Deflections and Slopes of Beams

Table 1. 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)



Table 2. 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 =