(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

I want to derive a formula for deflection w(x) from the Euler-Bernoulli beam equation. It's essentially only four integrations but I'm not sure about my boundary conditions, particularly wrt shear. The beam is a cantilever with a point load at the unsupported end.

And apologies in advance for the clumsy latex...

2. Relevant equations

P = load

w = deflection = 0 when x =0

[tex]\frac{dw}{dx}[/tex]= slope = 0 when x =0

EI[tex]\frac{d^{2}w}{dx^{2}}[/tex] = bending moment = 0 when x = L

-EI[tex]\frac{d^{3}w}{dx^{}3}[/tex] = shear force = 0 when?

I guess my question is: what boundary condition do I need to get rid of the C[tex]_{1}[/tex] after the first integration and I suppose if this is the right way to go about this at all!

3. The attempt at a solution

Here's what I've done so far:

EI[tex]\frac{d^{4}w}{dx^{4}}[/tex]=P

EIEI[tex]\frac{d^{3}w}{dx^{3}}[/tex]=Px + C[tex]_{1}[/tex]

I've left C[tex]_{1}[/tex] here and carried it through since I don't have a clue about the shear BC.

EI[tex]\frac{d^{2}w}{dx^{2}}[/tex]=P[tex]\frac{x^{2}}{2}[/tex] + C[tex]_{1}[/tex]x +C[tex]_{2}[/tex]

EI[tex]\frac{d^{2}w}{dx^{2}}[/tex] = bending moment = 0 when x = L, so

C[tex]_{2}[/tex]=-[tex]\frac{PL^{2}}{2}[/tex]-C[tex]_{1}[/tex]L

EI[tex]\frac{d^{2}w}{dx^{2}}[/tex]=P[tex]\frac{x^{2}}{2}[/tex] + C[tex]_{1}[/tex]x - [tex]\frac{PL^{2}}{2}[/tex]-C[tex]_{1}[/tex]L

EI[tex]\frac{dw}{dx}[/tex]=[tex]\frac{Px^{3}}{6}[/tex]+C[tex]_{1}[/tex][tex]\frac{x^{2}}{2}[/tex]-[[tex]\frac{PL^{2}}{2}[/tex]-C[tex]_{1}[/tex]L]x + C[tex]_{3}[/tex]

[tex]\frac{dw}{dx}[/tex] = 0 when x = 0 so C[tex]_{3}[/tex]=0

and finally

EIw=[tex]\frac{Px^{4}}{24}[/tex]+[tex]\frac{C_{1}x^{3}}{6}[/tex]-[[tex]\frac{PL^{2}}{2}[/tex]-C[tex]_{1}[/tex]L][tex]\frac{x^{2}}{2}[/tex] + C[tex]_{4}[/tex]

w=0 when x=0 so C[tex]_{4}[/tex]=0

so:

EIw=[tex]\frac{Px^{4}}{24}[/tex]+[tex]\frac{C_{1}x^{3}}{6}[/tex]-[[tex]\frac{PL^{2}}{2}[/tex]-C[tex]_{1}[/tex]L][tex]\frac{x^{2}}{2}[/tex]

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# Derivation of Deflection from Euler-Bernoulli Beam Equation

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