Flexural analysis of reinforced fibrous concrete members using the finite element method

A numerical procedure based on the finite element method is developed for the geometric and material nonlinear analysis of reinforced concrete members containing steel fibres and subjected to monotonic loads. The proposed procedure is capable of tracing the displacements, strains, stresses, crack propagation, and member end actions of these structures up to their ultimate load ranges. A frame element with a composite layer system is used to model the structure. An iterative scheme based on Newton-Raphson's method is employed for the nonlinear solution algorithm. The constitutive models of the nonlinear material behaviour are presented to take into account the nonlinear stress-strain relationships, cracking, crushing of concrete, debonding and pull-out of the steel fibres, and yielding of the reinforcement. The geometric nonlinearity due to the geometrical change of both the structure and its elements are also represented. The numerical solution of a number of reinforced fibrous concrete members are compared with published experimental test results and showed good agreement.     

Optimal design of structural steel frameworks

This paper reviews work conducted at the University of Waterloo during the 1980s concerning the computer-automated design of least-weight structural steel frameworks. First, design under static loads is considered whereby the members of the structure are automatically sized using commercial steel sections in full conformance with design standard provisions for elastic strength/stability and stiffness. This problem is illustrated for the least-weight design of a steel mill crane framework comprised of a variety of member types and subject to a number of load effects. Then, the design methodology is extended to the least-weight design of structural steel frameworks under both service and ultimate loading conditions. Here, acceptable elastic stresses and displacements are ensured at the service-load level while, simultaneously, adequate safety against plastic collapse is ensured at the ultimate-load level. This design problem is illustrated for the least-weight design of an industrial steel mill framework for which plastic behaviour is governed by conservative piecewise linear yield conditions. Finally, the computer-based design methodology is extended to the least-weight design of structural steel frameworks subjected to dynamic loading. Constraints are placed on dynamic displacements, dynamic stresses, natural frequencies and member sizes. The design problem is illustrated for the least-weight design of a steel trussed arch subjected to non-structural masses and an impulse force.     

Nonlinear analysis of steel frameworks through direct modification of member stiffness properties

This paper presents a computer-based method for nonlinear analysis of planar steel frameworks under monotonic loading that is directly based on the matrix displacement method of analysis. Stiffness degradation factors progressively deteriorate the post-elastic bending, shearing and axial stiffness properties of framework members over an incremental load history until failure of part or all of the structure occurs. The analytical procedure is based on Timoshenko beam theory to allow for the analysis of frameworks for which effects of shear deformation on elastic behavior may be significant, and employs transcendental stability functions to model the effect of axial force on the bending stiffness of members. Also accounted for is the influence of residual stresses on the initial-yield and full-yield capacities of members, and the effect of both combined moment plus axial force and combined moment plus shear force on the post-elastic behavior of members. The nonlinear analysis method is applied for two benchmark planar steel structures, and it is shown to give results comparable to both experimental and analytical results previously published in the literature. The method is readily implemented on a computer using conventional matrix structural analysis techniques, and is directly applicable for the efficient nonlinear analysis of frameworks of any scale or complexity.     

Numerical and analytical investigation of collapse loads of laced built-up columns

Built-up columns are often used in steel buildings and bridges providing economical solutions in cases of large spans and/or heavy loads. Two main effects should be taken into account in their design that differentiate them from other structural members. One is the significant influence of shear deformations due to their reduced shear rigidity. The second is the interaction between global and local buckling. These effects are addressed here from both a numerical and an analytical point of view for laced built-up columns. It is concluded that the largest loss of capacity occurs when the local and global Euler critical stresses and the yield stress all coincide. This reduction in capacity becomes more prominent in the presence of imperfections, reaching magnitudes in the order of 50%. Despite the detrimental effects of mode interaction many major design codes do not provide sufficient pertinent guidance. In order to address this issue, a simple analytical method is proposed for calculating the collapse load of laced built-up members taking into account the above effects as well as global imperfections, local out-of straightness and plasticity, which is then verified by means of nonlinear finite element analysis, using either beam or shell elements. The proposed method is found to provide improved accuracy in comparison to EC3 specifications in cases of global elastic failure.     

