TRANSPORT AND ROAD RESEARCH LABORATORY Department of Transport

RESEARCH REPORT 124

A MECHANISM PROGRAM FOR COMPUTING THE

STRENGTH OF M A S O N R Y ARCH BRIDGES

by M A Crisfield (TRRL) and A J Packham (British Rail Research)

The views expressed in this Report are not necessarily those of the Department of Transport.

Structural Analysis Unit Structures Group Transport and Road Research Laboratory Crowthorne, Berkshire, RG11 6AU 1987.

ISSN 0266-5247

Ownership of the Transport Research Laboratory was transferred from the Department of Transport to a subsidiary of the Transport Research Foundation on 1 st April 1996.

This report has been reproduced by permission of the Controller of HMSO. Extracts from the text may be reproduced, except for commercial purposes, provided the source is acknowledged.

CONTENTS

Abstract

1. Introduction

2. A modified mechanism method

2.1 The virtual work and equilibrium equations

2.2 Distribution of load through the fill

2.3 Resistance due to lateral earth pressure

2.4 The thrust-line and off-set loading

3. Discussion of the assumptions

3.1 Infinite Young's modulus and no effect of geometric non-linearity

3.2 Rigid abutments

3.3 No tensile strength

3.4 Unit thickness of arch

3.5 Assumptions relating to the lateral earth pressure

3.6 The assumption of 'no-backing'

3.7 'Yielding' at the hinges

4. Future developments

5. Applications

5.1 Problems with the thrust-line lying outside the arch

5.2 The results for the full-scale tests

5.2.1 Bargower bridge

5.2.2 Preston bridge

5.2.3 Prestwood bridge

5.2.4 Bridgemill bridge

5.2.5 Torksey bridge

5.2.6 The MEXE solutions

5.3 Solutions for a small-scale bridge

Page

1

1

1

9

11

12

12

12

12

12

12

12

6. Some comments on the use of scaled models

7. Conclusions

8. Acknowledgements

9. References

10. Appendix A: Lateral earth pressure

© CROWN COPYRIGHT 1985 Extracts from the text may be reproduced,

except for commercial purposes, provided the source is acknowledged.

° ,

14

16

17

17

18

A MECHANISM PROGRAM FOR COMPUTING THE STRENGTH OF M A S O N R Y ARCH BRIDGES

ABSTRACT

The Report describes the theory governing a computer program which uses a 'mechanism method' to compute the collapse loads for masonry arch bridges. The program is used to analyse five arch- bridges recently tested to collapse as part of TRRL's programme of research.

1. INTRODUCTION

About forty per cent of the bridge-stock in the UK consists of brick and stone arch-bridges. Current methods of assessment (Department of Transport 1984a, b) are largely based on the Military Engineering Experimental Establishment (MEXE) monograph. The latter was mainly derived from a permissible stress approach involving an elastic analysis of the arch (Pippard 1952, Pippard and Baker 1957). Because of the need to reassess many arch- bridges and in view of the possible limitations of the MEXE method, TRRL has undertaken a programme of research into masonry arch-bridges. This has involved both full-scale and model testing as well as theoretical analysis. The latter has used both the non-linear finite element method (Crisfield 1984, 1985a, b, Crisfield and Wills 1986) and a mechanism technique (Crisfield 1985a, b) that is closely related to both Heymans (1982) plastic methods and a mechanism procedure due to Pippard and co-workers (1952, 1957). Other work on mechanism methods is due to Davies (1985) and Harvey and Smith (1985).

The present Report gives a more detailed description of the theory and describes the application of the computer program to the analysis of a number of masonry-arch bridges (Hendry et al 1985, 1986, Page 1987, Page and Grainger 1987). In comparison with the original computer program, the following enhancements have been introduced:

(a) An allowance for material damage at the hinges by taking account of the compressive strength of the arch material

(b) By reading-in surveyed co-ordinates and using a curve-fitting technique, the program can be applied to the analysis of bridges of distorted shape. In addition, segmental, parabolic or elliptical arches can be automatically defined.

(c) Plotting facilities have been introduced so that the data can be simply checked and the collapse mode can be easily visualised. The plots show both the hinge positions and the line-of-thrust. The latter provides an important visual check on the validity of the solution.

(d) All four hinge positions (including the 'load

hinge' B, Figure 1) can be ' iterated' so that the program will give the minimum collapse load of the bridge with the load placed at the most vulnerable position.

2 A MODIFIED MECHANISM METHOD

It is usually assumed (Heyman 1982) that:

- - the E value of the arch-ring is infinite

-- the compressive strength of the arch-ring is infinite

- - the tensile strength of the arch-ring is zero

- - the abutments are rigid

- - there is no sliding between voussoirs

-- only a unit thickness of arch is analysed

-- the fill contributes weight but no other structural action

-- the spandrel walls do not contribute to the strength

Four hinges are then postulated (Figure 1, but with hinges A and C lying on the intrados and hinges B and D lying on the extrados) and the collapse load, P, can be computed from statics. In relation to the plastic theorems this gives an 'upper bound' collapse load. However, by investigating all the possible hinge configurations, one can obtain the lowest of the 'upper bound" solutions. The resulting thrust-line can be plotted and should not pass outside the arch. In these circumstances, plasticity theory can be used to argue that we have a 'safe' or ' lower bound' load because (Heyman, 1982) 'If a thrust line can be found, for the complete arch, which is in equilibrium with the external loading (including self weight), and which lies everywhere within the masonry of the arch ring, then the arch is safe'. Later in the Report, the terms 'upper bound' and ' lower bound' wil l be used in a looser engineering context that does not relate to plasticity theory.

In the remainder of this Section, we will describe in detail how the mechanism procedure has been implemented and the ways in which it has been modified from its basic form. The assumptions, on which the method is based, wil l be discussed in Section 3.

2.1 THE V I R T U A L W O R K A N D E Q U I L I B R I U M E Q U A T I O N S

In contrast to most previous mechanism or plastic procedures (Heyman 1982), the arch-ring is assumed

I Block 2

H

a Vd Fig.1 Mechanism with equilibrating forces

to have a finite compressive strength so that the hinge points A - D lie inside the arch as illustrated in Figure 1. This figure is similar to one given by Pippard (1952) and Pippard and Baker (1957) although, in contrast to the latter, stress-blocks associated with 'yielding' of the arch material are now included at the hinges. Consequently, al lowance is made for the compressive strength of the bridge via 'yielding' which implies infinite ductility. In reality, such ducti l i ty is not exhibited by brick and stone work. Hence, the adopted compressive strength, oy, should be lower than the full crushing strength (Department of Transport 1984a). In addition, it should reflect the strength of the stone or brick- work, which includes the mortar, rather than the signif icantly higher strength of the parent material on its own.

