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Shear design curves of unstiffened plate girder web panels at high temperatures
Journal of Constructional Steel Research 164 (2020) 105808
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Shear design curves of unstiffened plate girder web panels at high temperatures Gh Pourmoosavi a, S.A. Mousavi Ghasemi a, *, B. Farahmand Azar b, S. Talatahari b, c a
Department of Civil Engineering, Tabriz Branch, Islamic Azad University, Tabriz, Iran Department of Civil Engineering, University of Tabriz, Tabriz, Iran c Engineering Faculty, Near East University, North Cyprus, 10 Mersin, Turkey b
a r t i c l e i n f o
a b s t r a c t
Article history: Received 19 May 2019 Received in revised form 10 August 2019 Accepted 12 October 2019 Available online xxx
Owing to the lack of an analytical research on the evaluation of mode change of web panels in addition to the need for shear design curves of unstiffened plate girder web panels exposed to shear buckling especially at elevated temperatures, the present article considers the mentioned topics. To this end, the web panel shear design relationships mentioned in AISC360-16 are modified to be used in fire conditions. This is achieved by direct utilization of steel stress-strain reduction factors in EN1993-1-2 at elevated temperatures. Analytical equations and design curves are developed to estimate the ultimate shear strength, failure mode and critical limiting temperature for the web panels under the action of a specific shear load. The results based on the curves are compared to the findings of current paper numerical analysis, existing experimental and numerical studies, indicating a good agreement between the results. © 2019 Elsevier Ltd. All rights reserved.
Keywords: Fire condition Shear design Web panel Slenderness Design curve
1. Introduction
1.2. Literature review
1.1. Background
There have been extensive studies on the shear behavior of steel plates when the temperature is considered as normal. Fire design is another important aspect that should be addressed separately [6]. Steel structure and lifespan are affected by fire, and the shear buckling of web panels in addition to failure modes of girders by considering fire as a parameter has been an interesting subject in recent years [2]. This has been the primary aim of some studies. With regard to plate girders at elevated temperatures, a theoretical model was proposed by Vimonsatit et al. [7] to estimate the failure loads when a mixture of axial and shear or simply shear stresses are applied. The developed analysis could consider the fire resistance with respect to shear strength associated with isothermal criterion, or limiting temperature for known values of axial thermal restriction and constant shear force. The outputs indicated that web buckling becomes the dominant failure mode at such temperatures. Noteworthy is that the authors extended the normal temperature relations proposed in the BS 5950: Part 1 standard [8] so as to study axial stress and material conditions at elevated temperatures. In an empirical study under fire condition, Tan and Qian [9] investigated the shear characteristics of a number of steel plate girders with simply-supported boundary conditions and thermal restraints that reduced the ultimate shear strength (USS). The higher the thermal restraint and the more slender the web plate, the more significant
Steel plate I-girders have extensively been employed in the construction of bridges and buildings. A conventional method of acquiring notable strength-to-weight ratios is to consider deep slender webs for I-girders, although shear buckling is a threat to slender web plates [1e3]. By subjecting a plate to in-plane shear, failure is expected prior to or after the proportional limit of material [3]. Elastic buckling of unstiffened, slender web plates at the onset of exerting the load as well as different types of nonlinearity in the post-buckling state can be anticipated [4]. Yielding before the occurrence of buckling, however, happens for compact and short web plates. Simultaneous geometrical instability and material yielding are notable phenomena in non-compact web plates [5]. To gain a better understanding of the shear characteristics of unstiffened web plates, the following review is provided which is mainly aimed at studies at elevated temperatures.
* Corresponding author. E-mail address: [email protected] (S.A.M. Ghasemi). https://doi.org/10.1016/j.jcsr.2019.105808 0143-974X/© 2019 Elsevier Ltd. All rights reserved.
G. Pourmoosavi et al. / Journal of Constructional Steel Research 164 (2020) 105808
(1)
where E20 is the Young's modulus at 20 C, ʋ is the Poisson's ratio, tw and D are the web plate thickness and depth, respectively, and K is the ESB coefficient which is dependent on the boundary conditions and aspect ratio of the plate [13,27] and is found as.
Kss ¼ 4 þ
5:34 f2
Kss ¼ 5:34 þ
Ksf ¼
4 f2
f ¼ aD<1
(2a)
forf ¼ a D 1
(2b)
for
5:34 2:31 þ 8:39f 3:44 þ f f2
Ksf ¼ 8:98 þ
5:61 1:99 3 f2 f
for
for
f ¼ aD<1
f ¼ aD 1 =
(2c)
(2d)
where Kss is the ESB coefficient of plates with simply-supported conditions along all four edges, Ksf is the ESB coefficient of plates with two opposite simply-supported edges and two other fixed edges, f ¼ a/D is the web panel aspect ratio, and a is the transverse stiffeners' distance relative to each other. In Basler's model [17,21,26], the ESB coefficient of a web panel with four edges simply-supported conditions is obtained as Eqs. (2a) and (2b). To have a simple equation, AISC 360-16 [28] suggest a cover curve as described by Eq. (2e) to calculate the ESB coefficient of a stiffened web plate with transverse stiffeners. The real boundary condition of web panel was found to be closer to the clamped case in the range of practical design parameters of plate girders [29]. In these cases, Lee et al. [29] proposed Eqs. (2f) to (2g) to calculate the ESB coefficient. For unstiffened long web panels with a/D > 3, the ESB strength is obtained assuming the ESB coefficient KAISC ¼ 5.34 in accordance with AISC 360-16 [28] (Article G2-5).
KAISC ¼ 5 þ
5 f2
KLee ¼ Kss þ
tf 4 2 Ksf Kss 1 2 5 3 tw
for f ¼ a D 3
(2e)
=
Many studies have investigated the elastic shear buckling (ESB) behavior of plates [1,3,17,25,26]. Based on these researches, the ESB strength of unstiffened web panels at ambient temperature, Vcr,20, subjected to pure shear loading, is obtained using the classical plate buckling equation as:
p2 E20 t 3w 12 1 y2 D
=
1.3. Elastic shear strength of unstiffened web panels at ambient temperature
Vcr;20 ¼ K
=
the impact of thermal restraint. The same researchers presented an analytical approach to evaluate the deflection of web plates with simply-supported boundary conditions before and after buckling, backed up using empirical and finite element approaches [10]. A recent study by Aziz and Kodur [11] on steel bridge girders employed a nonlinear FEM method to estimate their residual strength, indicating that the highest temperature of fire (along with the corresponding steel temperature) is the most influential parameter. In another study on thin steel plates, Salminen [12] benefited from FEM to propose an analytical method capable of estimating the shear resistance at uneven high temperatures. The shear buckling characteristics of plate girder webs are among another topics of interest for researchers. The results demonstrate that one is not allowed to assume elastic-perfectly plastic behavior in FEM at high temperatures as this yields incorrect outputs compared to nonlinear models [13]. Scandella et al. [14] showed that the failure of steel plate girders as a result of shear web buckling when exposed to fire is a probable phenomenon for those parts with localized flange buckling caused by bending torques. Furthermore, the impact of shear on the behavior of I-girders when exposed to fire was examined for various values of web slenderness and excitation alongside the inspection of insulation on the response. Shear was determined to be the main failure mode when fire occurs. A streamlined method was also developed to investigate shear capacity degradation for beams made of steel and composite materials under fire. This approach considers various technical aspects for evaluation of shear capacity such as the temperature-dependent degradation of components [15,16]. This research, however, suffered from using iteration method and its second figure that was used to compute the ratio of critical web slenderness was obtained only for a single steel grade with respect to the slenderness parameter which was incapable of considering the impact of boundary conditions on the web panel's shear behavior [1e3,17]. In a study on beams of Class 1, Quan et al. [18] developed a theoretical model for analyzing the shear buckling, either at elevated or normal temperatures. The exactness of the model was verified using FEM. The shear-bending behavior of homogenous I-girders when exposed to fire aided by numerical method was the subject of another study by Reis et al. [19]. They showed that the EC3 [20] recommendation for they safety assessment of the studied problem is suitable for design purposes if stress-strain reduction factors of Ref. [21] are directly applied at high temperatures. For steel plate girders, the same researchers also provided novel design criteria so as to estimate the ultimate shear strength when fire occurs [22]. Using a numerical approach, Reis et al. [23] examined the effect of several parameters including residual stress, boundary conditions and geometric imperfection on the USS of steel plate girders under shear buckling for different conditions. Using experimental and numerical methods, Glassman et al. [24] studied the shear buckling behavior of web plate girders exposed to fire. They showed that the amount of post-buckling shear strength for a flange with simply-supported boundary conditions lies somewhere between 19% and 29% of approaches based on the explicit flange model. They also showed that the buckling coefficients and consequently critical buckling loads decrease with increasing aspect ratio.
=
2
for 0:5
tf <2 tw (2f)
KLee ¼ Kss þ
4 Ksf Kss 5
for
tf 2 tw
(2g)
1.4. Ultimate shear strength of unstiffened web panels at ambient temperature According to AISC specifications (Chapter G) [28] on the ultimate shear strength (USS) of unstiffened web plates at ambient temperatures (20ºCe100 C), Vu,20 ¼ 4vVn,20 is found as Eq. (3) with a shear resistance factor of 4v ¼ 0.9.
