# A Heuristic Approach to Identify the Steel Grid Direction of R/C Slabs Using the Yield-Line Method for Analysis

Authors: Luigi Fenu; Valeria Colasanti; Eleonora Congiu; Gian Felice Giaccu; Francesco Trentadue; Bruno Briseghella
Journal: Advances in Civil Engineering
 (2019)

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## Abstract

In the last few years, nonregular reinforced concrete (R/C) slabs have become more popular in buildings and bridges due to architectural or functional requirements. In these cases, an optimum design method to obtain the ultimate load capacity and the minimum reinforcement amount should be used. For simple R/C slabs, the yield-line method is extensively used in engineering practice. In addition to strength, the “true” failure mechanism is also obtained by identifying the parameters that define it and minimizing the collapse load. Unfortunately, when the mechanism is too complicated to be described or defined by several parameters (e.g., in slabs with complicated geometry), the method becomes more difficult because the system of nonlinear equations becomes harder to solve through traditional methods. In this case, an efficient and robust algorithm becomes necessary. In this paper, a structural analysis of R/C slabs is performed by using the yield-line method in association with a zero-th order optimization algorithm (the sequential simplex method) to avoid calculating gradients as well as any derivatives. The constraints that often limit these parameters are taken into account through the exterior penalty function method, leading to a successful solution of the problem. Considering that the direction of each yield-line is sought by minimizing the ultimate load and finding the parameters defining the collapse mechanism, another parameter concerned with the direction of an orthotropic reinforcement grid is introduced. In this way, the number of unknown parameters increases, but aside from obtaining the ultimate load and the parameters defining the collapse mechanism, the solution also finds both best and worst reinforcement orientations.


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## Body

## 1. Introduction

Reinforced concrete (R/C) slabs with complex geometry are becoming a characteristic feature of many modern buildings and bridges. The yield-line method is often used in the design of R/C slabs at their ultimate load carrying capacity. Its reliability has been proven by many experimental tests conducted so far and by its widespread use [1–7]. Its main advantage is that it uses realistic collapse mechanism models and allows easy calculation in many simple applications. In these cases, after determining the typology of the collapse mechanism, meaning the number and form of the slab regions defined by the yield lines at ultimate, the virtual-work method is generally adopted to find the actual position of the yield lines and to calculate the ultimate load.Unfortunately, when using the main mode of the virtual-work method, if the collapse mechanism is more complicated (for instance, when symmetry does not simplify the problem) and/or the number of parameters defining the collapse mechanism is high, complex nonlinear equations need to be solved to minimize the collapse load. Therefore, to use the yield-line method when a complex system of nonlinear equations is faced, efficient and robust zero-th order algorithms are required.Zero-th order algorithms are particularly robust because they only use function values with no derivatives during the minimization process. Haftka and Gürdal [8] and Vanderplaats [9] described some applications of zero-th order methods in structural engineering. In the examples proposed in this study, the sequential simplex method was used. It is a zero-th order method originally defined by Spendley et al. [10] and improved by Nelder and Mead [11]. An application of the sequential simplex method to the optimum design of R/C structures by using the stringer-and-panel method was proposed by Biondini et al. [12], who applied it to an R/C supported beam. The use of the sequential simplex method for minimizing the ultimate load qu is extensively described in [8].In nonregular R/C slabs, it is necessary to identify the best reinforcement direction to minimize the used amount. In these cases, it can be useful to use the concept of structural optimization. It is well known that structural optimization is an important tool, both for sizing structure members and for helping the designer find the most suitable structural form [13–28]. Today, structural optimization is common in mechanical and aeronautical engineering, and in recent years it has been progressively adopted for structural-engineering applications, such as sizing building and bridge members [29–36], detailing reinforced concrete structures [37–45], shaping bridges [46–50], domes [51–54], and other three-dimensional structures [55].Regarding concrete slabs, Kabir et al. [56] conducted an experimental and numerical study to investigate the effect of different directions of the reinforcement arrangements on the ultimate behaviour of skew slabs. Anderheggen [57] proposed a design approach for the reinforcement of concrete slabs and walls based on finite elements, plasticity theory, and linear programming. Lourenço and Figueras [58, 59] formulated a computational code based on equilibrium equations and an iterative procedure to design the reinforcement of plates and shells. Lourenço [60] referred to the three-layer sandwich model in Eurocode 2 to design plate and shell reinforcement with a linear analysis by finite elements [61, 62]. Mancini [63] and Bertagnoli et al. [64] studied the skew reinforcement design of two-dimensional elements in reinforced concrete by outlining two design problems to be resolved (choice of reinforcement direction and evaluation of reinforcement ratio for the chosen directions) and by using genetic algorithms for this purpose. Bertagnoli et al. [65, 66] also extended this optimum design approach based on genetic algorithms to concrete shells.In this paper, a heuristic design approach using the yield-line method for analysis has been defined to identify the optimum reinforcement direction.In Section2, the considered parameters (i.e., slab boundaries, rotation axes of the slab regions delimited by the yield lines, and the degree of indetermination in assessing some of the rotation axes) for finding the collapse mechanism in the yield analysis are defined. The values of these parameters are obtained using a zero-th order optimization algorithm, which allows for finding the minimum of the ultimate load qu without calculating its gradient. In so doing, a calculation that is not numerically robust can be avoided because the gradient calculation requires division by very small numbers close to zero. In Section 3, the method proposed to optimize the orientation of the reinforcement steel grid is introduced. A parameter describing its direction is defined, and the new objective function (i.e., the ultimate load) is minimized depending on this additional parameter, as well as on the parameters defining the yield line positions, the pattern of the yield lines, and the direction of the steel grid that further weaken the slab, which are also obtained. The stronger reinforcement is then superimposed in this direction, thus finding the best direction of the steel grid that better reinforces the slab and minimizes the ultimate load for the given optimum direction of the reinforcement grid. The method’s efficacy is been demonstrated in a case study, performing a parametric optimization design of the slab reinforcement according to the grid orientation. Finally, conclusions are drawn.


## 2. R/C Slab Analysis through the Yield-Line Method Using an Optimization Algorithm

R/C slab analysis can be performed using the yield-line method if, on increasing the loads, the ductility is sufficiently high to allow the slab to become a mechanism at ultimate when the yield lines have already been completely developed. For R/C slabs currently used in civil engineering, a complete development of the yield lines can be attained when the reinforcement geometric ratio is less than approximately 1%; that is, at ultimate, the lever arm over the effective depth ratio is close to 0.9 [2]. Moreover, when using the yield-line analysis, a yield criterion for the reinforcement is needed: Johansen’s criterion is herein assumed [67].Consider an R/C slab with two orthogonal reinforcements, whose bars are spaced Δ in both directions and whose cross-sectional areas and effective depths areAsx, Asy and dx, dy, respectively. Say fyd is the design yield strength of the reinforcement steel. The plastic moment in the two directions (the principal with stronger reinforcement, cross-sectional area Asx, effective depth dx, and orthogonal one) can then be defined as(1)mp=0.9AsxΔfyddx,μmp=0.9AsyΔfyddy,with μ = Asydy/(Asxdx).For the virtual-work method, the external and internal work must be equal, namely,(2)LEst=LInt.This equation relates the plastic moment of the stronger reinforcement and that in the orthogonal direction to the ultimate loadqu and the n parameters φi defining the collapse mechanism, that is, mp=fφi,qu or, in a dual way, qu=gφi,mp, i=1,…,n.Therefore, the problem is to find then parameters φi that minimize qu, that is, to solve the system:(3)∂qu∂φi=0,i=1,…,n.System (3) can easily be solved through currently used methods (i.e., the substitution method), but only for simple cases, especially if simplified by symmetry, because qu is a nonlinear function of φi, and system (3) is nonlinear as well. This system can instead become more complicated to be solved when the number n of parameters φi, and therefore the number of equations is increased. When the parameters are few but the objective function becomes more complex, as in the problem that will be investigated in the following section, the varying direction of the reinforcement grid makes system (3) more complicated to solve.When applying the yield-line method with the virtual works to an R/C slab, the effective collapse mechanism is identified by defining some parametersφi, i = 1, …, n, which depend on the collapse mechanism. The number n of parameters φi defining the collapse mechanism, and therefore, the pattern of the yield lines delimiting the slab regions, is determined by the slab boundaries, the rotation axes of the slab regions, as well as by the degree of indetermination in assessing some of the rotation axes.Therefore, having assigned the arrangement of the reinforcement grid (for example, according to two orthogonal directions), and after solving system (3), the exact values of φi are found (and therefore the exact geometry of the mechanism), together with the actual value of the plastic moment mp or, dually, of the ultimate load qu.Although the kinematic method provides an “upper bound” solution, its error is usually not very large, and its use is therefore allowed by many codes (i.e., Eurocode 2 [68]; ACI Code, see ACI Commentary 8.2.1 [69]).


