Ecuaciones Generales de Deflexiones en Vigas...

ANALYSIS, DESIGN AND CONSTRUCTION OF ARCH BRIDGES

© 2002, 2003 OSCAR MUROY 0

CONTENTS

1. EFFECTS OF CURVATURE ON THE STRESS-DEFORMATION EQUATIONS IN THE

CROSS SECTION OF A CURVED BEAM

2. GENERAL EQUATIONS OF DEFLECTIONS IN A CURVED BEAM

3. DIRECTRIX OF THE ARCH

4. DEPTH VARIATION OF THE ARCH

5. TYPICAL CONFIGURATIONS OF ARCH BRIDGES

6. CONSTRUCTION PROCEDURES OF ARCH BRIDGES

7. SECOND ORDER DEFLECTIONS IN ARCHES

8. COMPENSATING TECHNIQUES ON ARCHES

9. ARCH STABILITY

10. TEMPERATURE VARIATION, CONCRETE SHRINKAGE AND SUPPORT

DISPLACEMENT EFFECTS

11. GEOMETRY CONTROL OF THE ARCH DURING CONSTRUCTION

12. BIBLIOGRAPHY

Conference Themes at the Instituto de la Construcción y Gerencia:

“Análisis y Diseño de Puentes en Arco”, June 2,002

“Concepción Estructural de Puentes en Arco”, August 2,003

Published in the Journal of the ICG, N° PT-09 y PT-21, respectively


© 2002, 2003 OSCAR MUROY 1

1. EFFECTS OF CURVATURE ON THE STRESS-DEFORMATION EQUATIONS IN

THE CROSS SECTION OF A CURVED BEAM

In a straight beam as well as in a curved axis beam, it has been proved, by model testing and

verified by a rigorous analysis using the Theory of Elasticity, that cross sections initially plane,

remain plane after being subjected to axial forces and moments.

Let a segment of a curved beam with AB axis,

with a radius of curvature R and a subtended

angle d (Fig. N° 1.1)

The total deformation could be divided in an

axial deformation ds and an angular

deformation d.

Then, in the fibre a distance y from the axis,

total deformation will be:

dyds

And the original length is dyR

Hence stress will be:

dyR

dydsEE

(1.1)

Fig 1.1

a) Section subjected to an axial force N:

In this case we will have the equations

0 MdAyA (1.2)

NdAA

(1.3)

The Moment of Inertia of the cross section is defined as:

AdAyI

2,

And doing:

dA

R

y

yI

A

1

'2

,

Being the value y/R, normally very small, developing in series of (y/R) and neglecting

larger terms to 3th order, we get the following expression:

2

1

1

R

IAdA

R

yA


© 2002, 2003 OSCAR MUROY 2

Using these expressions and the Eq. (1.1) in developing the Eq. (1.2), we obtain the

relation of axial and angular deformation:

R

ds

I

Id

' (1.4)

Replacing this Eq. (1.4) in the Eq. (1.3), and taking into account that:

dRds , segment of the curved beam axis

Axial deformation will be: dsEA

Nds

(1.5)

And replacing Eq. (1.5) in the Eq. (1.4), we obtain the angular deformation:

dsEAR

N

I

Id

'

(1.6)

Replacing Eq. (1.5) and (1.6) in the Eq. (1.1), we obtain the axial stress:

A

N

b) Cross section subjected to a bending moment M:

In this case we have the equations:

0 NdAA (1.7)

MdAyA

(1.8)

From Eq. (1.7), with the same assumptions as before, we obtain the relation between

angular deformation and axial deformation:

R

ds

AR

I

I

ARd

2

2

1 (1.9)

Replacing Eq. (1.9) in the Eq. (1.8), we obtain the axial deformation:

dsEAR

M

I

Ids

'

(1.10)

Replacing this value, in the Eq. (1.9) we obtain the angular deformation:

dsAR

I

EI

Md

21

'

(1.11)

Lastly, replacing these two values in Eq. (1.1), we obtain the bending stress, in the cross

section:

R

yI

My

I

I

AR

M

I

I

R

y

y

I

M

AR

M

I

I

1

1

''1

'' ,

Hence, in the curved element, we will have axial stresses in the neutral axis of the cross

section, due to a bending moment M:


© 2002, 2003 OSCAR MUROY 3

AR

M

I

I

'

c) Section subjected to shearing force T:

In this case we will get a shearing strain equal to :

ds

GA

Tds

AG

Tds

GA

Tdy

1/

Making /1 AA , equivalent area to shear force,

being k, a factor depending on the sectional shape.

Fig 1.2

d) Summary:

Then for a section subjected to M, N and T, we will get the following deformations:

Angular deformations:

dsEAR

N

I

I

AR

I

EI

M

I

Ids

EAR

N

I

Ids

AR

I

EI

Md

'1

''1

' 22

(1.12)

Axial deformations:

dsEAR

M

I

I

EA

Nds

' (1.13)

Shearing deformations:

(1.14)

In Eq. (1.12) and (1.13), second terms EAR

N

I

I

' and

EAR

M

I

I

', are the effects of curvature in

the deformations of a curved beam

Next we have a table with the values for areas, inertia and the parameters I / I’, I / I’ ( 1+ I /

AR2

), I / I’ ( I / AR2

) y A / A1:

Section A I I’/I

21

' AR

I

I

I 2

' AR

I

I

I A/A1

Rectangular

bxh bh 3

12

1bh

2

2031

Rh

2

1511

Rh 2

121

Rh 3/2

Box

bxh - b1xh1 11hbbh 3

11

3

12

1hbbh

3

11

3

5

11

5

220

31hbbh

hbbh

R *

11

3

11

3

212

1

hbbh

hbbh

R

Circular

h

2

4

1h

4

64

1h

2

811

Rh

2

1611

Rh 2

161

Rh 4/3

dsGA

Tyd

1


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Tubular

h y h1

2

1

2

4

1hh 4

1

4

64

1hh

2

1

2

4

1

2

1

24

28

11hh

hhhh

R

2

1

2

4

1

4

216

11hh

hh

R 2

1

2

216

1 hhR

*

11

3

11

3

23

11

3

5

11

5

212

120

31hbbh

hbbh

Rhbbh

hbbh

R

Table 1.1 Typical Cross Section Parameters

In order to have an idea of the values (h/R), we refer to tables N° 1.2 and N° 1.3 and we can

see that in concrete Arch Bridges, it is between 1/30 to 1/70 and 1/50 to >1/100 in Steel Arch

Bridges. Then (h/R)2

<1/900~1/10,000 and therefore very small comparing to1.

N° Bridge

Name Type

Span

l(m)

Rise

f(m) l/f Directrix

hs

(m) hs/l

hc

(m) hc/l

Cross

Section

1 Nant Ffrwd built-in 64.9 7.9 8.2 parab. 0.89 1/72.9 0.61 1/106.4 rectang.

2 Kimitsu built-in 66 14.9 4.4 1.8 1/36.7 1.2 1/55 Hollow

3 Mannen Two-hinged 79 10.6 7.5 2.1 1/37.6 2.1 1/37.6 rectang.

4 Omokage built-in 85.0 17.0 5 cos hip 2.1 1/40.5 1.5 1/56.7 Box

5 Nant Hir built-in 85.5 11.3 7.6 parab. 0.91 1/93.9 0.61 1/140.2 rectang.

6 Araya built-in 88 15.0 5.9 2.4 1/36.7 1.5 1/58.7 Box

7 Yoshimi built-in 90.0 18.0 5 cos hip 2.7 1/33.3 1.7 1/52.9 Box

8 Miyakawa built-in 92 17.0 5.4 cos hip 2.0 1/46 1.5 1/61.3 rect

hollow

9 Taf Fechan built-in 119.5 11.4 10.5 parab. 1.1 1/108.6 0.76 1/157.2 rectang.

10 Yumeno built-in 124.0 18.0 6.9 2.5 1/49.6 1.8 1/68.9

11 Taishaku built-in 145.0 30.0 4.8 cos hip 3.8 1/38.2 2.4 1/60.4 Box

12 Hokawazu Two-hinged 170.0 26.5 6.4 parab 4th

o 3.0 1/56.7 2.4 1/70.8 Box

13 Beppu

Myoban

built-in 235.0 35.5 6.6 4.5 1/52.2 3.5 1/67.1 Box

14 Río Paraná built-in 290.0 53.0 5.5 4.8 1/60.4 3.20 1/90.6 Box

15 Gladesville built-in 304.8 40.8 7.5 6.9 1/44.2 4.3 1/70.9 Box

Table 1.2 Concrete Arch Bridges

N° Bridge

Name Type

Span

l(m)

Rise

f(m) l/f Directrix

hs

(m) hs/l

hc

(m) hc/l

Cross

Section

1 South Street Two-hinged 58.8 8.8 6.7 circular 1.00 1/58.8 1.00 1/58.8 box

2 Northfolk Two-hinged 84.1 15.5 5.4 parab. 0.61 1/138.0 0.61 1/138.0 box


© 2002, 2003 OSCAR MUROY 5

3 New

Scotswood

Two-hinged 100.0 19.5 5.1 0.76 1/131.6 0.76 1/131.6 box

4 Leavenworth Two-hinged 128.0 24.4 5.2 0.85 1/150.6 0.85 1/150.6 box

5 Smith Av. Two-hinged 158.5 33.3 4.8 2.44 1/65.0 2.44 1/65.0 box

6 Río

Colorado

built-in 167.6 27.4 6.1 2.13 1/78.7 2.13 1/78.7 box

7 Cold Spring Two-hinged 213.4 36.3 5.9 2.74 1/77.9 2.74 1/77.9 box

8 Glenfield Two-hinged 228.6 37.9 6.0 1.22 1/187.4 1.22 1/187.4 box

9 Fort Pitt Two-hinged 228.6 37.2 6.1 1.64 1/139.4 1.64 1/139.4 box

10 Lewiston built-in 304.8 48.5 6.3 parab 4th

o 4.13 1/73.8 4.13 1/73.8 box

11 Roosevelt built-in 329.2 70.1 4.7 parab. 6.1 1/54 2.44 1/135.2 box

12 Vltava

Valley

Two-hinged 330.0 42.5 7.8 5.0 1/66 5.0 1/66 box

13 Fremont Two-hinged 382.5 103.9 3.7 1.22 1/313.7 1.22 1/313.7 box

Table 1.3 Steel Arch Bridges


© 2002, 2003 OSCAR MUROY 6

2. GENERAL EQUATIONS OF DEFLECTIONS IN CURVED BEAMS

Fig 2.1

The Navier-Bresse Equations for the displacements in curved beams are given by:

Angular Displacement:

dsEI

Mww

s

s

0

0 (2.1)

Horizontal Displacement:

dsGA

Tds

EA

Ndsy

EI

Myywuu

s

s

s

s

s

s

sincos

0001

000 (2.2)

Vertical Displacement:

dsGA

Tds

EA

Ndsx

EI

Mxxwvv

s

s

s

s

s

s

cossin

0001

000 (2.3)

where:

dsEI

Md , is the angular deformation in a segment ds of the arch

dsEA

Nds , axial deformation

dsGA

Tdy

1

, shear deformation

In these equations it is not considered the effects of curvature of the arch

Then, to take into account these effects, we should substitute these values, with the values:

dsEAR

N

I

I

AR

I

I

I

EI

Md

'1

' 2 (2.4)

dsEAR

M

I

I

EA

Nds

' (2.5)


© 2002, 2003 OSCAR MUROY 7

dsGA

Tdy

1

(2.6)

The axis curvature, is obtained from the equation:

3

2/322/32

cos"1

"

'1

"1y

tg

y

y

y

R

(2.7)

Developing first the Navier-Bresse equations, we have:

dsEI

Mww

s

s

0

0 (2.8)

dsGA

Tds

EA

Nds

EI

Mywywuu

s

s

s

s

s

s

sincos

0001

000 (2.9)

dsGA

Tds

EA

Nds

EI

Mxwxwvv

s

s

s

s

s

s

cossin

0001

000 (2.10)

Replacing Eq. (2.4) in Eq. (2.8):

dsEAR

N

I

I

AR

I

I

I

EI

Mww

s

s

0'

1'

20

This Eq. could be written as follows:

dsM

NR

AR

I

I

I

AR

I

I

I

EI

Mww

s

s

0

220'

1'

(2.11)

Replacing Eq. (2.4) and (2.5) in Eq. (2.9):

dsEAR

N

I

I

AR

I

I

I

EI

Mywywuu

s

s

0'

1'

2000

s

s

s

s

dsGA

Tds

EAR

M

I

I

EA

N

00

sincos' 1


dsR

AR

I

I

I

AR

I

I

I

EI

Mywywuu

s

s

0

cos

'1

'22000

s

s

s

s

dsGA

Tds

RI

I

EA

N

00

sincos'

1cos1

(2.12)

Replacing Eq. (2.4) and (2.5) in Eq. (2.10):


© 2002, 2003 OSCAR MUROY 8

dsEAR

N

I

I

AR

I

I

I

EI

Mxwxwvv

s

s

0'

1'

2000

s

s

s

s

dsGA

Tds

EAR

M

I

I

EA

N

00

cossin' 1


dsR

AR

I

I

I

AR

I

I

I

EI

Mxwxwvv

s

s

0

sin

'1

'22000

s

s

s

s

dsGA

Tds

RI

I

EA

N

00

cossin'

1sin1

(2.13)

Doing:

21

' AR

I

I

I

2' AR

I

I

I

cosR

sinR

Lastly, we have the equations:

dsM

NR

EI

Mww

s

s

0

0 (2.14)

dsEI

Mywywuu

s

s

0

000

s

s

s

s

dsGA

Tds

I

I

EA

N

00

sin'

1cos1

(2.15)

dsEI

Mxwxwvv

s

s

0

000

s

s

s

s

dsGA

Tds

I

I

EA

N

00

cos'

1sin1

(2.16)


© 2002, 2003 OSCAR MUROY 9

It has been examined the values of parameters , , and , for two typical cases: built-in

arch of 65 m span and two-hinged arch of 92 m span, whence the following conclusions could

be drawn for these parameters

00.1

00.0

00.1

00.1

00.2)´

(1 I

I

20.1)´

(1 I

I

Lastly the Navier equations could be written, taking into account the curvature effects, in the

following way:

dsEI

Mww

s

s

0

0 (2.17)

s

s

s

s

s

s

dsGA

Tds

EA

Nds

EI

Mywywuu

000

sincos20.11

000 (2.18)

s

s

s

s

s

s

dsGA

Tds

EA

Nds

EI

Mxwxwvv

000

cossin0.21

000 (2.19)

That is, the effect of curvature could be incorporated in the normal Navier equations,

modifying the value of the section area for the axial force, with a reduction factor of 0,8 and

0,5 in the equations of horizontal and vertical deflections, respectively.

