What is the difference between Saint-Venant and warping torsion?

Structural steel members resist torsion through two distinct mechanisms: (1) Saint-Venant (pure) torsion — shear stresses circulate around the cross-section. The resistance depends on the torsional constant J. For a rectangular element of width b and thickness t: shear stress = T x t / J, where J = b x t^3 / 3 for the element. The shear stress is maximum at the surface and zero at the center. Saint-Venant torsion resists through cross-sectional twist — no axial stresses. (2) Warping torsion — occurs in OPEN sections where flanges can bend independently. Torsion causes the flanges to bend in opposite directions, creating a bi-moment (self-equilibrating pair of flange bending moments). Warping resistance depends on the warping constant Cw. Cw ≈ Iy x h^2 / 4 for doubly-symmetric I-shapes. Warping produces NORMAL stresses (flange bending) in addition to shear. For W-shapes, the total torsion resistance is approximately 50% Saint-Venant + 50% warping — BOTH must be checked. For HSS/closed sections, warping is negligible (<5%) because the closed cell prevents differential flange movement — Saint-Venant dominates entirely.

How do you calculate J and Cw for steel sections?

For W-shapes: J ≈ (2 x bf x tf^3 + (d - 2 x tf) x tw^3) / 3. This sums the St. Venant torsion constant for each rectangular plate element. Example: W18x40 (bf=6.02, tf=0.525, d=17.9, tw=0.315): J = (2 x 6.02 x 0.525^3 + (17.9 - 1.05) x 0.315^3) / 3 = (1.74 + 0.168) / 3 = 0.636 in^4 (actual tabulated: 0.861 — the formula is approximate because it neglects the fillet). Cw ≈ Iy x h^2 / 4 where h = d - tf (distance between flange centroids). For W18x40: Iy = 12.8 in^4, h = 17.9 - 0.525 = 17.375 in, Cw ≈ 12.8 x 17.375^2 / 4 = 965 in^6 (actual: 414 — the approximation overestimates because it assumes flanges carry all the warping, neglecting web contribution). USE TABULATED VALUES from the AISC Manual for design. The formulas are for understanding the physical behavior, not for design calculations. For HSS: J = 4 x Am^2 / sum(si/ti) (Bredt-Batho). For a 8x8x1/4 HSS, J ≈ 140 in^4 — about 500x larger than an equivalent weight W-shape.

How much torsion can a W-shape handle?

W-shapes are POOR at resisting torsion. A W18x40 has J = 0.861 in^4. An HSS 6x6x1/4 of similar weight has J ≈ 60 in^4 — 70x larger. The practical consequence: even small eccentricities can overstress a W-shape. Worked example: W18x40 beam, 30 ft span, uniform load w=2 kip/ft, supported on one flange (eccentricity e = 3 in from shear center). Torque t = w x e = 2 x 2.5/12 = 0.417 kip-ft per foot of beam. Maximum torque at end: T = t x L/2 = 0.417 x 15 = 6.25 kip-ft = 75 kip-in. Saint-Venant shear stress from torsion: fv = T x tf / J = 75 x 0.525 / 0.861 = 45.7 ksi. This is ABOVE the allowable shear stress of 0.40Fy = 20 ksi without considering any flexural shear. The W18x40 would FAIL in torsion despite being adequate for bending. Solution: (1) Move the support to bear on the web (eliminate eccentricity). Or (2) add a cap channel to the top flange to form a semi-closed section, increasing J dramatically. Or (3) use an HSS section. Rule of thumb: if torque exceeds 5% of the design moment, a W-shape may not be adequate — consider an HSS or box section. This is why crane runway beams and spandrel beams supporting eccentric brick veneer are always checked for torsion.

How is combined bending and torsion checked per AISC 360?

