Deflection of a Beam with Distributed and Point Load
Introduction
This application will derive an explicit expression for the deflection of a beam with a distributed load and a point load.
Governing Equations
restart:
The Euler-Bernoulli equation
de≔EI⋅ⅆ4ⅆx4wx=qx:
Initial and boundary conditions
ibc≔w0=0,wL=0,D@@2w0=0,D@@2wL=0:
Distributed load and point load
q≔x→Q⋅1−Heavisidex−a+F⋅Diracx−b:
Solution of the Differential Equation
Solve the differential equation together with the initial/boundary conditions and the load distribution to get an explicit expression for the beam deflection.
deSol≔dsolvede,ibc,wx:deflection≔simplifyrhsdeSol,symbolic
deflection≔−QxL−xL+xL−a4Dirac1,L−a+4FxL−xL+xL−b3Dirac1,L−b−8QxL−xL+xL−a3DiracL−a−6QxL−a2L2+2La−a2−2x2HeavisideL−a+24FxL−xL+xL−b2DiracL−b+48xF−x22+bL−b2L−bHeavisideL−b−6LQ−x+a4Heavisidex−a−24FL−x+b3Heavisidex−b−48a2QLx−14a2−12x2Heaviside−a2+FbLx−12b2−12x2Heaviside−b−QxLL2+Lx−x28L−x144EIL
Derive the moment and shear distribution.
moment≔EI⋅diffdeflection,x,x
moment≔2QxL−a4Dirac1,L−a−2QL−xL−a4Dirac1,L−a−144FL−x+bHeavisidex−b−24FL−x+b3Dirac1,x−b+48a2QL−xHeaviside−a+96FbL−xHeaviside−b−12QLL2+Lx−x2−12QxLL−2x+2QL+xL−a4Dirac1,L−a−8FL+xL−b3Dirac1,L−b+16QL+xL−a3DiracL−a+72QL−a2xHeavisideL−a−48FL+xL−b2DiracL−b−144FxL−bHeavisideL−b+48LQ−x+a3Diracx−a−72LQ−x+a2Heavisidex−a−6LQ−x+a4Dirac1,x−a+144FL−x+b2Diracx−b−48−Qa2Heaviside−a2−FbHeaviside−b−QLL−2x4+QxL4L−x+8FL−xL−b3Dirac1,L−b−8FxL−b3Dirac1,L−b−16QL−xL−a3DiracL−a+16QxL−a3DiracL−a+48FL−xL−b2DiracL−b−48FxL−b2DiracL−b144L
shear≔diffmoment,x
shear≔−144FbHeaviside−b−72Qa2Heaviside−a+6QL−a4Dirac1,L−a+144FLHeavisidex−b−36QLL−x−36QLL−2x−24FL−b3Dirac1,L−b+48QL−a3DiracL−a+72QL−a2HeavisideL−a−144FL−b2DiracL−b−144FL−bHeavisideL−b+72LQ−x+a3Dirac1,x−a−216LQ−x+a2Diracx−a+144LQ−x+aHeavisidex−a−6LQ−x+a4Dirac2,x−a−432FL−x+bDiracx−b−24FL−x+b3Dirac2,x−b+216FL−x+b2Dirac1,x−b+36QxL144L
Plot the Deflection, Moment, and Shear
Assign parameters.
Q≔−12:F≔−10 :L≔20:a≔3:b≔5:EI≔10000:
Plot deflection, moment, and shear.
plotdeflection,x=0..L,size=1000,400,axesfont=Calibri,title=Deflection,labels=Distance along beam,Deflection,labeldirections=horizontal,vertical,labelfont=Calibri,titlefont=Calibri,16,bold,background=ColorTools:-ColorRGB,218/255,223/255,225/255,axis=gridlines=color=ColorTools:-ColorRGB,1,1,1
plotmoment,x=0..L,size=1000,400,axesfont=Calibri,title=Moment,labels=Distance along beam,Moment,labeldirections=horizontal,vertical,labelfont=Calibri,titlefont=Calibri,16,bold,background=ColorTools:-ColorRGB,218/255,223/255,225/255,axis=gridlines=color=ColorTools:-ColorRGB,1,1,1
Shear Distribution
plotshear,x=0..L,size=1000,400,axesfont=Calibri,title=Shear,labels=Distance along beam,Shear,labeldirections=horizontal,vertical,labelfont=Calibri,titlefont=Calibri,16,bold,background=ColorTools:-ColorRGB,218/255,223/255,225/255,axis=gridlines=color=ColorTools:-ColorRGB,1,1,1
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