1.2 Beams With Uniform Load and End Moments
1.2 Beams With Uniform Load and End Moments
1.2 Beams With Uniform Load and End Moments
Description
This application computes the maximum positive bending moment, the maximum deflection,
and the points of inflection for a beam with a uniformly distributed load and applied end
moments.
The values calculated include the location of the point of zero shear, the maximum positive
bending moment, the rotations at each end of the beam, and the location of the point of zero
slope. These computations are made within the Calculations section of this document which
begins on page 4.
The user must enter the span length, the uniformly distributed load per unit length, the end
moments, the modulus of elasticity, and the moment of inertia for the beam.
A summary of input and calculated values is shown on pages 7 and 8. Plots of the moment
versus distance across the span and shear versus distance across the span are shown on page 5.
Input
Notation
Xo distance from the left end to the point of zero shear and maximum
positive bending moment
w ⋅ L ⎛ ML − MR ⎞
Left end reaction: RL ≔ ―― + ⎜―――⎟ RL = 22.878 kip
2 ⎝ L ⎠
w ⋅ L ⎛ MR − ML ⎞
Right end reaction: RR ≔ ――+ ⎜――― ⎟ RR = 28.962 kip
2 ⎝ L ⎠
Location of the
RL
point of zero shear Xo ≔ ―― Xo = 10.592
from the left end: w
Shear as a function
of distance x from the V (x) ≔ RL − w ⋅ x
left end:
Moment as a
1 2
function of distance M (x) ≔ −ML + RL ⋅ x − ―⋅ w ⋅ x
x from the left end: 2
Maximum positive
(or least negative)
moment at distance Mmax ≔ M ⎛⎝Xo⎞⎠ Mmax = 69.162 ⋅
x from the left end:
Plot of Moment M(x) versus x for N Points Across the Span
L
N ≔ 20 x≔0⋅ , ―‥ L
N
80
60
40
20
0
0
-20
M (x) ( ⋅ ) -40
-60
-80
-100
-120
-140
27.5
22
16.5
11
5.5
-5.5
-11
-16.5
-22
-27.5
-33
XL ≔ 0 ⋅
XL = 2.589 ft
The following computation locates the distance XR from the right end to the nearest point of
inflection:
XR ≔ L
XR = 5.406
3
w ⋅ L − 8 ⋅ ML ⋅ L − 4 ⋅ M R ⋅ L
θL ≔ ――――――――― θL = 0.0019
24 ⋅ E ⋅ I
3
w ⋅ L − 8 ⋅ MR ⋅ L − 4 ⋅ ML ⋅ L
θR ≔ ――――――――― θR = 0.0002
24 ⋅ E ⋅ I
Slope (x) along the length of the beam expressed as a function of distance x from the left end:
2
1 ⎛⎛ w ⋅ L ⋅ x ML ⋅ x ⎛ L − x ⎞ ML ⋅ x MR ⋅ x ⎞
2 3 3
w⋅x ⎞ w⋅x
θ (x) ≔ θL − ―― ⋅ ⎜⎜―――− ――⎟ + ――− ―― − ⎜――⎟ ⋅ ―― − ――― ⎟
E ⋅ I ⎝⎝ 4 4 ⎠ 12 2 ⎝ L ⎠ 2 2⋅L ⎠
Distance X from the left reaction to the point of zero slope and maximum deflection:
L
XΔ ≔ ―
2
XΔ = 10.93 ft
Beam deflection (x) expressed as a function of distance x from the left end reaction:
2 2 3
1 ⎛⎛ w ⋅ L ⋅ x MR ⋅ x ⎞
2 3 4
w⋅x ⎞ x w⋅x ML ⋅ x ⎛ L − x ⎞ ML ⋅ x
δ (x) ≔ θL ⋅ x − ―― ⋅ ⎜⎜―――− ――⎟ ⋅ ―+ ――− ――― − ⎜――⎟ ⋅ ――― − ――― ⎟
E ⋅ I ⎝⎝ 4 4 ⎠ 3 24 3 ⎝ L ⎠ 6 6⋅L ⎠
XΔ = 10.93 ft
Δmax ≔ δ ⎛⎝XΔ⎞⎠
Δmax = 0.206 in
Summary
Input
Span length: L = 24
4
Moment of inertia: I = 6987
Computed Variables