Nonlinear thermal and mechanical analysis of edge effects in angle-ply laminates

A method has been developed for the thermal analysis of materially nonlinear Carbon Fibre Reinforced Plastic (CFRP) laminated plates using a quasi-three-dimensional iso-parametric finite element. The variation of the linear expansitivity of CFRP lamina with temperature in the transverse direction is included. The nonlinear initial thermal stresses resulting from thermal cooling from the stress-free temperature of 132.22 to 20°C in the [0/90]_s and [±45]_s laminates were found. These two types of laminates were then subject to a uniform applied strain ε_x until failure was detected inside the laminate according to a maximum strain failure criterion. This nonlinear analysis was based on an initial stress iteration method formulated in a previous paper. The [±45]_s laminate analysis was carried out with and without resin layers between the laminae modelled. All results obtained were compared with those of previous investigators. It was found that with initial thermal stresses included the [0/90]_s laminate failed at an earlier stage than without initial thermal stresses included. The [±45], laminate with initial thermal stresses included failed at a later stage than that without initial thermal effects and the inclusion of resin layers delayed the failure even further.     

Influence of local buckling on global instability: Simplified,large deformation,post-buckling analyses of plane trusses

In this paper, the influences of local (individual member) buckling and minor variations in member properties on the global response of truss-type structures are studied. A simple and effective way of forming the tangent stiffness matrix of the structure and a modified arc length method are devised to trace the nonlinear response of the structure beyond limit points, etc. Several examples are presented to indicate: (i) the broad range of validity of the simple procedure for evaluating the tangent stiffness, (ii) the effect of buckling of individual members on global instability and post-buckling response and (iii) the interactive effects of member buckling and global imperfections.     

Materially and geometrically nonlinear analysis of laminated anisotropic plates by p-version of FEM

A p-version finite element model based on degenerate shell element is proposed for the analysis of orthotropic laminated plates. In the nonlinear formulation of the model, the total Lagrangian formulation is adopted with moderately large deflections and small rotations being accounted for in the sense of von Karman hypothesis. The material model is based on the Huber-Mises yield criterion and Prandtl-Reuss flow rule in accordance with the theory of strain hardening yield function, which is generalized for anisotropic materials by introducing the parameters of anisotropy. The model is also based on the equivalent-single layer laminate theory. The integrals of Legendre polynomials are used for shape functions with p-level varying from 1 to 10. Gauss-Lobatto numerical quadrature is used to calculate the stresses at the nodal points instead of Gauss points. The validity of the proposed p-version finite element model is demonstrated through several comparative points of view in terms of ultimate load, convergence characteristics, nonlinear effect, and shape of plastic zone.     

Nonlinear Formulations of a Four-Node Quasi-Conforming Shell Element

The quasi-conforming technique was introduced in the 1980’s to meet the challenge of inter-elements conforming problems and give a unified treatment of both conforming and nonconforming elements. While the linear formulation is well established, the nonlinear formulation based on the quasi-conforming technique that includes geometric and material nonlinearity is presented in this paper. The formulation is derived in the framework of an updated Lagrangian stress resultant, co-rotational approach. The geometric nonlinear formulation provides solutions to buckling and postbuckling behaviour while the material nonlinear formulation considers the spread of plasticity within the element while maintaining an explicit construction of element matrices. Aside from the elasto-plastic constitutive relation, formulations on laminate composites and reinforced concrete are also presented. The formulations of laminate composite and reinforced concrete material are present based on the layer concept, the material properties can vary throughout the thickness and across the surface of a shell element. The various failure criteria for laminate composite are included in the formulation which makes it possible to analyses the progressive failure of fibre and matrix. For the reinforced concrete material, the nonlinearities as a result of tensile cracking, tension stiffening between cracks, the nonlinear response of concrete in compression, and the yielding of the reinforcement are considered. The steel reinforcement is modeled as a bilinear material with strain hardening.     

A general nonlinear analysis of concrete structures and comparison with frame tests

A computational technique is described for the general nonlinear analysis of large planar frames under static loading. The analysis accommodates both material and geometric sources of nonlinearity in a highly stable numerical procedure. For material behavior the moment-thrust-curvature relationships are reduced to polynomials made to represent common structural members. The nonlinear influence of axial shortening and P - Δ effects from displaced joints are accommodated by the analysis. Beam-column effects that magnify moments by the influence of axial force acting through deformed members can be approximated by introducing nodal points (or joints) at points of maximum deformation between the ends of members.Accuracy of the analytic procedure is demonstrated by comparing computed results with eight reinforced concrete frames tested at the University of Texas at Austin.     

