Deflection of a Beam with Distributed and Point Load - Maple 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

 > $\mathrm{restart}:$

The Euler-Bernoulli equation

 > $\mathrm{de}≔\mathrm{EI}\cdot \frac{{ⅆ}^{4}}{ⅆ{x}^{4}}w\left(x\right)=q\left(x\right):$

Initial and boundary conditions

 > $\mathrm{ibc}≔\mathrm{w}\left(0\right)=0,w\left(L\right)=0,\left(\mathrm{D}@@2\right)\left(w\right)\left(0\right)=0,\left(\mathrm{D}@@2\right)\left(w\right)\left(L\right)=0:$

 > $q≔x\to Q\cdot \left(1-\mathrm{Heaviside}\left(x-a\right)\right)+F\cdot \mathrm{Dirac}\left(x-b\right):$

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.

 > $\mathrm{deSol}≔\mathrm{dsolve}\left(\left\{\mathrm{de},\mathrm{ibc}\right\},w\left(x\right)\right):\phantom{\rule[-0.0ex]{0.0em}{0.0ex}}\mathrm{deflection}≔\mathrm{simplify}\left(\mathrm{rhs}\left(\mathrm{deSol}\right),\mathrm{symbolic}\right)$
 ${\mathrm{deflection}}{≔}\frac{{-}{Q}{}{x}{}\left({L}{-}{x}\right){}\left({L}{+}{x}\right){}{\left({L}{-}{a}\right)}^{{4}}{}{\mathrm{Dirac}}{}\left({1}{,}{L}{-}{a}\right){+}{4}{}{F}{}{x}{}\left({L}{-}{x}\right){}\left({L}{+}{x}\right){}{\left({L}{-}{b}\right)}^{{3}}{}{\mathrm{Dirac}}{}\left({1}{,}{L}{-}{b}\right){-}{8}{}{Q}{}{x}{}\left({L}{-}{x}\right){}\left({L}{+}{x}\right){}{\left({L}{-}{a}\right)}^{{3}}{}{\mathrm{Dirac}}{}\left({L}{-}{a}\right){-}{6}{}{Q}{}{x}{}{\left({L}{-}{a}\right)}^{{2}}{}\left({{L}}^{{2}}{+}{2}{}{L}{}{a}{-}{{a}}^{{2}}{-}{2}{}{{x}}^{{2}}\right){}{\mathrm{Heaviside}}{}\left({L}{-}{a}\right){+}{24}{}{F}{}{x}{}\left({L}{-}{x}\right){}\left({L}{+}{x}\right){}{\left({L}{-}{b}\right)}^{{2}}{}{\mathrm{Dirac}}{}\left({L}{-}{b}\right){+}{48}{}{x}{}{F}{}\left({-}\frac{{{x}}^{{2}}}{{2}}{+}{b}{}\left({L}{-}\frac{{b}}{{2}}\right)\right){}\left({L}{-}{b}\right){}{\mathrm{Heaviside}}{}\left({L}{-}{b}\right){-}{6}{}{L}{}{Q}{}{\left({-}{x}{+}{a}\right)}^{{4}}{}{\mathrm{Heaviside}}{}\left({x}{-}{a}\right){-}{24}{}{F}{}{L}{}{\left({-}{x}{+}{b}\right)}^{{3}}{}{\mathrm{Heaviside}}{}\left({x}{-}{b}\right){-}{48}{}\left(\frac{{{a}}^{{2}}{}{Q}{}\left({L}{}{x}{-}\frac{{1}}{{4}}{}{{a}}^{{2}}{-}\frac{{1}}{{2}}{}{{x}}^{{2}}\right){}{\mathrm{Heaviside}}{}\left({-}{a}\right)}{{2}}{+}{F}{}{b}{}\left({L}{}{x}{-}\frac{{1}}{{2}}{}{{b}}^{{2}}{-}\frac{{1}}{{2}}{}{{x}}^{{2}}\right){}{\mathrm{Heaviside}}{}\left({-}{b}\right){-}\frac{{Q}{}{x}{}{L}{}\left({{L}}^{{2}}{+}{L}{}{x}{-}{{x}}^{{2}}\right)}{{8}}\right){}\left({L}{-}{x}\right)}{{144}{}{\mathrm{EI}}{}{L}}$ (3.1)

Derive the moment and shear distribution.

