In the last step, we met the second of our two stress tools. The member stress view applies beam theory formulas everywhere along a member, while a stress plane solves the stress field across the real cross-section by finite element analysis. In this step, we'll use both tools on torque (or torsion), the action where the shape of the cross-section matters more than anywhere else.
We'll load two cantilevers of almost exactly the same weight - an I-section and a box - with the same torque, and see why one of them twists so much more than the other.
Torsion: an open and a closed section. Orbit, read the results and hover the members; nothing you do here changes the model.
Members can copy this model into their own projects and work the step in it. Join or sign in.
1.0 How open and closed sections resist torque
Whether a section’s walls form a closed loop makes a big difference to its resistance to twisting. Both open and closed sections resist torque through St Venant torsion, but their shear-stress patterns are very different. The comparison below assumes thin walls, elastic behaviour and uniform torsion, with the section free to warp and its cross-sectional outline remaining undistorted.
A closed section, such as a box or tube carries torque through a shear flow around the loop. Shear flow is the shear force per unit length of wall, measured along the cross-sectional perimeter. In thin-wall theory,
q=
τ
t
.
Under pure torque, this flow is constant around the loop, even if the wall thickness varies. The circulating shear forces resist torque over the dimensions of the whole section. Bredt’s formula expresses this as T=2A0q, giving
τ=2A0tT,
where A0 is the area enclosed by the wall’s midline and t is the local wall thickness. The stress therefore depends on both the enclosed area and the wall thickness; for a given torque and enclosed area, thinner parts of the wall carry higher shear stress.
An open section, such as an I-section or channel, has no continuous loop enclosing an area. Within each thin plate, the torsional shear stresses act in opposite directions near the two faces. Their effective separation is governed by the plate thickness, which makes this stress pattern much less efficient at resisting torque.
The flanges and web contribute together, sharing the same rate of twist. For a plate of thickness ti, the peak surface shear stress is approximately
τmax,i≈JTti,
where J is the section’s torsion constant. For an assembly of thin rectangular plates,
J≈i∑3biti3,
with bi and ti denoting each plate’s width and thickness. Within this approximation, the thickest plate has the highest surface stress. Local stress concentrations at junctions are not included. The cubic dependence on thickness explains why thin open sections have such small torsion constants.
The difference also appears in the twist. For a uniform member of length L carrying a constant torque T under these assumptions, the relative rotation of its ends is
θ=GJTL,
where G is the material’s shear modulus. The product GJ is the torsional rigidity; the member’s rotational stiffness is T/θ=GJ/L. A smaller J therefore means more twist under the same torque.
For a thin-walled, single-cell closed section,
J≈∮tds4A02,
where the integral follows the wall’s midline. If the thickness is constant, this simplifies to J≈4A02t/p, where p is the midline perimeter. Therefore, enclosed area, perimeter and wall thickness all matter. For comparable overall dimensions and amounts of material, a closed section generally has a much larger J than an open section.
These formulas describe St Venant torsion alone. Warping is the longitudinal displacement of parts of the cross-section as the member twists. Restraining it introduces additional longitudinal and shear stresses and can substantially increase resistance to twisting, particularly in I-sections and channels. In that situation, the St Venant formulas alone do not describe the full response.
Our model has two 2 m cantilevers standing side by side. Each is fixed at its base and twisted by 1 kN m at its top. On the left is a UC 203 x 203 x 46, an open section; on the right is an SHS 200 x 200 x 8, a closed section. They're almost exactly the same size and weight: 46 kg/m against 47.7 kg/m. So, any difference in how they respond comes from the shape of the section, not from how much steel is in it.
A stress plane sits at mid-height on each member, P1 on the UC and P2 on the box. Both have already been computed, so you can open them straight away.
Things to try
Press Load the model, switch on Member stress and select the τ chip. In 3D, τ is the torsional and transverse shear stress combined, and the card lists σ, τ, σvM and U together whichever chip is on. Hover over the outside face of a flange on the UC, then anywhere on the box, and read the stress. (On the UC's thinner web, it's lower, because the stress in each plate of an open section is proportional to its thickness.)
Scroll down to the Displacements table and read rz, the rotation about the vertical axis, for the two top nodes, o-top and c-top. This is the twist at the top of each cantilever.
Switch to the σvM chip and compare the two members again. There's no axial force or bending here, so the von Mises stress on both members comes from shear alone.
Open the Stresses tab and click P1 (the UC), then P2 (the box). Select τ (T) and look at where the torsional shear stress concentrates in each section (hover over the section to read the values).
