Inelastic Lateral-Torsional Buckling Strength of Steel I-Beams under Non-Uniform Temperature Conditions

2.1 Development of FE Model

The numerical study consists of two types of analyses, the heat transfer analysis for the propagation of thermal loads; and the static riks analysis for the application of uniform moment. In the heat transfer analysis, beams were modeled as 4-noded DS4 heat-transfer quadrilateral shell. While in the static riks analysis, beams were modeled as S4R shell elements.

The beam section used in the analysis was a W30x148 beam, that were obtained from the AISC manual (2011) and generally used for beam system. The cross-section of the beam is shown in Fig. 1 The beam was categorized as compact section in the AISC manual (2011) and Class 1 cross-section based on Eurocode 3 Part 1-1 (2005). The material properties at the ambient temperature were elastic modulus of 210 GPa, yield stress of 275 MPa, and a Poisson’s ratio, μ, of 0.3. For this study, varying unbraced lengths of 5 m, 7 m, 9 m, 12 m, 15 m, and 18 m were considered.

Fig. 1

W30×148 Cross Section

The mechanical load was applied at each ends of the beam and was expressed as the tension and compression forces at the bottom flange and the top flange, respectively, generating a uniform bending moment. Fig. 2 demonstrates the boundary condition of the models, in which all the beam models were simply supported. Point A was assumed to have hinge support, where the displacements at x, y, and z directions and the rotation about the x axis were restrained. While point B was assumed to have roller support, where the displacement in directions y, z and the rotation about the x axis are restrained. Lastly, lines W and F were restrained against y and z displacements, respectively.

Fig. 2

Loading and Boundary Conditions

The application of thermal loads (ISO-834, 1999) were shown in Fig. 3 Beam models were divided into six types based on the temperature exposure area. They were temperature exposed over whole beam (F-WB), compression flange (F-CF), tension flange (F-TF), 50% of whole beam (50%F-WB), 50% of compression flange (50%F-CF), and 50% of tension flange (50%F-TF).

Fig. 3

Heat Transfer Boundary Conditions

As mentioned previously, properties of steel changes when exposed to elevated temperatures. Thus, the temperature-dependent properties of steel such as thermal and mechanical properties, must be considered in the analysis.

Figs. 4(a) and (b) illustrates the variation of thermal conductivity and specific heat that were obtained from formula as given in Eurocode 3 Part 1-2 (2005). In addition, the density of steel was assumed to be 7,850 kg/m³ regardless of the temperature.

Fig. 4

Material Properties of Steel

The mechanical properties of steel includes the elongation coefficient, the yield strength, Young’s modulus and stress-strain relationship. The elongation coefficient was described in Eurocode 3 Part 1-2 (2005) as shown in Fig. 4(c) Reduction factors were employed to depict the degradation of the proportional limit (k_p,θ=f_p,_θ/f_p), yield strength (k_y,θ=f_y,θ/f_y) and Young's modulus (k_E,θ=E_a,θ/E) at elevated temperature, as shown in Fig. 4(d) In this figure, f_p,θ, f_y,θ and E_a,θ represent the proportional limit, effective yield strength and slope of the linear elastic range, at a temperature θ, respectively. Lastly, Fig. 5 shows the stress-strain curve at elevated temperature for hot rolled steel taken from Eurocode 3 Part 1-2 (2005).

Fig. 5

Stress-strain Curve at Elevated Temperature

Initial imperfections of residual stress and geometric imperfection were introduced to the numerical models as illustrated in Fig. 6 The residual stress followed the pattern from ECCS (1984). While the initial imperfection is set at 1/1000 of the unbraced length which was taken from the study of Avery and Mahendran (2000).

Fig. 6

Initial Imperfections

2.2 Model Validation

A comparison between the of the FE analysis and calculated buckling strengths from Eurocode 3 Part 1-2 (2005) were conducted to validate the accuracy of the simulated models. The beams were assumed to be heated throughout the beam (Case F-WB). The results were plotted terms of the reduction factors for lateral-torsional buckling (χ_LT and χ_LT,fi at ambient and elevated temperature, respectively) and different slenderness values (λLT and λL,fi at ambient and elevated temperature, respectively).

At ambient temperature, the reduction factor for lateraltorsional buckling (χ_LT) and the non-dimensional slenderness (λLT) are calculated as

Where M_b,Rd is the design buckling resistance moment, M_c,Rd is the design resistance for bending, W_pl,y is the plastic section modulus of the cross-section and M_cr is the elastic critical moment for lateral-torsional buckling.

At elevated temperature, the reduction factor for lateral torsional buckling in the fire design situation (χ_LT,fi) and the non-dimensional slenderness values (λLT,fi) are

Where M_b,fi,t,Rd is the design lateral torsional buckling resistance moment and M_fi,θ,Rd is the fire design moment resistance.

