OF CONTENTS
Page
ABSTRACT ‘ il
1. INTRODUCTION AND SCOPE 1
2. ASSUMED MATERIAL PROPERTIES AND MEMBER GEOMETRY 1
3. LOADING CONDITION 2
4. EQUILIBRIUM 2
5. THE DIFFERENTIAL EQUATION FOR THE DEFLECTION CURVE 5
6. SOLUTION OF THE DIFFERENTIAL EQUATION FOR THEDEFLECTION CURVE 8
7. APPLICATION OF THE SOLUTION TO DERIVE CRITICALLOAD, MAXIMUM DEFLECTION AND MAXIMUM STRESS 10
8. SUMMARY OF THEORETICAL ANALYSIS 12
9. APPLICATION OF THE THEORY TO PREDICT THE RESULTSOF FULL-SCALE TESTS ON MASONRY WALLS 13
APPENDIX I - NOTATION 20
APPENDIX II - DERIVATION OF SOLUTION 22
APPENDIX III - REFERENCES 25
FIGURES
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Elastic Stability and Deflections of Eccentrically -
Loaded Compression Members Made of Material With NoTensile Strength
by
Felix Y. Yokel
A mathematical solution is derived, which permits the
computation of critical load, deflection and stresses foreccentrically loaded slender prismatic compression members,made of materials that develop compressive strength but haveno tensile strength. A graphical presentation of the solu-tion facilitates its application. In an example of applica-tion, the solution is used to compute the strength of masonrywalls which were tested by the Structural Clay ProductsInstitute. There is good agreement between computed and mea-sured strength.
Key Words: Buckling , compression members, deflection ,
equilibrium, load eccentricity, masonry ,
stability , section cracking, stress distribution,unreinforced concrete.
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1. Introduction and Scope
In this paper solutions are derived for the computation
of elastic deflections and stability of compression members
made of materials with compressive strength and no tensile
strength and with a linear relationship between compressive
stresses and strains. Computed values could be considered
upper limits for deflections and lower limits for critical
load of solid prismatic walls made of masonry or unrein-
forced concrete, since these materials have properties simi-
lar to those assumed in the paper. In Appendix III the solu-
tions are used to predict the results of 39 full-scale tests
on brick walls which were conducted by the Structural Clay
Products Institute [ 2 ]
,
and the correlation between test
results and theory is discussed.
2. Assumed Material Properties and Member Geometry
The material is assumed to be elastic, to have a linear
relationship between stress and strain in compression and to
have no tensile strength. The problem is analyzed for a
prismatic member with a solid rectangular cross section with
dimensions as shown in Figure 1.
_i/Numbers in brackets refer toin Appendix m.
literature references listed
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3. Loading Conditions
Loading conditions are shown in Fig. 2. The compres-
sive force P is a lineload which is evenly distributed
along the width b of the member. The force is applied at
an eccentricity equal to or greater than the kern- eccentric! ty
(e ^ t/6) . The line of action of force P is parallel to
the axis of the undeflected member and the force is applied
through hinges at both ends of the member. Thus the ends
of the member are not restrained from rotation. The member
is also not restrained from shortening in the direction of
the compressive force.
4. Equilibrium
4.1 Equilibrium at a Cross Section
Equilibrium at any cross section requires that the
resultant reaction force be equal to and collinear with the
applied force P. The corresponding stress distribution
over a cross section is illustrated in Fig. 3 for two cases.
Figure 3(a) illustrates the case where P acts at an
eccentricity equal to kern eccentricity (e = t/6) . Figure
3(b) illustrates the case where e > t/6.
In the case illustrated in Figure 3(a) the compressive
2 /stress (a) at one face (the tension face)— of the cross sec-
tion is zero. At the other (compression) face, the maximum
— ^ ^ .
—The term "tension face" is used here to identify theface that is subjected to zero stress and developstension cracks, even though no tensile stress can existin the member.
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compressive stress (a ) occurs:o
2P''o
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In Figure 3(b) a tensile crack appears at the tension
side of the cross section, since the material has no tensile
strength. The uncracked part of the cross section has a
triangular stress distribution similar to that shown in
Figure 3(a), where a = 0 at the origin of the crack.
