Title Meshfree Sequentially Linear Analysis of Concrete

3 New meshfree method employing the Node-based Smoothed Point Interpolation Method 4 (NS-PIM) is presented as an alternative to the non-linear finite element approach for concrete 5 members. The non-linear analysis is replaced by sequentially linear analyses (SLA), and 6 smeared, fixed concrete cracking model was used. A notched concrete beam was employed 7 for validation. Using a crack band width factor of 2.0 and a 10 mm nodal spacing, the peak 8 load differed by only 3.5% from experimental ones. Overall results were similar to experi9 mental ones, as well as to those published by researchers using finite element SLA. The ap10 proach provides two major advantages over finite element-based SLA: (1) nodal distortion 11 insensitivity and (2) nodal spacing insensitivity. 12 13 Introduction 14 15 The finite element method (FEM) is the most widely used numerical method to study linear 16 and non-linear behaviour (for both materials and geometric components) of structures. The 17 method, in its application to non-linear structural analysis, has matured sufficiently to be the 18 basis of many commercial software packages (ANSYS, Abacus, ATINA, etc.). Despite sig19 nificant progress in its theoretical and numerical aspects, some weaknesses persist. These can 20 be summarised as follows: 21 1 A. Salam Al-Sabah, Ph.D., Research Scientist, Urban Modelling Group, School of Civil, Structural and Environmental Engineering, University College Dublin, Newstead, room G67, Belfield, Dublin 4, Ireland, [email protected] 2 Debra F. Laefer, Ph.D., Professor, Head Urban Modelling Group, School of Civil, Structural and Environmental Engineering, University College Dublin, Newstead, room G25, Belfield, Dublin 4, Ireland, [email protected] (corresponding author)  Results are mesh-dependent, with good results requiring a high quality mesh and each 22 element’s geometry satisfying shape and aspect ratio limits. 23  Models are stiffer than the actual structures. Hence, displacements are underestimat24 ed. 25  In analysis of geometric non-linearity, elements can become distorted sufficiently to 26 compromise output accuracy. 27  Crack propagation usually requires re-meshing, and the robustness of automatic re28 meshers is questionable, particularly in three-dimensional problems. 29 30 Modelling of reinforced concrete is an important topic, as it is one of the most widely used 31 composite materials in construction. Predicting its behaviour is complicated by factors such 32 as reinforcement yielding, non-linear reinforcement-concrete bond behaviour, non-linear be33 haviour of concrete in compression, and tension cracking of the concrete. This last aspect 34 contributes most significantly to the early, non-linear behaviour of reinforced concrete beams 35 and slabs. The application of non-linear FEM in the analysis of reinforced concrete structures 36 can be traced back to the 1960s when the first reinforced concrete finite element model which 37 includes the effect of cracking was developed by Ngo and Scordelis (1967). 38 39 When loaded in tension, concrete fails suddenly after reaching its tensile limit. The heteroge40 neous nature of concrete results in a quasi-brittle behaviour that is greatly affected by soften41 ing damage (Bazant and Jirásek 2002). To represent this, several fracture models have been 42 proposed as summarized by Rots and Blaauwendraad (1989). An important component of 43 these models is the Fracture Process Zone (FPZ), defined as the zone ahead of the crack tip in 44 which concrete undergoes softening behaviour due to microcracking. Two widely used crack45 ing models are the Fictitious (or cohesive) Crack Model (FCM) introduced by Hillerborg et 46 al. (1976), and Crack Band Model (CBM) as proposed by Bazant and Oh (1983). In the first 47 model, the FPZ is represented as a fictitious line that can transmit normal stress. Fracture en48 ergy is then expressed as a function of critical crack separation (or opening width, w ) (Bazant 49 and Jirásek 2002). In the CBM, fracturing is modelled as a band of parallel, densely distribut50 ed microcracks in the FPZ that has a certain width, which is referred to as the crack band 51 width (Bazant and Oh 1983). The average strain over the FPZ can be related to its defor52 mation through the crack band width. The fracture energy can then be represented as a func53 tion of a stress-strain curve and the crack band width. 