In the practice, this could be done, calculating the results for each of these values of reduced

area to axial force, with the final result lying in between these two values


© 2002, 2003 OSCAR MUROY 10

Arch, parabolic axis, two-hinged, parabolic variation of h

x y h cos 0 sin 0 y" R I/I´

-46.000 0.000 1.500 0.8042 0.5944 -0.0161 -119.6694 1.0000 1.0000 0.0000

-41.400 3.230 1.595 0.8326 0.5539 -0.0161 -107.8244 1.0000 1.0000 0.0000

-36.800 6.120 1.680 0.8608 0.5090 -0.0161 -97.5807 1.0000 1.0000 0.0000

-32.200 8.670 1.755 0.8882 0.4595 -0.0161 -88.8297 0.9999 1.0000 0.0000

-27.600 10.880 1.820 0.9141 0.4054 -0.0161 -81.4704 0.9999 1.0000 0.0000

-23.000 12.750 1.875 0.9380 0.3467 -0.0161 -75.4112 0.9999 1.0000 0.0001

-18.400 14.280 1.920 0.9590 0.2835 -0.0161 -70.5711 0.9999 1.0000 0.0001

-13.800 15.470 1.955 0.9763 0.2165 -0.0161 -66.8813 0.9999 0.9999 0.0001

-9.200 16.320 1.980 0.9892 0.1462 -0.0161 -64.2864 0.9999 0.9999 0.0001

-4.600 16.830 1.995 0.9973 0.0737 -0.0161 -62.7460 0.9998 0.9999 0.0001

0.000 17.000 2.000 1.0000 0.0000 -0.0161 -62.2353 0.9998 0.9999 0.0001

4.600 16.830 1.995 0.9973 -0.0737 -0.0161 -62.7460 0.9998 0.9999 0.0001

9.200 16.320 1.980 0.9892 -0.1462 -0.0161 -64.2864 0.9999 0.9999 0.0001

13.800 15.470 1.955 0.9763 -0.2165 -0.0161 -66.8813 0.9999 0.9999 0.0001

18.400 14.280 1.920 0.9590 -0.2835 -0.0161 -70.5711 0.9999 1.0000 0.0001

23.000 12.750 1.875 0.9380 -0.3467 -0.0161 -75.4112 0.9999 1.0000 0.0001

27.600 10.880 1.820 0.9141 -0.4054 -0.0161 -81.4704 0.9999 1.0000 0.0000

32.200 8.670 1.755 0.8882 -0.4595 -0.0161 -88.8297 0.9999 1.0000 0.0000

36.800 6.120 1.680 0.8608 -0.5090 -0.0161 -97.5807 1.0000 1.0000 0.0000

41.400 3.230 1.595 0.8326 -0.5539 -0.0161 -107.8244 1.0000 1.0000 0.0000

46.000 0.000 1.500 0.8042 -0.5944 -0.0161 -119.6694 1.0000 1.0000 0.0000

Span=92.00m, rise=17.00m, hc=2.00m, ha=1.50m

+/ (+/) -/ (-/) 1+(I/I´)

1-

(I/I´)

-46.000 0.000 -0.0104 0.6467 1.0000 -46.0004 1.0012 0.0000 1.6467 1.0104

-41.400 3.230 -0.0360 0.6932 1.0000 -41.4005 1.0005 3.2316 1.6932 1.0360

-36.800 6.120 -0.0729 0.7409 1.0000 -36.8005 1.0003 6.1220 1.7409 1.0729

-32.200 8.670 -0.1099 0.7888 1.0000 -32.2005 1.0003 8.6723 1.7888 1.1099

-27.600 10.880 -0.1461 0.8357 1.0000 -27.6005 1.0003 10.8827 1.8356 1.1461

-23.000 12.750 -0.1802 0.8798 1.0000 -23.0004 1.0002 12.7531 1.8798 1.1802

-18.400 14.280 -0.2110 0.9196 1.0000 -18.4003 1.0002 14.2835 1.9195 1.2110

-13.800 15.470 -0.2369 0.9531 1.0000 -13.8002 1.0002 15.4738 1.9530 1.2369

-9.200 16.320 -0.2566 0.9786 1.0000 -9.2002 1.0002 16.3240 1.9785 1.2566

-4.600 16.830 -0.2690 0.9946 1.0000 -4.6001 1.0002 16.8341 1.9944 1.2689

0.000 17.000 -0.2732 1.0000 1.0000 0.0000 1.0002 17.0042 1.9998 1.2731

4.600 16.830 -0.2690 0.9946 1.0000 4.6001 1.0002 16.8341 1.9944 1.2689

9.200 16.320 -0.2566 0.9786 1.0000 9.2002 1.0002 16.3240 1.9785 1.2566

13.800 15.470 -0.2369 0.9531 1.0000 13.8002 1.0002 15.4738 1.9530 1.2369

18.400 14.280 -0.2110 0.9196 1.0000 18.4003 1.0002 14.2835 1.9195 1.2110

23.000 12.750 -0.1802 0.8798 1.0000 23.0004 1.0002 12.7531 1.8798 1.1802

27.600 10.880 -0.1461 0.8357 1.0000 27.6005 1.0003 10.8827 1.8356 1.1461

32.200 8.670 -0.1099 0.7888 1.0000 32.2005 1.0003 8.6723 1.7888 1.1099

36.800 6.120 -0.0729 0.7409 1.0000 36.8005 1.0003 6.1220 1.7409 1.0729

41.400 3.230 -0.0360 0.6932 1.0000 41.4005 1.0005 3.2316 1.6932 1.0360

46.000 0.000 -0.0104 0.6467 1.0000 46.0004 1.0012 0.0000 1.6467 1.0104

38.8605

24.3307

1.8505 1.1586


© 2002, 2003 OSCAR MUROY 11

Arch, parabolic axis 4th order, two-hinged, parabolic variation of h


-46.000 0.000 1.500 0.8140 0.5809 -0.0133 -139.4380 1.0000 1.0000 0.0000

-41.400 3.140 1.595 0.8381 0.5456 -0.0139 -121.9665 1.0000 1.0000 0.0000

-36.800 5.985 1.680 0.8629 0.5053 -0.0145 -107.3676 1.0000 1.0000 0.0000

-32.200 8.524 1.755 0.8880 0.4598 -0.0150 -95.2398 0.9999 1.0000 0.0000

-27.600 10.745 1.820 0.9127 0.4087 -0.0154 -85.2713 0.9999 1.0000 0.0000

-23.000 12.640 1.875 0.9360 0.3519 -0.0158 -77.2158 0.9999 1.0000 0.0000

-18.400 14.201 1.920 0.9571 0.2896 -0.0161 -70.8774 0.9999 1.0000 0.0001

-13.800 15.422 1.955 0.9750 0.2223 -0.0163 -66.1026 0.9999 0.9999 0.0001

-9.200 16.297 1.980 0.9886 0.1508 -0.0165 -62.7742 0.9999 0.9999 0.0001

-4.600 16.824 1.995 0.9971 0.0762 -0.0166 -60.8101 0.9998 0.9999 0.0001

0.000 17.000 2.000 1.0000 0.0000 -0.0166 -60.1608 0.9998 0.9999 0.0001

4.600 16.824 1.995 0.9971 -0.0762 -0.0166 -60.8101 0.9998 0.9999 0.0001

9.200 16.297 1.980 0.9886 -0.1508 -0.0165 -62.7742 0.9999 0.9999 0.0001

13.800 15.422 1.955 0.9750 -0.2223 -0.0163 -66.1026 0.9999 0.9999 0.0001

18.400 14.201 1.920 0.9571 -0.2896 -0.0161 -70.8774 0.9999 1.0000 0.0001

23.000 12.640 1.875 0.9360 -0.3519 -0.0158 -77.2158 0.9999 1.0000 0.0000

27.600 10.745 1.820 0.9127 -0.4087 -0.0154 -85.2713 0.9999 1.0000 0.0000

32.200 8.524 1.755 0.8880 -0.4598 -0.0150 -95.2398 0.9999 1.0000 0.0000

36.800 5.985 1.680 0.8629 -0.5053 -0.0145 -107.3676 1.0000 1.0000 0.0000

41.400 3.140 1.595 0.8381 -0.5456 -0.0139 -121.9665 1.0000 1.0000 0.0000

46.000 0.000 1.500 0.8140 -0.5809 -0.0133 -139.4380 1.0000 1.0000 0.0000


+/ (+/) -/ (-/) 1+(I/I´)

1-

(I/I´)

-46.000 0.000 -0.0088 0.5679 1.0000 -46.0004 1.0011 0.0000 1.5679 1.0088

-41.400 3.140 -0.0307 0.6222 1.0000 -41.4005 1.0005 3.1412 1.6221 1.0307

-36.800 5.985 -0.0646 0.6783 1.0000 -36.8005 1.0003 5.9867 1.6782 1.0646

-32.200 8.524 -0.1008 0.7353 1.0000 -32.2005 1.0003 8.5257 1.7353 1.1008

-27.600 10.745 -0.1381 0.7920 1.0000 -27.6005 1.0002 10.7476 1.7920 1.1381

-23.000 12.640 -0.1749 0.8465 1.0000 -23.0004 1.0002 12.6431 1.8464 1.1749

-18.400 14.201 -0.2093 0.8964 1.0000 -18.4004 1.0002 14.2047 1.8963 1.2093

-13.800 15.422 -0.2393 0.9391 1.0000 -13.8003 1.0002 15.4258 1.9390 1.2393

-9.200 16.297 -0.2626 0.9720 1.0000 -9.2002 1.0002 16.3016 1.9719 1.2626

-4.600 16.824 -0.2775 0.9929 1.0000 -4.6001 1.0003 16.8284 1.9927 1.2774

0.000 17.000 -0.2826 1.0000 1.0000 0.0000 1.0003 17.0043 1.9998 1.2825

4.600 16.824 -0.2775 0.9929 1.0000 4.6001 1.0003 16.8284 1.9927 1.2774

9.200 16.297 -0.2626 0.9720 1.0000 9.2002 1.0002 16.3016 1.9719 1.2626

13.800 15.422 -0.2393 0.9391 1.0000 13.8003 1.0002 15.4258 1.9390 1.2393

18.400 14.201 -0.2093 0.8964 1.0000 18.4004 1.0002 14.2047 1.8963 1.2093

23.000 12.640 -0.1749 0.8465 1.0000 23.0004 1.0002 12.6431 1.8464 1.1749

27.600 10.745 -0.1381 0.7920 1.0000 27.6005 1.0002 10.7476 1.7920 1.1381

32.200 8.524 -0.1008 0.7353 1.0000 32.2005 1.0003 8.5257 1.7353 1.1008

36.800 5.985 -0.0646 0.6783 1.0000 36.8005 1.0003 5.9867 1.6782 1.0646

41.400 3.140 -0.0307 0.6222 1.0000 41.4005 1.0005 3.1412 1.6221 1.0307

46.000 0.000 -0.0088 0.5679 1.0000 46.0004 1.0011 0.0000 1.5679 1.0088

38.0835

24.2953

1.8135 1.1569


© 2002, 2003 OSCAR MUROY 12

Arch, cosine axis, two-hinged, parabolic variation of h m=qa/qc=0.7


-46.000 0.000 1.500 0.8141 0.5808 -0.0133 -139.2979 1.0000 1.0000 0.0000

-41.400 3.139 1.595 0.8381 0.5455 -0.0139 -122.0012 1.0000 1.0000 0.0000

-36.800 5.983 1.680 0.8630 0.5052 -0.0145 -107.4684 1.0000 1.0000 0.0000

-32.200 8.521 1.755 0.8881 0.4597 -0.0150 -95.3488 0.9999 1.0000 0.0000

-27.600 10.743 1.820 0.9127 0.4087 -0.0154 -85.3601 0.9999 1.0000 0.0000

-23.000 12.638 1.875 0.9360 0.3520 -0.0158 -77.2731 0.9999 1.0000 0.0000

-18.400 14.200 1.920 0.9571 0.2897 -0.0161 -70.9020 0.9999 1.0000 0.0001

-13.800 15.421 1.955 0.9750 0.2224 -0.0163 -66.0985 0.9999 0.9999 0.0001

-9.200 16.297 1.980 0.9886 0.1509 -0.0165 -62.7485 0.9999 0.9999 0.0001

-4.600 16.824 1.995 0.9971 0.0762 -0.0166 -60.7709 0.9998 0.9999 0.0001

0.000 17.000 2.000 1.0000 0.0000 -0.0166 -60.1171 0.9998 0.9999 0.0001

4.600 16.824 1.995 0.9971 -0.0762 -0.0166 -60.7709 0.9998 0.9999 0.0001

9.200 16.297 1.980 0.9886 -0.1509 -0.0165 -62.7485 0.9999 0.9999 0.0001

13.800 15.421 1.955 0.9750 -0.2224 -0.0163 -66.0985 0.9999 0.9999 0.0001

18.400 14.200 1.920 0.9571 -0.2897 -0.0161 -70.9020 0.9999 1.0000 0.0001

23.000 12.638 1.875 0.9360 -0.3520 -0.0158 -77.2731 0.9999 1.0000 0.0000

27.600 10.743 1.820 0.9127 -0.4087 -0.0154 -85.3601 0.9999 1.0000 0.0000

32.200 8.521 1.755 0.8881 -0.4597 -0.0150 -95.3488 0.9999 1.0000 0.0000

36.800 5.983 1.680 0.8630 -0.5052 -0.0145 -107.4684 1.0000 1.0000 0.0000

41.400 3.139 1.595 0.8381 -0.5455 -0.0139 -122.0012 1.0000 1.0000 0.0000

46.000 0.000 1.500 0.8141 -0.5808 -0.0133 -139.2979 1.0000 1.0000 0.0000


+/ (+/) -/ (-/)

