Beams With Uniform Load & End Moments Description This worksheet computes the maximum positive bending moment, the maximum deflection and the points of inflection for a beam with a uniformly distributed load (UDL) 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. The user inputs: 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 output parameters is given at the end of this worksheet
Diagram & Input Variables Span length, L ≔ 24 Left end moment,
M L ≔ 52
⋅
Right end moment,
M R ≔ 125
UDL load,
w ≔ 2.16 ―
Moment of inertia,
I ≔ 6987
Modulus of elasticity,
E ≔ 3600
⋅
4
Calculations Left end reaction,
w ⋅ L ⎛ M L − M R ⎞ RL ≔ ― + ―― = 22.878 2 L ⎝ ⎠ Right end reaction,
R R ≔
w⋅L 2
+
Location of the point of zero shear from the left end,
⎛ M R − M L ⎞ ⎝
L
⎠
= 28.962
Shear as a function of distance x from the left end,
V x ≔ RL − w ⋅ x
X 0 ≔
RL w
= 10.592
Moment as a function of distance x from the left end,
M x ≔ −M L + RL ⋅ x −
Maximum positive (or least negative) moment at distance x from the left end,
M max ≔ M X 0 = 69.162
⋅
1 2
⋅w⋅x
2
Moment M(x) vs distance x across the beam span N ≔ 20 Number of plot points,
x≔0
Plot intervals,
,
L N
‥L
10.592 69.162
80 60 40 20 0
0
2.5
5
7.5
10
12.5
15
17.5
20
22.5
25
-20
M x
-40 -60 -80 -100 -120 -140
x
Shear V(x) against distance x across the beam span 10.592
27.5 22 16.5 11 5.5 0
0
2.5
5
7.5
10
12.5
15
17.5
20
22.5
25
V x
-5.5 -11 -16.5 -22 -27.5 -33
x
⋅
Further calculations The following calculation locates the distance from the left end to the nearest point of inflection,
X L ≔ 0
Initial guess value,
Using the root function, we can find the the first point of zero bending moment and hence the first point of inflection,
X L ≔
M X L , X L = 2.589
The following calculation locates the distance from the right end to the nearest point of inflection, and will follow the same workflow as above,
X R ≔ L X R ≔ L −
M X R , X R = 5.406
The following calculations calculate the beam rotations at both ends, 3
θL ≔
At the left end,
w ⋅ L − 8 ⋅ M L ⋅ L − 4 ⋅ M R ⋅ L 2 4⋅ E ⋅ I
= 0.002
3
At the right end,
θ R ≔
w ⋅ L − 8 ⋅ M R ⋅ L − 4 ⋅ M L ⋅ L 2 4⋅ E ⋅ I
= 2.07 ⋅ 10
−4
The following calculation finds the distance from the left reaction to the point of zero slope, and maximum deflection, Slope along the length of the beam as function of distance x is given by,
⎛⎛ w ⋅ L ⋅ x 2 − w ⋅ x 3 ⎞ w ⋅ x 3 M L ⋅ x ⎛ L − x ⎞ M L ⋅ x M R ⋅ x 2 ⎞ θ x ≔ θL − ― ⋅ ――――― + ――− ―― − ― ⋅ ―― − ―― E ⋅ I ⎝⎝ 4 12 2 2 2⋅L ⎠ ⎠ ⎝ L ⎠ 1
Initial point of zero slope guess value,
X Δ ≔
X Δ ≔
L 2
θ X Δ , X Δ = 10.93
Beam deflection from the left end reaction as a function of distance x is given by,
⎛⎛ w ⋅ L ⋅ x 2 − w ⋅ x 3 ⎞ x w ⋅ x 4 M L ⋅ x 2 ⎛ L − x ⎞ M L ⋅ x 2 M R ⋅ x 3 ⎞ δ x ≔ θL ⋅ x − ⋅ ⋅ + − − ⋅ − E ⋅ I ⎝⎝ 4 24 3 6 6⋅L ⎠ ⎠ 3 ⎝ L ⎠ 1
Maximum deflection at point of zero slope,
Δmax ≔ δ X Δ = 0.206
Variable Summary Span length,
L = 24
Left end moment,
M L = 52
Right end moment,
M R = 125
UDL load,
w = 2.16
Moment of inertia,
I = 6.987 ⋅ 10
Modulus of elasticity,
E = 3.6 ⋅ 10
⋅ ⋅
3
3
Left end reaction,
RL = 22.878
Right end reaction,
R R = 28.962
Location of the point of zero shear, Maximum positive moment,
4
X 0 = 10.592
M max = 69.162
⋅
Slope at the left end of the beam,
θL = 0.002
Slope at the right end of the beam,
θ R = 2.07 ⋅ 10
Distance from left end to the point of maximum deflection,
X Δ = 10.93
Maximum deflection,
Δmax = 0.206
Distance from left end to the closest point of inflection,
X L = 2.589
Distance from right end to the closest point of inflection,
X R = 5.406
−4
