Finite Element methods have been used by civil and structural engineers since the 1960s, and the theory behind this is well researched. However, there is still a lack of direction on how to use the information obtained from this type of analysis to practically design a structure for strength and serviceability criteria. Design codes are broadly based on simplified calibrated strength models and are consistent with simplified and practical detailing. In this paper traditional methods of analysis of a simple pad foundation are compared with the linear finite element method, and the results compared to experimental results. The following questions are answered:
FE analyses can be either linear or non-linear. Linear FE analysis is the most commonly used type, but is limited in its capabilities as it does not take cracking and softening of the concrete into account (Rombach 2004). This type of analysis is suitable for an ultimate limit state design check, but cannot be used to check serviceability deflection and cracking. Non-linear FE analyses model the cracked behaviour of the concrete by means of an iterative process, but are complicated and time consuming to set up, and the software cost is significantly more than a linear FE program. In practice, flat plate type structures are generally designed using a linear FE analysis, and serviceability compliance done with 'rule of thumb' span to effective depth ratio checks. The main criticisms of linear FE analyses are its use of elastic material properties, which result in overestimated support moments and underestimated deflections (Jones & Morrison 2005), and an impractical required reinforcement contour output. Figure 1 shows the typical transverse bending moment distribution in a pad footing.
The significant advance in computer software technology in recent years has resulted in a surge in the use of finite element software to analyse the load effects in structures, and in particular flat plate type structures. The finite element method is an approximation in which a continuum is replaced by a number of discreet elements (Zienkiewicz et al 1976). Each component representing the system as a whole is known as a finite element. Parameters and analytical functions describe the behaviour of each element and are then used to generate a set of algebraic equations describing the displacements at each node, which can then be solved. The elements have a finite size and therefore the solution to these equations is approximate; the smaller the element the closer the approximation is to the true solution (Brooker 2006). The output from a linear finite element flat slab analysis is in the form of contour plots of stresses and moments. At a pinned support a section through these contour plots shows very large peaks in the stresses and bending moments. These peak bending moments can vary considerably depending on how the support conditions are modelled, and the element size. It is the opinion of the authors that the basics of using a linear FEM to analyse flat slabs is commonly understood by most designers. However, the modelling of column to flat plate connections is still open to numerous forms of interpretation and designer preference. The most common support models listed by Rombach (2004) are shown in Figure 2.
Modern codes allow for nonlinear analysis of reinforced concrete structures, but in practice such a complex analysis is seldom justified due to the large amount of work required and the cost of suitable software. Designs are usually based on linear-elastic material behaviour, assuming that the ductile properties of reinforced concrete allow for a limited redistribution of forces. Rombach (2004) states that the accuracy of such a simplified approach is generally sufficient. A conservative design approach is to have two slab models, one where columns are assumed to be pinned supports to determine the worst case sagging moment, and the second where the column supports are fixed to determine the worst-case hogging moments. Eurocode 2 does not prescribe a specific analysis or dictate how to interpret FE method load effects, which are open to a wide range of interpretations depending on how the column supports are modelled. Most commonly used FE packages give no clear directive on how to detail the reinforcement for flat slabs designed using FE. In general it is accepted that the design engineer will use the required reinforcement contour plots to decide how to place the slab reinforcement. It is, however, obvious that if the FE reinforcement contours are followed exactly this would lead to a very impractical reinforcement layout.
PROF NICK DEKKER received the degrees BScEng, BEng Hons and MEng from the University of Pretoria and a PhDfrom the University of the Witwatersrand. He spent most of his professional career with BKS (now AECOM) where he was responsible forthe design of a wide range of structures, including bridges, industrial and commercial buildings, shopping centres, sports centres and process buildings. In 1996 he co-founded the practice Dekker & Gelderblom, and was also appointed as Professor of Structural Engineering at the University of Pretoria. He received an NRF (National Research Foundation) rating in 1997. His fields of interest include structural design in steel, pre-stressed concrete and reinforced concrete.
In general, an analysis of a structure using FEM tools consists of idealization of the real structure, choice of the finite elements for the analysis, selection of suitable material models, discretization/mesh generation, defining boundary condition, assigning loads/actions and calculation of load effect. FEM-Design 17 is user-friendly software, developed not only for the design and analysis of concrete structures, but also to model, analyze and design steel, timber and foundation structures in accordance with Eurocode with national annexes (StruSoft 2018). In the analysis, the characteristic compressive strength of concrete is considered as 35 MPa and 45 MPa, and the properties of pre-stressing steel are given in Table 1. Shrinkage strain of concrete has been estimated in accordance with NS EN 1992-1 (2011) (i.e., given in Table 2) and inserted into the software. The software calculates the specific normal force and bending moment causing the inserted shrinkage strain and applied to the flat plate as a load. Creep coefficient has been calculated as per NS EN 1992-1 (2011) (i.e., given in Table 2) and inserted into the software. Moreover, wobble coefficient of 0.01 per 1 m, anchorage slip of 4 mm and class 2 of relaxation of pre-stressing steel have been used in the analysis. In modeling the pre/stressed flat plate, shell elements with nine nodes (i.e., quadrilateral) and six nodes (i.e., triangular elements) were used, and the software automatically discretized the flat plate. The feature for modeling unbonded tendons was new as of January 2018 and is for analysis purposes only. The software converts the tendon profiles into equivalent loads, which are used in load combinations for the analysis. The software supports peak smoothing over singularity regions by calculating an average moment over a chosen distribution region.
A commonly used layout is one with concentrated tendons in one direction and distributed tendons in the other direction. Analysis shows that the use of banded tendons along the column center lines reduces the tensile stresses in the top fiber in a section over the column center line. A design engineer can take into account this finding while placing tendons in a flat plate design. Moreover, when considering the clashing of tendons, it can be seen that banded tendons will be placed close to each other, resulting in the most economical layout, due to placement costs. When tendons are distributed in both directions, it results in many clashing points, which leads to an increased unnecessary cost for weaving. Moreover, future research will focus on the analysis of layouts, using non-linear finite element analysis to study the optimization of the design.
Although ultra-high performance fiber reinforced concrete (UHPFRC) has been used recently as a sustainable construction technique for many precast segmental bridges (PSBs), no exhaustive numerical and experimental studies exist to assess the shear capacity and failure pattern of the joints in these bridges. Hence, to accurately investigate the shear behavior of the joints in UHPFRC precast segmental bridges, a numerical analysis model based on finite-element code was established in this study. Concrete damaged plasticity model was used to analyze the UHPFRC joint models by considering all the geometries, boundaries, interactions and constraints. In this paper, the numerical model was calibrated by two full-scale UHPFRC keyed dry and epoxy joints under confining pressure effect. The excellent agreement between the numerical results and experimental data demonstrated the reliability of the proposed numerical model. The validated numerical model was then utilized to investigate the parameters affecting shear behaviour of the joints in PSBs. For this purpose, 12 FE models were analyzed under different variable parameters namely, number of shear keys, confining stress, and types of joints (dry or epoxy). Furthermore, the numerical results were also compared with the five existing shear design provision models available in literature in terms of ultimate shear capacity.
An extensive review of the literature identified more than 300 completed bridges (i.e. pedestrian and motorway bridges combined) constructed worldwide using UHPFRC in one or more components (Binard 2017; Voo et al. 2018). In brief, UHPFRC is an ultra-high strength cementitious material that contains a high quantity of cement and silica fume, low quantity of water, incorporates large amounts of fibres and exhibits remarkable characteristics such as high fracture energy, low permeability, limited shrinkage and increase