In order to compute the collapse load, virtual work is adopted so that, in relation to Figure 2:

l da2y 1 2 P&P + VlA1 = V2A2 + V3~3 + HfA h + + ~. dbGy

2 1 2 (1 +A02)+ dc Gy (A02 "I-L~03) +'~'ddGyA03 . . . . . (1)

where: A~-&3 are the virtual displacements undertaken by the centroids of the dead weight of blocks 1 -3 respectively. The latter include the weight of both the arch and associated fill (see Figure 1).

Ae~-A83 are virtual rotations

d~-dd are the depths of the compressive 'yield- blocks'. In a traditional mechanism procedure, these are assumed to be zero. In the present implementat ion, the rigid-blocks 1 -3 (Figure 1) are assumed to rotate about hinge points lying on the inside of the compressive stress-blocks (Figure 2). Details on the computat ion of da-dd will be given later.

Hf is the total horizontal force assumed to be provided by the "passive resistance' of the fill. Details are given in Sections 2.3 and 10.0.

The relationships between A1-~3, Ah and A(~1-An3 are found from basic kinematics. In concept, equation (1) is applied for the complete range of possible hinge positions, until the minimum collapse load, Pro=n, is obtained. To this end, the arch barrel is divided into a set of 'elements' (Figure 3) which should strictly coincide with the stone voussoirs if hinges are only to occur at the mortar joints. However, this would clearly be impractical for brickwork-arches and current practise is to provide a large number (say 50) of such elements. For a given set of hinge positions, the weights of the blocks (V~-V3) are simply computed by summing the weights of the .included elements in conjunction with their associated vertical strips of fill.

For each trial hinge configuration, equation (1) is firstly applied with the depths d~-dd being zero. Having obtained the collapse load, P, equilibrium can be used to find the reactions, H, Va, V~ (Figure 1) and, by resolving tangentially at the hinge positions, da-d~ can be found. Hence, from Figures 1-3, for hinge A:

(~yd a = Vasin~ a + (H + Hf) cos~. a . . . . . . . . . . . . . . (2)

while for hinge B:

Gyd b = (V a - Vl) + (H + Hf) coso~ b . . . . . . . . . . . . . . (3)

for hinge C,

cydc = (Vd - V3) sin~c + (H + Hf) cOSec . . . . . . . . . (4)

and, at hinge D:

(~ydd = Vd sin(zd + Hcos(~d . . . . . . . . . . . . . . . . . . . . (5)

Having obtained da-dd, equation (1) is re-applied and a new collapse load is computed. This process is then repeated in an iterative fashion until after 2 -3 iterations, there is no further effective change in da-dd. The program then proceeds to change the hinge positions in a search for a lower load. A 'logic circuit' is used to minimise the number of trial

P

Instantaneous centre of rotation

v 2

d

a Fig.2 Mechanism with virtual deformations

D

Fig.3 Arch divided into elements

configurations. This circuit uses the gradient of the computed collapse load in relation to the hinge position. A further enhancement involves the use of da-dd from the previous hinge configuration as the starting point for the 'inner loop'.

2.2 D ISTRIBUTION OF LOAD T H R O U G H THE FILL

By the time the vertical stresses have spread through the surfacing and fill and reached the arch ring, they will act over a wider area than the area at the surface. If allowance is made for such a distribution, the load P (Figure 4) can be replaced by two loads P~ and P2 (approximately P/2 each) acting on either side of the hinge (Figure 4). It is clear (Figure 4) that the work done by P1 and P2 will be less than that done by the single load P. Therefore, if load- distribution is accounted for, the collapse load will be increased.

The program allows two options for the distribution of load through the fill (Figure 5). The first procedure

(Figure 5(a)) involves a uniform pressure applied over a horizontal line at the level of intersection with the arch directly under the load. The second procedure, which has been used for all the solutions given in this Report, involves a linearly varying pressure (Figure 5(b)) between the two points 1 and 2 (Figure 5(b)) at which the assumed 'spread' intersects the arch. The actual pressures at points 1 and 2 are assumed to be inversely proportional to the depth, h. The 'spread' may be taken to the extrados (as in most of the computations in Section 5) or to the centre-line of the arch (as in the 'upper-bound' computations in Section 5). If the distribution 'fan' takes part of the load beyond the abutment, this portion of load is omitted from the collapse calculations.

Earlier work (Crisfield 1985a, b) gives an example showing how such distribution may affect the computed collapse loads. For the particular example given, a value of e (Figure 5) of 1.0, lead to a 30 per cent increase in the collapse load. Further examples are given in Section 5.

P2

/ /

A

% B

\

Fig.4 The effect of load distribution

2.3 R E S I S T A N C E DUE TO LATERAL EARTH PRESSURE

It has been shown (Crisfield 1985a, b) that, for deep arches, unless some horizontal forces are applied to the arch above the springing remote from the load (Hf, Figure 2), the computed strengths can be • absurdly low. Such horizontal forces are assumed to be generated by the movement of the arch into the

fill or 'backing' (Figure 6). Alternatively, the effective springing can be assumed to rise above the real abutment (see Section 3.6). Because the presence or nature of such 'backing' is difficult to assess, it is considered safer to include an approximate allowance for the work done as the arch segment 3 (between C and D, Figures 1 and 3) moves into the fill. This is achieved by using the concept of 'passive resistance' (Pippard and Baker 1957) to introduce horizontal

w + 20Lh " ~

(a) Horizontal line distribution

(b) Distribution to inclined plane

Fig.5 Load distribution

4

stresses that are kn times the vertical (gravity) stresses. The factor k~ is given by:

k 1 + sin~ (6) P 1 -s in~

where ~ is the angle of friction. Cohesive effects are neglected as are 'active' and 'at-rest' soil pressures on blocks 1 (A-B) and 2 (B-C) in Figures 1 and 2. The latter are considered small in relation to other forces. The force Hf is found by summing the forces on the elements of block 3 and is assumed to act at the centre of pressure. Full details and a discussion of the adopted assumptions are given in Appendix A while the issue of 'backing' is discussed further in Section 3.6.

be repeated with the hinge B being moved to X and the load being offset to the left of the load-hinge (Figure 7(c)). A 'pre-processing' plot with only the loads and 'elements' (Crisfield and Packham 1987) will help to establish that the new load-position is correctly located in relation to the new 'load-hinge'.

For elliptical arches, once allowance is made for lateral-pressure, the thrust-l ine may again be found to pass outside the arch (Figures 8(a), (e)). This inconsistency has been found to be related to an outward horizontal reaction at the springing remote from the load. Ways to overcome this problem wil l be discussed in Section 5.1.

2.4 THE THRUST-LINE AND OFF-SET LOADING

The plotting sub-routines use the computed hinge positions and reactions in conjunction with the external dead and live loads to plot the thrust-line (Figure 7). If passive pressure-resistance is included, the resulting horizontal forces are allowed for in the computation of the thrust-line. The latter is derived directly from equilibrium. If the compressive 'yield- stress' is non-zero, the thrust-line passes through the centres of the compression-blocks (see Figure 7(d)).

