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© D.J.DUNN1MECHANICS OF SOLIDS -BEAMSTUTORIAL 3THE DEFLECTION OF BEAMSThis is the third tutorial on the bending of beams. You should judge your progress by completing the self assessment exercises. On completion of this tutorial you should be able to solve the slope and deflection of the following types of beams.A cantilever beam with a point load at the end.A cantilever beam with a uniformly distributed load.A simply supported beam with a point load at the middle.A simply supported beam with a uniformly distributed load.You will also learn and apply Macaulay’s method to the solution for beams with a combination of loads.Those who require more advanced studies may also apply Macaulay’s method to the solution of ENCASTRÉ.It is assumed that students doing this tutorial already know how tofind the bending moment in various types of beams.This information is contained in tutorial 2.
© D.J.DUNN2DEFLECTION OF BEAMS1. GENERAL THEORYWhen a beam bends it takes up various shapes such as that illustrated infigure 1. The shape may be superimposed on an x –y graph with the origin at the left end of the beam (before it is loaded). At any distance x metres from the left end, the beam will have a deflection y and a gradient or slope dy/dx and it is these that we are concerned with in this tutorial.We have already examined the equation relating bending moment and radius of curvature in a beam, namely REIMM is the bending moment.I is the second moment of area about the centroid.E is the modulus of elasticity andR is the radius of curvature.Rearranging we have EIMR1Figure 1 illustrates the radius of curvature which is defined as the radius of a circle that has a tangent the same as the point on the x-y graph. Figure 1Mathematically it can be shown that any curve plotted on x -y graph has a radius of curvature of defined as 2322dxdy1dxydR1