Analysis and design of reinforced concrete structures as a topology optimization problem

Technical codes for buildings deal with cracked reinforced concrete structures assuming concrete as a compression–only material, whereas rebar provides the structural component with the required tensile strength [1]. Numerical methods can handle reinforced concrete structures calling for demanding non–linear analysis. Indeed, well–known convergence issues arise when copying with concrete as a compression–only material. Recently, an alternative energy–based approach has been proposed to solve the equilibrium of a linear elastic no– tension medium exploiting its hyper–elasticity [2]. A topology optimization problem distributes an equivalent orthotropic material to minimize the strain energy of the no-tension body, thus avoiding more demanding non–linear analysis. This contribution provides an extension to the analysis and optimal design of reinforced concrete structures. Following [3], truss members are modeled within a two–dimensional no–tension continuum in order to model structural elements made of reinforced concrete. The solution of the equilibrium is straightforward within the approach proposed in [2], thus allowing performing analysis at the serviceability limit state with cracked sections. Also, introducing the areas of the reinforcement bars as an additional set of unknowns, a problem of size optimization is outlined to cope with the optimal rebar of r.c. structures. Preliminary numerical simulations are shown to assess the proposed procedure

Conceptual Design of Diagrids and Hexagrids by Distribution of Lattice Structures

The conceptual design of grid systems in tall buildings is addressed by combining optimization and multiscale analysis of lattice structures. Macroscopic properties of lattices with given cross-section are available in the literature for different cell topologies. A multi-material optimization problem is formulated to find the distribution of a prescribed discrete set of candidate cross-sections and shapes such that the structural weight of the grid is minimized under constraints on the lateral displacements of the building. Preliminary numerical simulations are shown, addressing the design of tall buildings that employ diagrids and hexagrids

Mixed variational formulations for micro-cracked continua in the multifield framework

Within the framework of multifield continua, we move from the model of elastic microcracked body introduced in (Mariano, P.M. and Stazi, F.L., Strain localization in elastic microcracked bodies, Comp. Methods Appl. Mech. Engrg. 2001, 190, 5657–5677) and propose a few novel variational formulations of mixed type along with relevant mixed FEM discretizations. To this goal, suitably extended Hellinger-Reissner principles of primal and dual type are derived. A few numerical studies are presented that include an investigation on the interaction between a single cohesive macrocrack and diffuse microcracks (Mariano, P.M. and Stazi, F.L., Strain localization due to crack–microcrack interactions: X–FEM for a multifield approach, Comp. Methods Appl. Mech. Engrg. 2004, 193, 5035–5062)

On the use of a truly–mixed formulation in topology optimization with global stress–constraints

The work refers to the field of topology optimization for bidimensional structures and addresses the case in which global stress–constraints are considered to improve final designs. Most of the previous research tackles this topic relying on classical displacement–based finite elements where stresses are recovered via post–processing techniques. The work conversely investigates the use of a truly–mixed formulation where stresses are independent variables of the problem while displacements play the secondary role of Lagrangian multiplier. The implemented discretization is based on a composite triangular element whose features may be advantageously exploited in stress–constrained topology optimization. The discretization is checkerboard–free and allows to tackle topology optimization with element–based constraints without introducing any additional filtering technique. The high accuracy in the evaluation of the average stresses is expected to improve the efficiency of the numerical procedure, especially in the case of a single global constraint that has to govern the whole domain. The adopted discretization also passes the robustness condition even in the case of incompressible materials and this allows to menage strength constraints also for rubber–like components. Basing on these ideas, numerical investigations are carried out to test preliminary applications of the truly–mixed technique coupled with topology optimization and global stress–constraints. To handle the well–known singularity problem, that affects the constraints imposition, an alternative scheme is herein adopted instead of a classical "–relaxation. An example where a homogenous stress distribution is expected is firstly tested, having the aim of pointing out the main features of the proposed procedure. Afterwards, numerical simulations address a classical L–shaped specimen, pointing out pros and cons of the approach

Analysis of no-tension structures under monotonic loading through an energy-based method

An approach is proposed to estimate the collapse load of linear elastic isotropic no-tension 2D solids. The material is replaced by a suitable equivalent orthotropic material with spatially varying local properties. A non-incremental energy-based algorithm is implemented to define the distribution and the orientation of the equivalent material, minimizing the potential energy so as to achieve a compression-only state of stress. The algorithm is embedded within a numerical procedure that evaluates the collapse mechanisms of structural elements under monotonic loading. The accuracy of the method is assessed through comparisons with the “exact” results predicted by limit analysi

