# Cantilever BEAM little problem. I have solved almost everything.

• wildleaf
In summary, the conversation discusses a problem involving a cantilever beam and its deflection and slope equations. The conversation also mentions the use of a bounding condition and the calculation of the modulus of elasticity using steel as a transformation material. The issue of which modulus of elasticity to use is also addressed, with the conclusion being to use E(steel).
wildleaf
Cantilever BEAM ! little problem. I have solved almost everything.

## Homework Statement

The link below has the problem:
http://i52.tinypic.com/2r3bcdc.jpg

## Homework Equations

M = E*I(d^2*w/dx^2)
slope = E*I (dw/dy) = integral of E*I(d^2*w/dx) + c1
deflection = E*I * (w) = double integral of E*I(d^2*w/dx) + c1x + c2

Bounding Condition: when x = 0 --> w = 0 and slope = 0
when x = L --> V = 0 and M = 0
Yb = ΣAY / ΣA

## The Attempt at a Solution

E*I(d^2*w/dx^2) = (-px^2/2) + (pLx) - (pL^2/2)
E*I (dw/dy) = (-px^3/6) +(pLx^2/2) - (pL^3/6) + c1 (c1 = 0 using when x = 0, slope=0)
w = (- p/(24*E*I)) * [x^4 - 4*L*x^3 + 6*L^2*x^2] + c2 (c2 = 0 using when x=0, w = 0)

I then solved for the I knowing that Yb = 9. I had to transform everything to steel, using n = 30 / 10 = 3. The new dimensions for steel become 4" by 18" (which is the same) and for Al = (4+4)*3 = 24" by 18". Then i calculated the I, and got I = 13608 in^4.

THE PROBLEM I HAVE IS THAT I DONT KNOW WHICH MODULUS OF ELASTIC TO USE FOR

w = (- p/(24*E*I)) * [x^4 - 4*L*x^3 + 6*L^2*x^2]

we know that p = 500, L = x = 20, I = 13609, E = ?

wildleaf: You can use E(steel). However, your n value currently looks wrong. Try again.

Ohhh... n = 1/3 ?? I always mess that up. If n = 1/3, then I = 3240.

Why do you use E(steel) and not E(Al) ??

wildleaf: You use E(steel) because you said you transformed everything to steel.

Your n value now looks correct.

Great job on solving the cantilever beam problem! It seems like you have found a solution for the deflection and slope equations, and have also correctly identified the bounding conditions. Regarding your question about which modulus of elasticity to use, it depends on the material you are using for the beam. The modulus of elasticity, or Young's modulus, is a material property that relates stress and strain and is different for different materials. For example, the modulus of elasticity for steel and aluminum are different. So, you will need to choose the appropriate modulus of elasticity for the material you are using in your problem. You can find the modulus of elasticity for different materials in material property tables or online. Good luck with your further analysis!

## What is a cantilever beam?

A cantilever beam is a type of structural element that is supported at only one end, with the other end projecting freely into space. It is commonly used in construction and engineering projects to support loads and distribute forces.

## How do you solve problems involving cantilever beams?

To solve problems involving cantilever beams, you will need to use principles from mechanics and structural analysis. This may involve calculating forces, moments, and deflections using equations and diagrams. It is important to have a good understanding of the properties and behavior of materials to accurately solve cantilever beam problems.

## What are some common challenges when dealing with cantilever beams?

Some common challenges when dealing with cantilever beams include ensuring structural stability, choosing appropriate materials and dimensions, and designing for specific load and environmental conditions. It is also important to consider potential sources of failure, such as bending, shear, and buckling.

## Can you give an example of a real-life application of a cantilever beam?

One example of a real-life application of a cantilever beam is a diving board. The board is supported at one end and the other end extends into the pool, allowing a person to jump off and create a momentary cantilever beam. The board needs to be designed to withstand the weight and force of a person jumping off, while also being flexible enough to provide a spring-like effect.

## How does the length of a cantilever beam affect its strength?

The length of a cantilever beam can have a significant impact on its strength. As the length increases, the beam experiences greater bending and deflection, which can lead to failure. The longer the beam, the greater the force needed to support it. This is why it is important to carefully consider the length when designing a cantilever beam.

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