Vn;20 ¼ 0:6syw;20 h tw for lw;20 1:1 Vn;20 ¼ 0:6syw;20 h tw
1:1
lw;20
for lw;20 > 1:1
(3a)
(3b)
G. Pourmoosavi et al. / Journal of Constructional Steel Research 164 (2020) 105808
lw;20 ¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi syw;20 D pffiffiffiffiffiffiffiffiffiffiffiffi tw K E20
3
2. Materials and methods
(3c)
where Vn,20 is the nominal shear strength at 20 C, syw,20 is the yield stress of web plate at 20 C, h signifies the overall depth of girder, tw and D are, respectively, the thickness and depth of web plate, E20 is the Young's modulus at 20 C. Also, lw,20 represents the nondimensional web slenderness (NWS) parameter at 20 C, and K is the coefficient of elastic shear buckling taken equal to 5.34 for simply-supported web panels across all edges. One should note that, based on the NWS parameter, web panels can be classified as slender, non-compact, and compact plates [1e3,5,30]. Accordingly, plates with lw,20 1.1 and lw,20 > 1.37 are considered compact and slender web panels, respectively. In addition, plates with 1.1 < lw,20 1.37 are considered non-compact web panels with inelastic shear buckling failure mode. 1.5. Aims and scopes It seems necessary to develop a simplified design approach in order to evaluate the shear strength of web panels under fire condition as the size of furnaces is limited, the cost of experiments aimed at testing the fire resistance of structures is quite high [23] and access to simulation software packages such as ANSYS and ABAQUS is not always guaranteed. It was shown that inelastic or elastic buckling of steel plate webs, that experience plastic buckling at normal temperatures, is a possible phenomenon at elevated temperatures owing to the deterioration of mechanical properties [2], i.e. a change in the failure mode is to be expected. There seems to be no theoretical research aimed at evaluating the web panel mode change that provides general shear design curves of unstiffened steel plate girders exposed to fire. This study seeks to fill this gap. In this regard, web panel shear design relationships of AISC360-16 specifications are exploited to be used in fire conditions. To this end, the stress-strain reduction factors provided in EN 19931-2 are directly applied. Afterwards, the design curves are proposed for prediction of the ultimate shear strength, failure mode, and limiting temperature of steel plate girders under fire by taking into account the strength degradation caused by high temperatures and the effects due to sectional instability. It must be noted that, limiting temperature is a temperature at which the web panel would fail when a constant shear load is exerted. The validation of the proposed curves is evaluated by comparing the results with numerical and experimental data.
2.1. Specifications of steel plate girder models In order to investigate the shear behavior of unstiffened web panels under pure shear at both ambient and elevated temperatures, 54 models with slender (S), non-compact (NC) and compact (C) webs at three temperatures of 20 C, 400 C, and 600 C were constructed using the finite element method (FEM) software package, ABAQUS. Selecting a wide range of thickness and NWS parameters provides the possibility for studying the elastic/plastic buckling behavior of plates [1]. The specifications of FEM models are presented in Table 1. The flange width, web depth, and the ratio of the flange thickness to the web thickness are fixed at bf ¼ 300 mm, D ¼ 1000 mm, and tf/tw ¼ 2, respectively. The models are named by PGi e j, in which i is the web panel aspect ratio (a/D), and j is the plate number that represents the web plate thickness. As an example, PG4-2 represents the PG4 girder with a girder length of a ¼ 4000 mm, an aspect ratio of a/D ¼ 4, and slender web plate with thickness of tw ¼ 10 mm. As another example, PG5-6 represents the PG5 girder with an aspect ratio of a/D ¼ 5, and compact web plate with thickness of tw ¼ 20 mm. Loading pattern and boundary conditions of the FEM models are shown in Fig. 1. Shear loads are applied to transverse edges and flanges using the shell edge load and surface traction load commands of ABAQUS, respectively. Both transverse edges of 1 and 2 are assumed to be simply supported. In order to consider realistic boundary conditions for web-flange junctures (edges 7 and 8), the top and bottom flanges are incorporated into FEM models [31]. In addition, to eliminate the lateral torsional buckling phenomenon, the entire mid-nodes of flanges are constrained in the y direction [1]. 2.2. Material properties In this article, the stress-strain reduction factors provided in EN 1993-1-2 [21] specific to steel for elevated temperatures are employed to consider fire effects. In fact, the design criteria at ambient temperatures are modified using the reduction factors such that they could be exploited for fire conditions [2,13,19]. At ambient temperature, yield stress, Young's modulus and Poisson's ratio of steel plates are considered 300 MPa, 200 GPa, and 0.3, respectively. Fig. 2 displays the curves of Eurocode reduction factors related to yield stress (sy), Young's modulus (E) and proportional limit stress (sp), ky,T, kE,T and kp,T, respectively with respect to temperature, where the superscript T refers to high temperature values.
Table 1 Specifications of the FEM models. Model
a/D
Temperature (ºC)
tw (mm)
tf (mm)
lw,20
Web plate classificationa
PG4-1 PG4-2 PG4-3 PG4-4 PG4-5 PG4-6 PG5-1 PG5-2 PG5-3 PG5-4 PG5-5 PG5-6 PG6-1 PG6-2 PG6-3 PG6-4 PG6-5 PG6-6
4
20, 400, 600
5
20, 400, 600
6
20, 400, 600
8 10 12 14 16 20 8 10 12 14 16 20 8 10 12 14 16 20
16 20 24 28 32 40 16 20 24 28 32 40 16 20 24 28 32 40
2.095 1.676 1.397 1.197 1.048 0.838 2.095 1.676 1.397 1.197 1.048 0.838 2.095 1.676 1.397 1.197 1.048 0.838
S S NC NC C C S S NC NC C C S S NC NC C C
a
S: Slender; NC: Non-Compact; C: Compact.
4
G. Pourmoosavi et al. / Journal of Constructional Steel Research 164 (2020) 105808
Fig. 3. Material properties of FEM models at both ambient and elevated temperatures.
Fig. 1. Boundary condition and pure shear loading of models in the FEM analysis.
Fig. 4. Convergence study and mesh sensitivity in the finite element models.
model with slender web panel which is more sensitive to mesh size than non-compact and compact web panels. Moreover, the element size of 20 mm was taken as the optimum requirement in the analysis with a maximum error percentage of 0.49%. Fig. 2. Stress-strain reduction factors of steel at high temperatures.
The stress-strain curves of steel used in FEM are shown in Fig. 3. It must be noted that strain hardening was not taken into account and the Von Mises yield criterion is used to evaluate the material nonlinearity. 2.3. Numerical modeling and validation 2.3.1. Numerical modeling The numerical modeling and analysis are performed using ABAQUS [32]. The S4R, four-node general-purpose linear shell element with reduced integration and hourglass control is used for modeling the girders. The modified Riks method is used for the nonlinear static analysis of unstiffened web panels. In this method, if the model is defined as a perfect plate without initial geometrical imperfections, the buckling phenomenon may not be observed and the system yields under a specific load [1,17]. Therefore, the first eigenmode of each girder is amplified with a very small magnitude of D/10000 and then employed as an initial girder shape for nonlinear analysis [31]. Afterwards, based on the eigenvalue and Riks analysis, the elastic shear buckling strength, Vcr, and ultimate shear strength, Vu, of each model is obtained, respectively. As shown in Fig. 4, concerning the mesh convergence study, a number of mesh sensitivity analyses are carried out based on the difference between the ESB load, Vcr, obtained via ABAQUS and the results of Basler's method using shear buckling coefficient proposed by Lee et al. (see Eq. (1)). This convergence study is done on the PG4-1
2.3.2. Verification of numerical modeling process Definitely the best way to corroborate any analytical analysis is to compare the results with experimental and numerical ones [17,23,33,34]. In 2007, Vimonsatit et al. [7] experimentally and numerically investigated 18 steel plate girders loaded predominantly in shear at both ambient and high temperatures of 400, 550, and 690 C. All simply supported test girders had a total length of 1.90 m. In this study, to verification of the numerical modeling process the test model TG3 was selected. The span measured from center-to-center of the support bearing blocks was 1.66 m. Other items including the geometrical and material properties, loading and boundary conditions are reported in Vimonsatit et al. [7]. The ultimate shear strength of the current finite element model and those of obtained by the test results is shown in Fig. 5. According to this figure, the USS of the FEM model at 400 and 690 C are equal to 67.41 and 16.64 kN, respectively. These results are close to 67.63 and 17.15 kN as USS predicted by the experimental work, with a maximum difference of 0.35% and 3%, respectively. Therefore, the proposed numerical modeling process has a good capability to predict the ultimate shear strength of steel plate girders at high temperatures. 3. The proposed shear design curves 3.1. Ultimate shear strength at elevated temperature Based on the direct application of the reduction factors of EN
G. Pourmoosavi et al. / Journal of Constructional Steel Research 164 (2020) 105808
5
3.2. Failure mode at elevated temperature
Fig. 5. Comparing the experimental and numerical analysis results.
1993-1-2 [21] for steel stress-strain relationship, the design shear strength of unstiffened web plates at elevated temperatures, Vu,T ¼ 4vVn,T, shall be determined as Eq. (4) with a shear resistance factor of 4v ¼ 0.9.
