Buckling load equation for buckling struts

[Nexus305]

I was trying to solve a question where they told me to find the buckling load of a wing strut in a plane that experiences a compressive load along its axis. The strut can be considered to be a pin-jointed at both ends. Dimensions of the strut is as follows: outside diameter is 150 mm and bore is 100 mm and length of 5 m.

For finding the buckling load i used Euler's equation for struts P = (Pi^2*EI)/L^2 and I got the correct answer but i don't understand what exactly I have to do in the second part of the question. Here it is:
(b) Find the buckling load if the wing is re-designed so that the strut is prevented from moving laterally in all planes at its centre.

For this my lecturer had used this equation P = (4*Pi^2*EI)/L^2
So basically 4 times the previous equation. I want to know how he got that equation and when should I use it.

Thanks!

 

[AlephZero]

Think about what shape the strut would be when it buckled, if the mid point was restrained.

(Hint: it is similar to two struts joined end to end).

 

[Nexus305]

In that case isn't it supposed to be P=(2*Pi^2*EI)/L^2, instead of P=(4*Pi^2*EI)/L^2

Because if the beam is like 2 struts joined end to end, there will be 2 deflections. The top half and the bottom half would deflect right? How does the equation have a 4 in it?

 

[greentlc]

I think it is because the effective length of the column is cut in half. By doing this and squaring it, it is like multiplying the equation by 4. You still need to use the actual length of the column in that equation. Someone please correct me if I am wrong.

 

[Nexus305]

thanks a lot guys!..makes sense now

 

[Travis_King]

The original length of the strut is: L
When it is secured at the center, each segment becomes: L/2
(L/2)^2 = L/4
which gives you your 4 in the numerator.