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Numerical investigation on ultimate shear strength of long steel plate girder web panels at high temperatures
Journal of Building Engineering 29 (2020) 101070
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Numerical investigation on ultimate shear strength of long steel plate girder web panels at high temperatures G.H. 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
Keywords: Fire condition Shear strength Long web panel Plate slenderness Finite element method
In this article, elastic shear buckling and ultimate shear strength of long steel plate girder web panels subjected to pure shear loading are investigated at both ambient and elevated temperatures by the finite element method (FEM). Ninety-six plate girders with compact, non-compact and slender long web panels are numerically analyzed and compared. From linear eigenvalue analysis, it is shown that AISC 360-16 predictions are more conservative than FEM results such that the difference is about 40% for all ranges of web panel slenderness. According to nonlinear analysis results at ambient temperature, there must be reduction factors applied to AISC 360-16 non-conservative predictions for considering the strength degradation caused by large slenderness values and effects due to initial geometrical imperfection. Furthermore, it is observed that the adopted equation of AISC 360-16 for fire situations yields values that are more non-conservative such that the difference reaches almost 18%. In this regard, the FEM results are used to develop a new equation for predicting the ultimate shear strength of long web panels by taking into account strength degradation caused by large slenderness values, high tem perature and initial geometrical imperfections.
1. Introduction 1.1. Background Steel plate I-girders have extensively been employed in the con struction of bridges and buildings. A conventional method of acquiring remarkable strength-to-weight ratios is to consider deep slender webs for I-girders, although shear buckling is a threat to slender web plates [1–3]. 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 thick web plates. Simultaneous geometrical instability and material yielding are notable phenomena in non-compact web plates [5]. There has been limited research on the postbuckling strength behavior of long web panels [6]. Basler [7] and Porter et al. [8] inde pendently developed a method to estimate the postbuckling reserve shear strength due to tension field action in steel plate girders. The
proposed methods, deal with web panels with aspect ratios of a/D � 3, and they are fundamentally based on the assumption that in order to the web panel to develop a complete tension field there must be anchor systems such as rigid flanges and adjacent panels [6]. where, a is the transverse stiffeners’ distance relative to each other, and D is the web €oglund [9] experimentally investigated the plate depth. In 1971, Ho shear behavior of long web panels and developed a theory to consider their shear strength [10]. According to test results, plate girders without intermediate transverse stiffeners showed substantial postbuckling strengths. Despite the former assumptions of existing classical tension field theories, Lee and Yoo [11–14] based on the analytical and exper imental studies found that even a simply supported panel without the flanges and adjacent panels are able to develop the postbuckling strength that is quantitatively equivalent to results which are observed in tests. Recently, these authors have emphasized the presence of post buckling strength in the long web panels. Although the postbuckling strength of plate girders with long web panels can be significant, how ever, it is not considered in many steel structures design codes due to lack of a comprehensive study [6]. AISC 360-10 limited the ultimate shear strength of slender web panels to their elastic shear buckling
* Corresponding author. E-mail address: [email protected] (S.A. Mousavi Ghasemi). https://doi.org/10.1016/j.jobe.2019.101070 Received 14 July 2019; Received in revised form 12 November 2019; Accepted 14 November 2019 Available online 19 December 2019 2352-7102/© 2019 Published by Elsevier Ltd.
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Journal of Building Engineering 29 (2020) 101070
strength. Currently, AISC 360–16, has improved the shear design equation for slender web panels. This development indicates that past research has emphasized existing the post buckling shear strength of the long web panels. In other words, the ultimate shear strength of slender web panels is not limited to their elastic shear strength. Given the newness of the topic, to gain a better understanding of the shear char acteristics of unstiffened web plates, the following review is provided with the aim of studies at elevated temperatures.
buckling, either at elevated or normal temperatures. The exactness of the model was verified using FEM. The shear-bending behavior of ho mogenous I-girders when exposed to fire with the aid of numerical methods was the subject of another study by Reis et al. [28]. They showed that the EC3 [29] recommendation for the safety assessment of the studied problem was suitable for design purposes if stress-strain reduction factors of Ref. [30] were directly applied at high tempera tures. For steel plate girders, the same researchers also provided novel design criteria so as to estimate the USS when fire occurs [31]. Using a numerical approach, Reis et al. [32] examined the effect of several pa rameters 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. [33] 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 was between 19% and 29% of approaches based on the explicit flange model. They also showed that the buckling coefficients and conse quently critical buckling loads decreased with increasing aspect ratio.