The thrust-line provides an important visual check on the solution. If it is found to pass outside the arch (Figure 7(b)), it is likely that the computer program has not found the true minimum load. This can occur because the true minimum load may involve the second hinge being displaced from the load position (Figure 7(c)). The standard analysis does not allow for this occurrence because, in general, the second hinge will be under the load (Figures 7(a)(d), 8-11). If an analysis results in a thrust-line such as that shown in Figure 7(b), it is easy to deduce that the 'load-hinge' should be displaced from the load (to a new point X in Figure 7(b)). The analysis can then

3 DISCUSSION OF THE ASSUMPTIONS

Most of the assumptions have already been mentioned. Some of them will now be amplified.

3.1 INFINITE Y O U N G ' S M O D U L U S A N D NO EFFECTS OF GEOMETRIC NON-L INEARITY

No account is taken of geometric non-linearities. This may be of particular concern in the analysis of shallow arches where there is some possibility of a 'snap-through' type of behaviour. Finite element studies (Crisfield 1985a, b Crisfield ~ Wills 1986) indicate that such effects may be significant. In this context, the appropriate Young's modulus is not that of the stone or bricks alone but rather that of the combined stone (or br ick) /mortar assembly. Some of the experimental results (Hendry et al 1985, Page and Grainger 1987) appear to indicate that, in such circumstances, the failure load can be reached before the full development of all four hinges. Consequently, the mechanism program may err on the unsafe side

. . . . . . . . . . . . : . . . ; . . . . : . . . - : . . . . . . . . . . . . . . . . . . . . . . . . • • • • • * • o e . o 0 o ° . . ° • . ° . o * . .

• . • . • • • • . , • o o ° ° ° o ° ° ~ = ° • . . = ° ° • • ° ° " • . . • . . . ' o . . o

• " ." •" "• "" ." • " • " " " ' ' " • " Ordinary fill . . ' . . - .'.: ~." • • ° °

• " . ° °

: . - - . - . . . . : . . . . . • .

• . • • .

- . . . . " . . - . . - Arch ring

Stone/masonry backing

Fig.6 Masonry arch with 'backing'

(a) Upper bound:O'y = 1000N/ram2; kp = 4.6

~_~Collapse load = 0.1 (N/mm width) i

(b) Lower bound, A:ery = 4N/ram2; kp = 0.01

~_~TCollapse load = 0.01 (N/ram width)

17.2%

(c) Lower bound, B:Oy = 4N/mm2; kp = 0.01

k•7Col lapse load = 662.6 (N/ram width)

(d) "Best estimate", A:Oy = 4N/mm2; kp = 3.0

Fig. 7 Mechanism solutions

for shallow arches. This deficiency could possibly be compensated for by using an artif icially low value for the compressive 'yield strength' of the arch material when analysing shallow arches.

3.2 R IGID A B U T M E N T S It has been assumed that the abutments are completely rigid. Particularly for shallow arches, this assumption is likely to be unsafe. For these arches, outward movement of the abutments may well induce the sort of 'snap-through' behaviour described in the previous Section. Franciosi (1986) has considered such effects in a simple 'model ' involving a rigid arch-ring and elastic movements at the abutments. In practice, the 'elasticity' in the ring itself is also likely to lead to a lowering of the collapse load.

3.3 NO TENSILE S T R E N G T H As a result of this assumption, the program errs on the safe-side. Experimental results by Pippard (1952, 1957) showed that the substitut ion of a weak lime-

for Bargower Bridge

mortar with stronger cement-mortar led to a significant increase in strength. Finite element results (Crisfield 1985a) have led to a similar conclusion. However, once allowance is made for tensile strength and the subsequent 'softening', the collapse load will be 'brittle' and will involve a sudden drop in load. In contrast, when the tensile strength is very low, the theoretical (and, in at least some cases, the experimental) load/deflection responses are 'ductile' and exhibit a significant plateau. Realistically, an arch will be subjected to cyclic loading prior to any dramatic overload. This will be likely to break down the tensile resistance. For these reasons, it is considered reasonable to neglect the tensile strength.

3.4 UNIT TH ICKNESS OF A R C H The calculations are based on a unit thickness of arch-ring so that no account is taken of the effect of the spandrel walls in stiffening and strengthening the arch. These walls are assumed to retain the fill transversely in a state of plain-strain but to serve no

6

)

0.0%

(a) Upper bound, A:Oy = 1000N/mm2; kp = 4.6 (b) Upper bound, B:Oy = 1000N/mm2; kp (reduced) = 2.7

Collapse load = 1616.9 (N/ram width)

(c) Upper bound, C: Oy =1000N/ram2;

kp = 4.6 (restricted Hinge 4)

~ llapse load = 26.9 (N/mm width)

5.7% (d) Lower bound,O'y= 4N/ram2; kp = 0.01

~ llapse load = 218.9 (N/mm width)

(e) "Best estimate", A :Oy = 4N/mm2; kp = 4.0

7.8%

(f) "Best estimate", B:O'y = 4N/ram2; kp (reduced) = 2.13

llapse load = 295.9 (N/mm width)

(g) "Best estimate", C:Oy = 4N/ram2;

kp (reduced) = 3.0 (restricted Hinge 4)

Fig. 8 Mechanism so lu t ions fo r br idge at P r e s t o n - u p o n - t h e - W e a l d m o o r s

other strengthening role. Clearly this is not true of structures in good condit ion where the arch barrel and spandrel walls are firmly bonded together. In practise, however, the arch barrel may well be detached or at least partially detached from the spandrel walls (Hendry et al 1986, Page and Grainger 1987). It is safer for assessment purposes to assume that such dislocation has occurred. For arches in

w h i c h the spandrels are clearly wel l-bonded to the barrel, an al lowance could be made for the extra strength by artificially increasing the value of the factor kp (see Section 2.3) for the passive fill resistance.

For real wheel Ioadings rather than hypothetical line- loads, a further diff iculty relates to the transverse distribution. If the stiffening effects of the spandrel walls are neglected, an approach similar to that of Section 2.2 could be applied to the transverse as well as longitudinal directions.

3.5 A S S U M P T I O N S RELATING TO THE LATERAL E A R T H PRESSURE

Section 2.3 has described a simple approximate method for including the lateral resistance in the mechanism calculations. A more detailed study is given in Appendix A. This study indicates that the present method may overestimate the resistance by as much as 25 per cent in some cases. It is possible that the more complex calculations may be included at a later stage. In the meantime, it is assumed that neglected strengthening effects (no-tension, unit thickness of arch, 'no-backing') can compensate for any overestimation of the lateral resistance. These effects are also assumed to compensate for the small destabilising effects of 'active' and 'at-rest' pressures which are ignored.

It will be shown in Section 5 that, for elliptical arches, the adopted technique for handling 'lateral pressure' may lead to an outward horizontal reaction at the springing remote from the load. In these circumstances, an option is included in the program (see Sect ion 5.1) for automatically reducing the passive pressure factor, kp.

