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HW06sol

# HW06sol - CE 121 Advanced Structural Analysis Homework 6...

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CE 121 - Advanced Structural Analysis Homework 6 Filip C. Filippou 1 2 3 4 5 1 2 3 4 5 6 7 8 9 16 12 deformations V 0.003704 0.0020833 0.0042130 0.0072111 0.0042130 0.007211 0.0020833 0.0018519 0.0018519 := Element lengths: L a 16 := L b 12 := L c 16 2 12 2 + := L c 20 = L d 8 2 12 2 + := L d 14.422 = L e L c := L f L d := L g 12 := L h 8 := L i 8 := The equilibrium equations (or free dofs) are numbered as follows: 1 2 3 4 5 6 7 1 2 3 4 5 6 7 8 9 Page 1

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CE 121 - Advanced Structural Analysis Homework 6 Filip C. Filippou We write the compatibility equations that express the relation between the element deformations and the free global dof displacements in the figure on the previous page. We have: V 1 U 1 = V 2 U 3 = V 3 U 1 16 L c U 2 16 L c U 3 12 L c + = V 4 U 4 8 L d U 5 12 L d + = V 5 U 6 16 L e U 7 12 L e + = V 6 U 1 8 L f U 4 8 L f U 5 12 L f + = V 7 U 7 = V 8 U 2 U 4 + = V 9 U 4 U 6 + = To show that the given deformations are compatible we select 7 deformations and determine the corresponding free dof displacements from the corresponding 7 compatibility relations. We can then substitute these displacements in the additional two equations and show that they are satisfied. While the first task seems to imply the solution of 7 equations in 7 unknowns, we realize that it is much simpler because three deformations suffice to locate the deformed position of the nodes of the triangle made up of the corresponding elements, as long as we know 3 translations of the 6 required. This is the case for this particular structural model. Take for example the triangle made up of elements a, b, and c (or 1, 2, 3). We know that three translations are zero at nodes 1 and 2. Thus, the three deformations suffice to determine the other three, i.e. the displacements at dofs 1, 2, and 3. From the first three compatibility relations we get U 1 V 1 := U 1 3.70410 3 = U 3 V 2 := U 3 2.08310 3 = U 2 L c 16 V 3 U 1 U 3 12 16 := U 2 2.25 10 7 × = Then, from compatibility relation 8 U 4 V 8 U 2 + := U 4 1.85210 3 = This completes the answer to question (2). Page 2
CE 121 - Advanced Structural Analysis Homework 6 Filip C. Filippou For question (1) we need to determine the displacements at the other three dofs. We continue From compatibility relation 4 we get U 5 L d 12 V 4 U 4 8 12 := U 5 7.43210 3 = Then from compatibility relation 9 we get U 6 V 9 U 4 + := U 6 3.70410 3 =

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