1.2. Literature review There have been extensive studies on the shear behavior of steel plates at ambient temperatures. Fire design is another important aspect that should be addressed separately [15]. 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. [16] to estimate the failure loads when a mixture of axial and shear or pure shear stresses were 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 became the dominant failure mode at such temperatures. Noteworthy is that the authors extended the ambient temperature relations proposed in the BS 5950: Part 1 standard [17] so as to study axial stress and material conditions at elevated temperatures. In an empirical study under fire condition, Tan and Qian [18] investigated the behavior of twelve simply-supported steel plate girders with thermal axial restraints. It was observed that the ultimate shear strength (USS) decreased significantly under a thermal restrain effect. This influence appeared to be more significant with higher thermal restraint and with greater web plate slenderness. 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 [19]. A recent study by Aziz and Kodur [20] on steel bridge girders employed a nonlinear finite element method (FEM) to estimate their residual strength, indicating that the highest fire temperature (along with the corresponding steel temperature) is the most influential parameter. In another study on thin steel plates, Salminen [21] benefited from FEM to propose an analytical method capable of esti mating the shear resistance at uneven high temperatures. The shear buckling characteristics of plate girder webs are among other 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 [22]. Scandella et al. [23] showed that the failure of steel plate girders as a result of shear web buckling when exposed to fire is a probable phe nomenon 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 eval uation of shear capacity such as the temperature-dependent degradation of components [24,25]. 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 web slenderness parameter which was inca pable of considering the impact of boundary conditions on the web panel’s shear behavior [1–3,26]. In a study on beams of Class 1, Quan et al. [27] developed a theoretical model for analyzing the shear
1.3. Elastic shear strength of long web panels at ambient temperatures Many studies have investigated the elastic shear buckling (ESB) behavior of plates [1,3,26,34,35]. Based on these researches, the ESB strength of web panels at ambient temperature, Vcr,20, subjected to pure shear loading, is obtained using the classical plate buckling equation as: Vcr;20 ¼ K
π2 E20 t3w 12 ð1 υ2 Þ D
(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 [13,22] and is found as � 5:34 Kss ¼ 4 þ 2 for ϕ ¼ a D < 1 (2a) ϕ Kss ¼ 5:34 þ
Ksf ¼
4 for ϕ ¼ a ϕ2
5:34 2:31 þ 8:39 ϕ þ ϕ ϕ2
Ksf ¼ 8:98 þ
5:61 ϕ2
� D�1
3:44 for ϕ ¼ a
� 1:99 for ϕ ¼ a D�1 ϕ3
(2b) � D<1
(2c) (2d)
where Kss is the ESB coefficient of plates with simply-supported condi tions along all four edges, Ksf is the ESB coefficient of plates with two opposite simply-supported edges and two other fixed edges, ϕ ¼ a/D is the web panel aspect ratio, and a is the transverse stiffeners’ distance relative to each other. In Basler’s model [13,22,36], 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, AASHTO [37] (Article 6.10.9.3.2–7) and AISC 360-16 [38] (Article G2-5) 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 [39]. In these cases, Lee et al. [39] proposed Eqs. (2f) to (2g) to calculate the ESB coefficient. For unstiffened long web panels (LWPs) with a/D > 3, the ESB strength is obtained assuming the ESB coefficient KAISC ¼ 5.34 and KAASHTO ¼ 5 in accordance with AISC 360-16 [38] (Article G2-5) and AASHTO [37] (Article 6.10.9.2), respectively. It must be noted that, according to AASHTO [37] (Article 6.10.9.1), interior web panels of I-shaped members (1) without a lon gitudinal stiffener and with a transverse stiffener spacing not exceeding 2
G.H. Pourmoosavi et al.
