Theory of Plates
Theory of Plates
Theory of Plates
Completed Version
The extension of Timoshenko beam theory to plates is the Reissner-Mindlin plate theory In Reissner-Mindlin plate theory the out-of-plane shear deformations are non-zero (in contrast to Kirchhoff plate theory) Almost all commercial codes (Abaqus, LS-Dyna, Ansys, ) use ReissnerMindlin type plate finite elements Assumed displacements during loading
deformed
reference undeformed and deformed geometries along one of the coordinate axis
Kinematic assumption: a plane section originally normal to the mid-surface remains plane, but in addition also shear deformations occur
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Kinematic equations
In plane-displacements
In this equation and in following all Greek indices only take values 1 or 2 It is assumed that rotations are small Rotation angle of normal: Angle of shearing: Slope of midsurface:
Out-of-plane displacements
The independent variables of the Reissner-Mindlin plate theory are the rotation angle and mid-surface displacement Introducing the displacements into the strain equation of three-dimensional elasticity leads to the strains of the plate
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Through-the-thickness strain:
The plate strains introduced into the internal virtual work of threedimensional elasticity give the internal virtual work of the plate
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Constitutive equations
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The independent variables in the weak form are corresponding test functions
and the
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The integrals are evaluated with numerical integration. If too few integration points are used, element stiffness matrix will be rank deficient.
The necessary number of integration points for the bilinear element are 2x2 Gauss points
The global stiffness matrix and global load vector are obtained by assembling the individual element stiffness matrices and load vectors
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As discussed for the Bernoulli and Timoshenko beams with increasing plate slenderness physics dictates that shear deformations have to vanish (i.e., )
Reissner-Mindlin plate and Timoshenko beam finite elements have problems to approximate deformation states with zero shear deformations (shear locking problem)
Bending moment and curvature constant along the beam Shear force and hence shear angle zero along the beam Displacements quadratic along the beam
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Deflection interpolation: Rotation interpolation: Shear angle: For the shear angle to be zero along the beam, the displacements and rotations have to be zero. Hence, a shear strain in the beam can only be reached when there are no deformations! Similarly, enforcing displacements and rotations! at two integration points leads to zero
In the following several techniques will be introduced to circumvent the shear locking problem
Use of higher-order elements Uniform and selective reduced integration Discrete Kirchhoff elements Assumed strain elements
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Constraint ratio is a semi-heuristic number for estimating an elements tendency to shear lock
Continuous problem
Number of equilibrium equations: 3 (two for bending moments + one for shear force) Number of shear strain constraints in the thin limit: 2
Constraint ratio:
Number of degrees of freedom per element on a very large mesh is ~3 Number of constraints per element for 2x2 integration per element is 8
Constraint ratio:
Number of constraints per element for one integration point per element is 2
Constraint ratio:
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Number of degrees of freedom per element on a very large mesh is ~ 4x3 =12 Number of constraints per element for 3x3 integration is 18 Constraint ratio:
Number of degrees of freedom per element on a very large mesh is ~ 9x3=27 Number of constraints per element for 4x4 integration is 32 Constraint ratio:
As indicated by the constraint ratio higher-order elements are less likely to exhibit shear locking
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The easiest approach to avoid shear locking in thin plates is to use some form of reduced integration
In uniform reduced integration the bending and shear terms are integrated with the same rule, which is lower than the normal In selective reduced integration the bending term is integrated with the normal rule and the shear term with a lower-order rule
Uniform reduced integrated elements have usually rank deficiency (i.e. there are internal mechanisms; deformations which do not need energy)
The global stiffness matrix is not invertible Not useful for practical applications
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Shear refers to the integration of the element shear stiffness matrix Bending refers to the integration of the element bending stiffness matrix
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The principal approach is best illustrated with a Timoshenko beam The displacements and rotations are approximated with quadratic shape functions
The inner variables are eliminated by enforcing zero shear stress at the two gauss points
Back inserting into the interpolation equations leads to a beam element with 4 nodal parameters
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It is assumed that the out-of-plane shear strains at edge centres are of higher quality (similar to the midpoint of a beam)
First, the shear angle at the edge centres is computed using the displacement and rotation nodal values
Subsequently, the shear angles from the edge centres are interpolated back
These reinterpolated shear angles are introduced into the weak form and are for element stiffness matrix computation used
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