Torsion Analysis — Steel Member Torsion

Steel member torsion analysis: St Venant torsion constant J, warping constant Cw, and combined torsion-shear stress distribution for open and closed sections. Educational use only.

This page documents the scope, inputs, outputs, and computational approach of the Torsion Analysis tool on steelcalculator.app. The interactive calculator runs in your browser; this documentation ensures the page is useful even without JavaScript.

What this tool is for

• Computing St Venant (pure) torsion and warping torsion stresses in steel members.
• Determining torsion constants J and Cw for common section types.
• Understanding the combined stress state when torsion is superimposed on bending and shear.

What this tool is not for

• It does not design the connections that must resist the torsional reaction.
• It does not handle complex loading paths, restrained warping boundary conditions, or bimoment effects in full generality.
• It does not check lateral-torsional buckling (which involves torsion but is a stability check, not a stress check).

Key concepts this page covers

• St Venant (uniform) torsion and shear stress tau = T t / J
• warping torsion and normal warping stresses
• torsion constant J for open and closed sections
• warping constant Cw

Inputs and outputs

Typical inputs: section type (W-shape, channel, HSS, angle), section dimensions, member length, end restraint conditions (free to warp, warping fixed), and applied torsional moment.

Typical outputs: torsion constant J, warping constant Cw, St Venant shear stress, warping normal stress, warping shear stress, and the combined stress check.

Computation approach

For open sections (W-shapes, channels, angles), J and Cw are taken from a built-in section database. The St. Venant shear stress is computed as tau = T*t / J, and the warping shear stress is estimated using simplified AISC Design Guide 9 approximations that account for support conditions (fixed, free, simply-supported). For closed sections (HSS, box), pure St. Venant torsion dominates and warping is negligible. The angle of twist is computed as phi = T*L / (G*J) for the pure torsion case. A utilization ratio is computed against 0.6*Fy per AISC 360.

Torsion Constants for Common Sections

Torsion constant J (St. Venant)

Rectangular plate: J = b × t³ / 3

W-shapes (approximate): J ≈ Σ (b_i × t_i³ / 3) for each plate element
  (AISC manual lists exact values)

HSS rectangular: J = 2 × t × (B-t)² × (H-t)² / (B+H-2t)

HSS round: J = pi × (D⁴ - (D-2t)⁴) / 32

J values for common steel sections

The 100-200x difference in J between open and closed sections is why HSS is preferred when torsion is significant.

Warping Torsion — Key Concepts

For open sections under torsion, two mechanisms resist twist:

1. St. Venant (pure) torsion: Uniform shear stress around the cross-section perimeter. Dominates for long members with free warping ends.

tau_sv = T × t / J
phi_sv = T × L / (G × J)

2. Warping torsion: The flanges bend laterally in opposite directions, creating both normal and shear stresses. Important for short members, concentrated torques, and warping-fixed ends.

Warping normal stress: sigma_w = B × omega_n / Cw
  Where B = bimoment, omega_n = normalized warping function

Warping shear stress: tau_w = T_w × S_w / (Cw × t)
  Where T_w = warping torsion, S_w = warping statical moment

Bimoment at midspan (torque T at midspan, both ends warping-fixed):
  B_max = T × L / 8 × tanh(aL/2) / (aL/2)
  a = sqrt(G × J / (E × Cw))

When warping matters vs. when St. Venant dominates

The parameter aL determines the regime: aL = L × sqrt(G×J/(E×Cw)). For W-shapes, a is typically 0.001-0.005 per inch, so aL > 5 for spans over 10 ft.

Worked Example — Torsion in a W-Shape Beam

Problem: A W16x36 beam (A992) spans 15 ft with fixed ends (warping-fixed). A concentrated torque of 20 kip-ft is applied at midspan from an eccentric facade load. Check torsional stress.

Step 1 — Section properties

W16x36: J = 0.54 in⁴, Cw = 3,540 in⁶, Sx = 56.5 in³
bf = 6.99 in, tf = 0.430 in, tw = 0.295 in
E = 29,000 ksi, G = 11,200 ksi

Step 2 — Torsional parameter

a = sqrt(G × J / (E × Cw)) = sqrt(11,200 × 0.54 / (29,000 × 3,540))
a = sqrt(6,048 / 102,660,000) = sqrt(5.89 × 10⁻⁵) = 0.00768 in⁻¹
aL = 0.00768 × 180 = 1.38

Since aL = 1.38 < 5: warping effects are significant.