 > $\mathrm{moment}≔\mathrm{EI}\cdot \mathrm{diff}\left(\mathrm{deflection},x,x\right)$
 ${\mathrm{moment}}{≔}\frac{{2}{}{Q}{}{x}{}{\left({L}{-}{a}\right)}^{{4}}{}{\mathrm{Dirac}}{}\left({1}{,}{L}{-}{a}\right){-}{2}{}{Q}{}\left({L}{-}{x}\right){}{\left({L}{-}{a}\right)}^{{4}}{}{\mathrm{Dirac}}{}\left({1}{,}{L}{-}{a}\right){-}{144}{}{F}{}{L}{}\left({-}{x}{+}{b}\right){}{\mathrm{Heaviside}}{}\left({x}{-}{b}\right){-}{24}{}{F}{}{L}{}{\left({-}{x}{+}{b}\right)}^{{3}}{}{\mathrm{Dirac}}{}\left({1}{,}{x}{-}{b}\right){+}{48}{}{{a}}^{{2}}{}{Q}{}\left({L}{-}{x}\right){}{\mathrm{Heaviside}}{}\left({-}{a}\right){+}{96}{}{F}{}{b}{}\left({L}{-}{x}\right){}{\mathrm{Heaviside}}{}\left({-}{b}\right){-}{12}{}{Q}{}{L}{}\left({{L}}^{{2}}{+}{L}{}{x}{-}{{x}}^{{2}}\right){-}{12}{}{Q}{}{x}{}{L}{}\left({L}{-}{2}{}{x}\right){+}{2}{}{Q}{}\left({L}{+}{x}\right){}{\left({L}{-}{a}\right)}^{{4}}{}{\mathrm{Dirac}}{}\left({1}{,}{L}{-}{a}\right){-}{8}{}{F}{}\left({L}{+}{x}\right){}{\left({L}{-}{b}\right)}^{{3}}{}{\mathrm{Dirac}}{}\left({1}{,}{L}{-}{b}\right){+}{16}{}{Q}{}\left({L}{+}{x}\right){}{\left({L}{-}{a}\right)}^{{3}}{}{\mathrm{Dirac}}{}\left({L}{-}{a}\right){+}{72}{}{Q}{}{\left({L}{-}{a}\right)}^{{2}}{}{x}{}{\mathrm{Heaviside}}{}\left({L}{-}{a}\right){-}{48}{}{F}{}\left({L}{+}{x}\right){}{\left({L}{-}{b}\right)}^{{2}}{}{\mathrm{Dirac}}{}\left({L}{-}{b}\right){-}{144}{}{F}{}{x}{}\left({L}{-}{b}\right){}{\mathrm{Heaviside}}{}\left({L}{-}{b}\right){+}{48}{}{L}{}{Q}{}{\left({-}{x}{+}{a}\right)}^{{3}}{}{\mathrm{Dirac}}{}\left({x}{-}{a}\right){-}{72}{}{L}{}{Q}{}{\left({-}{x}{+}{a}\right)}^{{2}}{}{\mathrm{Heaviside}}{}\left({x}{-}{a}\right){-}{6}{}{L}{}{Q}{}{\left({-}{x}{+}{a}\right)}^{{4}}{}{\mathrm{Dirac}}{}\left({1}{,}{x}{-}{a}\right){+}{144}{}{F}{}{L}{}{\left({-}{x}{+}{b}\right)}^{{2}}{}{\mathrm{Dirac}}{}\left({x}{-}{b}\right){-}{48}{}\left({-}\frac{{Q}{}{{a}}^{{2}}{}{\mathrm{Heaviside}}{}\left({-}{a}\right)}{{2}}{-}{F}{}{b}{}{\mathrm{Heaviside}}{}\left({-}{b}\right){-}\frac{{Q}{}{L}{}\left({L}{-}{2}{}{x}\right)}{{4}}{+}\frac{{Q}{}{x}{}{L}}{{4}}\right){}\left({L}{-}{x}\right){+}{8}{}{F}{}\left({L}{-}{x}\right){}{\left({L}{-}{b}\right)}^{{3}}{}{\mathrm{Dirac}}{}\left({1}{,}{L}{-}{b}\right){-}{8}{}{F}{}{x}{}{\left({L}{-}{b}\right)}^{{3}}{}{\mathrm{Dirac}}{}\left({1}{,}{L}{-}{b}\right){-}{16}{}{Q}{}\left({L}{-}{x}\right){}{\left({L}{-}{a}\right)}^{{3}}{}{\mathrm{Dirac}}{}\left({L}{-}{a}\right){+}{16}{}{Q}{}{x}{}{\left({L}{-}{a}\right)}^{{3}}{}{\mathrm{Dirac}}{}\left({L}{-}{a}\right){+}{48}{}{F}{}\left({L}{-}{x}\right){}{\left({L}{-}{b}\right)}^{{2}}{}{\mathrm{Dirac}}{}\left({L}{-}{b}\right){-}{48}{}{F}{}{x}{}{\left({L}{-}{b}\right)}^{{2}}{}{\mathrm{Dirac}}{}\left({L}{-}{b}\right)}{{144}{}{L}}$ (3.2)

