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### MCE466-15

Course: MCE 466, Fall 2009
School: Maryville MO
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Word Count: 374

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8 Chapter Linear Strain Triangle (Overview) Compare formulation of CST and LST Comparison of element performance 1 Comparison of CST and LST Formulations CST: Three nodes per element 6 DOF per element LST: 6 nodes per element 12 DOF per element 2 Displacement Interpolation CST: u ( x, y ) a1 + a 2 x + a3 y = v ( x , y ) a 4 + a 5 x + a 6 y LST: u ( x, y ) a1 + a 2 x + a3 y + a 4 x 2 + a5 xy +...

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8 Chapter Linear Strain Triangle (Overview) Compare formulation of CST and LST Comparison of element performance 1 Comparison of CST and LST Formulations CST: Three nodes per element 6 DOF per element LST: 6 nodes per element 12 DOF per element 2 Displacement Interpolation CST: u ( x, y ) a1 + a 2 x + a3 y = v ( x , y ) a 4 + a 5 x + a 6 y LST: u ( x, y ) a1 + a 2 x + a3 y + a 4 x 2 + a5 xy + a6 y 2 = v( x, y ) a7 + a8 x + a9 y + a10 x 2 + a11 xy + a12 y 2 3 Strains within each element ~ x {u ( x, y), v( x, y)} y Recall: CST: or ~ constant => Constant Strain Triangle (CST) LST: ~ at most linear in x & y => Linear Strain Triangle (LST) 4 Element Stiffness Matrix CST: [ k ] = tA [ B ] [ D][ B ] T 6x6 6x3 3x3 3x6 LST: 12x12 12x3 3x3 3x12 Since Bij terms depend on x & y, numerical integration is required (Chap. 10) 5 CST vs. LST Performance Comparison Consider the following plane stress analysis of a cantilever beam "4 x 16" mesh 6 CST vs. LST Performance Comparison (cont.) Test Cases: Mesh 4x16 4X16 8x32 2x8 Element CST LST CST LST Case A-1 B-2 A-2 B-1 7 CST vs. LST Performance Comparison (cont.) 8 Chapter 9 Axisymmetric Elements Examples of axisymmetric problems: Pressure vessels Cylindrical shaft with notch or filleted step Stresses near a spherical void Hertzian between contact spheres Use of axisymmetric elements provides computational efficiency as compared to full 3-D analysis 9 Axisymmetric Example Pressure Vessel 10 Axisymmetric Example Soil Foundation 11 Axisymmetric Example Valve Stem 12 Axisymmetric Volume Element 13 Stresses Axisymmetric Problems Non-zero stresses - r, , z, rz r = z = 0 ( due to axisymmetry ) 14 Strains Axisymmetric Problems Non-zero strains - r, , z, rz r = z = 0 ( due to axisymmetry ) Stress Strain Relations: 1 - r 1 - E z = 1 - (1 + )(1 - 2 ) 0 0 rz 0 0 r 0 z 0 1 - 2 2 rz 4x4 15 Formulation of 3 Node Triangular Axisymmetric Element 16 Axisymmetric Elements Displacement Fields Displacement fields: u ( r , , z ) = u (r , z ) v( r , , z ) = 0 w(...

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