Example 1.12 Mechanical Engineering - Bending of a beamFigure 1.3(a) illustrates an unloaded beam of length L. The beam is fixed at bothends. When a load (force) is applied the beam bends as illustrated in Figure 1.3(b).Let the weight of the load being supported per unit length be w. Referring toFigure 1.3(b) we see that, when loaded, the beam bends in the negative y direc-tion. The magnitude of the bending is the crucial quantity and this is illustratedin Figure 1.3(c). We refer to this as the displacement of the beam, denoted H(x).Note that in Figure 1.3(c) the H axis is positive in the downward direction. It ispossible to show thatdª HK= wdr(а)Figure 1.3(a) An unloadedbeam of length L;(b) beam bendingunder load (it isfixed at both ends);(c) displacement of(b)the beam.(c)where K is known as the structural rigidity of the beam and is assumed to be con-stant. It is the product of Young's modulus and the moment of inertia of the beamabout its central axis.(a) If w is constant, find an expression for the displacement of the beam.(b) The beam is fixed rigidly at both ends. Hence when x = 0 and x = L thedisplacement is zero, that is H(0) = H(L) = 0. Additionally at these end-pointsthe gradient of the beam remains zero, so H'(0) = H'(L) = 0. Find thedisplacement at any point x.(c) Show that the maximum displacement of the beam occurs half-way along its length.

w = Kd4Hdx4 where K = EIK = Flexural RigidityE = Young's modulus I = Moment of inertia

(a) If w is constant, find an expression for the displacement of the beam. (b) The beam is fixed rigidly at both ends. Hence when x = 0 and x = L the displacement is zero, that is H(0) = H(L) = 0. Additionally at these end-points the gradient of the beam remains zero, so H'(0) = H'(L) = 0. Find the displacement at any point x. (c) Show that the maximum displacement of the beam occurs half-way along its length. %3D %3D

(a) If w is constant, find an expression for the displacement of the beam. (b) The beam is fixed rigidly at both ends. Hence when x = 0 and x = L the displacement is zero, that is H(0) = H(L) = 0. Additionally at these end-points the gradient of the beam remains zero, so H'(0) = H'(L) = 0. Find the displacement at any point x. (c) Show that the maximum displacement of the beam occurs half-way along its length. %3D %3D

Structural Analysis

6th Edition

ISBN:9781337630931

Author:KASSIMALI, Aslam.

Publisher:KASSIMALI, Aslam.

Chapter2: Loads On Structures

Section: Chapter Questions

Problem 1P

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