AISC Chapter F Lateral Torsional Buckling Versus Eigenvalue Calculation Methods Compared

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1 Steel Beam | Lateral-Torsional Buckling Comparison Model

Introduction

Within the RF-STEEL AISC add-on module, lateral-torsional buckling (LTB) by default is considered when designing steel beams. There are a couple of stability analysis methods to choose from. The first method is to calculate LTB according to the AISC 360-16 [1] standard, Chapter F. The second method is to have RFEM perform an eigenvalue analysis to calculate the governing stability conditions and the elastic critical moment (M_cr). These methods all take place in Table 1.5 Effective Lengths – Members and can be changed within the drop-down menu.

2 Table 1.5 Effective Lengths – Members

Chapter F

In the AISC 360-16 [1] standard, Chapter F, the modification Factor (C_b) is calculated on the basis of the maximum moment at the midspan and quarter points along the beam using Eqn. F1-1. The unbraced length (L_r) and the limiting laterally unbraced length (L_p) must be calculated as well. For example, referring to F.1-2b taken from the AISC Verification problems [2], a W18X50 cross-section includes an applied uniform load. This, along with the loading criteria, can be viewed in Image 2. The material, Steel A992, will be used for the beam along with lateral restraints at the ends and third points. The self-weight of the beam will not be considered. Verified with hand calculations below, RF-STEEL AISC can be used to calculate the nominal flexural moment (M_n). This value is then compared to the required flexural strength (M_r,y).

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3 Example Beam for Multiple LTB Methods

First, the required flexural strength is calculated.

M_u = (ω ⋅ L²) / 8
M_u = 266.00 kip ⋅ ft.

Now, the lateral-torsional buckling modification factor (C_b) must be calculated for the center segment of the beam utilizing Eqn. F1-1 [1].

C_b = 1.01

The lateral-torsional buckling modification factor (C_b) must be calculated for the end-span beam utilizing Eqn. F1-1 [1].

C_b = 1.46

The higher required strength and lower C_b will govern. Now, the limiting laterally unbraced length (L_b) for the limit state of yielding can be calculated.

L_b = 69.9 in. = 5.83 ft.

Using Eqn. F2-6 [1] for a doubly symmetric I-shaped member, the limiting unbraced length for the limit state of inelastic lateral-torsional buckling is equal to:

L_r = 203 inches

Now, the flexural yielding limit state and the inelastic lateral-torsional buckling limit state must be compared to determine which is controlling. The lesser controls (L_p < L_b ≤ L_r) which is used in the nominal strength (M_n) calculation.

M_n = 339 kip-ft

Lastly, the resistance factor for flexural strength (φ_b) is multiplied by M_n to give the available flexural strength equal to 305 kip-ft.

Eigenvalue

The second analysis method to analyze LTB is according to an eigenvalue or Euler buckling analysis that predicts the theoretical buckling strength of an elastic structure, or in this case, a single beam member. When buckling takes place, eigenvalues are used to describe the values of loads. Then, eigenvectors are used to determine the shape of the eigenvalues that were calculated. When the resultant structure stiffness reaches zero, buckling takes place. The stress stiffness caused by a compressive load is removed from the elastic stiffness for this scenario. In most circumstances, the first few buckling modes are of the most interest. [3]

Since an eigenvalue buckling analysis is theoretical and predicts the buckling strength of an elastic structure, this method is a more exact approach and differs from the AISC 360-16 [1] leading to a less conservative critical moment (M_cr) value.

Comparison

When comparing the results between the RFEM add-on module RF-STEEL AISC and the verification example F.1-2B [2] from the AISC 360-16 [1], the values are almost exact. The results are compared below in Images 4 and 5, and the model can be downloaded below this article.

4 LTB Results in RF-STEEL AISC According to Chapter F

5 Nominal Flexural Strength Results from AISC 360-16 VE

With RF-STEEL AISC, it is possible to run an eigenvalue analysis for calculating LTB. Example F.1-2B [2], referenced above, was modeled in RFEM and the results were calculated. You can see in Image 6 the results from the eigenvalue analysis.

LTB Eigenvalue Analysis Result Comparison

The same value calculated from the AISC Design Examples came out as:
φ_bM_n = 305 kip-ft

M_n according to Chapter F [1] in RF-STEEL AISC varies when compared to M_cr from an eigenvalue analysis. Fundamentally, the AISC 360-16 [1] standard takes a more conservative approach with analytical calculations compared to an eigenvalue analysis, which is a more theoretical and exact approach. It is expected for M_cr to be a larger value, and you will see M_n is not equal to M_cr because if L.T.B is not controlling then M_n is equal to the controlling value between yielding or local buckling. Ultimately, it is up to the engineer's discretion which method or approach is suitable for their member design. Chapter F calculations are likely required, but an eigenvalue analysis can provide a second look at LTB design from a theoretical standpoint for additional member capacity.

The steel AISC verification problems from Chapter F can be found on Dlubal Software's website, where more details are shown comparing hand calculations to the results in RF-STEEL AISC. These are available in the link below with the model.

Knowledge Base

KB 001679 | AISC Chapter F Lateral Torsional Buckling Versus Eigenvalue Calculation Methods Compared