# Computationally Efficient Method for Steel Column Buckling in Fire

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Alternative Methods for Column Buckling in Fire

_{0.mid}. Lateral displacement (Equation (2)) develops with the increasing axial load N

_{e}, and the eccentricity of the axial load increases from y

_{0.mid}to y

_{tot.mid}= y

_{0.mid}+ y

_{1.mid}, where y

_{0.mid}and y

_{1.mid}are shown in Figure 1. This process obviously leads to an increase in the bending moment, as shown in Equation (3). The column remains stable until the increase in the external bending moment M

_{e}is fully compensated by the increase in the internal bending moment M

_{i}and the load N

_{e}is equal to the internal force N

_{i}. Stability criteria can be formulated as Equation (4). Generally, the internal bending moment M

_{i}and the internal axial force N

_{i}are interconnected. Section curvature χ links M

_{i}and y

_{tot.mid}. For a column with pinned ends, the mid-height section curvature and the maximum lateral deflection are linked with Equations (5) and (6), where ε

_{1}and ε

_{2}are the strains at the section extreme points and h is the section height. If the strains are available, the internal forces of the section can be calculated for elastic state, but it is not the case for steel in fire conditions. Using Equations (4)–(6), and using the dependences of M

_{i}and N

_{i}on the strains ε

_{1}and ε

_{2}, it is possible to define the buckling load limit of a column. Due to the complicated nature of the steel material law in fire conditions, the analytical model for Equation (4) is not so easy to establish. The Von Karman method can be effectively used in the numerical procedure with section discretisation, but computational efficiency of the method does not outperform finite element formulation.

## 3. Numerical Modelling

_{i}with the mean for the whole set of sections χ

_{gr.mean}(factor χ

_{i}/χ

_{gr.mean}). It is obvious that the responses of columns with different section profiles are rather similar. Buckling factors obtained by the FEM (χ

_{FEM}) and the EN 1993-1-2 methods (χ

_{EC}) are compared in Figure 2. Deviations between the non-linear FEM and EN 1993-1-2 results can be observed and are in line with similar results reported earlier in [4,6,8].

_{20°C}= 0.1) and for all temperatures, the stresses in the column at failure are above the proportionality limit; for a high slenderness value (λ

_{20°C}= 2.0) and for temperatures θ = 200 °C and θ = 500 °C, buckling took place when the maximum stresses approached the proportionality limit (first-yield criterion); while for θ = 900 °C, buckling takes place elastically; with a moderate slenderness value (λ

_{20°C}= 1.0), buckling took place when the section was partially in a plastic state, and the extent of plastification varied with temperature.

_{y}is the actual yield limit stress in ambient conditions; F

_{ult}is the ultimate load; σ

_{c}is the stress from the axial load; f

_{y.fi}is the yield limit stress corresponding to the temperature at failure calculated in accordance with EN 1993-1-2 [1]; χ

_{test}is the buckling factor according to the test result; χ

_{FEM}is the buckling factor calculated by FEM; and θ

_{test}is the average temperature at failure according to the test results. By analysing the results of the validation, it can be concluded that good correlations with the ETHZ [25] test were achieved.

## 4. Proposed Design Method

- the Navier–Bernoulli (plane sections) hypothesis is valid;
- the Eurocode [1] material model is valid;
- Point 1 is subjected to additional compression as a result of the lateral displacement, and, in contrast, Point 2 is subjected to unloading as a result of the lateral displacement.

_{c}at the centroid is equal to the mean value of stresses at Points 1 and 2 presented in (8). The internal axial force N

_{i}is split into components as shown in Equation (9): force in the flanges, according to Equations (10) and (11), and force in the wall according to Equation (12).

_{1}and α

_{2}are introduced as Equations (14) and (15).

_{i}) and the external axial forces (N

_{e}) are in equilibrium (Equation (22)). It is assumed that stresses at Point 1 follow the Eurocode [1] material model. Then, Equation (22) can be rewritten as Equation (23):

_{p.θ}—proportionality limit stress at temperature θ; ε

_{y}.