Vn;t kt;T 0:6syw;20 h tw
Vn;T ¼
for
lw;T 1:1
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1:1 ky;T kE;T 0:6syw;20 h tw
lw;20
(4a)
for
lw;T > 1:1
According to Eq. (4c), one notices that the limits of lw, 20 ¼ 1.1 and lw, 20 ¼ 1.37, which determine the web panel shear failure modes, change due to degradation of material properties under fire conditions. Variation of the NWS parameter in terms of steel temperature can be observed in Fig. 7. According to this figure, a compact plate at ambient temperatures (lw, 20 < 1.1) becomes a non-compact or slender plate at higher temperatures. As an example, plates with 0.827 < lw,20 < 1.1 at normal temperatures are prone to inelastic or plastic buckling at approximately 100e600 C. Furthermore, these plates are prone to elastic, inelastic, and plastic buckling at 600e750 C. On the other hand, these plates are collapsed as a result of plastic shear buckling at higher temperatures (T > 850 C). In general, if a structural designer intends to select a compact web panel with plastic shear buckling at every temperature value, it is necessary to choose a plate with lw, 20 < 0.827. However, in an economic design, one can choose the required NWS parameter according to this figure based on temperature ranges. As another example, plates with 1.1 < lw,20 < 1.37 are inclined to buckle (elastically or inelastically) at approximately 100e600 C. Also, these plates may experience elastic buckling at 600e750 C. In this case, if a designer has a maximum temperature criterion of 400 C, a web panel with 1.1 < lw,20 < 1.146 is suitable in order to avoid failure mode changes while offering inelastic shear buckling in all temperature ranges. 3.3. Limiting temperature
(4b) One
sffiffiffiffiffiffiffiffi ky;T lw;20 lw;T ¼ kE;T
of
the
most
important
parameters
needed
(4c)
where Vn,T is the nominal shear strength, and lw,T is the NWS parameter at constant temperature. Moreover, lw,20 is the NWS parameter at 20 C, and Vp,20 ¼ 0.6syw,20htw is the web plastic shear strength at ambient temperature. In this article, similar to the slenderness categorization of plates under the effect of shear loading at ambient temperatures, web panels with lw,T 1.1 and lw,T > 1.37 are considered as compact and slender plates, respectively. Also, web panels with 1.1 < lw,T 1.37 are considered to be non-compact plates with inelastic shear buckling failure mode. Therefore, according to Eq. (4), Fig. 6 can be used to determine the USS of unstiffened web panels at elevated temperatures in terms of web panel slenderness and steel temperature. As an example, the USS value of plates with lw,T 1.1 and lw,T > 1.1 at 600 C are Vu,T ¼ 0.423Vp,20 and Vu,T ¼ (0.37/lw,20)Vp,20, respectively.
Fig. 6. The web panel ultimate shear strength variation versus temperature.
Fig. 7. The variation of web panel slenderness parameter versus temperature.
Fig. 8. Web panels' shear strength reduction factor versus temperature.
for
6
G. Pourmoosavi et al. / Journal of Constructional Steel Research 164 (2020) 105808
understanding the structural behavior is the temperature it takes the structure to collapse. Nonetheless, an evaluation of the residual capacity of fire-exposed components is mandatory before application of load on structures. This helps the engineers with new techniques for adding appropriate structural members in a building or bridge. Therefore, in this section, Based on Eq. (4) and given that the NWS parameter at elevated temperatures (lw,T) is less than or greater than 1.1, the USS reduction factor (R ¼ Vu,T/Vu,20) of an unstiffened web panel can be derived from Fig. 8 in terms of steel temperature. It should be noted that Vu,T and Vu,20 are the USS values at elevated and normal temperatures, respectively. As an example, the USS reduction factor belonging to plates with lw,T 1.1 and lw,T > 1.1 at 400 C, are about 1 and 0.84, respectively. On the other hand, the proposed curves in Fig. 8 can be used to predict the limiting temperature, Tlim, at which the panel would fail when a constant shear load is exerted. For example, if, under a loading at ambient temperature, 60% of the shear capacity of an unstiffened web panel with an NWS parameter of lw,T 1.1 is used, the plate will reach its ultimate shear load at a temperature of 560 C. In other words, at a temperature of 560 C, this plate will reach its ultimate load by a shear strength reduction of 40%. 4. Validation of the proposed curves In this section, the accuracy of the developed method for evaluating the shear behavior of unstiffened web panels under fire conditions are examined by a comparison with the 1) current paper 54 plate girders with compact, non-compact and slender web panels, 2) the available eight experimental data [7] and, 3) the available four finite element analyses [16]. 4.1. Validation with current paper FEM data 4.1.1. Validation at ambient temperature In this section, the USS of unstiffened web panels is discussed at ambient temperatures. The USS strength of each model is obtained by the nonlinear Riks analysis using FEM. For example, Fig. 9 shows the failure mode shape corresponding to the PG4-1 model at 20 C. The USS strength of the finite element (Vu-FEM) and those of obtained by Fig. 6 is shown in Table 2. According to this table, the proposed curves are more accurate in compact plates with plastic shear buckling. However, by increasing the NWS parameter, the difference is increased. The maximum difference for compact, noncompact, and slender web plates is about 1.1%, 23%, and 28%, respectively.
Table 2 Ultimate shear strength at 20 C. Model
lw,20
Vu-FEM (1)
Eq. (4) or Fig. 6(2)
Ratio (2/1)
PG4-1 PG4-2 PG4-3 PG4-4 PG4-5 PG4-6 PG5-1 PG5-2 PG5-3 PG5-4 PG5-5 PG5-6 PG6-1 PG6-2 PG6-3 PG6-4 PG6-5 PG6-6
2.095 1.676 1.397 1.197 1.048 0.838 2.095 1.676 1.397 1.197 1.048 0.838 2.095 1.676 1.397 1.197 1.048 0.838
953.71 1523.96 2074.67 2431.23 2788.36 3535.08 941.55 1510.79 2072.31 2430.98 2783.09 3474.01 938.75 1507.15 2073.13 2428.85 2777.06 3470.02
702.25 1105.77 1604.56 2200.66 2757.89 3499.20 702.25 1105.77 1604.56 2200.66 2757.89 3499.20 702.25 1105.77 1604.56 2200.66 2757.89 3499.20
0.736 0.726 0.773 0.905 0.989 0.990 0.746 0.732 0.774 0.905 0.991 1.007 0.748 0.734 0.774 0.906 0.993 1.008
4.1.2. Validation at elevated temperature In this section, the USS of unstiffened web panels is discussed at elevated temperatures (400 C and 600 C). The USS of the finite element (Vu-FEM) and those of obtained by Fig. 6 are reported in Tables 3 and 4. According to this tables, the proposed curves are more accurate and in the safe region for compact plates. However, these methods lead to the unsafe condition of Ratio >1 for higher slenderness values. The difference at 400 C reaches to almost 3% and 7% for non-compact and slender web panels, respectively. In addition, by increasing the temperature to 600 C, especially for slender plates, proposed curves yield values that are more nonconservative for USS such that the difference is about 11%.
4.2. Validation with available experimental data In 2007, Nanyang Technological University conducted an experimental study on the shear buckling of steel plate girders at elevated and normal temperatures [7]. For the elevated temperature case, a uniform temperature was applied until the beam reached the desired temperature. Shear load was then applied slowly until the panel failed. In this article, in order to evaluate Eq. (4), two test models of TG3 and TG4 are utilized and the results based on Eq. (4) are compared with those of Ref. [7]. Table 5 lists the dimensions and material properties of the described girders. The
Table 3 Ultimate shear strength at 400 C.
Fig. 9. Failure mode shape of PG4-1model at 20 C.
Model
lw,400
Vu-FEM (1)
Eq. (4) or Fig. 6 (2)
Ratio (2/1)
PG4-1 PG4-2 PG4-3 PG4-4 PG4-5 PG4-6 PG5-1 PG5-2 PG5-3 PG5-4 PG5-5 PG5-6 PG6-1 PG6-2 PG6-3 PG6-4 PG6-5 PG6-6
2.504 2.003 1.669 1.431 1.252 1.002 2.504 2.003 1.669 1.431 1.252 1.002 2.504 2.003 1.669 1.431 1.252 1.002
560.83 868.78 1303.09 1797.33 2741.76 3444.49 555.22 864.91 1301.48 1795.23 2338.42 3427.82 565.59 865.57 1303.97 1797.25 2337.15 3525.11
587.55 925.16 1342.47 1841.20 2423.06 3499.20 587.55 925.16 1342.47 1841.20 2423.06 3499.20 587.55 925.16 1342.47 1841.20 2423.06 3499.20
1.048 1.065 1.030 1.024 0.884 1.016 1.058 1.070 1.031 1.026 1.036 1.021 1.039 1.069 1.030 1.024 1.037 0.993
G. Pourmoosavi et al. / Journal of Constructional Steel Research 164 (2020) 105808 Table 4 Ultimate shear strength at 600 C.
Table 7 Comparing the results of the proposed curves and the results of Ref. [16].
