# AISC Chapter F Lateral Torsional Buckling Versus Eigenvalue Calculation Methods Compared

### Technical Article

Utilizing the RF-STEEL AISC add-on module, steel member design is possible according to the AISC 360-16 standard. The following article will compare the results between calculating lateral torsional buckling according to Chapter F and Eigenvalue Analysis.

#### 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 dropdown menu.

#### Chapter F

In the AISC 360-16 [1] standard, Chapter F, the modification Factor (C_{b}) is calculated based on 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 Figure 2. The material, Steel A992, will be used for the beam along with lateral restraints at the ends and third points. 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}).

First, the required flexural strength is calculated.

M_{u} = (ω ⋅ L^{2}) / 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].

${\mathrm{C}}_{\mathrm{b}}=\frac{12.5{\mathrm{M}}_{\mathrm{max}}}{2.5{\mathrm{M}}_{\mathrm{max}}+3{\mathrm{M}}_{\mathrm{A}}+4{\mathrm{M}}_{\mathrm{B}}+3{\mathrm{M}}_{\mathrm{C}}}$

C_{b} |
Lateral-torsional buckling modification factor for nonuniform moment diagrams |

M_{max} |
Absolute value of maximum moment in the unbraced segment. |

M_{A} |
Absolute value of moment at quarter point of the unbraced segment. |

M_{B} |
Absolute value of moment at centerline of the unbraced segment. |

M_{C} |
Absolute value of moment at three-quarter point of the unbraced segment. |

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.

${\mathrm{L}}_{\mathrm{p}}=1.76\xb7{\mathrm{r}}_{\mathrm{y}}\xb7\sqrt{\frac{\mathrm{E}}{{\mathrm{F}}_{\mathrm{y}}}}$

L_{b} |
Limiting laterally unbraced length for the limit state of yielding |

r_{y} |
Radius of gyration about y-axis |

E |
Modulus of Elasticity |

F_{y} |
Yield Strength |

L_{b} = 69.9 in. = 5.83 ft.

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

${\mathrm{L}}_{\mathrm{r}}=1.95\xb7{\mathrm{r}}_{\mathrm{ts}}\xb7\frac{\mathrm{E}}{0.7{\mathrm{F}}_{\mathrm{y}}}\xb7\sqrt{\frac{\mathrm{J}\xb7\mathrm{C}}{{\mathrm{S}}_{\mathrm{x}}{\mathrm{h}}_{\mathrm{o}}}+\sqrt{{\left(\frac{\mathrm{Jc}}{{\mathrm{S}}_{\mathrm{x}}{\mathrm{h}}_{\mathrm{o}}}\right)}^{2}+6.76{\left(\frac{0.7{\mathrm{F}}_{\mathrm{y}}}{\mathrm{E}}\right)}^{2}}}$

E |
Modulus of Elasticity |

F_{y} |
Yield Strength |

J |
Torsional constant |

S_{x} |
Elastic section modulus taken about the x-axis |

h_{o} |
Distance between flange centroids |

L_{r} = 203 in.

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.

${\mathrm{M}}_{\mathrm{n}}={\mathrm{C}}_{\mathrm{b}}\left[{\mathrm{M}}_{\mathrm{p}}-({\mathrm{M}}_{\mathrm{p}}-0.7\xb7{\mathrm{F}}_{\mathrm{y}}\xb7{\mathrm{S}}_{\mathrm{x}})\left(\frac{{\mathrm{L}}_{\mathrm{b}}-{\mathrm{L}}_{\mathrm{p}}}{{\mathrm{L}}_{\mathrm{r}}-{\mathrm{L}}_{\mathrm{p}}}\right)\right]$

C_{b} |
Lateral-torsional buckling modification factor for nonuniform moment diagrams. |

M_{p} |
Plastic flexural strength. |

F_{y} |
Yield Strength |

S_{x} |
Elastic section modulus taken about the x-axis. |

L_{b} |
Distance between braces. |

L_{p} |
Limiting laterally unbraced length for the limit state of yielding. |

L_{r} |
Limiting laterally unbraced length for the limit state of inelastic lateral-torsional buckling. |

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 which 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 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 Figures 4 and 5 and the model can be downloaded below this article.

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 Figure 6 the results from the eigenvalue analysis.

The same value calculated from the AISC Design Examples came out to be:

φ_{b}M_{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 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's 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.

#### Author

#### Alex Bacon, EIT

Technical Support Engineer

Alex is responsible for customer training, technical support, and continued program development for the North American market.

#### Keywords

Steel Lateral torsional buckling AISC Eigenvalue

#### Reference

#### Links

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