Figs. 7 and 8 illustrate the comparison of results between FE and Eurocode at ambient temperature and elevated temperature, respectively. It can be observed that the simulated models and the calculated strengths from Eurocode behaved similarly with reasonable accuracy. These validate the FE models for investigating the inelastic buckling behavior of steel beams at elevated temperature.

Fig. 7

Comparison of the Numerical and Design Values for χ_LT and λ_LT at 20 °C

Fig. 8

Comparison of the Numerical and Design Values for χ_LT,fi and λ_LT,fi at Elevated Temperature

3.1 Critical buckling load

Geometric and material non-linear analysis with imperfections using shell finite element models were carried out. Eurocode 3 Part 1-6 proposed a method to define the buckling resistance on the load-strain curve shape. According to Eurocode 3 Part 1-6 (2007), a geometric and material non-linear analysis should be performed on the perfect structure to determine the perfect elastic-plastic buckling resistance ratio. The imperfect elastic-plastic buckling resistance should be found as the lowest load obtained from the three following criteria;

• Criterion C1: The maximum load factor on the loaddeformation-curve (limit load);

• Criterion C2: The bifurcation load factor when this occurs during the loading path prior to reaching the limit point of the load-deformation-curve;

• Criterion C3: The tolerable deformation when this occurs during the loading path prior to reaching a bifurcation load or a limit load.

The load-deformation curves at 20 ℃ and 600 ℃ at the points T1, T2, T3, B1, B2 and B3 (as shown in Fig. 9) with a length of 7m are presented in Figs. 10 and 11 The load-deformation curves are the most similar to the criterion of C1 which is selected as the critical buckling load.

Fig. 9

Points for the Load-deformation Curve

Fig. 10

Load–deformation Curve at 20°C with a Length of 7 m

Fig. 11

Load–deformation Curve at 600°C with a Length of 7m in all Cases

3.2 Flexural strength ratio at elevated temperature and buckling mode shape

Fig. 12 illustrates the reduction ratio of the critical load from the inelastic buckling analysis for the six elevated temperature scenarios. The simulated data were plotted as the ratio of the critical flexural strength at elevated temperature (M_T℃) and at ambient temperature (M_20℃) with respect to the unbraced length.

Fig. 12

Reduction Ratio in the Inelastic Buckling Strength

The analyzed beams with a lengths from 5m to 18m showed variations in the reduction of the flexural strength based on the temperature exposure area. The expected behavior was observed at F-WB, demonstrating fastest decrease in strength when the whole beam is heated. The strength degrades to approximately 52% to 60% at 400℃, and 22% to 24% at 600℃. While the least critical scenario was observed at case of F-TF and 50%F-TF. At 600℃, the flexural strength decreased to between 76% and 81% for F-TF, and 84% and 88% for 50%F-TF.

At 50%F-WB and F-CF, the reduction in flexural strength was relatively similar at lower temperature. However, as the temperature exceeds 400℃, 50%F-WB showed a greater decrease in strength than F-CF. The flexural strength decreased to at least 63% for both cases at 400℃. While at 600℃, the flexural strength was down to approximately 31% to 36% for 50%F-WB, and 43% to 55% for F-CF. Lastly, at 50%F-CF, the flexural strength gradually decreased to 73% to 80% in strength at 400℃ and 51% to 60% at 600℃.

The average reduction ratio are summarized in Table 1 For cases of partially elevated temperature in compression components, F-CF and 50%F-CF, the reduction in flexural strength was the lowest. At 600℃, the flexural strength decreased by an average of about 53% and 44%, respectively. In cases of partially elevated temperature in tension components, F-TF and 50%F-TF, the reduction in flexural strength decreased slowly. At 600℃, the flexural strength decreased by an average of about 21% and 13%, respectively. In cases of F-WB and 50%F-WB, the reduction in flexural buckling strength was the highest. At 600℃, the flexural strength decreased by an average of about 77% and 67%, respectively.

Table 1

The Average Reduction Ratio

Figs. 13 and 14 show the inelastic buckling mode shapes of the beam models with unbraced length of 7 m exposed to temperatures of 300℃ and 600℃. The buckling occurred at the center of the beams for F-WB, F-CF, F-TF, and 50%F-TF illustrating the location of the critical point. In addition, the buckling was observed to occur at the center of the beam at 300℃ for cases 50%F-WB and 50%F-CF but changes the location of the critical point near the heated area at 600℃. At 300℃, the yield strength of steel does not change and the elastic modulus decreases to 80% of its ambient temperature values. While at 600℃, the yield strength decreases about half of its ambient temperature value, and the elastic modulus decreases more rapidly about 30% of its ambient values. Therefore, at 600℃ the beam at the heated area will be weaker than the non-heated area of the beam that leads to buckling point near the heated area.

Fig. 13

Buckling Mode Shape of the Beam with a Length of 7 m at 300°C

Fig. 14

Buckling Mode Shape of the Beam with a Length of 7 m at 600°C