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1 CST ELEMENT STIFFNESS MATRIX Strain energy Element Stiffness Matrix: Different from the truss and beam elements, transformation matrix [T] is not required in the two-dimensional element because [k] is constructed in the global coordinates. The strain energy of the entire solid is simply the sum of the element strain energies assembly

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2 CST ELEMENT FORCES Potential energy of concentrated forces at nodes Potential energy of distributed forces along element edges Surface traction force {T} = [T x, T y ] T is applied on the element edge 1-2 TxTx TyTy 3 1 2 x y s {T}={T x,T y }

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3 CST ELEMENT FORCES cont. Rewrite with all 6 DOFs Constant surface traction Work-equivalent nodal forces Equally divided to two nodes

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4 CST ELEMENT FORCES cont. Potential energy of distributed forces of all elements TxTx TyTy 3 1 2 S hlT y /2 3 1 2 hlT x /2 hlT y /2 hlT x /2

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5 CST ELEMENT FORCES cont. Potential energy of body forces distributed over the entire element (e.g. gravity or inertia forces). Potential energy of body forces for all elements What is the simple rule for distributing forces to nodes for CST element?

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6 CST ELEMENT OVERALL Total Potential Energy Principle of Minimum Potential Energy Assembly and applying boundary conditions are identical to other elements (beam and truss). Stress and Strain Calculation Nodal displacement {q (e) } for the element of interest needs to be extracted Finite Element Matrix Equation for CST Element Stress and strain are constant for CST element

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7 EXAMPLE 6.2 Cantilevered Plate Thickness h = 0.1 in, E = 3010 6 psi and = 0.3. Element 1 Area = 0.51010 = 50. 50,000 lbs 20 15 10 5 E2E2 E1E1 N1N1 N2N2 N3N3 N4N4

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8 ELEMENT 1 cont. Matrix [B] Plane Stress Condition How do you check B for errors?

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9 STIFFNESS MATRIX Stiffness Matrix for Element 1 Element 2: Nodes 1-3-4

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10 ELEMENT 2 cont. Matrix [B] Stiffness Matrix

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11 Assembly R x1, R y1, R x4, and R y4 are unknown reactions at nodes 1 and 4 displacement boundary condition u 1 = v 1 = u 4 = v 4 = 0 ASSEMEBLY AND BC Symmetric

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12 SOLUTION OF UNCONSTRAINED DOFs Reduced Matrix Equation and Solution

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13 ELEMENT STRAINS AND STRESSES Element Results Element 1

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14 ELEMENT STRAINS AND STRESSES cont. Element Results Element 2

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15 DISCUSSION These stresses are constant over respective elements. large discontinuity in stresses across element boundaries

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16 BEAM BENDING EXAMPLE xx is constant along the x-axis and linear along y-axis Exact Solution: xx = 60 MPa Max deflection v max = 0.0075 m -F-F 1 m 5 m F 1 2 3 4 5 6 7 8 9 10 xx x Max v = 0.0018

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17 BEAM BENDING EXAMPLE cont. y-normal stress and shear stress are supposed to be zero. yy Plot xy Plot

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18 CST ELEMENT cont. Discussions CST element performs well when strain gradient is small. In pure bending problem, xx in the neutral axis should be zero. Instead, CST elements show oscillating pattern of stress. CST elements predict stress and deflection about of the exact values. Strain along y-axis is supposed to be linear. But, CST elements can only have constant strain in y-direction. CST elements also have spurious shear strain. How can we improve accuracy? What direction? u2u2 v2v2 1 2 3

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19 CST ELEMENT cont. Two-Layer Model xx = 2.32 10 7 v max = 0.0028