Automatic Yield-Line Analysis of Practical Slab Configurations via Discontinuity Layout Optimization

The yield-line method of analysis (Johansen 1943) is a long-established and extremely powerful tool for estimating the maximum load sustainable by a reinforced concrete slab. To apply the method successfully, traditionally users have had to know something about the rules governing the construction of viable yield-line patterns, though these rules can be memorized, and for simple problems a hand analysis is quick and easy to perform. The guidance document produced by the U.K. Concrete Centre (Kennedy and Goodchild 2004) discusses the many benefits of yield-line design, in particular highlighting the highly economic reinforcement layouts that can result from its application (though it should be noted that the method considers flexural failure only, and serviceability considerations, which will sometimes govern the design, are not considered). Furthermore, various other guidance documents are available to assist new users, some of which also include useful formulas covering standard cases.

However, in many practical cases it can be difficult to identify the critical yield-line pattern by hand. This is true when the slab under consideration has an unusual geometry, reinforcement configuration, or pattern of applied loading. The presence of fixed (or so-called clamped) edges can also cause difficulties since in reality complex yield-line patterns (e.g., involving corner fans) will often be critical in such cases, and these can be difficult to deal with in a hand analysis. Most importantly, it must be borne in mind that the yield-line method is an upper-bound method in the context of the fundamental theorems of plasticity, which means that an incorrectly chosen yield-line pattern will result in an unsafe estimate of the strength of the slab under consideration.

To address this issue, an automated method of identifying critical yield-line patterns was first proposed in the 1970s by Chan (1972), then working at the University of Oxford, and subsequently by Munro and Da Fonseca (1978), working at Imperial College, London. Both groups of researchers discretized the slab under consideration into rigid elements separated by potential yield-lines and then used linear programming (LP) techniques to identify the critical yield-line pattern. Unfortunately, when using rigid elements it can be observed that the solutions obtained depend on the layout of the mesh discretization employed. This means that in many cases progressively reducing the size of the mesh does not lead to convergence toward the exact solution [e.g., as demonstrated in the recent study by Bleyer and De Buhan (2013)]. Various groups of researchers, for example, Johnson and coworkers (Johnson 1994; Ramsay and Johnson 1998) and Thavalingam et al. (1999), attempted to address this through the use of a two-stage procedure. This involved supplementing the original rigid element procedure with a geometry-optimization phase, allowing the positions of nodes to be adjusted to try to improve the solution. The main drawback is that such procedures rely on the initial solution being of the same form as the true optimal layout. This is not necessarily the case, and, in mathematical optimization terms, such procedures will therefore be prone to identifying solutions that are locally rather than globally optimal [e.g., Johnson (1994) conceded that his proposed two-stage approach “does not directly generate likely critical collapse modes”].

In the absence of general tools, various automated hand calculation yield-line analysis methods have been developed, such as the COncrete BRidge ASsessment (Cobras) package developed by Middleton (1997) specifically for bridge assessment. This proved to be a very useful tool, showing that many existing reinforced concrete slab bridges possessed significantly greater capacity than indicated by elastic analysis methods. However, because the tool relies on the use of an in-built library of predefined yield-line patterns, it is only suitable for analyzing a restricted range of slab geometries.

In parallel, various methods that seek to identify lower bound solutions have been investigated, such as those presented by Anderheggen and Knöpfel (1972), Krabbenhoft and Damkilde (2003), and, more recently, by Le et al. (2010) and Maunder and Ramsay (2012). However, it should be noted that these methods are comparatively complex since they involve the use of a nonlinear yield function and in addition are incapable of identifying discrete yield-lines directly [though these can be manually inferred from the output, as undertaken in the two-step slab analysis procedure recently described by Jackson and Middleton (2013)].

Given the inherent limitations of existing techniques, the opportunity was recently taken to apply the discontinuity layout optimization (DLO) procedure (Smith and Gilbert 2007) to the analysis of reinforced concrete slabs. Although full details are provided by Gilbert et al. (2014), in the present paper key features of the procedure are briefly outlined. It is then demonstrated that the procedure may straightforwardly be extended to treat practical slab analysis problems, involving orthotropic reinforcement, a wider variety of support conditions, and slabs that are subject to point, line, and patch loads. Additionally, it is shown that a recently developed rationalization procedure (He and Gilbert 2016) can be used to enhance the solutions obtained. In this paper the DLO-based procedure is applied to both benchmark problems from the literature and to more practical slab configurations, the aim being to clearly demonstrate its accuracy and usefulness.

