Using this calculator you can visualise the shear force, bending moment and deflection of a simply supported beam when a triangular load is applied spanning distance 'a' to 'b' from the left support.
Calculations
Applied force is negative (-) in the downwards direction.
Type 1 loading condition
Simply Supported Beam with Triangular Load
Type 2 loading condition
Simply Supported Beam with Triangular Load
Free body diagram
Free body diagram
Inputs
Geometry and Loading
Length of beam, L
:7.00m
Distance from left support to the zero load end, a
:6.00m
Distance from left support to the maximum load end, b
:1.00m
Magnitude of maximum load, F
:-12.00kN/m
Beam Properties
Elastic Modulus, E
:200.00GPa
Second Moment of Inertia, I
:142,000,000.00mm4
Outputs
Geometry and Loading
Span of load, d
:5.00m
Distance, c
:1.00m
Distance, e
:6.00m
Distance, f
:2.67m
Maximum forces and deflection
Max Shear
:18.57kN
Max Moment
:34.94kN m
Max Deflection
:-5.98mm
Distance, g
:4.33m
Resultant force of triangular load, w
:-30.00kN
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
Simply Supported Beam with Triangular Load
Type 2 loading condition
Simply Supported Beam with Triangular Load
Free body diagram
Free body diagram
Distance formulas for Type 1
Distance formulas for Type 2
Beam support reactions (same for Type 1 and Type 2):
ΣM@x=0=0Rb×L=−w×fRb=L−w×fwhere: w=21F×d
ΣM@x=L=0Ra×L=−w×gRa=L−w×gwhere: w=21F×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 the right end of the beam, there are a few extra steps to consider:
Change the FBD so that the load does a wrap-around the beam and stops at
b
from the wall on the underside of the beam
Cut a section 'A' just before the right support reaction
Apply Macaulay's Theorem as normal to get the
V(x)
and
M(x)
equations considering only the forces present on the left side of the section cut
Find slope,
m
given by:
m=x2−x1y2−y1=b−aF−0=dF
Type 1 loading condition
Free body diagram adjusted for Macaulay's Theorem
Type 2 loading condition
Free body diagram adjusted for Macaulay's Theorem
Type 1 loading condition:
V(x)=Ra<x−0>0+2dF<x−a>2−F<x−b>1−2dF<x−b>2
M(x)=Ra<x−0>1+6dF<x−a>3−2F<x−b>2−6dF<x−b>3
Type 2 loading condition:
V(x)=Rb<x−0>0+2dF<x−c>2−F<x−e>1−2dF<x−e>2
M(x)=Rb<x−0>1+6dF<x−c>3−2F<x−e>2−6dF<x−e>3
Step 3: Perform the Double Integration Method to find the deflection equation.
Integrate the Bending moment equations once to get the Slope Equation.
Apply the Boundary Conditions to find the constants
C1
and
C2
i) Type 1 loading condition:
BC 1: @ x=0, Y(x)=00=EI1[6Ra<0−0>3+120dF<0−a>5−24F<0−b>4−120dF<0−b>5+C1(0)+C2]0=EI1[0+0+0+0+0+C2]C2=0
BC 2: @ x=L, Y(x)=00=EI1[6Ra<L−0>3+120dF<L−a>5−24F<L−b>4−120dF<L−b>5+C1(L)+C2]0=EI1[6RaL3+120dF(L−a)5−24F(L−b)4−120dF(L−b)5+C1L]C1=−[6RaL2+120dLF(L−a)5−24LF(L−b)4−120dLF(L−b)5]
ii) Type 2 loading condition:
BC 2: @ x=L, Y(x)=00=EI1[6Rb<L−0>3+120dF<L−c>5−24F<L−e>4−120dF<L−e>5+C1(L)+C2]0=EI1[6RbL3+120dF(L−c)5−24F(L−e)4−120dF(L−e)5+C1L]C1=−[6RbL2+120dLF(L−c)5−24LF(L−e)4−120dLF(L−e)5]
You are now ready to plot the curves to determine the overall shear force, bending moment and deflection of a simply supported beam with a triangular load!