Momentum and Force Balances in Beams

Let us consider (Fig. 17.3) a beam element under a small deflection in which two close sections are separated by an infinitesimal distance dx = dl. M and T in the figure are, respectively, the flexural momentum and the shear force, while q is the external applied force (including the weight of the beam) per unit length. From the momentum balance, and momentarily disregarding the vectorial character of the magnitudes, we obtain 

This equation represents, in fact, the momentum balance condition for a beam element located between two sections separated by a very small distance. However, this is not the only balance condition. The resulting force acting on a bar element is cfr + qdx, where dT is the difference between the forces acting on two limiting sections of the beam element. If 

In cases in which the external forces applied to the beam are concentrated in a specific section, then in the regions of the beam where the forces do not act, the balance equation is greatly simplified. Actually, if q = 0, then T is constant. Integration of Eq. (17.17) gives 

Moreover, if the flexion of the beam is produced by concentrated forces, T experiences a jump in the points at which the forces are applied. 

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