Journal of Building Engineering 29 (2020) 101070
3D, or (2) with one or more longitudinal stiffeners and with a transverse stiffener spacing not exceeding 1.5D, shall be considered as stiffened web panels. Otherwise, the panel shall be considered unstiffened. � 5 KAASHTO ¼ KAISC ¼ 5 þ 2 for φ ¼ a D � 3 (2e) ϕ KLee ¼ Kss þ
4 Ksf 5
Kss
KLee ¼ Kss þ
4 Ksf 5
Kss
�
� 2 2 3
� 1
�
tf tw
�� for 0:5 �
tf <2 tw
tf �2 tw
for
1.5. Aims and scopes According to the above literature review and to the best of our knowledge, there is a lack of comprehensive research on the shear behavior of long web panels (LWPs) with a/D between 3.0 and 6.0 at both ambient and elevated temperatures, where a/D is the web panel aspect ratio, a is the transverse stiffeners’ distance relative to each other, and D is the web plate depth. In addition, it was found that the LWP shear design relationships of AISC 360-16 [38] and Lee [6] do not take fire effects into consideration. Moreover, most researchers have dis cussed their results based on web plate slenderness ratio parameter, D/tw, which is unable to take the effect of web plate boundary conditions and material properties into account [1,3,26]. Therefore, in this article, firstly a new effective parameter, namely non-dimensional web plate slenderness (NWS), is introduced. This parameter is employed in a nonlinear finite element analysis to investigate the elastic shear buckling (ESB) and ultimate shear strength (USS) of LWPs at both ambient and elevated temperatures. Then, to examine the predictions of AISC 360-16 [38] and Lee et al. [6] in case of fire, 96 plate girders with compact, non-compact and slender LWPs are numerically analyzed and compared. To this end, the web panel shear design relationships mentioned in AISC 360-16 and those of Lee et al. [6] are modified to be used in fire con ditions. This is achieved by direct utilization of steel stress-strain reduction factors in EN1993-1-2 [30] at elevated temperatures. Finally, a new equation is proposed to obtain the USS of LWPs under pure shear by considering the strength degradation caused by large slenderness values, high temperatures and initial geometrical imperfection.
(2f) (2g)
1.4. Ultimate shear strength of LWPs at ambient temperatures 1.4.1. AISC design equation According to AISC specifications (Chapter G) [38], the design ulti mate shear strength (USS) of unstiffened web plates at ambient tem peratures (20� C–150� C), Vu-AISC,20 ¼ φvVn-AISC,20 is found as: � Vu AISC;20 ¼ φv 0:6σ yw;20 h tw for λw;20 � 1:1 (3a) Vu
AISC;20
λw;20
¼ φv 0:6σ yw;20 h tw
AISC ¼
D tw
�
�
1:1
λw;20
� for λw;20 > 1:1
AISC
(3b)
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
σ yw;20
(3c)
KAISC E20
where Vn-AISC,20 is the nominal shear strength at 20� C, φv ¼ 0.9 repre sents the shear resistance factor, σyw,20 is the yield stress of web plate at 20� C, and h signifies the overall depth of girder. Also, λw,20-AISC repre sents the non-dimensional web slenderness (NWS) parameter at 20� C which is calculated based on the AISC 360-16 ESB coefficient (KAISC). One should note that based on the NWS parameter, web panels can be classified as slender, non-compact, and compact plates [1–3,5,26,40]. Accordingly, plates with λw,20 � 1.1 and λw,20 > 1.37 are considered as compact and slender web panels, respectively. In addition, plates with 1.1 < λw,20 � 1.37 are considered as non-compact web panels with in elastic shear buckling failure mode.
2. Materials and methods 2.1. Specifications of steel plate girder models In order to investigate the shear behavior of LWPs under pure shear at both ambient and elevated temperatures, 96 models with slender (S), non-compact (NC) and compact (C) webs at four temperatures of 20� C, 400� C, 600� C and 800� C were constructed using the FEM software package, ABAQUS. Selecting a wide range of thickness and NWS pa rameters provides the possibility for studying the elastic/plastic buck ling behavior of slender and stocky 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 – 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, PG3-2 represents the PG3 girder with a girder length of a ¼ 3000mm, an aspect ratio of a/D ¼ 3, and slender web plate with thickness of tw ¼ 10mm. As another example, PG4-6 represents the PG4 girder with an aspect ratio of 4, and compact web plate with thickness of tw ¼ 20mm. 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 [2,6]. In addition, to eliminate the lateral torsional buckling phenomenon, the entire mid-nodes of flanges are constrained in the y direction [1,2,26].
