- 5.1 Introduction
- 5.2 Pure Bending of Beams of Symmetrical Cross Section
- 5.3 Pure Bending of Beams of Asymmetrical Cross Section
- 5.4 Bending of A Cantilever of Narrow Section
- 5.5 Bending of a Simply Supported Narrow Beam
- 5.6 Elementary Theory of Bending
- 5.7 Normal and Shear Stresses
- 5.8 Effect of Transverse Normal Stress
- 5.9 Composite Beams
- 5.10 Shear Center
- 5.11 Statically Indeterminate Systems
- 5.12 Energy Method for Deflections
- 5.13 Elasticity Theory
- 5.14 Curved Beam Formula
- 5.15 Comparison of the Results of Various Theories
- 5.16 Combined Tangential and Normal Stresses
5.11 Statically Indeterminate Systems
A large class of problems of considerable practical interest relate to structural systems for which the equations of statics are not sufficient (but are necessary) for determination of the reactions or other unknown forces. Such systems are called statically indeterminate, and they require supplementary information for their solution. Additional equations usually describe certain geometric conditions associated with displacement or strain. These equations of compatibility state that the strain owing to deflection or rotation must preserve continuity.
With this additional information, the solution proceeds in essentially the same manner as for statically determinate systems. The number of reactions in excess of the number of equilibrium equations is called the degree of statical indeterminacy. Any reaction in excess of that which can be obtained by statics alone is said to be redundant. Thus, the number of redundants is the same as the degree of indeterminacy.
Several methods are available to analyze statically indeterminate structures. The principle of superposition, briefly discussed next, is an effective approach for many cases. In Section 5.6 and in Chapters 7 and 10, a number of commonly employed methods are discussed for the solution of the indeterminate beam, frame, and truss problems.
5.11.1 The Method of Superposition
In the event of complicated load configurations, the method of superposition may be used to good advantage to simplify the analysis. Consider, for example, the continuous beam shown in Fig. 5.21a, which is then replaced by the beams shown in Fig. 5.21b and c. At point A, the beam now experiences the deflections (υA)p and (υA)R due to P and R, respectively. Subject to the restrictions imposed by small deformation theory and a material obeying Hooke′s law, the deflections and stresses are linear functions of transverse loadings, and superposition is valid:
Figure 5.21. Superposition of displacements in a continuous beam.
This procedure may, in principle, be extended to situations involving any degree of indeterminacy.