Structural mechanics

As a result, linear and angular deformations are calculated by the formulas:

Since z 0 , the generalized Hooke law, connecting stresses and strains, is written as:

where is the Poisson ratio.

The normal and tangential stresses caused by the bending of the plate linearly vary along the thickness of the plate and are calculated through

of the plate section are calculated through the corresponding stresses.

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Figure 15.19, c shows the distribution of forces along the faces of an elementary prism dx dy h . In order not to clutter the drawing, stresses,

bending moments and torques are not shown on the faces x 0 and y 0 . On these faces, the increments of stresses are equal to zero

15.13. Stiffness Matrix Formation Example

of the Rectangular Plate Element

In this example, the finite element of a rigid plate is used.

1. Each node of the plate finite element has three degrees of freedom: w – vertical displacement (deflection),

w – angle of rotation about the axis x ,

y

w – angle of rotation about the axis y .

x

In the directions of these displacements, we impose additional links and thus obtain the primary system of the displacement method (Figure 15.20).

Figure 15.20

It is necessary to form a matrix in the local coordinate system, which would make it possible to transform the vector of nodal displacements

Z into vector of nodal reactions RZ .

We define the deflections function w(x, y) of the element in the form of a polynomial with 12 arbitrary constants. It must identically satisfy

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the homogeneous (load acts in nodes) differential equation of the deformed plate surface:

where ai – unknown independent parameters, which in the future must be expressed in terms of Z .

L – coefficient matrix of dimension 3 by 12:

ing the expression (15.44), we can establish the relationship between the vectors Z and a :

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where H is the transformation matrix of dimension 12 by 12:

5. The displacements vector u using expressions (15.44) and (15.47) is represented in the form:

6. After determining the displacement vector, one can find the vector

of generalized relative deformations k , whose components are the curvature and torsion of the middle surface of the plate:

Performing the appropriate differentiation, we obtain:

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7. Linearly distributed (per unit length of the plate section) bending moments and torques for isotropic plates are calculated by the formulas:

where D denotes the value of linear bending stiffness of the plate, the so-called cylindrical stiffness:

Bending moments corresponding to positive curvatures are considered positive.

In the matrix form of notation, the relationship of the vector of gener-

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The expression of the vector of generalized internal forces M through generalized displacements Z can be obtained by substituting expression k from (15.50) in the dependence (15.52):

8. Variation of the potential energy density of the element deformation

taking into account expressions (15.50) and (15.53), can be written as follows:

A Z T H 1 T BT C B H 1 Z .

For the entire volume of the finite element, the variation of the potential strain energy will have the form:

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9. Virtual work of nodal forces F on changes (variations) of nodal displacements Z is:

In accordance with the principle of virtual displacements A W , therefore equalities (15.55) and (15.56) allow us to relate the vector of nodal forces F and the vector Z :

10. The stiffness matrix (reaction matrix) of the finite element RE allows us to express the force vector F through the vector Z :

Comparing expressions (15.58) and (15.59), we find the stiffness matrix of the finite element:

RE H 1 T BT CBd H 1 .

The matrix RE (lower triangle) for the rectangular element of the plate is presented in table 15.2.

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Table 15.2

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Table 15.2 (сontinuation)

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Table. 15.2 (ending)

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