Cantilever beam deflection calculator

Tanvesh
Masters in Structural Engineering | Research Interest - Artificial Intelligence and Machine learning in Civil Engineering | Youtuber | Teacher | Currently working as Research Scholar at NIT Goa

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Cantilever beam deflection calculator for point load, udl, uvl, trapezoidal, triangular load for deflection, slope, bending, fixed end moment and shear formula.

What is a cantilever beam

A beam which is restrained from linear translation and rotation in all directions at one end and the other end is free is called as a cantilever beam.

A cantilever beam in practical can be seen under a balcony constructed off a building.

A cantilever beam design can be such that the depth of the beam at fixed end is maximum and then over the free end it can be reduced gradually.

How to use calculator

This cantilever beam deflection calculator is programmed to calculate the deflection profile, slope, shear force diagram (sfd), bending moment diagram (bmd) and end reactions.

Required: Young’s Modulus (E) of the material, length (L) of the beam, area moment of inertia (I), load intensity (w1), distance at which w1 acts ‘a’, load intensity w2 and distance at which w2 acts ‘b’.

Cantilever with trapezoidal load can be converted to udl by keeping load intensity w1=w2. The same load can be converted into udl by keeping either of them to be zero.

A trapezoidal load can be converted to a udl, uvl, as well as combination of udl and uvl. In the following example a trapezoidal load has been converted to UDL + UVL by keeping distance ‘a’ = distance ‘b’ =0 and distance ‘c’ = distance ‘d’ = L.

Parameter

Values

Moment at A

M_{A}=\frac{w_{1} * L^{2}}{2}+\frac{\left(w_{2}-w_{1}\right) * L^{2}}{3}

Reaction at A

V_{A}=\left(w_{1} * L\right)+\left(\frac{\left(w_{2}-w_{1}\right) * L}{2}\right)

Moment Equation

M=V_{A} * x-M_{A}-\frac{w_{1} * x^{2}}{2}-\frac{(w_{2}-w_{1}) * x^{3}}{6 L}+\frac{w_{2} *(x-L)^{2}}{2}

Deflection

E I * \delta=\frac{V_{A} * x^{3}}{6}-\frac{M_{A} * x^{2}}{2}+\frac{w_{2} *(x-L)^{4}}{24}-\frac{w_{1} * x^{4}}{24}-\frac{(w_{2}-w_{1}) * x^{5}}{120 * L}

Slope

E I * \theta=\frac{V_{A} * x^{2}}{2}-M_{A} * x+\frac{w_{2} *(x-L)^{3}}{6}-\frac{w_{1} * x^{3}}{6}+\frac{(w_{2}-w_{1}) * x^{4}}{24 * L}

Cantilever beam with udl

Cantilever beam with udl can be calculated using ‘Calculator 1’ and selecting load type as UDL.

A cantilever beam carrying half udl will have distance ‘a’ = 0, distance ‘b’ = L/2 or distance ‘a’= L/2 and distance ‘b’ = L.

For a cantilever beam with uniformly distributed load for full length will have distance ‘a’ =0 and distance ‘b’ = L.

All units can be changed by the user.

A cantilever beam will have moment reaction at both ends to be 0 and will have vertical reactions at both ends. Slope at both end will not be 0.

Cantilever beam with udl on entire span formula

Parameter

Values

Moment at A

M_{A}=\frac{w* L^{2}}{2}

Reaction at A

V_{A}=\left(w * L\right)

Moment Equation

M=V_{A} * x-M_{A}-\frac{w* x^{2}}{2}+\frac{w* x^{3}}{6 L}+\frac{w*(x-L)^{2}}{2}

Deflection

E I * \delta=\frac{V_{A} * x^{3}}{6}-\frac{M_{A} * x^{2}}{2}+\frac{w*(x-L)^{4}}{24}-\frac{w* x^{4}}{24}

Slope

E I * \theta=\frac{V_{A} * x^{2}}{2}-M_{A} * x+\frac{w *(x-L)^{3}}{6}-\frac{w* x^{3}}{6}

Cantilever beam with triangular load can be converted into cantilever beam with uvl (left sided) or a uvl (right sided). Uniformly varying load can be achieved by keeping either distance ‘a’ = distance ‘b’ or distance ‘b’ = distance ‘c’.

