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

## Other Calculators ###### Fixed Beam Calculator ###### Propped Cantilever Calculator ###### Simply Supported beam Calculator ## What is simply supported beam

If a beam is restrained in translation in both directions at one end and only in single direction at other end and not restrained against rotation at both ends is called as simply supported beam.

In simple words, one end is hinged, other end is roller, this definition however can be changed and both ends being hinge support can also be considered as simply supported beam.

## How to use calculator

This simply supported beam with trapezoidal load 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’.

Simply supported beam 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.

## Simply supported beam with trapezoidal load

Keeping distance a=b=0 and distance c=d=L; let intensity 1 be w1 and intensity 2 be w2.

Parameter

Values

Reaction at A

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

Reaction at B

V_{B}=\frac{w_{1}L}{2}+\frac{(w_{2}-w_{1})*L}{3}

Moment Equation

M=V_{A} * x-\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{w_{2} *(x-L)^{4}}{24}-\frac{w_{1} * x^{4}}{24}+\frac{w_{1} * x^{5}}{120 * L}-\frac{V_{A}*x*L^{2}}{6}+\frac{w_{1}*x*L^{3}}{30}

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_{1} * x^{4}}{24 * L} -\frac{V_{A}*L^{2}}{6}+\frac{w_{1}*L^{3}}{30}

## Simply supported beam with udl

Simply supported beam with udl can be analyzed by ‘Calculator 1’, by selecting load type as ‘UDL’.

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

For a simply supported 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 simply supported 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.

### Simply supported beam with udl formula

Parameter

Values

Reaction at A

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

Reaction at B

V_{B}=\left(\frac{w L}{2}\right)

Moment Equation

M=V_{A} * x-\frac{w* x^{2}}{2}

Deflection

E I * \delta=\frac{V_{A} * x^{3}}{6}+\frac{w*(x-L)^{4}}{24}-\frac{w* x^{4}}{24}-\frac{V_{A}*x*L^{2}}{6}+\frac{w*x*L^{3}}{30}

Slope

E I * \theta=\frac{V_{A} * x^{2}}{2}-M_{A} * x+\frac{w *(x-L)^{3}}{6}-\frac{w* x^{3}}{6}+\frac{w * x^{4}}{24 * L} -\frac{V_{A}*L^{2}}{6}+\frac{w*L^{3}}{30}

## Simply supported beam with UVL

For simply supported beam with uvl, use ‘Calculator 1’ and select type of load as ‘Triangular’.

### Simply supported beam with uvl (left sided) formula

Parameter

Values

Reaction at A

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

Reaction at B

V_{B}=\frac{w*L}{3}

Moment Equation

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

Deflection

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

Slope

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

### Simply supported beam with uvl (Right sided) formula

Parameter

Values

Reaction at A

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

Reaction at B

V_{B}=\frac{w*L}{6}

Moment Equation

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

Deflection

E I * \delta=\frac{V_{A} * x^{3}}{6} -\frac{w * x^{4}}{24}+\frac{w* x^{5}}{120 * L}-\frac{V_{A}*x*L^{2}}{6}+\frac{w*x*L^{3}}{30}

Slope

E I * \theta=\frac{V_{A} * x^{2}}{2} -\frac{w * x^{3}}{6}+\frac{w* x^{4}}{24 * L} -\frac{V_{A}*L^{2}}{6}+\frac{w*L^{3}}{30}

## Simply supported beam with point load

A point load is considered to be idealization in engineering mechanics, as any physical load that has a very small contact area that can be idealized as a point loading.

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

### Simply supported beam with point load formula

Parameter

Values

Reaction at A

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

Reaction at B

V_{B}=\frac{w}{2}

Moment Equation

M=V_{A} * x-w*(x-0.5*L)

Deflection

E I * \delta=\frac{V_{A} * x^{3}}{6} -\frac{w*(x-0.5*L)^{3}}{6}-\frac{V_{A}*x*L^{2}}{6}+\frac{w*x*L^{2}}{48}

Slope

E I * \theta =\frac{V_{A} * x^{2}}{2}-\frac{w*(x-0.5L)^{2}}{2} -\frac{V_{A}*L^{2}}{6}+\frac{w*L^{2}}{48}

## Simply supported beam with moment

For simply supported beam with moment use calculator 2 and choose type of loading as Moment.

Case 1: For simply supported beam with moment at center put distance ‘a’ = L/2.

Case 2: For simply supported beam with moment load at one end put distance ‘a’= 0 or distance ‘a’ = L.

Case 3: For simply supported beam with moment at both ends you may algebraically add the results of case 2 by keeping distance ‘a’ = 0 and distance ‘a’ = L respectively.

### Simply supported beam with moment at center formula

Parameter

Values

Reaction at A

V_{A}=-V_{B}

Reaction at B

V_{B}=\frac{-M}{L}

Moment Equation

M=V_{A} * x-M

Deflection

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

Slope

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

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