Model
lw,600
Vu-FEM (1)
Eq. (4) or Fig. 6 (2)
Ratio (2/1)
PG4-1 PG4-2 PG4-3 PG4-4 PG4-5 PG4-6 PG5-1 PG5-2 PG5-3 PG5-4 PG5-5 PG5-6 PG6-1 PG6-2 PG6-3 PG6-4 PG6-5 PG6-6
2.580 2.064 1.720 1.474 1.290 1.032 2.580 2.064 1.720 1.474 1.290 1.032 2.580 2.064 1.720 1.474 1.290 1.032
275.73 381.32 565.63 798.08 1256.78 1598.13 288.18 380.85 566.54 799.22 1058.06 1594.36 242.85 382.13 569.92 802.70 1057.84 1682.60
268.05 422.08 612.47 840.01 1105.46 1644.62 268.05 422.08 612.47 840.01 1105.46 1644.62 268.05 422.08 612.47 840.01 1105.46 1644.62
0.972 1.107 1.083 1.053 0.880 1.029 0.930 1.108 1.081 1.051 1.045 1.032 1.104 1.105 1.075 1.046 1.045 0.977
results of the analysis performed on the selected girders at normal and elevated temperature can be seen in Table 6. As observed, the curves of Fig. 6 are capable of properly predicting the USS with an acceptable accuracy. 4.3. Validation with available FEM data In 2014, Kodur and Naser [16] evaluated the shear capacity of fire-exposed steel and composite beams. For their study, four noncomposite steel beams constructed from hot-rolled sections, specifically W16 26, W18 40, W24 55, and W40 167, were picked. The failure temperatures of beams (failure occurs when shear capacity drops below an assumed loading level of 40% of shear capacity) are provided in Table 7. To obtain the limiting temperature, Tlim, according to Fig. 8, one should note that web panels with lw,T 1.1 and lw,T > 1.1 reached a shear strength reduction factor of R ¼ 40% at a temperature of about 600 C. Therefore, the limiting temperature, Tlim, can easily be obtained. To have a more accurate estimate, according to Fig. 7, the web panel NWS parameter in all selected beams under fire condition is greater than 1.1 at 600 C. For instance, for W18 40 section, the NWS parameter is smaller than 1.1 for T < 300 C. In other words, for this beam with lw,20 ¼ 0.961 at ambient temperature, the NWS parameter of 1.1 (kE/ky)0.5 under fire condition will be smaller than Table 5 Dimensions and properties of specimens tested at elevated temperatures. Girder
h (mm)
tf (mm)
D (mm)
tw (mm)
D/tw
lw,20
syw,20 (MPa)
TG3 TG5
317 317
6 6
305 305
2 1.5
152.5 203.33
2.5 2.99
287.8 332
Table 6 Comparison between the results of Eq. (4) and Ref. [7]. Girder
Temperature (ºC)
lw,T
Vu (kN) Eq. (4) or Fig. 6 (1)
Vcr (kN) Test [7] (2)
Ratio (1)/(2)
TG3
20 400 565 690 20 400 550 700
2.5 2.99 2.97 3.23 2.99 3.58 3.51 3.97
43.3 36.22 21.11 8.38 21.86 18.29 11.66 3.78
53.35 30.08 19.87 7.05 21.05 17.63 13 4.5
0.81 1.20 1.06 1.19 1.04 1.04 0.90 0.84
TG5
7
Steel section
h (mm)
tf (mm)
D (mm)
tw (mm)
lw,20
Tlim (1)
TFEM [16] (2)
Ratio (1)/ (2)
W16 26 W18 40 W24 55 W40 167
398.78 454.66 599.44 980.44
8.76 13.335 12.83 26.16
381.25 427.99 573.78 928.12
6.35 8 10.03 16.51
1.079 0.961 1.028 1.010
594 594 594 594
595 617 604 610
1.00 0.96 0.98 0.97
0.961 for a temperature ranges about T > 300 C. Therefore, based on Fig. 8, for all selected beams with lw,T > 1.1, the value of Tlim is obtained equal to 594 C which is very close to the finite element values with a maximum error percentage of 3.72%. 5. Conclusions In the present work, the ultimate shear behavior of unstiffened steel web panels was investigated at elevated temperatures. In this regard, shear design relationships of AISC 360-16 specifications were adapted to fire conditions using the direct application of the proposed stress-strain reduction factors in EN 1993-1-2 for steel at high temperatures. Next, analytical equations and design curves were proposed to predict the ultimate shear strength, failure mode, and limiting temperature under a constant shear load at which the panel would fail. The validation of the proposed curves was evaluated by comparing the results with numerical and experimental data. In each section, engineering design examples were also presented. Such assessment is beneficial for developing novel strategies so that structural members can be retrofitted in buildings and bridges. Appendix A. Supplementary data Supplementary data to this article can be found online at https://doi.org/10.1016/j.jcsr.2019.105808. References [1] V. Broujerdian, P. Mahyar, A. Ghadami, Effect of curvature and aspect ratio on shear resistance of unstiffened plates, J. Constr. Steel Res. 112 (2015), https:// doi.org/10.1016/j.jcsr.2015.04.025. [2] A. Ghadami, V. Broujerdian, Flexureeshear interaction in hybrid steel I-girders at ambient and elevated temperatures, Adv. Struct. Eng. 22 (6) (2018) 1501e1516, https://doi.org/10.1177/1369433218817893. [3] A. Gheitasi, M.M. Alinia, Slenderness classification of unstiffened metal plates under shear loading, Thin-Walled Struct. 48 (2010) 508e518, https://doi.org/ 10.1016/j.tws.2010.02.004. [4] M.M. Alinia, H.R. Habashi, A. Khorram, Nonlinearity in the postbuckling behaviour of thin steel shear panels, Thin-Walled Struct. 47 (2009) 412e420, https://doi.org/10.1016/j.tws.2008.09.004. [5] M.M. Alinia, A. Gheitasi, S. Erfani, Plastic shear buckling of unstiffened stocky plates, J. Constr. Steel Res. 65 (2009) 1631e1643, https://doi.org/10.1016/ j.jcsr.2009.04.001. [6] A. Reis, N. Lopes, P.V. Real, Ultimate shear strength of steel plate girders at normal and fire conditions, Thin-Walled Struct. 137 (2019) 318e330. [7] V. Vimonsatit, K.H. Tan, S.K. Ting, Shear strength of plate girder web panel at elevated temperature, J. Constr. Steel Res. 63 (2007) 1442e1451, https:// doi.org/10.1016/J.JCSR.2007.01.002. [8] B. British, Standards Institution, Structural Use of Steelwork in Building: Code of Practice for Design-Rolled and Welded Sections, BSI, 1990. [9] K. Tan, Z. Qian, Experimental behaviour of a thermally restrained plate girder loaded in shear at elevated temperature, J. Constr. Steel Res. 64 (2007) 596e606, https://doi.org/10.1016/j.jcsr.2007.12.008. [10] Z. Qian, K. Tan, Deflection behaviour of plate girders loaded in shear at elevated temperatures, J. Constr. Steel Res. 65 (4) (2009) 991e1000. [11] E. Aziz, V. Kodur, An approach for evaluating the residual strength of fire exposed bridge girders, J. Constr. Steel Res. 88 (2013) 34e42. [12] M. Salminen, M. Heinisuo, Numerical analysis of thin steel plates loaded in shear at non-uniform elevated temperatures, J. Constr. Steel Res. 97 (2014) 105e113, https://doi.org/10.1016/j.jcsr.2014.02.002. [13] M.E.M. Garlock, J.D. Glassman, Elevated temperature evaluation of an existing steel web shear buckling analytical model, J. Constr. Steel Res. 101 (2014)
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395e406, https://doi.org/10.1016/j.jcsr.2014.05.021. [14] C. Scandella, M. Knobloch, M. Fontana, Numerical analysis on the fire behaviour of steel plate girders, in: Prog. Saf. Struct. Fire Proc. 8th Int. Conf. Struct. Fire, Tongji University Press, 2014, pp. 105e112. [15] V.K.R. Kodur, M.Z. Naser, Effect of shear on fire response of steel beams, J. Constr. Steel Res. 97 (2014) 48e58, https://doi.org/10.1016/ j.jcsr.2014.01.018. [16] V.K.R. Kodur, M.Z. Naser, Approach for shear capacity evaluation of fire exposed steel and composite beams, J. Constr. Steel Res. 141 (2018) 91e103. [17] A. Ghadami, V. Broujerdian, Shear behavior of steel plate girders considering variations in geometrical properties, J. Constr. Steel Res. 153 (2019) 567e577, https://doi.org/10.1016/j.jcsr.2018.11.009. [18] G. Quan, S. Huang, I. Burgess, An analytical approach to modelling shear panels in steel beams at elevated temperatures, Eng. Struct. 85 (2015) 73e82. [19] A. Reis, N. Lopes, P. Real, Shear-bending interaction in steel plate girders subjected to elevated temperatures, Thin-Walled Struct. 104 (2016) 34e43, https://doi.org/10.1016/j.tws.2016.03.005. [20] EN 1993-1-5, Eurocode3, Design of Steel Structures - Part 1-5: Plated Structural Elements, European Committee for Standardisation (CEN), 2006. [21] EN 1993-1-2, Eurocode3, Design of Steel Structures - Part 1-2: General Rules Structural Fire Design, European Committee for Standardisation (CEN), 2005. [22] A. Reis, N. Lopes, P.V. Real, Numerical study of steel plate girders under shear loading at elevated temperatures, J. Constr. Steel Res. 117 (2016) 1e12, https://doi.org/10.1016/j.jcsr.2015.10.001. [23] A. Reis, N. Lopes, E. Real, P. Real, Numerical modelling of steel plate girders at normal and elevated temperatures, Fire Saf. J. 86 (2016) 1e15, https://doi.org/ 10.1016/j.firesaf.2016.08.005. [24] J. Glassman, M. Garlock, E. Aziz, V. Kodur, Modeling parameters for predicting the postbuckling shear strength of steel plate girders, J. Constr. Steel Res. 121
(2016) 136e143, https://doi.org/10.1016/j.jcsr.2016.01.004. [25] M.M. Alinia, A. Gheitasi, M. Shakiba, Postbuckling and ultimate state of stresses in steel plate girders, Thin-Walled Struct. 49 (4) (2011) 455e464, https://doi.org/10.1016/j.tws.2010.12.008. [26] M. Amani, B. Edlund, M.M. Alinia, Buckling and postbuckling behavior of unstiffened slender curved plates under uniform shear, Thin-Walled Struct. 49 (8) (2011) 1017e1031, https://doi.org/10.1016/j.tws.2011.03.007. [27] S.C. Lee, C.H. Yoo, Strength of plate girder web panels under pure shear, J. Struct. Eng. 124 (1998) 184e194, https://doi.org/10.1061/(ASCE)07339445(1998)124:2(184). [28] ANSI/AISC 360-16, Specification for Structural Steel Buildings, American. Inst. Steel Constr. (2016), 1e612. [29] S.C. Lee, J.S. Davidson, C.H. Yoo, Shear buckling coefficients of plate girder web panels, Comput. Struct. 59 (1996) 789e795, https://doi.org/10.1016/00457949(95)00325-8. [30] M. Amani, M.M. Alinia, M. Fadakar, Imperfection sensitivity of slender/stocky metal plates, Thin-Walled Struct. 73 (2013) 207e215, https://doi.org/10.1016/ J.TWS.2013.08.010. [31] S.C. Lee, D.S. Lee, C.H. Yoo, Ultimate shear strength of long web panels, J. Constr. Steel Res. 64 (2008) 1357e1365, https://doi.org/10.1016/ J.JCSR.2008.01.023. [32] Abaqus 6.14. ABAQUS Analysis User's Manual, ABAQUS Inc., 2016. [33] V. Broujerdian, A. Ghamari, A. Ghadami, An investigation into crack and its growth on the seismic behavior of steel shear walls, Thin-Walled Struct. 101 (2016), https://doi.org/10.1016/j.tws.2015.09.016. [34] M. Prachar, M. Jandera, F. Wald, B. Zhao, Lateral torsional-buckling of class 4 steel plate beams at elevated temperature: experimental and numerical comparison, J. Struct. Fire Eng. 6 (2015) 223e236.