### 2.1. Application of the Yield-Line Method Using an Unconstrained Optimization Algorithm

Consider, for instance, the trapezoidal R/C slab of Figure1 with three supported edges, a free one, and a uniformly distributed load q.Figure 1
Trapezoidal R/C slab reinforced by a grid of steel bars with stronger reinforcement parallel to the free edge.This is a simple example of a nonsymmetrical concrete slab in which it is intuitive to conclude that the best direction of the reinforcement grid is not the one shown in Figure1; although in practice, an engineering designer would likely place the grid with this orientation, that is, with the stronger reinforcement parallel to the free edge (Figure 1).Therefore, it is assumed that an orthotropic grid of reinforcement bars (spaced Δ in both directions and with given reinforcement ratioμ) is placed along two orthogonal directions x and y. In this case, if the free edge and the other parallel sides are sufficiently long to not make the yield lines meet, the collapse mechanism is defined by determining the angles φ1 and φ2 formed between the direction orthogonal to the free edge and the two yield lines that define the three regions of the slab since there is no degree of indetermination in identifying their rotational axes.Aside from the reinforcement grid, Figure1 also shows the rotation axes of the three slab regions defined by the yield lines and the related rotation vectors ϑ1, ϑ2, and ϑ3. The distance of the centroid of each slab region from its rotation axis is(4)dG1=Δ13sinα,dG2=Δ23,dG3=B33A/B−2tanφ1+tanφ22A/B−tanφ1+tanφ2,with Δ1=Bcotα+tanφ1 and Δ2=Btanφ2.Therefore, for eachi-th slab region, say Si, δi, and LEsti=1/2qSiδi denote the area, the centroid shift, and the work done by the external load q, respectively. For a unit shift of the free edge of the slab central region (Figure 2), when adding up the three external works in each slab region, the total external work is(5)LEst=16qB23AB−tanφ1+cotα+tanφ2.Figure 2
Virtual displacements of the slab regions.The modules of the rotation vectors are(6)ϑ1=1Δ1sinα,ϑ2=1Δ2,ϑ3=1B,and therefore, the internal work of the plastic moments mp and μmp in each slab region are, respectively,(7)LInt1=mpl1cosφ1⋅ϑ1+μmpl1sinφ1⋅ϑ1,LInt2=mpl2cosφ2⋅ϑ2+μmpl2sinφ2⋅ϑ2,LInt3=mpl1cosφ1+l2cosφ2⋅ϑ3+μmpl1sinφ1+l2sinφ2⋅ϑ3,where l1=B/cosφ1 and l2=B/cosφ2 are the lengths of the yield lines.On summing these three contributions after calculating the scalar products, the total internal work is(8)LInt=mpBΔ1+1tanφ2+μmptanφ1BcotαΔ1−1−tanφ2.Therefore, the internal energy is dissipated in the slab to form the mechanism results, which are proportional to the plastic moment. Setting the total external work (5) equal to the total internal work (8) and extracting qu, the objective function qu=quφ1,φ2,mp is obtained, that is,(9)qu=6mpB2Btanφ2+Δ1−μtanφ1tanφ2Bcotα−Δ1−Δ1tanφ22Δ1tanφ23A/B−tanφ1+tanφ2−cotα,subject to the following constraints:−90°−α°≤φ1≤90°, 0≤φ2≤90°, and Δ1+Δ2≤A+Bcotα.It can be noted that while in this problem there are only two parameters, the nonlinear system (3) obtained through deriving equation (9) with respect to φ1 and φ2 is however not easy to solve. Therefore, traditional methods (such as the substitution method) cannot be adopted for this problem, and the use of an evolutionary optimization algorithm becomes usually necessary.


#### 2.1.1. The Use of Zero-th Order Algorithms to Minimise the Objective Function: The Case of the Sequential Simplex Method

In this research, different evolutionary algorithms with zero-th order, such as genetic and particle swarm algorithms, as well as the sequential simplex method, have been considered. In solving the nonlinear system (3), different zero-th order algorithms have been compared, obtaining, as expected, very similar results. As a matter of fact, this holistic approach to optimizing the reinforcing grid inclination is only based on the physical laws governing the failure of concrete plates. Of course, different algorithms have different efficiencies and levels of robustness in solving system (3) but without affecting this holistic optimization method.The sequential simplex method was originally proposed by Bhatt et al. [4] and improved by Ramsay and Johnson [5]. It is less robust than the genetic and particle swarm algorithms, but it is more efficient because, even if the gradient is not calculated (it is a zero-th order algorithm), the steepness of the objective function is considered.In the analysed problem, the higher levels of robustness of the genetic and particle swarm algorithms were not quite as important due to the quick convergence, while the higher efficiency of the sequential simplex method was useful for saving computation time. The flowchart in Figure3 shows how the sequential simplex method solves system (3) to minimize qu.Figure 3
Flowchart of the sequential simplex method.The method starts by defining in then-dimensional space O, φ1,…,φn a simplex with n+1 vertices x0, x1,…,xn, each of them defined by a vector xi with n coordinates; the solution consists in finding the coordinate vector φ = [φ1,φ2,…,φn]T minimizing the function qu and therefore maximizing the function mp.Considerqu=quφi in the n-dimensional space O, φ1,…,φn. In this space, a simplex is an n+1-faced polyhedral figure with n+1 vertices 0, 1, …, n. For example, in the three-dimensional space O, φ1, φ2, and φ3, it is a tetrahedron with 3+1 vertices and 3+1 sides. If all the vertices have the same distance, we have a regular simplex. This means that in the two-dimensional space, the simplex is a triangle, and a regular simplex is an equilateral triangle. If qu=quφi is defined in the O, φ1,…,φn space, i = 1, …, n, then it has a real value in each simplex vertex x0, x1, …, xn. Hence, if we find a simplex with a very small distance between these vertices, the smaller the better; the n+1 values of qu in the n vertices are very similar. This means that, if we implement an algorithm which finds these values very close to the minimum of the function qu, then the minimum of qu is considered as achieved, depending on the required precision, that is, on the maximum vertex distance which must be lower than a given small value.Consider, for instance, the contour-plotted function in theO, φ1, and φ2 space of Figure 4, where the two simplices a and b are also defined. The simplex b is smaller than the simplex a, and the values of the function in each vertex of the simplex b are closer to the minimum than those of the simplex a. Starting from a simplex like the simplex a, the sequential simplex method searches for another simplex like the simplex b, where the function values (e.g., the values of function qu) calculated at the simplex vertices are closer to the minimum than those of the vertices of the simplex a. Therefore, the smaller the simplex and the closer to the minimum function values at its vertices, the better is the approximation of the minimum function value.Figure 4
Contour-plotted function with the simplexa and the simplex b in a two-dimensional space.Therefore, it must be implemented an algorithm that starting from an assigned simplex likea finds an enough small simplex like b whose vertex coordinates are sufficiently close to those that make qu minimum. After the initial simplex has been defined, we evaluate qu at each vertex finding two vertices xl and xh where qu has the lowest and highest values (respectively, qul and quh), as well as a third vertex xs where qu has the second highest value qus. Therefore, the algorithm will find the minimum of qu discarding the vertex xhwhere qu is higher and replacing it with one point where it is lower. By doing so, a new simplex is defined whose new vertex xh is also ready to be discarded when the function qu takes on, in the subsequent simplex, a quh value lower than in the simplex of the previous step. The operations that search for a new vertex for the new simplex are named reflection, contraction, and expansion. Discarded xh, reflection needs to find the centroid x¯ of the remaining vertices and to look for the new vertex xr in the line joining xh to x¯, namely, xr=x¯+αx¯−xh, where α is usually assumed to be 1. Then, calculation of qur=quxr is needed. If qul<qur≤quh, this is good and xr replaces xh. If qur<qul, this is better and we can try to further decrease qu in a point xe taken in the same direction by expansion, namely, xe=x¯+βxh−x¯, where β is usually assumed to be 2. Now, if qus<qur, the expansion has attained its purpose and xe replaces xh; otherwise, xr will replace xh.Finally, ifqus<qur<quh, after replacing xh by xr, we make contraction finding xc=x¯+γxh−x¯, with 0<γ<1 and usually chosen to be 1/2; otherwise qur≥quh, we make contraction with no replacement. After contraction, if que>quh, the algorithm must be restarted after replacing all the vertices by a new set of points xl=xi+1/2xl−xi, i=0,1,…,n; otherwise, we simply replace xh by xc. After this whole process, we have defined a new simplex. In this way, at each step, the algorithm searches for a new simplex until the distance between the vertices is so small and quh so close to qul that convergence is reached when(10)11+n∑i=0nqui−qux¯21/2<ε,the lower ε, the better.By then considering the example of Figure1, having defined the geometry of the trapezoidal R/C slab through assigning A=5.00m, B=3.00 m, and α=60° and having assumed μ=0.7 for the ratio between weaker and stronger steel reinforcement and by using the optimization algorithm (i.e., the sequential simplex method), solving the system (3), one obtains φ1=16.61° and φ2=38.27° with Δ1=2.63 m and Δ2=2.37 m, and qu=0.606mp. The total internal energy LInt per unit mp dissipated in the slab to form the mechanism is 3.033. The plot of the unconstrained objective function (9) for A=5.00 m, B=3.00 m, α=60°, and μ=0.7 is shown in Figure 5.Figure 5
Contour plot (a) and 3D diagram (b) of the objective functionqu calculated for the slab of Figure 1.
(a)
(b)