1+(I/I´)

1-

(I/I´)

-46.000 0.000 -0.0088 0.5686 1.0000 -46.0004 1.0011 0.0000 1.5686 1.0088

-41.400 3.139 -0.0307 0.6221 1.0000 -41.4005 1.0005 3.1403 1.6221 1.0307

-36.800 5.983 -0.0645 0.6778 1.0000 -36.8005 1.0003 5.9850 1.6777 1.0645

-32.200 8.521 -0.1006 0.7346 1.0000 -32.2005 1.0003 8.5235 1.7345 1.1006

-27.600 10.743 -0.1379 0.7912 1.0000 -27.6005 1.0002 10.7453 1.7911 1.1379

-23.000 12.638 -0.1747 0.8457 1.0000 -23.0004 1.0002 12.6411 1.8456 1.1747

-18.400 14.200 -0.2092 0.8957 1.0000 -18.4004 1.0002 14.2031 1.8956 1.2092

-13.800 15.421 -0.2393 0.9387 1.0000 -13.8003 1.0002 15.4248 1.9386 1.2393

-9.200 16.297 -0.2627 0.9718 1.0000 -9.2002 1.0002 16.3011 1.9717 1.2627

-4.600 16.824 -0.2777 0.9928 1.0000 -4.6001 1.0003 16.8283 1.9927 1.2776

0.000 17.000 -0.2828 1.0000 1.0000 0.0000 1.0003 17.0043 1.9998 1.2827

4.600 16.824 -0.2777 0.9928 1.0000 4.6001 1.0003 16.8283 1.9927 1.2776

9.200 16.297 -0.2627 0.9718 1.0000 9.2002 1.0002 16.3011 1.9717 1.2627

13.800 15.421 -0.2393 0.9387 1.0000 13.8003 1.0002 15.4248 1.9386 1.2393

18.400 14.200 -0.2092 0.8957 1.0000 18.4004 1.0002 14.2031 1.8956 1.2092

23.000 12.638 -0.1747 0.8457 1.0000 23.0004 1.0002 12.6411 1.8456 1.1747

27.600 10.743 -0.1379 0.7912 1.0000 27.6005 1.0002 10.7453 1.7911 1.1379

32.200 8.521 -0.1006 0.7346 1.0000 32.2005 1.0003 8.5235 1.7345 1.1006

36.800 5.983 -0.0645 0.6778 1.0000 36.8005 1.0003 5.9850 1.6777 1.0645

41.400 3.139 -0.0307 0.6221 1.0000 41.4005 1.0005 3.1403 1.6221 1.0307

46.000 0.000 -0.0088 0.5686 1.0000 46.0004 1.0011 0.0000 1.5686 1.0088

38.0763 24.294

1.8132 1.1569


© 2002, 2003 OSCAR MUROY 13

Arch, circular axis, two-hinged, parabolic variation of h


-46.000 0.000 1.500 0.7597 0.6503 -0.0322 -70.7353 0.9999 1.0000 0.0000

-41.400 3.619 1.595 0.8108 0.5853 -0.0265 -70.7353 0.9999 1.0000 0.0000

-36.800 6.674 1.680 0.8540 0.5202 -0.0227 -70.7353 0.9999 1.0000 0.0000

-32.200 9.246 1.755 0.8904 0.4552 -0.0200 -70.7353 0.9999 1.0000 0.0001

-27.600 11.393 1.820 0.9207 0.3902 -0.0181 -70.7353 0.9999 1.0000 0.0001

-23.000 13.156 1.875 0.9457 0.3252 -0.0167 -70.7353 0.9999 1.0000 0.0001

-18.400 14.565 1.920 0.9656 0.2601 -0.0157 -70.7353 0.9999 1.0000 0.0001

-13.800 15.641 1.955 0.9808 0.1951 -0.0150 -70.7353 0.9999 0.9999 0.0001

-9.200 16.399 1.980 0.9915 0.1301 -0.0145 -70.7353 0.9999 0.9999 0.0001

-4.600 16.850 1.995 0.9979 0.0650 -0.0142 -70.7353 0.9999 0.9999 0.0001

0.000 17.000 2.000 1.0000 0.0000 -0.0141 -70.7353 0.9999 0.9999 0.0001

4.600 16.850 1.995 0.9979 -0.0650 -0.0142 -70.7353 0.9999 0.9999 0.0001

9.200 16.399 1.980 0.9915 -0.1301 -0.0145 -70.7353 0.9999 0.9999 0.0001

13.800 15.641 1.955 0.9808 -0.1951 -0.0150 -70.7353 0.9999 0.9999 0.0001

18.400 14.565 1.920 0.9656 -0.2601 -0.0157 -70.7353 0.9999 1.0000 0.0001

23.000 13.156 1.875 0.9457 -0.3252 -0.0167 -70.7353 0.9999 1.0000 0.0001

27.600 11.393 1.820 0.9207 -0.3902 -0.0181 -70.7353 0.9999 1.0000 0.0001

32.200 9.246 1.755 0.8904 -0.4552 -0.0200 -70.7353 0.9999 1.0000 0.0001

36.800 6.674 1.680 0.8540 -0.5202 -0.0227 -70.7353 0.9999 1.0000 0.0000

41.400 3.619 1.595 0.8108 -0.5853 -0.0265 -70.7353 0.9999 1.0000 0.0000

46.000 0.000 1.500 0.7597 -0.6503 -0.0322 -70.7353 0.9999 1.0000 0.0000


+/ (+/) -/ (-/) 1+(I/I´)

1-

(I/I´)

-46.000 0.000 -0.0186 1.0000 1.0000 -46.0003 1.0020 0.0000 1.9999 1.0186

-41.400 3.619 -0.0631 1.0000 1.0000 -41.4004 1.0006 3.6214 1.9999 1.0631

-36.800 6.674 -0.1105 1.0000 1.0000 -36.8003 1.0004 6.6762 1.9999 1.1105

-32.200 9.246 -0.1468 1.0000 1.0000 -32.2003 1.0003 9.2488 1.9999 1.1468

-27.600 11.393 -0.1749 1.0000 1.0000 -27.6003 1.0003 11.3963 1.9999 1.1749

-23.000 13.156 -0.1967 1.0000 1.0000 -23.0003 1.0003 13.1596 1.9999 1.1967

-18.400 14.565 -0.2132 1.0000 1.0000 -18.4002 1.0002 14.5684 1.9999 1.2132

-13.800 15.641 -0.2254 1.0000 1.0000 -13.8002 1.0002 15.6444 1.9999 1.2254

-9.200 16.399 -0.2338 1.0000 1.0000 -9.2001 1.0002 16.4029 1.9999 1.2338

-4.600 16.850 -0.2387 1.0000 1.0000 -4.6001 1.0002 16.8541 1.9999 1.2387

0.000 17.000 -0.2403 1.0000 1.0000 0.0000 1.0002 17.0038 1.9999 1.2403

4.600 16.850 -0.2387 1.0000 1.0000 4.6001 1.0002 16.8541 1.9999 1.2387

9.200 16.399 -0.2338 1.0000 1.0000 9.2001 1.0002 16.4029 1.9999 1.2338

13.800 15.641 -0.2254 1.0000 1.0000 13.8002 1.0002 15.6444 1.9999 1.2254

18.400 14.565 -0.2132 1.0000 1.0000 18.4002 1.0002 14.5684 1.9999 1.2132

23.000 13.156 -0.1967 1.0000 1.0000 23.0003 1.0003 13.1596 1.9999 1.1967

27.600 11.393 -0.1749 1.0000 1.0000 27.6003 1.0003 11.3963 1.9999 1.1749

32.200 9.246 -0.1468 1.0000 1.0000 32.2003 1.0003 9.2488 1.9999 1.1468

36.800 6.674 -0.1105 1.0000 1.0000 36.8003 1.0004 6.6762 1.9999 1.1105

41.400 3.619 -0.0631 1.0000 1.0000 41.4004 1.0006 3.6214 1.9999 1.0631

46.000 0.000 -0.0186 1.0000 1.0000 46.0003 1.0020 0.0000 1.9999 1.0186

41.9979 24.483

1.9999 1.1659


© 2002, 2003 OSCAR MUROY 14

Arch, parabolic axis, built-in, Strassner variation of h


-32.500 0.000 1.6500 0.7809 0.6247 -0.0246 -85.3216 0.9999 1.0000 0.0000

-29.300 2.434 1.5502 0.8111 0.5850 -0.0246 -76.1436 0.9999 1.0000 0.0000

-25.000 5.308 1.4407 0.8517 0.5241 -0.0246 -65.7654 0.9999 1.0000 0.0000

-20.000 8.077 1.3389 0.8972 0.4417 -0.0246 -56.2559 0.9999 1.0000 0.0000

-15.000 10.231 1.2575 0.9381 0.3464 -0.0246 -49.2097 0.9999 1.0000 0.0001

-10.000 11.769 1.1921 0.9710 0.2390 -0.0246 -44.3727 0.9999 1.0000 0.0001

-5.000 12.692 1.1403 0.9925 0.1222 -0.0246 -41.5516 0.9999 0.9999 0.0001

0.000 13.000 1.1000 1.0000 0.0000 -0.0246 -40.6250 0.9999 1.0000 0.0001

5.000 12.692 1.1403 0.9925 -0.1222 -0.0246 -41.5516 0.9999 0.9999 0.0001

10.000 11.769 1.1921 0.9710 -0.2390 -0.0246 -44.3727 0.9999 1.0000 0.0001

15.000 10.231 1.2575 0.9381 -0.3464 -0.0246 -49.2097 0.9999 1.0000 0.0001

20.000 8.077 1.3389 0.8972 -0.4417 -0.0246 -56.2559 0.9999 1.0000 0.0000

25.000 5.308 1.4407 0.8517 -0.5241 -0.0246 -65.7654 0.9999 1.0000 0.0000

29.300 2.434 1.5502 0.8111 -0.5850 -0.0246 -76.1436 0.9999 1.0000 0.0000

32.500 0.000 1.6500 0.7809 -0.6247 -0.0246 -85.3216 0.9999 1.0000 0.0000

Span=65.00m, rise=13.00m, ha=1.65m, hc=1.10m

+/ (+/) -/ (-/) 1+(I/I´) 1-(I/I´)

-32.500 0.000 -0.0150 0.6098 1.0000 -32.5009 1.0021 0.0000 1.6097 1.0150

-29.300 2.434 -0.0394 0.6578 1.0000 -29.3007 1.0008 2.4360 1.6578 1.0394

-25.000 5.308 -0.0948 0.7253 1.0000 -25.0006 1.0004 5.3098 1.7253 1.0948

-20.000 8.077 -0.1600 0.8049 1.0000 -20.0004 1.0003 8.0790 1.8048 1.1600

-15.000 10.231 -0.2216 0.8800 1.0000 -15.0003 1.0002 10.2328 1.8799 1.2216

-10.000 11.769 -0.2732 0.9429 1.0000 -10.0002 1.0002 11.7713 1.9428 1.2731

-5.000 12.692 -0.3078 0.9851 1.0000 -5.0001 1.0002 12.6943 1.9850 1.3077

0.000 13.000 -0.3200 1.0000 1.0000 0.0000 1.0001 13.0018 1.9999 1.3200

5.000 12.692 -0.3078 0.9851 1.0000 5.0001 1.0002 12.6943 1.9850 1.3077

10.000 11.769 -0.2732 0.9429 1.0000 10.0002 1.0002 11.7713 1.9428 1.2731

15.000 10.231 -0.2216 0.8800 1.0000 15.0003 1.0002 10.2328 1.8799 1.2216

20.000 8.077 -0.1600 0.8049 1.0000 20.0004 1.0003 8.0790 1.8048 1.1600

25.000 5.308 -0.0948 0.7253 1.0000 25.0006 1.0004 5.3098 1.7253 1.0948

29.300 2.434 -0.0394 0.6578 1.0000 29.3007 1.0008 2.4360 1.6578 1.0394

32.500 0.000 -0.0150 0.6098 1.0000 32.5009 1.0021 0.0000 1.6097 1.0150

27.2105 17.5432

1.8140 1.1695


© 2002, 2003 OSCAR MUROY 15

Arch, parabolic axis 4th order, built-in, Strassner variation of h


-32.500 0.0000 1.6500 0.7576 0.6527 -0.0341 -67.4729 0.9999 1.0000 0.0000

-29.300 2.5861 1.5466 0.7977 0.6030 -0.0320 -61.6449 0.9999 1.0000 0.0001

-25.000 5.5493 1.4344 0.8484 0.5294 -0.0294 -55.6247 0.9999 1.0000 0.0001

-20.000 8.3122 1.3321 0.9004 0.4350 -0.0270 -50.6880 0.9999 1.0000 0.0001

-15.000 10.3984 1.2521 0.9430 0.3328 -0.0251 -47.4323 0.9999 1.0000 0.0001

-10.000 11.8549 1.1890 0.9744 0.2249 -0.0238 -45.4233 0.9999 1.0000 0.0001

-5.000 12.7154 1.1391 0.9936 0.1133 -0.0230 -44.3472 0.9999 1.0000 0.0001

0.000 13.0000 1.1000 1.0000 0.0000 -0.0227 -44.0104 0.9999 1.0000 0.0001

5.000 12.7154 1.1391 0.9936 -0.1133 -0.0230 -44.3472 0.9999 1.0000 0.0001

10.000 11.8549 1.1890 0.9744 -0.2249 -0.0238 -45.4233 0.9999 1.0000 0.0001

15.000 10.3984 1.2521 0.9430 -0.3328 -0.0251 -47.4323 0.9999 1.0000 0.0001

20.000 8.3122 1.3321 0.9004 -0.4350 -0.0270 -50.6880 0.9999 1.0000 0.0001

25.000 5.5493 1.4344 0.8484 -0.5294 -0.0294 -55.6247 0.9999 1.0000 0.0001

29.300 2.5861 1.5466 0.7977 -0.6030 -0.0320 -61.6449 0.9999 1.0000 0.0001

32.500 0.0000 1.6500 0.7576 -0.6527 -0.0341 -67.4729 0.9999 1.0000 0.0000


+/ (+/) -/ (-/) 1+(I/I´) 1-(I/I´)

