Sequentially Linear Analysis (SLA), an event-by-event solution strategy in which a sequence of scaled linear analyses with decreasing secant stiffness is performed, representing local damage increments; is a robust alternative to nonlinear finite element analysis of quasi-brittle structures. Since it is based on a fixed smeared crack constitutive model, severe spurious stresses and inaccuracies may develop due to misalignment of the crack with the principal stress directions. To this end, the elastic-brittle fraction model was conceived. The model separates the continuum into several parallel fractions or layers, each with different properties, chosen in order to represent the overall constitutive softening behaviour as accurately as possible. The main idea is to mimick a rotating crack by a superposition of fractions, each with a fixed crack direction. In this article, the model is presented for both the 2-dimensional and 3-dimensional frameworks, with a general transition from any saw-tooth law to fraction material properties. The fraction models are then validated and compared against the fixed crack model with SLA: using single element and structural case studies. It is shown that the fraction model is able to mimick the rotating crack model, that it leads to lesser spurious cracks and narrower localisation bands, and in turn results in a more flexible post-peak response over all case studies compared to the fixed crack model.

In the finite element modelling of masonry structures, the micro-modelling technique of differentiating the continuum into a linear elastic bulk, and interfaces representing non-linear joints is common. However, this approach of simulating cracking-crushing-shearing failure possibilities in interfaces, typical of damage in masonry, also poses numerical stability issues due to the quasi-brittle nature of the failure. In this regard, the article proposes the use of numerically robust sequentially linear procedures and a suitable discretised tension-shear-compression failure model for interfaces. Sequentially linear solution procedures describe the nonlinear response of a specimen/structure through a sequence of scaled linear analyses, each of which represents locally applied damage increments, using secant-stiffness based discretised constitutive relations called saw-tooth laws. The constitutive formulation proposed herein includes a tension cut-off criterion combined with a uniaxial discretised softening law, a Coulomb friction criterion with a discretised cohesion softening law, and a compression cut-off criterion combined with a uniaxial discretised hardening–softening law. It is presented for both two-dimensional (2D) line interfaces and three-dimensional (3D) planar interfaces. The applicability of these formulations are illustrated using 2D and 3D models of a pushover analysis on a squat unreinforced masonry wall. The simulations are made using Sequentially Linear Analysis (SLA) and the Force-Release method, which are total (load-unload) and incremental sequentially linear methods respectively. The clear global softening in the force–displacement evolution and the localised brittle shear failure observed in the experiment are reproduced well and in a stable manner.

Sequentially linear solution procedures provide a robust alternative to their traditional incremental-iterative counterparts for finite element simulation of quasi-brittle materials. Sequentially linear analysis (SLA), one such non-incremental (total) approach, has been extended to non-proportional loading situations in the past few years. Although the process of damage propagation and localisation is often dynamic in nature, the simulation being quasi-static poses a fundamental problem. This article gives an overview of the different approaches to address non-proportional loading in SLA and other sequentially linear methods, and their corresponding redistribution methodologies to address the dynamic phenomenon. Furthermore, the inherent differences between two such methods: SLA (total) and the Force-Release method (incremental), and their suitability to structural continuum models involving non-proportional loading, are illustrated using real-life concrete and masonry experimental benchmarks tested up to and beyond brittle collapse. In each illustration, SLA is shown to enforce equilibrium during dynamic failure by load reduction, using the intermittent proportional loading, while allowing for active damage propagation resulting in a relaxed failure mechanism which manifests as snap-back(s). Contrarily, the Force-Release method is shown to describe the collapse through states of disequilibrium.

Sequentially linear analysis (SLA), an event-by-event procedure for finite element (FE) simulation of quasi-brittle materials, is based on sequentially identifying a critical integration point in the FE model, to reduce its strength and stiffness, and the corresponding critical load multiplier (λ _crit), to scale the linear analysis results. In this article, two strategies are proposed to efficiently reuse previous stiffness matrix factorisations and their corresponding solutions in subsequent linear analyses, since the global system of linear equations representing the FE model changes only locally. The first is based on a direct solution method in combination with the Woodbury matrix identity, to compute the inverse of a low-rank corrected stiffness matrix relatively cheaply. The second is a variation of the traditional incomplete LU preconditioned conjugate gradient method, wherein the preconditioner is the complete factorisation of a previous analysis step's stiffness matrix. For both the approaches, optimal points at which the factorisation is recomputed are determined such that the total analysis time is minimised. Comparison and validation against a traditional parallel direct sparse solver, with regard to a two-dimensional (2D) and three-dimensional (3D) benchmark study, illustrates the improved performance of the Woodbury-based direct solver over its counterparts, especially for large 3D problems.

Sequentially linear analysis (SLA), a non-incremental-iterative approach towards finite element simulation of quasi-brittle materials, is based on sequentially identifying a critical integration point in the model, to reduce its strength and stiffness, and the associated critical load multiplier (λ _crit ), to scale the linear analysis results. In this article, two novel methods are presented to enable SLA simulations for non-proportional loading situations in a three-dimensional fixed smeared crack framework. In the first approach, the cubic function in the load multiplier is analytically solved for real roots using trigonometric solutions or the Cardano method. In the second approach, the load multiplier is expressed as a function of the inclination of a potential damage plane and is deduced using a constrained optimization approach. The first method is preferred over the second for the validation studies due to computational efficiency and accuracy reasons. A three-point bending beam test, with and without prestress, and an RC slab tested in shear, with and without axial loads, are used as benchmarks. The proposed solution method shows good agreement with the experiments in terms of force-displacement curves and damage evolution.