Sunday, November 17, 2013

Beam Deflection - Cantilever Beams

I got some good info from the internet, which I have pasted here. The link is also posted at the bottom for you to refer the website. It is pretty informative.

Deflections and Slopes of Cantilever Beams

Deflections and slopes of cantilever beams
 [picture] [end of picture]
v = deflection in the y direction
  (positive upward)
[inline math]dv/dx =slope of the deflection curve
δB =–v(L) =deflection at end B of
  the beam (downward)
θB =[inline math]angle of rotation at end B
  of the beam (clockwise)
EI = constant
1[picture] [end of picture]
v = - (6L^2 - 4Lx + x^2)
vnull = - (3L^2 - 3Lx + x^2)
δB =
θB =
2[picture] [end of picture]
v = - (6a^2 - 4ax + x^2)(0 le x le a)
vnull = - (3a^2 - 3ax + x^2)(0 le x le a)
v = - (4x - a)(a le x le L)
v = - (a le x le L)
At x = a:
v = -
vnull = -
δB = (4L - a)
θB =
3[picture] [end of picture]
v = - (3L + 3a - 2x)(0 le x le a)
vnull = - (L + a - x)(0 le x le a)
v = - (x^4 - 4Lx^3 + 6L^2x^2 - 4a^3x + a^4)
(a le x le L)
vnull = - (x^4 - 3Lx^2 + 3L^2x - a^3)
(a le x le L)
At x = a:
v = - (3L + a)
vnull = -
δB = (3L^4 - 4a^3L + a^4)
θB = (L^3 - a^3)
4[picture] [end of picture]
v = - (3L - x)
vnull = - (2L - x)
δB =
θB =
5[picture] [end of picture]
v = - (3a - x)(0 le x le a)
vnull = - (2a - x)(0 le x le a)
v = - (3x - a)(a le x le L)
vnull = - (a le x le L)
At x = a:
v = - (3L + a)
vnull = -
δB = (3L - a)
θB =
6[picture] [end of picture]
v = -
vnull = -
δB =
θB =
7[picture] [end of picture]
v = - (0 le x le a)
vnull = - (0 le x le a)
v = - (2x - a)(a le x le L)
vnull = - (a le x le L)
At x = a:
v = - (2L - a)
vnull = -
δB = (2L - a)
θB =
8[picture] [end of picture]
v = - (10L^3 - 10L^2 - 5Lx^2 - x^3)
vnull = - (4L^3 - 6L^2x + 4Lx^2 - x^3)
δB =
θB =
9[picture] [end of picture]
v = - (20L^3 - 10L^2 - 5Lx^2 + x^3)
vnull = - (8L^3 - 6L^2x + 4Lx^2 + x^3)
δB =
θB =
10[picture] [end of picture]
v = - (48L^4)
vnull = - (2π^2Lx - π^2x^2 - 8L^2)
δB = (π^3 - 24)
θB = (π^2 - 8)
Source http://virtual.cvut.cz/beams/

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