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Der Artikel untersucht das thermische Verhalten von Brandschutzmörteln auf Gips-Basis und betont ihre Rolle bei der Verbesserung der Feuerbeständigkeit von Stahlstützen. Es stellt eine neuartige Mörtelzusammensetzung mit Glasfasern vor, die bessere Wärmedämmeigenschaften aufweist als herkömmliche Mörtel. Die Studie verwendet fortschrittliche numerische Modelle, um die thermomechanische Reaktion geschützter Stahlstützen unter normalen Brandbedingungen unter Berücksichtigung verschiedener Einflussfaktoren wie Mörtelzusammensetzung, Dicke und Säulengeometrie zu analysieren. Die Ergebnisse zeigen, dass eine Erhöhung der Brandschutzdicke und die Verwendung optimierter Mörtelzusammensetzungen die Feuerwiderstandsfähigkeit von Stahlstützen deutlich verbessern können. Darüber hinaus schlägt der Artikel Änderungen an den in der EN 1993-1-2 skizzierten aktuellen Konstruktionsstandards vor, die darauf abzielen, die Genauigkeit von Temperaturvorhersagen und Feuerwiderstandsbewertungen zu verbessern. Die vorgeschlagenen Änderungen berücksichtigen die temperaturabhängigen thermischen Eigenschaften von Mörteln auf Gipsbasis und bieten einen zuverlässigeren Ansatz für die Brandschutzgestaltung. Die Studie vergleicht auch die Leistung verschiedener Säulenquerschnitte und hebt die Vorteile der Verwendung von Abschnitten mit niedrigeren Querschnittsfaktoren für eine verbesserte Feuerbeständigkeit hervor. Insgesamt bietet der Artikel wertvolle Einblicke in die Optimierung von Brandschutzsystemen für Stahlkonstruktionen und trägt zur Entwicklung robusterer und effizienterer Konstruktionsmethoden bei.
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Abstract
The high thermal conductivity of steel, combined with its rapid degradation in mechanical properties with increasing temperature, makes it vulnerable to fire. Fire protection materials are effectively designed to control the temperature rise within steel members. This paper is a companion to a previous numerical analysis study on protected square hollow section (SHS) steel columns using thermally enhanced gypsum-based mortars. It offers a more detailed numerical investigation into the thermal performance of different gypsum-based mortar compositions used as a passive fire protection material for different types of steel columns. Firstly, finite element models for SHS steel columns were developed and verified against data from previous fire resistance tests. Then, a parametric study was conducted to explore how factors like fire protection thickness and composition, cross-section (square, rectangular, and H-shaped sections), steel tube thickness, column slenderness, and applied load level (serviceability load states) affect their fire performance under the ISO-834 standard fire curve. Comparisons were made between numerical results and current design methods from Eurocodes. It was observed that existing design methods excessively underestimate the actual fire resistance of protected columns, particularly for class-4 cross-sections especially when mortars with highest thermal insulation capacity are used. Moreover, the thermal properties of fire protection mortars should be considered in the structural steel temperature prediction as a function of temperature during fire conditions. Based on the study’s findings, modifications to current design methods for predicting the temperature evolution of columns as a function of the cross-sections and fire protection compositions, were presented with enhanced accuracy. These proposed modifications can potentially contribute to future development in Eurocode and improved fire resistance predictions.
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1 Introduction
To ensure the safety design for steel structures and minimize structural damage, it is essential to enhance the strength and stiffness of steel when exposed to temperatures exceeding \(500\,^{\circ} {{\hbox {C}}}\) [1‐3]. This improvement can be achieved through the application of insulating materials, which act as a protective barrier for the steel members, preventing them from reaching high temperatures during potential fire events [4]. Among the various insulating materials, including boards [5] sprayed-on coatings [6, 7], foams [8], intumescent coatings [9, 10], and lightweight concrete [11], gypsum-based fire protection mortars have recently gained significant attraction due to their low cost and excellent thermal insulation capacity [12]. Despite their high thermal insulation capacity, they have been rarely discussed in the literature as a means to enhance the fire resistance rating of steel members.
Thermal properties of gypsum-based fire protection mortars (e.g., thermal conductivity, specific heat, and bulk density) are of paramount importance. These parameters play a pivotal role in selecting effective thermal insulation materials because analytical and numerical models heavily rely on them [13]. Several tests have indicated that the thermal properties of gypsum-based mortars exhibit significant non-linearity as temperature rises. This non-linearity is influenced by a range of factors, such as the temperature level, moisture content, heating rate, porosity, thermal expansion of different particles and materials, dehydration and decomposition reactions, and the level of cracking [14, 15].
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Although some studies [12, 16‐18] have shown the importance and influence of the factors such as water content and thermal expansion on the thermal performance and failure mechanisms of gypsum-based mortars, these aspects were not intended to be explored in detail in this study. Instead, the focus is placed on the influence of mortar composition.
Furthermore, the current design standards in EN 1993-1-2 [19] and its upcoming revision, prEN 1993-1-2 [20], do not account for variations in fire-protection-related thermal properties when predicting temperature evolution within protected steel members. Therefore, to propose the changes to the regulations, it is necessary to consider different characteristics of steel member while predicting their temperature evolution and conduct thermal analysis tests on the thermal properties of newly developed gypsum-based mortar compositions. These proposed modifications play a crucial role in optimizing the fire protection weight for steel structures and reducing associated costs. In a previous study conducted by Abidi et al. [21], it was noted that the thermal conductivity of gypsum-based mortars may change with alterations in thicknesses. Besides that, the study highlighted that the composition of fire protection mortars and the size of aggregates are crucial factors in determining their thermal insulation capacity [21].
There are some previous studies, demonstrating the efficiency of gypsum-based mortars in providing thermal protection and improving the fire resistance of structural steel members [12, 15, 22]. Caetano et al. [15] conducted a comprehensive experimental investigation to evaluate the thermal insulation capacity of gypsum-based mortars applied to steel plates. Additionally, the fire resistance of short steel columns protected with such mortars under compression service load and subjected to elevated temperatures was assessed. Their findings revealed that the inclusion of nano- and micro-particles of silica slightly (less than 5%) enhanced the insulating capacity of these mortars, leading to higher fire resistance ratings and making them more effective in protecting steel structures compared to commercial gypsum-based mortars. Laím et al. [22] did further numerical investigation into the thermal response of the developed fire protection mortars for the improvement of fire resistance of steel SHS column members. The numerical results indicated that a change of around 10 mm in the thickness of fire protection material was necessary to shift from one fire resistance rating to another. Their study also included a sensitivity analysis of different parameters [fire protection thickness and type (composition), column slenderness, and load level] and proposed modifications to current design rules in Eurocodes for temperature prediction in protected SHS steel tubular columns under fire conditions.
This actual paper is a companion to the previous study conducted by Laím et al. [22], in which further numerical investigations were carried out on the thermal performance of gypsum-based mortars for the improvement of fire resistance rating of steel members when exposed to the ISO-834 standard fire curve [23]. The numerical analyses were based on recently developed numerical models, being verified through previous experimental test results carried out by the authors and presented and discussed in [14, 16]. Moreover, a new composition of gypsum-based mortar, including glass fiber, is also introduced. The thermal properties of this composition were determined using three distinct testing methods: the Transient Plane Source (hot disc or TPS method) [24], Thermal Gravimetric Analysis (TGA) method, and Porosimeter tests [25]. A comprehensive parametric study was then conducted to evaluate the thermo-mechanical response of steel columns, considering various influencing factors, such as fire protection thickness and composition, cross-section type and dimensions, steel tube thickness, column slenderness, and axial compression load ratio. Afterwards, the temperature evolution obtained through numerical simulations for protected steel members is compared with data from available design methods. Based on this comparison, some modifications were proposed to the prediction equations established by EN 1993-1-2:2005 [19] while addressing steel columns with different cross-sections and newly developed compositions with glass fiber.
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2 Development of Numerical Models
2.1 Definition of Numerical Models
Three-dimensional finite-element models (FEM) of the column specimens, which were coated with fire protection mortars and exposed to high temperatures, were created using the advanced Finite Element (FE) software ABAQUS (version 2021) [26], as shown in Fig. 1. These models were then verified against previous experimental data [22]. The column specimens were made of SHS \(150\times 150\times 8\) section. The inner radius \(r_{i}\) and outer radius \(r_{o}\) (see Fig. 1) were calculated using Eqs. 1 and 2, where t is the thickness of the steel hollow section.
$$\begin{aligned} r_{i}= & {\left\{ \begin{array}{ll} 2 \, t & {\text {if }} t \le 6 {\text { mm}}, \\ 2.5 \, t & {\text {if }} 6 < t \le 10 {\text { mm}}, \\ 3 \, t & {\text {if }} t > 10 {\text { mm}}, \\ \end{array}\right. } \end{aligned}$$
(1)
$$\begin{aligned} r_{o}= & r_{i} + t. \end{aligned}$$
(2)
The procedure for evaluating the fire resistance of the protected steel columns involved a three-step sequential analysis process, as described in the following sections: including (i) linear buckling analysis, (ii) heat transfer analysis, (iii) thermo-mechanical analysis (dynamic implicit quasi-static analysis) (see Fig. 2).
