Torsional Analysis of Steel Members

 Preface:

Structural analysis of steel members requires the engineer to assess the need for a torsional analysis.  If resultant loads applied to a member pass through the shear center of the shape, then torsional loads may be ignored.  However, if any eccentric loads are applied, torsion must be considered.  Torsion on closed shapes is relatively simple and straightforward.  Analysis of open sections, such as I-beams (W shapes, etc.) and channels requires a slightly more complicated review.

 The structural engineer typically designs steel members to be loaded such that torsion is not a concern.  Eccentric loads are typically avoided.  Typical industrial facilities will attach loads to the structural members that result in unforeseen torsion loads.  Many cases, the designers of the attachments don’t address the original design loads on the structural member to which they attach.  For example, when designing a power plant piping system, supports for the piping and equipment are typically attached to structural members.  Pipe supports that are cantilevered from the structural members will result in torsion on the member.  Support designers typically assume the structural member is a rigid point at which to start their design.  Structural designers may not be aware of the equipment loads later imposed upon their design.  Depending on the piping or equipment loads, the torsional loads may not be insignificant.  Likewise the structural designer may ignore small eccentricities in applied loads (i.e., loads only offset a few inches from the shear center) assuming torsion is not significant with only small moment arms.  However, large loads as small moment arms could still produce significant torsional loads on the beam.

 In straight beam sections, the torsional loads are typically added to the bending loads.  For structural members that are already close to allowable limits, addition of torsional stresses may result in exceeding design allowable limits.  In some cases, the stresses may be acceptable while the deflections are unacceptable.  This could be the case where the deflecting beam may affect walkways or equipment supported by the beam.

 Therefore structural designers and support designers both must be aware of the potential for torsional stresses on their designs.  Any engineer that is evaluating beam stresses and deflections must also consider the impact of any torsional loads imposed upon their designs.

Torsional Loads: 

Rotation:

A member undergoing torsion will rotate about its shear center through an angle of f as measured from each end of the member.  This rotational displacement function, f, and its derivatives with respect to member length are used to determine the torsional stresses of the member.

torsion1.gif (2176 bytes)

 

Fig. 1: Rotated Section

Warping:

Torsion on a member will result in the cross section rotating a given amount.  Non-circular sections will also experience warping of the cross section.  In an I-beam (W shape, etc.) this can be seen as one corner of the upper flange warping out of the plane of the cross section, while the other corner of the same flange warps into the plane of the cross section.  The top flange will warp opposite of the bottom flange  (see Figure 2)

torsion2.gif (2644 bytes)

In addition to circular cross sections, this warping will not occur on sections where the section is composed of plates and the centerline of the members forming the shape meet at a common point.  For example, a structural Tee is composed of two plate elements where their centerlines meet at a common point.  A tee will not experience the warping that the above I beam will.

The warping stresses of the member are obviously dependent upon any restraint of the cross section’s ability to deflect.

 Torsional Stresses:

 The stresses induced on a member as a result of torsion may be classified into three categories.  Torsional shear stress, Warping Shear stress and Warping normal stress.

 Pure torsional shear stress:

These stresses act in a direction that is parallel to the edges of the particular shape's elements.  These stresses vary linearly across the thickness of the element.   For a given shape, the pure torsional shear stress is greatest in the thickest element.  The behavior of the stress can be conceptualized by assuming a thin membrane is attached to the cross section of the member.  If the membrane is attached to the edges of the cross section and pressurized between the structural member and membrane, the membrane would bulge outward from the cross section.   The slope of the bulging membrane at any point along the surface is proportional to the torsional shear stress at that location.  The direction of these stresses is tangent to the shape of the bulging membrane. 

