Structural Analysis Methods I: Beam Elements - Mechanics of Structures: Variational and Computational Methods - page 231

References

Hetenyi, M., 1946, Beams on Elastic Foundation , University of Michigan Press, Ann Arbor, MI.
Pestel, E. and Leckie, F., 1963, Matrix Methods in Elastomechanics , McGraw-Hill, NY.
Pilkey, W.D., 1994, Formulas for Stress , Strain, and Structural Matrices , Wiley, NY.
Pilkey, W.D., 2002, Analysis and Design of Elastic Beams, Computational Methods , Wiley, NY.
Pilkey, W.D. and Chang, P.Y., 1978, Modern Formulas for Statics and Dynamics , McGraw-Hill, NY.
Pilkey, W.D. and Pilkey, O.H., 1986, Mechanics of Solids , Krieger Publishers, Melbourne, FL.
Wunderlich, W., 1967, On the analysis of shells of revolution by transfer matrices, Ing.-Arch. , No. 36,
pp. 262-279.
Wunderlich, W., 1966, Calculation of transfer matrices applied to the bending theory of shells of Revo-
lution, Proc. Int. Symposium The Use of Electronic Digital Computers in Structural Engineering,
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Zurm uhl, R. and Falk, S., 1986, Matrizen and ihre Anwendungen, Numerische Methoden, Teil 2 , 5th ed.,
Springer-Verlag, Berlin.

Problems

Stiffness matrices

4.1 Several stiffness coefficients for a beam were found in Section 4.4.1 using a direct
evaluation procedure. Complete the direct evaluation of the stiffness coefficients for
a beam using the configurations of Fig. 4.7.
4.2 Find the stiffness matrix for an extension bar element (Fig. P4.2) using the approximate
series approach. Begin with the series

u (ξ ) = u 1 + u 2 ξ , where ξ = x /.

FIGURE P4.2

Answer: See the stiffness matrix for extension in Example 4.1.

4.3 Repeat the previous problem, except add a new DOF at the point which lies halfway
between a and b, and use a quadratic series

2

u (ξ ) = u 1 + u 2 ξ + u 3 ξ

Answer:

7 − 81

EA
3

k i = − 8 6 − 8

1 − 87

4.4 Derive the 2 × 2 stiffness matrix of the stepped truss element (extension bar) shown

in Fig. P4.4.