-32.500 0.0000 -0.0196 0.7380 1.0000 -32.5009 1.0025 0.0000 1.7379 1.0196

-29.300 2.5861 -0.0526 0.7882 1.0000 -29.3007 1.0010 2.5886 1.7881 1.0526

-25.000 5.5493 -0.1176 0.8489 1.0000 -25.0005 1.0004 5.5516 1.8488 1.1176

-20.000 8.3122 -0.1821 0.9070 1.0000 -20.0003 1.0003 8.3145 1.9069 1.1821

-15.000 10.3984 -0.2325 0.9502 1.0000 -15.0002 1.0002 10.4005 1.9501 1.2325

-10.000 11.8549 -0.2678 0.9789 1.0000 -10.0001 1.0002 11.8569 1.9788 1.2678

-5.000 12.7154 -0.2886 0.9949 1.0000 -5.0001 1.0001 12.7173 1.9948 1.2886

0.000 13.0000 -0.2954 1.0000 1.0000 0.0000 1.0001 13.0017 1.9999 1.2954

5.000 12.7154 -0.2886 0.9949 1.0000 5.0001 1.0001 12.7173 1.9948 1.2886

10.000 11.8549 -0.2678 0.9789 1.0000 10.0001 1.0002 11.8569 1.9788 1.2678

15.000 10.3984 -0.2325 0.9502 1.0000 15.0002 1.0002 10.4005 1.9501 1.2325

20.000 8.3122 -0.1821 0.9070 1.0000 20.0003 1.0003 8.3145 1.9069 1.1821

25.000 5.5493 -0.1176 0.8489 1.0000 25.0005 1.0004 5.5516 1.8488 1.1176

29.300 2.5861 -0.0526 0.7882 1.0000 29.3007 1.0010 2.5886 1.7881 1.0526

32.500 0.0000 -0.0196 0.7380 1.0000 32.5009 1.0025 0.0000 1.7379 1.0196

28.4110 17.6168

1.8941 1.1745


© 2002, 2003 OSCAR MUROY 16

Arch, hyperbolic cosine axis, built-in, Strassner variation of h


-32.500 0.0000 1.6500 0.7579 0.6524 -0.0342 -67.1693 0.9999 1.0000 0.0001

-29.300 2.5835 1.5465 0.7980 0.6026 -0.0319 -61.6094 0.9999 1.0000 0.0001

-25.000 5.5435 1.4344 0.8486 0.5291 -0.0293 -55.7831 0.9999 1.0000 0.0001

-20.000 8.3051 1.3322 0.9004 0.4350 -0.0269 -50.8902 0.9999 1.0000 0.0001

-15.000 10.3925 1.2522 0.9429 0.3332 -0.0251 -47.5560 0.9999 1.0000 0.0001

-10.000 11.8516 1.1891 0.9743 0.2254 -0.0238 -45.4210 0.9999 1.0000 0.0001

-5.000 12.7145 1.1391 0.9935 0.1137 -0.0231 -44.2375 0.9999 1.0000 0.0001

0.000 13.0000 1.1000 1.0000 0.0000 -0.0228 -43.8593 0.9999 1.0000 0.0001

5.000 12.7145 1.1391 0.9935 -0.1137 -0.0231 -44.2375 0.9999 1.0000 0.0001

10.000 11.8516 1.1891 0.9743 -0.2254 -0.0238 -45.4210 0.9999 1.0000 0.0001

15.000 10.3925 1.2522 0.9429 -0.3332 -0.0251 -47.5560 0.9999 1.0000 0.0001

20.000 8.3051 1.3322 0.9004 -0.4350 -0.0269 -50.8902 0.9999 1.0000 0.0001

25.000 5.5435 1.4344 0.8486 -0.5291 -0.0293 -55.7831 0.9999 1.0000 0.0001

29.300 2.5835 1.5465 0.7980 -0.6026 -0.0319 -61.6094 0.9999 1.0000 0.0001

32.500 0.0000 1.6500 0.7579 -0.6524 -0.0342 -67.1693 0.9999 1.0000 0.0001


+/ (+/) -/ (-/) 1+(I/I´) 1-(I/I´)

-32.500 0.0000 -0.0196 0.7417 1.0000 -32.5009 1.0025 0.0000 1.7416 1.0196

-29.300 2.5835 -0.0525 0.7892 1.0000 -29.3007 1.0010 2.5860 1.7891 1.0525

-25.000 5.5435 -0.1171 0.8470 1.0000 -25.0005 1.0004 5.5459 1.8470 1.1171

-20.000 8.3051 -0.1812 0.9034 1.0000 -20.0004 1.0003 8.3073 1.9033 1.1812

-15.000 10.3925 -0.2318 0.9467 1.0000 -15.0002 1.0002 10.3946 1.9466 1.2318

-10.000 11.8516 -0.2678 0.9768 1.0000 -10.0001 1.0002 11.8536 1.9767 1.2678

-5.000 12.7145 -0.2893 0.9943 1.0000 -5.0001 1.0001 12.7163 1.9942 1.2893

0.000 13.0000 -0.2964 1.0000 1.0000 0.0000 1.0001 13.0018 1.9999 1.2964

5.000 12.7145 -0.2893 0.9943 1.0000 5.0001 1.0001 12.7163 1.9942 1.2893

10.000 11.8516 -0.2678 0.9768 1.0000 10.0001 1.0002 11.8536 1.9767 1.2678

15.000 10.3925 -0.2318 0.9467 1.0000 15.0002 1.0002 10.3946 1.9466 1.2318

20.000 8.3051 -0.1812 0.9034 1.0000 20.0004 1.0003 8.3073 1.9033 1.1812

25.000 5.5435 -0.1171 0.8470 1.0000 25.0005 1.0004 5.5459 1.8470 1.1171

29.300 2.5835 -0.0525 0.7892 1.0000 29.3007 1.0010 2.5860 1.7891 1.0525

32.500 0.0000 -0.0196 0.7417 1.0000 32.5009 1.0025 0.0000 1.7416 1.0196

28.3969 17.6149

1.8931 1.1743


© 2002, 2003 OSCAR MUROY 17

Arch, circular axis, built-in, Strassner variation of h


-32.500 0.000 1.6500 0.7241 0.6897 -0.0559 -47.1250 0.9998 0.9999 0.0001

-29.300 2.784 1.5377 0.7832 0.6218 -0.0442 -47.1250 0.9998 0.9999 0.0001

-25.000 5.822 1.4225 0.8477 0.5305 -0.0348 -47.1250 0.9999 0.9999 0.0001

-20.000 8.545 1.3219 0.9055 0.4244 -0.0286 -47.1250 0.9999 0.9999 0.0001

-15.000 10.549 1.2451 0.9480 0.3183 -0.0249 -47.1250 0.9999 1.0000 0.0001

-10.000 11.927 1.1851 0.9772 0.2122 -0.0227 -47.1250 0.9999 1.0000 0.0001

-5.000 12.734 1.1376 0.9944 0.1061 -0.0216 -47.1250 0.9999 1.0000 0.0000

0.000 13.000 1.1000 1.0000 0.0000 -0.0212 -47.1250 0.9999 1.0000 0.0000

5.000 12.734 1.1376 0.9944 -0.1061 -0.0216 -47.1250 0.9999 1.0000 0.0000

10.000 11.927 1.1851 0.9772 -0.2122 -0.0227 -47.1250 0.9999 1.0000 0.0001

15.000 10.549 1.2451 0.9480 -0.3183 -0.0249 -47.1250 0.9999 1.0000 0.0001

20.000 8.545 1.3219 0.9055 -0.4244 -0.0286 -47.1250 0.9999 0.9999 0.0001

25.000 5.822 1.4225 0.8477 -0.5305 -0.0348 -47.1250 0.9999 0.9999 0.0001

29.300 2.784 1.5377 0.7832 -0.6218 -0.0442 -47.1250 0.9998 0.9999 0.0001

32.500 0.000 1.6500 0.7241 -0.6897 -0.0559 -47.1250 0.9998 0.9999 0.0001


+/ (+/) -/ (-/) 1+(I/I´) 1-(I/I´)

-32.500 0.000 -0.0293 1.0000 1.0000 -32.5007 1.0034 0.0000 1.9998 1.0293

-29.300 2.784 -0.0754 1.0000 1.0000 -29.3005 1.0011 2.7871 1.9998 1.0754

-25.000 5.822 -0.1457 1.0000 1.0000 -25.0004 1.0005 5.8247 1.9999 1.1457

-20.000 8.545 -0.2003 1.0000 1.0000 -20.0003 1.0003 8.5478 1.9999 1.2002

-15.000 10.549 -0.2361 1.0000 1.0000 -15.0002 1.0002 10.5511 1.9999 1.2361

-10.000 11.927 -0.2590 1.0000 1.0000 -10.0001 1.0002 11.9287 1.9999 1.2590

-5.000 12.734 -0.2718 1.0000 1.0000 -5.0000 1.0001 12.7358 1.9999 1.2717

0.000 13.000 -0.2759 1.0000 1.0000 0.0000 1.0001 13.0017 1.9999 1.2758

5.000 12.734 -0.2718 1.0000 1.0000 5.0000 1.0001 12.7358 1.9999 1.2717

10.000 11.927 -0.2590 1.0000 1.0000 10.0001 1.0002 11.9287 1.9999 1.2590

15.000 10.549 -0.2361 1.0000 1.0000 15.0002 1.0002 10.5511 1.9999 1.2361

20.000 8.545 -0.2003 1.0000 1.0000 20.0003 1.0003 8.5478 1.9999 1.2002

25.000 5.822 -0.1457 1.0000 1.0000 25.0004 1.0005 5.8247 1.9999 1.1457

29.300 2.784 -0.0754 1.0000 1.0000 29.3005 1.0011 2.7871 1.9998 1.0754

32.500 0.000 -0.0293 1.0000 1.0000 32.5007 1.0034 0.0000 1.9998 1.0293

29.9981 17.7108

1.9999 1.1807


© 2002, 2003 OSCAR MUROY 18

3. DIRECTRIX OF THE ARCH

The directrix of an Arch should be the more approximate possible to the funicular curve for

the loadings applied to the arch.

As the funicular curve is derived, graphically, for an statically determined structure (a three-

hinged arch) and will give an approximate shape of the funicular configuration, as it not

considered, stresses produced by the elastic deformations of the structure.

Fig 3.1

Also equally, we have to take into account, that the funicular correspond to a certain loading

case. As the bridge is subjected to varying conditions of traffic live loads, we have to choose

the state of loading more representative or critical to find a funicular, such that in the other

cases of loading, the funiculars divert the least possible from the directrix chosen.