 > $\mathrm{shear}≔\mathrm{diff}\left(\mathrm{moment},x\right)$
 ${\mathrm{shear}}{≔}\frac{{-}{144}{}{F}{}{b}{}{\mathrm{Heaviside}}{}\left({-}{b}\right){-}{72}{}{Q}{}{{a}}^{{2}}{}{\mathrm{Heaviside}}{}\left({-}{a}\right){+}{6}{}{Q}{}{\left({L}{-}{a}\right)}^{{4}}{}{\mathrm{Dirac}}{}\left({1}{,}{L}{-}{a}\right){+}{144}{}{F}{}{L}{}{\mathrm{Heaviside}}{}\left({x}{-}{b}\right){-}{36}{}{Q}{}{L}{}\left({L}{-}{x}\right){-}{36}{}{Q}{}{L}{}\left({L}{-}{2}{}{x}\right){-}{24}{}{F}{}{\left({L}{-}{b}\right)}^{{3}}{}{\mathrm{Dirac}}{}\left({1}{,}{L}{-}{b}\right){+}{48}{}{Q}{}{\left({L}{-}{a}\right)}^{{3}}{}{\mathrm{Dirac}}{}\left({L}{-}{a}\right){+}{72}{}{Q}{}{\left({L}{-}{a}\right)}^{{2}}{}{\mathrm{Heaviside}}{}\left({L}{-}{a}\right){-}{144}{}{F}{}{\left({L}{-}{b}\right)}^{{2}}{}{\mathrm{Dirac}}{}\left({L}{-}{b}\right){-}{144}{}{F}{}\left({L}{-}{b}\right){}{\mathrm{Heaviside}}{}\left({L}{-}{b}\right){+}{72}{}{L}{}{Q}{}{\left({-}{x}{+}{a}\right)}^{{3}}{}{\mathrm{Dirac}}{}\left({1}{,}{x}{-}{a}\right){-}{216}{}{L}{}{Q}{}{\left({-}{x}{+}{a}\right)}^{{2}}{}{\mathrm{Dirac}}{}\left({x}{-}{a}\right){+}{144}{}{L}{}{Q}{}\left({-}{x}{+}{a}\right){}{\mathrm{Heaviside}}{}\left({x}{-}{a}\right){-}{6}{}{L}{}{Q}{}{\left({-}{x}{+}{a}\right)}^{{4}}{}{\mathrm{Dirac}}{}\left({2}{,}{x}{-}{a}\right){-}{432}{}{F}{}{L}{}\left({-}{x}{+}{b}\right){}{\mathrm{Dirac}}{}\left({x}{-}{b}\right){-}{24}{}{F}{}{L}{}{\left({-}{x}{+}{b}\right)}^{{3}}{}{\mathrm{Dirac}}{}\left({2}{,}{x}{-}{b}\right){+}{216}{}{F}{}{L}{}{\left({-}{x}{+}{b}\right)}^{{2}}{}{\mathrm{Dirac}}{}\left({1}{,}{x}{-}{b}\right){+}{36}{}{Q}{}{x}{}{L}}{{144}{}{L}}$ (3.3)

