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

: The yield-line method provides a powerful means of rapidly estimating the ultimate load that can be carried by a reinforced concrete slab. The method can reveal hidden reserves of strength in existing slabs and can lead to highly economic slabs when used in design. Originally conceived before the widespread availability of computers, the yield-line method subsequently proved difficult to computerize, limiting its appeal in recent years. However, it was recently demonstrated that the discontinuity layout optimization (DLO) procedure could be used to system-atically automate the method, and various isotropically reinforced, uniformly loaded slab examples were used to demonstrate this. The main purpose of this paper is to demonstrate that the DLO procedure can also be applied to a wide range of more practical slab problems, for example involving orthotropic reinforcement, internal columns, and point, line, and patch loads. The efficacy of the procedure is demonstrated via application to a variety of example problems from the literature; for all problems considered solutions are presented that improve upon existing numerical solutions. Furthermore, in a number of cases, solutions derived using previously proposed automated yield-line analysis procedures are shown to be highly nonconservative. This work is the terms of the


Introduction
The yield-line method of analysis (Johansen 1943) is a longestablished 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 geometryoptimization 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.

Discontinuity Layout Optimization Formulation
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 θ n along yield-line, twisting rotation θ t , and out-of-plane displacement δ. 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): Subject to Bd ¼ 0 ð1bÞ where d and p = vectors containing respectively the aforementioned displacement variables and corresponding nonnegative plastic multiplier variables. In the objective function of Eq. (1a), λ is a dimensionless load factor, here applied only to live loads; λf T L d and f T D d describe the external work done, respectively, by live and dead loads (calculated in DLO by considering the effects Step 2: nodal discretization (using four nodal divisions); (c) Step 3: nodes are connected by potential yield-lines; (d) Step 4: identify the subset of yield-lines present in the critical collapse mechanism using linear programming (LP); (e) Step 5 (optional): visualize deformed shape of loads lying in strips "above" each yield-line; the coefficients in f L and f D for the load types considered in this paper are provided in Fig. 3). Also, g T p describes the internal energy dissipation along yield-lines. In Eq. (1b), B is a compatibility matrix used to ensure that yield-line displacements are kinematically admissible; see also Fig. 4. In Eq. (1c), N is a plastic flow rule matrix describing the relation between the yield-line displacements in d and their associated plastic multipliers p. Also, in Eq. (1d), the external work done by a live load is normalized to ensure that λ 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.

Orthotropic Slab Reinforcement
In engineering practice, many slabs contain orthotropic reinforcement; such slabs were not considered in Gilbert et al. (2014). However, it will be shown here that orthotropic reinforcement can be py and m − px , m − py are, respectively, the x-and y-direction positive and negative plastic moments of resistance per unit length. Also, m þ p and m − p are the plastic moments of resistance per unit length along a yield-line inclined at an angle of ϕ to the x-axis. Using the stepped yield-criterion approach proposed by Johansen (1943), the latter can be calculated from In the DLO formulation (Gilbert et al. 2014), the preceding equations can be used when calculating the internal energy dissipation terms for a given yield-line [i.e., the coefficients in g, used in the objective function given by Eq. (1a)]. Note that since the orientation of a given yield-line connecting two nodes is known in advance, computing the energy dissipation terms is straightforward; hence, orthotropically reinforced slabs can be treated without difficulty in DLO.

Boundary Conditions
In Gilbert et al. (2014) and He and Gilbert (2016), only four boundary conditions were considered [free, symmetry, fixed, and simple (anchored) boundaries; see the first four columns in Fig. 6]. However, in engineering practice, other boundary conditions are frequently encountered; the remaining columns in Figs. 6 and 7 provide details of the additional boundary conditions considered here. Additionally, here an optional support strength factor i is applied in the case of fixed supports [i is then a multiplier in internal energy dissipation terms in the objective function given by Eq. (1a)]. Further explanation of the remaining boundary conditions follows: 1. Nonanchored simple supports An external simple support can be anchored or nonanchored; in the latter case, uplift may occur, which means that twisting and out-of-plane displacements can now be nonzero, with θ t l=2 AE δ ≥ 0 ensuring that uplift (only) can occur.

Knife-edge supports
These types of support may be located internally beneath the slab, with the slab above the support being continuous. This means that the slab can rotate along the support. If there exists no relative normal rotation at the support (i.e., θ n ¼ 0), then no yield-line is allowed to form at the support. If θ n ≠ 0, then a yield-line may develop along the support and internal energy dissipation needs to be accounted for. Regarding the twisting rotation θ t and out-of-plane displacement δ, both (a) anchored and (b) nonanchored knife-edge supports are considered here. In the latter case, twisting and out-of-plane displacements can now be nonzero, and uplift can be allowed to occur (as with the nonanchored simple supports).

