Monday, November 18, 2013

Beam Deflection - Simple Beams

Beam Deflection - Simple 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 Simple Beams

 [picture] [end of picture]
v = deflection in the y direction
  (positive upward)
[inline math]dv/dx =slope of the deflection curve
δC =–v(L/2) =deflection at midpoint C of
  the beam (downward)
x1 = distance from support A to
  point of maximum deflection
δmax =–vmax =maximum deflection (downward)
θA =[inline math]angle of rotation at left-hand
  end of the beam (clockwise)
θB =[inline math]angle of rotation at right-hand
  end of the beam (counterclockwise)
EI = constant
1[picture] [end of picture]
v = - (L^3 - 2Lx^2 + x^3)
vnull = - (L^3 - 6Lx^2 + 4x^3)
δC = δmax =
θA = θB =
2[picture] [end of picture]
v = - (9L^3 - 24Lx^2 + 16x^3)(0 le x le )
vnull = - (9L^3 - 72Lx^2 + 64x^3)(0 le x le )
v = - (8x^3 - 24Lx^2 + 17L^x - L^3)
( le x le L)
vnull = - (24x^2 - 48Lx + 17L^2)( le x le L)
δC =
θA =
θB =
3[picture] [end of picture]
v = - (a^4 - 4a^3L + 4a^2L^2 + 2a^2x^2-
- 4aLx^2 + Lx^3)(0 le x le a)
vnull = - (a^4 - 4a^3L + 4a^2L^2 + 6a^2x^2-
- 12aLx^2 + 4Lx^3)(0 le x le a)
v = - ( - a^2L + 4L^2x + a^2x - 6Lx^2 + 2x^3)
(a le x le L)
vnull = - (4L^2 + a^2 - 12Lx + 6x^2)
(a le x le L)
θA = (2L - a)^2
θB = (2L^2 - a^2)
4[picture] [end of picture]
v = - (3L^2 - 4x^2)(0 le x le )
vnull = - (L^2 - 4x^2)(0 le x le )
δC = δmax =
θA = θB =
5[picture] [end of picture]
v = - (L^2 - b^2 - x^2)(0 le x le a)
vnull = - (L^2 - b^2 - 3x^2)(0 le x le a)
θA =
θB =
If a ge b:
δC =
If a le b:
δC =
If a ge b:
x1 = and
δmax =
6[picture] [end of picture]
v = - (3aL - 3a^2 - x^2)(0 le x le a)
vnull = - (aL - a^2 - x^2)(0 le x le a)
v = - (3Lx - 3x^2 - a^2)(x le a le L - a)
vnull = - (L - 2x)(x le a le L - a)
δC = δmax = (3L^2 - 4a^2)
θA = θB =
7[picture] [end of picture]
v = - (2L^2 - 3Lx + x^2)
vnull = - (2L^2 - 6Lx + 3x^2)
δC =
θA =
θB =
x1 = L(1 - )and
δmax =
8[picture] [end of picture]
v = - (L^2 - 4x^2)(0 le x le )
vnull = - (L^2 - 12x^2)(0 le x le )
δC = 0
θA =
θB = -
9[picture] [end of picture]
v = - (6aL - 3a^2 - 2L^2 - x^2)
(0 le x le a)
vnull = - (6aL - 3a^2 - 2L^2 - 3x^2)
(0 le x le a)
At x = a:
v = - (2a - L)
vnull = - (3aL - 3a^2 - L^2)
θA = (6aL - 3a^2 - 2L^2)
θB = (3a^2 - L^2)
10[picture] [end of picture]
v = - (L - x)
vnull = - (L - 2x)
δC = δmax =
θA = θB =
11[picture] [end of picture]
v = - (7L^4 - 10L^2x^2 + 3x^4)
vnull = - (7L^4 - 20L^2x^2 + 15x^4)
δC =
θA =
θB =
x1 = 0.5193L
δmax = 0.00652
12[picture] [end of picture]
v = - (5L^2 - 4x^2)^2
(0 le x le )
vnull = - (5L^2 - 4x^2)(L^2 - 4x^2)
(0 le x le )
δC = δmax =
θA = θB =
13[picture] [end of picture]
v = -
vnull = -
δC = δmax =
θA = θB =
Source http://virtual.cvut.cz/beams/
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