Cantilever statically indeterminate beam with point load

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sara291
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hi
please if anyone can tell me how i can get the following equation (attachments) of deflection for statically indeterminate cantilever beam with point load (attachments). I'm not an engineering student and having difficulty in coming up with boundary conditions.
looking for help. thanks
 
Attachments
  • Two piecewise deflection equations for a beam. Top: if x < a, y(x) = P(L-a)^2 x / (12EIL^3) times (3aL^2 - 2Lx^2 - ax^2). Bottom: if x > a, y(x) = Pa / (12EIL^3) times (L-x)^2 times (3L^2x - a^2x - 2a^2L).
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  • Sketch labelled b) of a deflected beam fixed at the left wall, with a dashed horizontal reference line, a downward arrow marking a point load, and a filled triangle support at the right end.
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  • Deflected shape sketch of a fixed-support beam labelled c): a vertical wall at the left, a dashed horizontal reference line, a curve dipping below it, a downward arrow on the curve, and a triangular pin support at the right end.
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At the built-in end, the deflection and slope of the beam are both equal to zero. At the propped end, only the deflection is equal to zero.
 
thank you SteamKing. it means at clamped end y(x) & dy/dx = 0 and propped end only dy/dx=o.
 
sara291 said:
thank you SteamKing. it means at clamped end y(x) & dy/dx = 0 and propped end only dy/dx=o.

You got it half right. At the propped end, the deflection y = 0, not the slope dy/dx, as can be seen by inspection of the figure.
 
i tried these condition to find the deflection equation (eq image in thumbnail) using eq M=EI d^2y/dx^2, by integration method. but when i find the constants of integration by applying these BC and put their values i got zero instead of above equation. please help!
 
Attachments
  • Table of beam formulas, Figure 17: a beam fixed at one end and supported at the other with a concentrated load at any point, showing reactions, moments, shear, maximum deflection and deflection equations.
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Can you post your calculations? It might be easier than trying to go through the whole exercise from scratch.
 
thank you SteamKing, and sorry for late reply, part of my calculation is in attachment. i hope you understand my writing...
 
Attachments
  • Handwritten beam deflection derivation on lined paper: EI d²y/dx² = M, bending moment expressions with P and a, integrated twice to give EI dy/dx and EI y with constants A and B and boundary conditions.
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  • Handwritten derivation on lined paper: 'putting eq (2), (3), (4) into (1)', then EIy = P b^2 x^2 (a+2l)/12l^3 type expansion terms, followed by lines combining terms and simplifying to zero, with fraction terms involving P, a, b, l and x.
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