 The maximum torsional shear stress for the cross section can be determined from the equation:

                                            tt = Gtf'

Where G is the shear modulus, t is the element thickness and f' is the first derivative of the rotational displacement function.  The shear stresses will act as shown in figure 3:

torsion3.gif (6163 bytes)

Warping Shear Stresses

 When the member is restrained such that the cross section cannot warp freely, warping stresses will be induced.   This includes warping shear stresses as well as warping normal stresses. Warping shear stresses act in a direction that is parallel to the edges of the particular shape's elements.  These stresses are constant across the thickness of the element and vary along the length of the element.  The equation for calculating Warping Shear Stress is:

                                            tw = (-ESwf'')/t

Where E is Young's modulus, Sw is the warping statical moment at a point on the cross section, t is the element thickness and f'' is the second derivative of the rotational displacement function.     The distribution of these stresses are shown in figure 4:

torsion4.gif (6670 bytes)

Warping Normal Stresses:

 Warping Normal Stresses are direct tension and compression stresses resulting from bending of the element due to torsion.  These stresses act perpendicular to the surface of the cross section. They are constant across the thickness of an element of the cross section, but vary in magnitude along the length of the element.   These stresses are determined by:

                                  sw = EWnsf'''

 Where E is Young's Modulus, Wns is the normalized warping constant at a point s on the cross section, and f''' is the third derivative of the rotational displacement function.  The tension and compression warping normal stresses are depicted in figure 5.

torsion5.gif (7153 bytes)

Treatment of Torsional Stresses:

 As shown in figures 3 through 5, the stresses will add directly to the bending and shear stresses already imparted to a member.  The warping normal stresses act in the direction of bending stresses.  The maximum normal/bending stress for an I-beam will therefore be the sum of the maximum warping normal stress and bending stress.  The shear stresses will likewise add to the existing shear stresses of the member.  Since this is a direct addition to existing design stresses of a member, it is easy to see how an already highly loaded member may exceed allowable stress limits when torsion is applied.

 Torsional Constant-a common error:

J is used to describe the torsional constant.  Unfortunately, this same variable is used to describe the polar moment of inertia of a shape.  These are NOT the same thing.  To add to the confusion, in the case of a circular member they are numerically equal.  With other shapes, severe miscalculations result when the polar moment of inertia is used as the torsional constant.  The polar moment of inertia is the sum of the X and Y moments of inertia.  For an I-beam the torsional constant is equal to:

 torsion6.gif (1136 bytes)

Where t is the element thickness.  For a W8x24, the polar moment of inertia is approximately 101 in4 whereas the torsional constant is only 0.35 in4.  Since f is inversely proportional to J, this error could result in grossly under-calculating the stress.

 Evaluation of stresses:

The torsional stresses are all a function of the rotational displacement function f and its various derivatives.  The function f is dependent upon the type of loading as well as the end conditions of the beam.  AISC has compiled a series of tables to determine the various rotational functions and their derivatives for 12 load cases.  Using the AISC tables (ref. Torsional Analysis of Steel Members, AISC 1983 edition). The procedure for determining stresses is as follows:

 Identify the member loading and end conditions in relation to load cases 1 through 12.

  1. Identify member length.
  2. Determine a (point of load application divided by beam length) for the desired load case.
  3. With Load case and a, enter the tables for the closest value.  Interpolate between tables as necessary.
  4. From the tables the three derivatives of the rotational function are known at a point along the length of the beam as a function of J, G and the imposed load. 

 Stresses may now be calculated at the point of interest along the beam.

This method requires the user to first identify where the stresses will be of greatest concern.  The rotational derivatives (thus the associated stresses) are not necessarily a maximum value at the same point along the length of the beam.  This process could include numerous tedious iterations to determine the high stress and perform necessary interpolations between charts provided for cardinal values of a.

 A better alternative:

Archon Engineering, PC has performed this analysis many times by hand before deciding there must be a better way.  To reduce the time engineers were spending on such tedious, repetitive tasks Archon developed TASM.  This program includes 12 load cases that match the same 12 load cases of the AISC publication.   The member properties are identified from pull down lists.  Stresses may be evaluated at any point along the beam without having to do the tedious interpolations when the desired location is not a cardinal value of a.  In addition to know the stresses at the point of analysis, the program also evaluates the location where the max/min stresses occur.  A concise printout details the user’s input and gives all pertinent output relating the torsional analysis.

 For the Structural Engineer, the Pipe Support Engineer or any one who must evaluate the effects of torsion, a several hour hand calculation is only 10 seconds with this product.  TASM retails for only $30.00, which is less than most text books and less than an hour of an engineer’s time.  Compare that to several hours of hoping you read the right value from those numerous lines on those hard to follow tables.

 

 Archon Central