Fig. 1
FEA model developed for fire protected SHS steel column
A linear buckling analysis was performed to obtain the buckling modes of the column specimens, including local and flexural modes (see Fig. 2a) using eigenvalue analysis of the finite element model. The combination of buckling modes provided an initial imperfection in the geometry of the specimens with the maximum magnitude of L/1000 and b/200 for global and local imperfections, respectively, as recommended by EN 1993-1-5 [27]. This imperfection was used as input into thermo-mechanical analysis in order to predict the over-buckling behaviour of the protected SHS steel column specimens. Note that both the local and global buckling modes were determined to impose an equivalent final imperfection pattern.
2.1.2 Heat Transfer Analysis
A heat transfer analysis has been undertaken to predict the temperature distribution along the length of steel elements protected with mortar coating when all external surfaces were exposed to high temperature (see Fig. 2b). The four-node heat transfer quadrilateral shell element DS4 was employed to measure the thermal responses (temperature–time histories) of the steel tube column. The thermal boundary conditions were defined according to the recommendations in EN 1991-1-2 [28], EN 1992-1-2 [29], and EN 1996-1-2 [30] using two types of surfaces to obtain the temperature rise in the member. Radiation and convection were modelled as radiation to ambient and film conditions with the resultant emissivity coefficient taken as 0.63 and convective heat transfer coefficient equal to 25 \({\text {W}}/({\text {m}}^{2} \, {\text {K}})\), respectively. In addition, a thermal contact conductance of 150 \({\text {W}}/({\text {m}}^{2} \, {\text {K}})\) up to \(100\,^{\circ} {{\hbox {C}}}\) and of 80 \({\text {W}}/({\text {m}}^{2} \, {\text {K}})\) beyond that temperature was considered in order to simulate the thermal resistance to heat conduction at the mortar–steel interface [22, 31].
2.1.3 Thermo-mechanical Analysis
Using the temperature distribution calculated along the length of the steel column as an input for non-linear stress analysis, the imperfect column was subjected to thermo-mechanical actions, i.e., high temperatures and axial compression. This was done through static general and quasi-static dynamic implicit analysis, with the activation of geometric non-linear option (*NLGEOM=ON) in the ABAQUS program. This allowed the finite element analysis model to account for the large displacement effects.
A three-dimensional finite element model of a steel tube and mortar was developed using four-noded S4R shell elements with reduced integration. Furthermore, a general contact model was used to represent the interaction between the steel tube and the mortar protection. Two contact behaviours were defined in the general contact models, including tangential and normal behaviour. A friction coefficient equal to 0.3 was assumed for the tangential behaviour, and the normal behaviour was set to the hard contact mode.
The applied loading equalled 50% of the design load-bearing capacity, calculated following EN 1993-1-1 [32]. In order to apply the constraints and loads, semi-rigid and pin-ended supports were established using kinematic coupling constraints. The nodes within the end sections of the members were constrained to a reference point (RP) located in the cross-section plane where boundary conditions were defined. All degrees of freedom were constrained at the lower reference point (U1 = U2 = U3 = UR1 = UR2 = UR3 = 0) to simulate a fixed boundary condition. However, at the top of the column, where the concentrated load was also applied, three degrees of freedom were restrained (U1 = U2 = UR3) to simulate a pinned boundary condition, as depicted in Fig. 3.
Fig. 3
Numerical FE model, boundary conditions and loading positions
It is worth noting that while in the buckling mode, the reference points were located at the centroid of the cross-sections, in the thermo-mechanical model, both reference points were horizontally shifted by L/1000 to simulate eccentricity during the test, included specifically for the validation of the numerical models. Lastly, a mesh size of 10 mm (b/15) on the section of the columns was adopted based on a sensitivity analysis for all three mentioned analyses.
2.2 Definition of Material Properties
An accurate estimation of the thermo-mechanical behaviour of the protected steel column strongly depends on accurately characterizing the non-linear behaviour of their materials. For the steel material, the nominal mechanical property of steel for designing at ambient temperature can be found in EN 1993-1-1 [32]. Regarding the characterization of steel properties at elevated temperatures, well-defined thermal and mechanical properties for such conditions can be found in EN 1993-1-2:2005 [19]. A non-linear and isotropic mechanical behaviour was defined for steel, employing the Von Mises yield criterion. The stress-strain relationship was determined based on EN 1993-1-2:2005 [19], in which the yield strength and modulus of elasticity were affected by reduction factors, as specified in Table 3.1 of this standard.
The stress–strain curve for steel at elevated temperatures is derived by employing Young’s modulus (E) and the steel yield strength (\(f_{y}\)) equal to 210 GPa and 355 MPa (S355), respectively [22]. Furthermore, the finite element method (FEM) incorporated the large strain effects of steel into consideration, which means that true stress–strain curves were used in the FEM instead of nominal curves (as described in Eq. 3, where true stress (\(\sigma\)) and true strain (\(\epsilon\)) are used instead of engineering (nominal) stress (\(\sigma _{\text {nom}}\) and strain (\(\epsilon _{\text {nom}}\)), respectively) [33]. It was assumed that the mechanical properties of the steel remained uniform across the cross-section of the columns. This assumption was made because residual stresses resulting from the manufacturing process could be disregarded at high temperatures, particularly when critical temperatures exceed \(400\,^{\circ} {{\hbox {C}}}\) [34].
In addition to the mechanical properties of steel, EN 1993-1-2:2005 [19] has also introduced its thermal properties which are essential to be included in the numerical model. These thermal properties include bulk density, specific heat, and thermal conductivity. The thermal expansion was also considered the same as the ones established by EN 1993-1-2:2005 [19].
In this study, four types of mortars were carefully selected based on their thermal insulation capacity, following an extensive experimental campaign conducted at the laboratory of the Civil Engineering Department in the University of Coimbra (DEC-UC) [15]. The first three mortar types, including the most suitable commercial fire protection mortar (CFPM) and the most suitable developed fire protection mortars (DFPM), with and without silica micro and nanoparticles (DFPM_SMNP and DFPM, respectively), had previously been evaluated in terms of thermal and mechanical properties [15, 35, 36]. The fourth type, the most suitable developed fire protection mortar with glass fiber (DFPM_GF), was created during this study. An overview of the compositions of these developed mortars, primarily composed of gypsum (G) and perlite (P), is provided in Table 1. Their average mass density (\(\rho _{P}\)) and water-to-gypsum ratio (by weight) (W/G) were approximately 840 \({\text {kg}}/{\text {m}}^{3}\) and 1.21, respectively. In DFPM_SMNP, equal weight ratios of 0.5 were considered for both silica micro particles to gypsum (SMP/G) and silica nano particles-to-gypsum (SNP/G). In DFPM_GF, the fiber glass-to-gypsum ratio (FG/G) was also set at 0.5. The curing duration for all the specimens was about 2 months [22].
The thermal properties of these mortars, including thermal conductivity, specific heat, and bulk density, have been measured through thermal analysis tests like TGA, TPS, and Porosimeter tests for three different temperature levels: 25, 250, and \(350\,^{\circ} {{\hbox {C}}}\) [22]. Notably, due to the limitations of the laboratory’s facilities, measurements were not taken beyond \(350\,^{\circ} {{\hbox {C}}}\).
Table 2 presents the thermal property data, including bulk density (\(\rho _{P}\)), thermal conductivity (\(\lambda _{P}\)), and specific heat (\(C_{P}\)) at different temperature levels for different compositions of fire protection material. These measurements were conducted for the developed mortars within the laboratory of DEC-UC. The remaining thermal properties of the mortar can be predicted as a function of temperature based on experimental and numerical studies available in the literature [22, 37, 38].
Moreover, the developed fire protection mortars were estimated to have a modulus of elasticity of 1.3 GPa and a Poisson’s ratio of 0.2 [39, 40]. These values were used as input parameters for the thermo-mechanical analysis in the FE numerical model.