To find the locus of the funicular, we assumed the arch of Fig. N° 3.1

The triangle of forces corresponding to loads in the segment of arch x:

OAdx

dyQ

'OAxdx

dy

dx

d

dx

dyQ

The difference of these values OA’ and OA, gives a resultant of loads qx x

Then: xqOAOAxdx

dy

dx

dQ x

'

Lastly, the equation of the funicular is:

xqdx

ydQ

2

2

a) Funicular for an uniform load q:

Being qx = q, uniform along the length of the arch:

Then:


© 2002, 2003 OSCAR MUROY 19

qdx

ydQ

2

2

Solving the differential equation and the conditions, x = 0, y = 0 and y’ = 0:

Q

qxy

2

2

We have also, for x = l/2 and y = f:

Q

qlf

8

2

The isostatic thrust being: f

qlQ

8

2

,

Then, the funicular equation is a parabola:

2

2

4x

l

fy ,

b) Funicular for a load varying parabolically:

Let xq the parabolic varying load:

2

2/l

xqqqq cacx

2

24 x

l

qqq ca

c

Then, in the equation of the locus:

2

22

2

4 xl

qqq

dx

ydQ ca

c

Solving the differential equation and the

conditions, x = 0, y = 0 and y’ = 0:

22

232

xxQl

qq

Q

qy cac

Fig 3.2

From the condition, x = l/2 and y = f:

4432

22

2

ll

Ql

qq

Q

qf cac

the isostatic thrust is: ca qq

f

lQ 5

48

2

Replacing this value of Q in the funicular equation:

22

235

8

l

x

l

xqqq

qq

fy cac

ca

These equations are equally valid for the case of load diminishing to the springings


© 2002, 2003 OSCAR MUROY 20

c) Funicular for a load varying similarly to the directrix:

When the load increases toward the springing: 1c

a

q

q,

Let the load be similar to the directrix curve:

f

qqyqq ca

cx

Then, in the locus equation:

f

qqyq

dx

ydQ ca

c2

2

or: Q

qy

fQ

qq

dx

yd cca

2

2

Fig. 3.3

The general solution of this differential equation without the second term is:

xQf

qqCy ca cosh

And the particular solution of the equation with the second term:

1Cy

Then the general solution of the differential equation with the second term is:

1cosh CxQf

qqCy ca

From the condition: x = 0, y = 0 and y’ = 0, differentiating and replacing in the differential

equation, we obtain:

fqq

qCC

ca

c

1

1cosh x

Qf

qqf

qq

qy ca

ca

c

Doing:

1c

a

q

qm , and

2

l

Qf

qqk ca

From the condition: x = l/2, y = f, we obtain:

1cosh1

1

kf

mf , the relation between m and k being: km cosh


© 2002, 2003 OSCAR MUROY 21

Lastly, using these relations and: 2/l

x , we have:

1cosh1

1

kf

my

And the isostatic thrust:

f

qq

k

lQ ca

2

2

d) Funicular for a load varying similarly to the directrix:

When load diminishes to the springings: 1c

a

q

q,

Let it be the load similar to the directrix:

f

qqyqq ac

cx

Then, for the locus equation:

f

qqyq

dx

ydQ ac

c2

2

or: Q

qy

fQ

qq

dx

yd cac

2

2

Fig. 3.4

General solution of this differential equation without second member is:

xQf

qqCy ac cos

And the particular solution of the equation with the second member is:

1Cy

Then the general solution of this differential equation with second member is:

1cos CxQf

qqCy ac

From the condition: x = 0, y = 0 and y’ = 0, differential and replacing in the differential

equation, we obtain

fqq

qCC

ac

c

1

1cos x

Qf

qqf

qq

qy ac

ac

c

Doing:


© 2002, 2003 OSCAR MUROY 22

1c

a

q

qm , and

2

l

Qf

qqk ac

From the condition: x = l/2, y = f, we obtain:

1cos1

1

kf

mf , the relation between m and k is: km cos

Lastly, using these relations and: 2/l

x , we have:

kfm

y cos11

1

And the isostatic thrust:

f

qq

k

lQ ac

2

2

e) Circular funicular:

The circular funicular or circular segment

correspond to a state of uniform radial

pressure q:

Then, in this case, correspond to load

state with vertical loads: qv = q sin and

horizontal loads: qh = q sin

Fig 3.5

Next it is shown fig. 3.6 and 3.7, where it has been obtained for 2 arch bridges, the

directrixes for a parabolic, circular, 4th order parabolic and the hyperbolic cosine or

trigonometric cosine as the case maybe.

x y (cos hip) y (parab 4th

o) y (parab) y (circular)

-32.500 0.000 0.000 0.000 0.000

-29.300 2.584 2.586 2.434 2.784

-25.000 5.544 5.549 5.308 5.822

-20.000 8.305 8.312 8.077 8.545

-15.000 10.393 10.398 10.231 10.549

-10.000 11.852 11.855 11.769 11.927

-5.000 12.714 12.715 12.692 12.734

0.000 13.000 13.000 13.000 13.000

5.000 12.714 12.715 12.692 12.734

10.000 11.852 11.855 11.769 11.927

15.000 10.393 10.398 10.231 10.549

20.000 8.305 8.312 8.077 8.545

25.000 5.544 5.549 5.308 5.822


© 2002, 2003 OSCAR MUROY 23

29.300 2.584 2.586 2.434 2.784

32.500 0.000 0.000 0.000 0.000

Table 3.1 Built-in arch 65m spam and 13m rise

Fig. 3.6

x y (parab) y (parab 4th o) y (coseno) y (circular)

-46.000 0.000 0.000 0.000 0.000

-41.400 3.230 3.140 3.139 3.619

-36.800 6.120 5.985 5.983 6.674

-32.200 8.670 8.524 8.521 9.246

-27.600 10.880 10.745 10.743 11.393

-23.000 12.750 12.640 12.638 13.156

-18.400 14.280 14.201 14.200 14.565

-13.800 15.470 15.422 15.421 15.641

-9.200 16.320 16.297 16.297 16.399

-4.600 16.830 16.824 16.824 16.850

0.000 17.000 17.000 17.000 17.000

4.600 16.830 16.824 16.824 16.850

9.200 16.320 16.297 16.297 16.399

13.800 15.470 15.422 15.421 15.641

18.400 14.280 14.201 14.200 14.565

23.000 12.750 12.640 12.638 13.156

27.600 10.880 10.745 10.743 11.393

32.200 8.670 8.524 8.521 9.246

36.800 6.120 5.985 5.983 6.674

41.400 3.230 3.140 3.139 3.619

46.000 0.000 0.000 0.000 0.000

Table 3.2 Two-hinged Arch, 92m Span and 17m Rise

0.0

2.0

4.0

6.0

8.0

10.0

12.0

14.0

-35.0 -25.0 -15.0 -5.0 5.0 15.0 25.0 35.0

Y

X

Built-in arch 65m spam and 13m rise

cos hip

parab 4°g

parab

circular


© 2002, 2003 OSCAR MUROY 24

Fig. 3.7

0.0

2.0

4.0

6.0

8.0

10.0

12.0

14.0

16.0

18.0

-50.0 -40.0 -30.0 -20.0 -10.0 0.0 10.0 20.0 30.0 40.0 50.0

Y

X

Two-hinged Arch, 92m span and 17m rise

parab

parab 4° g

coseno

circular


© 2002, 2003 OSCAR MUROY 25

4. VARIATION OF THE ARCH DEPTH

The shape and the depth of the cross section of the arch are determined by the section design,

in such a way that the stresses produced by the most unfavourable combination of bending

moment and axial force acting on the section, would not exceed the permissible service load

stresses or satisfied the factor of safety at ultimate loads. The maximum and minimum

bending moments due to transit live loads and the corresponding axial force are the condition

which in most cases defines the dimensions and other parameters of the cross section.

Two of the most important and critical sections of the arch, are the springing and the crown

section. Then normally we start determining the parameters of the sections at these locations,

and for these purposes it is useful to take references to already built bridges, as it listed in

tables 1.2 and 1.3.

Having been defined these two sections, we can assume a progressive variation of the

sections between these two points, graphically, or by mean of equations, to complete the

geometry of the arch

a) Two-hinged Arch:

Among the different proposals to define the depth variation h, it could be mentioned the

followings:

Parabolic variation: 2pxhh cx , being: ac hh

lp

2

4

Variation according to Chalos from the Ecole de Ponts et Chaussées: 5

21

l

xk

II c

x

being: 1a

c

I

Ik

Variation proportional to the inertia of the arch section I:

1cos

l

xkII ax

being: 1a

c

I

Ik

Variation proportional to the depth of the arch section h:

1cos

l

xkhh ax

being: 1a

c

h

hk


© 2002, 2003 OSCAR MUROY 26

x h (parab) h (cosine Ix) h(cosine hx)

h(Chalos,

n=5)

-46.000 1.500 1.500 1.500 1.500

-41.400 1.595 1.600 1.578 1.641

-36.800 1.680 1.687 1.655 1.767

-32.200 1.755 1.762 1.727 1.866

-27.600 1.820 1.827 1.794 1.934

-23.000 1.875 1.880 1.854 1.972

-18.400 1.920 1.924 1.905 1.991

-13.800 1.955 1.957 1.946 1.998

-9.200 1.980 1.981 1.976 2.000

-4.600 1.995 1.995 1.994 2.000

0.000 2.000 2.000 2.000 2.000

4.600 1.995 1.995 1.994 2.000

9.200 1.980 1.981 1.976 2.000

13.800 1.955 1.957 1.946 1.998

18.400 1.920 1.924 1.905 1.991

23.000 1.875 1.880 1.854 1.972

27.600 1.820 1.827 1.794 1.934

32.200 1.755 1.762 1.727 1.866

36.800 1.680 1.687 1.655 1.767

41.400 1.595 1.600 1.578 1.641

46.000 1.500 1.500 1.500 1.500

Table 4.1 Two-hinged Arch, 92m Span and 17m Rise

Fig. 4.1

1.4

1.5

1.6

1.7

1.8

1.9

2.0

2.1

-50.0 -40.0 -30.0 -20.0 -10.0 0.0 10.0 20.0 30.0 40.0 50.0

h

X

Variation of h

parab

coseno Ix

coseno hx

Chalos, n=5


© 2002, 2003 OSCAR MUROY 27

b) Built-in arch:

Among the different proposals to define the variation of the depth h, it could be mention

the followings:

Parabolic variation: 2

pxhh cx , being ca hhl

p 2

4

Variation according to Chalos, family of equations n

cx

l

xk

II

21

, being a

c

I

Ik 1 y n

= 1, 2, 3 ó 4

Variation according to Strassner, the inertia I inversely proportional to cosine:

l

x

II c

x2

1cos , being

cos1

a

c

I

I

r

Variation according to Strassner, the depth h, inversely proportional to cosine:

, being

3

cos1

a

c

I

I

x h (parab) h (Strassner1) h (Strassner2)

h (Chalos,n=1)

-32.500 1.650 1.650 1.650 1.650

-29.300 1.547 1.547 1.544 1.538

-25.000 1.425 1.434 1.424 1.426

-20.000 1.308 1.332 1.314 1.329

-15.000 1.217 1.252 1.230 1.254

-10.000 1.152 1.189 1.169 1.193

-5.000 1.113 1.139 1.126 1.143

0.000 1.100 1.100 1.100 1.100

5.000 1.113 1.139 1.126 1.143

10.000 1.152 1.189 1.169 1.193

15.000 1.217 1.252 1.230 1.254

20.000 1.308 1.332 1.314 1.329

25.000 1.425 1.434 1.424 1.426

29.300 1.547 1.547 1.544 1.538

32.500 1.650 1.650 1.650 1.650

Table 4.2 Built-in for 65m span and 13m rise

3/1

21cos

l

x

hh c

x


© 2002, 2003 OSCAR MUROY 28

Fig. 4.2

Variation according to Chalos

x h (parab) h (n=2) h (n=3) h (n=4) h (n=1)

-32.500 1.6500 1.650 1.650 1.650 1.650

-29.300 1.5470 1.460 1.401 1.355 1.538

-25.000 1.4250 1.316 1.251 1.209 1.426

-20.000 1.3080 1.220 1.168 1.140 1.329

-15.000 1.2170 1.161 1.127 1.112 1.254

-10.000 1.1520 1.126 1.108 1.102 1.193

-5.000 1.1130 1.106 1.101 1.100 1.143

0.000 1.1000 1.100 1.100 1.100 1.100

5.000 1.1130 1.106 1.101 1.100 1.143

10.000 1.1520 1.126 1.108 1.102 1.193

15.000 1.2170 1.161 1.127 1.112 1.254

20.000 1.3080 1.220 1.168 1.140 1.329

25.000 1.4250 1.316 1.251 1.209 1.426

29.300 1.5470 1.460 1.401 1.355 1.538

32.500 1.6500 1.650 1.650 1.650 1.650

Table 4.3 Two-hinged Arch, 65m span and 13m rise

1.0

1.1

1.2

1.3

1.4

1.5

1.6

1.7

-35.0 -25.0 -15.0 -5.0 5.0 15.0 25.0 35.0

h

X

Variation of h

parab

Strassner 1

Strassner 2

Chalos, n=1


© 2002, 2003 OSCAR MUROY 29

Fig. 4.3

1.0

1.1

1.2

1.3

1.4

1.5

1.6

1.7

-35.0 -25.0 -15.0 -5.0 5.0 15.0 25.0 35.0

h

X

Variation of h

parab

Chalos, n=2

Chalos, n=3

Chalos, n= 4

Chalos, n=1


© 2002, 2003 OSCAR MUROY 30

5. TYPICAL CONFIGURATIONS OF ARCH BRIDGES

Arch Bridges are economically competitive, from the 50 m span upwards in concrete arches

and larger for steel bridges, due to a costlier construction procedure and the arch in itself is an

element more to build besides the bridge deck, so in lower limits of these spans, an

economical comparison should be made with beams or frames alternatives

The typical basic configurations of arch bridges that are constructed nowadays, belongs

largely to built-in and two-hinged arches and for the relative position of the bridge deck, in

upper, intermediate and lower deck arches. In the next figure sketches of these configurations

are shown.

a) Two-hinged Arch b) Built-in Arch

Fig 5.1

There is very little number of three-hinged arches, although one of best known and depicted

in any anthological review of Bridges is the Salginatobel Bridge, designed by the Swiss

Engineer Robert Maillart, built in 1930.

hinged

Fig 5.2

One-hinged arches do not represent any structural advantage respect to the other types and

there are not known bridge of this type.

However, these two last configurations, have been used as a temporary stage of construction,

before a construction technique called compensation of arches is applied. This technique will

be explained afterwards.

