MHB The Spring Constant for a Beam: How to Derive the Deflection Formula

Kaspelek
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Hi guys,

I've come across this problem and I'm not sure on where to start?

Any help would be greatly appreciated. :cool:
 

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From Wikipedia:
Center loaded beam

The elastic deflection (at the midpoint $C$) of a beam, loaded at its center, supported by two simple supports is given by:

$$\delta_C=\frac{FL^3}{48EI}$$ where:

$F$= Force acting on the center of the beam
$L$ = Length of the beam between the supports
$E$ = Modulus of elasticity
$I$ = Area moment of inertia

Using this, along with Hooke's Law and Newton's 3rd Law of Motion will give you the result you want.
 
Yeah that I can deduce, what I'm more unsure of is how to actually derive that deflection formula!

Any ideas?
 
Are you given any kind of IVP or ODE?
 
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?
 
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
 
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. :cool:

Kaspelek said:
Yeah that I can deduce, what I'm more unsure of is how to actually derive that deflection formula!

Any ideas?
Hi Kaspelek! Welcome to MHB! :)

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?
 
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