1 ECMM108: Linear elastic and limit analysis of plates Part 1: Linear elastic analysis of plates Use Abaqus (or equivalent finite element package) to model one of the plates given in the figure below. The model should be based on linear elastic behaviour with the properties tabulated in Table 1. The plate problems and the specific magnitude of loading for individual students are listed in Table 2 at the end of this document. You can assume that the plate is homogeneous and isotropic. A uniformly distributed load acts on the entire region of the plate in the direction normal to its plane. Table 1: Plate properties – dimensions and material properties Plate Young’s modulus Poisson’s ratio Thickness (mm) Floor slab 25 kN/mm2 0.2 150 6 m 4 m Problem 1: Plate simply supported on two edges 6 m 4 m Problem 2: Plate simply supported on three edges 2 Modelling with Abaqus Create a uniform mesh of rectangular elements having side dimensions 1/10th of the length of the shorter edges of the plate, i.e. split the initial rectangle into at least 10 divisions each way. Define material properties as given in Table 1. Use soft conditions for modelling supports, i.e. without restraining twist movements. Report structure The report submitted must include the following. Finite element model and results A brief outline of the model stating the level of mesh discretization (i.e. element size and mesh structure). A contour map of vertical deflection with the maximum value of deflection and its location indicated clearly; Distributions of reactions at the plate boundaries with maximum values and its locations indicated clearly; (5 marks) Checking of Abaqus results for ONE element as follows. Select the element with the largest moments (Mx or My). Indicate its location in the plate. From the output of nodal rotations, derive by hand the curvatures and twists at the nodes and the centre of the element. Be aware of the differences between x and y, and the rotations provided by Abaqus as illustrated in Figure 15.1-4 in Cook et al. (2001)1. (10 marks) Then using the DM matrix, derive the components of moment at these points and compare with the output from Abaqus; (10 marks) From the output of nodal deflections, derive by hand the gradient of deflection w at the nodes and centre of the element and the rotations. Hence derive the transverse shear strains at the nodes and centre and the corresponding shear forces. Compare the shear forces at the centre of the element as derived above and as output from Abaqus. (10 marks) 1 Cook et al. 2001. Concepts and Applications of Finite Element Analysis, 4th ed. John Wiley & Sons, Inc. 4 m 2 m Problem 3: Plate simply supported on two opposite edges 3 Refine the mesh incrementally by a factor of 2 (i.e. element sizes of 1/20th, 1/40th, etc. of the length of the plate). Investigate if there is improvement in results by examining deflections. (5 marks) Check whether results from finite element analysis for the different levels of mesh refinement satisfy equilibrium locally and explain why (or why not). (10 marks) Part 2: Yield line analysis of slabs For the same plate problem solved in Part 1, use the proposed yield line patterns to calculate the flexural strength, or yield bending moment MY kNm/m, required to support the specified load. Note that the yield line pattern that is provided defines only one possible collapse mechanism for the slab. There are an infinite number of possibilities! However many mechanisms are considered in practice, the one which minimises the collapse load for a given strength (least upper bound) gives the best solution. Report structure The report submitted must include the following. Parameterization of the yield line pattern (10 marks) Derivation of internal and external work equations in terms of the yield line parameters (20 marks) Derivation of the yield bending strength MY (5 marks) Derivation of the optimal MY value and corresponding sketch of pattern. (10 marks) A brief discussion on how the values from limit analysis for MY are different to bending moments derived from linear elastic analysis. (5 marks) Table 2: Problem/loading assignment for individual students Name Problem Load (kN/m2) Name Problem Load (kN/m2) LUCKHAM H 1 4 HUANG J 2 8 CHEUNG AW 2 5 ANGGANI V 3 5.5 KENNEDY M 3 4 JIAN H 1 6 LI L 1 4.5 NAKHJIRI A 2 9 DADGARNEJAD A 2 6 CHOI KC 3 6 LU H 3 4.5 MANTOGLOU F 1 6.5 MYAT A 1 5 EL MOUNAYAR A 2 10 GUO Z 2 7 NAAIM I 3 6.5 WEN W 3 5