### 2.2. Application of the Yield-Line Method Using a Constrained Optimization Algorithm

When evaluatingqu, the variable φi, i = 1, …, n, may have to be constrained. In fact, to respect the assigned mechanism, some φi could be defined only in a specific interval and/or an expression containing some variables could be limited.In general, the functionqu can be subject to both unequal constraints sjφ≤0, j=1,…,m and equal constraints hkφ=0, k=1,…,l. These constraints can be taken into account through a penalty function P(φ). Some constrained structural engineering problems solved by using penalty function methods are shown by Cetin et al. [20].Using virtual works in the yield-line method, the exterior penalty function method can be incorporated in the sequential simplex method. To this aim, a penalty functionPφ is added to the function quφ, thus obtaining the new function(11)Φφ,rp=quφ+rpPφ,where Φφ,rp is named the pseudo-objective function and rp is a scalar multiplier much higher than 1 (namely, in this kind of problems, chosen to be between 100 and 1000), while the m unequal constraints sjφ≤0 and the l equal constraints hkφ=0 are taken into account through the exterior penalty function Pφ:(12)Pφ=∑j=1mmax0,sjφ2+∑k=1lhkφ2.If the constraints are not violated at any point near the minimum, the penalty functionPφ is not active, its value is zero, and Φφ,rp=quφ. This is the case of unconstrained problems, where, depending on the mechanism typology, qu reaches its minimum without being constrained.On the contrary, ifΦφ,rp is constrained when reaching the minimum, the penalty function Pφ becomes active. This means that at the points where the constraints are violated, qu cannot reach its unconstrained minimum because the related mechanism is not possible and the problem is not defined (i.e., the mechanism of Figure 6(b)). In this case, the penalty function Pφ assumes a positive real value when violating the relative constraint, thus making Φφ,rp increase greatly, also depending on the chosen value of rp.Figure 6
Effect of the penalty function coupled to the sequential simplex method when using the yield-line method for analysis of a trapezoidal R/C slab. (a) Constrained solution. The pseudo-objective function value isΦφ1=10.53°;φ2=21.08°=0.347with P(x) = 0. (b) Unconstrained solution kinematically inadmissible. The pseudo-objective function value after incrementing the parameter values of one degree ⇒Pφ≠0 and Φφ1=11.53°;φ2=22.08°;rp=100=2.037.Actually, by assigning different dimensions to the slab of Figure1, qu do not always reach its unconstrained minimum with the previously assigned mechanism. Therefore, in this case, the constraints to the objective function qu become active.For instance, this occurs for a trapezoidal R/C slab, similar to the previous one withμ=0.7, but with A=2.00 m, B=3.50 m, and α=80° (Figure 6). In this case, using the sequential simplex method coupled with the exterior penalty function method, one obtains qu=2.170mp with φ1=10.53°, φ2=21.08° and with Δ1=1.27 m, Δ2=1.35 m. The length of the free edge A+Bcotα=2.62 m is equal to the sum Δ1+Δ2. In other words, a constrained minimum is reached with Φ=0.347 for a unitary value of mp and Pφi=0 (Figure 6(a)). This means that the penalty function has automatically discarded solutions with Δ1+Δ2 > A+Bcotα. In fact, disregarding the constraints, that is without coupling any penalty function to the objective function, the unconstrained minimum would have been reached with φ1=12.30°, φ2=21.32°, Δ1=1.38 m, Δ2=1.37 m, and Δ1+Δ2=2.75 m. However, this is impossible because the constraint Δ1+Δ2≤A+Bcotα would have been violated and another slab region, kinematically inadmissible, would have been generated (Figure 6(b)). By instead increasing the parameter values φ1=10.53° and φ2=21.08° obtained in the constrained solution (i.e., by adding one degree to each parameter), that is, setting φ1=11.53° and φ2=22.08°, the violation of the constraint makes Pφi≠0, thus greatly increasing the pseudo-objective function Φφi,rp. For instance, for rp=100, one obtains Φφ1=11.53°,φ2=22.08°,rp=100=2.037, which is almost six times the value Φ=0.347 obtained in the minimum, meaning that the solution found is actually a constrained solution.Therefore, having assigned the mechanism, ifqu reaches its unconstrained minimum and, on slightly increasing the parameters, we still have Pφ=0 and Φφ,rp=quφ; then, the collapse mechanism does not need to be constrained, and no penalty function would be necessary. Otherwise, when qu reaches a constrained minimum, a slight increase of the parameters leads to Pφ≠0. In this case, the penalty function becomes active, thus highly increasing Φφ,rp. This means that, in general, the collapse mechanism must be constrained in order to find the actual solution.It must be noted that, for the given mechanism with one inner trapezoidal slab region and two lateral triangular ones, the constrained minimum of Figure6(a) led to three triangular slab regions where the inner trapezoidal region diverged into a triangular one. This constrained minimum was reached after having seen that lower values of φ1 and φ2 led to higher values of the objective function qu.Nevertheless, doesqu attain the constrained minimum for this diverged mechanism with three triangular slab regions? No, it cannot, because this would be true for the given mechanism, but the kinematic analysis shows that the slab also admits another failure mechanism with an inner triangular slab region and two lateral trapezoidal ones having a side in common intersecting the free side of the slab (Figure 7). For kinematic reasons, this side in common must lie on a straight line passing on the point where the rotation axes of the two trapezoidal slab regions meet each other. This means that in this mechanism, the two yield lines defining the triangular region are always constrained to meet each other along the straight line passing through two points: the first one where these two yield lines meet and the second one where the lines overlapping the rotation axes of the two trapezoidal slab regions meet each other (Figure 7).Figure 7
Failure mechanism with an inner triangular slab region and two lateral trapezoidal ones having a side in common intersecting the free side.The kinematically admissible mechanisms of this problem are hence two, a first one with the inner trapezoidal region and two lateral triangular regions (i.e., in Figure1) and a second one with the inner triangular region and two lateral trapezoidal regions (Figure 7). Therefore, the minimum of qu must be searched for both mechanisms by varying φ1 and φ2 without violating the constraints. For both mechanisms, the optimization algorithm will search for the minimum of qu for φ1>0 and φ2>0 until, in both cases, the mechanism diverges forming three triangular slab regions.Similarly to the first mechanism and following the same procedure, when considering the second mechanism illustrated in Figure7, the total internal work is to be obtained by adding the internal works calculated for each of the three slab regions. For this aim, the direction of their rotation vectors ϑi, i = 1, …, 3, and their plastic moments mpi and μmpi are to be evaluated, as well as the magnitude of the rotation vectors ϑi.After identifying the centroid of each slab region, the three related external works are evaluated and added to each other to obtain the total external work. By then adding the total internal work to the total external work, the objective functionqu(φ1, φ2) is explicited and minimized through using the sequential simplex method.The results show that for the slab dimensions in Figures3 and 7, the minimum qu is obtained for φ1=30.71° and φ2=37.65° with Δ1 = Δ2 = 1.31 m. This yield-line pattern occurs for qu=1.891mp. This ultimate load is thus lower than 2.17mp, previously obtained from the other mechanism at its boundary for which also its inner trapezoidal slab region becomes triangular. This means that this first mechanism cannot occur for qu/mp=2.17 because for qu/mp=1.891, the slab would have already failed with the second mechanism shown in Figure 7.


## 2.1. Application of the Yield-Line Method Using an Unconstrained Optimization Algorithm

Consider, for instance, the trapezoidal R/C slab of Figure1 with three supported edges, a free one, and a uniformly distributed load q.Figure 1
Trapezoidal R/C slab reinforced by a grid of steel bars with stronger reinforcement parallel to the free edge.This is a simple example of a nonsymmetrical concrete slab in which it is intuitive to conclude that the best direction of the reinforcement grid is not the one shown in Figure1; although in practice, an engineering designer would likely place the grid with this orientation, that is, with the stronger reinforcement parallel to the free edge (Figure 1).Therefore, it is assumed that an orthotropic grid of reinforcement bars (spaced Δ in both directions and with given reinforcement ratioμ) is placed along two orthogonal directions x and y. In this case, if the free edge and the other parallel sides are sufficiently long to not make the yield lines meet, the collapse mechanism is defined by determining the angles φ1 and φ2 formed between the direction orthogonal to the free edge and the two yield lines that define the three regions of the slab since there is no degree of indetermination in identifying their rotational axes.Aside from the reinforcement grid, Figure1 also shows the rotation axes of the three slab regions defined by the yield lines and the related rotation vectors ϑ1, ϑ2, and ϑ3. The distance of the centroid of each slab region from its rotation axis is(4)dG1=Δ13sinα,dG2=Δ23,dG3=B33A/B−2tanφ1+tanφ22A/B−tanφ1+tanφ2,with Δ1=Bcotα+tanφ1 and Δ2=Btanφ2.Therefore, for eachi-th slab region, say Si, δi, and LEsti=1/2qSiδi denote the area, the centroid shift, and the work done by the external load q, respectively. For a unit shift of the free edge of the slab central region (Figure 2), when adding up the three external works in each slab region, the total external work is(5)LEst=16qB23AB−tanφ1+cotα+tanφ2.Figure 2
Virtual displacements of the slab regions.The modules of the rotation vectors are(6)ϑ1=1Δ1sinα,ϑ2=1Δ2,ϑ3=1B,and therefore, the internal work of the plastic moments mp and μmp in each slab region are, respectively,(7)LInt1=mpl1cosφ1⋅ϑ1+μmpl1sinφ1⋅ϑ1,LInt2=mpl2cosφ2⋅ϑ2+μmpl2sinφ2⋅ϑ2,LInt3=mpl1cosφ1+l2cosφ2⋅ϑ3+μmpl1sinφ1+l2sinφ2⋅ϑ3,where l1=B/cosφ1 and l2=B/cosφ2 are the lengths of the yield lines.On summing these three contributions after calculating the scalar products, the total internal work is(8)LInt=mpBΔ1+1tanφ2+μmptanφ1BcotαΔ1−1−tanφ2.Therefore, the internal energy is dissipated in the slab to form the mechanism results, which are proportional to the plastic moment. Setting the total external work (5) equal to the total internal work (8) and extracting qu, the objective function qu=quφ1,φ2,mp is obtained, that is,(9)qu=6mpB2Btanφ2+Δ1−μtanφ1tanφ2Bcotα−Δ1−Δ1tanφ22Δ1tanφ23A/B−tanφ1+tanφ2−cotα,subject to the following constraints:−90°−α°≤φ1≤90°, 0≤φ2≤90°, and Δ1+Δ2≤A+Bcotα.It can be noted that while in this problem there are only two parameters, the nonlinear system (3) obtained through deriving equation (9) with respect to φ1 and φ2 is however not easy to solve. Therefore, traditional methods (such as the substitution method) cannot be adopted for this problem, and the use of an evolutionary optimization algorithm becomes usually necessary.