Table 1
Constitutive composition of fire protection mortars [22]
Mortar reference
G
P
W/G
SMP/G
SNP/G
FG/G
\(\rho _{P}\)
(% v/v)
(% v/v)
(% w/w)
(% w/w)
(% w/w)
(% w/w)
\(({\text {kg}}/{\text {m}}^{3})\)
CFPM
N/A
N/A
\(0.60^{\text {a}}\)
–
–
–
757
DFPM
40%
60%
1.21
–
–
–
860
DFPM_SMNP
40%
60%
1.21
0.5
0.5
–
880
DFPM_GF
40%
60%
1.21
–
–
0.5
776
aRatio between water and all solid particles
Table 2
Measured thermal properties of fire protection mortars [22]
Mortar reference
Thermal properties
\(\rho _{P}\,({\text {kg}}/{\text {m}}^{3})\)
\(\lambda _{P} \,({\text {W}}/{\text {m K}})\)
\(C_{P}\, ({\text {J}}/{\text {kg K}})\)
\(20\,^{\circ }{\hbox {C}}\)
\(250\,^{\circ }{\hbox {C}}\)
\(350\,^{\circ }{\hbox {C}}\)
\(20\,^{\circ }{\hbox {C}}\)
\(250\,^{\circ }{\hbox {C}}\)
\(350\,^{\circ }{\hbox {C}}\)
\(20\,^{\circ }{\hbox {C}}\)
\(250\,^{\circ }{\hbox {C}}\)
\(350\,^{\circ }{\hbox {C}}\)
CFPM
757
597
597
0.25
0.15
0.15
961
1193
1193
DFPM
860
719
719
0.23
0.19
0.13
862
1168
681
DFPM_SMNP
880
736
736
0.24
0.19
0.12
863
1140
722
DFPM_GF
776
635
635
0.22
0.16
0.10
820
794
307
Figure 4 illustrates the apparent thermal properties for DFPM_GF as a function of temperature. The graph demonstrates the non-linear nature of mortar thermal properties with increasing temperature. This behaviour arises due to the factors such as temperature, water content, heating rate, porosity, thermal expansion of the different particles and materials, dehydration and decomposition reactions and level of cracking [37, 38].
Figures 5 and 6 compare the experimental and numerical mean surface temperature measured on the 1.25 m long SHS \(150\times 150\times 8\) steel tubular column, protected with different types of fire protection mortars for 20 mm and 30 mm mortar thicknesses, respectively. Note that, as described in [15], the experimental temperature data shown in these figures for the column surface were determined by calculating the average temperature of 12 thermocouples placed along the column. Similarly, the numerical temperature data from the FEA were derived by averaging the nodal temperatures at locations corresponding to the thermocouple positions. This approach ensures a strong correlation between experimental and numerical results. As shown in Figs. 5 and 6, the numerical models accurately predicted the temperature distribution on the column’s surface.
The figures also showed that the heating time required to increase the steel temperature to \(150\,^{\circ} {{\hbox {C}}}\) was longer than that for the temperature above \(150\,^{\circ} {{\hbox {C}}}\). The reduction in heating rate for temperatures below \(150\,^{\circ} {{\hbox {C}}}\) was due to the evaporation of the moisture and crystalline water in the gypsum-based mortar, leading to a reduction of thermal conductivity up to 50% [38]. In addition, when the heating rate of hot air is high enough equal to that recommended by the ISO-834 standard curve [23], a plateau in the steel temperature occurs. This is the reason for the difference observed in the behaviour of the curves up to \(100\,^{\circ} {{\hbox {C}}}\). This behaviour can be attributed to the increased vapour pressure that appeared inside the pores of the mortar as a result of the high heating rate and therefore evaporation of moisture or volatile substances present within the mortar. The increased vapor pressure within mortar’s pores produces a migration of water vapour in both ways across the fire protection thickness. While vapour water is present at the steel–mortar interface, this one acts as a thermal barrier, limiting the rate at which the steel temperature increases and resulting in the observed plateau in the temperature curve at \(100\,^{\circ} {{\hbox {C}}}\) [22]. As seen in Figs. 5 and 6, this phenomenon was not modelled in the FEA since the developed FEM did not take the pore vapour pressure of mortar or in general the fluid mechanics in the mortar micro-structure into account in the numerical analysis type.
Figure 7 shows a comparison of the axial displacement-temperature curve for both the finite element calculation and the test results for SHS steel column when protected with 20 mm commercial fire protection mortar. Analysis results showed that as the column is heated, the axial displacement increases (see Fig. 7) due to the thermal elongation of the column. Increasing the temperature leads to gradual degradation in the mechanical properties of steel until the point where the yield strength of the steel or the buckling stress of steel plate is reached. The column temperature at this point corresponds to the critical temperature. At this moment, the axial displacement is sharply reduced to zero, and the column no longer supports the applied load; either it buckles or fails in compression. The comparison showed that the developed finite element model could predict the high-temperature behaviour of the column specimen with reasonable precision, and the minor differences in failure time (less than 5%) are negligible. Therefore, the developed FEA model has been proven to be reliable for conducting a series of parametric studies to evaluate the role of each parameter in the structural fire performance of columns when exposed to the standard fire curve ISO-834 [23].
Fig. 5
Temperature evolution on steel at the mortar–steel interface when mortar thickness is 20 mm
Figure 8 illustrates a comparison between the final deformed configuration resulting from finite element analysis (FEA) in ABAQUS and the corresponding configuration obtained from test results. Note that both experimental and numerical configurations show that local buckling was the main failure mode observed at high temperatures for this type of columns.
Fig. 8
Deformed configuration of SHS steel column after fire test
3 Design and Numerical Parametric Results and Discussion
3.1 Influence of Key Parameters on the Fire Resistance of Tubular Steel Columns
After the development of accurate numerical models, an extensive parametric study was conducted to assess the influence of various parameters on the fire resistance rating and critical temperature of protected tubular steel columns. These parameters included mortar composition (CFPM, DFPM, DFPM_SMNP, and DFPM_GF), fire protection thickness (ranging from 12.5 to 35 mm), axial compression load ratio (15%, 30%, and 50% of the design value of the load-bearing capacity of the respective columns at ambient temperature, \(N_{b,Rd}\) [32]), column thickness (8, 12, 16 mm), column cross-section (HE-section and hollow sections), column width (150, 300, 450 mm), and column length (1, 1.25, 4, 7 m). Table 3 provides a detailed overview of the design parameter values used in this study. For HE-section columns, it should be noted that a contour encasement fire protection approach was adopted. The numerical results obtained for these parameters were compared with those derived from the design methods available in EN 1993-1-2 [19], in terms of critical temperature. Figure 9 presents a comparative analysis of the results for column slenderness values ranging from 0.15 to 1.09 with respect to their critical temperature.
Fig. 9
Comparison between the design and numerical predictions of the critical temperatures of protected columns
To begin with, a satisfactory agreement was observed between numerical and design predictions (Eqs. 4–8 [19]) concerning critical temperature. The average errors for the 50% and 30% load ratios were found to be 7.3% and 5.9%, respectively, with the lowest average error recorded at 4.5% for the 15% load ratio. However, the maximum errors were noted at 18.6% for the 50% load ratio, and 13.7% and 14.7% for the 30% and 15% load ratios, respectively (see Tables 3, 4, 5). It is important to note that the differences exceeding 10%, as shown in Fig. 9 and Tables 3, 4, and 5, originate from the HE-section columns classified as class 4 cross-sections under fire conditions. In addition, it was observed that shorter columns tend to exhibit better fire resistance and higher critical temperatures compared to their slender counterparts. Increasing the non-dimensional slenderness in SHS column sections protected with 12.5 mm of SMNP mortar from 0.16 to 1.09 resulted in a 7% and 5.7% decrease in critical temperature, along with a 7.4% and 10.9% reduction in critical failure time for the 50% and 15% load ratios, respectively.
Figure 10 shows the actual impact of fire protection (FP) thickness on the fire resistance of SHS \(150\times 150\times 8\) steel tubular columns, each with a length of 1.25 m. The columns are protected with different fire protection materials, namely the commercial one (CFPM), the developed one with perlite (DFPM), silica micro and nanoparticles (DFPM_SMNP), and glass fiber (DFPM_GF). A range of FP mortar thicknesses (from 12.5 to 35 mm) has been selected to protect the tubular steel columns exposed to the ISO-834 standard fire curve and various load levels (50%, 30%, and 15% load ratios). It can be observed that the FP thickness has a crucial role in determining the critical time for these columns. As depicted in Fig. 10, an increase in FP thickness significantly improves the fire resistance of the columns. For example, with a 20 mm thickness of DFPM, the fire resistance rating at the 50% load level is R90 (see Fig. 10b). On the other hand, a 30 mm thickness of the same composition in the protected column gives a fire resistance rating of R120 at the same load level, signifying a 30-min improvement. This underlies that increasing FP thickness from 12.5 mm to 30- or 35-mm leads to a significant improvement in the fire resistance rating for these columns. Regarding the critical temperature of steel columns, the fire protection thickness is independent on this parameter for the same serviceability level when the fire protection is homogeneous and uniformly distributed along the column, as is the case here.
