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The Spring Constant for a Beam: How to Derive the Deflection Formula
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In summary: If you don't know how to solve differential equations, you should consult with a physics professor or online resources.In summary, the problem is that the deflection of a beam is not known, but can be found using beam bending theory. The equation to solve for the deflection is found by finding the moment at various points along the beam. If you don't know how to solve differential equations, you should consult with a physics professor or online resources.
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MarkFL
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From Wikipedia:
Using this, along with Hooke's Law and Newton's 3rd Law of Motion will give you the result you want.
Using this, along with Hooke's Law and Newton's 3rd Law of Motion will give you the result you want.
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Kaspelek
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Yeah that I can deduce, what I'm more unsure of is how to actually derive that deflection formula!
Any ideas?
Any ideas?
- #4
MarkFL
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Are you given any kind of IVP or ODE?
- #5
Kaspelek
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Nope, just what's given in the question. I'm assuming you need to perhaps draw a free body diagram and find the reaction forces to start off?
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MarkFL
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From what I gather, one obtains a 4th order homogeneous IVP, but to be honest, I don't know how it is derived. Perhaps one of our physics folks can get you started in the right direction. :D
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I like Serena
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Kaspelek said:Hi guys,
I've come across this problem and I'm not sure on where to start?
Any help would be greatly appreciated.
Hi Kaspelek! Welcome to MHB! :)Kaspelek said:
From beam bending theory, we have that:
$$M(x) = -EI \frac{d^2w}{dx^2}$$
where $M(x)$ is the bending moment at a distance x along the beam, $E$ is the elastic modulus, $I$ is the moment area of inertia of a cross section, and $w$ is the deflection of the beam.
Can you find the bending moment $M$ at an arbitrary distance $x$ along the beam?
If you don't know, you should indeed start with a free body diagram and the reaction forces in it.
If you can find the bending moments, can you solve the resulting differential equation?
FAQ: The Spring Constant for a Beam: How to Derive the Deflection Formula
What is the spring constant for a beam?
The spring constant for a beam, also known as the stiffness coefficient, is a measure of the relationship between the force applied to a beam and the resulting displacement. It is represented by the symbol k and is measured in units of force per unit length (N/m).
How is the spring constant for a beam calculated?
The spring constant for a beam can be calculated by dividing the applied force by the resulting displacement. It can also be calculated by dividing the modulus of elasticity of the material by the cross-sectional area and length of the beam.
What factors affect the spring constant for a beam?
The spring constant for a beam is affected by several factors, including the material of the beam, its cross-sectional area, length, and boundary conditions. Other factors such as temperature, load distribution, and external forces can also affect the spring constant.
What is the relationship between spring constant and stiffness of a beam?
The spring constant and stiffness of a beam are directly proportional. This means that as the spring constant increases, the stiffness of the beam also increases. This relationship is described by Hooke's Law which states that the displacement of a material is directly proportional to the applied force.
How is the spring constant for a beam used in engineering applications?
The spring constant for a beam is an important parameter that is used in various engineering applications. It is used to design and analyze structures such as bridges, buildings, and machines. It is also used to calculate deflections and stresses in beams, and to determine the natural frequency of a structure.