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Shear design curves of unstiffened plate girder web panels at high temperatures Gh Pourmoosavi a, S.A. Mousavi Ghasemi a, *, B. Farahmand Azar b, S. Talatahari b, c a
Department of Civil Engineering, Tabriz Branch, Islamic Azad University, Tabriz, Iran Department of Civil Engineering, University of Tabriz, Tabriz, Iran c Engineering Faculty, Near East University, North Cyprus, 10 Mersin, Turkey b
a r t i c l e i n f o
a b s t r a c t
Article history: Received 19 May 2019 Received in revised form 10 August 2019 Accepted 12 October 2019 Available online xxx
Owing to the lack of an analytical research on the evaluation of mode change of web panels in addition to the need for shear design curves of unstiffened plate girder web panels exposed to shear buckling especially at elevated temperatures, the present article considers the mentioned topics. To this end, the web panel shear design relationships mentioned in AISC360-16 are modified to be used in fire conditions. This is achieved by direct utilization of steel stress-strain reduction factors in EN1993-1-2 at elevated temperatures. Analytical equations and design curves are developed to estimate the ultimate shear strength, failure mode and critical limiting temperature for the web panels under the action of a specific shear load. The results based on the curves are compared to the findings of current paper numerical analysis, existing experimental and numerical studies, indicating a good agreement between the results. © 2019 Elsevier Ltd. All rights reserved.
Keywords: Fire condition Shear design Web panel Slenderness Design curve
1. Introduction
1.2. Literature review
1.1. Background
There have been extensive studies on the shear behavior of steel plates when the temperature is considered as normal. Fire design is another important aspect that should be addressed separately [6]. Steel structure and lifespan are affected by fire, and the shear buckling of web panels in addition to failure modes of girders by considering fire as a parameter has been an interesting subject in recent years [2]. This has been the primary aim of some studies. With regard to plate girders at elevated temperatures, a theoretical model was proposed by Vimonsatit et al. [7] to estimate the failure loads when a mixture of axial and shear or simply shear stresses are applied. The developed analysis could consider the fire resistance with respect to shear strength associated with isothermal criterion, or limiting temperature for known values of axial thermal restriction and constant shear force. The outputs indicated that web buckling becomes the dominant failure mode at such temperatures. Noteworthy is that the authors extended the normal temperature relations proposed in the BS 5950: Part 1 standard [8] so as to study axial stress and material conditions at elevated temperatures. In an empirical study under fire condition, Tan and Qian [9] investigated the shear characteristics of a number of steel plate girders with simply-supported boundary conditions and thermal restraints that reduced the ultimate shear strength (USS). The higher the thermal restraint and the more slender the web plate, the more significant
Steel plate I-girders have extensively been employed in the construction of bridges and buildings. A conventional method of acquiring notable strength-to-weight ratios is to consider deep slender webs for I-girders, although shear buckling is a threat to slender web plates [1e3]. By subjecting a plate to in-plane shear, failure is expected prior to or after the proportional limit of material [3]. Elastic buckling of unstiffened, slender web plates at the onset of exerting the load as well as different types of nonlinearity in the post-buckling state can be anticipated [4]. Yielding before the occurrence of buckling, however, happens for compact and short web plates. Simultaneous geometrical instability and material yielding are notable phenomena in non-compact web plates [5]. To gain a better understanding of the shear characteristics of unstiffened web plates, the following review is provided which is mainly aimed at studies at elevated temperatures.
* Corresponding author. E-mail address: [email protected] (S.A.M. Ghasemi). https://doi.org/10.1016/j.jcsr.2019.105808 0143-974X/© 2019 Elsevier Ltd. All rights reserved.
G. Pourmoosavi et al. / Journal of Constructional Steel Research 164 (2020) 105808
(1)
where E20 is the Young's modulus at 20 C, ʋ is the Poisson's ratio, tw and D are the web plate thickness and depth, respectively, and K is the ESB coefficient which is dependent on the boundary conditions and aspect ratio of the plate [13,27] and is found as.
Kss ¼ 4 þ
5:34 f2
Kss ¼ 5:34 þ
Ksf ¼
4 f2
f ¼ aD<1
(2a)
forf ¼ a D 1
(2b)
for
5:34 2:31 þ 8:39f 3:44 þ f f2
Ksf ¼ 8:98 þ
5:61 1:99 3 f2 f
for
for
f ¼ aD<1
f ¼ aD 1 =
(2c)
(2d)
where Kss is the ESB coefficient of plates with simply-supported conditions along all four edges, Ksf is the ESB coefficient of plates with two opposite simply-supported edges and two other fixed edges, f ¼ a/D is the web panel aspect ratio, and a is the transverse stiffeners' distance relative to each other. In Basler's model [17,21,26], the ESB coefficient of a web panel with four edges simply-supported conditions is obtained as Eqs. (2a) and (2b). To have a simple equation, AISC 360-16 [28] suggest a cover curve as described by Eq. (2e) to calculate the ESB coefficient of a stiffened web plate with transverse stiffeners. The real boundary condition of web panel was found to be closer to the clamped case in the range of practical design parameters of plate girders [29]. In these cases, Lee et al. [29] proposed Eqs. (2f) to (2g) to calculate the ESB coefficient. For unstiffened long web panels with a/D > 3, the ESB strength is obtained assuming the ESB coefficient KAISC ¼ 5.34 in accordance with AISC 360-16 [28] (Article G2-5).