The maximum stress at the compression face of the cross
section is:
a = ^ (eq. 1)3bu
where u is the distance between the action line of force P
and the compression face of the cross section.
The uncracked depth of the section in figure 3(b) is
3u, and the depth of the crack is t-3u. The expression in
eq. 1 is valid for all cases where e ^ t/6. This includes
the case in figure 3(a), where u = t/3
,
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4.2 Equilibrium of the Structural Member
The stress distribution within the structural member,
when force P is applied at an eccentricity e ^ t/6, is
illustrated in Fig. 4. The rectangle outlined by broken
lines shows the undeflected shape of the member. The
deflected shape is shown by the heavy outline. The shaded
area within the deflected member shows the uncracked zone
which supports the load. The stress distribution at one
particular cross section is shown by the heavily -shaded
triangle
.
The distance between the compression face of the mem-
ber and the line of action of force P, u, varies along the
height of the member, because of member deflection. The
maximum distance, u^ , occurs at the two member ends. The
minimum distance, u , occurs at midheight. The line ofo
action of force P, which is shown in the figure by a dashed
line, passes through the edge of the middle third of the
compression zone. Since the stress at the compression face
of each section, a , is inversely proportional to u, theo
maximum compressive stress in the member, a occurs at^ max
midheight
:
2P^max 3bu
o
(eq. 2)
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5 . The Differential Equation for the Deflection Curve
Figure 5 shows the deflection curve of the compression
face of the member, together with the coordinate system used.
Since the member is symmetrically loaded the origin was
assumed at midheight. The x axis is parallel to the action
line of P, and is tangential to the deflection curve at the
origin. At each point, u = u + y, and, at x = h/2,o
u = u + y = u. .
o 1
Figure 6 shows a small element of the deflected member
between the compression face and the boundary of the uncracked
zone, where a = 0 at all times. The element itself is shown
by the solid outline. The average depth of the element is
3u. The initial undeflected length is d£.
The curvature of the compression face is caused by the
shortening of the face under compressive stress relative to
the boundary of the uncracked zone, which does not change in
length since a ~ 0 on the boundary.
As dJl-^0, the following expression can be written for
4), the change in the slope of the deflection curve over the
length of the element:
. e d)l
where e is the average strain over length d£,.
5
: noi# Ut:}rT&i^l%tQ srfT ^ .£
-.^.-' 4!
TIE) i>viuo ite.iSy*ifl'l&b BdT Hr^od^-Z oTugiH *
M ^ •
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..,^
( :
nisi 10 b*^t»iiof t* ei i^dascn adj aaitiS
. -^ii
^jH 9fi^ ot Jaitt?t ;>*7 zk bIaA x -^HT . Jriglfifrfbfti ' AdfrtxsBBB-S'.* ''’.-*>? ^
•
ifi ovTji’? t?4^ o^ ijt A 7 xr
-
>fti\ t ;;i bflii Id anilr
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1 /
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&r a/ti !^o a^il|!5-3V^ .aiiilumi ' rfoe j^iii \d
". M z'f Uaj':jaI^oI>nH; IcSSiHi ,al
aff7 ‘<1 i ^1 iiO 1 1 n %:> offt l^‘
:. «
>T ' v - jcftji WiBca Tqrq<j:*}V^al^iu^ -jtfit *:o arunat.iorfa'
li 9$g:s^ ion Be;o 6 dt* Tr\w
/
‘ " .yttbiu^diS 9jff * ?j 3Dnl 2
Ofii xc '. G nnxjD^i 1 al> l,o t»«ri- • ' Jii osti^nd-j ofU .
’
.’*
. • -A:'
;jfn!T( 9 fi sifj ^0 rff^i>,£ \ .
<e^
Vl'
:Jv*
** k.-*
•^- "C' r,
* t : ,^:iD
T'<v:? ni::»Tt 9! a4^£:*r'>^fc ?*c ^ at!!jKw•
* ‘3 ^“ _ ^ i
**J
n I
..* .'
