Deflection of a Beam with Distributed and Point Load - Maple Programming Help

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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

deEIⅆ4ⅆx4wx=qx:


Initial and boundary conditions

ibcw0=0,wL=0,D@@2w0=0,D@@2wL=0:


Distributed load and point load

qxQ1Heavisidexa+FDiracxb:

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.

deSoldsolvede,ibc,wx:deflectionsimplifyrhsdeSol,symbolic

deflectionQxLxL+xLa4Dirac1,La+4FxLxL+xLb3Dirac1,Lb8QxLxL+xLa3DiracLa6QxLa2L2+2Laa22x2HeavisideLa+24FxLxL+xLb2DiracLb+48xFx22+bLb2LbHeavisideLb6LQx+a4Heavisidexa24FLx+b3Heavisidexb48a2QLx14a212x2Heavisidea2+FbLx12b212x2HeavisidebQxLL2+Lxx28Lx144EIL

(3.1)

 

Derive the moment and shear distribution.

momentEIdiffdeflection,x,x

moment2QxLa4Dirac1,La2QLxLa4Dirac1,La144FLx+bHeavisidexb24FLx+b3Dirac1,xb+48a2QLxHeavisidea+96FbLxHeavisideb12QLL2+Lxx212QxLL2x+2QL+xLa4Dirac1,La8FL+xLb3Dirac1,Lb+16QL+xLa3DiracLa+72QLa2xHeavisideLa48FL+xLb2DiracLb144FxLbHeavisideLb+48LQx+a3Diracxa72LQx+a2Heavisidexa6LQx+a4Dirac1,xa+144FLx+b2Diracxb48Qa2Heavisidea2FbHeavisidebQLL2x4+QxL4Lx+8FLxLb3Dirac1,Lb8FxLb3Dirac1,Lb16QLxLa3DiracLa+16QxLa3DiracLa+48FLxLb2DiracLb48FxLb2DiracLb144L

(3.2)

 

sheardiffmoment,x

shear144FbHeavisideb72Qa2Heavisidea+6QLa4Dirac1,La+144FLHeavisidexb36QLLx36QLL2x24FLb3Dirac1,Lb+48QLa3DiracLa+72QLa2HeavisideLa144FLb2DiracLb144FLbHeavisideLb+72LQx+a3Dirac1,xa216LQx+a2Diracxa+144LQx+aHeavisidexa6LQx+a4Dirac2,xa432FLx+bDiracxb24FLx+b3Dirac2,xb+216FLx+b2Dirac1,xb+36QxL144L

(3.3)

Plot the Deflection, Moment, and Shear

Assign parameters.

Q12:F10 :L20:a3:b5:EI10000:

 

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