KAISC ¼ 5 þ
5 f2
KLee ¼ Kss þ
tf 4 2 Ksf Kss 1 2 5 3 tw
for f ¼ a D 3
(2e)
=
Many studies have investigated the elastic shear buckling (ESB) behavior of plates [1,3,17,25,26]. Based on these researches, the ESB strength of unstiffened web panels at ambient temperature, Vcr,20, subjected to pure shear loading, is obtained using the classical plate buckling equation as:
p2 E20 t 3w 12 1 y2 D
=
1.3. Elastic shear strength of unstiffened web panels at ambient temperature
Vcr;20 ¼ K
=
the impact of thermal restraint. The same researchers presented an analytical approach to evaluate the deflection of web plates with simply-supported boundary conditions before and after buckling, backed up using empirical and finite element approaches [10]. A recent study by Aziz and Kodur [11] on steel bridge girders employed a nonlinear FEM method to estimate their residual strength, indicating that the highest temperature of fire (along with the corresponding steel temperature) is the most influential parameter. In another study on thin steel plates, Salminen [12] benefited from FEM to propose an analytical method capable of estimating the shear resistance at uneven high temperatures. The shear buckling characteristics of plate girder webs are among another topics of interest for researchers. The results demonstrate that one is not allowed to assume elastic-perfectly plastic behavior in FEM at high temperatures as this yields incorrect outputs compared to nonlinear models [13]. Scandella et al. [14] showed that the failure of steel plate girders as a result of shear web buckling when exposed to fire is a probable phenomenon for those parts with localized flange buckling caused by bending torques. Furthermore, the impact of shear on the behavior of I-girders when exposed to fire was examined for various values of web slenderness and excitation alongside the inspection of insulation on the response. Shear was determined to be the main failure mode when fire occurs. A streamlined method was also developed to investigate shear capacity degradation for beams made of steel and composite materials under fire. This approach considers various technical aspects for evaluation of shear capacity such as the temperature-dependent degradation of components [15,16]. This research, however, suffered from using iteration method and its second figure that was used to compute the ratio of critical web slenderness was obtained only for a single steel grade with respect to the slenderness parameter which was incapable of considering the impact of boundary conditions on the web panel's shear behavior [1e3,17]. In a study on beams of Class 1, Quan et al. [18] developed a theoretical model for analyzing the shear buckling, either at elevated or normal temperatures. The exactness of the model was verified using FEM. The shear-bending behavior of homogenous I-girders when exposed to fire aided by numerical method was the subject of another study by Reis et al. [19]. They showed that the EC3 [20] recommendation for they safety assessment of the studied problem is suitable for design purposes if stress-strain reduction factors of Ref. [21] are directly applied at high temperatures. For steel plate girders, the same researchers also provided novel design criteria so as to estimate the ultimate shear strength when fire occurs [22]. Using a numerical approach, Reis et al. [23] examined the effect of several parameters including residual stress, boundary conditions and geometric imperfection on the USS of steel plate girders under shear buckling for different conditions. Using experimental and numerical methods, Glassman et al. [24] studied the shear buckling behavior of web plate girders exposed to fire. They showed that the amount of post-buckling shear strength for a flange with simply-supported boundary conditions lies somewhere between 19% and 29% of approaches based on the explicit flange model. They also showed that the buckling coefficients and consequently critical buckling loads decrease with increasing aspect ratio.
=
2
for 0:5
tf <2 tw (2f)
KLee ¼ Kss þ
4 Ksf Kss 5
for
tf 2 tw
(2g)
1.4. Ultimate shear strength of unstiffened web panels at ambient temperature According to AISC specifications (Chapter G) [28] on the ultimate shear strength (USS) of unstiffened web plates at ambient temperatures (20ºCe100 C), Vu,20 ¼ 4vVn,20 is found as Eq. (3) with a shear resistance factor of 4v ¼ 0.9.
Vn;20 ¼ 0:6syw;20 h tw for lw;20 1:1 Vn;20 ¼ 0:6syw;20 h tw
1:1
lw;20
for lw;20 > 1:1
(3a)
(3b)
G. Pourmoosavi et al. / Journal of Constructional Steel Research 164 (2020) 105808
lw;20 ¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi syw;20 D pffiffiffiffiffiffiffiffiffiffiffiffi tw K E20
3
2. Materials and methods
(3c)
where Vn,20 is the nominal shear strength at 20 C, syw,20 is the yield stress of web plate at 20 C, h signifies the overall depth of girder, tw and D are, respectively, the thickness and depth of web plate, E20 is the Young's modulus at 20 C. Also, lw,20 represents the nondimensional web slenderness (NWS) parameter at 20 C, and K is the coefficient of elastic shear buckling taken equal to 5.34 for simply-supported web panels across all edges. One should note that, based on the NWS parameter, web panels can be classified as slender, non-compact, and compact plates [1e3,5,30]. Accordingly, plates with lw,20 1.1 and lw,20 > 1.37 are considered compact and slender web panels, respectively. In addition, plates with 1.1 < lw,20 1.37 are considered non-compact web panels with inelastic shear buckling failure mode. 1.5. Aims and scopes It seems necessary to develop a simplified design approach in order to evaluate the shear strength of web panels under fire condition as the size of furnaces is limited, the cost of experiments aimed at testing the fire resistance of structures is quite high [23] and access to simulation software packages such as ANSYS and ABAQUS is not always guaranteed. It was shown that inelastic or elastic buckling of steel plate webs, that experience plastic buckling at normal temperatures, is a possible phenomenon at elevated temperatures owing to the deterioration of mechanical properties [2], i.e. a change in the failure mode is to be expected. There seems to be no theoretical research aimed at evaluating the web panel mode change that provides general shear design curves of unstiffened steel plate girders exposed to fire. This study seeks to fill this gap. In this regard, web panel shear design relationships of AISC360-16 specifications are exploited to be used in fire conditions. To this end, the stress-strain reduction factors provided in EN 19931-2 are directly applied. Afterwards, the design curves are proposed for prediction of the ultimate shear strength, failure mode, and limiting temperature of steel plate girders under fire by taking into account the strength degradation caused by high temperatures and the effects due to sectional instability. It must be noted that, limiting temperature is a temperature at which the web panel would fail when a constant shear load is exerted. The validation of the proposed curves is evaluated by comparing the results with numerical and experimental data.
2.1. Specifications of steel plate girder models In order to investigate the shear behavior of unstiffened web panels under pure shear at both ambient and elevated temperatures, 54 models with slender (S), non-compact (NC) and compact (C) webs at three temperatures of 20 C, 400 C, and 600 C were constructed using the finite element method (FEM) software package, ABAQUS. Selecting a wide range of thickness and NWS parameters provides the possibility for studying the elastic/plastic buckling behavior of plates [1]. The specifications of FEM models are presented in Table 1. The flange width, web depth, and the ratio of the flange thickness to the web thickness are fixed at bf ¼ 300 mm, D ¼ 1000 mm, and tf/tw ¼ 2, respectively. The models are named by PGi e j, in which i is the web panel aspect ratio (a/D), and j is the plate number that represents the web plate thickness. As an example, PG4-2 represents the PG4 girder with a girder length of a ¼ 4000 mm, an aspect ratio of a/D ¼ 4, and slender web plate with thickness of tw ¼ 10 mm. As another example, PG5-6 represents the PG5 girder with an aspect ratio of a/D ¼ 5, and compact web plate with thickness of tw ¼ 20 mm. Loading pattern and boundary conditions of the FEM models are shown in Fig. 1. Shear loads are applied to transverse edges and flanges using the shell edge load and surface traction load commands of ABAQUS, respectively. Both transverse edges of 1 and 2 are assumed to be simply supported. In order to consider realistic boundary conditions for web-flange junctures (edges 7 and 8), the top and bottom flanges are incorporated into FEM models [31]. In addition, to eliminate the lateral torsional buckling phenomenon, the entire mid-nodes of flanges are constrained in the y direction [1]. 2.2. Material properties In this article, the stress-strain reduction factors provided in EN 1993-1-2 [21] specific to steel for elevated temperatures are employed to consider fire effects. In fact, the design criteria at ambient temperatures are modified using the reduction factors such that they could be exploited for fire conditions [2,13,19]. At ambient temperature, yield stress, Young's modulus and Poisson's ratio of steel plates are considered 300 MPa, 200 GPa, and 0.3, respectively. Fig. 2 displays the curves of Eurocode reduction factors related to yield stress (sy), Young's modulus (E) and proportional limit stress (sp), ky,T, kE,T and kp,T, respectively with respect to temperature, where the superscript T refers to high temperature values.
Table 1 Specifications of the FEM models. Model
a/D
Temperature (ºC)
tw (mm)
tf (mm)
lw,20
Web plate classificationa
PG4-1 PG4-2 PG4-3 PG4-4 PG4-5 PG4-6 PG5-1 PG5-2 PG5-3 PG5-4 PG5-5 PG5-6 PG6-1 PG6-2 PG6-3 PG6-4 PG6-5 PG6-6
4
20, 400, 600
5
20, 400, 600
6
20, 400, 600
8 10 12 14 16 20 8 10 12 14 16 20 8 10 12 14 16 20
16 20 24 28 32 40 16 20 24 28 32 40 16 20 24 28 32 40
2.095 1.676 1.397 1.197 1.048 0.838 2.095 1.676 1.397 1.197 1.048 0.838 2.095 1.676 1.397 1.197 1.048 0.838
S S NC NC C C S S NC NC C C S S NC NC C C
a
S: Slender; NC: Non-Compact; C: Compact.
4
G. Pourmoosavi et al. / Journal of Constructional Steel Research 164 (2020) 105808
Fig. 3. Material properties of FEM models at both ambient and elevated temperatures.
Fig. 1. Boundary condition and pure shear loading of models in the FEM analysis.
Fig. 4. Convergence study and mesh sensitivity in the finite element models.
model with slender web panel which is more sensitive to mesh size than non-compact and compact web panels. Moreover, the element size of 20 mm was taken as the optimum requirement in the analysis with a maximum error percentage of 0.49%. Fig. 2. Stress-strain reduction factors of steel at high temperatures.
