I have a very long beam which is subjected to a torque (I have the moments and shear forces with respect to X along the beam). I can calculate the critical load when the beam would bend if it was loaded axially, however I don't know how to calculate the critical torque on the beam before it "buckles" by twisting. How would I go about calculating this critical load/torque?

There is no buckling under torsion in a beam, there is lateral web buckling under vertical loads under certain circumstances.

A beam under pure torque will gradually twist due to St Venant and warping torsion until it yields with no sudden loss of strength as in buckling under axial load, however, under combined axial stress and torque it will buckle and needs to be analyzed.

this is the section torque moment under pure torsion.

$$T=G*I_t \phi^ \prime -EI\phi^ { \prime \prime \prime }$$

where

T is the torsional moment at the cross-section.

$$\phi^ \prime \ and\ \phi^ { \prime \prime \prime }$$ are first and third derivateves WRT X.

$$I_w$$ is the warping constant

$$I_t$$ is ST Venant's torsional constant.

• what is phi here? Also, is this applicable to a long thinn beam which would bend if it was twisted under load? – Beacon of Wierd Feb 7 at 19:58
• $phi$ is angle of rotation of the beam. First derivative is ratio of rotation/ unit length, third derivative is the shear on the flange due to warping. Long thin beam is the same. But if the high torque causes start of yield and plastic hardening, it may not be visible. So the beam will not behave as expected and may coil up or flex into erregular spiral depending on manufacturing defects. – kamran Feb 7 at 21:34
• Ok, thank you :) and * here is multiplication, right? Not convolution or something else? And I realize now that my question is poorly worded, basically I have a long thinn beam (Second area of moment in y axis is much greater than in the x axis) which is put under load and I am wondering how I calculate when it will fail due to the beam twisting under the load. This equation is applicable to that situation, right? :) – Beacon of Wierd Feb 8 at 23:07
• yes, multiplication. if you need more detail let me know to edit my answer and add some diagrams and equations. but put simply extruded thin plates like steel studs take torque by warping moment, whereas thick beams and closed sections take it by St Venant shear. and again yes this equation is applicable to your case. you need to pre-calc a couple of factors though. – kamran Feb 9 at 0:25