1.4.2. Lee’s modified design equation Lee et al. [6] investigated the validity of an arbitrary limit imposed by Basler on the maximum aspect ratio of a transversely stiffened web panel and developed a new equation for predicting the USS of web panels with high aspect ratios (3 � a/D � 6) at ambient temperatures. According to this research, the USS of an LWP at 20� C, Vu-Lee,20, without considering the effects of residual stress, and large initial geometrical imperfections, is calculated as: � Vu Lee;20 ¼ Rλ;20 0:58σ yw;20 D tw ð0:6C þ 0:4Þ (4a) 8 for < Rλ;20 ¼ 1 for Rλ;20 ¼ 1:35C þ 0:6 : Rλ;20 ¼ 5:62C þ 0:145 for 8 C¼1 > > > > > 1 :12 > > > > 1:57 > >C ¼ : ðλw;T AASHTO Þ2 λw;20
D AASHTO ¼ tw
C � 0:3 0:1 < C < 0:3 C � 0:1
for
λw;T
AASHTO
< 1:12
for
1:12 � λw;T
AASHTO
for
λw;T
> 1:40
AASHTO
(4b)
� 1:40
(4c)
2.2. Material properties
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
σyw;20
KAASHTO E20
(4d)
In this article, the stress-strain reduction factors provided in EN 1993-1-2 [30] specific to steel for elevated temperatures are employed to consider fire effects. In fact, the design criteria at ambient tempera tures are modified using the reduction factors such that they could be exploited for fire conditions [2,22,28]. At ambient temperature, yield
where Rλ,20 is the shear strength reduction factor due to high slenderness at 20� C, and λw,20-AASHTO represents the NWS parameter at 20� C which is calculated based on the AASHTO ESB coefficient (KAASHTO). 3
G.H. Pourmoosavi et al.
Journal of Building Engineering 29 (2020) 101070
Table 1 Specifications of the FEM models (1 inch ¼ 25.4 mm). Model
a/D
Temperature (� C)
tw (mm)
tf (mm)
D/tw
λw,20-AASHTO
λw,20-AISC
Web plate classificationa
PG3-1 PG3-2 PG3-3 PG3-4 PG3-5 PG3-6
3
20, 400, 600, 800
8 10 12 14 16 20
16 20 24 28 32 40
125 100 83.33 71.42 62.50 50
2.055 1.644 1.370 1.174 1.027 0.822
2.055 1.644 1.370 1.174 1.027 0.822
S S NC NC C C
PG4-1 PG4-2 PG4-3 PG4-4 PG4-5 PG4-6
4
20, 400, 600, 800
8 10 12 14 16 20
16 20 24 28 32 40
125 100 83.33 71.42 62.50 50
2.165 1.732 1.443 1.237 1.083 0.866
2.095 1.676 1.397 1.197 1.048 0.838
S S NC NC C C
PG5-1 PG5-2 PG5-3 PG5-4 PG5-5 PG5-6
5
20, 400, 600, 800
8 10 12 14 16 20
16 20 24 28 32 40
125 100 83.33 71.42 62.50 50
2.165 1.732 1.443 1.237 1.083 0.866
2.095 1.676 1.397 1.197 1.048 0.838
S S NC NC C C
PG6-1 PG6-2 PG6-3 PG6-4 PG6-5 PG6-6
6
20, 400, 600, 800
8 10 12 14 16 20
16 20 24 28 32 40
125 100 83.33 71.42 62.50 50
2.165 1.732 1.443 1.237 1.083 0.866
2.095 1.676 1.397 1.197 1.048 0.838
S S NC NC C C
a
S: Slender; NC: Non-compact; C: Compact.
Fig. 2. Stress-strain reduction factors of steel at high temperatures.
Fig. 1. Boundary condition and pure shear loading of models in the FEM analysis.
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 diagrams of Eurocode reduction factors related to yield stress (σy), Young’s modulus (E) and proportional limit stress (σ p), ky,T, kE,T and kp,T, respectively with respect to temperature, where the superscript T refers to high temper ature values. 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.
Fig. 3. Material properties of FEM models at ambient and elevated temperatures.
2.3. Numerical modeling and validation 2.3.1. Numerical modeling The numerical modeling and analysis are performed using ABAQUS 4
G.H. Pourmoosavi et al.
Journal of Building Engineering 29 (2020) 101070
[41]. 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 LWPs. In this method, if the model is defined as a perfect plate without initial geometrical imperfections, the buckling phenome non may not be observed and the system yields under a specific load [1, 26]. 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 [6]. 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 PG3-1 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 0.42%.
