Transcript of 1 CST ELEMENT STIFFNESS MATRIX Strain energy –Element Stiffness Matrix: –Different from the...
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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