Optimization of singular problems

A new optimization method is presented that optimizes singular structures. An example of a singular problem is deleting an inefficient member from a structure. As the member is deleted, the stresses in the member may increase above the allowables. When a member is deleted the nature of the analysis changes because the member stiffness becomes zero. This causes a local optima because stress constraints prevent inefficient members from zeroing. The problem is reformulated using the percent method so that the appropriate stress constraints are deleted as the member is deleted. Several examples show that the global optimal design is reached. Other methods to reach the global optima are appropriate only if the optimal structure is statically determinate. The percent optimization is also useful for optimization of discrete problems.     

Finite element analysis of out-of-plane buckling of reinforced concrete walls

In this paper a FE model for the study of the out-of-plane buckling of reinforced concrete walls is derived. The concrete is modelled using non-linear orthotropic 16 d.f. plate bending elements; the reinforcing steel using elasto-plastic beam elements. In the plane of the structure stresses are calculated using either four or eight-node membrane elements with bar elements used for the reinforcing steel. The buckling load is calculated by determining when the determinant of the out-of-plane tangent stiffness becomes zero. Comparison of the FE model with available experimental results shows good agreement.     

Optimum least-cost design of a truss roof system

An algorithm is presented encompassing the application of optimization methods to the least-cost elastic design of roof systems composed of rigid steel trusses, web joists and steel roof deck. The method is capable of designing rigid trusses that can be fabricated from various grades of steel and several types of standard sections. The selection of open web joist is presently limited to standard H-series, and decking material is standard 22 gage. The design is based upon AISC allowable values where combined stresses resulting from axial forces and secondary bending moments are considered. The effective column lengths are computed using the characteristic buckling equation for a member whose ends are elastically restrained against rotation. The procedure developed considers changes in the mechanical properties of the members, geometric variations in the truss configuration and changes in topology. Selected sets of members may be chosen to be identical, and chord members may be defined as continuous over several panels. Also investigated is the problem of finding the design containing the optimum number of trusses. A number of examples are presented which demonstrate the flexibility and generality of the design approach developed.     

On the machining induced residual stresses in IN718 nickel-based alloy: Experiments and predictions with finite element simulation

Residual stresses after machining processes on nickel-based super alloys is of great interest to industry in controlling surface integrity of the manufactured critical structural components. Therefore, this work is concerned with machining induced residual stresses and predictions with 3-D Finite Element (FE) based simulations for nickel-based alloy IN718. The main methods of measuring residual stresses including diffraction techniques have been reviewed. The prediction of machining induced stresses using 3-D FE simulations and comparison of experimentally measured residual stresses for machining of IN718 have been investigated. The influence of material flow stress and friction parameters employed in FE simulations on the machining induced stress predictions have been also explored. The results indicate that the stress predictions have significant variations with respect to the FE simulation model and these variations can be captured and the resultant surface integrity can be better represented in an interval. Therefore, predicted residual stresses at each depth location are given in an interval with an average and standard deviation.     

Nonlinear free vibration of fixed off-shore framed structures

The dynamical behavior of fixed off-shore framed structures is studied using the Wittrick-Williams algorithm to solve the nonlinear eigenvalue problem. The effects of shear deformation and rotary inertia as well as axial static loading are considered in this study of nonlinear free vibration.

The members are assumed to be rigidly connected and the added water mass is assumed equal to the mass of the water displaced. The structural modeling is based on a two-dimensional representation of the three-dimensional tower assuming a constant dimension equal to the base length perpendicular to the plane. The distributed masses of the members in the plane of the frame are computed by summing up the structural mass, the mass of the water contained in the tube, and the mass of the water displaced. The member masses in the plane perpendicular to the frame are assumed to be lumped at the horizontal cross-brace levels.

The results of the study indicate that while the first two frequencies obtained from the nonlinear and linear eigenvalue solutions agree closely, the effect of nonlinear eigenvalue solution is significant for the higher frequencies. The results also highlight the significant effects of the axial static force in the dynamic tangent stiffness matrix in the free vibration study of the off-shore structure. Fields for further research include (i) soil-structure interaction studies for gravity off-shore structures, buried pipelines, and (ii) nuclear power plant structures.     