The results indicate that, at the 50% load level, achieving a fire resistance rating of 150 minutes (R150) for the column requires 30 mm of DFPM with glass fiber, while using 30 mm of other compositions such as DFPM, CFPM, and DFPM_SMNP only results in a fire resistance rating of 120 min (R120). Therefore, the advantage of using FPM with glass fiber becomes more pronounced as the required fire resistance rating increases.
Figure 11 describes how the studied parameters affect the critical time of these protected columns. As it is evident from the results, the thickness of materials plays a crucial role in the critical time of columns. For instance, the critical time of protected SHS steel column increases by approximately 35% when the steel tube thickness changes from 8 to 16 mm, regardless of whether it is protected with a 12.5- or 25-mm thickness of SMNP mortar. However, in the case of RHS (rectangular hollow section) steel columns, the increase in fire resistance achieved by increasing the steel tube thickness to 2 mm (from 6 to 8mm) is reported to be around 20% and 10% for width-to-depth ratios equal to 2 and 3, respectively.
In addition to material thickness, the section factor strongly affects the critical time of protected steel columns. For example, with rectangular hollows sections (RHS), increasing the width-to-depth ratio from 2 to 3 has a marginal effect (under 10%) on both the critical mean temperature and critical time, as long as the section factor remains almost constant. However, for HE sections, changing from an HE200A to an HE500A cross-section significantly lowers the section factor, from 221 to 108 \({\text {m}}^{-1}\), a reduction of about 51%. This decrease in section factor leads to a notable increase in the critical time to failure, ranging from about 45.3 to 59.2% for DFPM_SMNP, and from 57.3 to 69.7% for DFPM_GF, under load levels of 50% and 15%. These findings indicate that using cross-sections with lower section factors can substantially enhance fire resistance. For instance, shifting from the section with the lowest section factor [e.g., C-72-SHS-DFPM(S)-25] to the section with the highest section factor [e.g., C-41-I-DFPM(S)-25] doubled the fire resistance rating, increasing the critical time to 2 h and 18 min at a 15% load level. Therefore, adopting sections with lower section factors rather than higher ones can greatly improve the fire resistance rating of the steel columns.
The load level is also another critical factor that must be taken into account in the fire design of steel columns. It is observed that changing the load level from 50 to 15%, can even lead to rising the elevated critical mean temperature and critical time up to 40% and 65% increase in protected steel tubular columns. This change directly translates to a higher fire-resistant rating in the design for fire safety.
3.2 Comparison Between Numerical and Design Results
While simplified design methods may not be fully sufficient for optimizing fire protection materials in steel structures, advanced design methods offer a highly effective solution for achieving this purpose. To illustrate, Figs. 12 and 13 compare numerical (FEA) and analytical (EN 1993-1-2 [19]) results for the temperature evolution of various steel columns protected with 25 mm and 12.5 mm layers of DFPM_SMNP and DFPM_GF, respectively, when exposed to the ISO standard fire curve [23]. It is worth noting that all these figures focus on steel temperatures beyond \(200\,^{\circ} {{\hbox {C}}}\), given the previously mentioned inaccuracies in FEA predictions up to this level due to the presence of high-water vapour inside the mortars at the beginning of a standard fire event.
According to Figs. 12 and 13, the prediction equations (Eqs. 9 and 10 from EN 1993-1-2:2005 [19]), used for estimating the temperature evolution of a protected steel member under uniform temperature, yield inaccurate results over time. The temperature predictions tend to be overly conservative, particularly for cross-sections classified as class 4. Conversely, unreliable temperature predictions may occur when the FPM degradation level is high at elevated temperatures (Fig. 12).
The difference observed in temperature predictions between the two methods, EN 1993-1-2:2005 [19] and FEA, can be attributed to different definitions used for thermal properties of FPM when predicting temperatures in protected steel members. EN 1993-1-2:2005 [19] assumes the thermal properties of FPM to be constant, maintaining their values at ambient temperature through the temperature evolution due to the absence of temperature estimation of the FPM. In contrast, the FEA approach employs temperature-dependent thermal properties for the FPM. This difference in temperature prediction may also result from the inaccurate presumption of no water content in fire protection mortars.
To address this limitation and improve the accuracy of design predictions, a refined approach was implemented. This approach incorporates temperature-dependent thermal properties of the FPM into the Eurocode 1993-1-2 design predictions. It assumes the FPM temperature as an average between the gas temperature (according to the ISO-834 standard) and the steel section temperature, using linear interpolation applied for each thermal property at each time step. As illustrated in Figs. 12 and 13, this refined approach, denoted as EC3-1-2(v2), improved temperature prediction accuracy, reducing average mean errors from 44.3 to 6.7%, and aligning more closely with FEA prediction results. However, certain cases still exhibited under-conservative predictions, raising concerns about safety and reliability. Therefore, it is evident that while the refined approach offers notable improvement, a more comprehensive analytical method is needed to reliably predict temperature development in the protected steel elements.
It is important to note that the temperature evolution based on EN 1993-1-2:2005 [19] can lead to both conservative and non-conservative temperature predictions throughout the fire duration, as depicted in Figs. 12 and 13. Consequently, this approach may result in overly conservative or under conservative fire resistance predictions, as demonstrated in Figs. 12, 13, and 14.
Based on the findings of this study, it is observed that protected columns with HE and rectangular cross-sections under a 50% load level (LL) exhibit the highest average error (52%) in fire resistance predictions when using the EC3 design approach, compared to columns with square cross-sections, across various fire protection compositions and thicknesses (see Fig. 14).
Moreover, a slight improvement in design predictions is observed as the load level decreases from 50 to 15%. Notably, the design predictions fit better with CFPM composition (mortars with the lowest thermal insulation capacity). Conversely, the predictions are less accurate for the new composition with glass fiber (DFPM_GF), featuring mortars with the highest thermal insulation capacity.
Fig. 12
Different temperature predictions for different types of steel columns, with each column being protected with a 25 mm layer of DFPM_SMNP
Based on the numerical findings from parametric studies highlighting inaccuracies in design predictions, it is required to propose modifications to the design formula for estimating temperatures in protected steel columns. In this regard, a study by Laím et al. [22] introduced some additional parameters (Eqs. 11 and 12) to the design prediction equations established by EN 1993-1-2:2005 [19] (Eqs. 9 and 10), aiming to improve their accuracy. These parameters include the dry thermal properties of FPMs at \(350\,^{\circ} {{\hbox {C}}}\) (\(\lambda _{p,350}\),\(\rho _{p,350}\),\(c_{p,350}\)), adjustments in the thermal conductivity of FPMs for higher temperatures (\(\lambda _{p}^{*}\)), and accounting for the delay in temperature rise due to the moisture content of the FPMs (\(t^{*}\)). In this study, these previously proposed formulations (Eqs. 11‐15) [22], originally developed for SHS columns protected with two different mortar compositions (DFPM_SMNP and CFPM), are evaluated and corrected to include RHS columns protected with a broader range of gypsum-based mortar compositions like DFPM_GF and DFPM (see Eqs. 16‐19). Additionally, a newly modified formula (Eqs. 19‐20) is proposed for protected HE-shaped (I-section) columns to complement the design prediction equations specified in EN 1993-1-2:2005 [19].
In the new proposal, the impact of different mortar compositions is taken into account through the application of the factor, \(\phi\), which depends on both fire protection thickness and the thermal conductivity limits of FPMs (at \(350\,^{\circ} {{\hbox {C}}}\) and \(1200\,^{\circ} {{\hbox {C}}}\)) (Eqs. 16–17). The gas temperature corresponding to the deploit of the thermal degradation of such mortars is estimated by Eq. 19 (\(\theta _{d}\)). This parameter is adjusted accordingly based on the cross-sectional type, and it depends on the fire protection thickness, column thickness and dimensions.
As noted in Eurocode standard EN 1993-1-2 [19], the equations for estimating steel temperatures rely on a mass lumped approach, which assumes that the exposed insulation surface temperature equals the surrounding fire gas temperature. This assumption can lead to inaccuracies in temperature estimations for heavily insulated sections, especially when the temperature gradient within the insulation is significant [41, 42]. This was also evident from the results presented in Sect. 3.2 of this paper, where a higher error was observed for the steel temperature evolution between Eurocode-based calculations and numerical simulations. To address this limitation, the improved formulas (Eqs. 13–17) for SHS, RHS, and I-sections in this study were developed using a data-driven approach, with scalar values derived through linear regression analysis of data obtained from numerical simulations. By minimizing mean absolute percentage error, the formulas predict steel temperatures within the column, ranging from 400 to \(900\,^{\circ} {{\hbox {C}}}\) (the typical failure temperature range for steel members [43‐45]) with high accuracy.