Analysis of masonry vaults as a topology optimization problem

Congreso celebrado en la Escuela de Arquitectura de la Universidad de Sevilla desde el 24 hasta el 26 de junio de 2015.An innovative approach is proposed to analyze 3D masonry vaults, assuming masonry to behave as a linear elastic no‐tension material. Masonry is replaced by a suitable equivalent orthotropic material with spatially varying elastic properties and negligible stiffness in any direction along which tensile stresses must be prevented. An energy‐based algorithm is implemented to define the distribution and the orientation of the equivalent material for a given load, minimizing the potential energy so as to achieve a purely compressive state of stress. The algorithm is embedded within a numerical procedure that performs a non‐incremental analysis under given loads. The collapse load of masonry structural elements can also be predicted running a sequence of independent analyses. The capabilities of the approach in predicting the crack pattern in typical masonry vaults are also shown

Simple Homogenization-Topology Optimization Approach for the Pushover Analysis of Masonry Walls

A topology optimized rigid triangular FE macro-model with non-linear homogenized interfaces for the pushover analysis of in plane loaded masonry is presented. The shape of the mesh and the position of the interfaces is evaluated through a topology optimization approach that detects the main compressive stress fluxes in the structure. Different values of the horizontal action are considered to derive an adaptive mesh or an optimal discretization that is suitable for multiple loads. Masonry properties are calibrated by means of a homogenization approach in the nonlinear range. To tackle elastic and inelastic deformations, interfaces are assumed to behave as elasto-plastic with softening in both tension and compression, with orthotropic behavior. The two-step procedure competes favorably with classic equivalent frame approaches because it does not require a-priori assumptions on the mesh and on the length of the rigid offsets. An example of technical relevance is discussed, relying into a multi-story masonry wall loaded up to failure

Optimal strengthening of masonry arch bridges with externally bonded reinforcing layers

Strengthening is a natural step following a failed bridge assessment. Referring to masonry bridges, a numerical tool is presented to find the optimal distribution of reinforcement to be externally bonded to two-dimensional elastic no-tension structural elements, with the aim of maximizing their overall stiffness. Notwithstanding the non-linearity of the adopted material model, no incremental procedure is needed to prescribe equilibrium of the strengthened element. Indeed, the same minimization procedure handles both the energy-based solution of the no-tension elastic body and the topology optimization problem that distributes the optimal reinforcement. A few numerical simulations are presented to assess the capabilities of the proposed procedure in defining the optimal reinforcement layouts for masonry arches and arch bridges, subjected to gravity loads and resting on fixed or elastic foundations. Designers can exploit the tool to sketch a preliminary layout of the FRP strengthening, which should be subsequently detailed according to technical codes

On the Virtual Element Method for Topology Optimization on polygonal meshes: a numerical study

It is well known that the solution of topology optimization problems may be affected both by the geometric properties of the computational mesh, which can steer the minimization process towards local (and non-physical) minima, and by the accuracy of the method employed to discretize the underlying differential problem, which may not be able to correctly capture the physics of the problem. In light of the above remarks, in this paper we consider polygonal meshes and employ the virtual element method (VEM) to solve two classes of paradigmatic topology optimization problems, one governed by nearly-incompressible and compressible linear elasticity and the other by Stokes equations. Several numerical results show the virtues of our polygonal VEM based approach with respect to more standard methods

Assessment and Reduction of the Seismic Vulnerability of a Stone Masonry Vault

A numerical approach is presented to assess the seismic vulnerability of barrel masonry vaults and evaluate the eectiveness of a traditional retrofitting intervention consisting in the reinforcement of the extrados. A linear elastic no–tension model is adopted to cope with the negligible strength in tension of ancient brick and stone masonry and perform a two–dimensional finite element analysis of arch–like sections. Instead of implementing conventional load history analysis or limit load analysis, the minimization of the relevant strain energy function is implemented to solve the non–linear equilibrium under the effect of dierent load scenarios. A segmental barrel vault made of stone masonry is investigated in an ancient building under static and seismic loads. The collapse load of the structural element is computed before and after the intervention and the reduction achieved in terms of seismic vulnerability is evaluated as prescribed by technical codes