_{θ}—yield strain; a

_{EC}, b

_{EC}and c

_{EC}—identical to the parameters a, b and c of the EN 1993-1-2 [1] material model. For a pin-ended column with sinusoidal initial curvature, the total lateral displacement at the mid-height can be expressed as a function of the section curvature—Equation (24):

_{e}is expressed in Equation (26). Dependency of the stress at Point 2 and the strain at Point 2 on the strain at Point 1 is established in Equation (27):

_{2}and the secant modulus E

_{s.fi}is approximated as in Equation (28):

_{s.fi}is formulated as Equations (29) and (30):

_{fi}is simply an inverse function to the EN 1993-1-2 [1] stress function in the inelastic range. Equation (23) can be rewritten into Equation (31):

_{0}–ϒ

_{8}are used for convenience according to Equations (33)–(42):

_{1}in Equation (44):

_{4}:

_{s.fi}and ϒ

_{0}are both functions of N

_{e}. Critical force is introduced as Equation (51) and the stability criteria is presented as Equation (52):

_{i}, while the actual section area is denoted by A. In case the design load is denoted by N

_{fi}, N

_{e}is defined as Equation (53):

_{cr.fi.max}was obtained numerically using Equation (51), satisfying condition in Equation (52). The buckling factor χ

_{fi}was derived using Equation (54):

_{fi}is compared to the buckling factor obtained by FEM χ

_{FEM}. Performance was satisfactory for small-to-moderate initial imperfection values: y

_{0}= L/1000 − L/500. The performance of the method for a relatively large initial imperfection of y

_{0}= L/250 was judged to be unsatisfactory. According to the accepted approach, the initial imperfection of y

_{0}= L/1000 is used for buckling problems in fire conditions [24]. As the proposed model is intended to be used in reliability analysis where initial imperfection is one of the important parameters, the method was improved by introducing a fitting procedure. It was assumed that the main reason why the proposed method diverged from the non-linear FEM is the assumption of linear stress variation, as shown in Equation (8). The assumption was modified as follows: the factors α

_{1}and α

_{2}(Equations (14) and (15)) were redefined as Equations (55) and (56), where parameter g was introduced as a function of initial imperfection and slenderness:

_{1}and α

_{2}was fitted to a non-linear simulation data array. The non-linear FEM data array was formed from the results for temperature range θ = 200–900 °C, the slenderness range λ

_{20°C}= 0.1–2.0, the yield limit range 235–460 MPa and the initial imperfection range L/y

_{0}= 250–1000. The results of the fitting procedure were compiled into Equations (58)–(62):

^{−5}s compared to the 2 s time for non-linear FEM procedure (8 cores 3.6 GHz). The computation time for the capacity verification was even shorter, approaching 10

^{−6}s.

- Calculate the parameter u using Equation (57);
- Calculate the parameters p
_{1}, p_{2}, p_{3}, q_{1}and q_{2}using Equations (59)–(62); - Calculate the parameter g using Equation (58);
- Calculate the parameters α
_{1}and α_{2}using Equations (55) and (56); - Calculate the parameter β using Equation (20);
- Calculate the secant modulus E
_{s.fi}using Equations (29) or (30) as:

- 7.
- Calculate the model parameters χ
_{0}and γ_{0}using Equations (25) and (33); - 8.
- 9.
- Calculate the critical force using Equation (52) as:

- 10.
- Check the buckling capacity:

## 5. Example

_{y}= 355 MPa, E = 210,000 MPa

_{eff}= 3010 mm -> λ

_{20°C}= 0.5

^{2}, I

_{y}= 2.83 × 10

^{7}mm

^{4}

_{fi}= 720 kN

_{0}= 6.0 mm

_{y,θ}= 0.780, k

_{p,θ}= 0.360, k

_{E,θ}= 0.600, ε

_{p,θ}= 0.0010143, ε

_{y,θ}= 0.02, f

_{y,θ}= 276.9 MPa

_{p,θ}= 127.8 MPa, E

_{a,θ}= 126,000 MPa

_{1}, p

_{2}, p

_{3}, q

_{1}and q

_{2}using Equations (59)–(62):