A complete DLO analysis comprises several steps (Fig. 1). First, the slab is discretized using nodes spatially distributed across the problem domain [Fig. 1(b)], which are then interconnected with potential yield-lines [Fig. 1(c)]. Each yield-line employs the variables shown in Fig. 2: normal rotation ${\theta }_n$ along yield-line, twisting rotation ${\theta }_t$, and out-of-plane displacement $\delta$. With respect to these displacement variables, a linear programming (LP) problem comprising an objective function and constraints can be formulated as follows (after Gilbert et al. 2014):where $\mathbf{d}$ and $\mathbf{p}$ = vectors containing respectively the aforementioned displacement variables and corresponding nonnegative plastic multiplier variables. In the objective function of Eq. (1a), $\lambda$ is a dimensionless load factor, here applied only to live loads; $\lambda {\mathbf{f}}_L^T\mathbf{d}$ and ${\mathbf{f}}_D^T\mathbf{d}$ describe the external work done, respectively, by live and dead loads (calculated in DLO by considering the effects of loads lying in strips “above” each yield-line; the coefficients in ${\mathbf{f}}_L$ and ${\mathbf{f}}_D$ for the load types considered in this paper are provided in Fig. 3). Also, ${\mathbf{g}}^T\mathbf{p}$ describes the internal energy dissipation along yield-lines. In Eq. (1b), $\mathbf{B}$ is a compatibility matrix used to ensure that yield-line displacements are kinematically admissible; see also Fig. 4. In Eq. (1c), $\mathbf{N}$ is a plastic flow rule matrix describing the relation between the yield-line displacements in $\mathbf{d}$ and their associated plastic multipliers $\mathbf{p}$. Also, in Eq. (1d), the external work done by a live load is normalized to ensure that $\lambda$ directly defines the load factor. By solving the linear optimization problem given by Eqs. (1a)–(1e), the load factor at collapse and the associated yield-line pattern can be obtained. The deformed shape can also be plotted [Fig. 1(e)] to clearly indicate the form of the predicted failure mechanism.

Whereas the example shown in Fig. 1 contains very few nodes, in practice much denser nodal grids can be employed to obtain more accurate solutions. However, a side effect of this is that the resulting yield-line patterns can become quite complex in form. To simplify these, a postprocessing rationalization step, which involves adjusting the positions of the nodes via geometry optimization, can optionally be performed (He and Gilbert 2016). Unlike previously proposed methods that require a manual interpretation step (e.g., Johnson 1995; Jackson and Middleton 2013), here the rationalization is performed automatically following completion of a standard DLO analysis, generating yield-line patterns that are both simplified (i.e., contain fewer nodes and yield-lines) and more critical (i.e., the solutions are better). The extensions to the mathematical derivations described in He and Gilbert (2016) required to enable treatment of the practical slabs considered in the present paper are provided in Appendix I.

Steps in the DLO procedure are shown in Fig. 1; a MATLAB script implementing the basic procedure is provided as supplemental data for interested readers. Dense nodal grids can be employed using a modern desktop computer, so that highly accurate numerical solutions can, if necessary, be obtained. To solve standard DLO problems, including the largest problems, involving approximately 10,000 nodes, the LimitState:SLAB software was used; this software is freely available for academic use. To obtain rationalized solutions, the postprocessing step described by He and Gilbert (2016) was used, programmed in a MATLAB script. All results were obtained using an Intel i5-2310-based desktop PC with 6 GB RAM and running Microsoft Windows 7.

Numerical results are summarized in Table 1, which contains both DLO solutions obtained using dense nodal grids involving 10,000 nodes, to provide highly accurate solutions, and rationalized DLO solutions, which are easier to interpret visually. For problems with known analytical solutions, the margin of error was always found to be well within 1%. For other problems the results obtained in the present paper were found to be more accurate (i.e., lower) than those obtained using the numerical methods described in the existing literature. Additional details of each problem considered are provided in the following sections.

Discontinuity layout optimization (DLO) provides a powerful means of automating the yield-line method. Also, a given DLO solution, which can be complex in form, can be rationalized to aid visual interpretation and improve accuracy if required. In the present paper, it has been demonstrated that DLO can be applied to a wide variety of problems incorporating practical features (e.g., orthotropic reinforcement and a wide variety of support conditions and loading types). For all the example problems considered in the paper, DLO solutions have been found which are more accurate than those obtained using previously proposed upper bound numerical analysis techniques; in some cases this has shown that literature solutions are highly nonconservative.

Fig. S1, Table S1, MATLAB scripts, and an explanatory document are available online in the ASCE Library (www.ascelibrary.org).

In geometry optimization, nodal positions ($\mathbf{x}$ and $\mathbf{y}$) are considered as optimization variables, in addition to $\mathbf{d}$ and $\mathbf{p}$ in optimization problem (1). Therefore, the coefficient matrices and vectors in (1) contain the optimization variables, which are continuously updated during the optimization process. For example, whilst $\mathbf{g}$ comprises constants coefficient values when slabs are isotropically reinforced, when orthotropic reinforcement is present the coefficient values are affected by the yield-line angles $\varphi$ [see also Fig. 5 and yield-criterion (3)] that are determined by $\mathbf{x}$ and $\mathbf{y}$; hence $\mathbf{g}$ is now a function of the optimization variables.

In addition, functions representing the load effect terms ${\mathbf{f}}_L$ and ${\mathbf{f}}_D$ can become nonsmooth with respect to nodal positions, which can cause problems. In this paper extra constraints are added to prevent these functions from becoming nonsmooth. Thus in Fig. 23(a) node A is made nonmovable since it coincides with a point load; in Fig. 23(b), node B is restrained so as to only be able to move in the direction of the line load. For patch loads, domain decomposition (which divides a slab domain into several separate sub-domains; see He and Gilbert 2016 for details) can be used, and a subdomain can be created in the patch load area. Note that the above approaches restrict nodal movements and will therefore potentially somewhat reduce the accuracy of the numerical solutions obtainable using geometry optimization.

Vertices of the floor slab (Fig. 24) are given in Table 2. Dimensions of the holes are given in Table 3 and those of the blade columns in Table 4.

Initial work in the area of this study was undertaken while the second author was in receipt of an EPSRC Advanced Research Fellowship, Grant No. GR/S53329/01. The authors wish to thank Charles H. Goodchild from the U.K. Concrete Centre and John Sestak from Powell Tolner & Associates for providing the geometric data required for the analysis of the slab shown in Fig. 18.