Post on 09-Sep-2018
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Plane Problems: Constitutive Equations■ Constitutive equations for a linearly elastic and isotropic material in plane stress (i.e., σz=τxz=τyz=0):
where the last column has the initial (thermal) strains which are 0 , xy000 === γαεε Tyx ∆
• Rewriting in a compact form and solving for the stress vector,
where
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Plane Problems: Approximate Strain-Displacement Relations
• From the above, by definition
, ,xv
yu
yv
xu
xyyx ∆∆
∆∆
∆∆
∆∆ +≈≈≈ γεε
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Plane Problems: Strain-Displacement Relations
■ As the size of the rectangle goes to zero, in the limit,
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Plane Problems: Displacement Field Interpolated■ Interpolating the displacement field, u(x,y) and v(x,y), in the plane finite element from nodal displacements,
where entries of matrix N are the shape (interpolation) functions Ni.
■ From the previous two equations,
where B is the strain-displacement matrix.
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Stiffness Matrix and strain energy■ Strain energy density of an elastic material (energy/volume)
εε ET21
• Integrating over the element volume, the total strain energy is
( )dEBBdE 21
21
∫∫ == dVdVU TTT εε
where the term in parantheses is identified to be the element stiffness matrix.
■ The strain energy then becomes
where the term on the right is the total work done on the element.
1 1 r2 2
T TU = =d kd d
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Important Note on interpolation (shape) functions
■ Observe that, for a given material, stiffness matrix k (and, therefore, the behavior of an element) depends solely on N, the interpolation functions, and ∂. The latter prescribes differentiations which define strains in terms of displacements.
NBEBBk ∂=∫= , dVT
■ The variation of the shape functions in the element compared toactual variations of the true displacements determines element size required for good accuracy. Low-0rder shape functions will require smaller elements than higher-order shape functions.
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Loads and Boundary Conditions
■ Surface tractions: distributed loads on a boundary of a structure; e.g., pressure.■ Body forces: loads acting on every particle of the structure; e.g., acceleration (gravitaionalor otherwise), magnetic forces.■Concentrated forces and moments.
• Boundary conditions on various segments of the surface:
A to B: free. B to C: normal traction (pressure)
C to D: shear traction. D to A: zero displacements (dofs=0)
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Constant Strain Triangle (CST)
■ The sequence 123 in node numbers must go counterclockwisearound the element.■ Linear displacement field in terms of generalized coordinates βi:
■ Then, the strains are(constant within the element!!)
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Constant Strain Triangle (CST): Stiffness Matrix■Strain-displacement relation, ε=Bd, for the CST element
where 2A is twice the area of the triangle and xij=xi- xj , etc.
■ From the general formula
where t: element thickness (constant)
■ NOTE: To represent high strain gradient will require very largenumber of small CST elements
tATEBBk =
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Linear Strain Triangle (LST)
■ The element has six nodes and 12 dof. Not available in Genesis!
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Linear Strain Triangle (LST)■ The displacement field in terms of generalized coordinates:
which are quadratic in x and y.
■ The strain field:
which are linear in x and y.