Bilinear Quadrilateral (Q4): CQUAD4 in GENESIS

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1 Bilinear Quadrilateral (Q4): CQUAD4 in GENESIS The Q4 element has four nodes and eight nodal dof. The shape can be any quadrilateral; we ll concentrate on a rectangle now. 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! 1

2 Q4: The strain fields The strain field in the element: Observation 1: ε x f(x) Q4 cannot exactly model the following beam where ε x x 2

3 Q4: Behavior in Pure Bending of a Beam Observation 2: When β 4 0, ε x varies linearly in y and γ xy 0. The former is a desirable characteristic of Q4 if a beam in pure bending is to be modeled because normal strain varies linearly along the depth coordinate of a beam in pure bending but γ xy 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! 3

4 Q4: The displacement field Generalized coordinates β i 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 4

5 Q4: The Shape (Interpolation) Functions Note: N 1 =1, N 2 =N 3 =N 4 =0 at x=-a, y=-b, which is node 1, so that u= N 1 u 1 = u 1 at that node. Similarly N i =1 while all other Ns are zero at node i. Fig See Eqn 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. 5

6 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: 6

7 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, ε x to vary linearly in x so that pure bending can be represented exactly and bending due to a transverse loading can be modeled accurately with rectangular Q6 elements. The addition of quadratic terms also allows a vanishing shear strain γ xy as is proper for pure bending. 7

8 Modeling Bending with the Q6 Element g 1 through g 4 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: 8

9 Quadratic Quadrilateral (Q8): None in GENESIS Q8 has 4 corner nodes and 4 side nodes and 16 nodal dof. 9

10 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) 10

11 The strain field: Quadratic Quadrilateral (Q8): Strains 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. 11

12 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. 12

13 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. 13

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