Step 3 — St. Venant shear stress

tau_sv = T × t_max / J = 20 × 12 × 0.430 / 0.54 = 191 psi

This is very low — St. Venant shear stress is negligible for open sections.

Step 4 — Warping normal stress (maximum at flange tips)

B_max at midspan: B = T×L/8 × tanh(aL/2)/(aL/2)
  = 20×12×180/8 × tanh(0.69)/(0.69)
  = 5,400 × 0.60 / 0.69 = 4,700 kip-in²

sigma_w = B × omega_max / Cw
omega_max ≈ bf × (d-tf) / 4 = 6.99 × (15.86-0.43) / 4 = 27.0 in²

sigma_w = 4,700 × 27.0 / 3,540 = 35.8 ksi

Combined stress (bending + warping):
fb = M / Sx (say M = 100 kip-ft for gravity loads)
fb = 100 × 12 / 56.5 = 21.2 ksi

fb + sigma_w = 21.2 + 35.8 = 57.0 ksi > Fy = 50 ksi → FAILS

The warping stress alone is 35.8 ksi, exceeding 0.6×Fy = 30 ksi.

Step 5 — Remedy

Options:
1. Use HSS section (no warping): HSS10x6x3/8, J = 71.4 in⁴
   tau = 20×12×0.375/71.4 = 1.26 ksi < 30 ksi ✓ (by huge margin)

2. Add torsional bracing at midspan to reduce unbraced length
3. Eliminate the eccentricity by moving the load to the shear center
4. Use a heavier W-section with larger J and Cw

Option 1 (HSS) is the most effective and common solution.

Extended Torsional Properties Table

The following table provides torsional constants J, warping constants Cw, and shear center locations for a wide range of steel sections. For open sections, the warping constant Cw is critical for evaluating warping normal stresses. For closed sections, Cw is zero because warping is restrained by the closed cross-section.

Key observations from the table:

• Channels have their shear center located outside the channel web, meaning any load applied at the centroid or at the face of the channel (such as a slab bearing on the top flange) will induce torsion. This is why channels used as spandrel or edge beams require special torsion analysis.

• Angles have the lowest torsional stiffness of any common section. An L4x4x1/4 has J = 0.05 in^4, making it extremely vulnerable to torsion from even small eccentric loads.

• HSS sections have J values 50-200 times larger than comparable W-shapes and zero warping, making them the optimal choice when torsion is a significant design consideration.

Torsion in Spandrel Beams

Spandrel beams (edge beams at the perimeter of a floor slab) are among the most common members subject to significant torsion. The torsion arises because the floor slab loads the beam eccentrically -- the slab bears on the inward flange, and the resulting force acts at an offset from the beam shear center.

Source of torsion in spandrel beams:

For a W-shape spandrel beam supporting a slab on the inward flange, the slab reaction acts at a distance e from the beam centroid (the shear center for doubly-symmetric W-shapes). The torsional moment per unit length is t = w_slab x e, where w_slab is the slab reaction per foot and e is the eccentricity from the beam web to the slab load point.

For a 6 in. slab bearing on a W16x36 spandrel:
  Slab reaction (tributary): w = 2.5 kip/ft (approximate)
  Load eccentricity from web: e = bf/2 + slab_bearing/2 = 6.99/2 + 3.0/2 = 5.0 in
  Torsion per unit length: t = 2.5 × 5.0 = 12.5 kip-in/ft

Over a 20 ft span: T_total = 12.5 × 20 = 250 kip-in = 20.8 kip-ft

This level of torsion produces significant warping stresses in a W-shape and typically requires either a closed section (HSS or built-up box) or torsional bracing from the slab or deck.

Preventing Torsion in Floor Framing

Several practical strategies can reduce or eliminate torsion in floor framing:

1. Detail the slab-to-beam connection to resist torsion: Metal deck and concrete slabs can provide significant rotational restraint to the beam top flange if properly connected. Welded studs through the metal deck create a composite section that resists twist. AISC Design Guide 9 provides guidance on the rotational stiffness provided by slab connections.

2. Use torsional bracing: Cross-frames, diaphragms, or lateral braces between adjacent beams prevent rotation. For spandrel beams, a diagonal brace from the bottom flange to the slab or deck at regular intervals (8-10 ft) effectively prevents twist.