The stress-strain curves of steel used in FEM are shown in Fig. 3. It must be noted that strain hardening was not taken into account and the Von Mises yield criterion is used to evaluate the material nonlinearity. 2.3. Numerical modeling and validation 2.3.1. Numerical modeling The numerical modeling and analysis are performed using ABAQUS [32]. The S4R, four-node general-purpose linear shell element with reduced integration and hourglass control is used for modeling the girders. The modified Riks method is used for the nonlinear static analysis of unstiffened web panels. In this method, if the model is defined as a perfect plate without initial geometrical imperfections, the buckling phenomenon may not be observed and the system yields under a specific load [1,17]. Therefore, the first eigenmode of each girder is amplified with a very small magnitude of D/10000 and then employed as an initial girder shape for nonlinear analysis [31]. Afterwards, based on the eigenvalue and Riks analysis, the elastic shear buckling strength, Vcr, and ultimate shear strength, Vu, of each model is obtained, respectively. As shown in Fig. 4, concerning the mesh convergence study, a number of mesh sensitivity analyses are carried out based on the difference between the ESB load, Vcr, obtained via ABAQUS and the results of Basler's method using shear buckling coefficient proposed by Lee et al. (see Eq. (1)). This convergence study is done on the PG4-1
2.3.2. Verification of numerical modeling process Definitely the best way to corroborate any analytical analysis is to compare the results with experimental and numerical ones [17,23,33,34]. In 2007, Vimonsatit et al. [7] experimentally and numerically investigated 18 steel plate girders loaded predominantly in shear at both ambient and high temperatures of 400, 550, and 690 C. All simply supported test girders had a total length of 1.90 m. In this study, to verification of the numerical modeling process the test model TG3 was selected. The span measured from center-to-center of the support bearing blocks was 1.66 m. Other items including the geometrical and material properties, loading and boundary conditions are reported in Vimonsatit et al. [7]. The ultimate shear strength of the current finite element model and those of obtained by the test results is shown in Fig. 5. According to this figure, the USS of the FEM model at 400 and 690 C are equal to 67.41 and 16.64 kN, respectively. These results are close to 67.63 and 17.15 kN as USS predicted by the experimental work, with a maximum difference of 0.35% and 3%, respectively. Therefore, the proposed numerical modeling process has a good capability to predict the ultimate shear strength of steel plate girders at high temperatures. 3. The proposed shear design curves 3.1. Ultimate shear strength at elevated temperature Based on the direct application of the reduction factors of EN
G. Pourmoosavi et al. / Journal of Constructional Steel Research 164 (2020) 105808
5
3.2. Failure mode at elevated temperature
Fig. 5. Comparing the experimental and numerical analysis results.
1993-1-2 [21] for steel stress-strain relationship, the design shear strength of unstiffened web plates at elevated temperatures, Vu,T ¼ 4vVn,T, shall be determined as Eq. (4) with a shear resistance factor of 4v ¼ 0.9.
Vn;t kt;T 0:6syw;20 h tw
Vn;T ¼
for
lw;T 1:1
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1:1 ky;T kE;T 0:6syw;20 h tw
lw;20
(4a)
for
lw;T > 1:1
According to Eq. (4c), one notices that the limits of lw, 20 ¼ 1.1 and lw, 20 ¼ 1.37, which determine the web panel shear failure modes, change due to degradation of material properties under fire conditions. Variation of the NWS parameter in terms of steel temperature can be observed in Fig. 7. According to this figure, a compact plate at ambient temperatures (lw, 20 < 1.1) becomes a non-compact or slender plate at higher temperatures. As an example, plates with 0.827 < lw,20 < 1.1 at normal temperatures are prone to inelastic or plastic buckling at approximately 100e600 C. Furthermore, these plates are prone to elastic, inelastic, and plastic buckling at 600e750 C. On the other hand, these plates are collapsed as a result of plastic shear buckling at higher temperatures (T > 850 C). In general, if a structural designer intends to select a compact web panel with plastic shear buckling at every temperature value, it is necessary to choose a plate with lw, 20 < 0.827. However, in an economic design, one can choose the required NWS parameter according to this figure based on temperature ranges. As another example, plates with 1.1 < lw,20 < 1.37 are inclined to buckle (elastically or inelastically) at approximately 100e600 C. Also, these plates may experience elastic buckling at 600e750 C. In this case, if a designer has a maximum temperature criterion of 400 C, a web panel with 1.1 < lw,20 < 1.146 is suitable in order to avoid failure mode changes while offering inelastic shear buckling in all temperature ranges. 3.3. Limiting temperature
(4b) One
sffiffiffiffiffiffiffiffi ky;T lw;20 lw;T ¼ kE;T
of
the
most
important
parameters
needed
(4c)
where Vn,T is the nominal shear strength, and lw,T is the NWS parameter at constant temperature. Moreover, lw,20 is the NWS parameter at 20 C, and Vp,20 ¼ 0.6syw,20htw is the web plastic shear strength at ambient temperature. In this article, similar to the slenderness categorization of plates under the effect of shear loading at ambient temperatures, web panels with lw,T 1.1 and lw,T > 1.37 are considered as compact and slender plates, respectively. Also, web panels with 1.1 < lw,T 1.37 are considered to be non-compact plates with inelastic shear buckling failure mode. Therefore, according to Eq. (4), Fig. 6 can be used to determine the USS of unstiffened web panels at elevated temperatures in terms of web panel slenderness and steel temperature. As an example, the USS value of plates with lw,T 1.1 and lw,T > 1.1 at 600 C are Vu,T ¼ 0.423Vp,20 and Vu,T ¼ (0.37/lw,20)Vp,20, respectively.
Fig. 6. The web panel ultimate shear strength variation versus temperature.
Fig. 7. The variation of web panel slenderness parameter versus temperature.
Fig. 8. Web panels' shear strength reduction factor versus temperature.
for
6
G. Pourmoosavi et al. / Journal of Constructional Steel Research 164 (2020) 105808
understanding the structural behavior is the temperature it takes the structure to collapse. Nonetheless, an evaluation of the residual capacity of fire-exposed components is mandatory before application of load on structures. This helps the engineers with new techniques for adding appropriate structural members in a building or bridge. Therefore, in this section, Based on Eq. (4) and given that the NWS parameter at elevated temperatures (lw,T) is less than or greater than 1.1, the USS reduction factor (R ¼ Vu,T/Vu,20) of an unstiffened web panel can be derived from Fig. 8 in terms of steel temperature. It should be noted that Vu,T and Vu,20 are the USS values at elevated and normal temperatures, respectively. As an example, the USS reduction factor belonging to plates with lw,T 1.1 and lw,T > 1.1 at 400 C, are about 1 and 0.84, respectively. On the other hand, the proposed curves in Fig. 8 can be used to predict the limiting temperature, Tlim, at which the panel would fail when a constant shear load is exerted. For example, if, under a loading at ambient temperature, 60% of the shear capacity of an unstiffened web panel with an NWS parameter of lw,T 1.1 is used, the plate will reach its ultimate shear load at a temperature of 560 C. In other words, at a temperature of 560 C, this plate will reach its ultimate load by a shear strength reduction of 40%. 4. Validation of the proposed curves In this section, the accuracy of the developed method for evaluating the shear behavior of unstiffened web panels under fire conditions are examined by a comparison with the 1) current paper 54 plate girders with compact, non-compact and slender web panels, 2) the available eight experimental data [7] and, 3) the available four finite element analyses [16]. 4.1. Validation with current paper FEM data 4.1.1. Validation at ambient temperature In this section, the USS of unstiffened web panels is discussed at ambient temperatures. The USS strength of each model is obtained by the nonlinear Riks analysis using FEM. For example, Fig. 9 shows the failure mode shape corresponding to the PG4-1 model at 20 C. The USS strength of the finite element (Vu-FEM) and those of obtained by Fig. 6 is shown in Table 2. According to this table, the proposed curves are more accurate in compact plates with plastic shear buckling. However, by increasing the NWS parameter, the difference is increased. The maximum difference for compact, noncompact, and slender web plates is about 1.1%, 23%, and 28%, respectively.
Table 2 Ultimate shear strength at 20 C. Model
lw,20
Vu-FEM (1)
Eq. (4) or Fig. 6(2)
Ratio (2/1)
PG4-1 PG4-2 PG4-3 PG4-4 PG4-5 PG4-6 PG5-1 PG5-2 PG5-3 PG5-4 PG5-5 PG5-6 PG6-1 PG6-2 PG6-3 PG6-4 PG6-5 PG6-6
2.095 1.676 1.397 1.197 1.048 0.838 2.095 1.676 1.397 1.197 1.048 0.838 2.095 1.676 1.397 1.197 1.048 0.838
953.71 1523.96 2074.67 2431.23 2788.36 3535.08 941.55 1510.79 2072.31 2430.98 2783.09 3474.01 938.75 1507.15 2073.13 2428.85 2777.06 3470.02
702.25 1105.77 1604.56 2200.66 2757.89 3499.20 702.25 1105.77 1604.56 2200.66 2757.89 3499.20 702.25 1105.77 1604.56 2200.66 2757.89 3499.20
0.736 0.726 0.773 0.905 0.989 0.990 0.746 0.732 0.774 0.905 0.991 1.007 0.748 0.734 0.774 0.906 0.993 1.008
4.1.2. Validation at elevated temperature In this section, the USS of unstiffened web panels is discussed at elevated temperatures (400 C and 600 C). The USS of the finite element (Vu-FEM) and those of obtained by Fig. 6 are reported in Tables 3 and 4. According to this tables, the proposed curves are more accurate and in the safe region for compact plates. However, these methods lead to the unsafe condition of Ratio >1 for higher slenderness values. The difference at 400 C reaches to almost 3% and 7% for non-compact and slender web panels, respectively. In addition, by increasing the temperature to 600 C, especially for slender plates, proposed curves yield values that are more nonconservative for USS such that the difference is about 11%.
4.2. Validation with available experimental data In 2007, Nanyang Technological University conducted an experimental study on the shear buckling of steel plate girders at elevated and normal temperatures [7]. For the elevated temperature case, a uniform temperature was applied until the beam reached the desired temperature. Shear load was then applied slowly until the panel failed. In this article, in order to evaluate Eq. (4), two test models of TG3 and TG4 are utilized and the results based on Eq. (4) are compared with those of Ref. [7]. Table 5 lists the dimensions and material properties of the described girders. The
Table 3 Ultimate shear strength at 400 C.
