# Fixed beam deflection calculator – Free

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

## What is a Fixed beam

When both ends of a beam are restrained from vertical movement, horizontal movement or rotation, then that beam is termed as a fixed beam.

A fixed beam is also called as an Encaster beam or built in beam, since in usual construction, the ends of the beams which are built integrally with columns or other structures are capable of developing moments, and develop vertical and horizontal reaction.

A fixed beam has 6 degrees of restraint at both ends. in a 3 dimensional case and 3 restraints at each end in a 2 dimensional case.

## How to Use Calculator​

Calculator 1 : – It is a fixed beam calculator which can find deflection, slope, moment and shear for a uniformly distributed load (udl), uniformly varying load (uvl), triangular load and trapezoidal loading.

Calculator 2 :- It is a fixed beam calculator which can find deflection, slope, moment and shear for a point load and concentrated moment loading.

Step1 is to select the units either as metric units or imperial units.

User is given the option to assign all the input quantities units separately. The units for each quantity are from metric system of units and imperial system of units.

Required Quantities:

• Length of beam (L)
• Youngs Modulus of material (E) –> to calculate deflection and slope.
• Area moment of inertia (I) for slope and deflection calculation.
• Flexural rigidity is automatically calculated, however user is given the option to input custom flexural rigidity. Also the units of flexural rigidity can be changed.
• Distance of load intensity 1 from left support and/or distance of load intensity 2 from left support.

The inputs for loading intensity and distance can be seen in the top most diagram changing in the real time.

Sign Conventions:

• Downward loads are considered to be positive.
• Anticlockwise moment and anticlockwise reaction is positive.
• Upward reaction is positive, downward reaction is negative.
• Downward deflection is considered negative.
• Negative slope is clockwise rotation and is measured in radians. Positive slope is anticlockwise rotation measured in radians.

Assumptions:

• The material is homogeneous and isotropic.
• Cross section remains same throughout the length.

## Fixed beam with trapezoidal load Formula

A fixed beam with trapezoidal loading can be used to analyze fixed beam with any type of continuous distributed loading. In the Calculator 1 user can select type of load as ‘Trapezoidal’

• By keeping load intensity 1 and 2 equal and distance ‘b’ = distance ‘a’ and distance ‘d’ = distance ‘c’, trapezoidal load can be converted to UDL.
• By keeping load intensity w1=0 and keeping distance ‘b’ = distance ‘a’ and distance ‘d’ = distance ‘c’, trapezoidal load can be converted to a left sided right angled triangle.
• By keeping w2=0 and other settings from previous point, trapezoidal load can be converted to a right sided right angled triangle.

Find fixed beam deflection formula, slope equation, moment, end support reactions and shear equation in following table:-

Kindly note: Downward loading is positive, upward reactions are positive, anticlockwise moment is positive.

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

Parameter

Values

Moment at A

M_{A}=\frac{w_{1} * L^{2}}{2}+\frac{\left(w_{2}-w_{1}\right) * L^{2}}{3}-\frac{2 * w_{1} * L^{2}}{3}-\frac{\left(w_{2}-w_{1}\right) * L^{2}}{2} +\frac{w_{1} * L^{2}}{4}+\frac{\left(w_{2}-w_{1}\right) * L^{2}}{5}

Moment at B

M_{B}=\frac{w_{1} * L^{2}}{3}+\frac{\left(w_{2}-w_{1}\right) * L^{2}}{4}-\frac{w_{1} * L^{2}}{4}-\frac{\left(w_{2}-w_{1}\right) * L^{2}}{5}

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

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}

## Fixed beam calculator with UDL formula

Fixed beam calculator with udl is programmed to find deflection, rotation or slope, moment and shear of the fixed beam with UDL. User has to select type of load as ‘UDL’ to calculate for uniformly distributed loading.

Fixed beam with udl loading can be assumed to be a continuous wall load on the beam, or a continuous load put on the beam.

Fixed beam with udl spanning entire length will have distance a to be equal to zero and distance b should be equal to length of beam.

Fixed beam carrying half udl will have the distance a=0 or a= L/2 and distance b=L/2 or b=L respectively for two cases.

Fixed beam with udl in any other position needs to be input with appropriate distances.

Equations for fixed beam carrying UDL for entire span are given in following table:-

Kindly note: Downward loading is positive, upward reactions are positive, anticlockwise moment is positive.

### Fixed beam deflection formula for UDL​

Parameter

Values

Moment at A

M_{A}=\frac{w* L^{2}}{12}

Moment at B

M_{B}=-\frac{w* L^{2}}{12}

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-M_{A}-\frac{w* x^{2}}{2}+\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}

Fixed beam with triangular loading calculator can be used by selecting type of load as ‘Triangular’ in Calculator 1

For fixed beam with uvl of left sided right angled triangle, distance b= distance c. For fixed beam with uvl of right sided right angled triangle distance a = distance b.

### Fixed beam with triangular load formula​

Parameter

Values

Moment at A

M_{A}=\frac{5*w*L^{2}}{96}

Moment at B

M_{B}=-\frac{5*w*L^{2}}{96}

Reaction at A

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

Reaction at B

V_{B}=\frac{w* L}{4}-\frac{M_{A}+M_{B}}{L}

Deflection

EI*\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

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

### Fixed beam with UVL (left sided)​

Parameter

Values

Moment at A

M_{A}=\frac{w* L^{2}}{30}

Moment at B

M_{B}=\frac{w* L^{2}}{20}

Reaction at A

V_{A}=\frac{wL}{2}-V_{B}

Reaction at B

V_{B}=\left(\frac{w* L}{3}\right)-\frac{\left(M_{A}+M_{B}\right)}{L}

Moment Equation

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

Deflection

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

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}

### Fixed beam with UVL (Right sided)​

Parameter

Values

Moment at A

M_{A}=\frac{w* L^{2}}{20}

Moment at B

M_{B}=\frac{w* L^{2}}{30}

Reaction at A

V_{A}=\left(\frac{w* L}{3}\right)-\frac{\left(M_{A}+M_{B}\right)}{L}

Reaction at B

V_{B}=\frac{wL}{2}-V_{A}

Moment Equation

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

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}

## Fixed beam with point load

Fixed beam with point load is analyzed using ‘Calculator 2’.

Required quantities for calculation: Beam geometry information, Load intensity (w) and distance at which the load acts (a). Please note that a < = L

Fixed beam carrying central point load can be found by keeping a=L/2.

Fixed beam with eccentric point load is found for any arbitrary value of ‘a’ such that it is always less than or equal to length of beam.

Kindly note that this fixed beam calculator is programmed only for single concentrated point load, for fixed beam with two point loads it can be analyzed using principle of superposition, and hence the calculator can be used twice for both the loads and the results can be added.

### Fixed beam deflection formula for central Point load

Parameter

Values

Moment at A

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

Moment at B

M_{B}=-\frac{w* a^{2}*b}{L^{2}}

Reaction at A

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

Reaction at B

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

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}

Fixed beam with moment calculator is analyzed using ‘Calculator 2’ by selecting load type as ‘Moment’

For fixed beam with moment at centre, distance ‘a’ should be kept half the length of the beam.

### Fixed beam with moment at centre formula

For fixed beam with moment at centre put a=L/2 in the following equations.

Parameter

Values

Moment at A

M_{A}=\frac{M*b*(2a-b)}{L^{2}}

Moment at B

M_{B}=-\frac{M*a*(2b-a)}{L^{2}}

Reaction at A

V_{A}=-V_{B}

Reaction at B

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

Moment Equation

M=V_{A} * x-M_{A}-M

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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