'“rl.ds* i'i v.>'>•
3
a
but c - 2P 1
3bu • E(see eq . 1)
where a is the average stress at the compression face overo
length dil and E is Young’s modulus of elasticity,
thus 0 =_£ dil _ 2P dil
3u 9Eb•
The length of the compression face of the deflected
element, d£ (1 - e) ,can be expressed in terms of t)ie radius
of curvature p:
p 4> = diil(l - e)
ord£ (1 - e)
2P 1
9Eb u^(l - e)
generally2 2
d y/dx
[1 + (dy /dx)^]^/2
The following equation can therefore be written for
d^y/dx^:
3 /
d^y _ 2P [1 + (dy/dx)^] ^
di? " 9Eb n^(l - e)
6
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(l .po 99S)^S
•'jur
m,
, f•’ .'
•s^vo itolgBOtqwo.’:) a*!:J od'i zi d ^Tsdw• !'
.
®
,viiax3^8^o Tio ^^ulufcr? n 2i J hnn Sh
J5';'-- . .V 'H-'E
^ XV- - i
* ' -*/, >
^IS ^ >Xi> 3' .tL- I» o.i»t»,j,»« I 19 A
J -. ;,4
'
im orn.
«K,/
hata^f^oh ^dS lo ijoi ajla^^o HjjRnoX oKT. •
• -ii ,’
,• 4\ i- 4
awlf>R*r or(5 *ig enrwj r'i *.l3 - I) -tb ^irrami^IsD
./ i 4
. 'aa%: Q 9TU:JRVTUD lo
r"!':•'* 4fiP* ^
..
•- I)4t» *» #a
«*
.
X
f5"-'IT^ S? (9 - t> ib
• , »
'. t"*«•
•, ,
•* .
j • , . .•/•
_I-W ,>•
+ J)
1'
Q
mTrS
ifll- nsJJtTw o<! 9io5 9t3ifJ (u‘3 JioiiBupo gfliwoliol: arfT
I
3^'
,
- - fPl, <<3^ ^dae xb
\,
,K -y'jSf t _:- •,
* lii’-
This derivation is envisioned to apply to materials
which develop relatively small strain (e < 0.005). The
quantity 1 - e will therefore approximately equal unity.
The maximum deflection developed will also be small rela-
tive to the height of the member, and the slope of the
deflection curve dy/dx, will be much smaller than 1 even
at the point of maximum slope. The following approximation
will therefore cause errors which are of lower-order
magnitude and do not appreciably affect the result:
[1 - (dy/dx)^ ^
1 ^ e
A similar assumption is traditionally made in derivations
of elastic deflections and critical loads for conventional
structural members.
2 2thus the expression for d y/dx reduces to:
d^y _ 1 _ 2P_ 1
7^ P" 9Eb • 2
dx ^ u
2PSince the value is constant for any particular member,
the constant:
2P^1
~ substituted into the equation.
7
ei sf, 1* . . 1,0 -0 -vb* 5 r«ij,dvir«5 -i-iT
•rrfT .fvdO-C > »j ‘hftx^i* f|oi^veb /foS
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-
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B ia7U^'y(j\ja
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- — . / -r.
. .^o j jrj:ut>A cfft ctHnl .' i-. JT^^"
a• -' -
furthermore: u = u + y*o
The following differential equation can therefore be written:
d y _ 1—
2
” 1dx (u +y)
where : K, = 2PTCF
= constant
(eq. 3)
The following three boundary conditions apply to this equa-
tion (refer to Fig. 5)
:
at X = 0, II o (1)
at X = 0, II o (2)
at X = h/2; y = u - uo
(3)
6. Solution to the Differential Equation for the DeflectionCurve
An exact solution for eq . 3 has been derived. Detailed
steps of that derivation are presented in Appendix II. The
following equation has been derived for the deflection curve:
= ±fu. In
n/v" + y/u.
-)](eq. 4)
8
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n ” ; S\rf • X JS
vi
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-rAAJ
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'jc'lj 'fo3 cio^kffVpi^ i f i j ft’Q # o^ noiJ»i^o2 jB.devitoD
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: ov*Tiso SO
t
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\ * »«"
; -#
• •*
'
if boundary condition (3): at h/2; y = u. - uo
is substituted into this equation, the following relationship
between P, u^ , and u^ can be derived (See Appendix II):
P
Pec
0.40528 Cl 1 - a + a Zn
where
:
Pec
tt^EIe
h2
The term P^^ is an "equivalent" critical load, computed from an
"equivalent" I , which is based on an "equivalent"
thickness of 3u^
.