3. Select sections with adequate torsional properties: HSS sections (square or rectangular) eliminate warping stresses entirely. For open sections, choosing a heavier W-shape increases J and Cw, but the improvement is modest compared to switching to a closed section.

4. Reduce the load eccentricity: Bearing the slab directly on the beam web (through a bearing angle or shelf angle) eliminates torsion. Alternatively, designing the slab edge to bear symmetrically about the web reduces the eccentricity.

5. Design for torsion explicitly: When torsion cannot be eliminated, calculate the warping normal and shear stresses and include them in the combined stress check. Use the torsion analysis calculator to determine whether the section is adequate.

Combined Torsion and Bending Interaction per AISC

AISC 360 does not provide a single closed-form interaction equation for combined torsion and bending. Instead, the approach is to compute all stress components at the critical point and check them against the appropriate limit:

Normal stress check:

f_total = f_bending + f_warping_normal

For the flange tip (governed by warping + bending):
f_total = M/Sx + B × omega_max / Cw

Check: f_total <= phi × Fy = 0.90 × 50 = 45 ksi (for A992, LRFD)

Shear stress check:

tau_total = tau_st_venant + tau_warping_shear

tau_st_venant = T × t_max / J
tau_warping = T_w × S_w / (Cw × t)

Check: tau_total <= phi × 0.6 × Fy = 0.90 × 30 = 27 ksi (LRFD)

Combined normal and shear (von Mises equivalent):

f_eq = sqrt(f_total² + 3 × tau_total²)

Check: f_eq <= phi × Fy (LRFD)

This approach is described in AISC Design Guide 9 (Torsional Analysis of Structural Steel Members). The key insight is that for open sections, the warping normal stress at the flange tip often governs the design -- not the St. Venant shear stress. For closed sections (HSS), there is no warping, so only the St. Venant shear stress and the bending stress need to be combined.

Frequently Asked Questions

Why do open sections have such low torsional stiffness? The torsional stiffness of a section depends on J, which for open sections is approximately the sum of bt^3/3 for each plate element. Since t (plate thickness) is cubed, thin plates contribute very little. A W14x30 has J approximately 0.4 in^4, while a comparable HSS10x6x3/8 has J approximately 80 in^4 -- a 200x difference. This is why open sections like W-shapes should be loaded through the shear center or braced to prevent torsion.

What is warping torsion and when is it important? Warping torsion occurs in open sections when the flanges resist torsion through differential bending (one flange deflects laterally one way, the other the opposite way). This creates normal stresses in the flanges (warping normal stresses) in addition to shear stresses. Warping is important for concentrated torsional loads, short members, or members with restrained warping at the ends. For long members with free warping ends, St Venant torsion dominates and warping effects are small.

How do I avoid torsion in design? The most effective strategy is to load beams through or near the shear center, which eliminates the torsional moment. For W-shapes, the shear center coincides with the centroid. For channels and angles, the shear center is offset from the centroid, so loads applied at the centroid will induce torsion. Practical measures include using bracing to prevent rotation, loading through the web plane, or selecting closed sections (HSS) when torsion cannot be avoided.

Why do spandrel beams experience torsion? Spandrel beams (edge beams at the building perimeter) support the floor slab on only one side. The slab reaction acts eccentrically relative to the beam shear center, producing a torsional moment per unit length equal to the slab reaction times the eccentricity. For example, a slab bearing on the inward flange of a W16 spandrel applies the load at 4-6 inches from the web, producing a distributed torque of 10-15 kip-in/ft. Over a 20 ft span, this creates a significant torsional demand that often exceeds the capacity of an open section.

How do I prevent torsion in floor framing? The most effective strategies are: (1) use composite action by welding studs through the metal deck, which creates rotational restraint at the top flange; (2) provide torsional bracing from the beam bottom flange to the slab or adjacent framing at regular intervals; (3) select HSS or closed sections for spandrel beams where torsion is unavoidable; (4) reduce the load eccentricity by using a bearing angle that places the slab load closer to the beam web; and (5) use a wider beam flange to reduce the eccentricity of the slab bearing. For most buildings, a combination of composite action and selective bracing is sufficient.

