SDC Torsion Design

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Warping and Rotational Properties Program (WARPP)

SDC announces a our new WARPP computer program geared for calculating the both the flexural and torsion-related section properties of any arbitrarily shaped open section for structural members having either a constant or variable profile geometry. WARPP calculates the following mechanical properties to facilitate steel design per the AISC Manual of Steel Construction 13th Edition:

  • Elastic Centroid Location (Cx, Cy)
  • Plastic Centroid Location (ex, ey)
  • Plastic Modulus (Zx, Zy)
  • Principal-axis Moments of Inertia (Ixp, Iyp)
  • Pure Torsion Constants (J)
  • Shear Center location
  • Warping Constant (Cw)
  • Torsion characteristic parameter (β)
  • Statical Moments (Qx, Qy)
  • Warping Static Moment (Sw)

Attached is a sample output for an arbitrary open shape Detailed Open Section Properties.pdfWe chose this unusual shape to help validate a number of logical and numerical procedures/subroutines that:

  1. automate the shear flow scheme
  2. calculate the shear center-based properties of the section, for which the shear flow pattern is not obvious.

The shear flow application is correct only when the resulting warping static moment (Sw) vanishes at the end of all branch terminals. Otherwise the warping constant (Cw) would definitely be in error due to a flawed shear flow pattern.

AISC Code and Commentary

Many changes have been made to the 13th Edition of the AISC Code (black book) from the 9th Edition or "green book". As many municipalities or facility owners increasingly adopt (with legal and liability implications) the latest edition of International Building Code, structural engineers are compelled to understand more about various failure modes to update their design approaches adapting to the new code intent.

Besides failure due to general material yielding in bending and shear, all other modes of failure deal with buckling (structural stability). From a global failure point of view, "flexural" buckling can occur at a lower or higher critical stress than "torsional" buckling. The failure mode depends on the cross section geometry, member slenderness, unbraced length against swaying and how far apart the compression elements (flange and/or web) are braced against rotation. From a local failure point of view, some flanges or webs may buckle at a lower stress depending on the slenderness of the compression elements. To evaluate structural stability in general we must deal with:

  1. local buckling
  2. pure global flexural buckling
  3. pure global torsional bucking
  4. global flexural-torsional buckling
  5. global lateral-torsional buckling.

This information is not new. Formulas specific to each failure mode already exist in the "green book" and earlier AISC editions except that they appear in different form. We may have gotten so used to using buckling-related coefficients or formulas in our design, hardly associating them with the word "buckling".

Among all five (5) failure modes notice that the word "global" is used to distinguish it from "local" to make a point that is normally omitted. In some "global" cases the word "pure" is used to indicate that the member would fail in translation along one of the cross section’s principal axis or rotation about the member longitudinal axis through the shear center. "Pure" implies "ideal" or "perfect" in terms of (a) member geometry, (b) material properties and (c) loading application. The AISC the critical stress applicable to pure flexural buckling (Fex and Fey) is given in equation E4-9 and E4-10 for bending about the x- and y-axis, respectively and for pure torsional buckling the critical stress Fez is given in E4-11.

In real life applications, the theoretical "pure" hardly exists but "imperfections" are found to be everywhere. To name a few, imperfections can be attributed from the design (unsymmetrical section), manufacturing/fabrication (residual stress, tolerance, non-homogeneous material properties, weld defects, straightness, out-of-plumb) or from load applications (eccentricity). When a member with imperfections is under axial compression, the pure flexural buckling phenomenon would couple (inevitably) with pure torsional buckling thus inducing a combined "flexural-torsional buckling" if the member is "weak" against torsion. The flexural-torsional buckling Fcr for members with a generic profile geometry can be calculated from equation E4-6. When the same member is under pure flexural loading, some element (flange) in the compression zone can suddenly buckle with rotation about the local strong axis of the element causing "lateral torsional buckling" when the applied moment reaches a critical value. R&D in this area is still evolving. Its application is limited to symmetric sections only.

The worst outcome of a structure under load is a buckling failure. Safety and stability is gauged by the amount of load the structure can sustain without collapsing. This safe load is determined by the lowest value of critical stress Fcr enveloped from all probable failure modes. Many of the Code equations require calculation of the warping constant (Cw). The calculation of Fcr for standard sections is fairly simple since the warping constant (Cw) along with all the other standard properties are provided in the Shape Tables. However, for built-up sections, the calculation of Cw is difficult for most engineers. Unfortunately many heavy industries utilize built-up sections for beam columns and long span crane girders.

The analysis of unsymmetrical built-up shapes is the most difficult part of the new AISC Code to understand. The Fcr appearing in equations F12-3 and F12-4 represents two different types of buckling/critical stress. There is no formula for either one of the Fcr’s except that the code requires that they be "determined by analysis". The Fcr in Chapter F should not be confused with the Fcr noted in other section(s) of the Code dedicated to the critical stress due to flexural-torsional buckling.

The Code generalizes the critical stress resulting from all buckling modes as "elastic buckling stress". As stated in the Code Commentary: "the stresses are to be limited by the yield stress or the elastic buckling stress. The stress distribution and/or the elastic buckling stress must be determined from principles of structural mechanics, text books or handbooks, such as SSRC (Galambos, 1998), papers in journals, or finite element analyses". Much as we tried per AISC recommendation, SDC has not been able to acquire any literature dealing with "lateral torsional buckling" for unsymmetrical sections. In lieu of referencing papers in journals, our last resort is to honor the AISC yield stress limit by keeping all the calculated stresses below the yield stress. SDC is undertaking major modification of our automated crane girder design tools including the following:

  • Accept user-defined "effective girder component" based on width-thickness ratio of each element up to the AISC non-compact limit.
  • Use "gross section properties" to calculate flexural shear- and all torsion-related stresses.
  • Use "effective section properties" per AIST guideline to calculate flexural fiber stresses.
  • For unsymmetrical sections, use SRSS combination of all ASD (bending plus warping normal) fiber stress with (flexural horizontal shear plus pure torsion plus warping torsion) shear stress and then limit the SRSS value to the smaller of: (a) material allowable stress of Fy / Ω or 0.6 Fy and (b) flexural torsional buckling stress. This interim scheme has the concurrence of Prof. Galambos. 

Torsional Warping Constant (Cw) Sample Calculation

SDC has performed detailed hand calculations to verify our new computer program to determining the torsional warping constant (Cw) for any arbitrary open section. Attached is a hand calculation for a typical 1940's style crane girder found in many older steel mills. The channel riveted/bolted to the web of the girder is connected to the girder tie-back angle which is bolted to the column flange.

SDC has spent years solving the torsional properties of open sections to upgrade or repair crane girders. The 13th Edition of the AISC Steel Construction Manual requires the calculation of Cw for any unsymmetrical built-up open sections. Hopefully, this detailed calculation will help other structural engineers address their unique designs as required by AISC.

OpenWarpingConst - Rev 1.pdf