The rigid block analysis method (a linear programming formulation)

Basic method

A brief description of a modern (rigid block) formulation of the upper bound or 'mechanism’ method of analysis follows. The method described is of general applicability and is particularly suitable for computer use.

Consider a loaded arch rib comprising n voussoirs [Figure 1], which are rigid, infinitely strong and have surfaces which are rough enough to prevent sliding failure occurring. Assume that constituent blocks within the arch may be subject to both dead loading p and live loading q. At the ultimate limit state the problem to be solved is: what load factor l applied to the live loading will lead to global collapse of the arch?

Figure 1 Loaded arch rib containing n voussoirs

Using an approach based on virtual work, the problem becomes:

Minimise

(1)

Where the whole structure live load, dead load and displacement vectors are denoted respectively qT={q1, q2.. qn+1}, pT={p1, p2.. pn+1}and dT = {d1, d2.. d n+1}. And where the block live load, dead load and displacement vectors are denoted respectively , and ; individual block displacement and force components are shown on Figure 2.

(a)

(b)

Figure 2 Block displacement components (a), and force components (b)

Subject to constraints which:

(a) stipulate that the right hand abutment (notional block n+1) is fixed in space. i.e:

(2)

(b) displace the structure according to the magnitude of the live load. Thus in general the required constraint is:

(3)

This simplifies to (4) if only given block m is subject to a vertical load qy:

(4)

Alternatively displacements of block j, dj, can be expressed as a function of the relative rotations between adjacent units. i.e:

dj = Ajr                                                                                 (5)

Where Aj is a transformation matrix derived from the geometry of the structure [see Appendix 1 for derivation], rT={}, and where represent rotations about the intrados and extrados respectively (e.g as shown on Figure 3).

In fact it is convenient to express the displacement terms in equations (1) to (3) in terms of the relative rotations between blocks, , and to select these as the problem variables. This is because the necessary stipulation that these variables must take on only non-negative values is a standard feature of linear programming solution algorithms (note that are not restricted from simultaneously taking on positive values; this represents separation).

Any standard linear programming technique, such as the Simplex method [1], can then be used to obtain a solution for l . There are, however, two important reasons why it may not be possible to obtain a value for l :

1. There might be no feasible solution to the problem, i.e. there may be no combination of values for the problem variables ('s) that satisfy all constraining equations. If this is the case then it follows that the arch is geometrically locked - the arch will not fail in a hinged mechanism. For example, this outcome might arise in the case of a very thick or flat arch.
2. The solution for l may be unbounded. This will occur if the arch is already unstable under its own dead weight. For example, this outcome might arise in the case of a very thin semicircular arch.

As an upper-bound problem is the linear-programming dual of the corresponding lower-bound problem, the values of the variables involved in the dual (lower-bound) problem are obtainable from the final LP tableau. Thus in the case of an arch bridge the magnitude of the abutment thrusts at failure become available without further calculation.

Removing 'no-sliding' assumption

Removal of the 'no-sliding' restriction increases the generality of the method by permitting adjacent blocks greater freedom of movement.

Inclusion of Coulomb frictional sliding is difficult from a mathematical point of view. However, these difficulties disappear if the principle of normality is assumed to hold true (i.e. if an associated flow rule is assumed). In this case sliding movement between blocks is treated as ‘plastic shearing’, also termed dilatant or interlocking friction. Drucker [2] has shown that results from an analytical model in which frictional interfaces are modelled as plastic shearing will be upper bounds on the 'exact' values.

In terms of the formulation of the rigid-block method, additional variables can be introduced to allow sliding movements to be controlled. Thus variables yj+, yj- can be defined, respectively, as the positive and negative shearing displacement of block j relative to block j-1 parallel to the interface between the blocks (yj+,yj->0). Movement perpendicular to the interface (separation) can be stipulated to be sb(yj++yj-) where sb is the coefficient of dilatant friction between blocks. Clearly the parameter sb must normally always take on some (small) non-zero value, otherwise the arch will be found to be unstable under its own dead weight.

To include the additional variables in the analysis (5) is be rewritten as:

(6)

Where Bj is a suitable transformation matrix [see appendix for derivation] and where sT={}

Figure 4 shows the full range of feasible movements of adjacent blocks when the sliding capability is added. Figure 10 shows the influence of the value of the coefficient of dilatant friction on the predicted failure load and associated failure mechanism for the 23-block arch tested originally by Pippard and Ashby [13].

Figure 4 Feasible relative movements of adjacent blocks

Including crushing of the masonry in the analysis

The assumption that the masonry is incompressible implies that infinite stresses can be sustained by the masonry at hinges. Although this is clearly unrealistic, the assumption will often lead to only small overestimates in collapse load, depending of course on the compressive strength of the masonry and on the bridge span and shape. Removing the assumption from the analysis transforms the problem from a linear to a non-linear one. However, using a simple iterative solution strategy, such as the one given below, it is typically found that convergence can be obtained in three or four iterations [3]:

1. Compute the collapse load of the bridge, and, from the final LP tableau, deduce the magnitudes of the thrust transmitted across each joint.
2. Determine the depth of masonry required to transmit the thrust across each joint and then locally reduce the depth of masonry in the arch by half this depth+ (from the intrados or extrados as appropriate).
3. Update the transformation matrices Aj and Bj.
4. Repeat from step 1 until specified convergence criteria are met.