3.6 THE A S S U M P T I O N OF "NO-BACKING"

It is assumed that the arch structure consists solely of the arch ring wi th the only other strengthening effects being due to the we igh t and lateral resistance of the soil fill. In reality, many arches have considerable 'backing' (Rankine 1898, Howe 1897, Alexander E~ Thompson 1900, 1916, Page 1987) provided in their construct ion (Figure 6). This 'backing' may consist of properly laid bricks or stones in some cases being bonded wi th mortar. Rankine (1898) wrote 'The backing of an arch consists of block-in-course, coursed rubble, or random rubble and sometimes of concrete' . Strong

8

'backing' is particularly likely for semi-circular or elliptical arches which would probably otherwise have been very unstable during construction. Rankine (1898) also wrote 'many semi-elliptical arches may be treated as approximately hydrostatic arches' as well as 'To give the greatest possible security to a hydrostatic arch, especially if the span is great compared with the rise, the backing ought to be built of solid rubble masonry up to the level of the crown of the extrados'. Only that portion of the arch above the 'joints of rupture' was treated as the real arch and analysed with the aid of the middle-third rule. Further 'it is below these joints that conjugate (horizontal) pressure from without is required to sustain the arch, and that consequently the backing must be built with squared side-joints'. Unfortunately the procedure for computing these 'joints of rupture' is a little obscure. However, for 'most circular arches, the angles of rupture (which relates to a line normal to the arch ring at the joint of rupture) lie between 44 ° and 55 ° to the horizontal'. Unfortunately, in practise, we cannot usually be sure that circular arch bridges have been constructed in such a manner and that solid backing has been provided below this level so that the effective arch may be considered as springing from these 'joints'.

However, if it is known that good backing exists, it may be reasonable to reduce the effective span of the arch. This can be achieved computationally by preventing the hinge, remote from the load (i.e. D in Figures 1-3) from forming below the backing level (see Section 5). Another analytical device would involve maintaining the full arch with no restrictions on the hinge positions but increasing the passive pressure factor (Sections 2.3, 3.5 and 10.0).

3.7 "YIELDING" AT THE HINGES We have already commented (Section 2.1) on the need to be conservative in estimating the compressive yield strength, oy, because of the limited ductil ity of the stone or brick-work. At the end hinges, the line of thrust may not lie in the tangential direction as has been assumed in the derivation of equations (2)-(5). However, the analysis assumes that the yielding can only occur in the tangential direction and that, in other directions, the arch is infinitely strong. At the abutments, any overestimation of strength resulting from this assumption is likely to be compensated by the extra strength of the material in bi-axial compression.

4 FUTURE D E V E L O P M E N T S

At present the program can only handle a single load. In the future, it is hoped to introduce extensions to enable the treatment of multiple axles. In concept, this is easily achieved by generalising equation (1) to give:

1 2 1 2 ~'L--'PLAP+ x '~PD~P=~ "da ~ Y + z db Oy(1 +&e2)+

1 2 1 2 ~ 'd c Cy (Ae 2 + Ae3)+ ~" ddayAe 3 . . . . . . . . . . . . . (6)

where the summations are performed over all the elements, Ap are virtual displacements at the centres of these elements, Po are the equivalent dead loads and PL the equivalent fixed live loads. The factor represents the amplification (or load factor) on the live loads, PL and the objective is to find Zr~n. In practise, It would be difficult to provide an efficient algorithm based on equation (6) which included the automatic computation of the worst loading position, particularly if the axles were widely spaced. It would probably be better to adopt a semi-interactive approach whereby the user iterated the axle positions in order to produce the minimum collapse load. To this end, the plotted thrust-line would be of considerable help.

An easy development would involve the introduction of a slope to the road surface which would alter the dead loading on the arch 'elements'. This facility would have been valuable when analysing Prestwood Bridge (Page 1987) which had a noticeably sloping surface. In addition, the size of the program could be reduced and its efficiency improved. The former would be necessary in order to run the program on a micro. There is plenty of scope for improvement. Finally, following further parametric studies, more advice should be given on the use of the program in an assessment environment. In particular, thought must be given to a 'load factor' as well, possibly, as a 'geometric factor' (Heyman 1982) for reducing the effective depth of the arch ring.

5 APPLICATIONS

The program has been used to compute the collapse strength of a range of bridges tested in TRRL's recent program of research (Hendry et al 1985, 1986, Page 1987, Page and Grainger 1987). Before detailing these solutions, it should be emphasised that there are many factors that can significantly influence the experimental results. In particular, the direct contribution of the spandrel walls can either be large or negligible. In practise, a tool such as the current mechanism program, would be used to compute realistic but safe lower bounds. It is rather a different matter to predict the results from an experiment. The material and other factors used in the following comparisons should not be taken as a guide-line for future assessment purposes but rather as a first attempt to investigate the possibilities of the computer program.

Figures 7-11 show the plotted mechanism solutions for which the arch-dimensions, properties and results are summarised in Table 1. Before discussing the overall trend of the results, we will consider some difficulties that were encountered with the semi- elliptical 'Preston-arch' (Page 1987).

5.1 PROBLEMS WITH THE THRUST- LINE LYING OUTSIDE THE ARCH

We have already discussed one instance in which the

thrust-line may lie outside the arch (Section 2.4 and Figure 7(b)) and have indicated (Section 2.4 and Figure 7(c)) how the problem may be overcome. The only other instances where the thrust-line has been found to lie seriously outside the arch were encountered with the semi-elliptical 'Preston-arch' (Page 1987). In addition they only occurred once allowance was made for the passive resistance of the fill. Examples are given in Figures 8(a) and 8(e). Clearly these thrust-lines are unsatisfactory. Clues were obtained from the right-hand horizontal reaction. The solution plotted in Figure 8(a) involved a negative reaction indicating an outward rather than an inward external force at the springing so that the thrust-line is, in the last element, entering the fill before returning to the hinge. Hence, the lateral- pressure (Sections 2.3, 3.5 and Appendix A) would appear to be overestimated. A simple strategy to overcome this problem involves retaining the shape of the lateral pressure Figure A - l ) but reducing automatical ly the passive pressure-factor, kp, until the longitudinal reaction is zero. Details of the required input data are given by Crisfield and Packham (1987). In these circumstances, the solution of Figures 8(a) is changed to that of Figure 8(b). The latter solution involves a signif icantly lower collapse load and a thrust-line that (almost) lies within the arch.

The 'best-estimate' solution (see Section 5.2) shown in Figure 8(e), did not involve a negative reaction although the thrust-l ine lay well outside the arch. However, during the search for the hinge- configuration giving the minimum load, solutions were obtained in which the RHS reaction was negative and hence the message in Figure 8(e). There are other cases (ie Figure 7(d)) in which a similar warning message is output and yet the solution is perfectly satisfactory. Consequently, the message should only be of concern when the thrust- line indicates an unsatisfactory solution.