### 2.1.1. The Use of Zero-th Order Algorithms to Minimise the Objective Function: The Case of the Sequential Simplex Method

In this research, different evolutionary algorithms with zero-th order, such as genetic and particle swarm algorithms, as well as the sequential simplex method, have been considered. In solving the nonlinear system (3), different zero-th order algorithms have been compared, obtaining, as expected, very similar results. As a matter of fact, this holistic approach to optimizing the reinforcing grid inclination is only based on the physical laws governing the failure of concrete plates. Of course, different algorithms have different efficiencies and levels of robustness in solving system (3) but without affecting this holistic optimization method.The sequential simplex method was originally proposed by Bhatt et al. [4] and improved by Ramsay and Johnson [5]. It is less robust than the genetic and particle swarm algorithms, but it is more efficient because, even if the gradient is not calculated (it is a zero-th order algorithm), the steepness of the objective function is considered.In the analysed problem, the higher levels of robustness of the genetic and particle swarm algorithms were not quite as important due to the quick convergence, while the higher efficiency of the sequential simplex method was useful for saving computation time. The flowchart in Figure3 shows how the sequential simplex method solves system (3) to minimize qu.Figure 3
Flowchart of the sequential simplex method.The method starts by defining in then-dimensional space O, φ1,…,φn a simplex with n+1 vertices x0, x1,…,xn, each of them defined by a vector xi with n coordinates; the solution consists in finding the coordinate vector φ = [φ1,φ2,…,φn]T minimizing the function qu and therefore maximizing the function mp.Considerqu=quφi in the n-dimensional space O, φ1,…,φn. In this space, a simplex is an n+1-faced polyhedral figure with n+1 vertices 0, 1, …, n. For example, in the three-dimensional space O, φ1, φ2, and φ3, it is a tetrahedron with 3+1 vertices and 3+1 sides. If all the vertices have the same distance, we have a regular simplex. This means that in the two-dimensional space, the simplex is a triangle, and a regular simplex is an equilateral triangle. If qu=quφi is defined in the O, φ1,…,φn space, i = 1, …, n, then it has a real value in each simplex vertex x0, x1, …, xn. Hence, if we find a simplex with a very small distance between these vertices, the smaller the better; the n+1 values of qu in the n vertices are very similar. This means that, if we implement an algorithm which finds these values very close to the minimum of the function qu, then the minimum of qu is considered as achieved, depending on the required precision, that is, on the maximum vertex distance which must be lower than a given small value.Consider, for instance, the contour-plotted function in theO, φ1, and φ2 space of Figure 4, where the two simplices a and b are also defined. The simplex b is smaller than the simplex a, and the values of the function in each vertex of the simplex b are closer to the minimum than those of the simplex a. Starting from a simplex like the simplex a, the sequential simplex method searches for another simplex like the simplex b, where the function values (e.g., the values of function qu) calculated at the simplex vertices are closer to the minimum than those of the vertices of the simplex a. Therefore, the smaller the simplex and the closer to the minimum function values at its vertices, the better is the approximation of the minimum function value.Figure 4
Contour-plotted function with the simplexa and the simplex b in a two-dimensional space.Therefore, it must be implemented an algorithm that starting from an assigned simplex likea finds an enough small simplex like b whose vertex coordinates are sufficiently close to those that make qu minimum. After the initial simplex has been defined, we evaluate qu at each vertex finding two vertices xl and xh where qu has the lowest and highest values (respectively, qul and quh), as well as a third vertex xs where qu has the second highest value qus. Therefore, the algorithm will find the minimum of qu discarding the vertex xhwhere qu is higher and replacing it with one point where it is lower. By doing so, a new simplex is defined whose new vertex xh is also ready to be discarded when the function qu takes on, in the subsequent simplex, a quh value lower than in the simplex of the previous step. The operations that search for a new vertex for the new simplex are named reflection, contraction, and expansion. Discarded xh, reflection needs to find the centroid x¯ of the remaining vertices and to look for the new vertex xr in the line joining xh to x¯, namely, xr=x¯+αx¯−xh, where α is usually assumed to be 1. Then, calculation of qur=quxr is needed. If qul<qur≤quh, this is good and xr replaces xh. If qur<qul, this is better and we can try to further decrease qu in a point xe taken in the same direction by expansion, namely, xe=x¯+βxh−x¯, where β is usually assumed to be 2. Now, if qus<qur, the expansion has attained its purpose and xe replaces xh; otherwise, xr will replace xh.Finally, ifqus<qur<quh, after replacing xh by xr, we make contraction finding xc=x¯+γxh−x¯, with 0<γ<1 and usually chosen to be 1/2; otherwise qur≥quh, we make contraction with no replacement. After contraction, if que>quh, the algorithm must be restarted after replacing all the vertices by a new set of points xl=xi+1/2xl−xi, i=0,1,…,n; otherwise, we simply replace xh by xc. After this whole process, we have defined a new simplex. In this way, at each step, the algorithm searches for a new simplex until the distance between the vertices is so small and quh so close to qul that convergence is reached when(10)11+n∑i=0nqui−qux¯21/2<ε,the lower ε, the better.By then considering the example of Figure1, having defined the geometry of the trapezoidal R/C slab through assigning A=5.00m, B=3.00 m, and α=60° and having assumed μ=0.7 for the ratio between weaker and stronger steel reinforcement and by using the optimization algorithm (i.e., the sequential simplex method), solving the system (3), one obtains φ1=16.61° and φ2=38.27° with Δ1=2.63 m and Δ2=2.37 m, and qu=0.606mp. The total internal energy LInt per unit mp dissipated in the slab to form the mechanism is 3.033. The plot of the unconstrained objective function (9) for A=5.00 m, B=3.00 m, α=60°, and μ=0.7 is shown in Figure 5.Figure 5
Contour plot (a) and 3D diagram (b) of the objective functionqu calculated for the slab of Figure 1.
(a)
(b)


## 2.1.1. The Use of Zero-th Order Algorithms to Minimise the Objective Function: The Case of the Sequential Simplex Method