Ie
2 7bu?T2
—
therefore Pec
27 7T^ EbUi^
12 • T1h
P becomes the Euler load P when the section isec c
loaded at the edge of the kern (3u^ = t).
uo
a = — is a measure^1
deflection.
of the magnitude of the maximum
9
4) i„,o - y Ja ; (t)' non: fijrto:) y;'tKl>.iuo4*3j
guknm^lol t?iiJr ,a;i4^y^ «Iiil edw^fe*> 1 .
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7. Application of the Solution to Derive Critical Load,Maximum Deflection and Maximum Stress
Figure 7 shows a graphical presentation of equation 5.
The plot is non dimensional and can be used to solve any
particular numerical problem.
Values of P/P are plotted along the ordinate. On the0 c
abscissa, values of u /u. are plotted. It can be seen thato -L
each value of P/P__ corresponds to two values of u /u, ,0 C 0-1-
except for the highest value of P/P that can be maintained
by the member. The plot of the solution is interpreted as
follows
:
I
The value u /u, = 1 corresponds to an undeflectedO J-
section. Thus this corresponds to no axial load or P/P = 0
Obviously, no axial load could be maintained if u vanisheso
or u /u, = 0, since the mid section would be cracked over theo -*
full width, t. Thus the value P/P = 0 also corresponds to0 C
u /u, = 0.
When P is applied to an undeflected member and increased
deflections would increase correspondingly and the value of
u /u, would decrease. This corresponds to the portion of
the curve between u /u. = 1 and u /u, = 0.625, which repre-o JL o -L
sents a condition of stable equilibrium. At the value of
u /u, = 0.625 the maximum load is applied that the member
can support, provided compressive strength was not exceeded
at a lower value of P/P . Any further increase of P beyond
this point would cause the member to collapse.
10
V I- 4.
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x‘- ¥’
Thus the important conclusion is reached, that elastic
instability will occur at the critical load of:
P = 0.285 P ^cr ec
where Pec
27it^ Ebu,^
T~12 h^
Ebu.^
or P = 0.64 >— (eq. 6)cr
The values of P/P„_ corresponding to values of u /u, ine c o -L
the range: 0.625 > u /u, > 0 have little physical signifi-
cance. They correspond to a second deflection that can be
maintained by loads smaller than the critical load in a state
of unstable equilibrium.
Maximum stresses in the member, corresponding to any
load smaller than the critical load (P ) , can also becr
computed using the plot in Fig. 7 to compute u , and eq. 2o
to compute maximum stress:
2P^max ~ 3bu
o
11
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r
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8. Summary of Theoretical Analysis
For the prismatic member shown in Fig. 1, made of
material with no tensile strength, and subjected to the
loading condition shown in Fig. 2. The following values
can be computed from an exact solution to the differential
equation for the deflection curve:
* The critical load:
2 3TT^Ebu,
Prr = ^cr
* The maximum compressive stress:
- 2P r
^max ” 3bu (^q. )
o
where u can be determined from Fig. 7.o
* The maximum deflection can be derived from Fig. 7.
12
4:
I
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V
9. .Application of the Theory to Predict the Results of
Full-Scale Tests on Masonry Walls
9.1 Source of Test Results and Test Specimens
Only few test data are available which fit the boundary
conditions and material properties assumed in this paper.
The test results chosen for comparison were published by the
Structural Clay Products Institute (SCPI) [2] ,and represent
39 tests on full-scale brick walls of various slenderness
ratios (h/t),loaded as shown in Fig. 2. Loads were applied
at two eccentricities: e = t/6, and e = t/3. The walls were
solid walls of single- and double-wythe construction, built
in running bond. Slenderness ratios varied from h/t =6.6
to h/t = 46.1, Mortar was ASTM type S [1] and the compres-
sive strength of the masonry (f^J^) varied from 4,160 psi to
5 ,100 psi
.
9.2 Comparison of the Assumed Material Properties with the
Properties of the Test Specimens.
While brick masonry has material properties similar to
those assumed in the solution presented in this paper, several
differences between the properties of the test specimens and
the assumed idealized properties should be noted:
13
I
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nw^J ...