Fig. 9. Failure mode shape of PG4-1model at 20 C.
Model
lw,400
Vu-FEM (1)
Eq. (4) or Fig. 6 (2)
Ratio (2/1)
PG4-1 PG4-2 PG4-3 PG4-4 PG4-5 PG4-6 PG5-1 PG5-2 PG5-3 PG5-4 PG5-5 PG5-6 PG6-1 PG6-2 PG6-3 PG6-4 PG6-5 PG6-6
2.504 2.003 1.669 1.431 1.252 1.002 2.504 2.003 1.669 1.431 1.252 1.002 2.504 2.003 1.669 1.431 1.252 1.002
560.83 868.78 1303.09 1797.33 2741.76 3444.49 555.22 864.91 1301.48 1795.23 2338.42 3427.82 565.59 865.57 1303.97 1797.25 2337.15 3525.11
587.55 925.16 1342.47 1841.20 2423.06 3499.20 587.55 925.16 1342.47 1841.20 2423.06 3499.20 587.55 925.16 1342.47 1841.20 2423.06 3499.20
1.048 1.065 1.030 1.024 0.884 1.016 1.058 1.070 1.031 1.026 1.036 1.021 1.039 1.069 1.030 1.024 1.037 0.993
G. Pourmoosavi et al. / Journal of Constructional Steel Research 164 (2020) 105808 Table 4 Ultimate shear strength at 600 C.
Table 7 Comparing the results of the proposed curves and the results of Ref. [16].
Model
lw,600
Vu-FEM (1)
Eq. (4) or Fig. 6 (2)
Ratio (2/1)
PG4-1 PG4-2 PG4-3 PG4-4 PG4-5 PG4-6 PG5-1 PG5-2 PG5-3 PG5-4 PG5-5 PG5-6 PG6-1 PG6-2 PG6-3 PG6-4 PG6-5 PG6-6
2.580 2.064 1.720 1.474 1.290 1.032 2.580 2.064 1.720 1.474 1.290 1.032 2.580 2.064 1.720 1.474 1.290 1.032
275.73 381.32 565.63 798.08 1256.78 1598.13 288.18 380.85 566.54 799.22 1058.06 1594.36 242.85 382.13 569.92 802.70 1057.84 1682.60
268.05 422.08 612.47 840.01 1105.46 1644.62 268.05 422.08 612.47 840.01 1105.46 1644.62 268.05 422.08 612.47 840.01 1105.46 1644.62
0.972 1.107 1.083 1.053 0.880 1.029 0.930 1.108 1.081 1.051 1.045 1.032 1.104 1.105 1.075 1.046 1.045 0.977
results of the analysis performed on the selected girders at normal and elevated temperature can be seen in Table 6. As observed, the curves of Fig. 6 are capable of properly predicting the USS with an acceptable accuracy. 4.3. Validation with available FEM data In 2014, Kodur and Naser [16] evaluated the shear capacity of fire-exposed steel and composite beams. For their study, four noncomposite steel beams constructed from hot-rolled sections, specifically W16 26, W18 40, W24 55, and W40 167, were picked. The failure temperatures of beams (failure occurs when shear capacity drops below an assumed loading level of 40% of shear capacity) are provided in Table 7. To obtain the limiting temperature, Tlim, according to Fig. 8, one should note that web panels with lw,T 1.1 and lw,T > 1.1 reached a shear strength reduction factor of R ¼ 40% at a temperature of about 600 C. Therefore, the limiting temperature, Tlim, can easily be obtained. To have a more accurate estimate, according to Fig. 7, the web panel NWS parameter in all selected beams under fire condition is greater than 1.1 at 600 C. For instance, for W18 40 section, the NWS parameter is smaller than 1.1 for T < 300 C. In other words, for this beam with lw,20 ¼ 0.961 at ambient temperature, the NWS parameter of 1.1 (kE/ky)0.5 under fire condition will be smaller than Table 5 Dimensions and properties of specimens tested at elevated temperatures. Girder
h (mm)
tf (mm)
D (mm)
tw (mm)
D/tw
lw,20
syw,20 (MPa)
TG3 TG5
317 317
6 6
305 305
2 1.5
152.5 203.33
2.5 2.99
287.8 332
Table 6 Comparison between the results of Eq. (4) and Ref. [7]. Girder
Temperature (ºC)
lw,T
Vu (kN) Eq. (4) or Fig. 6 (1)
Vcr (kN) Test [7] (2)
Ratio (1)/(2)
TG3
20 400 565 690 20 400 550 700
2.5 2.99 2.97 3.23 2.99 3.58 3.51 3.97
43.3 36.22 21.11 8.38 21.86 18.29 11.66 3.78
53.35 30.08 19.87 7.05 21.05 17.63 13 4.5
0.81 1.20 1.06 1.19 1.04 1.04 0.90 0.84
TG5
7
Steel section
h (mm)
tf (mm)
D (mm)
tw (mm)
lw,20
Tlim (1)
TFEM [16] (2)
Ratio (1)/ (2)
W16 26 W18 40 W24 55 W40 167
398.78 454.66 599.44 980.44
8.76 13.335 12.83 26.16
381.25 427.99 573.78 928.12
6.35 8 10.03 16.51
1.079 0.961 1.028 1.010
594 594 594 594
595 617 604 610
1.00 0.96 0.98 0.97
0.961 for a temperature ranges about T > 300 C. Therefore, based on Fig. 8, for all selected beams with lw,T > 1.1, the value of Tlim is obtained equal to 594 C which is very close to the finite element values with a maximum error percentage of 3.72%. 5. Conclusions In the present work, the ultimate shear behavior of unstiffened steel web panels was investigated at elevated temperatures. In this regard, shear design relationships of AISC 360-16 specifications were adapted to fire conditions using the direct application of the proposed stress-strain reduction factors in EN 1993-1-2 for steel at high temperatures. Next, analytical equations and design curves were proposed to predict the ultimate shear strength, failure mode, and limiting temperature under a constant shear load at which the panel would fail. The validation of the proposed curves was evaluated by comparing the results with numerical and experimental data. In each section, engineering design examples were also presented. Such assessment is beneficial for developing novel strategies so that structural members can be retrofitted in buildings and bridges. Appendix A. Supplementary data Supplementary data to this article can be found online at https://doi.org/10.1016/j.jcsr.2019.105808. References [1] V. Broujerdian, P. Mahyar, A. Ghadami, Effect of curvature and aspect ratio on shear resistance of unstiffened plates, J. Constr. Steel Res. 112 (2015), https:// doi.org/10.1016/j.jcsr.2015.04.025. [2] A. Ghadami, V. Broujerdian, Flexureeshear interaction in hybrid steel I-girders at ambient and elevated temperatures, Adv. Struct. Eng. 22 (6) (2018) 1501e1516, https://doi.org/10.1177/1369433218817893. [3] A. Gheitasi, M.M. Alinia, Slenderness classification of unstiffened metal plates under shear loading, Thin-Walled Struct. 48 (2010) 508e518, https://doi.org/ 10.1016/j.tws.2010.02.004. [4] M.M. Alinia, H.R. Habashi, A. Khorram, Nonlinearity in the postbuckling behaviour of thin steel shear panels, Thin-Walled Struct. 47 (2009) 412e420, https://doi.org/10.1016/j.tws.2008.09.004. [5] M.M. Alinia, A. Gheitasi, S. Erfani, Plastic shear buckling of unstiffened stocky plates, J. Constr. Steel Res. 65 (2009) 1631e1643, https://doi.org/10.1016/ j.jcsr.2009.04.001. [6] A. Reis, N. Lopes, P.V. Real, Ultimate shear strength of steel plate girders at normal and fire conditions, Thin-Walled Struct. 137 (2019) 318e330. [7] V. Vimonsatit, K.H. Tan, S.K. Ting, Shear strength of plate girder web panel at elevated temperature, J. Constr. Steel Res. 63 (2007) 1442e1451, https:// doi.org/10.1016/J.JCSR.2007.01.002. [8] B. British, Standards Institution, Structural Use of Steelwork in Building: Code of Practice for Design-Rolled and Welded Sections, BSI, 1990. [9] K. Tan, Z. Qian, Experimental behaviour of a thermally restrained plate girder loaded in shear at elevated temperature, J. Constr. Steel Res. 64 (2007) 596e606, https://doi.org/10.1016/j.jcsr.2007.12.008. [10] Z. Qian, K. Tan, Deflection behaviour of plate girders loaded in shear at elevated temperatures, J. Constr. Steel Res. 65 (4) (2009) 991e1000. [11] E. Aziz, V. Kodur, An approach for evaluating the residual strength of fire exposed bridge girders, J. Constr. Steel Res. 88 (2013) 34e42. [12] M. Salminen, M. Heinisuo, Numerical analysis of thin steel plates loaded in shear at non-uniform elevated temperatures, J. Constr. Steel Res. 97 (2014) 105e113, https://doi.org/10.1016/j.jcsr.2014.02.002. [13] M.E.M. Garlock, J.D. Glassman, Elevated temperature evaluation of an existing steel web shear buckling analytical model, J. Constr. Steel Res. 101 (2014)
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