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 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 shapes is relatively simple and straightforward. Analysis of open 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 designing a power plant piping system, system, supports supports for for the piping piping and and equipment equipment are typically typically attached attached to structur structural al members. Pipe supports that are cantilevered from the structural members will result in torsion on the member. member. Support designers typically assume the structural member is a rigid point at which to start start their design. Structural designers may not be aware of the equipment loads later imposed upon their design. Depending on the piping 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 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 acceptable while while the deflection deflectionss are unacceptab unacceptable. le. This could could be be the case 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 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 φ of φ as measured from each end of the member. This rotational displacement function, φ , and its derivatives with respect to member length are used to determine the torsional stresses of the member.
Fig. 1: Rotated Section Warping:
Torsion on a member will result in the cross section rotating a given given amount. Noncircular 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) 2)
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 composed of two plate elements where their centerlines meet at a common point. A tee will 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 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 edges of the cross section and pressurized pressurized between between the the structural structural member member and membrane, membrane, the membrane membrane would would bulge bulge outward from the cross section. The slope of the bulging bulging membrane at any point along along the surface is proportional 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: τ t = Gtφ Gtφ ' Where G is the shear modulus, t is the element thickness and ' is the first derivative of the rotational displacement function. The shear stresses will act as shown shown in figure 3:
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 particular shape's shape's elements. elements. These These stresses stresses are constant constant across across the the thickness thickness of of the element and vary along the length of the element. The equation for calculating Warping Shear Stress is: τ w = (-ESwφ '')/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 '' is the second derivative of the rotational displacement function. The distribution of these stresses are shown in figure 4:
Warping Normal Stresses: Warping Normal Stresses are direct tension and compression stresses resulting from bending of the element element due due to torsion torsion.. These stresses stresses act perpendicu perpendicular lar to the surface of 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: σw = EWnsφ ''' Where E is Young's Modulus, Wns is the normalized warping constant at a point s on the cross section, and ''' is the third derivative of the the rotational displacement function. The tension and compression warping normal stresses are depicted in figure 5.
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 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 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:
Where t is the element thickness. For a W8x24, W8x24, the polar moment of inertia is 4 approximately 101 in whereas the torsional constant is only 0.35 in 4. Since φ 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 φ and its various derivatives. The function φ is dependent upon the type of loading as well as the end conditions of the beam. beam. AISC has compiled 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. Ident Identify ify memb member er leng length. th. 2. Determine α (point of load application divided by beam length) for the desired load case. 3. With With Load Load case case and and α, enter the tables for the closest value. Interpolate between tables as necessary. 4. From the the tables tables the three three derivatives derivatives of of the rotational rotational function function are are known known at a point along along the the length length of the the beam as a function function of of J, G and the imposed 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 α.
