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MatlabPlaneStressExample

# MatlabPlaneStressExample - Matlab Plane Stress...

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Page 1 of 18. Copyright J.E. Akin. All rights reserved. Matlab Plane Stress Example (Draft 2, April 9, 2007) Introduction Here the Matlab closed form element matrices for the T3 element (3 node triangle, constant stress) is illustrated for a square plate, 2 by 2 inches. It is fixed at the top left corner, is restrained from horizontal (but not vertical) displacement at its bottom lect corner. It is loaded only by a horizontal body force load acting to the right. Away from the stress concentrations at the two corners it is essentially a 1-D problem. If the whole left edge were restrained againts horizontal motion it would correspond to a axial bar hanging under its own weight. Then the free end deflection is δ = W L x /(2 A E ) and the axial stress varies from σ max = W / A at the support to zero at the free end. Here L x is the horizontal length, W the weight, E the elastic modulus, A = L x t is the area for a given thickness (t=0.005 here). Due to Poisson’s ratio, the solution here has a vertical (y) displacement. The above plot shows the plate and a finer mesh solution for the resultant displacements, from CosmosWorks. The stress in the y-direction should be zero except at the two stress concentrations.

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Page 2 of 18. Copyright J.E. Akin. All rights reserved. Matlab solution Here a crude mesh is formed by using the two diagonal lines to form four elements that are three node (constant strain) triangles, CST. There are two displacement components (degrees of freedom, dof) at each node. There are three strains and thress stresses to be determined. The execution and sample plots will be shown first. The modular source script is listed at the end of this document, and as a downloadable file on the class web site. The nodal data are stored in the file msh_bc_xyz.tmp , the element type and connectivity in msh_typ_nodes.tmp , the essential (displacement component) boundary condition data in msh_ebc.tmp . If point loads or sources existed then file msh_load_pt.tmp would also be present. Those files are used for data validation plots as well as input to the stress calculations. The current example starts in Unix by invoking Matlab and running mesh plotting options to check the data. % Matlab >> addpath /net/course-a/mech517/public_html/Matlab_Plots >> mesh_shrink_plot Read 5 mesh coordinate pairs,4 elements with 3 nodes each >> bc_flags_plot This displays the packed binary code for each node that has an essential boundary condition. Since there are two displacement components here there are two digits in the packed integer. A one denotes true (a restraint exists) while a zero denotes a free displacement. They are ordered as Ux (horizontal) and Uy (vertical) components. For this application the allowed flags are 00, 10, 01, or 11.