Cantilever related questions and clarifications

[NeoPhysics]

Hi,

I am trying to understand the force and stress calculation in a cantilever beam, with point load at free end and fixed end at other.

So, Bending Force F=3EI.d/L3, E-Elastic Modulus, I Second moment of Inertia along XX axis, d deflection, L distance between point of consideration and point of force application. ...(1)

Stress = F.L.y/I, y being distance from neutral axis. .......(2)

From 1 and 2, stress = 3.E.y.d/L2............(3)


So here is my confusion. Pls help.
1. Let us say I am suspending a weight of 1kg, which is approx 10N force. From 1, L at the point of force application is zero. Does that mean F is inifinite where L=0?

2. From 2, stress is inversely proportional to Moment of Inertia. I beam has less MI than a retangular beam of same hieght and weight. So I select a 3x3 retangular beam and carve out a I beam, I reducing the MI therefore increasing the stress, Does not sound right...Pls help.

3. Materials have different tensile and compressive strengths. Let's say I have a material with compressive strength half the tensile strength. When I bend, the top portion is subject to tension and the lowest is subject to compression. From a design view, should I consider the compressive strength to decide the cross section area?

Thoroughly confused. Pls enlighten.
Neophysicist

 

[SteamKing]

1. The equation for 'bending force' is a rearranged formula for the deflection in a cantilever beam resulting when a force F is applied at the free end. Thus,
deflection = F*L^3 / (3 E I). L is the length of the beam, not the location of the force or the position where the deflection is calculated.

For the deflection at a location 'x' when F is applied at the free end, the formula is
deflection at 'x' = F * L^2 * (3*L - x) / (6 E I), and x is measured from the fixed end.
You can see that when x = L, the first formula above is obtained.

2. If you take a 3x3 beam and carve out an I-beam section from that, the inertia of the I-beam section will be less than the original rectangular section. The deflection of the beam is dependent on the actual cross section of the beam, not what it might have been at some other time in the past.

3. Most materials used for construction of beams have similar strength properties in tension and compression. If you were trying to construct a cantilever beam from concrete which had not been reinforced, then it would be proper to consider the effect of compression and tension separately when designing the beam.

 

[NeoPhysics]

Mr Steamking,


much. You clarified my doubts like sun shine on mist!. My heartfelt thanks for your time and kind responses.

For #2, you are essentially saying "Hey if you use a I beam, the deflection is function of the current beam's MI". I understand. But then for a given 3X3 perimeter an I beam fitting into it will have less MI and there fore more stress. If then, why do we say I beam is better?

As to 3, I am trying to build a model with Balsa. Balsa has 4400 PSI in compression and 8700 PSI in tension. So when I subject a beam made of Balas, the top part is in tension and lower part is in compression. For design purpose, I assume I should go with compression strength. Am I right?

Thanks,

 

[SteamKing]

There are different factors which come into play when selecting the cross section for a given beam. Often, it is using a cross section which has a certain moment of inertia but which also has the least weight per unit length. Other times, it could be which cross section provides access for inspection and maintenance.

The I-beam section locates material at the upper and lower surfaces where bending stresses are a maximum. It is also an open section, which allows for ease of inspection and maintenance of the structure. This is not to say that all beams must be constructed using open sections.

Since your project is using a material which has different strength properties in tension and compression, yes, I would use the compressive stress as a guide for determining the best cross section to use. I don't know what your design will encompass, but if the compressive stress is the limiting factor, the you should also check for buckling if loads will be applied axially along the length of the member, in addition to any bending loads.

 

[rezeew]

Hello Neophysicist,

I am happy to assist you with your questions about cantilever beams and their force and stress calculations. Here are some clarifications and explanations to help you better understand the concepts:

1. In equation (1), the distance L refers to the distance between the point of consideration and the point of force application. It is not the length of the beam itself. So, when L=0, it means that the point of consideration is at the point of force application and the beam is experiencing a point load. This does not mean that the force F is infinite, but rather that it is concentrated at a single point on the beam.

2. In equation (2), the moment of inertia (I) is a measure of a beam's resistance to bending. It is not related to the beam's height or weight. So, if you were to carve out an I beam from a rectangular beam, you would not necessarily be reducing the moment of inertia. It depends on the specific dimensions and shape of the I beam and how it compares to the original rectangular beam in terms of strength and stiffness. In general, I beams are designed to have a higher moment of inertia compared to rectangular beams, which allows them to withstand higher bending forces without experiencing excessive stress. So, your approach of using a 3x3 rectangular beam may not necessarily result in a stronger and stiffer beam compared to an I beam with the same height and weight.

3. When designing for bending, it is important to consider both the tensile and compressive strengths of the material. The cross section area should be chosen based on the material's lower strength, as it is the weakest point of the beam. In your example, if the material has a compressive strength that is half its tensile strength, then you would need to design for the lower compressive strength to ensure the beam can withstand both tension and compression forces. This may result in a larger cross section area compared to using the tensile strength alone.

I hope this helps clarify some of your confusion. Let me know if you have any further questions or need more clarification. I am always happy to help others understand complex concepts. Keep learning and exploring!