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Bilinear Quadrilateral (Q4): CQUAD4 in GENESIS

■ The displacement field in terms of generalized coordinates:

■ So, u and v fields are bilinear in x and y (i.e., product of two linear polynomials). Because of form, sides are stiffer than diagonals-artificial anisotropy!

■ The Q4 element has four nodes and eight nodal dof.

■ The shape can be any quadrilateral; we’ll concentrate on a rectangle now.

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Q4: The strain fields■ The strain field in the element:

■Observation 1: εx ≠ f(x) ⇒ Q4 cannot exactly model the following beam where εx ∝ x

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Q4: Behavior in Pure Bending of a Beam■ Observation 2: When β4 ≠ 0, εx varies linearly in y and γxy≠0. The former is a desirable characteristic of Q4 if a beam in purebending is to be modeled because normal strain varies linearly along the depth coordinate of a beam in pure bending but γxy≠0 is undesirable because there is no shear strain.

■ Fig. (a) is the correct deformation in pure bending while (b) is the deformation of Q4 (the sides remain straight).■ Physical interpretation: applied moment is resisted by a spurious shear stress as well as flexural (normal) stresses.

⇒ the Q4 element is too stiff in bending!

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Q4: The displacement field■ Generalized coordinates βi can be computed in terms of nodal dof and the displacement field can be expressed as

where matrix N is 2x8 and the shape functions are

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Q4: The Shape (Interpolation) Functions■ Note: N1=1, N2=N3=N4=0 at x=-a, y=-b, which is node 1, so that u= N1 u1= u1 at that node.

■ Similarly Ni=1 while all other Ns are zero at node i.

Fig 3.4.3

■See Eqn. 3.4-4 for the expression of strains in terms of the shape functions (ε=∂ Nd).

■ All in all, Q4 converges properly with mesh refinement and works better than CST in most problems.

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Improved Bilinear Quadrilateral (Q6): None in GENESIS■ Q4 element is overstiff in bending. For the following problem, it gives deflections and flexural stresses smaller than the exact values and the shear stresses are greatly in error:

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Q6 Element: Displacement Field■ The problem with Q4 element can be overcome when a new element (Q6) is defined with the following displacement field:

where the summation terms alone define Q4 and ξ=x/a, η=y/b. The additional terms are often called “bubble modes”

■ The addition of quadratic terms in ξ and η allows, for example, εx to vary linearly in x so that pure bending can be represented exactly and bending due to a transverse loading can be modeledaccurately with rectangular Q6 elements.

■ The addition of quadratic terms also allows a vanishing shear strain γxy as is proper for pure bending.

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Modeling Bending with the Q6 Element■ g1 through g4 are additional dof but not nodal dof. They are called internal dof. Q6 thus has 12 dof.

■ Modeling the previous bending problem with Q6 elements gives the following stresses:

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Quadratic Quadrilateral (Q8): None in GENESIS

■ Q8 has 4 corner nodes and 4 side nodes and 16 nodal dof.

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Quadratic Quadrilateral (Q8): Displacement field

■The displacement field, which is quadratic in x and y:

■ Two shape functions:

■Observe from the displacement field: the edges x=±a deform into a parabola (i.e., quadratic displacement in y) (same for y=±b)

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Quadratic Quadrilateral (Q8): Strains

■ The strain field:

■ Strains have linear and quadratic terms. Hence, Q8 can represent a good deal of strain states exactly.

For example, states of constant strain, bending strain, etc.

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Quadratic Quadrilateral (Q8): Curved Elements

■ Q8 is useful when curved boundaries such as at a hole are to be represented, which are approximated better with elements having side nodes.

■ But, curving a side distorts an element and is undesirable unless necessary.

■ Therefore, internal element sides should be straight.

■ A side node may be shifted toward a corner where there is stress concentration such as at a crack tip. Hence Q8 is useful for such problems also.

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Elements with Drilling DOF■ Drilling dof: rotational dof about an axis normal to the plane.

■ An LST with these added to each node has 9 dof.

■ This dof allows twisting and bending rotations of shells under some loads to be represented.