It has already been pointed-out in Section 3.6 that semi-elliptical arches are likely to have substantial 'backing'. For the current 'Preston-arch' (Page 1987), this backing consisted of sand-stone to the level indicated in Figure 8. As a result of this 'strong- backing', it might be reasonable to re-analyse the Preston bridge with the RHS hinge being lifted above the springing but below the top of the strong backing (see Crisfield and Packham 1987 for details of the required data). If, as part of an 'upper-bound' solution, this hinge is set to node 41 (of the 51 nodes), which is near the top of the 'backing', the computer program gives no solution. This implies that there is no valid 'three-bar mechanism' (Figure 4) and hence the 'mechanism strength' is infinite.* On the other hand, if the RHS hinge is constrained not to be below node 45, which is approximately half-way up the 'backing' (Figures 8(c)

* F o o t n o t e : This is certainly true of an arch with infinite compressive strength. For an arch with finite compressive strength, it is possible that the computer program abandons its search prematurely and that a valid solution with hinges within the arch may exist.

Plate 1 The five bridges

(a) Bargower Bridge

(b) Bridgemill Bridge (c) Preston Bridge

(d) Prestwood Bridge

i

(e) Torksey Bridge

10

and (g)), valid solutions are obtained. The 'extended- limits' warnings in these Figures merely implies that there would be a lower solution for different 'restricted' hinge-positions. However, in this instance, the restriction is deliberate.

5.2 THE RESULTS FOR THE FULL- SCALE TESTS

Plates l (a)-(e) show the five bridges for which the dimensions and some properties are summarised in the Table 1. This Table also compares the solutions from the mechanism program with both experimental results (Hendry et al 1985, 1986, Page 1987, Page and Grainger 1987) and a factored 'MEXE' (Department of Transport 1984a, b) assessment. The

latter was obtained by comput ing the maximum servicability axle-load over the width of the bridge and multiplying by an assumed 'factor of safety' of 3.0. In every case, except the narrow Prestwood bridge (Page 1987), this involved two single-axles lying side-by-side.

The values of the passive fill factor, kp, compressive strength in the arch, oy, and load-distribution factor, ~, used in the 'upper bound analyses' were 4.6 (~)=40°), 1 000 N/mm 2 (effectively infinite) and 0.5 respectively. For the ' lower bound solutions', these values were 0.01 (approximately zero)), 4 N /mm 2 and 0 respectively. Values use for the 'best estimate' solutions are given at the bottom of Table 1.

TABLE 1

Results from mechanism pro Iram, destructive tests and the 'MEXE' assessment method

Structure Bargower Bridgemill Preston Prestwood Torksey

Type Span (mm) Rise (mm) Span/rise Ring thickness

(mm) Span/thickness Fill depth at crown (mm)

Width (mm) Approx. shape Densities (kN/m3)

Ring: fill Load posn.*

Fig. nos.

Experimental collapse load (kN)

Program upper bound (kN)

Program lower bound (kN)

Factored (3) MEXE (kN)****

Program 'best estimate' (kN)

Fill factor, kp ~y Distribution

factor,

Stone 10 360 5 180 2.0 558

18.6 1 200

8 680 semi-circ.

2.7:2.1 1/3 7

5 600 13 873

0.10

3 854 (4.4)

5 751:7 726

3.0 4.0:10.0

0.5

Stone 18 290 2 840 6.44 711

25.7 203

8 300 segmental

2 .1 :2 .2 1/4 10

3 100 5 495

1 816

1 089 (8.5)

2 545 :3 640

3.0 4 .0 :10.0

0.5

Stone 5 180 1 636 3.17 360

14.4 380

5 700 semi-ellip.

2 .3 :2 .2 1/3 8

2 100 1 873!: 9 216"*

1 259***

154

1 790 (3.5)

1 248!: 1 761"* 1 686*** 4 .0 :3 .0

4.0

0.5

Brick 6 550 1 428 4.59 220

29.8 165

3 80O distorted

2 .0 :2 .0 1/4 9

228 168

42

53 (12.9)!!

177

3.0 4.0

0.5

Brick 4 900 1154 4.28 343

14.2 241

7 160 segmental

2 .1 :2 .0 1/4 11

1 060 2 668

705

877 (3.6)

1 415

3.0 4.0

0.5

* As approximate ratio of internal span ** Position of hinge D limited to account for affect of 'backing' * * * Solutions with automatically reduced kp * * * * The figures in brackets give the achieved factor of safety in relation to the MEXE assessment ! Thrust-line clearly outside the arch l! Modified MEXE values (Section 5.2.3) because the arch is distorted

11

5.2.1 Bargower bridge This bridge was semi-circular (Figure 7 and Hendry et al, 1986) and attained a collapse load of 5 600 kN. Because the bridge was so deep, the upper and lower bound results show a massive range (Table 1). This is typical of the solutions for deep arches for which the strengths are very dependant on the fill or backing and, in terms of the mechanism program, on the adopted kp value [Crisfield 1985 a, b]. Assuming a compressive strength for the stone-work, oy, of 10 N / m m 2, which is of the order of one-third of the strength of the parent stone (33 N/mm2-- [Hendry et al 1986]), the 'best estimate' solution was 7 726 kN which is 38 per cent higher than the experimental collapse load. However, during the testing, problems were encountered with the application and maintenance of the load at the large deflections that were encountered. As a result, the bridge was cycled at high load several t imes before the maximum load was reached. The final failure was associated with compressive material damage [Hendry et al 1986] and it is possible that the previous cycling reduced the strength. This possibil ity is reflected in an alternative 'best estimate' mechanism solution (see Table 1) which, with oy set to 4 N/ram 2 gave a collapse load (5 751 kN) that was very close to the experimental value.

5.2.2 Preston bridge This bridge was semi-elliptical (Figure 8 and Page, 1987) and attained a collapse load of 2 100 kN. Much discussion on the analysis has already been given (Section 5.1). As a first attempt to al low for its backing, the 'best estimate' solution involved a high

.kp factor of 4.0 (equivalent to $=37° ) . However this solution involved the thrust line of Figure 8(e) that lies outside the arch. A 'best-estimate' solution that was 20 per cent too low was obtained (Figure 8(9)) when the right-hand hinge D was constrained not to form below hinge 45 (half-way up the backing (Figure 8(g)). A similar constraint on the right-hand hinge was required in order to produce an upper- bound solution that lay above the experimental value.

5.2.3 Prestwood bridge This narrow bridge was severely distorted (Figure 9 and Page, 1987) and attained a collapse load of 228 kN. To account for the distortion, a 'curve- f i t t ing' sub-routine [Crisfield and Packham 1987] was used to input the geometry. The MEXE method is not strictly applicable to such distorted bridges. Hence, the MEXE value given in Table 1 for this bridge was obtained by reducing the full MEXE value by a factor of 0.64 which was the reduction ratio which the mechanism-program solutions indicated would result from the distortion when the bridge was loaded at the internal quarter point. The mechanism results given in Table 1 relate to the measured distorted shape (Figure 9). The results relating to a perfect segmental shape were about 56 per cent stronger.