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_ ^ D
rt a j .» a ")cf bliictde cj, X boW^
^
4- '
.
• <>r V Iy
(1) The masonry used in the specimens tested should be
expected to develop tensile strength of an order of
magnitude of 2-3 percent of its compressive strength.
This property would somewhat change the equilibrium,
conditions illustrated in Fig. 3, which were assumed in
the derivation. The change should result in somewhat
greater-than-predicted capacities. The discrepancy
would be greatest where stability failure occurs at
relatively low stress levels (large slenderness ratios
and eccentricities)
.
(2) The stress-strain curve for brick is not exactly
linear. The tangent modulus of elasticity at failure
tends to be about 70 percent of the initial, tangent
modulus of elasticity [3]
.
If deflections are predicted
on the basis of the modulus of elasticity at low
stress levels, deflections at high stress levels would
probably be greater than the predicted deflections.
This change in stiffness also slightly modifies the
stress distribution shown in Fig. 3. These effects
will increase with increasing stress levels at failure.
(Small slenderness ratios and eccentricity) .
(3) The brick units themselves have greater strength
and stiffness than the mortar beds connecting the units.
This discontinuity results in a stress distribution
which is much more complex than the idealized stress
14
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.Vi®
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/ ,% .(vt |3lT:rn-f^#S% lifiR aoibftl IlBm3)ff
/tf^n5.3^E 79.io^5r ^vftff savX^^f^adX eJiw ^faiid odT
?L_ ^ fV
noiJ'udtYjsib r,:.£?ai a ni aJXozsi \-5 ivniJno.ieib eidw»^
^ r ^I
boiilaohi rod^* itE'fU ;<^Jq<noD >-ioi:i fi'yim ei d^ldw^A
distribution assumed in the solution. The net effect
of these discontinuities on strength and stiffness
has not been evaluated.
9.3 Equations for Strength Calculations
The test results presented by SCPI were obtained from
a variety of brick walls, and for the sake of comparison,
individual tests were normalized by dividing the vertical
load carried by each wall by , the load capacity of a
cross section subjected to axial compression: P^ = f^ bt
.
The value of f^, the compressive strength under axial load,
was in each case derived from the average strength of short
brick prisms which were constructed from similar brick and
mortar at the time the corresponding test specimens were
constructed and which were cured under similar conditions.
It has been observed for brick masonry [3] that compressive
strength in flexure exceeds the compressive strength in
axial compression and that the strength increases with
increasing strain gradients. In this case it has been deter-
mined that cross sectional capacity of walls loaded at t/6
and t/3 eccentricities can be approximately computed by using
the stress distribution in Fig. 3 and assuming a compressive
strength in flexure equal to 1.6 times the strength in axial
compression: af’ = 1.6f’.^ mm
15
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It was also determined, that for the walls tested, the
modulus of elasticity is predicted with sufficient accuracy
_ 3by assuming the fixed ratio of: f'/E = 1.215 x 10
The following equations can therefore be derived:
Cross Sectional Capacity:
at e = t/6:
at e = t/3:
Wall Capacity:
Walls may either fail when the flexural compressive
strength is exceeded or when elastic instability occurs.
For the case of a compression failure, wall capacity may
be computed from eq. 2:
P = 0.8 Pu o
P = 0.4 Pu o
at e = t/6:u
P = 0.8 Pu o u
o
and =0.8—P u.o 1
uat e = t/3: P = 0.4 P -2.
u o u^
P uand ^ = 0.4
uwhere — can be determined from Fig. 7.
Ui
uo
For a stability failure the following equations can be
derived from eq . 6:
TT Ebu
p„ = 0.64 ]_ = 0.6427 h \ ^ /
16
ftl,
yIf
• *
9ittf .()•»? a^j ^I(rw 4»dt lot Ss^t ;b okIc ?i>w sT
ysETiOii. Jasu^aiTlii? i!;fw bw^afbsiq «f 5o eufobom
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> 9
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9vi223tq«oo XflTOxan orfj a^iiw Ii*>' V -9
4. M
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oij
:2 .po i«ot1 boiuqmoo, 9d
fe- 8. a *- :d\j * d iB
jj3.0
....