AISC 360-22 Section H3.3 provides the combined stress check: (fa/Fa) + (fb/Fb) + (fv/Fv) <= 1.0 for ASD, or the interaction equation for LRFD. For combined flexure + torsion: (1) Calculate normal stresses: fa from axial (if any), fb from flexure (M/S) + warping normal stress (sigma_w = B x omega / Cw, where B is the bi-moment). The warping normal stress can be significant near fixed ends where warping is restrained. (2) Calculate shear stresses: fv = flexural shear (VQ/It) + Saint-Venant torsion shear (T x t / J). These ADD in one web and subtract in the other. (3) For open sections: the bi-moment B must be calculated along the beam — it is maximum at fixed ends (warping restrained) and zero at free ends (warping free). The warping normal stress at a fixed end: sigma_w = B_max x omega_n / Cw, where omega_n is the normalized warping function. For a W-shape, omega_n at the flange tip ≈ bf x h / 4. (4) Combined stress check at the critical point (typically flange tip at the fixed end, where both flexural normal stress and warping normal stress are maximum). Most engineers use AISC Design Guide 9 for torsion — the calculations are too involved for hand methods beyond the simplest cases.

When should I use a box section vs a W-shape for torsion?

The decision rule: if the design torque T* exceeds 5-10% of the design moment M*, a W-shape is likely inadequate and a box section should be considered. Quantitatively: (1) Calculate the required J_req = T* x t_max / (phi x 0.6Fy), where t_max is the maximum element thickness (typically flange). (2) Compare to available J for W-shapes at the required Sx. (3) If no W-shape has adequate J, switch to HSS or box. For comparison: A W18x40 (Sx=68.4, J=0.861) supports M* ≈ 250 kip-ft but only T* ≈ 3 kip-ft. A box section 18x12x5/16 (similar Sx) has J ≈ 200+ in^4, supporting T* ≈ 700 kip-ft. Box sections are 100-500x stiffer in torsion PER UNIT FLEXURAL CAPACITY. Practical guidance: crane runway beams (eccentric wheel load, T ≈ 0.1-0.2x M) — always use box or cap channel on W-shape. Spandrel beams with eccentric cladding (T ≈ 0.05-0.1x M) — check torsion, may need HSS. Offset columns on transfer beams (T ≈ 0.2-0.5x M) — always use box girder. Curved beams (torsion from curvature) — box or HSS mandatory.

Steel Torsion Design — Saint-Venant, Warping, and AISC Design Guide 9

Q: How do you calculate J and Cw for steel sections?

For W-shapes: J ≈ (2 x bf x tf^3 + (d - 2 x tf) x tw^3) / 3. This sums the St. Venant torsion constant for each rectangular plate element. Example: W18x40 (bf=6.02, tf=0.525, d=17.9, tw=0.315): J = (2 x 6.02 x 0.525^3 + (17.9 - 1.05) x 0.315^3) / 3 = (1.74 + 0.168) / 3 = 0.636 in^4 (actual tabulated: 0.861 — the formula is approximate because it neglects the fillet). Cw ≈ Iy x h^2 / 4 where h = d - tf (distance between flange centroids). For W18x40: Iy = 12.8 in^4, h = 17.9 - 0.525 = 17.375 in, Cw ≈ 12.8 x 17.375^2 / 4 = 965 in^6 (actual: 414 — the approximation overestimates because it assumes flanges carry all the warping, neglecting web contribution). USE TABULATED VALUES from the AISC Manual for design. The formulas are for understanding the physical behavior, not for design calculations. For HSS: J = 4 x Am^2 / sum(si/ti) (Bredt-Batho). For a 8x8x1/4 HSS, J ≈ 140 in^4 — about 500x larger than an equivalent weight W-shape.

Q: How much torsion can a W-shape handle?

W-shapes are POOR at resisting torsion. A W18x40 has J = 0.861 in^4. An HSS 6x6x1/4 of similar weight has J ≈ 60 in^4 — 70x larger. The practical consequence: even small eccentricities can overstress a W-shape. Worked example: W18x40 beam, 30 ft span, uniform load w=2 kip/ft, supported on one flange (eccentricity e = 3 in from shear center). Torque t = w x e = 2 x 2.5/12 = 0.417 kip-ft per foot of beam. Maximum torque at end: T = t x L/2 = 0.417 x 15 = 6.25 kip-ft = 75 kip-in. Saint-Venant shear stress from torsion: fv = T x tf / J = 75 x 0.525 / 0.861 = 45.7 ksi. This is ABOVE the allowable shear stress of 0.40Fy = 20 ksi without considering any flexural shear. The W18x40 would FAIL in torsion despite being adequate for bending. Solution: (1) Move the support to bear on the web (eliminate eccentricity). Or (2) add a cap channel to the top flange to form a semi-closed section, increasing J dramatically. Or (3) use an HSS section. Rule of thumb: if torque exceeds 5% of the design moment, a W-shape may not be adequate — consider an HSS or box section. This is why crane runway beams and spandrel beams supporting eccentric brick veneer are always checked for torsion.