Column supports
Columns are frequently used in engineering practice; these can effectively be modeled using a combination of the aforementioned line support types. An external column is normally modeled using a simple support, while internal columns can now be modeled using either enclosed knife-edge supports or, if an internal column passes through the slab, by fixed supports (since the slab region is now discontinuous). In the latter case an optional support strength factor i can be applied. A summary of all column support types is shown in Fig. 7.

Application of Automated Method
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.

Singly Reinforced Slabs
Slabs that are singly reinforced (i.e., having no hogging resistance) will be considered first; these are of particular interest because the critical failure mechanism is likely to involve so-called corner levers or corner fans [which are often ignored by practicing engineers, who may instead pragmatically apply a margin of safety of, e.g., 10% (Kennedy and Goodchild 2004) to account for this and other simplifications].

Isotropic Six-Sided Slabs
The six-sided slabs shown in Figs. 8 and 9 were previously analyzed by Wüst and Wagner (2008). In both cases, , there is no resistance to hogging moments). It is evident that the yield-line patterns identified by Wüst and Wagner (2008), shown in Figs. 8(b) and 9(b), involve yieldlines that intersect each of the corners and, hence, do not include corner levers or corner fans. In contrast, corner fans are evident in both the standard DLO solutions [Figs. 8(c) and 9(c)] and the rationalized DLO solutions [Figs. 8(d) and 9(d)]. Comparing the solutions obtained by Wüst and Wagner (2008) and those obtained using DLO, it is evident that the former are more than 10% higher than the latter (using 10,000 node DLO solutions provided here as benchmarks, differences for the two problems are as follows: ð17.75=15.953 − 1Þ × 100% ¼ 11.3% and ð54.4=47.424 − 1Þ × 100% ¼ 14.7%), indicating that using the yield-line patterns identified by Wüst and Wagner (2008) would be nonconservative, even if the usual 10% margin of safety were applied.

Orthotropic Trapezoidal Slab
The trapezoidal slab shown in Fig. 10 was previously analyzed by Balasubramanyam and Kalyanaraman (1988). The slab is orthotropically reinforced and simply supported on three sides. The relatively simple yield-line pattern identified by Balasubramanyam and Kalyanaraman (1988), shown in Fig. 10(b), does not include corner levers or corner fans and corresponds to a required moment capacity of m þ px ¼ 11.84q. In contrast, the DLO solution shown in Fig. 10(c) is more complex and corresponds to a required moment capacity that is 18% higher [using the most accurate DLO solution available here, ð13.97=11.84 − 1Þ × 100% ¼ 18%]. This again indicates that application of the usual 10% margin of safety would be insufficient to ensure a safe design.

Roof Slab
The slab shown in Fig. 11 has two external column supports and two knife-edge supports. The problem was originally analyzed by Bäcklund (1973), then by Munro and Da Fonseca (1978), and subsequently by Balasubramanyam and Kalyanaraman (1988). The rationalized DLO solution is shown in Fig. 11(c). A local failure can be observed near one external column; additionally, a fan-type mechanism is developed near the knife-edge supports. Though the slab is not anchored to the knife-edge supports, the failure mechanism shows no uplift of the slab.
Johansen's Slab with Point Supports Johansen (1943) investigated a slab having two point supports [ Fig. 12(a)]. Ideally a point support acts as a fulcrum, providing no rotational restriction and permitting uplift without necessitating plastic deformation of the slab. However, in DLO a point support is most conveniently modeled as a column of finite, though small, size [ Fig. 12(b)]. This means that care must be exercised since, if a  column restricts translational displacement of the slab along all its edges (e.g., a Type I or III column in Fig. 7), plastic deformation must occur near the column when uplift behavior occurs. In contrast, a Type II column behaves similarly to a point support, except that the fulcrum has been offset from the initial position [cf. Fig. 12(a)]. The influence of the support type and column size μ on the critical yield-line pattern is shown in Fig. 13. The following observations may be made: • First, when a relatively small column size is used (μ ¼ 0.001), the resulting global yield-line pattern is largely insensitive to column type. However, local mechanisms are found close to Types I and III columns, and nonanchored (Type II) columns provide the best approximation of a point support; • Second, when larger columns are used (μ ¼ 0.2), it is clear that the size of the column affects the failure mode significantly: a fan-type mechanism is developed near the Types I and III columns, and the uplift disappears. In contrast, uplift remains when Type II columns are involved, though the failure mechanism is different from that observed when a very small column size is present; and • Third, when relatively large columns are used (μ ¼ 0.5), yieldlines develop along the column edges and no uplift is observed regardless of which of the three support types are used. In this case, the same failure mechanism is observed irrespective of whether Type I or II columns are present.