In relation of Arch Bridges, it should be distinguished when the arch is of a truss or lattice

construction, which could be considered as a pseudo-arch, because although its shape

correspond to an arch, structurally it is analysed more properly as a truss. Fig. 5.3


© 2002, 2003 OSCAR MUROY 31

Fig 5.3

With the extraordinary advancement of the Structural of Analysis, which has broaden the

scope of computable structural types, have emerged a large number of variants of these basic

configurations.

Usually, the structure of the Arch Bridge is composed of 2 parallel arches in the width of the

deck or it a slab type arch, with the width of the deck.

A variant, in this respect, are the configurations with 2 arches, in sloped planes approaching

or converging at the crown zone.

Fig 5.4


© 2002, 2003 OSCAR MUROY 32

When for reasons of poor soil or being an intermediate span over elevated supports, that has

no capacity to take large lateral thrusts of the lower deck arch, it is convenient to adopt the

structural scheme of a tied arch.

Variants of the tied arch are the use of lateral semi arches and compression struts, which

reduces the thrust or this is transferred far apart to a safer zone

Fig 5.5

Lastly, the hangers or columns of the arch are in most cases vertical. Variants in this aspect,

are the sloped hangers or even interlaced hangers and columns with triangular arrangement of

the columns.

Fig 5.6

c


© 2002, 2003 OSCAR MUROY 33

For smaller than 40m span, it has been designed arch bridges with filled spandrels, of

reinforced concrete arches, although in these cases an economic comparison should be made

with the frame or beam solutions.

Fig 5.7


© 2002, 2003 OSCAR MUROY 34

6. CONSTRUCTION PROCEDURES FOR ARCH BRIDGES

In a large proportion of cases, arch bridges are built over deep ravine or over permanent water

courses, with the additional problem of being a waterway which will make costlier or even

unviable for conventional construction using false work supported on the terrain.

Fig 6.1

From these situations, it emerges naturally the idea to construct from above. This type of

construction procedure, has gained a general acceptance for many years, and the most spread

that could be adapted to the national realities could be mentioned; it is the use of cable stays

to support the structure or false work temporally during the construction procedure and the

use of construction travellers, while advancing the construction stages.


© 2002, 2003 OSCAR MUROY 35

Fig 6.2

The use of these construction procedures implies a tight involvement to them with the

analysis and design process, as the structure should be designed for the different construction

stages and at the same time the construction procedure must be executed so as to agree to the

foreseen behaviour for the structure, in its different stages of the construction.

Fig 6.3


© 2002, 2003 OSCAR MUROY 36

7. SECOND ORDER DEFLECTIONS IN ARCHES

Deflections w, u and v in the Navier – Bresse equations, are obtained from an undeformed

geometry of the arch, by assuming that the deflections are small and can be neglected, and

being as yet unknown the arch deformation.

With span surpassing the 100 m, (at this time Arch Bridges larger than a 500 m span have

already been constructed), it becomes necessary to calculate the real deformations, from the

deformed shape of the arch, when applying loads. This is particularly significant in arches, as

when deforming the arch, rise diminishes and consequently the compensating moment due to

horizontal thrust, deriving into larger bending moments

Angular Displacement:

dsEI

Mww

s

s

0

0 (7.1)

Horizontal Displacement:

s

s

s

s

s

s

dsGA

Tds

EA

Nds

EI

Mywywuu

000

sincos20.11

000 (7.2)

Vertical Displacement:

s

s

s

s

s

s

dsGA

Tds

EA

Nds

EI

Mxwxwvv

000

cossin0.21

000 (7.3)

The real deformations determination could be made by successive approximations as follows:

We shall call x0(x), y0(x), 0(x) to the initial geometry, from which we obtain the forces N0,

T0 y M0

Applying the equations 7.1 to 7.3, we obtain the elastic deformations, which we will call

w1(x), u1(x) y v1(x)

Then the first approximation of the deformed shape, would be:

)()()( 101 xvxyxy

)()()( 101 xuxxxx

)()()( 101 xwxx

Deformations in the x direction can be negligible, in comparison to the element dimension of

the arch and it won’t be taken into account further. With the arch deformed geometry; we

would obtain the corrected values N1, T1 and M1

With this new geometry and the applied forces, we would get a second set of deformations for

the structures w2(x), u2(x) and v2(x)

The second approximation of the deformed shape would be:

)()()( 202 xvxyxy

)()()( 202 xwxx

With this new deformed geometry of the arch, we correct again the values N2, T2 and M2

Proceeding in this way, we would get after n iterations wn(x), un(x) and vn(x):


© 2002, 2003 OSCAR MUROY 37

The nth time approximation of the deformed shape would be:

)()()( 0 xvxyxy nn

)()()( 0 xwxx nn

So we would get a series of values y1(x), y2(x), y3(x),........., yn-1(x), yn(x) and of 1(x), 2(x),

3(x),........., n-1(x), n(x)

In a stable structure, for the loading that is subjected, these series of values are convergent to

the final values of the deformed shape.

And lastly the forces, taken into account the deformed shape of the structure would be Nn, Tn

and Mn

As an example, we shall examine the case of an arch of 60m span, steel with 60cm depth,

subjected to concentrated loads of dead weight, as per fig. N° 7.1:

Fig 7.1

For the case of two-hinged arch, the successive deformations obtained are as shown in table

N° 7.1 and Fig. N° 7.2

Computations for the final deflections, has been repeated until relative error of 0.001, is

reached, which has been obtained after 3 iterations.

Final deflections are in this case, therefore, larger:

In the maximum positive deflection, at 7.5m from springing:

5.099/3.338=1.53

In the maximum negative deflection, at crown

15.956/10.911=1.46


© 2002, 2003 OSCAR MUROY 38

Fig 7.2

Fig 7.3

Final successive Bending Moments are as shown in Table N° 7.1 and Fig. N° 7.3

Final Bending Moments are in this case, therefore, larger:

In the maximum negative bending moment at 9.0m from springing:

48.85/34.81=1.40

In the maximum positive bending moment at the crown:

39.53/28.13=1.41

-18

-16

-14

-12

-10

-8

-6

-4

-2

0

2

4

6

0 6 12 18 24 30 36 42 48 54 60

v (cm

)

X (m)

VERTICAL DEFLECTIONS DUE TO DEAD WEIGHT

'v1'

'v2'

'v3'

'v4'

-60

-50

-40

-30

-20

-10

0

10

20

30

40

50

0 6 12 18 24 30 36 42 48 54 60

Mf (T

.m

)

X (m)

BENDING MOMENTS DUE TO DEAD WEIGHT

'MF1'

'MF2'

'MF3'

'MF4'


© 2002, 2003 OSCAR MUROY 39

X v1 v2 v3 v4 X Mz1 Mz2 Mz3 Mz4

(m) (cm) (cm) (cm) (cm) (m) (T.m) (T.m) (T.m) (T.m)

0 0 0 0 0 0 0 0 0 0

1.5 1.25 1.593 1.701 1.738 1.5 -24.25 -27.09 -27.85 -28.11

3 2.188 2.836 3.042 3.113 3 -35.21 -40.36 -41.81 -42.29

4.5 2.847 3.753 4.044 4.145 4.5 -33.79 -40.74 -42.79 -43.49

6 3.195 4.296 4.652 4.776 6 -20.88 -29.11 -31.65 -32.51

7.5 3.338 4.56 4.96 5.099 7.5 -32.92 -42.06 -44.96 -45.96

9 3.126 4.375 4.787 4.932 9 -34.81 -44.28 -47.38 -48.45

10.5 2.611 3.801 4.199 4.338 10.5 -27.3 -36.59 -39.75 -40.85

12 1.797 2.825 3.172 3.294 12 -10.54 -19.14 -22.19 -23.27

13.5 0.857 1.637 1.904 1.998 13.5 -18.28 -25.95 -28.75 -29.75

15 -0.312 0.135 0.29 0.345 15 -17.65 -24.01 -26.42 -27.29

16.5 -1.654 -1.608 -1.591 -1.584 16.5 -8.82 -13.55 -15.46 -16.16

18 -3.107 -3.526 -3.672 -3.723 18 7.92 5.02 3.73 3.24

19.5 -4.492 -5.411 -5.736 -5.851 19.5 0.47 -0.66 -1.27 -1.51

21 -5.892 -7.32 -7.83 -8.011 21 0.48 1.2 1.29 1.29

22.5 -7.265 -9.196 -9.89 -10.137 22.5 7.96 10.51 11.31 11.57

24 -8.502 -10.897 -11.763 -12.073 24 22.74 26.98 28.43 28.93

25.5 -9.44 -12.239 -13.258 -13.623 25.5 13.49 19 21.02 21.74

27 -10.179 -13.281 -14.414 -14.822 27 11.41 17.92 20.38 21.26

28.5 -10.705 -14.006 -15.215 -15.651 28.5 16.19 23.43 26.17 27.17

30 -10.911 -14.277 -15.511 -15.956 30 28.13 35.66 38.5 39.53

31.5 -10.661 -13.959 -15.169 -15.605 31.5 16.19 23.36 26.1 27.1

33 -10.183 -13.288 -14.425 -14.835 33 11.41 17.93 20.39 21.28

34.5 -9.47 -12.273 -13.296 -13.665 34.5 13.49 19.04 21.07 21.8

36 -8.513 -10.915 -11.788 -12.103 36 22.74 26.99 28.45 28.97

37.5 -7.237 -9.173 -9.873 -10.126 37.5 7.96 10.47 11.27 11.54

39 -5.907 -7.347 -7.866 -8.053 39 0.48 1.22 1.33 1.35

40.5 -4.526 -5.457 -5.791 -5.913 40.5 0.47 -0.6 -1.2 -1.43

42 -3.128 -3.562 -3.719 -3.777 42 7.92 5.05 3.78 3.31

43.5 -1.649 -1.618 -1.613 -1.614 43.5 -8.82 -13.56 -15.44 -16.13

45 -0.338 0.093 0.235 0.283 45 -17.65 -23.97 -26.35 -27.21

46.5 0.82 1.585 1.841 1.928 46.5 -18.28 -25.89 -28.67 -29.66

48 1.766 2.776 3.112 3.227 48 -10.54 -19.09 -22.12 -23.18

49.5 2.595 3.766 4.152 4.284 49.5 -27.3 -36.56 -39.69 -40.78

51 3.09 4.322 4.725 4.863 51 -34.81 -44.23 -47.3 -48.36

52.5 3.291 4.498 4.89 5.023 52.5 -32.92 -41.99 -44.87 -45.85

54 3.153 4.241 4.59 4.709 54 -20.88 -29.05 -31.56 -32.42

55.5 2.819 3.713 3.997 4.094 55.5 -33.79 -40.7 -42.73 -43.42

57 2.14 2.779 2.981 3.05 57 -35.21 -40.28 -41.72 -42.2

58.5 1.156 1.489 1.594 1.629 58.5 -24.25 -26.95 -27.7 -27.95

60 -0.049 -0.049 -0.049 -0.049 60 0 0.07 0.07 0.07

Table 7.1 Vertical Deflections and bending moments in the two-hinged arch

For the case of built-in arch, successive deflections obtained are as shown in table N° 7.2 and

Fig. N° 7.4

Computations for the final deflections, have been repeated until a relative error of 0.001,

which was obtained with only 2 iterations.


© 2002, 2003 OSCAR MUROY 40

Fig 7.4

Fig 7.5

Final deflections are in this case, therefore, larger:

In the maximum positive deflection at 9.00m from springing:

1.265/0.807=1.57

In the maximum negative deflection at crown

8.061/6.678=1.21

Final successive Bending Moments are as it is shown in Table N° 7.2 and Fig. N° 7.5

Final Bending Moments are in this case, therefore, larger:

In the maximum negative bending moment at 9.00m at springing

23.32/20.01=1.17

In the maximum positive bending moment at crown

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

0 6 12 18 24 30 36 42 48 54 60

v (cm

)

X (m)

VERTICAL DEFLEXIONS DUE TO DEAD WEIGHT

'v1'

'v2'

'v3'

-30

-20

-10

0

10

20

30

40

50

0 6 12 18 24 30 36 42 48 54 60

Mf (T

.m

)

X (m)

BENDING MOMENT DUE TO DEAD WEIGHT

'MF1'

'MF2'

'MF3'


© 2002, 2003 OSCAR MUROY 41

24.83/20.75=1.20

In the maximum positive bending moment at springing

47.04/40.87=1.15

X v1 v2 v3 X Mz1 Mz2 Mz3

(m) (cm) (cm) (cm) (m) (T.m) (T.m) (T.m)