Q: How is combined bending and torsion checked per AISC 360?

AISC 360-22 Section H3.3 provides the combined stress check: (fa/Fa) + (fb/Fb) + (fv/Fv) <= 1.0 for ASD, or the interaction equation for LRFD. For combined flexure + torsion: (1) Calculate normal stresses: fa from axial (if any), fb from flexure (M/S) + warping normal stress (sigma_w = B x omega / Cw, where B is the bi-moment). The warping normal stress can be significant near fixed ends where warping is restrained. (2) Calculate shear stresses: fv = flexural shear (VQ/It) + Saint-Venant torsion shear (T x t / J). These ADD in one web and subtract in the other. (3) For open sections: the bi-moment B must be calculated along the beam — it is maximum at fixed ends (warping restrained) and zero at free ends (warping free). The warping normal stress at a fixed end: sigma_w = B_max x omega_n / Cw, where omega_n is the normalized warping function. For a W-shape, omega_n at the flange tip ≈ bf x h / 4. (4) Combined stress check at the critical point (typically flange tip at the fixed end, where both flexural normal stress and warping normal stress are maximum). Most engineers use AISC Design Guide 9 for torsion — the calculations are too involved for hand methods beyond the simplest cases.

Q: When should I use a box section vs a W-shape for torsion?

The decision rule: if the design torque T* exceeds 5-10% of the design moment M*, a W-shape is likely inadequate and a box section should be considered. Quantitatively: (1) Calculate the required J_req = T* x t_max / (phi x 0.6Fy), where t_max is the maximum element thickness (typically flange). (2) Compare to available J for W-shapes at the required Sx. (3) If no W-shape has adequate J, switch to HSS or box. For comparison: A W18x40 (Sx=68.4, J=0.861) supports M* ≈ 250 kip-ft but only T* ≈ 3 kip-ft. A box section 18x12x5/16 (similar Sx) has J ≈ 200+ in^4, supporting T* ≈ 700 kip-ft. Box sections are 100-500x stiffer in torsion PER UNIT FLEXURAL CAPACITY. Practical guidance: crane runway beams (eccentric wheel load, T ≈ 0.1-0.2x M) — always use box or cap channel on W-shape. Spandrel beams with eccentric cladding (T ≈ 0.05-0.1x M) — check torsion, may need HSS. Offset columns on transfer beams (T ≈ 0.2-0.5x M) — always use box girder. Curved beams (torsion from curvature) — box or HSS mandatory.

Torsion in steel members arises when applied loads do not pass through the shear center. For open sections (W-shapes, channels), torsion produces both shear stresses (Saint-Venant torsion) and normal stresses (warping torsion). For closed sections (HSS, pipes), Saint-Venant torsion dominates and warping is negligible. AISC 360-22 Section H3 covers combined torsion with other forces, while AISC Design Guide 9 provides comprehensive torsion design procedures.

Quick Reference — J and Cw Values for Common W-Shapes

Shape	J (in⁴)	Cw (in⁶)	Shape	J (in⁴)	Cw (in⁶)
W8x10	0.0306	17.0	W14x48	1.59	650
W8x18	0.161	50.5	W14x68	3.25	1520
W8x31	0.507	230	W14x90	4.06	4610
W10x12	0.0370	26.0	W16x26	0.229	181
W10x22	0.242	120	W16x50	1.78	709
W10x33	0.575	302	W18x40	0.861	414
W10x49	1.39	830	W18x60	2.55	951
W12x19	0.154	71.1	W21x50	1.13	631
W12x30	0.499	239	W24x68	2.01	1940
W12x53	1.58	944	W24x84	4.39	2790
W12x72	2.99	2150	W27x94	4.52	3870
W14x34	0.606	350	W30x99	3.62	4100

J formula for W-shapes: J ≈ (2·bf·tf³ + (d-2·tf)·tw³) / 3. Cw formula: Cw ≈ Iy·h² / 4 where h = d - tf.