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:v.'^'
ifi o
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P.T rro"^> bon rental ‘jb ->cf mu — o-roii'
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,
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^ - Vj , :
>' V ji.5, 5:V
. V '
,^.:fi’*tr
0 • ^ , 0 ® ^
E = (fVl.215) . 10
and P = £’ bto m
P = 192 Pcr
cr _ 192
Pq (h/t)
for e = t/6;
and P TOOcr _ 192
r4i^)
3u^ = t
Pq Ch/t)
for e = t/3;
and P
=2
cr _ 24
Pq (h/t)"
The following general equations for wall capacity as
a fraction of P^ therefore apply:
at e = t/6:
P upH = 0.8 —
whichever is smaller
at e = t/3:
P u= 0.4
o
ou.
(eq. 7)
oru 24
Pq (h/t)"
whichever is smaller (eq. 8)
17
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,
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'
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S ItiA.f
9.4 Comparison of Theory with Test Results
Curves based on equations 7 and 8 are plotted in Fig.
8. Note that for the t/6 eccentricity compression failure
occurs at slenderness ratios smaller than 22 (h/t < 22)
.
For the t/3 eccentricity compression failure occurs at
slenderness ratios smaller than 11 (h/t < 11)
.
In Fig. 9 the theoretical curves are compared with
test results. Test results at t/6 eccentricity are shown
by the triangles and should be compared with the correspond-
ing curve. Test results at the t/3 eccentricity are shown
by the circles.
In general the trend of the test results is correctly
predicted by theory and the magnitude of the load capacity
is approximately predicted. For the t/6 eccentricity capa-
city was overestimated for h/t = 14 by about 101 . At
h/t = ^6 the capacity was underestimated by a substantial
margin, probably because at this low stress level, the
effect of the tensile strength is significant. For all
other slenderness ratio the average capacity was approxi-
mately predicted. For the t/3 eccentricity the capacity
also tended to be underestimated at very low failure-stress
levels, however, in the range of stability failure, the
theoretical curve tends to approach the lower limit of the
measured capacities.
18
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3??;«3
9.5 Conclusion From the Comparison of the Theory with
Test Results
The theory correctly predicted the trend of the test
results and approximately predicted the magnitude of mea-
sured capacities. Since equations 7 and 8 are simple and
relatively easy to use, their practical application should
be seriously considered.
19
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E^Iwao>5 "t>eT v ^3
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baB a^cimli 8 bfr« t efrai^c/t^a <J3|iia ba5u«
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, .
?a ,1^
‘ * am .rfeafebl^rtoa ylaiioiTae ad
Appendix I. Notation
af ’
m
b
e
di
E
£’m
h
I
Ie
P
Pc
Pec
Pcr
Po
u
t
u
ui
u.
flexural compressive strength of masonry
width of member
load eccentricity
incremental length
Young^s modulus of elasticity
compressive strength of masonry under axial load
height of member
2 Pconstant (
9^)b t
moment of inertia
2 7bu'equivalent moment of inertia (
—
compressive force applied to member
tt^EIcritical (Euler) load ( j) ^
h Ti El"equivalent” critical load (
h^critical load of member
cross sectional axial-load capacity
load capacity under eccentric vertical compressive load
thickness of member
distance between line of action of compressive load
and compression face of member
distance between line of action of compressive load and
compression face of member at midheight
distance between line of action of compressive load and
compression face of member at member support
20
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k > *1 *
vyi •£.: j
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bBj>l fsixE lobnu \inoiBm ’io ©vi4
'• -rji
'm '. i^{ jr
yrf
qirciifm Jo xf
f| C - r '
1
fneJ 57103 ,;i3
, C’^rr) lo JSJwaom}-f-4 I,t^ ^ *1
„..y*
fr ‘‘f r 'V*— ) fiijrfanJt irt!||da3T TO^iJBviup^ I
» *
#
r, .-V
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tryn -H . Vi]^
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/I a
1 1 df ' 51# J 0 0£ Oi I. so l : 1 1
3
.. 1
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T:djp^9« iu |S i
bfiol 9 ‘i\ip^,j»iqiuoo *Vo a 6itps anl? ‘ vf
^if''"'.' *; lo ixOX 4Te5^qflr?io bms >j
.