The proposed method accurately predicts steel temperature–time curves with mean errors averaging around 5.8%, as depicted in Figs. 15 and 16. Figure 17 highlights significant improvements when comparing results based on the fire resistance of different columns. At a 50% load level (LL), the proposed fire resistance prediction resulted in average errors of 8.2% for protected SHS columns (a reduction of 75.3% relative to design predictions, as indicated in Table 6), 14.0% for protected RHS columns (a 70.1% decrease compared to design predictions), and 23.4% for protected HE-section columns (representing a 55% decrease from design predictions). At a 15% load level (LL), the method yielded average errors in fire resistance rating of 6.7%, 16% and 21.3% for protected SHS, RHS, and HE-sections, respectively, representing reductions of approximately 50.4%, 59.8%, and 49% from design predictions, as calculated using the proposed formula.
Tables 6 and 9 also present fire resistance predictions for SHS columns using the approach by Laím et al. [22], allowing for the comparison with the new proposal in this paper. The results demonstrate improved accuracy in time-temperature predictions compared to the previous model by Laím et al. [22], which was limited to square hollow sections and two compositions of gypsum based mortar compositions (DFPM_SMNP and CFPM). Nevertheless, both methods yielded fire resistance ratings with minimal differences of less than 3% for SHS columns. In addition, comparing these tables reveals greater accuracy in terms of critical time between numerical and design predictions for thinner thicknesses of FPMs. This trend has also been observed in the results obtained from both the newly proposed formula and Laím’s proposal.
This suggested approach can be effectively used for the fire design of different column types when protected with gypsum-based fire protection mortars, within the conditions specified by Eqs. 21 and 22. Without further studies, this method is particularly suitable for columns exposed to the ISO-834 standard fire curve [23], which is due to the temperature-dependent nature of the thermal properties in the mortar-based fire protection materials, as demonstrated earlier. Without further studies, this method is particularly suitable for columns exposed to the ISO-834 standard fire curve [23], owing to the temperature-dependent nature of the thermal properties in the mortar-based fire protection materials, as demonstrated earlier. Besides that, fire resistance standards in Eurocodes heavily rely on nominal temperature–time curves, particularly the ISO-834 standard fire curve [23].
This paper presents a numerical study investigating the thermal behaviour of newly fire protection mortars with different compositions. The aim was to enhance the fire resistance prediction of different column types, including SHS, RHS, and HE, of steel column elements. At the end, an analytical methodology was improved to predicts steel temperature–time curves of protected steel columns.
It was observed that thermo-mechanical simulations effectively replicated the structural response of steel columns protected with gypsum-based mortars under standard fire conditions. This accuracy was achieved by directly or indirectly incorporating the degradation of fire protection materials as a function of temperature into the modelling process.
The numerical findings demonstrated that increasing the fire resistance rating by one level requires approximately a 10 mm increase in the thickness of fire protection materials. This highlights the importance of these models and their potential to optimize protected steel structures by reducing their weight.
The comparison of various compositions of the gypsum-based fire protection mortars revealed that the newly formulated composition containing glass fiber offers a notable advantage over others. Specifically, this formulation has the potential to enhance the fire resistance rating by 30 min in comparison to other compositions.
The numerical analysis indicated that changing the cross-sectional type of the column from square/rectangular hollow sections to I-sections (like HEA sections) could significantly increase the fire resistance rating of the structural steel columns by as much as 30–90 min.
The EN 1993-1-2 [19] methodology for predicting temperature evolution in protected steel members during fire exposure may not be suitable enough, as indicated by this study. A conservative discrepancy in the fire resistance of the studied column was observed, with even higher errors for columns with HE and rectangular cross-sections. These findings suggest that there might be an underestimated potential in protected steel members based on the design predictions of the EN 1993-1-2 [19] methodology, particularly when high fire resistance ratings are required, thus favouring precast concrete solutions.
The results suggest potential improvement to the design methodologies outlined in EN1993-1-2 [19] for steel structures. Unlike the Eurocode method, the proposed formulas produced more accurate results, with average differences below 5% while incorporating column geometric parameters and the thermal properties of different fire protection mortar compositions as a function of temperature. However, it is important to note that these proposed formulas are limited to standard fire conditions, as recommended by current fire standards [23, 28, 46].
Given that the present study is exclusively focused on ISO-834 standard fire [23], and considering the possibilities of other types of fire (e.g., parametric curves or traveling fires) as described in Eurocode, there is a need to do further experimental and numerical research. As an example, the refined approach implemented in this paper could be evaluated for fires with both heating and cooling phases. Future works should extend the proposed design method modifications to encompass different types of fires and contribute to a comprehensive revision of EN 1993-1-2 [19]. Such a revision would aim to incorporate non-reversible thermal properties of gypsum-based mortars into the fire design of protected steel structural elements.