Fig. 5. Comparing the experimental and numerical analysis results.
3.1. Elastic shear buckling strength In this section, the ESB strength of models is discussed at both ambient and elevated temperatures. The ESB strength of each model is obtained by linear eigenvalue analysis using FEM. For example, Fig. 6 shows the first mode shapes of PG3-1, PG4-1, PG5-1, and PG6-1 models at 20� C. The ESB strength of the finite element (Vcr-FEM), AISC (Vcr-AISC), and Lee et al. (Vcr-Lee) models are shown in Table 2. According to this table, AISC prediction is more conservative than FEM results such that the difference is about 40% for all ranges of web panel slenderness. On the other hand, the ESB coefficient of Lee et al. which considers the effects of flange-to-web stiffness ratio, tf/tw, on the ESB strength of girders, leads to more accurate results with a maximum difference of 2%. It must be noted that similar results with a similar error percentage should be obtained at high temperatures.
2.3.2. Verification of numerical modeling process Definitely the best way to corroborate any numerical analysis is to compare the results with experimental ones [26,31,32]. In 2007, Vimonsatit et al. [16] 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 verify of the nu merical modeling process the test model TG3 is 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. [16]. 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 dif ference 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.
3.2. Ultimate shear strength at ambient temperatures In this section, the USS of LWPs is discussed at ambient temperatures. The USS strength of each model is obtained by the nonlinear Riks analysis using FEM. For example, Fig. 7 shows the failure mode shapes corresponding to the PG3-1, PG4-1, PG5-1, and PG6-1 models at 20� C. The USS strength of the finite element (Vu-FEM), AISC (Vu-AISC), and Lee et al. (Vu-Lee) models is shown in Table 3. The ratio of predicted values based on AISC (Eq. (3)) and Lee et al. [6] (Eq. (4)) to the ones obtained by the FEM model, Vu-FEM, with respect to the NWS parameter, λw,20-AISC, is plotted in Fig. 8. � (7a) Vu;T proposed ¼ Rλ;T RT RImp ky;T 0:58σyw;20 D tw ð0:6C þ 0:4Þ
3. Results and discussion In this section, the results are investigated in three parts. In the first part, the ESB strength of the FEM models is discussed. In the second part, the USS of models is investigated at ambient temperature. Finally, the third part is devoted to the USS of LWPs considering the reduction factors due to large web panel slenderness (Rλ,T), high temperatures (RT), and initial geometric imperfections (RImp).
8 for < Rλ;T ¼ 1 for Rλ;T ¼ 1:35C þ 0:6 : Rλ;T ¼ 5:62C þ 0:145 for 8 > < RT ¼ 1 > : RT ¼
1 1:03 � ðλw;T
0:3 AASHTO Þ
C � 0:3 0:1 < C < 0:3 C � 0:1
(7b)
for
λw;T
AASHTO
< 0:9
for
λw;T
AASHTO
� 0:9
� �0:05 8 tw > > � ð1:15 0:3λw;Lee Þ � 1 > RImp ¼ 1:03 � < Imp � �0:05 > > > : RImp ¼ 1:03 � tw � ð0:34 þ 0:3λw;Lee Þ � 1 Imp
(7c)
for
λw;Lee < 1:37
for
λw;Lee � 1:37 (7d)
Fig. 4. Convergence study and mesh sensitivity in the finite element models. 5
Journal of Building Engineering 29 (2020) 101070
G.H. Pourmoosavi et al.
