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MIT2_092F09_sol6

# MIT2_092F09_sol6 - 2.092/2.093 FINITE ELEMENT OF SOLIDS AND...

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2.092/2.093± F INITE E LEMENT OF S OLIDS AND F LUIDS F ALL 2009± Homework 6- solution Instructor: Prof. K. J. Bathe Assigned: TA: Seounghyun Ham Due: Problem 1 (20 points):± Figure 1: Stress distributions with 1×4 mesh± Session 14 Session 16

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Figure 2: Stress distributions with 2×8 mesh
Figure 3: Stress distributions with 4×8 mesh

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Figure 4: Stress distributions with 4×16 mesh
· Euler-Bernoulli Beam Theory d 4 w p Governing equation: EI =p(x)=- 0 x . dx 4 L dw ( 0) d 2 w ( L ) d 3 w ( L ) Boundary conditions: w (0) = 0 ; = 0 ; = 0 ; = 0 . dx dx 2 dx 3 After solving the governing equation with boundary conditions, we are able to obtain d 2 w p 0 x 3 2 2 2p L 2 τ xx =-Ez =Ez ⎟⎜ -L x+ L 3 and τ xx = 0 =3.2×10 9 2 at x=0, dx 2 2EIL 3 3 h 2 N m z= h . 2 Table 1: Numerical results for maximum stresses τ Mesh Max. stress yy Max. stress τ yz Max. stress τ zz 1×4 3.37E+09 9.71E+06 1.25E+09 2×8 3.71E+09 1.10E+08 1.42E+09 4×8 3.88E+09 1.38E+08 1.46E+09 4×16 4.17E+09 1.87E+08 1.60E+09 The finite analysis stress, τ yy

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