Acknowledgements
This work is financed by national funds through FCT—Portuguese Foundation for Science and Technology, under Grant Agreement 2022.10619.BD attributed to the 1st author. The authors gratefully acknowledge the FCT for its support under the framework of the Research Projects PTDC/ECI-EGC/31850/2017 (NANOFIRE) and CENTRO-01-0247-FEDER-047136 (Switch2Steel). The authors also thank the FCT/MCTES for its partial funding (through national funds, PIDDAC) under the R&D Unit Institute for Sustainability and Innovation in Structural Engineering (ISISE), under Reference UIDB/04029/2020, and under the Associate Laboratory Advanced Production and Intelligent Systems ARISE, Portugal under Reference LA/P/0112/2020.
Declarations
Conflict of interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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Main design parameters for parametric studies of the steel column specimens under 50% load level
Column (C–N–Type–FP–\(t_{FP}\))
\(\dfrac{A_{p}}{V}\) (m-1)
L (m)
\(W\times H \times t_{s}\)\(({{\text {mm}}}^{3})\)
\(t_{FP}\) (mm)
A \(({{\text {mm}}}^{2})\)
\(N_{cr}\) (kN)
\(\lambda\) (–)
\(N_{b,Rd}\) (kN)
\(0.50 N_{b,Rd}\)(50% load level)
\(\theta _{cr,EC}\) (\(^\circ\)C)
\(\theta _{cr,FEA}\) (\(^\circ\)C)
Difference (%)
C–1–SHS–CFPM–12.5
131
1.25
\(150\times 150\times 8\)
12.5
4214.4
39,402.3
0.19
1496.1
569
608.1
\(-6.4\)
C–2–SHS–CFPM–20
131
1.25
\(150\times 150\times 8\)
20
4214.4
39,402.3
0.19
1496.1
569
607.6
\(-6.4\)
C–3–SHS–CFPM–25
131
1.25
\(150\times 150\times 8\)
25
4214.4
39,402.3
0.19
1496.1
569
607.1
\(-6.3\)
C–4–SHS–CFPM–30
131
1.25
\(150\times 150\times 8\)
30
4214.4
39,402.3
0.19
1496.1
569
614.3
\(-7.4\)
C–5–SHS–DFPM–12.5
131
1.25
\(150\times 150\times 8\)
12.5
4214.4
39,402.3
0.19
1496.1
569
608
\(-6.4\)
C–6–SHS–DFPM–20
131
1.25
\(150\times 150\times 8\)
20
4214.4
39,402.3
0.19
1496.1
569
607.5
\(-6.3\)
C–7–SHS–DFPM–30
131
1.25
\(150\times 150\times 8\)
30
4214.4
39,402.3
0.19
1496.1
569
614.1
\(-7.3\)
C–8–SHS–DFPM(S)–8
131
1.25
\(150\times 150\times 8\)
8
4214.4
39,402.3
0.19
1496.1
569
607.0
\(-6.3\)
C–9–SHS–DFPM(S)–12.5
131
1.25
\(150\times 150\times 8\)
12.5
4214.4
39,402.3
0.19
1496.1
569
607.9
\(-6.4\)
C–10–SHS–DFPM(S)–15
131
1.25
\(150\times 150\times 8\)
15
4214.4
39,402.3
0.19
1496.1
569
607.5
\(-6.3\)
C–11–SHS–DFPM(S)–20
131
1.25
\(150\times 150\times 8\)
20
4214.4
39,402.3
0.19
1496.1
569
607.6
\(-6.4\)
C–12–SHS–DFPM(S)–25
131
1.25
\(150\times 150\times 8\)
25
4214.4
39,402.3
0.19
1496.1
569
607.0
\(-6.3\)
C–13–SHS–DFPM(S)–30
131
1.25
\(150\times 150\times 8\)
30
4214.4
39,402.3
0.19
1496.1
569
613.8
\(-7.3\)
C–14–SHS–DFPM(S)–35
131
1.25
\(150\times 150\times 8\)
35
4214.4
39,402.3
0.19
1496.1
569
607.0
\(-6.3\)
C–15–SHS–DFPM(G)–12.5
131
1.25
\(150\times 150\times 8\)
12.5
4214.4
39,402.3
0.19
1496.1
569
607.8
\(-6.4\)
C–16–SHS–DFPM(G)–20
131
1.25
\(150\times 150\times 8\)
20
4214.4
39,402.3
0.19
1496.1
569
607.5
\(-6.3\)
C–17–SHS–DFPM(G)–30
131
1.25
\(150\times 150\times 8\)
30
4214.4
39,402.3
0.19
1496.1
569
612.2
\(-7.1\)
C–18–SHS–DFPM(S)–12.5
131
1.0
\(150\times 150\times 8\)
12.5
4214.4
61,566.1
0.16
1496.1
573
617.3
\(-7.2\)
C–19–SHS–DFPM(S)–12.5
131
4.0
\(150\times 150\times 8\)
12.5
4214.4
3847.9
0.62
1154.0
554
543.1
2.0
C–20–SHS–DFPM(S)–12.5
131
7.0
\(150\times 150\times 8\)
12.5
4214.4
1256.5
1.09
731.5
539
573.8
\(-6.1\)
C–21–SHS–DFPM(S)–25
131
1.0
\(150\times 150\times 8\)
25
4214.4
61,566.1
0.16
1496.1
573
613.5
\(-6.6\)
C–22–SHS–DFPM(S)–25
131
4.0
\(150\times 150\times 8\)
25
4214.4
3847.9
0.62
1154.0
554
586.0
\(-5.5\)
C–23–SHS–DFPM(S)–25
131
7.0
\(150\times 150\times 8\)
25
4214.4
1256.5
1.09
731.5
539
574.6
\(-6.2\)
C–24–SHS–DFPM(G)–12.5
131
7.0
\(150\times 150\times 8\)
12.5
4214.4
1256.5
1.09
731.5
539
573.7
\(-6.1\)
C–25–SHS–DFPM(G)–25
131
7.0
\(150\times 150\times 8\)
25
4214.4
1256.5
1.09
731.5
539
574.6
\(-6.2\)
C–26–SHS–DFPM(S)–8
172
1.25
\(150\times 150\times 6\)
8
3301.5
30,807.0
0.20
1172.0
569
579.6
\(-1.8\)
C–27–RHS–DFPM(S)–8
170
1.25
\(150\times 300\times 6\)
8
4492.0
54,878.5
0.17
1594.7
496
513.9
\(-3.5\)
C–28–RHS–DFPM(S)–12.5
170
1.25
\(150\times 300\times 6\)
12.5
4492.0
54,878.5
0.17
1594.7
496
513.3
\(-3.4\)
C–29–RHS–DFPM(S)–25
170
1.25
\(150\times 300\times 6\)
25
4492.0
54,878.5
0.17
1594.7
496
513.9
\(-3.5\)
C–30–RHS–DFPM(S)–8
169
1.25
\(150\times 450\times 6\)
8
4770.6
78,949.9
0.15
1693.6
490
532.6
\(-8.0\)
C–31–RHS–DFPM(S)–12.5
169
1.25
\(150\times 450\times 6\)
12.5
4770.6
78,949.9
0.15
1693.6
490
532.2
\(-7.9\)
C–32–RHS–DFPM(S)–25
169
1.25
\(150\times 450\times 6\)
25
4770.6
78,949.9
0.15
1693.6
490
531.1
\(-7.7\)
C–33–RHS–DFPM(S)–8
129
1.25
\(150\times 300\times 8\)
8
6614.4
70,627.2
0.18
2348.1
502
535.5
\(-6.3\)
C–34–RHS–DFPM(S)–12.5
129
1.25
\(150\times 300\times 8\)
12.5
6614.4
70,627.2
0.18
2348.1
502
534.3
\(-6.0\)
C–35–RHS–DFPM(S)–25
129
1.25
\(150\times 300\times 8\)
25
6614.4
70,627.2
0.18
2348.1
502
533.4
\(-5.9\)
C–36–RHS–DFPM(S)–8
128
1.25
\(150\times 450\times 8\)
8
7404.6
101,852.1
0.16
2628.6
491
515.1
\(-4.7\)
C–37–RHS–DFPM(S)–12.5
128
1.25
\(150\times 450\times 8\)
12.5
7404.6
101,852.1
0.16
2628.6
491
512.5
\(-4.2\)
C–38–RHS–DFPM(S)–25
128
1.25
\(150\times 450\times 8\)
25
7404.6
101,852.1
0.16
2628.6
491
512.6
\(-4.2\)
C–39–I–DFPM(S)–8
221
1.25
HE200A
8
5137.5
34,386.1
0.23
1795.7
568
584.9
\(-2.9\)
C–40–I–DFPM(S)–12.5
221
1.25
HE200A
12.5
5137.5
34,386.1
0.23
1795.7
568
588.5
\(-3.5\)
C–41–I–DFPM(S)–25
221
1.25
HE200A
25
5137.5
34,386.1
0.23
1795.7
568
