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Beam Deflections and Slopes
from Mechanics of Materials [1012468]
Table F
Beam and Loading Elastic Curve Maximum Deflection Slope at End Equation of Elastic Curve
−PL33EI -\frac{PL^3}{3EI} −PL22EI -\frac{PL^2}{2EI} y=P6EI(x3−3Lx2) y= \frac{P}{6EI}(x^3-3Lx^2)
−wL48EI -\frac{wL^4}{8EI} −wL36EI -\frac{wL^3}{6EI} y=w24EI(x4−4Lx3+6L2x2) y= \frac{w}{24EI}(x^4-4Lx^3+6L^2x^2)
−ML22EI -\frac{ML^2}{2EI} −MLEI -\frac{ML}{EI} y=−M2EIx2 y= -\frac{M}{2EI}x^2
−PL348EI -\frac{PL^3}{48EI} ±PL216EI \pm \frac{PL^2}{16EI} For x≤12L: For  x\leq \frac{1}{2}L:

y=P48EI(4x3−3L2x) y = \frac{P}{48EI}(4x^3-3L^2x)
For a > b:

−Pb(L2−b2)3/293EIL -\frac{Pb(L^2-b^2)^{3/2}}{9\sqrt{3}EIL}

 

 θA=−Pb(L2−b2)6EIL θ_A = -\frac{Pb(L^2-b^2)}{6EIL}

 

 For x<a: For  x<a:

y=Pb6EIL[x3−(L2−b2)x] y = \frac{Pb}{6EIL}[x^3-(L^2-b^2)x]

 
at xm=L2−b23 at  x_m= \sqrt{\frac{L^2-b^2}{3}} θB=+Pa(L2−a2)6EIL θ_B = +\frac{Pa(L^2-a^2)}{6EIL} For x = a:

y=−Pa2b23EIL y = -\frac{Pa^2b^2}{3EIL}
−5wL4384EI -\frac{5wL^4}{384EI} ±wL324EI \pm \frac{wL^3}{24EI} y=−w24EI(x4−2Lx3+L3x) y = -\frac{w}{24EI}(x^4-2Lx^3+L^3x)
ML293EI \frac{ML^2}{9\sqrt{3}EI} θA=+ML6EI θ_A = +\frac{ML}{6EI} y=−M6EIL(x3−L2x) y = -\frac{M}{6EIL}(x^3-L^2x)
θB=−ML3EI θ_B = -\frac{ML}{3EI}