In this research, different evolutionary algorithms with zero-th order, such as genetic and particle swarm algorithms, as well as the sequential simplex method, have been considered. In solving the nonlinear system (3), different zero-th order algorithms have been compared, obtaining, as expected, very similar results. As a matter of fact, this holistic approach to optimizing the reinforcing grid inclination is only based on the physical laws governing the failure of concrete plates. Of course, different algorithms have different efficiencies and levels of robustness in solving system (3) but without affecting this holistic optimization method.The sequential simplex method was originally proposed by Bhatt et al. [4] and improved by Ramsay and Johnson [5]. It is less robust than the genetic and particle swarm algorithms, but it is more efficient because, even if the gradient is not calculated (it is a zero-th order algorithm), the steepness of the objective function is considered.In the analysed problem, the higher levels of robustness of the genetic and particle swarm algorithms were not quite as important due to the quick convergence, while the higher efficiency of the sequential simplex method was useful for saving computation time. The flowchart in Figure3 shows how the sequential simplex method solves system (3) to minimize qu.Figure 3
Flowchart of the sequential simplex method.The method starts by defining in then-dimensional space O, φ1,…,φn a simplex with n+1 vertices x0, x1,…,xn, each of them defined by a vector xi with n coordinates; the solution consists in finding the coordinate vector φ = [φ1,φ2,…,φn]T minimizing the function qu and therefore maximizing the function mp.Considerqu=quφi in the n-dimensional space O, φ1,…,φn. In this space, a simplex is an n+1-faced polyhedral figure with n+1 vertices 0, 1, …, n. For example, in the three-dimensional space O, φ1, φ2, and φ3, it is a tetrahedron with 3+1 vertices and 3+1 sides. If all the vertices have the same distance, we have a regular simplex. This means that in the two-dimensional space, the simplex is a triangle, and a regular simplex is an equilateral triangle. If qu=quφi is defined in the O, φ1,…,φn space, i = 1, …, n, then it has a real value in each simplex vertex x0, x1, …, xn. Hence, if we find a simplex with a very small distance between these vertices, the smaller the better; the n+1 values of qu in the n vertices are very similar. This means that, if we implement an algorithm which finds these values very close to the minimum of the function qu, then the minimum of qu is considered as achieved, depending on the required precision, that is, on the maximum vertex distance which must be lower than a given small value.Consider, for instance, the contour-plotted function in theO, φ1, and φ2 space of Figure 4, where the two simplices a and b are also defined. The simplex b is smaller than the simplex a, and the values of the function in each vertex of the simplex b are closer to the minimum than those of the simplex a. Starting from a simplex like the simplex a, the sequential simplex method searches for another simplex like the simplex b, where the function values (e.g., the values of function qu) calculated at the simplex vertices are closer to the minimum than those of the vertices of the simplex a. Therefore, the smaller the simplex and the closer to the minimum function values at its vertices, the better is the approximation of the minimum function value.Figure 4
Contour-plotted function with the simplexa and the simplex b in a two-dimensional space.Therefore, it must be implemented an algorithm that starting from an assigned simplex likea finds an enough small simplex like b whose vertex coordinates are sufficiently close to those that make qu minimum. After the initial simplex has been defined, we evaluate qu at each vertex finding two vertices xl and xh where qu has the lowest and highest values (respectively, qul and quh), as well as a third vertex xs where qu has the second highest value qus. Therefore, the algorithm will find the minimum of qu discarding the vertex xhwhere qu is higher and replacing it with one point where it is lower. By doing so, a new simplex is defined whose new vertex xh is also ready to be discarded when the function qu takes on, in the subsequent simplex, a quh value lower than in the simplex of the previous step. The operations that search for a new vertex for the new simplex are named reflection, contraction, and expansion. Discarded xh, reflection needs to find the centroid x¯ of the remaining vertices and to look for the new vertex xr in the line joining xh to x¯, namely, xr=x¯+αx¯−xh, where α is usually assumed to be 1. Then, calculation of qur=quxr is needed. If qul<qur≤quh, this is good and xr replaces xh. If qur<qul, this is better and we can try to further decrease qu in a point xe taken in the same direction by expansion, namely, xe=x¯+βxh−x¯, where β is usually assumed to be 2. Now, if qus<qur, the expansion has attained its purpose and xe replaces xh; otherwise, xr will replace xh.Finally, ifqus<qur<quh, after replacing xh by xr, we make contraction finding xc=x¯+γxh−x¯, with 0<γ<1 and usually chosen to be 1/2; otherwise qur≥quh, we make contraction with no replacement. After contraction, if que>quh, the algorithm must be restarted after replacing all the vertices by a new set of points xl=xi+1/2xl−xi, i=0,1,…,n; otherwise, we simply replace xh by xc. After this whole process, we have defined a new simplex. In this way, at each step, the algorithm searches for a new simplex until the distance between the vertices is so small and quh so close to qul that convergence is reached when(10)11+n∑i=0nqui−qux¯21/2<ε,the lower ε, the better.By then considering the example of Figure1, having defined the geometry of the trapezoidal R/C slab through assigning A=5.00m, B=3.00 m, and α=60° and having assumed μ=0.7 for the ratio between weaker and stronger steel reinforcement and by using the optimization algorithm (i.e., the sequential simplex method), solving the system (3), one obtains φ1=16.61° and φ2=38.27° with Δ1=2.63 m and Δ2=2.37 m, and qu=0.606mp. The total internal energy LInt per unit mp dissipated in the slab to form the mechanism is 3.033. The plot of the unconstrained objective function (9) for A=5.00 m, B=3.00 m, α=60°, and μ=0.7 is shown in Figure 5.Figure 5
Contour plot (a) and 3D diagram (b) of the objective functionqu calculated for the slab of Figure 1.
(a)
(b)


## 2.2. Application of the Yield-Line Method Using a Constrained Optimization Algorithm

When evaluatingqu, the variable φi, i = 1, …, n, may have to be constrained. In fact, to respect the assigned mechanism, some φi could be defined only in a specific interval and/or an expression containing some variables could be limited.In general, the functionqu can be subject to both unequal constraints sjφ≤0, j=1,…,m and equal constraints hkφ=0, k=1,…,l. These constraints can be taken into account through a penalty function P(φ). Some constrained structural engineering problems solved by using penalty function methods are shown by Cetin et al. [20].Using virtual works in the yield-line method, the exterior penalty function method can be incorporated in the sequential simplex method. To this aim, a penalty functionPφ is added to the function quφ, thus obtaining the new function(11)Φφ,rp=quφ+rpPφ,where Φφ,rp is named the pseudo-objective function and rp is a scalar multiplier much higher than 1 (namely, in this kind of problems, chosen to be between 100 and 1000), while the m unequal constraints sjφ≤0 and the l equal constraints hkφ=0 are taken into account through the exterior penalty function Pφ:(12)Pφ=∑j=1mmax0,sjφ2+∑k=1lhkφ2.If the constraints are not violated at any point near the minimum, the penalty functionPφ is not active, its value is zero, and Φφ,rp=quφ. This is the case of unconstrained problems, where, depending on the mechanism typology, qu reaches its minimum without being constrained.On the contrary, ifΦφ,rp is constrained when reaching the minimum, the penalty function Pφ becomes active. This means that at the points where the constraints are violated, qu cannot reach its unconstrained minimum because the related mechanism is not possible and the problem is not defined (i.e., the mechanism of Figure 6(b)). In this case, the penalty function Pφ assumes a positive real value when violating the relative constraint, thus making Φφ,rp increase greatly, also depending on the chosen value of rp.Figure 6
Effect of the penalty function coupled to the sequential simplex method when using the yield-line method for analysis of a trapezoidal R/C slab. (a) Constrained solution. The pseudo-objective function value isΦφ1=10.53°;φ2=21.08°=0.347with P(x) = 0. (b) Unconstrained solution kinematically inadmissible. The pseudo-objective function value after incrementing the parameter values of one degree ⇒Pφ≠0 and Φφ1=11.53°;φ2=22.08°;rp=100=2.037.Actually, by assigning different dimensions to the slab of Figure1, qu do not always reach its unconstrained minimum with the previously assigned mechanism. Therefore, in this case, the constraints to the objective function qu become active.For instance, this occurs for a trapezoidal R/C slab, similar to the previous one withμ=0.7, but with A=2.00 m, B=3.50 m, and α=80° (Figure 6). In this case, using the sequential simplex method coupled with the exterior penalty function method, one obtains qu=2.170mp with φ1=10.53°, φ2=21.08° and with Δ1=1.27 m, Δ2=1.35 m. The length of the free edge A+Bcotα=2.62 m is equal to the sum Δ1+Δ2. In other words, a constrained minimum is reached with Φ=0.347 for a unitary value of mp and Pφi=0 (Figure 6(a)). This means that the penalty function has automatically discarded solutions with Δ1+Δ2 > A+Bcotα. In fact, disregarding the constraints, that is without coupling any penalty function to the objective function, the unconstrained minimum would have been reached with φ1=12.30°, φ2=21.32°, Δ1=1.38 m, Δ2=1.37 m, and Δ1+Δ2=2.75 m. However, this is impossible because the constraint Δ1+Δ2≤A+Bcotα would have been violated and another slab region, kinematically inadmissible, would have been generated (Figure 6(b)). By instead increasing the parameter values φ1=10.53° and φ2=21.08° obtained in the constrained solution (i.e., by adding one degree to each parameter), that is, setting φ1=11.53° and φ2=22.08°, the violation of the constraint makes Pφi≠0, thus greatly increasing the pseudo-objective function Φφi,rp. For instance, for rp=100, one obtains Φφ1=11.53°,φ2=22.08°,rp=100=2.037, which is almost six times the value Φ=0.347 obtained in the minimum, meaning that the solution found is actually a constrained solution.Therefore, having assigned the mechanism, ifqu reaches its unconstrained minimum and, on slightly increasing the parameters, we still have Pφ=0 and Φφ,rp=quφ; then, the collapse mechanism does not need to be constrained, and no penalty function would be necessary. Otherwise, when qu reaches a constrained minimum, a slight increase of the parameters leads to Pφ≠0. In this case, the penalty function becomes active, thus highly increasing Φφ,rp. This means that, in general, the collapse mechanism must be constrained in order to find the actual solution.It must be noted that, for the given mechanism with one inner trapezoidal slab region and two lateral triangular ones, the constrained minimum of Figure6(a) led to three triangular slab regions where the inner trapezoidal region diverged into a triangular one. This constrained minimum was reached after having seen that lower values of φ1 and φ2 led to higher values of the objective function qu.Nevertheless, doesqu attain the constrained minimum for this diverged mechanism with three triangular slab regions? No, it cannot, because this would be true for the given mechanism, but the kinematic analysis shows that the slab also admits another failure mechanism with an inner triangular slab region and two lateral trapezoidal ones having a side in common intersecting the free side of the slab (Figure 7). For kinematic reasons, this side in common must lie on a straight line passing on the point where the rotation axes of the two trapezoidal slab regions meet each other. This means that in this mechanism, the two yield lines defining the triangular region are always constrained to meet each other along the straight line passing through two points: the first one where these two yield lines meet and the second one where the lines overlapping the rotation axes of the two trapezoidal slab regions meet each other (Figure 7).Figure 7
Failure mechanism with an inner triangular slab region and two lateral trapezoidal ones having a side in common intersecting the free side.The kinematically admissible mechanisms of this problem are hence two, a first one with the inner trapezoidal region and two lateral triangular regions (i.e., in Figure1) and a second one with the inner triangular region and two lateral trapezoidal regions (Figure 7). Therefore, the minimum of qu must be searched for both mechanisms by varying φ1 and φ2 without violating the constraints. For both mechanisms, the optimization algorithm will search for the minimum of qu for φ1>0 and φ2>0 until, in both cases, the mechanism diverges forming three triangular slab regions.Similarly to the first mechanism and following the same procedure, when considering the second mechanism illustrated in Figure7, the total internal work is to be obtained by adding the internal works calculated for each of the three slab regions. For this aim, the direction of their rotation vectors ϑi, i = 1, …, 3, and their plastic moments mpi and μmpi are to be evaluated, as well as the magnitude of the rotation vectors ϑi.After identifying the centroid of each slab region, the three related external works are evaluated and added to each other to obtain the total external work. By then adding the total internal work to the total external work, the objective functionqu(φ1, φ2) is explicited and minimized through using the sequential simplex method.The results show that for the slab dimensions in Figures3 and 7, the minimum qu is obtained for φ1=30.71° and φ2=37.65° with Δ1 = Δ2 = 1.31 m. This yield-line pattern occurs for qu=1.891mp. This ultimate load is thus lower than 2.17mp, previously obtained from the other mechanism at its boundary for which also its inner trapezoidal slab region becomes triangular. This means that this first mechanism cannot occur for qu/mp=2.17 because for qu/mp=1.891, the slab would have already failed with the second mechanism shown in Figure 7.