589.5
\(-3.6\)
C–42–I–DFPM(G)–8
221
1.25
HE200A
8
5137.5
34,386.1
0.23
1795.7
568
585.2
\(-2.9\)
C–43–I–DFPM(G)–12.5
221
1.25
HE200A
12.5
5137.5
34,386.1
0.23
1795.7
568
588.3
\(-3.5\)
C–44–I–DFPM(G)–25
221
1.25
HE200A
25
5137.5
34,386.1
0.23
1795.7
568
589.3
\(-3.6\)
C–45–I–DFPM(S)–8
140
1.25
HE340A
8
12,800.0
191,486.5
0.15
4544.0
507
579.5
\(-12.5\)
C–46–I–DFPM(S)–12.5
140
1.25
HE340A
12.5
12,800.0
191,486.5
0.15
4544.0
507
592.3
\(-14.4\)
C–47–I–DFPM(S)–25
140
1.25
HE340A
25
12,800.0
191,486.5
0.15
4544.0
507
590.6
\(-14.2\)
C–48–I–DFPM(G)–8
140
1.25
HE340A
8
12,800.0
191,486.5
0.15
4544.0
507
578.9
\(-12.4\)
C–49–I–DFPM(G)–12.5
140
1.25
HE340A
12.5
12,800.0
191,486.5
0.15
4544.0
507
590.9
\(-14.2\)
C–50–I–DFPM(G)–25
140
1.25
HE340A
25
12,800.0
191,486.5
0.15
4544.0
507
589.7
\(-14.0\)
C–51–I–DFPM(S)–8
108
1.25
HE500A
8
18,672.3
267,011.3
0.16
6628.7
507
622.8
\(-18.6\)
C–52–I–DFPM(S)–12.5
108
1.25
HE500A
12.5
18,672.3
267,011.3
0.16
6628.7
507
623.0
\(-18.6\)
C–53–I–DFPM(S)–25
108
1.25
HE500A
25
18,672.3
267,011.3
0.16
6628.7
507
616.4
\(-17.8\)
C–54–I–DFPM(G)–8
108
1.25
HE500A
8
18,672.3
267,011.3
0.16
6628.7
507
619.7
\(-18.2\)
C–55–I–DFPM(G)–12.5
108
1.25
HE500A
12.5
18,672.3
267,011.3
0.16
6628.7
507
618.2
\(-18.0\)
C–56–I–DFPM(G)–25
108
1.25
HE500A
25
18,672.3
267,011.3
0.16
6628.7
507
610.9
\(-17.0\)
C–57–RHS–DFPM(G)–8
170
1.25
\(150\times 300\times 6\)
8
4492.0
54,878.5
0.17
1594.7
496
513.8
\(-3.5\)
C–58–RHS–DFPM(G)–12.5
170
1.25
\(150\times 300\times 6\)
12.5
4492.0
54,878.5
0.17
1594.7
496
513.7
\(-3.4\)
C–59–RHS–DFPM(G)–25
170
1.25
\(150\times 300\times 6\)
25
4492.0
54,878.5
0.17
1594.7
496
513.9
\(-3.5\)
C–60–RHS–DFPM(G)–8
169
1.25
\(150\times 450\times 6\)
8
4770.6
78,949.9
0.15
1693.6
490
533.1
\(-8.1\)
C–61–RHS–DFPM(G)–12.5
169
1.25
\(150\times 450\times 6\)
12.5
4770.6
78,949.9
0.15
1693.6
490
532.8
\(-8.0\)
C–62–RHS–DFPM(G)–25
169
1.25
\(150\times 450\times 6\)
25
4770.6
78,949.9
0.15
1693.6
490
532.3
\(-7.9\)
C–63–RHS–DFPM(G)–8
129
1.25
\(150\times 300\times 8\)
8
6614.4
70,627.2
0.18
2348.1
502
535.7
\(-6.3\)
C–64–RHS–DFPM(G)–12.5
129
1.25
\(150\times 300\times 8\)
12.5
6614.4
70,627.2
0.18
2348.1
502
534.4
\(-6.1\)
C–65–RHS–DFPM(G)–25
129
1.25
\(150\times 300\times 8\)
25
6614.4
70,627.2
0.18
2348.1
502
533.8
\(-6.0\)
C–66–RHS–DFPM(G)–8
128
1.25
\(150\times 450\times 8\)
8
7404.6
101,852.1
0.16
2628.6
491
515.1
\(-4.7\)
C–67–RHS–DFPM(G)–12.5
128
1.25
\(150\times 450\times 8\)
12.5
7404.6
101,852.1
0.16
2628.6
491
512.5
\(-4.2\)
C–68–RHS–DFPM(G)–25
128
1.25
\(150\times 450\times 8\)
25
7404.6
101,852.1
0.16
2628.6
491
512.6
\(-4.2\)
C–69–SHS–DFPM(S)–12.5
90
1.25
\(150\times 150\times 12\)
12.5
5758.7
54,307.7
0.19
2044.3
569
608.7
\(-6.5\)
C–70–SHS–DFPM(S)–12.5
70
1.25
\(150\times 150\times 16\)
12.5
7037.7
66,405.2
0.19
2498.4
569
607.2
\(-6.3\)
C–71–SHS–DFPM(S)–25
90
1.25
\(150\times 150\times 12\)
25
5758.7
54,307.7
0.19
2044.4
569
608.9
\(-6.6\)
C–72–SHS–DFPM(S)–25
70
1.25
\(150\times 150\times 16\)
25
7037.7
66,405.2
0.19
2498.4
569
608.5
\(-6.5\)
C–73–SHS–DFPM(G)–25
90
1.25
\(150\times 150\times 12\)
25
5758.7
54,307.7
0.19
2044.4
569
609
\(-6.6\)
Table 4
Main design parameters for parametric studies of the steel column specimens under 30% and 50% load levels
Column (C–N–Type–\(t_{FP}\))
\(\dfrac{A_{p}}{V}\) (m-1)
\(\lambda\) (–)
\(N_{b,Rd}\) (kN)
\(0.30 N_{b,Rd}\)(30% load level)
\(0.50 N_{b,Rd}\)(50% load level)
\(\theta _{cr,EC}\) (\(^\circ\)C)
\(\theta _{cr,FEA}\) (\(^\circ\)C)
Difference (%)
\(\theta _{cr,EC}\) (\(^\circ\)C)
\(\theta _{cr,FEA}\) (\(^\circ\)C)
Difference (%)
C–6–SHS–DFPM–20
131
0.19
1496.1
653
677.3
\(-3.6\)
569
607.5
\(-6.3\)
C–9–SHS–DFPM(S)–12.5
131
0.19
1496.1
653
677.6
\(-3.6\)
569
607.9
\(-6.4\)
C–12–SHS–DFPM(S)–25
131
0.19
1496.1
653
677.0
\(-3.5\)
569
607.0
\(-6.3\)
C–16–SHS–DFPM(G)–20
131
0.19
1496.1
653
677.3
\(-3.6\)
569
607.5
\(-6.3\)
C–41–I–DFPM(S)–25
221
0.23
1795.7
652
665.8
\(-2.1\)
568
589.5
\(-3.6\)
C–47–I–DFPM(S)–25
140
0.15
4544.0
596
671.5
\(-11.2\)
507
590.6
\(-14.2\)
C–53–I–DFPM(S)–25
108
0.16
6628.7
596
690.4
\(-13.7\)
507
616.4
\(-17.8\)
Table 5
Main design parameters for parametric studies of the steel column specimens under 15% and 50% load levels
Column (C–N–Type–\(t_{FP}\))
\(\dfrac{A_{p}}{V}\) (m-1)
\(\lambda\)\((-)\)
\(N_{b,Rd}\) (kN)
\(0.15 N_{b,Rd}\)(15% load level)
\(0.50 N_{b,Rd}\)(50% load level)
\(\theta _{cr,EC}\) (\(^{\circ} {{\hbox {C}}}\))
\(\theta _{cr,FEA}\) (\(^{\circ} {{\hbox {C}}}\))
Difference (%)
\(\theta _{cr,EC}\) (\(^{\circ} {{\hbox {C}}}\))
\(\theta _{cr,FEA}\) (\(^{\circ} {{\hbox {C}}}\))
Difference (%)
C–1–SHS–CFPM–12.5
131
0.19
1496.1
749
765.5
\(-2.2\)
569
608.1
\(-6.4\)
C–2–SHS–CFPM–20
131
0.19
1496.1
749
765.2
\(-2.1\)
569
607.6
\(-6.4\)
C–3–SHS–CFPM–25
131
0.19
1496.1
749
764.3
\(-2.0\)
569
607.1
\(-6.3\)
C–4–SHS–CFPM–30
131
0.19
1496.1
749
768.5
\(-2.5\)
569
614.3
\(-7.4\)
C–5–SHS–DFPM–12.5
131
0.19
1496.1
749
765.4
\(-2.1\)
569
608.0
\(-6.4\)
C–6–SHS–DFPM–20
131
0.19
1496.1
749
765.2
\(-2.1\)
569
607.5
\(-6.3\)
C–7–SHS–DFPM–30
131
0.19
1496.1
749
768.5
\(-2.5\)
569
614.1
\(-7.3\)
C–8–SHS–DFPM(S)–8
131
0.19
1496.1
749
750.3
\(-0.2\)
569
607.0
\(-6.3\)
C–9–SHS–DFPM(S)–12.5
131
0.19
1496.1
749
765.3
\(-2.1\)
569
607.9
\(-6.4\)
C–10–SHS–DFPM(S)–15
131
0.19
1496.1
749
765.1
\(-2.1\)
569
607.5
\(-6.3\)
C–11–SHS–DFPM(S)–20
131
0.19
1496.1
749
765.2
\(-2.1\)
569
607.6
\(-6.4\)
C–12–SHS–DFPM(S)–25
131
0.19
1496.1
749
765.0
\(-2.1\)
569
607.0
\(-6.3\)
C–13–SHS–DFPM(S)–30
131
0.19
1496.1
749
772.7
\(-3.1\)
569
613.8
\(-7.3\)
C–14–SHS–DFPM(S)–35
131
0.19
1496.1
749
765.1
\(-2.1\)
569
607.0
\(-6.3\)
C–15–SHS–DFPM(G)–12.5
131
0.19
1496.1
749
765.3
\(-2.1\)
569
607.8
\(-6.4\)
C–16–SHS–DFPM(G)–20
131
0.19
1496.1
749
765.2
\(-2.1\)
569
607.5
\(-6.3\)
C–17–SHS–DFPM(G)–30
131
0.19
1496.1
749
767.2
\(-2.4\)
569
612.2
\(-7.1\)
C–18–SHS–DFPM(S)–12.5
