8/12/2019 12 - Page 137.pdf

1/1

95.13 Slope and Deflection of Beams 119Equations 5.28 and 5 29 are the finite diyerence approximations of the standard beamdeflection differential equations and, because they are written in terms of ordinates on eitherside of the central point i, they are known as central diferences. Alternative expressions whichcan be formed to contain only ordinates at, or to the right of i or ordinates at, or to the left of iare known as forward and backward differences, respectively but these will not be consideredhere.Now from eqn. 5.2)

. . At position i , combining eqn. 5.2)and 5.29).5.30)

A solution for any of the deflection (y) values can then be obtained by applying the finitedifference equation at a series of points along the beam and solving the resultingsimultaneous equations ee Example 5.8.

The higher the number of points selected the greater the accuracy of solution but the morethe number of equations which are required to be solved. The method thus lends itself tocomputer-assisted evaluation.

In addition to the solution of statically determinate beam problems of the type treated inExample 5.8 the method is also applicable to the analysis of statically indeterminate beams,i.e. those beam loading conditions with unknown or redundant) quantities such as proploads or fixing moments-see Example 5.9.

The method is similar in that the bending moment is written in terms of the applied loadsand the redundant quantities and equated to the finite difference equation at selected points.Since each redundancy is usually associated with a known or assumed)condition of slopeordeflection, e.g. zero deflection at a propped support, there will alwaysbe sufficient equationsto allow solution of the unknowns.

The principal advantages of the finite difference method are thus:a) that it can be applied to statically determinate or indeterminate beams,b) that it can be used for non-prismatic beams,c) that it is amenable to computer solutions.

5 13 eflections due to temperature effectsIt has been shown in 2.3 that a uniform temperature increaset on an unconstrained bar of

length will produce an increase in lengthL aLt

where a is the coefficiento inear expansion of the material of the bar. Provided that the barremains unconstrained, i.e. is free to expand, no stresses will result.Similarly, in the case of a beam supported in such a way that longitudinal expansion canoccur freely, no stresses are set up and there will be no tendency for the beam to bend. If,

however, the beam is constrained then stresseswill result, their values being calculated using