SandorR
Student
- Dec 17, 2020
- 16
Hello!
I am studying basic design of timber members to Eurocode 5.
Section 6.3.3 gives the basic equation for the critical bending stress of members, considering lateral-torsional buckling:
sigma_m,crit = pi * sqrt(E * I_z * G * I_tor) / (L_eff * W_y) (eqn 6.31)
where E is the elastic modulus, G the shear modulus, I_z the second moment of inertia, I_tor the torsional moment of inertia, L_eff the effective length and W_y the section modulus.
There is also another formula given for a softwood solid rectangular cross section:
sigma_m,crit = (0,78 * b ^ 2 * E) / (h * L_eff) (eqn 6.32)
where b is the width of the beam and h the heigth.
My question is: why are there two formulas? Why do we require a separate formula for a solid rectangular section? Curiously, the equation 6.32 does not include shear modulus.
I did a test calculation: taking a square cross section with h = b = 10mm with E = 7400 N/mm^2 and G = 460 N/mm^2 (the material values I found online for European wood) and L_eff = 200mm. With these values, formula 6.31 gives me around 19 N/mm^2 for the critical stress while formula 6.32 gives me around 290 N/mm^2 ! Huge difference, so the formulas are not equivalent at least. I calculated I_tor using formulas from this Wikipedia page:
So why are the formulas different? The code seems to suggest that 6.31 is more general, but shouldn't it then yield same results as the simpler formula? The lack of shear modulus G in 6.32 seems to suggest that for this failure mode, no rotation/torsion is taking place. It would seem intuitive to assume no lateral torsional buckling for a square cross section (as my calculation example) but the code talks about rectangular sections, not just square ones. So I suppose this formula should also be used even if we have a very thin and tall cross section this that would count as a rectangle as well?
Thank you!
I am studying basic design of timber members to Eurocode 5.
Section 6.3.3 gives the basic equation for the critical bending stress of members, considering lateral-torsional buckling:
sigma_m,crit = pi * sqrt(E * I_z * G * I_tor) / (L_eff * W_y) (eqn 6.31)
where E is the elastic modulus, G the shear modulus, I_z the second moment of inertia, I_tor the torsional moment of inertia, L_eff the effective length and W_y the section modulus.
There is also another formula given for a softwood solid rectangular cross section:
sigma_m,crit = (0,78 * b ^ 2 * E) / (h * L_eff) (eqn 6.32)
where b is the width of the beam and h the heigth.
My question is: why are there two formulas? Why do we require a separate formula for a solid rectangular section? Curiously, the equation 6.32 does not include shear modulus.
I did a test calculation: taking a square cross section with h = b = 10mm with E = 7400 N/mm^2 and G = 460 N/mm^2 (the material values I found online for European wood) and L_eff = 200mm. With these values, formula 6.31 gives me around 19 N/mm^2 for the critical stress while formula 6.32 gives me around 290 N/mm^2 ! Huge difference, so the formulas are not equivalent at least. I calculated I_tor using formulas from this Wikipedia page:
So why are the formulas different? The code seems to suggest that 6.31 is more general, but shouldn't it then yield same results as the simpler formula? The lack of shear modulus G in 6.32 seems to suggest that for this failure mode, no rotation/torsion is taking place. It would seem intuitive to assume no lateral torsional buckling for a square cross section (as my calculation example) but the code talks about rectangular sections, not just square ones. So I suppose this formula should also be used even if we have a very thin and tall cross section this that would count as a rectangle as well?
Thank you!