## 3. Choice of the Best Reinforcement Direction

Since the yield-line method locates the narrow bands in the slab where plastic deformations are concentrated, and the yield lines are therefore spontaneously selected by the mechanism as “weakness” lines in the slab at yield, this method can be suitably extended to determine the most unfavourable reinforcement arrangement, that is, the “weakest” slab reinforcement arrangement at yield with the lowest ultimate loadqu.A heuristic optimization method for determining the best direction of the reinforcement grid is hence proposed. The grid orientation is indicated through a further parameter whose most unfavourable value is, in general, unknown. Since the mechanism is identified throughn unknown parameters φi, i = 1, …, n, this further unknown parameter φn+1 increases the number of unknowns by 1. Therefore, n + 1 equations are needed to identify the values of all the n + 1 unknown parameters.Therefore, through a heuristic point of view, once the parameterφn+1 is known, the stronger reinforcement Asx should replace the weaker reinforcement Asy and vice versa, thus giving the best orientation to the reinforcement grid. The simple example described in the following section illustrates the application of this heuristic optimization method.


### 3.1. An Example of the Application of the Proposed Heuristic Optimization Method for Identifying the Best Grid Orientation by Using the Yield-Line Method for Analysis

Consider another simple example of a slab similar to the previous one in Figure1, only differing from it by the orientation of the reinforcement grid that, in general, can be at an angle to the free edge, identified by an additional parameter φ3. Therefore, φ3 is also the slope to the direction orthogonal to the free edge of the weaker reinforcement with area Asy and direction y (Figure 8).Figure 8
Trapezoidal R/C slab with reinforcement grid at an angle to the free edge. The angle between directiony of the weaker reinforcement and the direction at a right angle to the free edge is φ3, assumed as an additional parameter in the optimization algorithm.Figure9 shows the three slab regions of the slab of Figure 8 separately, their respective rotation axes with the related rotation vectors ϑ1, ϑ2, and ϑ3, as well as the plastic moments mp and μmp along the yield lines. The distance of the centroid of each slab region from its rotation axis, the modulus of the rotation vectors, and therefore the total external work is also, of course, the same as that of the slab in Figure 1 with the given reinforcement direction. For each of the three slab regions in Figure 9, the internal works of the plastic moments mp and μmp are(13)LInt1=mpl1cosφ1+φ3⋅ϑ1+μmpl1sinφ1+φ3⋅ϑ1,LInt2=mpl2cosφ2−φ3⋅ϑ2+μmpl2sinφ2−φ3⋅ϑ2,LInt3=mpl1cosφ1+φ3+l2cosφ2−φ3⋅ϑ3+μmpl1sinφ1+φ3+l2sinφ2−φ3⋅ϑ3.Figure 9
Slab regions with the reinforcement at an angle to the free edge.On summing up these three contributions, the total internal work is(14)
L
Int
=
m
p
cos
φ
1
+
φ
3
cos
φ
1
sin
α
+
φ
3
Δ
1
sin
α
−
sin
φ
3
B
+
cos
φ
2
−
φ
3
cos
φ
2
cos
φ
3
Δ
2
+
sin
φ
3
B
B
+
μ
m
p
sin
φ
1
+
φ
3
cos
φ
1
cos
φ
3
B
−
cos
α
+
φ
3
Δ
1
sin
α
+
sin
φ
2
−
φ
3
cos
φ
2
cos
φ
3
B
−
sin
φ
3
Δ
2
B
.By setting up the total external work equal to the total internal work and extractingqu, the objective function qu=quφ1,φ2,φ3,mp, subject to the three constraints −90°−α°≤φ1, 0≤φ2, and Δ1+Δ2≤A+Bcotα, is obtained.Minimizingqu, one obtains φ1=9.67°, φ2=33.18°, and φ3=107.69° with Δ1=2.24 m, Δ2=1.96 m, and qu=0.486mp. The total internal energy LInt per unit mp is 2.625 (Figure 10).Figure 10
Most unfavourable reinforcement arrangement.This means that the mechanism has spontaneously selected the slab’s “weakness” lines (yield lines), thus identifying their positions through the anglesφ1 and φ2, as well as the position of the weaker reinforcement Asy. Therefore, supposing an absurd scenario where someone wants to weaken the slab, the weaker reinforcement should be placed at an angle φ3=107.69° to the direction orthogonal to the free edge (Figure 10).Once the reinforcement grid orientation that would weaken the slab is identified, if one instead wants to better reinforce the slab, the stronger reinforcementAsx must be positioned in place of the weaker (that is, at an angle of 107.69° in the direction at a right angle to the free edge), thus positioning the weaker reinforcement Asy with an inclination φ3=17.69°, that is, shifted 90∘ from the y direction (φ3=107.69°), previously obtained through the sequential simplex method.In other words, contrary to what is required by the algorithm in order to weaken the slab, to better reinforce the slab, the reinforcementAsx must be positioned in place of Asy , that is, with an inclination of 17.69° to the free edge and the slab side orthogonal to the free edge (see Figure 11). The ultimate load related to this best reinforcement arrangement is then obtained by minimizing the objective function qu=quφ1,φ2,φ3=17.69°,mp by using the optimization algorithm but with only the two variables, φ2 for a given φ3. It is thus found that φ1=17.39° and φ2=37.44∘ (with Δ1=2.67 m, Δ2=2.30 m), when qu=0.620mp. The ratio LInt over mp becomes 3.112.Figure 11
Most favourable reinforcement arrangement.This simple example demonstrates that, as expected, the best reinforcement arrangement has the stronger reinforcementAsx, not parallel to the free edge but at an angle of 17.69°.The main aim of the simple example herein illustrated is to better explain the proposed heuristic optimization method. It is worth highlighting that when the algorithm is used with three parameters, it searches for the minimum ofqu, while also choosing the most unfavourable orientation φ3 of the reinforcement grid. Conversely, when two parameters are used, the algorithm searches for the minimum of qu for a given grid orientation, that is, for a given value of φ3. By then allowing φ3 to vary in the interval [0°, 180°] or symmetrically in the interval [0°, −180°] (the two variation intervals refer to the same grid orientations, see Figure 12), a parametric design of the orientation of the reinforcement grid can be performed. In so doing, after evaluating the ultimate load for any given value of φ3 (that is, the ratio qu/mp as a function of φ3), the optimum grid orientation can be determined. The results of the parametric design are shown in Figure 12. As previously obtained through the proposed optimization method, the minimum of qu/mp actually occurs for φ3 = 107.69° and, symmetrically, for φ3 = −71.31∘, i.e., for the same grid orientation. Conversely, the maximum of qu/mp is obtained for φ3 shifted by 90°, that is, for φ3=17.69°, meaning that the parametric optimization confirms the results obtained through the proposed heuristic optimization method.Figure 12
Variation of thequ/mp ratio as a function of φ3.The other grid orientations considered in Figure12 lead to intermediate values of the ratio qu/mp, shown to vary sinusoidally with φ3. The diagram shows that by taking a unit value of mp, the maximum value of qu is approximately 27.5% higher than the minimum. The reinforcement arrangement in Figure 1 with φ3=0 and a stronger steel parallel to the free edge demonstrates a good efficiency, with qu/mp being only 2.5% lower than the maximum (Figure 12).Finally, the cross-sectional area and the spacing of the grid reinforcement bars can be calculated. For this aim, let us assume a slab thickness at 14 cm (for instance,dx = 10 cm and a cover of 4 cm). The design value of the steel yield strength can be assumed to be fyd = 391 MPa that can be referred to as fyk = 450 MPa, B450 for Eurocode 2. The design factored load qud= 12.35 kN/m2 is also assumed.Considering the optimum direction of the steel grid (φ3=17.69°), the required cross-sectional area per unit length of the stronger reinforcement is determined to be Asxreq =  qud/(0.620·fyd·0.9·dx) = 5.66 cm2/m so that an actual stronger reinforcement Asx = ϕ12/20 cm can be provided. By instead considering the reinforcement grid in Figure 1 (φ3=0°), a reinforcement Asxreq = 5.85 cm2/m parallel to the free edge is required so that the bar spacing must be reduced to 19 cm for the same bar diameter ϕ12. Therefore, having assumed a reinforcement ratio μ = 0.7, since the ratio between the cross-sectional area of bars ϕ12 and bars ϕ10 is 0.695 (that is, almost 0.7), the provided weaker reinforcement Asy is then ϕ10/20 cm and ϕ10/19 cm for φ3=17.69°and φ3=0°, respectively. As expected, by comparing the steel weights of these two reinforcement arrangements, the weight of the optimum one is approximately 2.5% lower than that of the other, as it happens for the ratio qu/mp.The same goal is achieved for the trapezoidal slab of Figure7 with variable orientation of the reinforcement grid failing by the second mechanism considered in the previous section. Of course, the rotation axes position and direction are not affected by the variable orientation of the reinforcement grid. Moreover, for any φ3 including the grid orientation considered in the slab of Figure 7, the magnitude of the rotation vectors ϑ1, ϑ2, and ϑ3 does not vary by varying φ1 and φ2. The directions of the plastic moments mpi and μmpiare instead affected by the variation of the grid orientation. After evaluating the external and the internal works, following the procedure of the previous example of this section, the objective function qu=quφ1,φ2,φ3,mp is obtained. It is subject to the constraints −90°−α°≤φ1, 0≤φ2, and Δ1+Δ2≤A+Bcotα. For the dimensions of the slab of Figure 7, by minimizing qu, one obtains that the grid orientation more weakening the slab is φ3=95.30°, meaning that the weaker reinforcement Asy, initially parallel to the y-axis, rotates of an angle φ3=95.30 in the clockwise direction to weaken the slab as much as possible. The stronger reinforcement Asx is then positioned in place of the weaker Asy, that is, at an angle of 5.30° in the clockwise direction with respect to the x-axis (Figure 13). The inclination to the y-axis of the weaker reinforcement Asy is thus 5.30°, that is, shifted 90° from the direction previously obtained through minimizing the function qu=quφ1,φ2,φ3,mp. For given φ3 = 5.30°, the ultimate load qu related to this best grid orientation is then obtained by minimizing the objective function qu=quφ1,φ2,φ3=5.30°,mp with only the two variables φ1 and φ2 for given φ3 = 5.30°. One obtains qu=1.906mp with the yield-line pattern identified by φ1=30.56°, φ2=37.06° (with Δ1=1.30 m, Δ2=1.32 m). It can be noted that, as expected, in this case, the ratio of the ultimate load qu=1.906mp of the slab with optimum grid orientation φ3 = 5.30° over the ultimate load qu=1.891mp of the slab with φ3 = 0° is 0.99, that is, higher than in the previous case of the slab of Figure 1. Of course, this ratio tends to 1 for α tending to 90°, that is, when the trapezoidal slab tends to become a rectangular slab.Figure 13
Optimum grid orientation for the slab of Figure7.Regarding the saving of construction materials, the results obtained in the example (approximately 2-3%) are related to the analysed geometry and are comparable to other studies, for example, by Mancini [63] and by Bertagnoli et al. [66]. In large structures, even a small improvement could yield a significant saving of construction materials. A greater material saving from 5 to 10% can be obtained for different geometries, especially in complex buildings that are becoming quite frequent in contemporary architecture. This was confirmed by the research performed by Kabir et al. [56], whose work both numerically and experimentally investigated the effect of varying the direction of the reinforcements of skew slabs. They found that the best reinforcement orientation can improve the ultimate load up to 10%, thus leading to a proportional increase of the plastic moment, which means for a given ultimate load, there is an equivalent saving of the reinforcement cross-sectional area and steel weight.The structural design of contemporary architecture will likely require procedures for designing the reinforcement of concrete slabs with complex geometry, with an increasing amount of concrete slabs needing not only optimization of the reinforcement direction and of steel weight but also optimization of the production method, as in innovative tailor-made concrete structures [70] with all slabs having different reinforcement arrangements.