_{1}and α

_{2}using Equations (55) and (56):

_{EC}, b

_{EC}and c

_{EC}:

_{s.fi}using Equations (29) or (30):

_{0}and γ

_{0}using Equations (25) and (33):

_{cr.fi}itself is dependent on the axial force N

_{e}, the maximum buckling capacity N

_{cr.fi.max}can be calculated iteratively or by increasing stepwise the axial load N

_{e}until N

_{e}= N

_{cr.fi}. In this example, the maximum buckling capacity calculated using the proposed method is N

_{cr.fi.max}= 724,610 N, which is very close to the buckling capacity calculated using nonlinear FEM, N

_{fi.FEM}= 736,260 N (the difference is 1.6%).

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 2.**Buckling factors obtained by FEM vs. EN 1993-1-2 for different steel grades and heating temperatures.

**Figure 3.**Stress distributions at buckling of SHS160x6 S355 for different slenderness ratios and heating temperatures.

**Figure 4.**Mid-height section strain paths for RHS160x6 S355 with different slenderness ratios at θ = 500 °C.

**Figure 7.**Performances of the proposed method against non-linear FEM for different steel grades, heating temperatures and initial imperfection values.

**Figure 8.**Performances of the proposed method against non-linear FEM after modification for different steel grades, heating temperatures and initial imperfection values.

Section Type | Profile | Axis |
---|---|---|

RHS | SHS50 × 3 | - |

SHS160 × 6 | - | |

SHS300 × 12 | - | |

RHS80 × 60 × 4 | Strong | |

RHS150 × 100 × 6 | Strong | |

RHS300 × 150 × 10 | Strong |

**Table 2.**Descriptive statistics for individual buckling factor vs. average buckling factor for the whole set.

Steel Grade | S235 | S355 | S460 |
---|---|---|---|

Mean | 1.00008 | 1.00008 | 1.00005 |

Standard Error | 0.00010 | 0.00009 | 0.00010 |

Standard Deviation | 0.00057 | 0.00064 | 0.00066 |

Minimum | 0.99013 | 0.99203 | 0.99119 |

Maximum | 1.00986 | 1.00912 | 1.00949 |

**Table 3.**Validation of the numerical results against the test series from ETHZ [25].

ID | Profile | Axis | L_{eff}, mm | f_{y.fi}, MPa | θ_{test}, °C | F_{ult}, kN | χ_{test} | χ_{FEM} | χ_{FEM}/χ_{Test} |
---|---|---|---|---|---|---|---|---|---|

L2 | SHS 160 × 5.0 | strong | 1 981 | 284 | 400 | 760 | 0.82 | 0.76 | 0.93 |

L5 | SHS 160 × 5.0 | strong | 1 983 | 155 | 550 | 467 | 0.92 | 0.86 | 0.93 |

L6 | SHS 160 × 5.0 | strong | 1 983 | 43 | 700 | 130 | 0.92 | 0.88 | 0.96 |

L08 | HEA 100 | strong | 1 921 | 356 | 400 | 608 | 0.76 | 0.71 | 0.94 |

L07 | HEA 100 | strong | 1 920 | 198 | 550 | 395 | 0.88 | 0.81 | 0.92 |

L01 | HEA 100 | strong | 1 921 | 73 | 700 | 152 | 0.92 | 0.87 | 0.94 |

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**MDPI and ACS Style**

Kervalishvili, A.; Talvik, I.
Computationally Efficient Method for Steel Column Buckling in Fire. *Buildings* **2023**, *13*, 407.
https://doi.org/10.3390/buildings13020407

**AMA Style**

Kervalishvili A, Talvik I.
Computationally Efficient Method for Steel Column Buckling in Fire. *Buildings*. 2023; 13(2):407.
https://doi.org/10.3390/buildings13020407

**Chicago/Turabian Style**

Kervalishvili, Andrei, and Ivar Talvik.
2023. "Computationally Efficient Method for Steel Column Buckling in Fire" *Buildings* 13, no. 2: 407.
https://doi.org/10.3390/buildings13020407