131
0.16
1496.1
753
771.1
\(-2.3\)
573
617.3
\(-7.2\)
C–19–SHS–DFPM(S)–12.5
131
0.62
1154.0
729
721.1
1.1
554
543.1
2.0
C–20–SHS–DFPM(S)–12.5
131
1.09
731.5
697
727.2
\(-4.1\)
539
573.8
\(-6.1\)
C–21–SHS–DFPM(S)–25
131
0.16
1496.1
753
769.0
\(-2.1\)
573
613.5
\(-6.6\)
C–22–SHS–DFPM(S)–25
131
0.62
1154.0
729
733.3
\(-0.6\)
554
586.0
\(-5.5\)
C–23–SHS–DFPM(S)–25
131
1.09
731.5
697
728.0
\(-4.3\)
539
574.6
\(-6.2\)
C–24–SHS–DFPM(G)–12.5
131
1.09
731.5
697
727.3
\(-4.2\)
539
573.8
\(-6.1\)
C–25–SHS–DFPM(G)–25
131
1.09
731.5
697
728.0
\(-4.3\)
539
574.6
\(-6.2\)
C–26–SHS–DFPM(S)–8
172
0.20
1172.0
749
745.2
0.5
569
579.6
\(-1.8\)
C–27–RHS–DFPM(S)–8
170
0.17
1594.7
682
710.4
\(-4.0\)
496
513.9
\(-3.5\)
C–28–RHS–DFPM(S)–12.5
170
0.17
1594.7
682
710.2
\(-4.0\)
496
513.3
\(-3.4\)
C–29–RHS–DFPM(S)–25
170
0.17
1594.7
682
709.5
\(-3.9\)
496
513.9
\(-3.5\)
C–30–RHS–DFPM(S)–8
169
0.15
1693.6
680
720.6
\(-5.6\)
490
532.6
\(-8.0\)
C–31–RHS–DFPM(S)–12.5
169
0.15
1693.6
680
720.0
\(-5.6\)
490
532.2
\(-7.9\)
C–32–RHS–DFPM(S)–25
169
0.15
1693.6
680
720.6
\(-5.6\)
490
531.1
\(-7.7\)
C–33–RHS–DFPM(S)–8
129
0.18
2348.1
684
720.6
\(-5.1\)
502
535.5
\(-6.3\)
C–34–RHS–DFPM(S)–12.5
129
0.18
2348.1
684
721.3
\(-5.2\)
502
534.3
\(-6.0\)
C–35–RHS–DFPM(S)–25
129
0.18
2348.1
684
721.0
\(-5.1\)
502
533.4
\(-5.9\)
C–36–RHS–DFPM(S)–8
128
0.16
2628.6
681
715.8
\(-4.9\)
491
514.9
\(-4.6\)
C–37–RHS–DFPM(S)–12.5
128
0.16
2628.6
681
711.5
\(-4.3\)
491
511.6
\(-4.0\)
C–38–RHS–DFPM(S)-25
128
0.16
2628.6
681
710.0
\(-4.1\)
491
511.3
\(-4.0\)
C–39–I–DFPM(S)–8
221
0.23
1795.7
748
747.1
0.1
568
584.9
\(-2.9\)
C–40–I–DFPM(S)–12.5
221
0.23
1795.7
748
749.7
\(-0.2\)
568
588.5
\(-3.5\)
C–41–I–DFPM(S)–25
221
0.23
1795.7
748
752.4
\(-0.6\)
568
589.5
\(-3.6\)
C–42–I–DFPM(G)–8
221
0.23
1795.7
748
746.9
0.1
568
585.2
\(-2.9\)
C–43–I–DFPM(G)–12.5
221
0.23
1795.7
748
748.7
\(-0.1\)
568
588.3
\(-3.5\)
C–44–I–DFPM(G)–25
221
0.23
1795.7
748
751.6
\(-0.5\)
568
589.3
\(-3.6\)
C–45–I–DFPM(S)–8
140
0.15
4544.0
686
752.4
\(-8.8\)
507
579.5
\(-12.5\)
C–46–I–DFPM(S)–12.5
140
0.15
4544.0
686
764.1
\(-10.2\)
507
592.3
\(-14.4\)
C–47–I–DFPM(S)–25
140
0.15
4544.0
686
759.9
\(-9.7\)
507
590.6
\(-14.2\)
C–48–I–DFPM(G)–8
140
0.15
4544.0
686
751.1
\(-8.7\)
507
578.9
\(-12.4\)
C–49–I–DFPM(G)–12.5
140
0.15
4544.0
686
759.5
\(-9.7\)
507
590.9
\(-14.2\)
C–50–I–DFPM(G)–25
140
0.15
4544.0
686
757.1
\(-9.4\)
507
589.7
\(-14.0\)
C–51–I–DFPM(S)–8
108
0.16
6628.7
685
799.3
\(-14.3\)
507
622.8
\(-18.6\)
C–52–I–DFPM(S)–12.5
108
0.16
6628.7
685
802.7
\(-14.7\)
507
623.0
\(-18.6\)
C–53–I–DFPM(S)–25
108
0.16
6628.7
685
794.2
\(-13.8\)
507
616.4
\(-17.8\)
C–54–I–DFPM(G)–8
108
0.16
6628.7
685
796.6
\(-14.0\)
507
619.7
\(-18.2\)
C–55–I–DFPM(G)–12.5
108
0.16
6628.7
685
795.1
\(-13.9\)
507
618.2
\(-18.0\)
C–56–I–DFPM(G)–25
108
0.16
6628.7
685
783.7
\(-12.6\)
507
610.9
\(-17.0\)
C–57–RHS–DFPM(G)–8
170
0.17
1594.7
682
710.5
\(-4.0\)
496
513.8
\(-3.5\)
C–58–RHS–DFPM(G)–12.5
170
0.17
1594.7
682
710.9
\(-4.1\)
496
513.7
\(-3.4\)
C–59–RHS–DFPM(G)–25
170
0.17
1594.7
682
710.8
\(-4.1\)
496
513.9
\(-3.5\)
C–60–RHS–DFPM(G)–8
169
0.15
1693.6
680
721.0
\(-5.7\)
490
533.1
\(-8.1\)
C–61–RHS–DFPM(G)–12.5
169
0.15
1693.6
680
721.9
\(-5.8\)
490
532.8
\(-8.0\)
C–62–RHS–DFPM(G)–25
169
0.15
1693.6
680
720.6
\(-5.6\)
490
532.3
\(-7.9\)
C–63–RHS–DFPM(G)–8
129
0.18
2348.1
684
721.1
\(-5.1\)
502
535.7
\(-6.3\)
C–64–RHS–DFPM(G)–12.5
129
0.18
2348.1
684
721.7
\(-5.2\)
502
534.4
\(-6.1\)
C–65–RHS–DFPM(G)–25
129
0.18
2348.1
684
722.0
\(-5.3\)
502
533.8
\(-6.0\)
C–66–RHS–DFPM(G)–8
128
0.16
2628.6
681
715.9
\(-4.9\)
491
515.1
\(-4.7\)
C–67–RHS–DFPM(G)–12.5
128
0.16
2628.6
681
712.3
\(-4.4\)
491
512.5
\(-4.2\)
C–68–RHS–DFPM(G)–25
128
0.16
2628.6
681
712.7
\(-4.4\)
491
512.6
\(-4.2\)
C–69–SHS–DFPM(S)–12.5
90
0.19
2044.3
749
765.8
\(-2.2\)
569
608.7
\(-6.5\)
C–70–SHS–DFPM(S)–12.5
70
0.19
2498.4
749
764.2
\(-2.0\)
569
607.3
\(-6.3\)
C–71–SHS–DFPM(S)–25
90
0.19
2044.4
749
766.2
\(-2.2\)
569
608.9
\(-6.6\)
C–72–SHS–DFPM(S)–25
70
0.19
2498.4
749
765.3
\(-2.1\)
569
608.5
\(-6.5\)
C–73–SHS–DFPM(G)–25
90
0.19
2044.4
749
766.3
\(-2.3\)
569
609.0
\(-6.6\)
Table 6
Critical time of the SHS steel tubular columns protected with different compositions, slendernesses, and FPMs and steel thicknesses under fire conditions
\({|\varvec{\mu} |}\) represents the absolute average error (in percentage), relative to FEA results. *\(|{\varvec{\epsilon}}_{min}|\) denotes the absolute maximum error (in percentage), relative to FEA. \(|{\varvec{\epsilon}}_{min}|\) denotes the absolute minimum error (in percentage), relative to FEA
Table 7
Critical time of different types of steel columns protected with 25 mm layer of DFPM_SMNP and DFPM_GF
\({|\varvec{\mu} |}\) represents the absolute average error (in percentage), relative to FEA results. *\(|{\varvec{\epsilon}}_{min}|\) denotes the absolute maximum error (in percentage), relative to FEA. \(|{\varvec{\epsilon}}_{min}|\) denotes the absolute minimum error (in percentage), relative to FEA
Table 8
Critical time of different types of steel columns protected with 12.5 mm layer of DFPM_SMNP and DFPM_GF
\({|\varvec{\mu} |}\) represents the absolute average error (in percentage), relative to FEA results. *\(|{\varvec{\epsilon}}_{min}|\) denotes the absolute maximum error (in percentage), relative to FEA. \(|{\varvec{\epsilon}}_{min}|\) denotes the absolute minimum error (in percentage), relative to FEA
Table 9
Critical time of different types of steel columns protected with 8 mm layer of DFPM_SMNP
\({|\varvec{\mu} |}\) represents the absolute average error (in percentage), relative to FEA results. *\(|{\varvec{\epsilon}}_{min}|\) denotes the absolute maximum error (in percentage), relative to FEA. \(|{\varvec{\epsilon}}_{min}|\) denotes the absolute minimum error (in percentage), relative to FEA
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