Fig. 6. First mode shapes of PG3-1, PG4-1, PG5-1, and PG6-1 models at 20� C. Table 2 Elastic shear buckling strength at ambient temperature (20� C). a/D
Model
D/tw
λw,20-AASHTO
λw,20-AISC
Vcr-FEM (1)
Vcr-AISC (2)
Vcr-Lee (3)
Ratio (2/1)
Ratio (3/1)
3
PG3-1 PG3-2 PG3-3 PG3-4 PG3-5 PG3-6
125 100 83.33 71.42 62.50 50
2.055 1.644 1.370 1.174 1.027 0.822
2.055 1.644 1.370 1.174 1.027 0.822
816.00 1591.80 2746.60 4354.50 6488.40 12620.0
513.66 1003.23 1733.59 2752.87 4109.25 8025.87
812.59 1587.10 2742.50 4355.00 6500.75 12696.78
0.629 0.630 0.631 0.632 0.633 0.636
0.996 0.997 0.999 1.000 1.002 1.006
4
PG4-1 PG4-2 PG4-3 PG4-4 PG4-5 PG4-6
125 100 83.33 71.42 62.50 50
2.165 1.732 1.443 1.237 1.083 0.866
2.095 1.676 1.397 1.197 1.048 0.838
795.85 1552.40 2678.70 4246.80 6327.70 12370.0
494.22 965.27 1667.99 2648.71 3953.76 7722.19
791.96 1546.79 2672.85 4244.39 6335.64 12374.30
0.621 0.622 0.623 0.624 0.625 0.624
0.995 0.996 0.998 0.999 1.001 1.000
5
PG5-1 PG5-2 PG5-3 PG5-4 PG5-5 PG5-6
125 100 83.33 71.42 62.50 50
2.165 1.732 1.443 1.237 1.083 0.866
2.095 1.676 1.397 1.197 1.048 0.838
786.82 1534.80 2648.30 4198.50 6255.90 12168.0
494.22 965.27 1667.99 2648.71 3953.76 7722.19
782.05 1527.45 2639.43 4191.31 6256.42 12219.57
0.628 0.629 0.630 0.631 0.632 0.635
0.994 0.995 0.997 0.998 1.000 1.004
6
PG6-1 PG6-2 PG6-3 PG6-4 PG6-5 PG6-6
125 100 83.33 71.42 62.50 50
2.165 1.732 1.443 1.237 1.083 0.866
2.095 1.676 1.397 1.197 1.048 0.838
787.64 1539.90 2657.10 4212.50 6276.80 12209.0
494.22 965.27 1667.99 2648.71 3953.76 7722.19
776.64 1516.87 2621.15 4162.29 6213.10 12134.97
0.627 0.627 0.628 0.629 0.630 0.632
0.986 0.985 0.986 0.988 0.990 0.994
Note: 1 inch ¼ 25.4 mm, 1 kip ¼ 4.45 kN, 1 psi ¼ 6.895 kPa.
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Journal of Building Engineering 29 (2020) 101070
Fig. 7. Failure mode shapes of PG3-1, PG4-1, PG5-1, and PG6-1 models at 20� C. Table 3 Ultimate shear strength at ambient temperature (20� C). a/D
Model
D/tw
λw,20-ASHTO
λw,20-
Vu-FEM (1)<
Vu-AISC (2)
Vu-Lee (3)
Ratio (2/1)
Ratio (3/1)
3
PG3-1 PG3-2 PG3-3 PG3-4 PG3-5 PG3-6
125 100 83.33 71.42 62.50 50
2.055 1.644 1.370 1.174 1.027 0.822
2.055 1.644 1.370 1.174 1.027 0.822
977.57 1555.19 2077.87 2432.45 2785.40 3485.95
715.93 1127.31 1635.81 2243.51 2757.89 3499.20
867.31 1302.46 1859.39 2368.44 2784.00 3480.00
0.732 0.725 0.787 0.922 0.990 1.004
0.887 0.837 0.895 0.974 0.999 0.998
4
PG4-1 PG4-2 PG4-3 PG4-4 PG4-5 PG4-6
125 100 83.33 71.42 62.50 50
2.165 1.732 1.443 1.237 1.083 0.866
2.095 1.676 1.397 1.197 1.048 0.838
953.71 1523.96 2074.67 2431.23 2788.36 3535.08
702.25 1105.77 1604.56 2200.66 2757.89 3499.20
836.54 1242.36 1779.31 2297.56 2784.00 3480.00
0.736 0.726 0.773 0.905 0.989 0.990
0.877 0.815 0.858 0.945 0.998 0.984
5
PG5-1 PG5-2 PG5-3 PG5-4 PG5-5 PG5-6
125 100 83.33 71.42 62.50 50
2.165 1.732 1.443 1.237 1.083 0.866
2.095 1.676 1.397 1.197 1.048 0.838
941.55 1510.79 2072.31 2430.98 2783.09 3474.01
702.25 1105.77 1604.56 2200.66 2757.89 3499.20
836.54 1242.36 1779.31 2297.56 2784.00 3480.00
0.746 0.732 0.774 0.905 0.991 1.007
0.888 0.822 0.859 0.945 1.000 1.002
6
PG6-1 PG6-2 PG6-3 PG6-4 PG6-5 PG6-6
125 100 83.33 71.42 62.50 50
2.165 1.732 1.443 1.237 1.083 0.866
2.095 1.676 1.397 1.197 1.048 0.838
938.75 1507.15 2073.13 2428.85 2777.06 3470.02
702.25 1105.77 1604.56 2200.66 2757.89 3499.20
836.54 1242.36 1779.31 2297.56 2784.00 3480.00
0.748 0.734 0.774 0.906 0.993 1.008
0.891 0.824 0.858 0.946 1.002 1.003
AISC
Note: 1 inch ¼ 25.4 mm, 1 kip ¼ 4.45 kN, 1 psi ¼ 6.895 kPa.