Prestwood was the only bridge for which the experimental collapse load did not lie between the computer upper and lower bounds. It was also the only bridge tested with the line-load spanning across, rather than inside, the spandrel walls. This coupled with narrowness of the bridge, would have led to a more significant contribution from the spandrel walls and could have accounted for the extra (about 36 per cent) strength.

5.2.4 Bridgemill bridge This shallow bridge was approximately segmental (Figure 10 and Hendry et al, 1985) and attained a collapse load of 3 100 kN. The best of the 'best estimate' collapse loads was again obtained with a low compressive strength (4 N/mm2). In this case, the justification relates to the extreme shallowness of the arch. As discussed in Section 3.1, an artificially low compressive strength may provide a compensation for the inability for the mechanism program to include the 'snap-through' effects of gemoetric non-linearity.

5.2.5 Torksey bridge This segmental bridge (Figure 11 and Page and Grainger 1987) attained a collapse load of 1 060 kN. The 'best estimate' mechanism solution was about 33 per cent too large. Experimentally, however, there was some evidence of some form of 'snap-through' behaviour before the formation of all four hinges [Page and Grainger 1987]. For this bridge, the spandrel walls were completely detached from the arch-barrel so that no additional stiffening effect would be expected.

5.2.6 The MEXE solutions Only for the Preston bridge, did the MEXE assessment method, when combined with a factor of safety of 3.0, give a collapse load that lay closer to the experimental value than the mechanism solutions. For both this bridge and the Torksey bridge, the MEXE method indicated factors-of-safety of about 3.5. For the other bridges (see Table 1), the inferred factors of-safety were significantly higher.

5.3 SOLUTIONS FOR A SMALL-SCALE BRIDGE

As part of a joint SERC/TRRL sponsored research programme, a metre-span small-scale bridge (Figure 12) was constructed and tested at Bolton Institute of Higher Education [Walker ~1- Melbourne 1987]. The bridge was made from model bricks and was 500 mm in width with 10 ram-down limestone aggregate being used for the fill. The rise was 500 ram, the arch thickness 53 mm and the depth of fill at the crown was 100 ram. The compressive strength of the brickwork was estimated at 4.0 N/ram 2. From the analysis view-point, the interesting features of this model are that (i) the load was applied directly to the arch-ring rather than onto the fill and (ii) there were

12

no spandrel walls although the fill was constrained from moving laterally.

The experimental collapse load was 4.4 kN. However, following this maximum, the load dropped sharply until it levelled off at a plateau at 3.5 kN. It is assumed that the maximum load corresponded to cracking of the mortar and that the plateau load of 3.5 kN should be treated as the collapse load because it would probably have been the maximum load if there was no tension in the mortar.

If this arch is analysed with allowance for the weight of the fill but with no allowance for any horizontal

resistance, the computed collapse load is (Figure 12(a)) only 0.096 kN or about 3 per cent of the experimental load! However, when allowance is made for horizontal resistance by introducing a passive pressure factor (kp) of 5.83, the computed collapse load increased to 3.34 kN (Figure 12(b)) which is very close to the experimental value. The computed hinge positions (Figure 12(b)) also compared reasonably with the experimental positions. This agreement was obtained when the angle of fr iction, ~, for the fill was set to 45 ° (to give kp = 5.83). The draft report [Walker ~ Melbourne 1987] quotes a very high value of 55 ° which was obtained from a shear-box test. Although there was no direct contribution to the

~ (N/mm width)

o 0 0 %

0.0%

o.1%

(a) Upper bound:Cry= 1000N/mm2; kp = 4.6

Collapse load = 12.5 (N/mm width)

Hinge 1 9.2% 10.3%

(b) Lower bound:Cry= 4N/turn2; kp = 0.01

= .

Hinge 1 13.3%

(c) "Best estimate":Oy= 4N/ram2; kp = 3.0

Fig. 9 Mechanism solutions for Prestwood Bridge

13

strength f rom any spandrel walls, it is possible that the strength of the model was influenced by friction between the fill and the retaining side-walls.

6 SOME COMMENTS ON THE USE OF SCALED MODELS

It was mentioned in Section 5.3 that for the small- scale bridge (Figure 12), the maximum load of 4.4 kN was fol lowed by a drop to a plateau of 3.5 kN. The plateau was assumed to relate to a no-tension mechanism while the maximum load was assumed to

involve tensile cracking of the mortar which has a non-zero tensile strength. Hendry et al [1986] tested a 5.42:1 scale model (s= 5.42) of Bargower bridge which also failed due to tensile failure (in an abutement block) and also gave a load/deflection response that fell sharply beyond the maximum load. It is interesting to compare the experimental and mechanism solutions for both the model and prototype (see Table 2).

When the compressive strength of the ring is set to be infinite, the mechanism program indicates that the full-scale bridge should be 159 times stronger than the model while the experimental ratio was only 62. The figure of 159 is almost exactly s 3 where s is the

1~_ Collapse load = 662.1 (N/mm width)

V- 7

Hinge 1 0.2%

(a) Upper bound:O'y = 1000N/mm2; kp = 4.6

Collapse load = 218.8 (N/ram width)

Hinge 4 ~ , Hinge. 1 27.7% 30.8%

(b) Lowerbound:Oy =4N/mm2; k p = 0 . 0 1

~__~Collapse load = 306.7 (N/mm width)

~ ~ - - ~ H i n g e ~ .in e 2 28.7%

29.80/0

Hinge 1 36.6%

(c) "Best estimate":Oy = 4N/ram2; kp = 3.0

Fig. 10 Mechanism solutions for Bridgemill Bridge

14

Collapse load = 372.6 (N/mm width)

\ /

(a) Upper bound:Oy = 1000N/ram2; kp = 4.6

Collapse load = 98.4 (N/mm width)

/

12.1% 8 . 2 %

(b) Lower bound:Oy = 4N/mm2; kp = 0.01

Collapse load = 197.7 (N/mm width)

\ 7

~ ~ n ~ e 1 8.4% 20%

(c) "Best estimate":Oy = 4N/mm2; kp = 3.0

Fig. 11 Mechanism solutions for Torksey Bridge

15

= .

(a) Without passive pressure

= ,

( b ) W i t h passive pressure

Fig. 12 Mechanism solutions for Bolton model

scaling parameter. This is the ratio that one would expect for-a no-tensile arch with the resistance largely afforded by fr ict ion in the fill. It would therefore appear that the relative strength of the model in relation to the strength of the prototype has been signif icantly increased by the tensile strength of the mortar.

