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Spring constant for a beam

Kaspelek

New member
Apr 21, 2013
26
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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MarkFL

Administrator
Staff member
Feb 24, 2012
13,775
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:

\(\displaystyle \delta_C=\frac{FL^3}{48EI}\)


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

New member
Apr 21, 2013
26
Yeah that I can deduce, what I'm more unsure of is how to actually derive that deflection formula!

Any ideas?
 

MarkFL

Administrator
Staff member
Feb 24, 2012
13,775
Are you given any kind of IVP or ODE?
 

Kaspelek

New member
Apr 21, 2013
26
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

Administrator
Staff member
Feb 24, 2012
13,775
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
 

Klaas van Aarsen

MHB Seeker
Staff member
Mar 5, 2012
8,776
Hi guys,

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

Any help would be greatly appreciated. :cool:
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?