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?

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 q^T={q₁, q₂.. q_n+1}, p^T={p₁, p₂.. p_n+1}and d^T = {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)

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 q_y:

                                                                 (4)

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

d_j = A_jr                                                                                 (5)

Where A_j is a transformation matrix derived from the geometry of the structure [see Appendix 1 for derivation], r^T={}, 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.

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 y_j⁺, y_j^- 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 (y_j⁺,y_j^->0). Movement perpendicular to the interface (separation) can be stipulated to be s_b(y_j⁺+y_j^-) where s_b is the coefficient of dilatant friction between blocks. Clearly the parameter s_b 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 B_j is a suitable transformation matrix [see appendix for derivation] and where s^T={}

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

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 A_j and B_j.
4. Repeat from step 1 until specified convergence criteria are met.