0 0 0 0 0 40.87 46.01 47.04

1.5 0.172 0.198 0.203 1.5 11.35 14.87 15.66

3 0.358 0.439 0.456 3 -4.49 -2.49 -1.98

4.5 0.566 0.724 0.759 4.5 -7.56 -7.02 -6.81

6 0.707 0.947 1 6 1.22 0.49 0.42

7.5 0.849 1.166 1.239 7.5 -14.62 -16.55 -16.89

9 0.807 1.178 1.265 9 -20.01 -22.75 -23.32

10.5 0.606 1.008 1.105 10.5 -15.71 -18.95 -19.69

12 0.238 0.635 0.733 12 -1.86 -5.28 -6.12

13.5 -0.168 0.19 0.282 13.5 -12.24 -15.71 -16.59

15 -0.742 -0.463 -0.388 15 -14.01 -17.19 -18.04

16.5 -1.446 -1.279 -1.23 16.5 -7.32 -9.95 -10.71

18 -2.238 -2.219 -2.205 18 7.53 5.64 5.04

19.5 -2.965 -3.117 -3.146 19.5 -1.58 -2.77 -3.16

21 -3.743 -4.078 -4.155 21 -2.99 -3.34 -3.51

22.5 -4.536 -5.061 -5.188 22.5 3.28 3.86 3.94

24 -5.266 -5.973 -6.149 24 17.09 18.55 18.88

25.5 -5.778 -6.647 -6.867 25.5 7.09 9.14 9.7

27 -6.208 -7.199 -7.453 27 4.46 7.05 7.77

28.5 -6.534 -7.607 -7.884 28.5 8.92 11.92 12.76

30 -6.678 -7.777 -8.061 30 20.75 23.95 24.83

31.5 -6.494 -7.563 -7.839 31.5 8.92 11.85 12.69

33 -6.211 -7.2 -7.453 33 4.46 7.04 7.76

34.5 -5.809 -6.673 -6.891 34.5 7.09 9.18 9.73

36 -5.277 -5.979 -6.153 36 17.09 18.55 18.88

37.5 -4.512 -5.03 -5.153 37.5 3.28 3.8 3.87

39 -3.757 -4.088 -4.162 39 -2.99 -3.35 -3.52

40.5 -3 -3.145 -3.171 40.5 -1.58 -2.74 -3.15

42 -2.258 -2.233 -2.217 42 7.53 5.63 5.03

43.5 -1.442 -1.27 -1.217 43.5 -7.32 -10 -10.76

45 -0.767 -0.482 -0.404 45 -14.01 -17.2 -18.05

46.5 -0.21 0.155 0.249 46.5 -12.24 -15.69 -16.58

48 0.207 0.609 0.71 48 -1.86 -5.29 -6.13

49.5 0.591 0.997 1.095 49.5 -15.71 -18.99 -19.72

51 0.772 1.147 1.236 51 -20.01 -22.76 -23.33

52.5 0.794 1.113 1.186 52.5 -14.62 -16.53 -16.87

54 0.665 0.908 0.962 54 1.22 0.48 0.41

55.5 0.532 0.69 0.724 55.5 -7.56 -7.04 -6.82

57 0.31 0.392 0.409 57 -4.49 -2.49 -1.98

58.5 0.067 0.091 0.095 58.5 11.35 14.95 15.74

60 -0.049 -0.049 -0.049 60 40.87 45.99 47.03

Table 7.2 Vertical Deflections and bending moment in a built-in arch


© 2002, 2003 OSCAR MUROY 42

8. COMPENSATION OF ARCHES

This is a construction procedure aimed to incorporate a favourable state of stresses to improve

the structural behaviour of the arch. In the past it has been used to decentre or remove the

false work for its disassembly.

For an arch which will become a build-in type, it could be embedded one or two joints. In an

arch which will lastly become a two-hinged type, it could be embedded a joint at the crown.

There are also two ways to execute these temporary joints: one is to effectively build a hinge,

in a stage of construction and then afterwards restore the monolithism of the hinge and so the

capacity to withstand the bending moments.

The second way is to insert flat jacks in the joint, and jacking up to introduce controlled

compressive forces, to generate a favourable state of forces for the improved behaviour of the

structure.

Fig. 8.1

Applying these concepts for a build-in arch bridge of 65m span, we shall examine the

variation of moments and the eccentricity of the axial forces due to permanent loads (self-

weight + dead weight)

a) When it is built temporary hinges in the springings and we have therefore a two hinged

arch temporarily for the permanent loads.

BENDING MOMENTS AXIAL FORCE

X PP PM PP+PM X PP PM PP+PM exc

(m) (T.m) (T.m) (T.m) (m) (T) (T) (T) (cm)

0 0 0 0 0 166.25 316.77 483.02 0.0

1.25 9.94 -13.38 -3.44 1.25 162.17 316.95 479.12 -0.7

2.5 17.65 -15.69 1.96 2.5 158.38 316.98 475.36 0.4

3.75 23.47 -34.16 -10.69 3.75 154.87 304.11 458.98 -2.3

5 27.45 -41.8 -14.35 5 151.62 304.23 455.85 -3.1

6.25 30.01 -38.62 -8.61 6.25 148.62 304.18 452.8 -1.9

7.5 31.11 -25.12 5.99 7.5 145.85 303.96 449.81 1.3

8.75 31.16 -36.5 -5.34 8.75 143.3 289.23 432.53 -1.2

10 30.16 -37.55 -7.39 10 140.96 289.24 430.2 -1.7

11.25 28.42 -28.28 0.14 11.25 138.82 289.08 427.9 0.0

12.5 26.11 -8.7 17.41 12.5 136.86 288.72 425.58 4.1

13.75 23.2 -20.49 2.71 13.75 135.08 274.48 409.56 0.7


© 2002, 2003 OSCAR MUROY 43

15 19.9 -22.22 -2.32 15 133.46 274.51 407.97 -0.6

16.25 16.44 -13.89 2.55 16.25 132 274.35 406.35 0.6

17.5 12.7 4.02 16.72 17.5 130.68 273.98 404.66 4.1

18.75 8.95 -5.74 3.21 18.75 129.51 263.96 393.47 0.8

20 5.26 -5.67 -0.41 20 128.46 263.96 392.42 -0.1

21.25 1.66 3.95 5.61 21.25 127.53 263.75 391.28 1.4

22.5 -1.76 23.15 21.39 22.5 126.72 263.33 390.05 5.5

23.75 -4.78 10.86 6.08 23.75 126.02 255.9 381.92 1.6

25 -7.61 7.9 0.29 25 125.42 255.97 381.39 0.1

26.25 -10.16 14.24 4.08 26.25 124.92 255.82 380.74 1.1

27.5 -12.25 30.15 17.9 27.5 124.52 255.45 379.97 4.7

28.75 -13.83 18.09 4.26 28.75 124.21 252.18 376.39 1.1

30 -15.01 15.34 0.33 30 123.99 252.24 376.23 0.1

31.25 -15.75 21.91 6.16 31.25 123.86 252.09 375.95 1.6

32.5 -16.04 37.79 21.75 32.5 123.82 251.71 375.53 5.8

33.75 -15.75 21.91 6.16 33.75 123.86 252.09 375.95 1.6

35 -15.01 15.34 0.33 35 123.99 252.24 376.23 0.1

36.25 -13.83 18.09 4.26 36.25 124.21 252.18 376.39 1.1

37.5 -12.25 30.15 17.9 37.5 124.52 251.89 376.41 4.8

38.75 -10.17 14.24 4.07 38.75 124.92 255.82 380.74 1.1

40 -7.61 7.9 0.29 40 125.42 255.97 381.39 0.1

41.25 -4.79 10.86 6.07 41.25 126.02 255.9 381.92 1.6

42.5 -1.76 23.15 21.39 42.5 126.72 255.62 382.34 5.6

43.75 1.65 3.95 5.6 43.75 127.53 263.75 391.28 1.4

45 5.26 -5.67 -0.41 45 128.46 263.96 392.42 -0.1

46.25 8.93 -5.74 3.19 46.25 129.51 263.96 393.47 0.8

47.5 12.7 4.02 16.72 47.5 130.68 263.76 394.44 4.2

48.75 16.42 -13.89 2.53 48.75 132 274.35 406.35 0.6

50 19.9 -22.22 -2.32 50 133.46 274.51 407.97 -0.6

51.25 23.17 -20.49 2.68 51.25 135.08 274.48 409.56 0.7

52.5 26.11 -8.7 17.41 52.5 136.86 274.26 411.12 4.2

53.75 28.39 -28.28 0.11 53.75 138.82 289.08 427.9 0.0

55 30.16 -37.55 -7.39 55 140.96 289.24 430.2 -1.7

56.25 31.13 -36.5 -5.37 56.25 143.3 289.23 432.53 -1.2

57.5 31.11 -25.12 5.99 57.5 145.85 289.04 434.89 1.4

58.75 29.97 -38.62 -8.65 58.75 148.62 304.18 452.8 -1.9

60 27.45 -41.8 -14.35 60 151.62 304.23 455.85 -3.1

61.25 23.43 -34.16 -10.73 61.25 154.87 304.11 458.98 -2.3

62.5 17.65 -15.69 1.96 62.5 158.38 303.85 462.23 0.4

63.75 9.89 -13.38 -3.49 63.75 162.17 316.95 479.12 -0.7

65 0 0 0 65 166.25 316.77 483.02 0.0

Table 8.1 Bending Moments and eccentricity of the axial forces in a two-hinged arch


© 2002, 2003 OSCAR MUROY 44

Fig. 8.2

b) When it is constructed temporary joints: one at crown or two in the fourth span, inserting

hydraulic jacks to generate a total horizontal displacement of 0.6cm

BENDING MOMENTS AXIAL FORCES

X PP PM TEMP (*) CALIB X PP PM TEMP (*) CALIB exc

(m) (T.m) (T.m) (T.m) (T.m) (m) (T) (T) (T) (T) (cm)

0 -55.11 24.6 94.9 -0.5 0 162.39 318.5 7.5 483.26 -0.1

1.25 -40.1 8.96 85.08 -4.3 1.25 158.24 318.71 7.63 479.36 -0.9

2.5 -27.55 4.49 75.69 0.9 2.5 154.38 318.77 7.76 475.60 0.2

3.75 -17.09 -16.05 66.72 -12.1 3.75 150.8 305.93 7.89 459.22 -2.6

5 -8.7 -25.66 58.17 -16.0 5 147.48 306.08 8.01 456.09 -3.5

6.25 -1.95 -24.36 50.03 -10.5 6.25 144.42 306.06 8.14 453.05 -2.3

7.5 3.14 -12.64 42.29 3.9 7.5 141.58 305.87 8.26 450.06 0.9

8.75 6.97 -25.7 34.96 -7.7 8.75 138.97 291.16 8.38 432.78 -1.8

10 9.54 -28.35 28.03 -10.0 10 136.57 291.21 8.5 430.47 -2.3

11.25 11.17 -20.59 21.49 -2.6 11.25 134.37 291.07 8.62 428.16 -0.6

12.5 12.02 -2.41 15.36 14.5 12.5 132.36 290.74 8.73 425.86 3.4

13.75 12.08 -15.53 9.6 -0.4 13.75 130.52 276.52 8.84 409.83 -0.1

15 11.54 -18.5 4.23 -5.6 15 128.85 276.57 8.94 408.25 -1.4

16.25 10.65 -11.3 -0.76 -0.9 16.25 127.33 276.43 9.04 406.62 -0.2

17.5 9.28 5.54 -5.37 13.1 17.5 125.97 276.09 9.14 404.95 3.2

18.75 7.71 -5.19 -9.6 -0.5 18.75 124.74 266.09 9.23 393.75 -0.1

20 6 -6.01 -13.45 -4.3 20 123.65 266.11 9.31 392.70 -1.1

21.25 4.2 2.82 -16.94 1.7 21.25 122.68 265.92 9.38 391.56 0.4

22.5 2.38 21.3 -20.05 17.3 22.5 121.84 265.51 9.45 390.34 4.4

23.75 0.76 8.39 -22.78 2.0 23.75 121.1 258.1 9.51 382.21 0.5

25 -0.85 4.87 -25.15 -3.9 25 120.48 258.18 9.57 381.68 -1.0

26.25 -2.36 10.76 -27.17 -0.2 26.25 119.96 258.04 9.61 381.04 0.0

27.5 -3.61 26.29 -28.81 13.6 27.5 119.54 257.68 9.65 380.27 3.6

28.75 -4.54 13.94 -30.08 -0.1 28.75 119.21 254.41 9.68 376.68 0.0

30 -5.25 10.98 -30.98 -4.1 30 118.98 254.48 9.71 376.53 -1.1

31.25 -5.71 17.42 -31.53 1.7 31.25 118.85 254.33 9.72 376.25 0.5

32.5 -5.9 33.26 -31.72 17.3 32.5 118.8 253.95 9.72 375.82 4.6

-60

-40

-20

0

20

40

60

80

100

0 5 10 15 20 25 30 35 40 45 50 55 60 65

Mf (T

.m

)

X (m)

BENDING MOMENT CALIBRATION

PP

PM

CAL


© 2002, 2003 OSCAR MUROY 45

33.75 -5.71 17.42 -31.53 1.7 33.75 118.85 254.33 9.72 376.25 0.5

35 -5.25 10.98 -30.98 -4.1 35 118.98 254.48 9.71 376.53 -1.1

36.25 -4.54 13.94 -30.08 -0.1 36.25 119.21 254.41 9.68 376.68 0.0

37.5 -3.6 26.29 -28.81 13.6 37.5 119.54 254.12 9.65 376.71 3.6

38.75 -2.36 10.76 -27.17 -0.2 38.75 119.96 258.04 9.61 381.04 0.0

40 -0.84 4.87 -25.15 -3.9 40 120.48 258.18 9.57 381.68 -1.0

41.25 0.76 8.39 -22.78 2.0 41.25 121.1 258.1 9.51 382.21 0.5

42.5 2.39 21.3 -20.05 17.4 42.5 121.84 257.8 9.45 382.63 4.5

43.75 4.2 2.82 -16.94 1.7 43.75 122.68 265.92 9.38 391.56 0.4

45 6.02 -6.01 -13.45 -4.2 45 123.65 266.11 9.31 392.70 -1.1

46.25 7.71 -5.19 -9.6 -0.5 46.25 124.74 266.09 9.23 393.75 -0.1

47.5 9.3 5.54 -5.37 13.1 47.5 125.97 265.87 9.14 394.73 3.3

48.75 10.64 -11.3 -0.76 -0.9 48.75 127.33 276.43 9.04 406.62 -0.2

50 11.56 -18.5 4.23 -5.6 50 128.85 276.57 8.94 408.25 -1.4

51.25 12.07 -15.53 9.6 -0.4 51.25 130.52 276.52 8.84 409.83 -0.1

52.5 12.05 -2.41 15.36 14.5 52.5 132.36 276.27 8.73 411.39 3.5

53.75 11.16 -20.59 21.49 -2.6 53.75 134.37 291.07 8.62 428.16 -0.6

55 9.56 -28.35 28.03 -9.9 55 136.57 291.21 8.5 430.47 -2.3

56.25 6.96 -25.7 34.96 -7.7 56.25 138.97 291.16 8.38 432.78 -1.8

57.5 3.17 -12.64 42.29 3.9 57.5 141.58 290.94 8.26 435.13 0.9

58.75 -1.96 -24.36 50.03 -10.5 58.75 144.41 306.06 8.14 453.04 -2.3

60 -8.67 -25.66 58.17 -15.9 60 147.48 306.08 8.01 456.09 -3.5

61.25 -17.11 -16.05 66.72 -12.1 61.25 150.8 305.93 7.89 459.22 -2.6

62.5 -27.51 4.49 75.69 0.9 62.5 154.38 305.64 7.76 462.47 0.2

63.75 -40.12 8.96 85.08 -4.3 63.75 158.23 318.71 7.63 479.35 -0.9

65 -55.07 24.6 94.9 -0.5 65 162.39 318.5 7.5 483.26 -0.1

Table 8.2 BendingMoments and eccentricity of the Axial Force in two-hinged arch

(*) The temperature effects correspond to a restrained expansion of lx = .t.lx= 0.0117x10-

