448 CHAPTER 15 Bending of Open and Closed, Thin-Walled Beams
Fig.15.19
Deflection of a simply supported beam carrying a concentrated load at midspan (Example 15.8).
IntegratingEq.(iii),wehave
EIv=W
16
(
4 z^3
3−L^2 z)
+C 2
andwhenz=0,v=0sothatC 2 =0.Theequationofthedeflectioncurveistherefore
v=W
48 EI
( 4 z^3 − 3 L^2 z) (iv)Themaximumdeflectionoccursatmidspanandis
vmidspan=−WL^3
48 EI
(v)Notethatinthisproblem,wecouldnotusetheboundaryconditionthatv=0atz=Ltodetermine
C 2 ,sinceEq.(i)appliesonlyfor0≤z≤L/2;itfollowsthatEqs.(iii)and(iv)forslopeanddeflection
applyonlyfor0≤z≤L/2,althoughthedeflectioncurveisclearlysymmetricalaboutmidspan.
Examples15.5through15.8arefrequentlyregardedas“standard”casesofbeamdeflection.
15.3.1 Singularity Functions
ThedoubleintegrationmethodusedinExamples15.5through15.8becomesextremelylengthywhen
evenrelativelysmallcomplicationssuchasthelackofsymmetryduetoanoffsetloadareintroduced.
For example, the addition of a second concentrated load on a simply supported beam would result
in a total of six equations for slope and deflection, producing six arbitrary constants. Clearly, the
computationinvolvedindeterminingtheseconstantswouldbetedious,eventhoughasimplysupported
beamcarryingtwoconcentratedloadsisacomparativelysimplepracticalcase.Analternativeapproach
istointroduceso-calledsingularityorhalf-rangefunctions.Suchfunctionswerefirstappliedtobeam