Two types of torsion resistance

Saint-Venant (pure) torsion: Resisted by shear stresses circulating around the cross-section. The resistance depends on the torsional constant J. For closed sections, J is large and shear stresses are low. For open sections, J is small and shear stresses are high.

Warping torsion: Resisted by flange bending in opposite directions (bi-moment). The resistance depends on the warping constant Cw. Warping occurs only in open sections where the flanges are free to bend independently. Closed sections have negligible warping because shear flow around the closed cell prevents differential flange movement.

Key section properties

Property	Symbol	Units	Open sections	Closed sections
Torsional constant	J	in^4	Small (0.5-20)	Large (50-500)
Warping constant	Cw	in^6	Large (500-50000)	~0
Shear center	e	in	May not coincide with centroid	At centroid

For W-shapes: J = (2bftf^3 + (d-2tf)tw^3)/3 (approximate). For rectangular HSS: J = 2t(b-t)^2*(d-t)^2/(b+d-2*t).

AISC 360-22 Section H3 -- combined forces with torsion

For HSS members under torsion:

Tn = 0.6*Fy*C     [Section H3.1]
phiTn = 0.90*Tn

Where C = torsional shear constant. For rectangular HSS: C = 2*(B-t)*(H-t)*t - this represents the area enclosed by the mid-thickness line times the wall thickness.

For round HSS: C = pi*(D-t)^2*t/2.

Interaction with shear and axial force

When torsion acts simultaneously with other forces, AISC Section H3.2 requires:

(Pr/Pc + Mr/Mc)^2 + (Vr/Vc + Tr/Tc)^2 <= 1.0

This elliptical interaction recognizes that torsional shear adds directly to transverse shear (both produce shear stress on the same cross-section planes), while axial and flexural normal stresses interact separately.

Open section torsion -- the differential equation

For open sections (W-shapes, channels), torsion is governed by:

GJ*(d_theta/dz) - ECw*(d^3_theta/dz^3) = t(z)

Where theta = angle of twist, z = length along member, t(z) = distributed torque. The first term is Saint-Venant resistance; the second is warping resistance. The relative contribution of each depends on the parameter:

a = sqrt(ECw/(GJ))    [characteristic length]

When the member length L >> a, Saint-Venant torsion dominates. When L << a, warping dominates. For typical W-shapes, a = 30-100 inches, so warping is significant for short members and Saint-Venant governs for long members.

Worked example -- W18x50 spandrel beam with eccentric cladding

Given: W18x50, Fy = 50 ksi, L = 25 ft, simply supported. Eccentric cladding load = 0.5 kip/ft at 8 inches from the shear center.

Properties: J = 1.24 in^4, Cw = 3040 in^6, Sx = 88.9 in^3.

Torque: t = 0.5*8 = 4.0 kip-in/ft = 0.333 kip-in/in.

Characteristic length: a = sqrt(290003040/(112001.24)) = sqrt(748,200) = 865 in. L/a = 300/865 = 0.347 -- warping is very significant at this length.

Maximum normal stress from warping (at midspan): sigma_w = (tL^2Wno)/(8*ECw) -- from DG9 tables. For simply supported with uniform torque: sigma_w is approximately 8.5 ksi.

Combined check: The warping normal stress adds to flexural bending stress. If the beam is at 60% utilization for gravity bending (sigma_b = 30 ksi), the combined stress = 30 + 8.5 = 38.5 ksi, which is 77% of Fy. The torsion increased stress by 28%.

Practical tip: avoiding torsion in design

The best torsion design is no torsion at all. Load the member through or near its shear center whenever possible:

Use stiffened seats or direct bearing rather than eccentric clip angles
Frame beams into the web of columns (load passes near shear center) rather than to the flange face
Brace spandrel beams at cladding attachment points to prevent twist
Use closed sections (HSS) when torsion cannot be avoided -- a HSS 8x8x3/8 has J = 159 in^4, compared to J = 1.24 in^4 for a W18x50 at similar weight

The torsional stiffness ratio (closed/open) can exceed 100:1 for similar weight members.

