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

[Kaspelek]

Hi guys,

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

Any help would be greatly appreciated.

 

[MarkFL]

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.

 

[Kaspelek]

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

Any ideas?

 

[MarkFL]

Are you given any kind of IVP or ODE?

 

[Kaspelek]

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?

 

[MarkFL]

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

 

[I like Serena]

Hi guys,

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

Any help would be greatly appreciated.

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?