' bnn buQS ‘^•vler.orqmi^zt lo noslos io 9 /fr^saib^ u• o
,r-
I>E
, 1
^ tn mbm noi cic*^\r4moo
ivl^^OTqraoO lo froxt'^i .' Q BJCihiaib .u:
4», .. . X-
J'T 0q e 1 3<k3 t44iT } i. fo d c; 5fa 1 b 5 cl n
o
i ^ s qnr€»
o
ft ./
•A^ ^*LM .%0
\
*1
a dimensionless parameter —G strain
4) change in slope of de flection
P radius of curvature
a compressive stress
ao
maximum compressive stress in
amaxmaximum compressive stress in
curve
cross section
member
21
'. 7 ^
't
-2- ^a^lftoi ^aOTib 7 '
t>
•-I 1 >
f,-,
» «tiC)
’,.'' t Wv1 1'.-,'
>
nifiTje
0VTi-'3 ffOxJoi^Iieb lo oqolE nx ognsH^‘
i.'-.; .- !•>; ( it
, '5 ^ !'
VdifutsvtUD lo q
*^vi ^ t>
noi^Dae ;^8otD ai 2a3t^« avi2^o^qmo:> o
? :
' v;»' iii,
' '>{
iad®ofli nl EebT^e avf220tqmo3 nujorixam ' Vt>* xaui
^'
i ^ ‘ ^ . 'a*
APPENDIX II
DERIVATION OF SOLUTION
Kl_y)
2 > where2P_9bE
Boundary Conditions :
22
It „ . . _fvl
' O
rtabcLung
‘""' »#®r'' .' \ i'
.t A» I
^.Vi
'!ik\ li- , xb .,^025^,-.?(^ / xb xb^ ^ n(b xb^
Ir-0 '*' ‘“"‘^ rXS*^ V *f O DLi
O J|_T,^,. -
-'
1
“
<0 ' i« >
xb1B
»SJJii.
(It( + x;
o ('
- Jft
/_1_A‘'" HT\ + ^. . 3^,;>'
\'T 's'"'/ 0
'-’' -^“ ^^
i dfliixv-i
“« V. -v r.,,
.,-3
> 7,.fx—
\
;->
• C.Oii^D.-'^^
4^
^ Jf {k.±^,f,+ >.-*b -^a'S
"iJ •' \'^' "
Ti,
y m >*3‘.-:i-S^l
ay - Vy Wo + y) + fJvy I ^ * y>
= >/y (^Uq + y) + Uq Zn (Vy^ + \/Uq + y)
•*
• X = ± [Vy (uq + y) + Uq Zn (V7 + >/% + y)]
at X = Q, y = 0;
• • C2 = -
^ = ± ^"o + y) + Uq ^ ^)]
Substituting K2
X = ±1 [j v (u, + y) + u,
Solving for u^:
at X = -, y = ui - Uq;
Assume a = ^̂1
„ 9Eu, 3
P = —o-^ a yj! - a + atn(nF?
!a(P
23
t ( * V '?* *
I If
’'("-Hlii
H *• ^ kI o“ tv <- o«>) tV r ']
Xi>
+ [<V + o"V + '’S + ^(V + o'’^ -
:0 « X ,0 X 3a
,D + o*'
nrL.t<5 -o** v^ * ^ ^
a.)„5 , TTf^-lJ 1
1
%^1 ,/ .f
'-
^m
gxiJtJu3l:3a<fl/3
• l‘f
:^u 3oi '^aivIoS
,'*'S
[(i^V
K '-4
r^oU !* fU *• Y X 3^
+ - I^)<5 l±
[(?V*"' » T TO
- ’.. C^'l
'"''!.<
't*. "^''T'^virC
:'./.• v4:
• * » .n
V-
p vr*j - iV
1-
it •><
* y--^ V
Let P "equivalent" critical load
then
or
ec
27tt^ Eu^12i?“
P
Pec
0.40528 a 1 “ a + a Zn2
— = fMec
24
» ^*1* p>.'- ~'
Appendix m - References
[1] American Society for Testing Material, ”ASTM C270-68,
Standard Specifications for Mortar for Unit Masonry”,
Philadelphia, Pa, , 1968
[2] Structural Clay Products Institute, Recommended
Practice for Engineered Brick Masonry , Tables 5-20
and 5-21, McLean, Va., November 1969
[3] Yokel, F. Y. , Mathey, R. G. and Dikkers ,R. D.