3x25x6500 = 1.9 cm for a temperature variation of 25°C. As we need only 0.6 cm, for the

forced displacement of the jacks, we shall be multiplying these effects by 0.6/1.9 = 0.316

Fig. 8.3

-60

-40

-20

0

20

40

60

80

100

0 5 10 15 20 25 30 35 40 45 50 55 60 65

Mf (T

.m

)

X (m)

CALIBRATION OF BENDING MOMENTS

PP

PM

TEMP

CAL


© 2002, 2003 OSCAR MUROY 46

9. ARCH STABILITY

There is no analytical procedure to lead to a general formulation, suitable to any arch with

whichever support conditions.

a) Two-hinged arch b) Built-in arch

Fig. 9.1

Due to the complexity of the problem, this analysis has been carried out case by case,

assuming a successive simplifying hypothesis, using the Engesser-Vianello method of

successive approximations, to obtain delimited values for the critical load of horizontal thrust

which will drive to the buckling of the structure and afterward it has been checked by model

testing

So to get the critical load qcr:

3l

EIqcr

For circular arches, constant section and constant radial pressure, value is:

f/l Built-in 1 hinge 2 hinge 3 hinge

0.1 58.9 33.0 28.4 22.2

0.2 90.4 50.0 39.3 33.5

0.3 93.4 52.0 40.9 34.9

0.4 80.7 46.0 32.8 30.2

0.5 64.0 37.0 24.0 24.0

Table 9.1

For parabolic arches, constant section and uniform load, value is given by:


0.1 60.7 33.8 28.5 22.5

0.2 101.0 59.0 45.4 39.6

0.3 115.0 ---- 46.5 46.5

0.4 111.0 96.0 43.9 43.9

0.5 97.4 ---- 38.4 38.4

Table 9.2

For parabolic arches, section hx=hc/cos and uniform load, value is given by:


0.1 65.5 36.5 30.7 24.0

0.2 134.0 75.8 59.8 51.2

0.3 204.0 ---- 81.1 81.1


© 2002, 2003 OSCAR MUROY 47

0.4 277.0 187.0 101.0 101.0

0.5 ---- ---- ---- ----

Table 9.3

For hyperbolic cosine arches, constant section and uniform load, value is given by:


0.1 59.4 28.4

0.2 96.4 43.2

0.3 112.0 41.9

0.4 92.3 35.4

0.5 80.7 27.4

Table 9.4

When we want to obtain the critical thrust Hcr:

2l

EIH c

cr

For parabolic arches, constant section and uniform load, values is given by:


0.1 75.8

----

---- 35.6

36.2

28.5

---

0.2 63.1

58.5

---- 28.4

28.2

24.9

22.7

0.3 47.9

43.8

---- 19.4

19.8

20.2

18.8

0.4 34.8

34.2

---- 13.7

13.6

15.4

13.6

0.5 ----

30.4

---- 9.6

---

----

11.2

Table 9.5

For parabolic arches I=Ic/cos and uniform load, value is given by:


0.1 78.2

78.4

---- 37.2 29.4

0.2 71.0

70.8

---- 31.6 27.7

0.3 61.3

61.1

---- 25.1 25.3

25.1

0.4 51.1

51.1

---- 19.4 22.6

19.4

0.5 41.9

41.8

---- 15.0 19.8

15.0

Table 9.6


© 2002, 2003 OSCAR MUROY 48

With the development of matrix method for the structural analysis, it could make a more

general set out on the buckling of elements subjected to an axial load N, introducing the

concept of geometric stiffness KG.

The set out consist in representing a structural member, connected to an auxiliary structure of

rigid articulated bars, where the axial forces are acting, as it is shown in the next Fig. 9.2

Fig. 9.2

When the real structure deforms, the same will happen to the auxiliary structure with a set of

axial forces to stabilize the auxiliary system fG

In an element of the structure, we have:

Fig. 9.3

Which could be written in matrix form as:

Extending this set out to the whole structure


© 2002, 2003 OSCAR MUROY 49

=

… …

………………………………………………………

………………………………………………………

Symbolically it could be written as fG = KG.ν, where KG is a symmetrical matrix, called

geometric stiffness

This is the first approximation of the stiffness matrix. With a closer approximation of higher

order or more refined, we can get the consistent geometric stiffness.

The static equilibrium equation is: K.ν - KG .ν = P

For the static stability or considering the bucking problem, we have: K.ν - KG.ν = 0, which

constitute a problem of eigen values: K – KG = 0

Solving this equation, we obtain the values: l1, l2, l3, ………, lN, that satisfy this equation:

K – l KG = 0, and represent the safety factors for the different buckling modes. The

significant value is the first l1.


© 2002, 2003 OSCAR MUROY 50

10. EFFECTS OF TEMPERATURE VARIATION, SHRINKAGE SHORTENING AND

SUPPORT DISPLACEMENTS

Besides the actions of self-weight, dead loads and the traffic surcharges, we must study the

effects of temperature variation, in relation to temperature during construction; in the case of

concrete arches: shrinkage shortening and creep; and the support displacements due to soil

settlements.

a) With the temperature variation, we have a restrained expansion or shortening of the

structure, because the supports do not allow a free movement, causing, therefore, tension

or compression forces and bending moments in the arch.

Fig. 10.1

Then in an arch with a span l, between supports, for a temperature variation t, we would

have a free expansion of: l = .t.l, or in its orthogonal components:

lx = .t.lx

ly = .t.h

And the bending moment :

M = Ma – H.y – V.x

Being Ma, H and V, the bending moment, horizontal thrust and vertical reaction at the

springing. Ma and Mb are 0 for the two-hinged arch and the vertical reactions V are 0 for

arch supports at the same level.

b) For the shrinkage shortening in concrete arches, we have a restrained contraction, causing

therefore, traction forces in the arch.

Free contraction: l = sh.l, or in its orthogonal components

lx = sh.lx

ly = sh.h

And the bending moment:

M = Ma – H.y – V.x

c) In the case of support displacements, we would get the possible settlements:

x, y y

And the bending moment:

M = Ma – H.y – V.x


© 2002, 2003 OSCAR MUROY 51

d) The creep or plastic flow of concrete, produces a gradual increase in the deflections of the

arch axis, without stress variation. The deflection at the end of the plastic flow is estimated

by the formulae

1

of

EE , being Eo, the initial modulus of Elasticity and Ef, the final

modulus of Elasticity. For a normal condition climate and the concrete preparation, =2.0.

Plastic flow will develop in a period of one to two years

Fig. 10.2 Built-in Arch

-140

-120

-100

-80

-60

-40

-20

0

20

40

60

80

100

0 5 10 15 20 25 30 35 40 45 50 55 60 65

Mz (T

.m

)

X (m)

BENDING MOMENTS DUE TO TEMP, DX and DY

Mz-temp

Mz-dx

Mz-dy

-16

-14

-12

-10

-8

-6

-4

-2

0

2

4

6

8

10

12

14

0 5 10 15 20 25 30 35 40 45 50 55 60 65

Fa (T

)

X (m)

AXIAL FORCES DUE TO TEMP, DX and DY

Fa-temp

Fa-dx

Fa-dy

-10

-8

-6

-4

-2

0

2

4

6

8

10

0 5 10 15 20 25 30 35 40 45 50 55 60 65

Fc (T

)

X (m)

SHEARING FORCES DUE TO TEMP, DX and DY

Fc-temp

Fc-dx

Fc-dy


© 2002, 2003 OSCAR MUROY 52

Fig. 10.3 Two-hinged Arch

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

0 4.6 9.2 13.8 18.4 23 27.6 32.2 36.8 41.4 46 50.6 55.2 59.8 64.4 69 73.6 78.2 82.8 87.4 92

Fa (T

)

X (m)

AXIAL FORCES DUE TO TEMP, DX and DY

Fa-temp

Fa-dx

Fa-dy

-30

-25

-20

-15

-10

-5

0

5

10

15

20

25

30

0 4.6 9.2 13.8 18.4 23 27.6 32.2 36.8 41.4 46 50.6 55.2 59.8 64.4 69 73.6 78.2 82.8 87.4 92

Mz (T

.m

)

X (m)

BENDING MOMENTS DUE TO TEMP, DX and DY

Mz-temp

Mz-dx

Mz-dy

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0 4.6 9.2 13.8 18.4 23 27.6 32.2 36.8 41.4 46 50.6 55.2 59.8 64.4 69 73.6 78.2 82.8 87.4 92

Fc (T

)

X (m)

SHEARING FORCES DUE TO TEMP, DX and DY

Fc-temp

Fc-dx

Fc-dy


© 2002, 2003 OSCAR MUROY 53

11. ARCH GEOMETRY CONTROL DURING CONSTRUCTION

As much as for the correct set-out of the arch geometry and to verify the structural behaviour

of the arch, it should be exercised strict control of the arch geometry all through the main

stages of the construction.

a-1) During the arch construction, immediately after pouring the concrete, when it is

constructed with complete false work or at every stage of arch construction, when it is

done by stages:

Fig. 11.1

yp = yc + vp

being :

yc, false work geometry of the arch.

vp, elastic deflection due to arch self-weight

yp, arch geometry, due to self-weight or deformed shape due to self-

weight

a-2) In the construction method by cantilevering-out, the cycle of operations that is repeated

shall be:

1. Cable tensioning to withstand the traveller weight in its new position.

2. Displacement of the traveller to its new position

3. Formwork, fixing the reinforcing bars, and pouring of concrete in its new position.

4. Cable tensioning, after the construction of the new stage.

5. Loosening of traveller to move to new position.

At stage i At stage i+1

Cable

Tensioning

Arch

Deflection

Cable

Tensioning

Arch

Deflection

Ti1 v

i1 T

i+11 v

i+11

Ti2 v

i2 T

i+12 v

i+12

Ti3 v

i3 T

i+13 v

i+13

.... .... .... ....

Tin-1 v

in-1 T

i+1n-1 v

i+1n-1

Ti+1

n vi+1

n


© 2002, 2003 OSCAR MUROY 54

being :

vij, elastic deflection in the abscisa j, with the arch and the traveller

up to stage i

vi+1

j, elastic deflection in the abscisa j, with the arch and the traveller

up to stage i + 1

Tij, tension at the cable j, up to stage i

b) After the whole bridge construction:

Fig. 11.2

ym = yp + vm + i.vp

being :

vm, elastic deflection due to dead load

i , plastic flow coefficient due to self-weight, in this stage.

ym, arch geometry, with self-weight and dead load or deformed shape

due to self-weight and dead load.

c) After bridge construction and when concrete plastic flow has concluded:

ym, = yp + (1+).vm + .vp

being :

, final plastic flow coefficient (=2.0)

ym,, arch geometry, with self-weight, dead load after conclusion of

plastic flow

d) During load test or with surcharge:

Fig. 11.3

yf = ym, + vt

being :


© 2002, 2003 OSCAR MUROY 55

vt, elastic deflection due to surcharge

yf, arch geometry, under test truck or surcharge load or deformed

shape under surcharge


© 2002, 2003 OSCAR MUROY 56

12. BIBLIOGRAPHY

1) Theory of Structures, S. Timoshenko, Mc Graw Hill, 1945

2) Resistencia de Materiales, J. Courbon, Aguilar, 1968

3) Análisis de Estructuras Indeterminadas, J. S. Kinney, CECSA, 1963

4) Specifications of Highway Bridges, Japan Road Association, 1984

5) The Heads of the Valley Road, A. S. Coombs y L.W. Hinch, Proceedings of The

Institution of Civil Engineers, Oct. 1969

6) Concrete Bridges, A. C. Liebenberg, Longman Scientific and Technical, 1992

7) Theorie und Berechnung der Stahlbrücken, A. Hawranek y O. Steinhardt, Springer

Verlag, 1958

8) Estructuras de Hormigón Armado, Tomo VI, Bases para la construcción de puentes

monolíticos, F. Leonhardt, El Ateneo, 1992 (orig. 1979)

9) Theory of Elastic Stability, S. Timoshenko, Mc Graw Hill, 1961

10) Structural Steel Designer’s Handbook, R. Brockenbrough y F. Merritt, Mc Graw Hill,

1999

11) Bridges/Brücken, F. Leonhardt, Deutsche Verlags-Anstalt, 1994

12) Bridges in Japan, Dobuku Gakkai, several years

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