3.0 What you'll see
What you should see
UC 203 x 203 x 46 (open)
SHS 200 x 200 x 8 (closed)
Torsion constant J
22.2 cm⁴
5780 cm⁴
τ, member stress view
49.55 MPa
1.70 MPa
σvM, member stress view
85.82 MPa
2.94 MPa
Twist at the top (rz)
111.5 mrad
0.43 mrad
τ (T) peak, stress plane
74.0 MPa
2.8 MPa
The same torque, on two sections of the same weight, produces about 29 times the shear stress and, in this analysis, 260 times the twist in the open section. The table shows the twist the same way the Displacements table does, in milliradians. The UC's 111.5 mrad is 0.1115 rad, a little over 6° at its top, which you'd easily see on a real member. The box's 0.43 mrad is about a fortieth of a degree.
So where does the 260 come from? It's simply the ratio of the two torsion constants. The torque, the length and the material are the same for both members, so the twist depends on J alone. The box's walls form a loop enclosing almost 370 cm2, and the shear flow works on all of it. The UC's plates can only use their own thickness as a lever arm.
This is why, when a member has to carry torque - an edge beam supporting a slab on one side, say, or a beam curved on plan - the first question you should ask is whether it can be a closed section.
The stress planes tell us more. For the box, the result along each wall varies a little through the thickness, from about 1.6 MPa on the inside face to 1.8 MPa on the outside, averaging Bredt's 1.70 MPa. Bredt's formula assumes the stress is uniform through the wall, so a small variation like this is what we'd expect from a wall of real thickness. Around the corners, the stress dips to 1.2 MPa at the outside and rises to about 2.8 MPa at the tight inside corners.
For the UC, the flange faces carry the 49.5 MPa the formula gives, but the peak torsional shear stress is about 74 MPa, roughly 50% higher. Look at where it is: at the root fillets, where the web meets the flanges. The thin-plate formula treats the section as three separate plates. The real rolled section has extra material at the junctions, where the plates meet through a curved fillet, and the stress concentrates there.
This is exactly the kind of place where we'd expect the two tools to disagree. The member stress view uses the thin-plate formula and doesn't model the fillets, while the stress plane meshes the real shape, fillets included. So, when they differ like this, the stress plane's peak is the one to use in your check.
Hand check for both sections
The UC. In an open section, the largest stress is in the thickest plate. For the UC, that's the flange, with tf = 11.0 mm, and the section tables give J = 22.2 cm4. We'll also need the shear modulus to find the twist. With G=E/[2(1+ν)] = 80.8 GPa, the stress and the twist at the top are,
τ=JT
The stress is worked in N and mm, and the twist in N and m. Both agree with the member stress view and the Displacements table.
The SHS. For the box, we use Bredt's formula. The middle line of the wall sits half a wall thickness in from each face, so it's a square of side 200 - 8 mm. It encloses A0=(200−8)2 = 36,864 mm, so,
τ=2A0tT
which is the member stress view's value for the box.
The von Mises stress. In both cases, there's no normal stress at all - no axial force and no bending - so the von Mises stress is simply 3τ. For the UC, 3 MPa. For the box, MPa. Both match the row in the table.
4.0 Warping, which neither tool models
What neither tool shows: warping
When an open section twists, its flanges don't just rotate. They also bend in their own planes, in opposite directions, so the cross-section no longer stays flat. This is called warping. Our analysis treats every member as twisting freely. If the base fully prevents the flanges from warping, three things change that the analysis doesn't model.
First, the restraint makes the member stiffer. For this UC, the warping torsion theory (Vlasov's) gives a twist at the top of about 0.046 rad, around 40% of the 0.1115 rad above. So, the open section would twist about 100 times as much as the box, not 260.
Second, the flanges carry part of the torque by bending sideways, so less is left for St Venant torsion. At mid-height, where P1 sits, only about half the torque is carried by St Venant torsion, and the torsional shear stress there would be about half what both tools show.
Third, that sideways bending puts normal stress into the flanges where none appears in our results: about 31 MPa at the flange tips at mid-height and 80 MPa at the base, under just 1 kN m. The member stress view shows no normal stress on this member at all.
A real base plate gives only part of that restraint, so the real behaviour lies somewhere between the two pictures: the free warping our analysis assumes, and the full restraint described above. Just how much restraint does a real base plate provide? That depends heavily on its detailing, and it isn't something the analysis can tell you.
Closed sections barely warp, so none of this matters for the box. For an open section carrying real torque, warping has to be checked separately. This is one more reason to avoid carrying torque in open sections wherever you can.
5.0 What to take from this step
Closed sections carry torque as a shear flow round the whole section, working on the full area their walls enclose. As a result, they're many times stiffer and stronger in torsion than open sections of the same weight. Open sections twist their thin plates individually, concentrate stress at the root fillets (which only the stress plane shows), and warp, and neither tool captures the warping stresses.
So, if a member has to carry torque, you should make it a closed section wherever you can. If it has to be open, remember that the member stress view is only part of the picture, and check warping separately.
Next, we'll look at two members whose stresses are comfortably low and which would still fail.