,
'^Strength of Masonry Walls Under Compressive and
Transverse Load”, Building Science Series 54 , National
Bureau of Standards, Washington, D.C., January 1971
25
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L'W'
8d<)l 4 •«*! 4 airfalsb^Xiiit
sW®I
^ ^ ^ IHp
' bobnoirnnopg^H ^oiutxJgnl ^jrolaboiJ? yslD XatxiJDiiTJ'^ [S]
; ,
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’’ '^
' tP '
*'
€dOI TadmdYoH , * sV y, nsavI^M tlX-S
,.Cl .« ,;*Tca3(iia bna ,0 .51 ,
xarfJBM". ,Y *"4 .'IojIqW f5]
V.'
lV' •».'«% bna Qvi aaojqmoD T ^b/tU^iXaVI xtnoesM lo rftTjft6
*
r :J 3 !
-.1 Lssctoi SzA ^ aal t oHa 5pit9’|3 ? fifti b I lufi ^ ’'bsbJ ^aidvenstti
TGI YTCunst ,^D.0 ‘^no:t^aida£W\abTAbmltP^ lo ubstiMt ,
J t =1.]
PliM n'
^ 'W vT .‘
.
1 / ;‘r '^- >.
t j I '.
'*«'^
ixr - .-. .AV , V’'t
1
if
I
fr '
,V-. (';
^ .>T :v «' -v
'
^
r . :/*r*nn »
'S ': vm
i|bW A'N'’"'
v.^.
1-^ t -H
Figure 1. Geometry and Dimensions of Member
Figure 2. Loading Conditions
*r-
V
V "
« *,^
' •*
; 'i^yl
•'«fir
tif
• 'm.'‘‘4 ,'t' /'
*
'• ^P'lCi*
fi feS
^ '' r*' ' ‘HVv
\ rk. ‘iV,-». j'‘ |i .
,|i
,:M;
/. -S
P '.'.' '.«• ^'- %.* ny«jejQKii.3 t ^ !!\'t e^.- ' ^
..
' fi-W
Figure 3 Equilibrium at a Cross Section
p
Stress Distribution
Cracked Zone
Figure 4. Equilibrium of Member
x=o x=h/2
Figure 5. Deflection Curve
Compression face
Boundary of uncracked zone
Figure 6, Curvature
^-*n
9f1t to 9V1UD noitoaltsGrs^
soot nol829iqmoo )
^!3
,r jr ‘ft!
i.vir
S\r1 = x
V*
"V
0=X
9vxoD Rol:r^9Xi9Q
>
.i fi
.JL ' ,5 ’1‘
eoot noieaeiqmoD
t - 'h >-j>-^
r )1 LJ /1 /H /1/*-
-it
lb
L 8rtbttb3)laDionu lo ^obriLtc8
7.'
r ;i^rV?vt 'V','
iX' Jii .-ltl
aTiotfviyO ,,..i
P.'W
p
U|
Figure 7. Graphical Presentation of Solution
9
Figure 8. Theoretical Strength of the TestedWalls as a Function of the Slenderness Ratio
w'
’
V.’
'* ‘^*'
- Mr «» ,(',
r" !•••'' 4 .*^
'}'^y f ^ ' i' ' -V*rt'4*
,5j,
/ *.;* • '
!
’•
I * • •»’ 1 • m>
4’; V:' .V
\a !.
?t:. .:'
f': ''<'.
4 8\t=0' Hi
3flu^iA^‘ noieeaj^qMoo
.I'",,.. < v.U i? \
,?. -'
. £_ ao-* V
/
3P»JJ|A^ YTI.ilaAT8-^/
ao:
5\f=a >0.^1
MOI2«3fiaMOO3AUJ1AT
‘VYVI^ 06 fj® QS
(S' ,• ,•
t\d
'^'*(. JSt
iir
1. 16,,
.v»9ih«
.{jiinuU Si* siirf^
Figure 9, Comparison of the Theory withTest Results
>>
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41
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