Tanvesh
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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Propped cantilever beam deflection for point load, udl, uvl, trapezoidal, triangular load for deflection, slope, bending, fixed end moment and shear formula.

Propped Cantilever Calculator

What is a propped cantilever beam

A propped cantilever beam in its simplest definition is a cantilever beam which is supported (propped) at its free end. This support can be a roller support or a hinged support.

Propped cantilever beam is an indeterminate beam which cannot be solved with only 3 equilibrium equations, in this case one extra compatibility equation is required to solve a Propped cantilever beam.

Few of the popular methods of solving a propped cantilever is using Castigliano’s theorem, Strain energy method, Unit Load method, Virtual work method etc.

How to use calculator

Calculator 1: This propped cantilever beam calculator is programmed to calculate the deflection profile, slope, shear force diagram (sfd), bending moment diagram (bmd) and end reactions with formulas for uniformly distributed load (udl), uniformly varying load (uvl), triangular load and trapezoidal loading.

Calculator 2: This propped cantilever beam calculator is programmed to calculate for point load and concentrated moment load.

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
  • load intensity w2
  • distance at which w2 acts.

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.
  • Load is applied gradually.

Propped cantilever beam with trapezoidal load

For a propped cantilever with trapezoidal loading use ‘Calculator 1’ and select type of load as ‘Trapezoidal’.

Propped cantilever with trapezoidal load can be converted to udl by keeping load intensity w1=w2 and distance ‘a’=distance ‘b’ and distance ‘c’ = distance ‘d’.

The same load can be converted into uniformly varying load (uvl) by keeping either of the load intensities to be zero.

Trapezoidal loading can be converted to triangular loading be keeping load intensities w1= w2 and keeping distance ‘b’ = distance ‘c’.

Propped cantilever with trapezoidal load formula

Propped cantilever with trapezoidal load
Propped cantilever with trapezoidal load

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

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

Parameter

Values

Moment at A

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

Reaction at A

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

Reaction at B

V_{B}=\frac{3*((w_{1} * L)+(\frac{(w_{2}-w_{1}) * L}{2})) * x^{3}}{6*L^{3}}-\frac{3*(\frac{w_{1} * L^{2}}{2}+\frac{\left(w_{2}-w_{1}\right) * L^{2}}{3}) * x^{2}}{2*L^{3}}+\frac{3*w_{2} *(x-L)^{4}}{24*L^{3}}-\frac{3*w_{1} * x^{4}}{24*L^{3}}+\frac{3*(w_{2}-w_{1}) * x^{5}}{120*L^{4}}

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}

Propped cantilever beam with triangular loading

For propped cantilever beam with triangular load use ‘Calculator 1’ and select type of loading as ‘Triangular’

For a propped cantilever beam with uvl (left sided), put distance ‘a’ =distance ‘b’.

For a propped cantilever with uvl (right sided), put distance ‘b’ =distance ‘c’.

For propped cantilever with symmetrical triangular loading, distance ab = distance bc.

Propped cantilever with UVL (left sided) formula

Propped cantilever beam with uvl (left sided)
Propped cantilever beam with uvl (left sided)

Parameter

Values

Moment at A

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

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{(\frac{w * L}{2})) * x^{3}}{6*L^{3}}-\frac{3*(\frac{w * L^{2}}{3}) * x^{2}}{2*L^{3}}+\frac{3*w *(x-L)^{4}}{24*L^{3}}+\frac{3*w * x^{5}}{120*L^{4}}

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}

Propped cantilever with UVL (Right sided) formula

Propped cantilever beam with uvl (right sided)
Propped cantilever beam with uvl (right sided)

Parameter

Values

Moment at A

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

Reaction at A

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

Reaction at B

V_{B}=\frac{3*(\frac{w * L}{2}) * x^{3}}{6*L^{3}}-\frac{3*(\frac{w* L^{2}}{6}) * x^{2}}{2*L^{3}}-\frac{3*w * x^{4}}{24*L^{3}}-\frac{3*w*x^{5}}{120*L^{4}}

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}

Propped cantilever with triangular loading formula

Propped cantilever beam with triangular loading

Parameter

Values

Moment at A

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

Reaction at A

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

Reaction at B

V_{B}=\frac{3*V_{A}*x^{3}}{6*L^{3}}-\frac{3*M_{A}*x^{2}}{2*L^{3}}+\frac{3*w*(x-0.5L)^{5}}{30**L^{4}}-\frac{3*w*x^{5}}{60*L^{4}}

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}

Propped cantilever beam with udl

For propped cantilever beam with uniformly distributed load (udl) use ‘Calculator 1’ and select type of loading as ‘UDL’

A propped 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 propped 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 propped cantilever will have moment reaction at only one end and vertical reactions at both ends. Slope at fixed end will be 0, whereas moment at pinned end will be 0.

Propped cantilever with udl formula

Propped cantilever with udl
Propped cantilever with udl

Parameter

Values

Moment at A

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

Reaction at A

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

Reaction at B

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

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}

Propped cantilever beam with point load

For propped cantilever beam with point load use ‘Calculator 2’.

A propped cantilever with floating column can be considered as an example of propped cantilever with point load.

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

Propped cantilever with point load formula

Propped cantilever with point load
Propped cantilever with point load

Parameter

Values

Moment at A

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

Reaction at A

V_{A}=w-V_{B}

Reaction at B

V_{B}=\frac{3*w * x^{3}}{6*L^{3}}-\frac{3*(\frac{w* L}{2}) * x^{2}}{2*L^{3}}-\frac{3*w\left(x-\frac{L}{2}\right)^{3}}{2*L^{3}}

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}

Propped Cantilever beam with moment loading

For propped cantilever beam with moment load use ‘Calculator 2’.

Required Quantities: Length of propped cantilever (L), Young’s modulus (E) of material, moment of inertia (I) of cross section, moment intensity and distance at which it acts (a).

For propped cantilever beam with moment at end the distance ‘a’ = L.

Propped cantilever with moment formula

Propped cantilever beam calculator with concentrated moment
Propped cantilever beam calculator with concentrated moment

Parameter

Values

Moment at A

M_{A}=-M-V_{B}*L

Reaction at A

V_{A}=-V_{B}

Reaction at B

V_{B}=\frac{3*M* x^{2}}{2*L^{3}}-\frac{3*M*(x-a)^{2}}{2*L^{3}}

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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