54 55 Concrete fracture models, combined with non-linear models for concrete and steel are typi56 cally combined with the FEM to produce numerical procedures for non-linear analysis of re57 inforced concrete. Early efforts to overcome this encountered two main challenges. The first 58 was the numeric instability due to tensile cracking. The second related to the softening por59 tion of the behaviour. The first was solved by adopting the incremental-iterative solution 60 method (Crisfield 1996), where the unbalanced forces were allowed to dissipate through solu61 tion iterations. Since the second resulted from the negative tangent stiffness of the softening 62 part of behaviour, it generated an unstable equilibrium with associated numerical issues in 63 solving the stiffness equation. To surmount this, several methods were initially proposed to 64 control the load or the displacement (Crisfield 1996). Prominent amongst these were the arc 65 length method (Crisfield 1996; Riks 1979) and its variations, the minimum residual dis66 placement method (Chan 1988), and the line search method (Crisfield 1996). Yet challenges 67 remained. These non-linear solution methods required the specification of many control pa68 rameters, which depended upon user experience and did not guarantee convergence. Inherent 69 to this are expectations that the user is a highly knowledgeable and experienced practitioner 70 and that the results are obtained after many mesh and parameter refinement attempts. This is 71 particularly true for concrete, where the sudden release of strain energy due to tensile crack72 ing can cause the numerical solution to fail. As such, the aim of this paper was to implement 73 an alternative to non-linear FEM in its application to concrete members. 74 75 Methodology 76 77 The following paragraphs describe the background and details of the particular meshfree 78 method adopted for this analysis, as well as the sequentially linear analysis method that was 79 employed. 80 81 Mesh free methods 82 When the FEM was introduced in the 1950s, the most widely used numerical method for 83 solving differential equations was the finite difference method (Courant et al. 1967). This 84 strong-form method had a simple mathematical foundation and was easy to implement nu85 merically. The main previous limitation was the need for a regular grid of points to define the 86 analysis domain. These limitations added to the general acceptance of the FEM as a better 87 and more flexible alternative. Although further research related to finite difference overcame 88 the necessity of a regular grid (Liszka and Orkisz 1980), the FEM came to dominate popular 89 usage because of its ability to define complicated geometries, its basis on a robust mathemat90 ical foundation, and its ease in conducting error analyses (Thomée 2001). 91 92 A fundamental alternative came in the form of meshfree methods. The first member of the 93 group was the Smoothed-Particle Hydrodynamics (SPH) (Gingold and Monaghan 1977; Lucy 94 1977) in 1977, which was initially applied in solving astrophysical problems. Since then, 95 multiple meshfree methods have been proposed (e.g. Li and Mulay 2013; Liu 2009). These 96 vary in their formulation procedure (strong, weak, weakened weak, or boundary integral) and 97 their local function approximations (moving least square, integral, differential, point, or parti98 tion of unity). Despite its name, most meshfree methods still require background cells to con99 duct the numerical integration of the system matrices. However, the meshfree methods that 100 are based on a strong formulation (e.g. the irregular finite difference method, the finite point 101 method, and local point collocation methods) do not usually require background cells. Unfor102 tunately, most of these methods suffer from reduced accuracy and instability due to node ir103 regularity (Atluri and Zhu 1998; Liu 2009). 104 105 In the work presented herein, a special meshfree Point Interpolation Method (PIM) called the 106 meshfree Node-based Smoothed Point Interpolation Method (meshfree NS-PIM) is used. The 107 method was first developed by Liu et al. (2005) under the name Linearly Conforming Point 108 Interpolation Method (LC-PIM). This was later changed to the Nodal Smoothing Operation 109 (2009) to distinguish it from the Edge-based Smoothed Point Interpolation Methods (ES110 PIM). Further details on NS-PIM are presented by Liu and Zhang (2013). 111 112 Meshfree NS-PIM was formulated using polynomial basis functions that have the Kronecker 113 delta function property, which allowed straightforward implementation of the essential 114 boundary conditions. Furthermore, the Generalized Smoothed Galerkin (GS-Galerkin) weak 115 form was used, which allowed use of incompatible assumed displacement functions. The 116 method is linearly conforming, with upper bound results that are free from volumetric locking 117 (Liu, 2009). 118 119 In NS-PIM, as in PIM, the displacement, h u , of any domain point, x , is approximated using 120 a shape (interpolation) function, ) (x I  . This function operates within a small local domain 121 around x (the support domain). The function interpolates the nodal displacement, I u , of the 122 nodes within the support domain of x (or the support nodes, n S ): 123