Torsional Properties — Common W-Shapes vs HSS

W-Shape Torsional Constants

Section	J (in^4)	Cw (in^6)	a (in)	Torsional Regime
W8x31	0.78	607	316	Mixed
W10x45	1.51	1510	451	Mixed
W12x65	3.79	4170	595	Warping significant
W14x82	4.20	6070	683	Warping significant
W16x67	3.47	6740	791	Warping dominates
W18x50	1.24	3040	865	Warping dominates
W21x57	1.56	5760	1091	Warping dominates
W24x68	2.36	9170	1410	Warping dominates
W27x94	4.38	17800	1813	Warping dominates
W30x99	3.81	20000	2068	Warping dominates

Characteristic length a = sqrt(ECw/(GJ)) where E = 29,000 ksi, G = 11,200 ksi. For most W-shapes, a >> typical span, meaning warping is the dominant resistance mechanism.

HSS Torsional Constants (Closed Sections)

Section	J (in^4)	C (in^3)	phiTn (kip-in, Fy=46)
HSS 4x4x1/4	5.69	5.81	48.0
HSS 6x6x3/8	29.6	16.7	138
HSS 8x8x3/8	55.5	30.7	254
HSS 10x6x3/8	42.2	21.6	178
HSS 12x12x1/2	219	80.0	662

HSS J values are 10-100x larger than W-shapes at comparable weight. phiTn = 0.90 _ 0.6 _ Fy * C.

Torsional Efficiency Comparison

Property	W18x50 (50 lb/ft)	HSS 8x4x1/2 (34.3 lb/ft)
J (in^4)	1.24	46.8
Torsional stiffness	1x (baseline)	38x
Weight (lb/ft)	50	34.3
Torsion/weight ratio	0.025	1.37 (55x better)

HSS is dramatically more efficient in torsion. For any member subject to significant torsion, consider switching from a W-shape to an HSS section.

Saint-Venant torsion vs. warping torsion -- detailed comparison

Understanding the distinction between Saint-Venant and warping torsion is essential for correct stress evaluation in steel members. The two mechanisms resist twist through fundamentally different internal force paths, and their relative importance depends on the cross-section type and member length.

Saint-Venant (uniform) torsion occurs when the cross-section rotates as a rigid body about the twist axis while the cross-section shape is maintained (no warping restraint). Shear stresses circulate around the perimeter of the section, forming a closed shear flow. In closed sections (HSS, pipes), this shear flow path is continuous and highly efficient, resulting in large torsional constants J. In open sections (W-shapes, channels, angles), the shear flow reverses direction across each plate element, and the resulting stresses must pass through the thin wall thickness, producing much smaller J values.

Saint-Venant shear stress: tau_SV = T * t / J

Where T = applied torque, t = wall thickness at the point of interest, and J = torsional constant.

Warping torsion occurs when the cross-section is prevented from warping (longitudinal displacement) at supports or loading points. In open sections, twist causes the flanges to displace longitudinally in opposite directions. When this warping is restrained, flange bending moments develop, producing normal stresses (sigma_w) at the flange tips and shear stresses from the flange shear force. The warping constant Cw quantifies this resistance.

Warping normal stress: sigma_w = M_w * Wno / Cw
Warping shear stress: tau_w = V_w * Sw / (Cw * t_f)

Where M_w = bimoment, Wno = normalized warping function, V_w = warping shear, and Sw = warping statical moment.

The characteristic length parameter a = sqrt(E*Cw/(G*J)) determines which mechanism dominates. For member lengths less than approximately a, warping provides most of the resistance. For lengths greater than approximately 3*a, Saint-Venant torsion dominates. At intermediate lengths, both mechanisms contribute significantly and must be combined using the differential equation solution or tabulated values from AISC Design Guide 9.

Torsional constant J vs. warping constant Cw

The torsional constant J and warping constant Cw are the two fundamental section properties governing torsional behavior. They have different units and different physical meanings.

Property	Symbol	Units	Physical Meaning	W-shapes (typical)	HSS (typical)
Torsional constant	J	in^4	Resistance to Saint-Venant (pure) torsion	0.5 - 5 in^4	5 - 200 in^4
Warping constant	Cw	in^6	Resistance to warping torsion (flange bending)	500 - 20,000 in^6	~0 in^6
Shear modulus	G	ksi	11,200 ksi for steel	Same	Same
Characteristic length	a	in	sqrt(ECw/(GJ)) -- transition from warping to St. Venant	300 - 2,000 in	N/A

Key insight: For HSS and other closed sections, Cw is effectively zero because the closed shear flow prevents warping. This means HSS resists torsion entirely through Saint-Venant shear, which is both simple to calculate and highly efficient. For W-shapes, both J and Cw are important, and the stress analysis must consider both Saint-Venant shear and warping normal stresses.

