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HW7soln

# HW7soln - Problem 10.1 Given A uniform beam of flexural...

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Problem 10.1 : Given : A uniform beam of flexural stiffness and length 300 in has one end fixed and the other simply supported. 9 10 lb-in EI = 2 Required : Determine the system stiffness matrix considering three beam segments and the nodal coordinates indicated. Solution : Start by assigning “extra” global DOF to the beam – these will help in the assigning of element stiffnesses to the global DOF and will be eliminated once the stiffness matrix has been assembled. Also, introduce the element numbering: Next, we obtain the element-level stiffness matrices from Equation 10.21, which corresponds to Figure 10.1 in the text. [] 22 3 2 12 6 12 6 46 2 12 6 4 e LL L EI k L L L ⎡⎤ ⎢⎥ = ⎣⎦ Since all of the elements are identical, we get the following results when we evaluate this matrix: 333 36 33 3 1 7 8 4 1 12 10 600 10 12 10 600 10 600 10 40 10 600 10 20 10 12 10 600 10 12 10 600 10 600 10 20 10 600 10 40 10 k ×× = −× − × × 7 8 ; 4 1 3 3

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[] 333 36 33 3 2 4 1 5 2 12 10 600 10 12 10 600 10 600 10 40 10 600 10 20 10 12 10 600 10 12 10 600 10 600 10 20 10 600 10 40 10 k ⎡⎤ ×× ⎢⎥ = −× − × × ⎣⎦ 4 1 ; 5 2 3 3 3 3 5 2 6 3 12 10 600 10 12 10 600 10 600 10 40 10 600 10 20 10 12 10 600 10 12 10 600 10 600 10 20 10 600 10 40 10 k = × 5 2 ; 6 3 3 3 The numbers on each row and column indicate which global DOF corresponds to the local DOF used to compute the given element of the stiffness matrix. Now, we assemble the local stiffness matrices into the global stiffness matrix. At this point, I’ll use all 8 global DOF: 66 666 3 3 3 3 3 3 3 3 80 10 20 10 0 0 600 10 0 600 10 20 10 20 10 80 10 20 10 600 10 0 600 10 0 0 0 20 10 40 10 0 600 10 600 10 0 0 0 600 10 0 24 10 12 10 0 12 10 600 10 600 10 0 600 10 12 10 24 10 12 10 0 0 0 600 10 600 10 0 K × ××× × × × × × × = × × 3 3 6 12 10 12 10 0 0 600 10 0 0 12 10 0 0 12 10 600 10 20 10 0 0 600 10 0 0 600 10 40 10 × ×− × × × × 6 × 6 × × Because of the constraints, the last three rows and columns of this matrix are not needed, since they correspond to DOF that are forced to equal zero and hence do not need to be solved for. Thus, the actual stiffness matrix for this problem comes from the upper five rows and columns of [ ] K : 3 80 10 20 10 0 0 600 10 6663 20 10 80 10 20 10 600 10 0 3 . 0 20 10 40 10 0 600 10 3 0 600 10 0 24 10 12 10 3 3 6 0 01 0 0 6 0 0 1 21 0 2 41 0 K × ×××× = × × × × Answer
Problem 10.2/4/6 : Given : A uniform beam of flexural stiffness and length 300 in has one end fixed and the other simply supported. The beam carries a uniform weight per unit length 9 10 lb-in EI = 2 3.86 lb/in. q = Required : (a) Determine the system mass matrix corresponding to the lumped mass formulation. (b) Use static condensation to eliminate the massless DOF and determine the transformation matrix and the reduced mass and stiffness matrices.

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HW7soln - Problem 10.1 Given A uniform beam of flexural...

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