Table 2 shows that when the compressive yield stress, or, in the model was reduced from infinity to

4.0 N/ram 2, the ratio between the strength of the prototype to the strength of the model was, according to the mechanism procedure, reduced from 159.4 to 105.4. Hence the reduced compressive strength has a more damaging effect on the prototype than on the model.

In summary, if scaled-models are to be used to represent the behaviour of prototypes, both the compressive and tensile strengths must be reduced. In particular, very weak mortar should be used for the models. If not, both the observed collapse modes and loads may be unrepresentative of the prototype. Theoretically, the only alternative to providing lower strengths for the models is to provide higher densities. If the densities of both arch and fill are increased by a factor of s, and the same compressive and tensile (zero) strengths are used, the mechanism program will give the same collapse mode for model and prototype with the latter giving a strength that is s 2 higher (see Table 2 for Oy=4 N/mm2).

7 CONCLUSIONS

When analysing four out of the five full-scale bridges, the mechanism method computed upper and lower bound solutions that bracketed the experimental collapse loads. Unfortunately, for a deep semi-circular arch and a semi-elliptical arch, the range between these bounds was very wide. 'Best estimate' solutions lay, with one exception, closer to the experimental results than those obtained from the factored (by 3.0) MEXE method. This exception involved a semi-elliptical arch for which the mechanism program had diff iculty in producing sensible solutions unless the effective right-hand springing was moved up into the 'backing'. In contrast to the MEXE method, the mechanism method can be used to estimate the strength of distorted arches. The program is easy to use and only requires a minute or so of computer t ime (on TRRL's Cyber 720). It could easily be modified to run on micros.

Many factors influence the strength of a masonry arch and it could be fortuitous if a computer program gives reasonable agreement with the experimental

Experiment Mech, oy~

~lech, oy= 10 N /mm 2

Mech, or=4 N/mm 2

rVlech, oy=4 N/mm2

T A B L E 2

Results from mechanism program, model and prototype for Bargower Bridge Colla )se loads in kN

S 3

Full-scale Model ratio = r r

56OO 9686

7726

5751

p m o d = S p p r

90 60.75

58.24

54.54

195.52

62.2 159.4

132.6

105.4

29.4

2.2 1.0

1.2

1.51

5.44--~ s

For mechanism solutions, e = 0 . 5

16

results. Such agreement does not, on its own, ensure that a program may be safely used for assessment. For example, assessment methods should probaby not account directly for the strength of spandrel walls: yet, for at least three of the tested bridges, the strength of the spandrels did appear to influence the results.

The deep semi-circular model-bridge was tested without spandrel walls. When no allowance was made for any horizontal resistance in the fill, the computed collapse strength was only 3 per cent of the experimental value while reasonable agreement was obtained when allowance was made for a passive resistance. It is, however, possible that the experimental strength was influenced by friction between the fill and the retaining side-walls.

Only one of the tested arches was very shallow and this appeared to have firm abutments. This will not always be the case. The mechanism program may be severly limited for such arches (see Section 3.2) and further work on shallow arches is clearly required. Nonetheless, it is felt that the computer program will eventually provide the basis of a useful assessment tool. However, because of the limitations of the theory, its final use for assessment will inevitably involve various empirical factors and much further work is required.

The mechanism program has been used to highlight some of the difficulties associated with the use of scaled models. In particular, it is essential that reduced strengths (particularly for the mortar) be used if the models are to be representative of full- scale bridges.

8 ACKNOWLEDGEMENTS

The work described in this report was carried out in the Structural Analysis Unit of the Structures Group of the TRRL. A significant portion of the work was performed while the second author was on secondment at TRRL from British Rail. Thanks are due to British Rail for participating in this collaborative venture. Thanks are also due to Mr John Page from Bridges Division, TRRL, who is organising the experimental side of TRRL's research work on arches, and has contributed significantly to this Report. In addition, Dr James Cheung of Structural Analysis Unit, TRRL, has made an important contribution.

9 REFERENCES

ALEXANDER, T and THOMPSON, A W (1900). The scientific design of masonry arches with numerous examples. Dublin Univ. Press.

ALEXANDER, T and THOMPSON, A W (1916). The scientific design of masonry arches, Chapt. 12 of Elementary Applied Mechanics. Macmillan.

CRISFIELD, M A (1984). A finite element computer program for the analysis of masonry arches. Department of the Environment Department of Transport TRRL Report LRl115: Transport and Road Research Laboratory, Crowthorne.

CRISFIELD, M A (1985a). Finite element and mechanism methods for the analysis of masonry and brickwork arches. Department of Transport TRRL Report RR19: Transport and Road Research Laboratory, Crowthorne.

CRISFIELD, M A (1985b), Computer methods for the analysis of masonry arches, Proc. 2nd. Int. Conf. on Civil and Structural Engineering Computing, Vol. 2, CiviI-Comp. Press, Edinburgh, pp. 213-220.

CRISFIELD, M A and WILLS, J (1986). Nonlinear analysis of concrete and masonry structures, in: Finite element methods for nonlinear problems, ed. P G Bergan et al., Springer-Verlag, pp. 639-652.

CRISFIELD, M A and PACKHAM, A J (1987). A mechanism program for masonry arches, TRRL Working Paper WP/SA/1/87 Transport ~ Road Research Laboratory, (unpublished paper available on direct personal application only).

DAVIES, S R (1985). The assessment of load carrying capacity of masonry arch bridges, Proc. 2nd. Int. Conf. on Civil and Structural Engineering Computing, Vol. 2, CiviI-Comp. Press, Edinburgh, pp. 203-206.

DEPARTMENT OF TRANSPORT ROADS AND LOCAL TRANSPORT DIRECTORATE (1984a). The assessment of h ighway bridges and structures. Departmental Standard BD 21/84.

DEPARTMENT OF TRANSPORT ROADS AND LOCAL TRANSPORT DIRECTORATE (1984b). The assessment of h ighway bridges and structures. Advice Note BA 16/84.

FRANCIOSI, C (1986). Limit behaviour of masonry arches in the presence of finite displacements, Int. J. Mech. Sci., Vol. 28, No. 7, pp. 463-471.

HARVEY, W. J and SMITH F W (1985). Assessment and design of masonry arch structures using a microcomputer, Proc. 2nd. Int. Conf. on Civil and Structural Engineering Computing, Vol. 2, Civil- Comp. Press, Edinburgh, pp. 207-212.

HENDRY, A W, DAVIES, S R and ROYLES, R (1985). Test on stone masonry arch at Bridgemill- Girvan, Department of Transport TRRL Report CR 7: Transport and Road Research Laboratory, Crowthorne.

HENDRY, A W, DAVIES, S R, ROYLES, R, PONNIAH, D A, FORDE, M C and KOMEYLI- BIRJANDI, F (1986). Test on masonry arch bridge at

17

Bargower, Department of Transport TRRL Report CR26: Transport and Road Research Laboratory, Crowthorne.

HEYMAN, J (1982). The masonry arch, Ellis Horwood Ltd., Chichester, 1982.