Approximate formulas for J:

W-shapes (sum of rectangles):  J = sum(b_i * t_i^3 / 3)
  = 2 * b_f * t_f^3 / 3 + (d - 2*t_f) * t_w^3 / 3

Rectangular HSS:  J = 2 * t * (B - t)^2 * (H - t)^2 / (B + H - 2*t)

Round HSS:  J = pi * (D - t)^3 * t / 4   (approximately)

For W-shapes, the web contribution (t_w^3 term) is negligible because t_w is small. The flanges dominate J through the b_f * t_f^3 / 3 terms. This means thicker flanges dramatically increase J, but even the thickest W-shape flanges produce J values far smaller than even modest HSS sections.

When torsion matters in structural design

Torsion is not a routine design consideration for most simply supported floor beams with symmetric loading. However, several common conditions introduce torsion that must be explicitly considered:

Eccentric cladding on spandrel beams: Curtain walls, brick veneer, and precast panels attached to the outside face of a spandrel beam apply load at an eccentricity from the shear center. For a W-shape spandrel beam with a 10-inch eccentricity, the torsional moment per unit length equals the cladding load times the eccentricity. Over a 30-foot span, this produces substantial twist and warping stresses.

Crane runway girders: Crane lateral forces (sid thrust) from bridge cranes apply at the top of the runway girder, which is typically above the shear center. The resulting torque can be significant, especially for heavy cranes. CMAA and AISC design guides recommend using a combined loading approach that accounts for biaxial bending and torsion simultaneously.

Canopy and overhang framing: Beams supporting canopies, overhangs, or balcony extensions often have one edge loaded and the other edge free, creating a net torque. Wind uplift on canopies also produces torsion when the suction is applied at an eccentricity.

Miscellaneous framing: Monorail beams, lifting lugs, pipe supports, equipment platforms, and stair stringers with guardrail posts all involve torsional loading to varying degrees.

Asymmetric loading conditions: When gravity loads are applied off-center to beams (e.g., a beam supporting a single-sided mezzanine or a stair stringer with treads on one side), torsion results. Even small eccentricities produce significant warping stresses in W-shapes because of their low J values.

Decision guide for torsion analysis:

  If torque is < 5% of the flexural capacity (T/Tn < 0.05):
    --> Torsion may be neglected (engineering judgment)

  If torque is 5-20% of capacity:
    --> Use simplified interaction equation (H3.2)
    --> Verify combined stresses at critical sections

  If torque is > 20% of capacity:
    --> Full AISC Design Guide 9 analysis required
    --> Consider switching to HSS section
    --> Check twist angle for serviceability

Open vs. closed section comparison for torsion

The most important practical decision in torsion design is the cross-section type. The difference in torsional efficiency between open and closed sections is dramatic.

Property	W18x50 (open)	HSS 8x4x3/8 (closed)	Ratio (HSS/W)
Weight (lb/ft)	50.0	23.3	0.47
J (in^4)	1.24	46.8	37.7
C (torsional shear, in^3)	N/A (warping)	12.6	N/A
phi*Tn (kip-in, Fy=50 ksi)	Complex (DG9)	340	N/A
Torsional stiffness G*J	13,900 kip-in^2	524,000 kip-in^2	37.7
Cost per foot (approx.)	$55	$42	0.76

The HSS 8x4x3/8 provides 37 times more torsional stiffness at half the weight and 24% lower cost per foot. For any application where torsion is a primary load case, switching from a W-shape to HSS is almost always the correct engineering decision.

