where *M* is the vector of internal bending moments, *w* is the local displacement component perpendicular to the element plane, and are the strain and curvature vectors, and *D** ^{ps}*,

For a homogeneous, isotropic material model, the two Hook matrices for the shell element are as follows:

where *E* is the elastic modulus and *v* is the Poisson's ratio, *h* is the thickness of the element. Comparing (C.3) and (C.4) one can see that these two matrices are proportional. This is only true for a transversely homogeneous, isotropic material.

where *E*_{1}, *E*_{2} are the elastic moduli in the first and second principal directions, respectively, *v*_{12} is the Poisson's ratio for the principal plane, *G*_{12} is the shear modulus. In this case, the Hook matrix for the plate-bending remains proportional to the matrix for the plane-stress problem.

The plate-bending stresses vary linearly across the thickness, and their maximum values are given by

The total stresses are the summation of these two contributions,

The strain and curvature calculations are based on the displacement solution, and the element local deformation matrix *B** ^{ps}* and

and *u** _{i}*,

The expressions of the deformation matrices *B** ^{ps}* and

The above stress calculations are first calculated at three Gauss integration points in an element, and then averaged into element center for the elemental values of the stresses. All the stress calculations are based on local coordinate system, thus the stress components are expressed in the local coordinate system. In order to avoid the coordinate-dependency of the components, principal stresses are calculated for output.

The stress state at a point can be represented by the well known Mohr's circle as shown in Figure C-1:

in which _{1} represents the maximum normal stress and _{2} the minimum normal stress.

The corresponding directions of the principal stresses are called the principal directions. The angle for these is calculated using the following equation:

The maximum shear stress is the extreme value of the shear stress. It is in ±45° from the principal directions. The value of the maximum shear stress is:

The mean stress is calculated by averaging the principal stresses:

The Mises-Hencky stress is calculated by using the following formulation:

The highest of the above mentioned stress values in a part is often used to check against the yield stress, to help the design engineer make sure the part does not fail. The distributions of these stresses are used to predict the performance of the design.

2. Kudryavtsev, L., *Structural Analysis of Injection-Molded Parts*, CIMP Progress Report No. 17, Feb. 1993

3. Zienkiewicz, O. C., *The Finite Element Method*, The third edition, 1977

4. Beer, F. P. and Johnson, E. R. Jr., *Mechanics of Materials*, McGraw-Hill, Inc., 1981, 1979

5. Spotts, M. F., *Design of Machine Elements*, Prentice-Hall, Inc. Englewood Cliffs, N.J., 1978, 1971, 1961, 1953, 1948

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