## 3.1. An Example of the Application of the Proposed Heuristic Optimization Method for Identifying the Best Grid Orientation by Using the Yield-Line Method for Analysis

Consider another simple example of a slab similar to the previous one in Figure1, only differing from it by the orientation of the reinforcement grid that, in general, can be at an angle to the free edge, identified by an additional parameter φ3. Therefore, φ3 is also the slope to the direction orthogonal to the free edge of the weaker reinforcement with area Asy and direction y (Figure 8).Figure 8
Trapezoidal R/C slab with reinforcement grid at an angle to the free edge. The angle between directiony of the weaker reinforcement and the direction at a right angle to the free edge is φ3, assumed as an additional parameter in the optimization algorithm.Figure9 shows the three slab regions of the slab of Figure 8 separately, their respective rotation axes with the related rotation vectors ϑ1, ϑ2, and ϑ3, as well as the plastic moments mp and μmp along the yield lines. The distance of the centroid of each slab region from its rotation axis, the modulus of the rotation vectors, and therefore the total external work is also, of course, the same as that of the slab in Figure 1 with the given reinforcement direction. For each of the three slab regions in Figure 9, the internal works of the plastic moments mp and μmp are(13)LInt1=mpl1cosφ1+φ3⋅ϑ1+μmpl1sinφ1+φ3⋅ϑ1,LInt2=mpl2cosφ2−φ3⋅ϑ2+μmpl2sinφ2−φ3⋅ϑ2,LInt3=mpl1cosφ1+φ3+l2cosφ2−φ3⋅ϑ3+μmpl1sinφ1+φ3+l2sinφ2−φ3⋅ϑ3.Figure 9
Slab regions with the reinforcement at an angle to the free edge.On summing up these three contributions, the total internal work is(14)
L
Int
=
m
p
cos
φ
1
+
φ
3
cos
φ
1
sin
α
+
φ
3
Δ
1
sin
α
−
sin
φ
3
B
+
cos
φ
2
−
φ
3
cos
φ
2
cos
φ
3
Δ
2
+
sin
φ
3
B
B
+
μ
m
p
sin
φ
1
+
φ
3
cos
φ
1
cos
φ
3
B
−
cos
α
+
φ
3
Δ
1
sin
α
+
sin
φ
2
−
φ
3
cos
φ
2
cos
φ
3
B
−
sin
φ
3
Δ
2
B
.By setting up the total external work equal to the total internal work and extractingqu, the objective function qu=quφ1,φ2,φ3,mp, subject to the three constraints −90°−α°≤φ1, 0≤φ2, and Δ1+Δ2≤A+Bcotα, is obtained.Minimizingqu, one obtains φ1=9.67°, φ2=33.18°, and φ3=107.69° with Δ1=2.24 m, Δ2=1.96 m, and qu=0.486mp. The total internal energy LInt per unit mp is 2.625 (Figure 10).Figure 10
Most unfavourable reinforcement arrangement.This means that the mechanism has spontaneously selected the slab’s “weakness” lines (yield lines), thus identifying their positions through the anglesφ1 and φ2, as well as the position of the weaker reinforcement Asy. Therefore, supposing an absurd scenario where someone wants to weaken the slab, the weaker reinforcement should be placed at an angle φ3=107.69° to the direction orthogonal to the free edge (Figure 10).Once the reinforcement grid orientation that would weaken the slab is identified, if one instead wants to better reinforce the slab, the stronger reinforcementAsx must be positioned in place of the weaker (that is, at an angle of 107.69° in the direction at a right angle to the free edge), thus positioning the weaker reinforcement Asy with an inclination φ3=17.69°, that is, shifted 90∘ from the y direction (φ3=107.69°), previously obtained through the sequential simplex method.In other words, contrary to what is required by the algorithm in order to weaken the slab, to better reinforce the slab, the reinforcementAsx must be positioned in place of Asy , that is, with an inclination of 17.69° to the free edge and the slab side orthogonal to the free edge (see Figure 11). The ultimate load related to this best reinforcement arrangement is then obtained by minimizing the objective function qu=quφ1,φ2,φ3=17.69°,mp by using the optimization algorithm but with only the two variables, φ2 for a given φ3. It is thus found that φ1=17.39° and φ2=37.44∘ (with Δ1=2.67 m, Δ2=2.30 m), when qu=0.620mp. The ratio LInt over mp becomes 3.112.Figure 11
Most favourable reinforcement arrangement.This simple example demonstrates that, as expected, the best reinforcement arrangement has the stronger reinforcementAsx, not parallel to the free edge but at an angle of 17.69°.The main aim of the simple example herein illustrated is to better explain the proposed heuristic optimization method. It is worth highlighting that when the algorithm is used with three parameters, it searches for the minimum ofqu, while also choosing the most unfavourable orientation φ3 of the reinforcement grid. Conversely, when two parameters are used, the algorithm searches for the minimum of qu for a given grid orientation, that is, for a given value of φ3. By then allowing φ3 to vary in the interval [0°, 180°] or symmetrically in the interval [0°, −180°] (the two variation intervals refer to the same grid orientations, see Figure 12), a parametric design of the orientation of the reinforcement grid can be performed. In so doing, after evaluating the ultimate load for any given value of φ3 (that is, the ratio qu/mp as a function of φ3), the optimum grid orientation can be determined. The results of the parametric design are shown in Figure 12. As previously obtained through the proposed optimization method, the minimum of qu/mp actually occurs for φ3 = 107.69° and, symmetrically, for φ3 = −71.31∘, i.e., for the same grid orientation. Conversely, the maximum of qu/mp is obtained for φ3 shifted by 90°, that is, for φ3=17.69°, meaning that the parametric optimization confirms the results obtained through the proposed heuristic optimization method.Figure 12
Variation of thequ/mp ratio as a function of φ3.The other grid orientations considered in Figure12 lead to intermediate values of the ratio qu/mp, shown to vary sinusoidally with φ3. The diagram shows that by taking a unit value of mp, the maximum value of qu is approximately 27.5% higher than the minimum. The reinforcement arrangement in Figure 1 with φ3=0 and a stronger steel parallel to the free edge demonstrates a good efficiency, with qu/mp being only 2.5% lower than the maximum (Figure 12).Finally, the cross-sectional area and the spacing of the grid reinforcement bars can be calculated. For this aim, let us assume a slab thickness at 14 cm (for instance,dx = 10 cm and a cover of 4 cm). The design value of the steel yield strength can be assumed to be fyd = 391 MPa that can be referred to as fyk = 450 MPa, B450 for Eurocode 2. The design factored load qud= 12.35 kN/m2 is also assumed.Considering the optimum direction of the steel grid (φ3=17.69°), the required cross-sectional area per unit length of the stronger reinforcement is determined to be Asxreq =  qud/(0.620·fyd·0.9·dx) = 5.66 cm2/m so that an actual stronger reinforcement Asx = ϕ12/20 cm can be provided. By instead considering the reinforcement grid in Figure 1 (φ3=0°), a reinforcement Asxreq = 5.85 cm2/m parallel to the free edge is required so that the bar spacing must be reduced to 19 cm for the same bar diameter ϕ12. Therefore, having assumed a reinforcement ratio μ = 0.7, since the ratio between the cross-sectional area of bars ϕ12 and bars ϕ10 is 0.695 (that is, almost 0.7), the provided weaker reinforcement Asy is then ϕ10/20 cm and ϕ10/19 cm for φ3=17.69°and φ3=0°, respectively. As expected, by comparing the steel weights