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Hugh_Struct
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Homework Statement
Hi,
I am trying to complete my MSc in Structural Engineering and being an engineer my maths sometimes let's me down. Please help me to solve this 3rd order differential equation...it relates to solving the later torsional buckling of a beam. Here goes!
So far i am at this point
(3) GIt dr/dx - EIw d3r/dx3 = M2 / (PI2EIz / L2) dr/dx
I assume there is a homogeneous solution to this where i could use the following boundary conditions
(v)o = (v)L = 0
(r)o = (r)L = 0
(d2r/dx2)0 = (d2r/dx2) = 0
Homework Equations
The problem begins with these two equations...
(1) EIz d2v/dv2 = -M r(X)
and
(2) GIt dr/dx - EIw d3r/dx3 = M dv/dx
Trial solutions of
(4) v(x) = w sin (PI x /L) and r(x) = r sin (PI x /L)
so differentiating (4) you can obtain
(5) v(x) = M / (PI2EIz / L2) r(x)
The Attempt at a Solution
Apparently substituting equation (5) into equation (2) and using the boundary conditions you can resolve the problem to find
3...below is the solution
M = Square root [ (PI2EIz / L2) * (GIt + PI2EIw / L2)]
I say apparently as i can only manage the substation part! I know you could also differentiate equation (2) to a 4th order differential and then substitute d^2v/dv^2 of equation (1) but i really need to know the other method.
I am SO sorry about the awful text...first time poster...long time reader!
Please help!
Thanks
Hugh
Much better to check this link out...page 2 and half of page 3 is what I'm after!
https://noppa.tkk.fi/noppa/kurssi/rak-54.3600/luennot/Rak-54_3600_lateral-torsional_buckling.pdf"
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