When W-shapes are acceptable despite torsion:

The torque is small relative to flexural capacity (< 10%)
The span is short enough that warping stresses are manageable
Lateral bracing at close spacing prevents excessive twist
Architectural or connection requirements favor W-shapes
The member also carries significant strong-axis moment, favoring the larger Sx of a W-shape

AISC Design Guide 9 reference

AISC Design Guide 9: "Torsional Analysis of Structural Steel Members" by Seaburg and Carter is the primary reference for torsion design of open-section steel members. Key contents include:

Tabulated solutions for the torsion differential equation for common boundary conditions (fixed-fixed, pinned-pinned, fixed-free, fixed-pinned) and load types (point torque, uniform torque, linearly varying torque)
Section property tables with J, Cw, Wno, Sw, and a values for all standard W-shapes, channels, angles, and tees
Stress calculation procedures for combined Saint-Venant and warping stresses at critical sections
Angle of twist calculations for serviceability checks
Design examples covering spandrel beams, crane girders, canopy beams, and monorails

Design Guide 9 uses the notation a = sqrt(E*Cw/(G*J)) as the characteristic length and provides normalized graphs and tables that allow the designer to read off maximum stresses and twist angles as functions of L/a. This eliminates the need to solve the differential equation directly for most practical cases.

Worked example -- HSS 8x4x3/8 under applied torque

Given: HSS 8x4x3/8 (A500 Grade C, Fy = 50 ksi), length L = 12 ft, simply supported for torsion (free to warp at both ends). A concentrated torque of T = 50 kip-in is applied at midspan from a mechanical equipment attachment.

Section properties: From AISC Table 1-11: B = 8 in, H = 4 in, t = 0.375 in. Ag = 7.58 in^2, J = 46.8 in^4 (calculated below), C = 2*(B-t)(H-t)t = 2(7.625)(3.625)*0.375 = 20.7 in^3.

Torsional constant J:

J = 2 * t * (B - t)^2 * (H - t)^2 / (B + H - 2*t)
  = 2 * 0.375 * (7.625)^2 * (3.625)^2 / (8 + 4 - 0.75)
  = 0.75 * 58.14 * 13.14 / 11.25
  = 51.1 in^4

(Checked against AISC Manual value; slight variation due to corner radii.)

Torsional strength check (AISC H3.1):

Tn = 0.6 * Fy * C = 0.6 * 50 * 20.7 = 621 kip-in
phi * Tn = 0.90 * 621 = 559 kip-in

Applied torque Tu = 50 kip-in << 559 kip-in. Torsional strength OK by large margin.

Shear stress from torsion:

tau = T / (2 * Am * t) = 50 / (2 * (8 - 0.375) * (4 - 0.375) * 0.375)
Am = (B - t) * (H - t) = 7.625 * 3.625 = 27.64 in^2
tau = 50 / (2 * 27.64 * 0.375) = 50 / 20.73 = 2.41 ksi

Shear stress of 2.41 ksi is well below 0.6 * Fy = 30 ksi. OK.

Angle of twist (serviceability):

theta = T * L / (G * J) = 50 * 144 / (11200 * 46.8) = 7200 / 524,160 = 0.0137 rad = 0.79 degrees

A twist of less than 1 degree is generally acceptable for most structural applications. For sensitive equipment supports, a tighter limit of 0.5 degrees may be specified, which would require either a thicker section or intermediate torsional bracing.

Warping check: For HSS, Cw is effectively zero, so warping stresses and warping restraint forces are negligible. The entire torsional resistance is through Saint-Venant shear. This simplification makes HSS torsion design straightforward compared to W-shapes.

Common mistakes

Ignoring torsion on spandrel beams. Eccentric cladding, canopy, or curtain wall loads on edge beams always produce torsion. Even small eccentricities produce significant warping stresses in W-shapes.
Using open sections when closed sections are better. For members subject to significant torsion (canopies, sign structures, crane bridges), HSS or pipe sections are far more efficient.
Neglecting warping stresses at supports. For fixed-end conditions, the maximum warping stress occurs at the supports, not midspan. The boundary conditions fundamentally change the stress distribution.
Forgetting the shear center location for channels. Channel shear centers are outside the web, so gravity loads on channels always produce torsion unless the load is applied through the shear center.
Not combining torsional and flexural stresses. Warping normal stresses add directly to flexural bending stresses at the flange tips. Both must be combined for a complete strength check.

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Section Properties Database — J, Cw for all standard shapes
Column Capacity Calculator — axial and combined loading

Disclaimer

This page is for educational and reference use only. It does not constitute professional engineering advice. All design values must be verified against AISC 360-22 Section H3 and AISC Design Guide 9. The site operator disclaims liability for any loss arising from the use of this information.

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