Plot the Deflection, Moment, and Shear

Assign parameters.

 >

Plot deflection, moment, and shear.

 > $\mathrm{plot}\left(\mathrm{deflection},x=0..L,\mathrm{size}=\left[1000,400\right],\mathrm{axesfont}=\left[\mathrm{Calibri}\right],\mathrm{title}="Deflection",\mathrm{labels}=\left["Distance along beam","Deflection"\right],\mathrm{labeldirections}=\left[\mathrm{horizontal},\mathrm{vertical}\right],\mathrm{labelfont}=\left[\mathrm{Calibri}\right],\mathrm{titlefont}=\left[\mathrm{Calibri},16,\mathrm{bold}\right],\mathrm{background}=\mathrm{ColorTools}:-\mathrm{Color}\left("RGB",\left[218/255,223/255,225/255\right]\right),\mathrm{axis}=\left[\mathrm{gridlines}=\left[\mathrm{color}=\mathrm{ColorTools}:-\mathrm{Color}\left("RGB",\left[1,1,1\right]\right)\right]\right]\right)$

 > $\mathrm{plot}\left(\mathrm{moment},x=0..L,\mathrm{size}=\left[1000,400\right],\mathrm{axesfont}=\left[\mathrm{Calibri}\right],\mathrm{title}="Moment",\mathrm{labels}=\left["Distance along beam","Moment"\right],\mathrm{labeldirections}=\left[\mathrm{horizontal},\mathrm{vertical}\right],\mathrm{labelfont}=\left[\mathrm{Calibri}\right],\mathrm{titlefont}=\left[\mathrm{Calibri},16,\mathrm{bold}\right],\mathrm{background}=\mathrm{ColorTools}:-\mathrm{Color}\left("RGB",\left[218/255,223/255,225/255\right]\right),\mathrm{axis}=\left[\mathrm{gridlines}=\left[\mathrm{color}=\mathrm{ColorTools}:-\mathrm{Color}\left("RGB",\left[1,1,1\right]\right)\right]\right]\right)$

Shear Distribution

 > $\mathrm{plot}\left(\mathrm{shear},x=0..L,\mathrm{size}=\left[1000,400\right],\mathrm{axesfont}=\left[\mathrm{Calibri}\right],\mathrm{title}="Shear",\mathrm{labels}=\left["Distance along beam","Shear"\right],\mathrm{labeldirections}=\left[\mathrm{horizontal},\mathrm{vertical}\right],\mathrm{labelfont}=\left[\mathrm{Calibri}\right],\mathrm{titlefont}=\left[\mathrm{Calibri},16,\mathrm{bold}\right],\mathrm{background}=\mathrm{ColorTools}:-\mathrm{Color}\left("RGB",\left[218/255,223/255,225/255\right]\right),\mathrm{axis}=\left[\mathrm{gridlines}=\left[\mathrm{color}=\mathrm{ColorTools}:-\mathrm{Color}\left("RGB",\left[1,1,1\right]\right)\right]\right]\right)$
 >