

Chegg Guided Solutions for An Introduction to the Finite Element Method 3rd Edition: Chapter 11

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    Solution to Problem 11.1: The specified primary degrees of freedom (i.e., displacements) are:

    U1 = V1 = 0, U43 = V43 = 0, U8 = 0, U15 = 0, U22 = 0, U29 = 0, U36 = 0

    The specified secondary degrees of freedom (i.e., forces) with zero magnitudes are:

    The nonzero (horizontal) forces at nodes 7, 14, 21, 28, 35, 42 and 49 can be computed as follows. The procedure to calculate the nodal forces is the same as that used for the calculation of nodal sources in Chapter 8 for single-variable problems, except that the nodal values must be decomposed into the components. Since the distributed force is along the coordinate, all nodal computed nodal forces are along the coordinate. Assume that have the units of N/m (if they are taken as N/m2, the final nodal values should be multiplied with the factor

    Note that a linearly varying force of the type

    over an element 'e' of length acting perpendicular to the length of the element, results in the nodal values of

    The above result can be used to find the nodal forces of the problem at hand. First note that the variation of which can be used to determine of each line element For example, the element between global nodes 7 and 14 has the values: Similarly, the next element has the values: Hence, the horizontal forces at nodes 7 and 14, for example, are

    Similarly, other values can be calculated

    Alternatively, the nonzero (horizontal) forces at nodes 7, 14, 21, etc. can be computed using the definition

    where denotes the nodal force at node of the element 'e', denote the interpolation functions of the element, and is the global coordinate (with origin at node 7). In the global coordinate system, with origin at node 7, the interpolation functions are given by

    First, we note that the horizontal traction for the problem at hand is given by Then we have


    The connectivity matrix is given by (all that matters is the counterclockwise local node numbering; the elements are numbered as in FEM2D mesh generator)

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