Small deflection of beams

The deflection of a beam under a vertical distribution of force, as shown in Fig. F.3, is a very basic problem in the theory of elasticity. Under the condition that the deflection is small and there is no force parallel to the main axis, the deflection m as a function of distance z satisfies a simple differential equation, as we will show in this section. 

To start with, we show two simple relations in statics see Fig. F.3. First, consider the equilibrium of the net force in the vertical direction. At each cross section of the beam, there is a vertical force V(z), which should compensate the external force f(z)  

Second, at each cross section of the beam, the normal stress forms a torque M z). The equilibrium condition of a small section of the beam dz with respect to rotation requires (see Fig. F.3) 

Under the influence of a torque, the beam deforms. In other words, the slope tan 6 changes with z, as shown in Eq. (F.4). For small deflections, tan 6 6. In other words, 

Consider a small section Az of the beam. The total force horizontal must be zero. For symmetrical cross sections, such as rectangles, circular bars, and tubes, symmetry conditions require that the neutral line of force must be in the median plane, denoted as jc = 0 (see Fig. F.4). The distribution of normal strain is then 

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