Using this calculator you can visualise the shear force, bending moment and deflection of a simply supported beam when a trapezoidal load is applied spanning distance 'a' to 'b' from the fixed support.
Calculations
Applied force is negative (-) in the downwards direction.
Type 1 loading condition
Cantilever beam with Trapezoidal load
Free body diagram
Type 2 loading condition
Cantilever beam with Trapezoidal load
Free body diagram
Inputs
Geometry and Loading
Length of beam, L
:8.00m
Distance from fixed support to minimum load end, a
:3.00m
Distance from fixed support to maximum load end, b
:7.00m
Magnitude of minimum load, F1
:-3.00kN/m
Magnitude of maximum load, F2
:-5.00kN/m
Beam Properties
Elastic Modulus, E
:200.00GPa
Second Moment of Inertia, I
:142,000,000.00mm4
Outputs
Geometry and Loading
Span of Loading, d
:4.00m
Distance, c
:1.00m
Distance, e
:0.00m
Maximum forces and deflection
Max Shear
:16.00kN
Max Moment
:-82.67kN m
Max Deflection
:-48.27mm
Distance, f
:5.17m
Resultant force of trapezoidal load, w
:-16.00kN
Output Diagrams
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Explanation
Using Macaulay's Theorem and the Double Integration Method, we can create the equations for shear force, bending moment and deflection as follows:
Want to know how to derive the equations? Keep reading!
Derivation
Step 1: Find the beam support reactions by taking moments at each end.
Type 1 loading condition
Cantilever beam with Trapezoidal load
Free body diagram
Type 2 loading condition
Cantilever beam with Trapezoidal load
Free body diagram
Distance formulas for Type 1
Distance formulas for Type 2
Beam support reactions (same for Type 1 and Type 2):
ΣFy=0V=−wwhere: w=21(F1+F2)d
ΣM0=0M=−w×fwhere: w=21(F1+F2)d
Step 2: Find the shear force
V(x)
and bending moment
M(x)
equations by using the table of Macaulay's Singularity Functions on the homepage. Because the load is not applied up to an end of the beam, there are a few extra steps to consider:
Change the FBD so that the distributed load extends all the way to an end of the beam. Make sure the resultant directly gives the original FBD.
Apply Macaulay's Theorem as normal to get the
V(x)
and
M(x)
equations considering all loads along the beam for Type 1 while only considering loads on the right side of section A-A for Type 2
Find slope,
m
given by:
m=x2−x1y2−y1=b−aF2−F1=dF2−F1
Type 1 loading condition
Free body diagram adjusted for Macaulay's Theorem
Type 2 loading condition
Free body diagram adjusted for Macaulay's Theorem (Only consider up to Section A-A)
Apply the Boundary Conditions to find the constants
C1
and
C2
Type 1 loading condition:
BC 1: @ x=0, θ(x)=00=EI1[−M<0−0>1+2V<0−0>2+6F1<0−a>3+24d(F2−F1)<0−a>4−6F2<0−b>3−24d(F2−F1)<0−b>4+C1]0=EI1[0+0+0+0+0+0+C1]C1=0
BC 2: @ x=0, Y(x)=00=EI1[−2M<0−0>2+6V<0−0>3+24F1<0−a>4+120d(F2−F1)<0−a>5−24F2<0−b>4−120d(F2−F1)<0−b>5+C2]0=EI1[0+0+0+0+0+0+C2]C2=0
Type 2 loading condition:
BC 1: @ x=L, θ(x)=00=EI1[6F1<L−c>3+24d(F2−F1)<L−c>4−6F2<L−e>3−24d(F2−F1)<L−e>4+C1]C1=−[6F1(L−c)4+24d(F2−F1)(L−c)4−6F2(L−e)3−24d(F2−F1)(L−e)4]
BC 2: @ x=L, Y(x)=00=EI1[24F1<L−c>4+120d(F2−F1)<L−c>5−24F2<L−e>4−120d(F2−F1)<L−e>5+C1(L)+C2]C2=−[24F1(L−c)4+120d(F2−F1)(L−c)5−24F2(L−e)4−120d(F2−F1)(L−e)5+C1L]