Slabs with Point and Line Loads
Several slab problems involving point loads considered by Johansen (1943) are now revisited using DLO. The resulting yield-line patterns shown in Fig. 14 resemble closely the analytical solutions given by Johansen, with the margin of error being less than 1% (see also Table 1). When line loads are present, fan-type mechanisms will often develop in the vicinity of the load. Examples originally considered by Johansen (1943Johansen ( , 1968 are analyzed here using DLO; the resulting yield-line patterns shown in Fig. 15 match Johansen's results closely, though the DLO solutions are more accurate.

Slabs with Patch Loads
The slab problem shown in Fig. 16 is taken from Ramsay and Johnson (1998); this involves self-weight and a patch load Q. Using geometry optimization, Ramsay and Johnson (1998) obtained a relatively accurate solution of Q ¼ 49.5 kN (which is only 3.5% higher than the 10,000 node DLO solution Q ¼ 47.85 kN). However, the process used by Ramsay and Johnson (1998) was cumbersome in that it involved identifying a suitable yield-line pattern for use in the geometry optimization stage. In contrast, the yield-line pattern shown in Fig. 16(c) was identified using DLO in a matter of seconds, without human intervention.

Real-World Slab
To further demonstrate the efficacy of DLO, the floor slab employed in a real-world building (Fig. 17) is now considered. The floor slab in question was also considered by Kennedy and Goodchild (2004), though for the purposes of this study full dimensions of the slab have been obtained from the original designers, Powell Tolner & Associates Ltd. This has allowed an accurate slab model to be created; full details are provided in Appendix II. A piecewise linear representation of the curved balconies is used and the walls and 'blade' columns are modeled using Type III internal columns. Following the lead of Kennedy and Goodchild (2004), here a support strength of i ¼ 1 is used to represent the walls around the core, and i ¼ 0 for the wall around the stairs. To model the 215 mm wide blade columns, two scenarios are considered: (1) i ¼ 1; and (2) i ¼ 0 for a more conservative design.
The rationalized yield-line patterns obtained using DLO are shown in Fig. 18. The first pattern matches very closely with the 'folding plate' mechanism assumed to be critical in Kennedy and Goodchild (2004). However, in the DLO solution fan-type mechanisms also develop around certain columns, leading to 2.8% increase in the required moment capacity [comparing m p ¼ 48.6 kNm=m from Kennedy and Goodchild (2004) to the 10,000 node DLO solution: ð49.9492=48.6 − 1Þ × 100% ¼ 2.8%, assuming a uniform applied pressure load of 21.7 kN=m 2 ]. Although the hand calculation result described in Kennedy and Goodchild (2004) appears remarkably accurate, it is worth noting that the process involved manually analysing 11 potential yield-line patterns, each postulated by an experienced engineer. In contrast when using DLO the yield-line patterns are identified automatically. Regarding the second, more conservative scenario, taking i ¼ 0 around the blade columns, the resulting yield-line pattern is similar, though there is now a 7.9% increase in the required moment capacity [comparing the Kennedy and Goodchild (2004) solution to the 10,000 node DLO solution: ð52.4439=48.6 − 1Þ× 100% ¼ 7.9%].

Characteristic Features of Yield-Line Patterns
Currently, an engineer analysing a slab using the yield-line method can draw upon well-established rules when postulating the critical yield-line pattern. As presented by Jones and Wood (1967), the basic rules are • Yield-lines are straight except where a region may become completely plastic at any point; • The yield-line between two or more bordering rigid regions must pass through the intersection of the axes of rotation of these regions; • Axes of rotation usually lie along supported edges or column lines; and • Yield-lines may only change direction when intersecting another yield-line. Gilbert et al. (2014), used observed features in DLO solutions, and then Mohr's circle analysis, to point out additional characteristic features of yield-line patterns for isotropically reinforced slabs: • Yield-lines of opposite signs should intersect at 90°, whether in the interior of a slab or at a fixed support; • Yield-lines of opposite signs should intersect simple supports and free edges at between 45°and 135°; and • Yield-lines of the same sign can intersect at any angle.
However, in the context of the present paper it is of interest to point out that the above relations can be generalized for orthotropically reinforced slabs by using the affine transformation method (Johansen 1943(Johansen , 1968Kennedy and Goodchild 2004;Nielsen and Hoang 2011); an example is shown in Fig. 19.

Problems Identifying Critical Yield-Line Patterns by Other Means
It is now clear that the DLO method provides a very efficient means of identifying critical yield-line patterns in the case of slabs with complex geometries, boundary conditions and/or load conditions. On the other hand it might be presumed that either hand-analysis or one of the previously proposed numerical procedures can be expected to provide reasonable solutions for simple slab analysis problems. However, as noted by Ramsay and Johnson (1997), this is not necessarily the case. Here two simple example problems encountered during the course of the present study will be used to investigate this issue.