How does AISC handle combined torsion and bending? AISC does not provide a single interaction equation for combined torsion and bending. Instead, per AISC Design Guide 9, compute all stress components at the critical point: bending normal stress (M/Sx), warping normal stress (B x omega / Cw), St. Venant shear stress (T x t / J), and warping shear stress. Check the total normal stress against phi x Fy and the total shear stress against phi x 0.6Fy. A von Mises equivalent stress check combining normal and shear stresses may also be used. For open sections, the warping normal stress at the flange tip typically governs. For closed sections, only St. Venant shear and bending stress need to be combined.

What sections are best for resisting torsion? Closed sections (HSS rectangular, HSS round, and built-up box sections) are far superior for torsion resistance. An HSS10x6x3/8 has J = 71.4 in^4 compared to J = 0.54 in^4 for a W16x36 -- over 130 times greater. Closed sections also have zero warping constant, meaning there are no warping normal stresses. Among open sections, heavier W-shapes have somewhat better torsional properties, but the improvement is modest because J scales with t^3. Channels and angles have the worst torsional properties due to low J, low Cw, and offset shear centers that make torsion unavoidable under typical loading.

How do I calculate the torsion constant J for a built-up section? For built-up open sections (plate girders, crane runway beams with cap channels), J is approximated by summing the bt^3/3 contribution of each rectangular element. For a W24x55 with a C12x20.7 cap channel welded to the top flange, J_total = J_W-shape + J_channel + additional contribution from the weld connecting them. The AISI and AISC manuals provide J values for standard rolled shapes. For custom built-up sections, finite element analysis (using CUFSM, MASTAN, or similar software) provides the most accurate torsional properties including the warping constant.

What is the shear center and why does it matter for torsion? The shear center (also called the flexural center) is the point on the cross-section through which a transverse load must pass to produce bending without torsion. For doubly-symmetric sections (W-shapes, HSS), the shear center coincides with the centroid. For channels, the shear center is located outside the web, meaning any load applied at the centroid will induce torsion. For angles, the shear center is at the intersection of the legs (the heel). Loading a member through a point other than the shear center always produces a torsional moment equal to the applied force times the eccentricity from the shear center.

How does torsion interact with lateral-torsional buckling? Lateral-torsional buckling (LTB) is a stability limit state where a beam under bending simultaneously deflects laterally and twists. Torsional loading reduces the LTB capacity because it pre-loads the member in twist, bringing it closer to the buckling threshold. AISC Chapter F LTB checks do not explicitly account for applied torsion; if significant torsion is present, the designer should either eliminate the torsion (through bracing or section selection) or use a reduced effective LTB capacity based on the combined torsion-bending interaction from AISC Design Guide 9.

What is AISC Design Guide 9 and when do I need it? AISC Design Guide 9 (Torsional Analysis of Structural Steel Members) provides the authoritative methodology for computing torsional stresses and deformations in steel members. It covers St. Venant torsion, warping torsion, bimoment calculations, and combined stress checks for W-shapes, channels, angles, and HSS under various support and loading conditions. It is needed whenever an open-section steel member carries significant torsional loads (spandrel beams, crane runway beams, members with eccentric connections, or any member where the applied load does not pass through the shear center).

How does torsion affect connection design? Torsional moments at member ends must be resisted by the connections, which transfer the torque as a force couple to the supporting members. For a W-shape spandrel beam with 20 kip-ft of torsion, the end connections must resist both the shear force and the torsional reaction. Simple shear tabs and single-angle connections provide negligible torsional restraint; end-plate connections, stiffened seated connections, or full-depth web connections with top and bottom flange clips are required to transfer torsion. The connection designer must ensure that both the torsional moment and the direct shear can be transferred without exceeding the capacity of the bolts, welds, or connecting elements.

What is the difference between uniform torsion and non-uniform torsion? Uniform (St. Venant) torsion occurs when the rate of twist is constant along the member length, as in a circular shaft or an open section with free warping at both ends under uniform torque. Non-uniform (warping) torsion occurs when the warping deformation varies along the length, typically due to concentrated torques, warping-fixed supports, or varying cross-sections. In steel design, most practical torsion problems involve both mechanisms: the St. Venant shear stress resists part of the torque while warping bending of the flanges resists the remainder. The relative contribution depends on the torsional parameter aL = L x sqrt(GJ/ECw).

Related pages

• Beam capacity calculator
• Column capacity calculator
• Moment of inertia calculator
• Section properties database
• HSS connections calculator
• Tools directory
• How to verify calculator results
• Disclaimer (educational use only)
• steel shear modulus and material properties
• Mohr circle for combined stress analysis

Disclaimer (educational use only)

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