Hwrk_6_Solution - UNIVERSITY OF CALIFORNIA, BERKELEY Spring...

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1 UNIVERSITY OF CALIFORNIA, BERKELEY Dept. of Civil and Environmental Engineering Spring Semester 2008 Instructor: Filip C. Filippou CE 121 – Advanced Structural Analysis Homework Set #6 (due March 7, 2008) 1. Problem (5 points) Determine the deformed shape of the statically determinate structure in the figure for the case that element b is heated up differentially between bottom and top of the section so that a thermal curvature of 3 2 10 rad unit of length results. To answer this question set up the structure compatibility matrix f A which expresses the relation between element deformations and global displacements at the free dof's in the form f = f A VU and solve for f U given the element deformations V (try to use as few dofs and element deformations as possible). Compare the structure compatibility matrix f A with the appropriate equilibrium matrix f B of homework problem 2-1. Draw the deformed shape of the structure . a b cd 10 10 5 10 1 2 3 4 5
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CE 121 - Advanced Structural Analysis Homework 6 Filip C. Filippou Problem 1 a b cd 10 5 1 2 3 4 5 Geometric properties: L a 10 := L b 10 := L c 5 := L d 10 := The smallest number of free dofs for this structural model is 4 (actually we could also have managed to get it down to 2 in this case, i.e. dofs 1 and 2), but we declare ourselves satisfied with 4. The change of angle between the "whiskers" attached to the nodes and the line connecting the nodes are reported at the four locations that are numbered in the following figure. a b c a b c a b c a b c d d d d 2 1 3 4 dof 1 dof 2 dof 3 dof 4 1 1 Page 6-1
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CE 121 - Advanced Structural Analysis Homework 6 Filip C. Filippou The structure compatibility matrix A f for this structure is A f 1 L a 1 L a 1 L a 0 1 1 0 0 0 0 1 1 0 0 0 1 L c := Note: the compatibility matrix is the transpose of the equilibrium matrix B f from homework problem 2-2 If the element deformations are supposed to match the angles of the compatibility matrix, then the following compatibility relations result V 3 U 1 L a U 3 + = V 1 U 1 L a U 2 + = V 2 U 1 L a U 2 + = V 4 U 3 U 4 L c = with these compatibility relations satisfied, the deformed shape of the structure for each dof is as follows a b c a b c a b c a b c d d d d 2 1 3 4 dof 1 dof 2 dof 3 dof 4 Page 6-2
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CE 121 - Advanced Structural Analysis Homework 6 Filip C. Filippou Element b is heated up so that a positive curvature results (bottom fiber heated up and top fiber is cooled down by the same amount). The corresponding element deformations are V 1 0 := V 2 210 3 L b 2 := V 3 3 L b 2 := V 4 0 := We can either solve the 4 compatibility relations for the four unknown global dof displacement values (just to show off the power of our computer tools!) U f lsolve A f V , () := U f 50 5 15 75 10 3 = or, we can gain insight by solving the compatibility relations "by hand" from the first two equations we get U 2 V 1 V 2 + 2 := U 2 5 10 3 = U 1 V 2 V 1 2 L a := U 1 50 10 3 =
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This note was uploaded on 01/04/2010 for the course CE 121 taught by Professor Filippou during the Fall '09 term at University of California, Berkeley.

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Hwrk_6_Solution - UNIVERSITY OF CALIFORNIA, BERKELEY Spring...

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