Cantilever beam with triangular load formula

Parameter

Values

Moment at A

M_{A}=\frac{w_{1} * L^{2}}{4}

Reaction at A

V_{A}=\frac{w* L}{2}

Deflection

E I*\delta=\frac{V_{A}*x^{3}}{6}-\frac{M_{A}*x^{2}}{2}+\frac{w*(x-0.5L)^{5}}{30L}-\frac{w*x^{5}}{60L}

Slope

E I*\theta=\frac{V_{A}*x^{2}}{2}-M_{A}*x+\frac{w*(x-0.5L)^{4}}{6L}-\frac{w*x^{4}}{12L}

Cantilever beam with UVL (Left side) formula

Parameter

Values

Moment at A

M_{A}=\frac{w * L^{2}}{3}

Reaction at A

V_{A}=\frac{w*L}{2}

Moment Equation

M=V_{A} * x-M_{A}-\frac{w * x^{3}}{6 L}+\frac{w *(x-L)^{2}}{2}

Deflection

E I * \delta=\frac{V_{A} * x^{3}}{6}-\frac{M_{A} * x^{2}}{2}+\frac{w *(x-L)^{4}}{24}-\frac{w*x^{5}}{120*L}

Slope

E I * \theta=\frac{V_{A} * x^{2}}{2}-M_{A} * x+\frac{w *(x-L)^{3}}{6}+\frac{w*x^{4}}{24*L}

Cantilever beam with UVL (Right side) formula

Parameter

Values

Moment at A

M_{A}=\frac{w*L^{2}}{6}

Reaction at A

V_{A}=\frac{w*L}{2}

Moment Equation

M=V_{A} * x-M_{A}-\frac{w * x^{2}}{2}+\frac{w * x^{3}}{6 L}

Deflection

E I * \delta=\frac{V_{A} * x^{3}}{6}-\frac{M_{A} * x^{2}}{2}-\frac{w * x^{4}}{24}+\frac{w * x^{5}}{120 * L}

Slope

E I * \theta=\frac{V_{A} * x^{2}}{2}-M_{A} * x-\frac{w * x^{3}}{6}-\frac{w * x^{4}}{24 * L}

Following is a case presented for cantilever beam with point load acting at center or midspan. For this the distance ‘a’ = L/2.

For cantilever beam with point load at free end, put distance ‘a’ = L.

Cantilever beam with point load formula

Parameter

Values

Moment at A

M_{A}=\frac{w* L}{2}

Reaction at A

V_{A}=w

Moment Equation

M=V_{A} * x-M_{A}-w\left(x-\frac{L}{2}\right)

Deflection

E I * \delta=\frac{V_{A} * x^{3}}{6}-\frac{M_{A} * x^{2}}{2}-\frac{w\left(x-\frac{L}{2}\right)^{3}}{2}

Slope

E I * \theta =\frac{V_{A} * x^{2}}{2} -M_{A} * x-\frac{w*x^{2}}{2}

For cantilever beam with moment load use ‘Calculator 2’ and select type of load as ‘Moment’

Cantilever beam with moment at free end is calculated by putting distance ‘a’ = L.

Cantilever beam with moment at midspan is calculated by putting distance ‘a’ = L/2.

Cantilever beam with moment at midspan formula

Parameter

Values

Moment at A

M_{A}=-M

Reaction at A

0

Moment Equation

M=M_{A}+M*(x-0.5L)

Deflection

E I * \delta=\frac{V_{A} * x^{3}}{6}-\frac{M_{A} * x^{2}}{2}-\frac{M*(x-a)^{2}}{2}

Slope

E I * \theta =\frac{V_{A} * x^{2}}{2} -M_{A} * x-M(x-a)

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