B

FIGURE 2.6 Continued.

2.2.4 Continuous Beams

Continuous beams like fixed-ended beams are statically indeterminate. Bending moments in these beams are functions of the geometry, moments of inertia, and modulus of elasticity of individual members besides the load and span. They may be determined by Clapeyron's theorem of three moments, the moment distribution method, or the slope deflection method.

An example of a two-span continuous beam is solved by Clapeyron's theorem of three moments. The theorem is applied to two adjacent spans at a time and the resulting equations in terms of unknown support moments are solved. The theorem states that

where MA, MB, and MC are the hogging moments at the supports A, B, and C, respectively, of two adjacent spans of lengths Lj and L2 (Figure 2.9); Aj and A2 are the areas of the bending moment diagrams produced by the vertical loads on the simple spans AB and BC, respectively; x1 is the centroid of Aj from A, and x2 is the distance of the centroid of A2 from C. If the beam section is constant within a span but remains different for each of the spans Equation 2.9 can be written as

where I1 and 12 are the moments of inertia of the beam section in spans L1 and L2, respectively.

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