94 Mechanics of Materials §5.1 Both the straightforward integration method and Macaulay's method are based on the d2y relationship M=EI dx2(sce§5.2andS5.3 Clapeyron's equations ofthree moments for continuous beams in its simplest form states that for any portion of a beam on three supports 1,2 and 3,with spans between of L and L2,the bending moments at the supports are related by -M1L1-2M2(L1+L2)-M3L2= 6A+A22 LLL2】 where A,is the area of the B.M.diagram,assuming span L simply supported,and is the distance of the centroid of this area from the left-hand support.Similarly,A2 refers to span L2,with 2 the centroid distance from the right-hand support(see Examples 5.6 and 5.7).The following standard resultsare usefu for 6Ax (a)Concentrated load W,distance a from the nearest outside support 6Ax Wa LT(L2-d2) (b)Uniformly distributed load w 6AX wL3 (see Example 5.6) L 4 Introduction In practically all engineering applications limitations are placed upon the performance and behaviour of components and normally they are expected to operate within certain set limits of,for example,stress or deflection.The stress limits are normally set so that the component does not yield or fail under the most severe load conditions which it is likely to meet in service. In certain structural or machine linkage designs,however,maximum stress levels may not be the most severe condition for the component in question.In such cases it is the limitation in the maximum deflection which places the most severe restriction on the operation or design of the component.It is evident,therefore,that methods are required to accurately predict the deflection of members under lateral loads since it is this form of loading which will generally produce the greatest deflections of beams,struts and other structural types of members. 5.1.Relationship between loading,S.F.,B.M.,slope and deflection Consider a beam AB which is initially horizontal when unloaded.If this deflects to a new position A'B'under load,the slope at any point C is dy i= dx94 Mechanics of Materials #5.1 Both the straightforward integration method and Macaulay’s method are based on the relationship M = El, d2Y (see 5 5.2 and 0 5.3). dx Clapeyron’s equations of three moments for continuous beams in its simplest form states that for any portion of a beam on three supports 1,2 and 3, with spans between of L, and L,, the bending moments at the supports are related by where A, is the area of the B.M. diagram, assuming span L, simply supported, and X, is the distance of the centroid of this area from the left-hand support. Similarly, A, refers to span L,, with f2 the centroid distance from the right-hand support (see Examples 5.6 and 5.7). The following standard results are useful for -: 6Af L (a) Concentrated load W, distance a from the nearest outside support 6Af Wa LL -- (L2 - a2) -~ (b) Uniformly distributed load w 6Af wL3 L 4 -- -- (see Example 5.6) Introduction In practically all engineering applications limitations are placed upon the performance and behaviour of components and normally they are expected to operate within certain set limits of, for example, stress or deflection. The stress limits are normally set so that the component does not yield or fail under the most severe load conditions which it is likely to meet in service. In certain structural or machine linkage designs, however, maximum stress levels may not be the most severe condition for the component in question. In such cases it is the limitation in the maximum deflection which places the most severe restriction on the operation or design of the component. It is evident, therefore, that methods are required to accurately predict the deflection of members under lateral loads since it is this form of loading which will generally produce the greatest deflections of beams, struts and other structural types of members. 5.1. Relationship between loading, S.F., B.M., slope and deflection Consider a beam AB which is initially horizontal when unloaded. If this deflects to a new position A‘B under load, the slope at any point C is dx