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Elastic Bending of Beams and Panels

When a beam is loaded by a force F or moments M, the initially straight axis is deformed into a curve. If the beam is uniform in section and properties, long in relation to its depth and nowhere stressed beyond the elastic limit, the deflection d, and the angle of rotation, q, can be calculated using elastic beam theory. The basic differential equation describing the curvature of the beam at a point x along its length is

[Equation]

Figure A2.1

where y is the lateral deflection and M(x) is the bending moment at the point x on the beam, E is Young's modulus and I is the second moment of area (Appendix A1). When M is constant, this becomes

[Equation]

Figure A2.2

where Ro is the radius of curvature before applying the moment and r the radius after it is applied. Deflections d and rotations q are found by integrating these equations along the beam. Equations for the deflection, d, and end slope, q, of beams, for various common modes of loading are shown in Fig. A2.

The stiffness of the beam is defined by

[Equation]

Figure A2.3

It depends on Young's modulus, E, for the material of the beam, on its length and on the second moment of its section, I. Values of C1 are listed in Fig. A2.

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Figure A2 Deflection of Beams