of these two reinforcement arrangements, the weight of the optimum one is approximately 2.5% lower than that of the other, as it happens for the ratio qu/mp.The same goal is achieved for the trapezoidal slab of Figure7 with variable orientation of the reinforcement grid failing by the second mechanism considered in the previous section. Of course, the rotation axes position and direction are not affected by the variable orientation of the reinforcement grid. Moreover, for any φ3 including the grid orientation considered in the slab of Figure 7, the magnitude of the rotation vectors ϑ1, ϑ2, and ϑ3 does not vary by varying φ1 and φ2. The directions of the plastic moments mpi and μmpiare instead affected by the variation of the grid orientation. After evaluating the external and the internal works, following the procedure of the previous example of this section, the objective function qu=quφ1,φ2,φ3,mp is obtained. It is subject to the constraints −90°−α°≤φ1, 0≤φ2, and Δ1+Δ2≤A+Bcotα. For the dimensions of the slab of Figure 7, by minimizing qu, one obtains that the grid orientation more weakening the slab is φ3=95.30°, meaning that the weaker reinforcement Asy, initially parallel to the y-axis, rotates of an angle φ3=95.30 in the clockwise direction to weaken the slab as much as possible. The stronger reinforcement Asx is then positioned in place of the weaker Asy, that is, at an angle of 5.30° in the clockwise direction with respect to the x-axis (Figure 13). The inclination to the y-axis of the weaker reinforcement Asy is thus 5.30°, that is, shifted 90° from the direction previously obtained through minimizing the function qu=quφ1,φ2,φ3,mp. For given φ3 = 5.30°, the ultimate load qu related to this best grid orientation is then obtained by minimizing the objective function qu=quφ1,φ2,φ3=5.30°,mp with only the two variables φ1 and φ2 for given φ3 = 5.30°. One obtains qu=1.906mp with the yield-line pattern identified by φ1=30.56°, φ2=37.06° (with Δ1=1.30 m, Δ2=1.32 m). It can be noted that, as expected, in this case, the ratio of the ultimate load qu=1.906mp of the slab with optimum grid orientation φ3 = 5.30° over the ultimate load qu=1.891mp of the slab with φ3 = 0° is 0.99, that is, higher than in the previous case of the slab of Figure 1. Of course, this ratio tends to 1 for α tending to 90°, that is, when the trapezoidal slab tends to become a rectangular slab.Figure 13
Optimum grid orientation for the slab of Figure7.Regarding the saving of construction materials, the results obtained in the example (approximately 2-3%) are related to the analysed geometry and are comparable to other studies, for example, by Mancini [63] and by Bertagnoli et al. [66]. In large structures, even a small improvement could yield a significant saving of construction materials. A greater material saving from 5 to 10% can be obtained for different geometries, especially in complex buildings that are becoming quite frequent in contemporary architecture. This was confirmed by the research performed by Kabir et al. [56], whose work both numerically and experimentally investigated the effect of varying the direction of the reinforcements of skew slabs. They found that the best reinforcement orientation can improve the ultimate load up to 10%, thus leading to a proportional increase of the plastic moment, which means for a given ultimate load, there is an equivalent saving of the reinforcement cross-sectional area and steel weight.The structural design of contemporary architecture will likely require procedures for designing the reinforcement of concrete slabs with complex geometry, with an increasing amount of concrete slabs needing not only optimization of the reinforcement direction and of steel weight but also optimization of the production method, as in innovative tailor-made concrete structures [70] with all slabs having different reinforcement arrangements.


## 4. Conclusions

The use of the virtual-work method within the yield-line approach is generally advantageous because while providing an “upper bound” solution, it is usually sufficiently precise. In addition to the ultimate load, it also provides the exact collapse mechanism of the slab. Unfortunately, even when there is not a prohibitive number of unknown parameters defining the collapse mechanism, the nonlinear equation system defining the problem is difficult to solve through traditional methods.Therefore, a robust and efficient algorithm is needed. These difficulties have been avoided by using the sequential simplex method, a zero-th order algorithm that is proven not only to be robust but also very efficient. Since this algorithm is efficient and robust and is slightly influenced by the increase in the number of parameters defining the collapse mechanism, in this paper, the yield-line method was extended by adding a further parameter defining the direction of the reinforcement grid. In fact, since the yield lines are narrow bands in the slab where plastic deformations are concentrated and whose actual position minimizes the ultimate load with respect to any other possible position, when the ultimate load is minimized as a function of both the position of the yield lines and the orientation of the reinforcement grid, this additional parameter identifies the most unfavourable orientation of the reinforcement grid corresponding to the minimum of the ultimate load. This means that the yield-line method can identify not only the orientation of the yield lines that minimize the ultimate load (that, therefore, further “weakens” the slab) but also the orientation of the reinforcement grid that, by contributing to minimizing the ultimate load, must be the most unfavourable among all the possible grid orientations, thus “weakening” the slab as much as possible.In so doing, it was possible to heuristically optimize the grid’s orientation through positioning the stronger reinforcement in place of the weaker one by rotating the reinforcement grid by 90°. Therefore, the optimization algorithm (i.e., any optimization algorithm as the sequential simplex method used in this study) searches for the worst reinforcement grid orientation (the one for which the ultimate load is minimum), while the heuristic optimization method herein proposed finds the best grid orientation, which is the one for which the ultimate load is at a maximum.The method has shown to be effective at optimizing the reinforcement direction in R/C slabs, as confirmed by the subsequently performed parametric design. In fact, for any possible orientation of the reinforcement grid, the yield-line method was used for the analysis of the R/C slab by assigning the grid orientation at each time. The results confirm those obtained through the heuristic optimization method herein proposed. Since at each stage and for each grid orientation, this parametric optimization procedure is performed using the yield-line method for analysis in its currently used application without any additional parameters accounting for the reinforcement direction; the results of the parametric optimization reliably confirm those of the heuristic optimization procedure.



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*Source: 6017146-2019-11-18.xml*