Steel beam with torsion

A steel beam loaded off its shear center: a W, S, M or HP shape, a channel, a welded I-girder of plates, or a rectangular or round HSS or pipe. The calculation finds the torsional rotation and its derivatives along the span for the end restraints you choose, combines the torsional stresses with the bending stresses point by point, and checks them to AISC 360-22 Chapter H, following AISC Design Guide 9, Torsional Analysis of Structural Steel Members (Seaburg and Carter, 1997), with ASCE 7-22 load combinations.

Inputs of the steel beam with torsion #

• Member: a rolled shape from the AISC Shapes Database v16.0 or a doubly symmetric welded I-girder (depth, flange width and thickness, web thickness; its torsional properties come from DG9 Eqs. 3.4 to 3.11). The steel grade must suit the section: A992, A572 Gr. 50 or A36 for open shapes, A500 or A1085 for HSS, A53 Gr. B for pipe. Self-weight is optional; it acts at the centroid, which for a channel is eo + x̄ from the shear center and so adds a torque.
• Length and end conditions: each end is set twice, because the two are independent. Flexure: pinned, fixed or free. Torsion: pinned (twist prevented, warping free), fixed (twist and warping prevented) or free. This is your judgement as the engineer: DG9 Appendix C.2 warns that a torsionally fixed end is harder to achieve than a flexurally fixed one, and that simply welding to an end plate or a column flange may not provide it. The calculation does not design the end detail. For a closed section the torsional pinned and fixed ends act alike (no warping).
• Loads: point loads and full or partial line loads per load type (D, L, Lr, S, R, W, E), each with its eccentricity e from the shear center (for a channel, positive toward the flange tips; a load at the centroid has e = eo + x̄). The torque is P e or w e. Every ASCE 7-22 2.3.1 (LRFD) or 2.4.1 (ASD) combination of the load types present is analysed; W and E act in both directions. See Loads and combinations.
• Lateral bracing: of the compression flange, continuous, at the ends only, at a spacing or at given points. Cb is computed (Eq. F1-1) or entered.
• Load points free to displace laterally and twist: on by default where the beam is not continuously braced. On, the warping normal stress is amplified as DG9 4.7.3 suggests (below). Clear it where the load points are braced against lateral displacement and twist. This too is your judgement.
• Rotation and deflection limits: DG9 4.8 gives no rotation limit: it depends on what the beam supports (masonry, curtain wall). The prefilled 0.5° is only a starting value; set your own, or 0 to turn the check off. Live load 360 and total load 240 are the IBC Table 1604.3 floor values; 0 turns a check off.

Auto-design finds the lightest shape of the same family (W, S, M, HP; C or MC; rectangular HSS; round HSS; pipe) that passes every check, with an optional depth limit. A welded I-girder is sized by you.

Checks performed #

The torsional analysis. Open sections solve E Cw θ'''' − G J θ'' = t (DG9 Eq. C.4) exactly, piecewise between the load points, with θ, θ' and θ'' continuous and the internal torque T = G J θ' − E Cw θ''' falling by each concentrated torque (DG9 C.2): pinned ends θ = θ'' = 0, fixed ends θ = θ' = 0, free ends θ'' = 0 and T = 0. DG9 instead reads the same functions from the charts of its Appendix B; the module needs no charts and handles any combination of loads and end restraints, each load type solved once and superposed. Closed sections have no significant warping (DG9 4.6.2) and carry St. Venant torsion only, T = G J θ'.

Stresses at every station (DG9 Chapter 4): τt = G t θ' (Eq. 4.1), τw = E Sw θ'''/t (Eq. 4.2a), σw = E Wn θ'' (Eq. 4.3a), σb = M / Sx (Eq. 4.5), τb = V Q / (I t) (Eq. 4.6). For an I-shape the maxima always add (DG9 4.6.1). A channel has no such rule, so the signs are tracked at the flange tip (point 0) and the flange-web junction (point 2), Wn0 = (h/2)(b' − E0) and Wn2 = E0 h/2 with E0 = eo + tw/2; a channel's shear stresses at the flange points are added as absolute values, with the flexural shear at point 1 included (conservative: DG9 Example 5.5 adds them with their signs and takes none at point 1). The maximum stress is sought along the whole span, not only at the stations a hand calculation would pick.

