__W__arping and __R__otational __P__roperties
__P__rogram (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 (I
_{xp}, I_{yp}) - Pure Torsion Constants (J)
- Shear Center location
- Warping Constant (C
_{w}) - Torsion characteristic parameter (β)
- Statical Moments (Q
_{x}, Q_{y}) - Warping Static Moment (S
_{w})

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

- automate the shear flow scheme
- 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 (S_{w}) vanishes at the end of all
branch terminals. Otherwise the warping constant (C_{w})
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:

- local buckling
- pure global flexural buckling
- pure global torsional bucking
- global flexural-torsional buckling
- 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 (F_{ex} and F_{ey})
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 F_{ez} 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 F_{cr} 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 F_{cr}
enveloped from all probable failure modes. Many of the Code
equations require calculation of the warping constant (C_{w}).
The calculation of F_{cr} for standard sections is fairly
simple since the warping constant (C_{w}) along with all the
other standard properties are provided in the Shape Tables. However,
for built-up sections, the calculation of C_{w} 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 F_{cr}
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 F_{cr}’s except that the code requires that they
be "determined by analysis". The F_{cr} in Chapter F should
not be confused with the F_{cr} 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 F
_{y}/ Ω or 0.6 F_{y}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.