Nonlinear and postbuckling analyses of curved composite panels subjected to combined temperature change and edge shear

The results of a detailed study of the nonlinear and postbuckling responses of curved unstiffened composite panels with central circular cutouts are presented. The panels are subjected to uniform temperature change and an applied in-plane edge shear loading. The analysis is based on a first-order shear-deformation Sanders-Budiansky type theory with the effects of large displacements, moderate rotations, transverse shear deformation and laminated anisotropic material behavior included. A mixed formulation is used with the fundamental unknowns consisting of the generalized displacements and the stress resultants of the panel. The nonlinear displacements, strain energy, transverse shear stresses, transverse shear strain energy density, and their hierarchical sensitivity coefficients are evaluated. Numerical results are presented for cylindrical panels with central circular cutouts and are subjected to uniform temperature change and an applied in-plane edge shear loading. The results show the effects of variations in the panel curvature, hole diameter, laminate stacking sequence and fiber orientation, on the nonlinear and postbuckling panel responses, and their sensitivity to changes in the various panel, layer and micromechanical parameters.     

Composite laminate optimization program suitable for microcomputers

The design of a composite panel requires some way of finding the minimum thickness laminate which will withstand the load requirements without failure. The mathematical complexity of this problem dictates the use of nonlinear optimization techniques. Although there are sophisticated optimization programs available capable of solving for the ply ratios, these programs are not often used in preliminary design because they require a large computer and some knowledge of the program's operation. As an alternative, specialized laminate optimization programs were developed which are compact and efficient enough to run on microcomputers. Only stresses at a point and inplane loads and deflections are considered. The programs are simple to use and require no knowledge of optimization. Techniques are developed in this paper that find minimum thickness laminates with either ply ratios or ply angles as design variables. Many test cases were run with these programs to demonstrate the weight savings possible over quasi-isotropic laminates. Particular interest is directed toward performance of the laminates under multiple independent loads. Initial orientations for the programs to operate on were studied, and 0/90/45/-45 laminates were found to be an effective starting point for design.     

Exact analytical theory of topology optimization with some pre-existing members or elements

This note deals with topological optimization of structures in which some members or elements of given cross-section exist prior to design and new members are to be added to the system. Existing members are costless, but new members and additions to the cross-section of existing members have a non-zero cost. The added weight is minimized for given behavioural constraints. The proposed analytical theory is illustrated with examples of least-weight (Michell) trusses having (a) stress or compliance constraints, (b) one loading condition and (c) some pre-existing members. Different permissible stresses in tension and compression are also considered. The proposed theory is also confirmed by finite element (FE)-based numerical solutions.     

Three-dimensional finite element analysis of interlaminar stresses in thick composite laminates

The three-dimensional finite element computer program has been developed to investigate interlaminar stresses in thick composite laminates. The finite element analysis is based on displacement formulation employing curved isoparametric 16-node elements. By using substructure technique, the program developed is capable of handling any composite laminates which consist of any number of orthotropic laminae and any orientations. In this paper, solid laminates and laminates with a circular hole were taken to study interlaminar stresses at the straight edge and the curved edge, respectively. Various solid laminates such as [45_n/0_n − 45_n/90_n]_s, [45/0/ − 45/90]_ns, and [45/0/ − 45/90]_sn (n = 1˜4) were analyzed. Also, [45/0/ − 45/90]_sn laminates with a circular hole were studied for n = 1 ˜ 20. The effect of laminate thickness and stacking sequence on the interlaminar stresses near the free edge was investigated. Interlaminar stresses were governed by stacking sequence rather than laminate thickness. The boundary layer width did not increase with laminate thickness but with the number of plies in the repeating unit.     

Nonlinearities in the analysis of frames

The material nonlinearity is introduced in the analysis through the representation of the stress-strain variations by continuous functions. The stress function can be adjusted to describe the stress-strain diagrams for various materials such as: steel, aluminum, concrete, etc., by varying the parameters related to the material properties. The continuous variation of the stresses from linear to nonlinear elastic, to plastic states is included in the formulation. For members exhibiting material nonlinearity within some parts of its span, the calculation for the locations of the transition of the stresses from the linear to nonlinear stages is not necessary. The geometrical nonlinearity analysis is also included in the derivations. The analysis can be performed either by an incremental or by an iterative approach, or by using an iterative process within each increment. The third approach is faster and better for structures subjected to high magnitude of nonlinearity. The stability of the structures due to nonlinearities is investigated. A software is modified to include the new formulations. Illustrative examples are provided for the comparisons of the results obtained with those solved previously.