Calculating Bending Stress & Moment of Inertia for Tapered Cantilever Beam

In summary: Deflection equation is still determined by the cross-section, and the deflection at the tip will still be determined by the unit force acting at the tip. You can use the deflection equation to determine the force needed to produce a given deflection at the tip.
  • #1
d.saldana
3
0
Hello all,

I need to calculate the bending stress of a tapered cantilever beam. However, I would like to know how to calculate the moment of inertia.

For a uniform shape beam, I know the Ix=(bh3)/12, but since h changes throughout the length of the beam how would inertia, I, be calculated?

I was able to figure out the height of the beam as a function of x, H(x).

Equation for H(x)= [(a-b)/L](x)+a
where;
a= height of beam at fix wall
b= height of beam at end
L= total length of beam
x= point along length of beam.

Any help would be greatly appreciated!
 

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  • #2
d.saldana said:
Hello all,

I need to calculate the bending stress of a tapered cantilever beam. However, I would like to know how to calculate the moment of inertia.

For a uniform shape beam, I know the Ix=(bh3)/12, but since h changes throughout the length of the beam how would inertia, I, be calculated?

I is still calculated in the usual manner, but instead of being a constant w.r.t. the length of the beam, I is now a function of the length coordinate.

[itex]I(x) = \frac{b(x)h(x)^{3}}{12}[/itex]

and I(x) will still be about the centroid of the section.

To calculate bending stress, you can calculate the section modulus of the beam at the point or points where you want to determine the bending stress and use the formula σ = M / SM as before.

I was able to figure out the height of the beam as a function of x, H(x).

Equation for H(x)= [(a-b)/L](x)+a
where;
a= height of beam at fix wall
b= height of beam at end
L= total length of beam
x= point along length of beam.

Any help would be greatly appreciated!

All you have to do to find the height is plug in a value for 'x'.
 
  • #3
Hi SteamKing,

Thank you for your help. Taking the cross-section at the wall I can calculate the bending stress at that point.

However, this led to more questions.

1) What would be the equation to determine the deflection at the very tip? I don't believe
δ=(FL3)/(3EI) since beam is tapered..
2) If I know the deflection needed, can I manipulate the deflection equation to obtain the force?
3) Would the moment of inertia change since taken at the very tip?

Again, thank you for your help..
 
  • #4
d.saldana said:
Hi SteamKing,

Thank you for your help. Taking the cross-section at the wall I can calculate the bending stress at that point.

However, this led to more questions.

1) What would be the equation to determine the deflection at the very tip? I don't believe
δ=(FL3)/(3EI) since beam is tapered..
2) If I know the deflection needed, can I manipulate the deflection equation to obtain the force?
3) Would the moment of inertia change since taken at the very tip?

Again, thank you for your help..

Now you have found where things get more complicated with varying cross-section beams.

When I is no longer constant w.r.t. the length of the beam, the deflection equation must be re-derived from first principles, knowing the bending moment in the beam as a function of the beam loading and the position of the loads. In other words,

[itex]θ(x) = \frac{1}{E}\int^{L}_{0}\frac{M(x)}{I(x)}dx + C_{1}[/itex]

[itex]δ(x) = \int^{L}_{0}θ(x)dx + C_{1}x + C_{2}[/itex]

assuming that E, Young's Modulus, is still constant, and M(x) represents the bending moment as a function of the length coordinate. The two unknown constants of integration, [itex]C_{1}[/itex] and [itex]C_{2}[/itex] are determined by applying the boundary conditions of the cantilever known at the fixed end, namely the slope and deflection there are both equal to zero.

Now, you can assume a unit force acting at the tip of the beam and work out the deflection at the tip based on that force. Once you have calculated the deflection due to a unit force, you can determine the force needed to produce a given deflection at the tip using a simple ratio.

The two integrations need not be carried out symbolically unless you are interested; numerical integration can be used if you are only interested in what happens say, at the the tip of the beam.
 
  • #5
OK, I see. I will work it out and see what I can come up with.. Thanks!
 
  • #6
Hi d.saldana,
Greetings to you. I have the same query as yours. I tried a lot, but could not get the defection and bending stress equations.
I would like to know whether you got the solution.
 
  • #7
I need help with this also, I have a tapered beam, and need to work out the theoretical deflection along intervals of x across the beam. I have the beam length, the butt diameter, the tip diameter, and the beam thickness. After that I have not a clue where to start, this forum seems to make sense to me but then i get lost when trying to implement what has been said into Excel
 
  • #8
Please start a new thread for this . If this is homework then start your new thread in homework section .

Anyway we need to know more about the problem . Try sketching the beam so that we can see what shape it is and where the loads act .
 

1. What is a tapered cantilever beam?

A tapered cantilever beam is a structural element that is fixed at one end and free at the other end, with a varying cross-sectional area along its length. This design allows for a gradual decrease in the beam's thickness and width from the fixed end to the free end, resulting in a change in the beam's stiffness and strength.

2. What are the advantages of using a tapered cantilever beam?

One advantage of using a tapered cantilever beam is that it can distribute stress and load more efficiently compared to a uniform beam. This can result in a lighter and more cost-effective design. Additionally, the varying cross-sectional area allows for greater flexibility in design and can help improve the overall performance of the beam.

3. What factors should be considered when designing a tapered cantilever beam?

The design of a tapered cantilever beam should take into account factors such as the desired stiffness and strength, the material properties, the loads and forces it will be subjected to, and the desired deflection and bending moment. The taper ratio, or the ratio of the beam's width at the fixed end to its width at the free end, is also an important factor to consider.

4. How is the taper ratio determined for a tapered cantilever beam?

The taper ratio for a tapered cantilever beam is typically determined by the designer based on the desired deflection, bending moment, and stress distribution along the beam. It can also be calculated using mathematical equations or determined through experimental testing.

5. What are some common applications of tapered cantilever beams?

Tapered cantilever beams are commonly used in various structural and mechanical applications, such as in bridges, buildings, aircraft wings, and crane booms. They can also be found in smaller-scale applications, like in bicycle frames and diving boards. Their efficient load distribution and customizable design make them suitable for a wide range of uses.

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