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# Force and Torque Balance

Force and torque balance on a section of the plate in a force balance on a section of the plate, as shown in Figure 3–15. Since P = 0 and since we have established V = 0, Equation (3–59) requires that M = constant. The constant must be Ma, the applied torque, as shown by a moment balance on an arbitrary section of the plate. To determine the deflection of the plate, we could integrate Equation (3–74) with q = P = 0. However, since we already know M ≡Ma, it is simpler to integrate Equation (3–73), the twice integrated form of the fourth-order differential equation. The boundary conditions are w = 0 at x = 0 and dw/dx = 0 at x = 0. These boundary conditions at the left end of the plate clarify what is meant by an embedded plate; the embedded end of the plate cannot be displaced, and its slope must be zero. The integral of Equation (3–73) subject to these boundary conditions is w = −Max 2 2D . (3.75)

The bent plate has the shape of a parabola. w is negative according to the convention we established if M is positive; that is, the plate is deflected upward.

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What is the displacement of a plate pinned at both ends (w = 0 at x = 0, L) with equal and opposite bending moments applied at the ends? The problem is illustrated. As a second example we consider the bending of a plate embedded at its left end and subjected to a concentrated force Va at its right end, as illustrated in Figure 3–17. In this situation, q = 0, except at the point x = L, and Equation (3–57) gives V = Bending of a plate pinned at both ends.

An embedded plate subjected to a concentrated load.

Va, as shown by the vertical force balance on the plate sketched in Figure 3–18. With P also equal to zero, Equation (3–59) for the bending moment simplifies to dM dx = Va. (3.76) This equation can be integrated to yield M = Vax+ constant, (3.77) and the constant can be evaluated by noting that there is no applied torque at the end x = L; that is, M = 0 at x = L. Thus we obtain M = Va(x− L). (3.78)

The bending moment changes linearly from −VaL at the embedded end to zero at the free end. A simple torque balance on the section of the plate shown in Figure 3–18 leads to Equation (3–78), since M must balance the torque of the applied force Va acting with moment arm L− x.

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