Influence of Material Models on Predicting the Fire Behavior of Steel Columns

Abstract

Finite-element (FE) analysis was used to compare the high-temperature responses of steel columns with two different stress–strain models: the Eurocode 3 model and the model proposed by National Institute of Standards and Technology (NIST). The comparisons were made in three different phases. The first phase compared the critical buckling temperatures predicted using forty-seven column data from five different laboratories. The slenderness ratios varied from 34 to 137, and the applied axial load was 20% to 60% of the room-temperature capacity. The results showed that the NIST model predicted the buckling temperature as or more accurately than the Eurocode 3 model for four of the five data sets. In the second phase, thirty unique FE models were developed to analyze the W8 × 35 and W14 × 53 column specimens with the slenderness ratio about 70. The column specimens were tested under steady-heating conditions with a target temperature in the range of 300°C to 600°C. The models were developed by combining the material model, temperature distributions in the specimens, and numerical scheme for non-linear analyses. Overall, the models with the NIST material properties and the measured temperature variations showed the results comparable to the test data. The deviations in the results from two different numerical approaches (modified Newton–Raphson vs. arc-length) were negligible. The Eurocode 3 model made conservative predictions on the behavior of the column specimens since its retained elastic moduli are smaller than those of the NIST model at elevated temperatures. In the third phase, the column curves calibrated using the NIST model was compared with those prescribed in the ANSI/AISC-360 Appendix 4. The calibrated curve significantly deviated from the current design equation with increasing temperature, especially for the slenderness ratio from 50 to 100.

Appendix

The general equation (Eq. A1) of the NIST true stress–true strain (σ–ɛ) model for steel accounts for temperature dependence of elastic modulus, yield strength, strain-hardening behavior, and strain sensitivity. The model was developed from a survey of many reported values of high-temperature structural steel behavior. The NIST technical report [31] contains supporting data, details of regression analyses and approaches, and estimates of the uncertainties in the fitted parameters.

$$ \begin{aligned} \sigma = E\left( T \right)\varepsilon \quad for\quad \varepsilon < F_{\text{y}} (T)/E(T) \hfill \\ \sigma = \left( {RF_{{{\text{y}}0}}^{{}} + \left( {k_{3} - k_{4} F_{\text{y0}}^{{}} } \right)\exp \left( { - \left( {\frac{T}{{k_{2} }}} \right)^{k1} \left( {\varepsilon - \frac{{RF_{{{\text{y}}0}}^{{}} }}{E(T)}} \right)^{n} } \right)} \right)\left( {\frac{{\varepsilon^{\prime}}}{{\varepsilon_{0}^{\prime } }}} \right)^{m} \quad for\quad \varepsilon \ge F_{\text{y}} (T)/E(T) \hfill \\ \end{aligned} $$

(A1)

where T is temperature, E is temperature-dependent steel modulus (Eq. A2), R is retained yield strength (Eq. A3), ɛ′/ɛ ₀ ′ is normalized strain rate, and m is strain-rate sensitivity (Eq. A4).

1.1 Elastic Modulus

The decrease in elastic modulus with increasing temperature was taken from reported literature data from tensile tests of structural steel.

$$ E\left( T \right) = E_{0} \exp \left( { - \frac{1}{2}\left( {\frac{{T^{*} }}{{e_{3} }}} \right)^{{e_{1} }} - \frac{1}{2}\left( {\frac{{T^{*} }}{{e_{4} }}} \right)^{{e_{2} }} } \right) $$

(A2)

where E ₀ is the elastic modulus at T = 20°C, and T ^* = T − 20°C in °C. The elastic modulus function decays smoothly from room temperature.

1.2 Yield Strength Behavior

The normalized retained yield strength, R, as a function of temperature for 42 individual structural steels from 16 different literature sources described in [3].

$$ R = \frac{{F_{y} \left( T \right)}}{{F_{yo}^{{}} }} = r_{5} + (1 - r_{5} )\exp \left( { - \frac{1}{2}\left( {\frac{{T^{*} }}{{r_{3} }}} \right)^{{r_{1} }} - \frac{1}{2}\left( {\frac{{T^{*} }}{{r_{4} }}} \right)^{{r_{2} }} } \right) $$

(A3)

Like the elastic modulus, it decays smoothly from the room-temperature value to a small minimum value with increasing temperature. The parameters of the equation were specially chose to have the greatest fidelity to the reported literature data in the range (400 < T < 650)°C.

1.3 Yield Strength Dependence of the Strain Hardening

For structural steel, the amount of strain hardening after yield generally decreases with increasing room-temperature yield strength. The parameters of the strain hardening behavior (k ₁, k ₂, k ₃, k ₄ and n) were determined by a non-linear regression of the stress–strain curves of eight different structural steels taken from the NIST WTC investigation [3]. This form joins the elastic portion of the stress strain curve.

1.4 Temperature Dependence of the Strain-Rate Sensitivity

The strain-rate sensitivity, m, of structural steels is nearly zero at room temperature and increases to a maximum at high temperature.

$$ m = m_{0} + m_{3} \left[ {1 - \exp \left( { - \left( {\frac{T}{{m_{2} }}} \right)^{m1} } \right)} \right] $$

(A4)

Table 6 shows the values of the individual parameters in Eq. A1 for various structural steels. All were developed from literature data that had been evaluated for completeness and quality.

Table 6 Values of the Parameters in the NIST Stress–Strain Model for Structural, Fire-Resistive (FR), Quenched-and-Tempered Plate (Q&T), and Bolt Steels