Advances in Civil Engineering

Volume 2019, Article ID 6017146, 15 pages

https://doi.org/10.1155/2019/6017146

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

Correspondence should be addressed to Bruno Briseghella; nc.ude.uzf@onurb

Received 26 April 2019; Revised 6 October 2019; Accepted 17 October 2019; Published 18 November 2019

Academic Editor: Hayri Baytan Ozmen

Copyright © 2019 Luigi Fenu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### 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.

#### 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 *q*_{u} 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 Section 2, 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 *q*_{u} 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 are , and , , respectively. Say *f*_{yd} is the design yield strength of the reinforcement steel. The plastic moment in the two directions (the principal with stronger reinforcement, cross-sectional area , effective depth *d*_{x}, and orthogonal one) can then be defined aswith *μ* = *d*_{y}/(*d*_{x}).

For the virtual-work method, the external and internal work must be equal, namely,

This equation relates the plastic moment of the stronger reinforcement and that in the orthogonal direction to the ultimate load *q*_{u} and the *n* parameters defining the collapse mechanism, that is, or, in a dual way, , .

Therefore, the problem is to find the *n* parameters that minimize , that is, to solve the system:

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 is a nonlinear function of , and system (3) is nonlinear as well. This system can instead become more complicated to be solved when the number *n* of parameters , 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 parameters , *i* *=* 1, …, *n*, which depend on the collapse mechanism. The number of parameters 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 are found (and therefore the exact geometry of the mechanism), together with the actual value of the plastic moment or, dually, of the ultimate load .

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 Figure 1 with three supported edges, a free one, and a uniformly distributed load .