Rectangular Slab Simply Supported on Two Adjacent Edges
When carrying out a yield-line analysis by hand, usually a yieldline pattern is postulated by hand [e.g., with the aid of the rules by Jones and Wood (1967)], and then the work method is used to obtain a solution. When using the work method various geometric parameters of a yield-line pattern can be adjusted to seek the minimum collapse load. For this reason, it might appear that the prescribed yield-line pattern need only be 'near correct'. Whilst in many cases use of a simple postulated yield-line pattern can lead to a reasonably accurate solution, an issue is that it can be very difficult to judge whether a prescribed yield-line pattern is truly 'near correct'. For example, consider a seemingly simple example problem provided in Kennedy and Goodchild (2004), a rectangular slab which is simply supported on two adjacent edges [ Fig. 20(a)]. Yield-line patterns provided in Johansen (1968) and Kennedy and Goodchild (2004) are shown in Figs. 20(b and c) respectively, whilst the yield-line pattern identified using DLO is shown in Fig. 20(d). Note that in the case of the pattern shown in Fig. 20(c), geometric parameters of the pattern have been adjusted to find the minimum collapse load. The pattern shown in Fig. 20(b) leads to a relatively accurate solution, very similar to that derived using DLO, shown in Fig. 20(d). However, the solution shown in Fig. 20(c) is clearly very inaccurate; the reason for this will therefore now be investigated further.   Table 1 for more accurate, 10,000 node, DLO solutions) The geometry of the yield-line pattern given in Kennedy and Goodchild (2004) and shown in Fig. 20(c) is controlled by two geometric parameters ξ and η, as shown in Fig. 21(a). The required moment capacity m p can be obtained by varying these parameters, as shown in Fig. 21(b). It can be noticed that for 0 < ξ ≤ 9, the maximum value of m p is 168.0 kNm=m, which actually needs to be at least 52.9% higher for a safe design [calculated using ð256.9=168.0 − 1Þ × 100%]. However, it is evident in Fig. 21 that from ξ > 0 to ξ ¼ 0, the m p values rise suddenly, showing the presence of a singularity. This implies that pattern in Fig. 20(c) does not transform smoothly to that shown in Fig. 20(b) simply by optimizing its geometric parameters (for example by using the 'Solver' optimization tool found in Microsoft Excel). This means that the yield-line pattern shown in Fig. 20(c) cannot really be considered to be "near correct". This clearly demonstrates that large errors can easily be encountered when using the yield-line method, even if the problem appears very simple. [Note that although the pattern shown in Fig. 20(c) was depicted in Kennedy and Goodchild (2004), the formula provided in that document actually corresponds to the pattern shown in Fig. 20(b)].

L-Shaped Slab
The L-shaped slab in figure Fig. 22(a) was investigated in Ramsay and Johnson (1998), using a rigid finite element based numerical analysis procedure. A relatively coarse mesh was used, leading to the yield-line pattern shown in Fig. 22(b) being identified. Although this pattern may appear qualitatively reasonable, the problem was recently revisited by Ramsay et al. (2015), who showed that the yield-line pattern is actually far from critical. The corresponding DLO yield-line pattern for this problem is shown in Fig. 22(c), which corresponds to a required m p value of 2.3743 kNm=m. This is some 40% higher  Table 1 for more accurate, 10,000 node, DLO solutions) than the solution given in the 1998 paper (1.70 kNm=m), showing that the latter solution was highly non-conservative. This indicates that, even for simple slab problems, the use of previously proposed numerical methods may lead to highly inaccurate solutions being obtained. It would also appear to show that it is difficult to qualitatively judge whether a given yield-line pattern is correct or otherwise (the highly inaccurate solution appears to have gone unnoticed in the literature for some 17 years, despite sustained interest in the field over that period). Fortunately the new DLO based automated yield-line analysis procedure overcomes accuracy issues associated with previously proposed numerical procedures (e.g., those utilizing rigid finite elements).

Conclusions
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  (c) alternative yield-line pattern, obtained by "optimization," m p ¼ 168.0 kNm=m; (d) DLO yield-line pattern (rationalized), m p ¼ 256.9 kNm=m 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.

Appendix I. Extra Considerations in Geometry Optimization
In geometry optimization, nodal positions (x and y) are considered as optimization variables, in addition to d and 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 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 ϕ [see also Fig. 5 and yield-criterion (3)] that are determined by x and y; hence g is now a function of the optimization variables.
In addition, functions representing the load effect terms f L and 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.

Appendix II. Geometric Data of the Real-World Slab
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.