HW10_Solution - EML 4507 FEA & DESGN Fall 2010 HW10...

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EML 4507 FEA & DESGN Fall 2010 HW10 Solution 3. Using two CST elements, solve the simple shear problem described in the figure and determine whether the CST elements can represent the simple shear condition accurately or not. Material properties are given as E = 10 GPa, = 0.25, and thickness is h = 0.1 m. The distributed force f = 100 kN/m 2 is applied at the top edge. Solution: Using Eq. (6.28), the element stiffness matrix can be calculated. For Element 1, 1 1 2 ( ) 9 2 3 3 5.33 0 5.33 1.33 0 1.33 0 2 2 2 2 0 5.33 2 7.33 3.33 2 1.33 [ ] 10 1.33 2 3.33 7.33 2 5.33 0 2 2 2 2 0 1.33 0 1.33 5.33 0 5.33 e u v u v u v k For Element 2, 1 1 3 (2) 9 3 4 4 2 0 0 2 2 2 0 5.33 1.33 0 1.33 5.33 0 1.33 5.33 0 5.33 1.33 [ ] 10 2 0 0 2 2 2 2 1.33 5.33 2 7.33 3.33 2 5.33 1.33 2 3.33 7.33 u v u v u v k According to the given displacement boundary conditions, only u 3 and u 4 are the unknown DOFs. Thus, we assemble the stiffness matrix only for those free DOFs. Then, we have 3 4 7.33 5.33 5000 5.33 7.33 5000 u u f 1 2 3 4 x y 1 m 1 m
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Note that the distributed force is equally divided into Nodes 3 and 4. The solution of the above equation provides non-zero displacements. By combining with zero displacements, we have nodal displacements, as 5 , 0, 0, 0.25, 0, 0.25, 0 { } 10 {0, 0 } m s Q The element strains can be calculated using Eq. (6.25), as (1) (1) (1) 5 (2) (2) (2) 5 { } [ ]{ } 10 {0, 0, 0.25} { } [ ]{ } 10 {0, 0, 0.25} Bq Thus, there are no normal strains and shear strains are same for both elements. The element stress can be calculated using the stress-strain relation for plane stress, as (1) (1) 5 (2) (2) 5 { } [ ]{ } {0, 0, 10 } Pa { } [ ]{ } {0, 0, 10 } Pa T T C C Note that only shear stress exists, which satisfy the pure shear condition. Since distributed force f = 10 kN/m 2 is applied at the top edge, the above shear stress is exact. Thus, the CST element can represent the pure shear condition accurately. The figure below shows the deformed and undeformed shape of the elements. 5. A structure shown in the figure is approximated with one triangular element. Plane strain assumption is used.
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This note was uploaded on 06/07/2011 for the course EML 4507 taught by Professor Kim during the Fall '11 term at University of Florida.

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HW10_Solution - EML 4507 FEA & DESGN Fall 2010 HW10...

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