# Deflection of a cantilever beam with applied tension at the free end

• mrbec
In summary: I'll try to work my way with this formula and see where it gets me. Thank you very much.Yes, but I would rather call ##w## force per unit length (N/m).
mrbec
TL;DR Summary
Hello. I would like to know if applying tension to a cantilever beam at the free end affects the deflection it would have otherwise due to only its self weight, and if it does, how to calculate the deflection depending on the tension applied.
Not much to add since the question is fairly simple, but again I'm wondering if applying tension at the free end of a cantilever beam affects the deflection it would have if only itself weight is considered. Intuitively, tension should tend to straighten the beam, and if it does, how to calculate the resulting deflection? Please find below a simple illustration of the problem. Maybe I should also note that tension is always applied and not only after the beam is bent. Sorry for the simplicity of the question and thanks in advance.

Welcome!
The original and final directions and magnitude of the tension vector (respect to the stiffness of the beam) are very important.
In other words, how much could the natural deflection of the beam modify the vector tension?

From Roark's Formulas for Stress and Strain: $$y_{max}=\frac{-w}{k^{2}P} \left( \frac{sinhkl \cdot \left( sinhkl-kl \right) }{coshkl} - \left( \left( coshkl-1 \right) - \frac{k^{2}}{2} l^{2}\right) \right)$$ where: $$k=\left( \frac{P}{EI} \right)^{\frac{1}{2}}$$
##w## - distributed load, ##P## - axial tensile load (there are separate formulas for compressive load case), ##l## - beam length, ##E## - Young’s modulus, ##I## - area moment of inertia.

Last edited:
Lnewqban said:
Welcome!
The original and final directions and magnitude of the tension vector (respect to the stiffness of the beam) are very important.
In other words, how much could the natural deflection of the beam modify the vector tension?
I'm quite confused tbh, but the magnitude of the tension vector is constant, and its final direction depends on the final deflection of the beam since it is along its axis and what i am interested in is how much could the tension vector modify the natural deflection of the beam. Maybe a better way to word my question would be: how does a beam behave when subjected to axial and transverse loads simultaneously? is it possible to use the euler-bernoulli equation to solve the problem? Thanks

FEAnalyst said:
From Roark's Formulas for Stress and Strain: $$y_{max}=\frac{-w}{k^{2}P} \left( \frac{sinhkl \cdot \left( sinhkl-kl \right) }{coshkl} - \left( \left( coshkl-1 \right) - \frac{k^{2}}{2} l^{2}\right) \right)$$ where: $$k=\left( \frac{P}{EI} \right)^{\frac{1}{2}}$$
##w## - distributed load, ##P## - axial tensile load (there are separate formulas for compressive load case), ##l## - beam length, ##E## - Young’s modulus, ##I## - area moment of inertia.

Correct me if I'm wrong but:
-Ymax is the deflection at the free end of the beam,
-w is the weight per length,
-P is the magnitude of the tension applied, and
-l is the length of the beam.
Thanks again.

mrbec said:
Correct me if I'm wrong but:
-Ymax is the deflection at the free end of the beam,
-w is the weight per length,
-P is the magnitude of the tension applied, and
-l is the length of the beam.
Thanks again.
I did not see that you added the nomenclature at the end, please ignore what i just said. Anyhow, the equation you provided gives the deflection at the free end of the beam, but is it possible to get the solution for every point along the beam since i want to know the position at which the deflection reaches a certain value? Thanks again.

mrbec said:
I did not see that you added the nomenclature at the end, please ignore what i just said. Anyhow, the equation you provided gives the deflection at the free end of the beam, but is it possible to get the solution for every point along the beam since i want to know the position at which the deflection reaches a certain value? Thanks again.

Yes, but I would rather call ##w## force per unit length (N/m).

Actually, the book also gives a formula for the deflection at any given point of the beam (with ##x## coordinate denoting distance from free end): $$y=y_{max}+ \frac{\theta_{max}}{k} sinhkx+LT_{y}$$ $$\theta_{max}=\frac{w}{kP} \cdot \frac{sinhkl-kl}{coshkl}$$ $$LT_{y}=\frac{-w}{Pk^{2}} \cdot (coshkx-1)- \frac{k^{2}}{2} x^{2}$$

mrbec
FEAnalyst said:
Actually, the book also gives a formula for the deflection at any given point of the beam (with ##x## coordinate denoting distance from free end): $$y=y_{max}+ \frac{\theta_{max}}{k} sinhkx+LT_{y}$$ $$\theta_{max}=\frac{w}{kP} \cdot \frac{sinhkl-kl}{coshkl}$$ $$LT_{y}=\frac{-w}{Pk^{2}} \cdot (coshkx-1)- \frac{k^{2}}{2} x^{2}$$
I'll try to work my way with this formula and see where it gets me. Thank you very much.

FEAnalyst said:
From Roark's Formulas for Stress and Strain: $$y_{max}=\frac{-w}{k^{2}P} \left( \frac{sinhkl \cdot \left( sinhkl-kl \right) }{coshkl} - \left( \left( coshkl-1 \right) - \frac{k^{2}}{2} l^{2}\right) \right)$$ where: $$k=\left( \frac{P}{EI} \right)^{\frac{1}{2}}$$
##w## - distributed load, ##P## - axial tensile load (there are separate formulas for compressive load case), ##l## - beam length, ##E## - Young’s modulus, ##I## - area moment of inertia.

What edition of Roark would that be, please?

Dr.D said:
What edition of Roark would that be, please?

I’ve found this formula in eighth edition of Roark’s, page 270 (table 8.9. Shear, Moment, Slope, and Deflection Formulas for Beams Under Simultaneous Axial Tension and Transverse Loading).

Thanks for the specific citation in Roark. That helps.

## 1. What is a cantilever beam?

A cantilever beam is a type of structural element that is supported at only one end, and the other end is free to move. It is commonly used in construction and engineering projects to support loads and resist bending and deflection.

## 2. What is deflection of a cantilever beam?

Deflection of a cantilever beam refers to the bending or displacement of the beam when a load is applied at the free end. It is a measure of how much the beam will bend under a given load.

## 3. How is deflection of a cantilever beam calculated?

The deflection of a cantilever beam can be calculated using the Euler-Bernoulli beam theory, which takes into account the material properties, beam dimensions, and applied load. There are also online calculators and software programs available for more accurate calculations.

## 4. How does applied tension at the free end affect the deflection of a cantilever beam?

Applied tension at the free end of a cantilever beam will cause the beam to bend downwards, increasing the deflection. The amount of deflection will depend on the magnitude of the applied tension and the properties of the beam.

## 5. How can the deflection of a cantilever beam be minimized?

The deflection of a cantilever beam can be minimized by using a stiffer material, increasing the beam's cross-sectional area, or reducing the applied load. Adding additional supports along the length of the beam can also help to reduce deflection.

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