+ This is the method put forward by Livesley [4]. Alternatively the masonry depth can be reduced by the full depth required to transmit the force. In this case the work equation (1) must then be modified to take account of the work done in subsequently opening the hinge, which now has an effective plastic moment of resistance. This latter method was suggested by Crisfield and Packham [5].

Multi-span arches

Due to the general nature of the rigid block analysis method modelling multi-span masonry arches presents no special difficulties. In a multi-span problem each span will have its own prescribed boundary constraints (2), leading to a total of 3q+1 constraining equations and a single objective function, where q is the number of spans in the structure under consideration [6]. Figure 6 shows typical output from the rigid block analysis for a twin-span arch bridge (with finite masonry crushing strength).

Figure 6 Output from rigid-block analysis program: bridge containing two 10m spans, crushing strength 2N/mm² (depth of masonry required to carry compressive force indicated by thickness of thrustline)

Multi-ring arches

In cases where an arch barrel is built up from a number of individual rings of bricks, connected only by a weak layer of mortar, it will frequently be found that these rings become debonded; consequently bridge carrying capacity will be reduced [7].

Bridges with debonded arch rings can be modelled using the rigid block analysis method. Firstly, each ring is modelled as a separate arch with its own prescribed boundary constraints (2). Secondly only the top ring will normally be subject to live loading q. Thirdly, as adjacent rings of blocks will in reality interact with one another, additional constraints must be added to those already described in order to prevent blocks overlapping and, if required, to properly account for friction at the interface between rings. Further details of the method are provided in [8]. Figure 7 shows the predicted collapse mechanism for Bolton bridge 5-2 (the bridge actually failed at a load of 500kN).

Figure 7 Output from rigid-block analysis program: Bolton model single-span model bridge 5-2

Because of the number of additional constraints required to avoid inter-penetration of adjacent blocks it is found that multi-ring arch problems are considerably more computationally expensive than single ring problems (although many problems still take only a matter of a few seconds to solve on a modern desktop PC).

APPENDIX

A. Derivation of transformation matrices for rigid block analysis

Constituent block j of an undisplaced arch is shown on Fig. A1. Suppose some point is defined as the local origin of the block, for example the midpoint of the interface closest to the left-hand abutment. Using polar co-ordinates the position of Oj relative to Oj+1 can be defined by [bj, ßj].

Fig. A1 Geometry of block in undisplaced arch

When an arch is subjected to a small (virtual) displacement, a number of hinges will form, at the intrados and extrados of the arch. The angle opened by a hinge on the extrados or intrados between block nos. j-1 and j of a given span are denoted as and respectively ( ³ 0).

Using the assumptions of small deflection theory, and noting that the left-hand abutment is fixed in space, it can be shown that that the horizontal displacement of the local origin of block j can be written as:

(A1)

Where:

(A2)

Similarly the vertical displacement of point Oj can be written as:

(A3)

Thus it follows that the transformation matrix Aj will be of the form:

(A4)

Where ck=akcosa k, dk=bkcosb k , ek=aksina 1, fk=bksinb k

Suppose that sliding can also occur. Variables control respectively the positive and negative shearing displacements of block j relative to block j-1 parallel to the interface between the blocks (³ 0); refer to Figure A2. Movement perpendicular to the interface (separation) is given by sb, where sb is termed the coefficient of dilatant friction (or separation) between blocks.

Figure A2 Relative sliding between blocks (assuming associated flow rule)

The additional horizontal displacement of block j due to sliding can be shown to be:

(A5)

Similarly the additional vertical displacement of block j due to sliding can be shown to be:

(A6)

Thus it follows that the transformation matrix Bj will be of the form:

(A7)

References

[1] Dantzig, G.B.,"Computational algorithm of the revised simplex method", Rand Corporation reprint RM-1266, 1953.

[2] Drucker, D.C.: "Coulomb friction, plasticity and limit loads", Transactions American Society Mechanical Engineers, 76, 71-74, 1953.

[3] Gilbert, M., "On the analysis of multi-ring brickwork arch bridges", 2nd International Arch Bridges Conference, Venice, 109-118, 1998.

[4] Livesley, R.K., "The collapse analysis of masonry arch bridges", Proc. conf. applied solid mechanics 4, Elsevier, 1992.

[5] Crisfield, M.A., Packham, A., "A mechanism program for computing the strength of masonry arch bridges", TRRL research report 124, DoT, UK, 1987.

[6] Melbourne, C., Gilbert, M., Wagstaff, M.,"The collapse behaviour of multi-span brickwork arch bridges", The Structural Engineer, 75(17), 297-305, 1997.

[7] Melbourne, C., Gilbert, M., "The behaviour of multiring brickwork arch bridges", The Structural Engineer, 73, 39-47, 1995.

[8] Gilbert, M., Melbourne, C., "Rigid-block analysis of masonry structures", The Structural Engineer, 72(21), 356-361, 1994.