Amplification (DG9 4.7.3). Where the load point can displace laterally and twist, σw is multiplied by φFe / (φFe − σb) in LRFD (Eq. 4.22), and by the ASD equivalent with Fe / Ω (Eq. 4.23), with Fe from Eq. F2-4. The guide calls this conservative and applies it at the load point; the module applies it along every unbraced segment, which is more conservative still. The check above stops the calculation from reporting a stress once σb reaches the buckling limit.

Specification. DG9 was written for the 1993 LRFD and 9th Edition ASD specifications. The module uses the limit states of AISC 360-22 H3 (φT = 0.90, ΩT = 1.67 on all of them).

The drawings show the elevation with the loads by type, the section with the shear center and the governing load at its eccentricity (dimensioned), and the moment, warping stress or torque, and rotation diagrams of the governing combinations.

Limitations #

• A single prismatic span. Continuous beams, cross-section changes, axial load and minor-axis bending are not covered. Torsion from a load at a varying eccentricity, or a linearly varying torque, must be entered in uniform pieces.
• Open sections need a compact web (F2 and F3 scope); slender-web girders (F4, F5) are refused with a message. Single angles, tees and Z-shapes are out of scope (the DG9 methods for them differ).
• The stress on an HSS in torsion is taken as T / C, and the section is assumed not to warp. Slotted or open-ended closed sections are not closed sections.
• The end connections must provide the restraint you assume: their design is not part of this calculation. Warping restraint at a real connection lies between pinned and fixed; bracket it by running both.
• The analysis is first order. The DG9 4.7.3 factor is the only second-order allowance, and Fe of Eq. F2-4 is the only stability value it uses.
• Local effects at the load point (web crippling, bracket design) are not checked.
• Shear buckling under torsion: DG9 4.7.1 notes that no such limit state has been defined; none is checked.

Verification examples #

Verification status: 122 values compared in 8 examples (8 published, 0 hand calculations from the cited equations): 102 within tolerance, 20 recorded deviations. Code basis: partial. Only some provisions were read in the target edition; the rest rests on older editions, examples or summaries.

How the torsion was verified. The published torsion values of DG9 Chapter 5 and of AISC Design Examples H.6 are chart readings: the guide reads the torsional functions from its Appendix B graphs to two digits. A comparison with them is a weak check, so:

• the solver is verified independently in the test suite against closed-form solutions of the differential equation (pinned ends with a concentrated and a uniform torque, fixed ends with both, a cantilever with an end torque, a very long member, the closed-section case), and against a separate finite-element integration by energy for every combination of pinned, fixed and free ends under mixed loads, to better than 0.01 %. These are hand calculations and a numerical integration, not published values;
• the published chart-based values are compared with a stated tolerance of 3 % instead of the usual 1 %. Where the reading is further off than that, the value is listed as a deviation with the exact function next to the reading.

DG9 Chapter 5 (printed pages 19 to 29). Loads entered as D and L giving the printed factored and service loads, no self-weight:

The deviations in plain words:

• Chart readings (DG9-CHART): the module uses the exact functions; it is higher or lower than the guide by what the two-digit reading leaves out, up to about 10 % on a stress that is itself a small difference.
• Example 5.2 torsional shear (DG9-5.2-TAU): the guide divides the whole 90 kip-in. torque by 2 t Ao, though the same example gives each support half of it when it computes the rotation. The module uses the 45 kip-in. each side actually carries, over C = 48.7 in.³.
• Example 5.2 properties (DG9-5.2-J): J = 187 in.⁴ of the 1997 tube against 176 in.⁴ in the current database.
• Example 5.5 stations (DG9-5.5-STATIONS): the guide looks at z/l = 0, 0.2 and 0.5; the module looks everywhere and finds a higher shear stress at z/l = 0.12.
• Example 5.5 flange shear (DG9-5.5-SIGNS): at points 1 and 2 at z/l = 0.2 the guide gets 3.44 and 3.12 ksi with signs; the module gets 4.00 and 4.10 ksi with absolute values, conservative and below the web value there.

AISC Design Examples v16.0, Chapter H (AISC 360-22), both design methods where the example shows both:

H.6 buckling stress (AISC-H6-FCR): the example multiplies the tabulated Mn at Cb = 1 by Cb = 1.32 without the Mp limit of Eq. F2-2; the module keeps Mn at or below Mp. The result is lower and on the safe side; yielding under normal stress governs in both.

Not compared with a published example, because none was found: the warping check under partial line loads, the amplification of DG9 4.7.3 (checked against the equation by hand), welded flanges noncompact or slender (checked against F3 and Table B4.1b by hand), and free end restraints on an open section with mixed loads (checked against the finite-element integration above).