HOWE, M A (1987). A treatise on arches, John Wiley 8 Sons, New York; Chapman 8 Hall Ltd., London.

PAGE, J (1987). Load tests to collapse in two arch- bridges at Preston, Shropshire and Prestwood, Staffordshire, Department of Transport TRRL Report RR110: Transport and Road Research Laboratory, Crowthorne.

PAGE, J and GRAINGER, J W (1987), Load test to collapse on a brick arch bridge at Torksey, Working Paper WP/B/134/87, 1987. Transport ~t Road Res. Lab., (Unpublished paper available on direct personal application only).

PIPPARD, A J S (1952). Studies in elastic structures, E Arnold Et Co., London.

PIPPARD, A J S and BAKER, J F (1957). The analysis of engineering structures. E Arnold, Ltd., London.

RANKINE, W J M (1898). A manual for civil engineering. Charles Griffin ~t Co., Ltd., London.

WALKER, P J and MELBOURNE, C (1987). Draft Report to TRRL on the load tests to collapse of three model brickwork masonry arches. Bolton Institute of Higher Education.

10 APPENDIX A: LATERAL EARTH PRESSURE

It can be seen from the simplified mechanism of Figure 2 that, as the arch deforms, work must be done moving the portion CD into the fill. In practice, this fill may partly consist of 'properly laid stones' which can even be mortared. In some circumstances, however, no special backing will be provided. In these circumstances, it is safest to assume no special backing and to treat the fill as a uniform near- cohesionless granular material with an angle-of- fr ict ion, ¢, of the order of 30 °. Hence, the work done in moving the portion CD (Figure 2) of the arch into the fill may be considered as mobilising passive pressure so that, adopting Rankine's theory [1898], the horizontal stresses are kn times the vertical, gravi ty stresses. The factor kp is given by:

kp= 1 + s i n @ . . . . . . . . . . . . . . . . . . . . . . . . . . (A.1) 1 - sin~

Alexander and Thompson [1916] indicate a value for ~) of 37 ° (kp=4) with 'crushed stone' and 30 ° (kp=3) for "dry, granular earth spread in layers'.

Returning to Figure 2, work will be done on the portion AB of the arch by the 'active pressure'. However, with the active pressure being significantly less than the passive pressure, it is reasonable, in the light of all the other assumptions, to neglect this effect. Hence in the mechanism program, we only consider the work done against the fill in section CD of Figure 2. In addition, it is assumed that the spandrel walls effectively prevent transverse movements in the fill so that it is in a state of 'plain strain'.

Rankine's method relates to a vertical, frictionless wall. Hence, considering for the present a straight arch-section CD from Figure 2 as shown in Figure A.1, the work is assumed to be generated by an imaginary vertical wall BD where DCB moves as a rigid-block and BD is assumed frictionless. Consequently, the work done is:

W = H~Ae 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (A.2)

where:

Hi = 1 2 2 -~ pkp (d0- de) . . . . . . . . . . . . . . . . . . . . . . (A.3)

and:

= do

3 3

2 (do- de) (A.4) 3 2 2

(do - dc)

In the mechanism program, we effectively apply this approach with the work being summed over the elements. It should be emphasised that this is an approximate method. Firstly, the deflections need to be large to mobilise the full passive resistance. Nonetheless, we have adopted a form of 'rigid- plastic' analysis and, within this context, the forces will be mobilised. However, the limitations of this 'rigid-plastic' approach (Section 3) should be borne in mind.

A further approximation involves the imaginary vertical wall BD in Figure A-1. In practise the work is performed by the rotation of the curved surface between C and D in Figure 3. As a simplification, we will now apply an approximate equilibrium analysis to a straight element such as CD in Figure A-2. and will assume that the fill is level with the top point C. Figure A-2 also shows a 'slip-line' DE which is assumed to be straight. Because we know that the tangential force on DE equals tan¢ times the normal force N, we can set up equilibrium equations for the block CDE which is of weight V. These equations are:

Pcoso~ = Ncosl3 + Ntan~ sinl3 . . . . . . . . . . . . . (A.6)

2

= - ~ - (tanoc + tanl3) . . . . . . . . . . . . . . . . . . . . V (A.7)

or V = Psino~ + Nsinl3 - Ntan~cosl3 . . . . . . . . . . . (A.8)

From which,

2

P = ~ (tans + tanl3) . . . . . . . . . . . . . . . . . . . . (A.9)

18

where: coscz (sinl3 - tan~cosl3) . . . . . (A.10)

x = sinc~ + (cosl3 + tan~sin~)

and we require the angle/3 to be such that P is a minimum. In the simple case of a vertical wall with a=O, eqns. (A.9) and (A.10) give:

p = pd 2 (1 +tan~tanl~) (A.11) 2 (1 -- lan~ cotl3)

so that to minimise the load,

dP =0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . (A.12)

d13 and:

tan21~ = - cot~ . . . . . . . . . . . . . . . . . . . . . . (A.13)

from which:

13= + . . . . . . . . . . . . . . . . . . . . . . . . (A.14)

and:

p = p d 2 (1 +sin S (A.15) 2 (1 - sin~)

which coincides wi th equations (A.1) and (A.3) for de--0.

In the more general case of an inclined wal l , w i th as0, such explici t di f ferent iat ion is di f f icul t and, consequent ly, a small computer program was wr i t ten to f ind the min imum value of P. Assuming a tr iangular distr ibut ion of stress against the wal l so that P acts at ~¢/3 (Figure A-2), the work done in rotat ing DC through a small angle A83 (as in Figures 2 and A - l ) is:

l W = Pmin~A03 . . . . . . . . . . . . . . . . . . . . . . . . (A.16)

If, instead, we adopt the approach used in the mechanism program,

d3 (A.17) W m = kp p y A03 . . . . . . . . . . . . . . . . . . . . . .

Assuming e = 30 °, the results are plotted in Figure A-3 wh ich also shows two results from a non-l inear f inite element program. From this figure, it wou ld appear that the method used in the mechanism program can be up to 25 per cent on the unsafe side for the work done against the fill. However, it is arguable that, for the situat ion in Figure A - l , where deS0, some lateral work wi l l be required in the fill above B. In addit ion, fur ther lateral work could be required in the fill above the 'sagging' hinge B in Figures 1-3. Further analysis is cont inuing especially using the non-l inear f inite element program.

d C

C ~ _i

d o Ae3

B

83

Fig.A1 Assumed 'passive action'

p kp d c

_1

\

\

p kp d o I

19

C E

Fig.A2 Forces on soil wedge D-C-E

I ~ Non-linear finite element solutions I

m

0.8 -- -- 80

m u

0.6

0.4

0.2

o.o I I I I I I 0 10 20 30 40 50 60

Angle of wall, (~ o

60 %

40 ~

20

I I o 70 80 90

Fig.A3 Solutions for 'inclined Wall'

20