7
G.H. Pourmoosavi et al.
Journal of Building Engineering 29 (2020) 101070
where Vn,T is the nominal shear strength, λw,T is the NWS parameter at a constant temperature of T, λw,20 is the NWS parameter at 20� C and Vp,20 ¼ 0.6σyw,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 λw, T � 1.1 and λw,T > 1.37 are considered as compact and slender plates, respectively. Also, web panels with 1.1 < λw,T � 1.37 are considered to be non-compact plates with inelastic shear buckling failure mode. On the other hand, Lee’s modified design equation for unstiffened long web plates at elevated temperatures, Vu-Lee,T, shall be determined as � (6a) Vu Lee;T ¼ Rλ;T ky;T 0:58σyw;20 D tw ð0:6C þ 0:4Þ 8 for < Rλ;T ¼ 1 Rλ;T ¼ 1:35C þ 0:6 for : Rλ;T ¼ 5:62C þ 0:145 for
Fig. 8. The ratio of Vu-AISC and Vu-Lee to the FEM results vs. λw-AISC at 20� C.
8 C¼1 > > > > > 1 :12 > > > > 1:57 > > :C ¼ ðλw;T AASHTO Þ2
λw;T
for
λw;T
AASHTO
< 1:12
for
1:12 � λw;T
AASHTO
for
λw;T
> 1:40
sffiffiffiffiffiffiffi ky;T � λw;20 AASHTO ¼ kE;T
AASHTO
� 1:40
8 C¼1 > > > > > 1 :12 > > > > 1:57 > > :C ¼ ðλw;T AASHTO Þ2
(7e)
λw;T
(7f)
AASHTO
3.3.1. Evaluation method Part 1–5 of EC3 [29] defines no specific criterion for USS verification under fire conditions. In this article, the stress-strain reduction factors provided in EN 1993-1-2 [30] 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 [2,22,28]. Based on the direct application of the reduction factors of EC3 for steel stress-strain rela tionship, the modified design shear strength of AISC for unstiffened web plates at elevated temperatures, Vu,T ¼ φvVn,T, shall be determined as: � Vu;T ¼ ky;T : φv 0:6σ yw;20 h tw for λw;T � 1:1 (5a)
λw;T
AISC ¼
sffiffiffiffiffiffiffi ky;T : λw;20 kE;T
AISC
�
1:1
λw;20
AISC
� for λw;T > 1:1
λw;T
AASHTO
< 1:12
for
1:12 � λw;T
AASHTO
for
λw;T
> 1:40
sffiffiffiffiffiffiffi ky;T : λw;20 kE;T
AASHTO
AASHTO
� 1:40
(6c)
(6d)
3.3.2. Ultimate shear strength In this section, the USS of LWPs is discussed at elevated temperatures (400� C, 600� C and 800� C). The USS of the finite element (Vu-FEM), AISC (Vu-AISC), and Lee et al. (Vu-Lee) models are reported in Tables 4–6. The ratio of predicted values based on AISC (Eq. (5)) and Lee et al. (Eq. (6)) to the ones obtained by FEM, Vu-FEM, with respect to the NWS parame ters of λw,400-AISC, λw,600-AISC, and λw,800-AISC are plotted in Figs. 9–11, respectively. According to Figs. 9 and 10, both AISC and Lee equations are more accurate and in the safe region for compact plates. However, these equations lead to the unsafe condition of Ratio >1 for higher slenderness values. The difference between the USS obtained via ABAQUS and the results of modified AISC equation (Eq. (5)) at 400� C reaches almost 4% and 7% for non-compact and slender LWPs, respectively. However, this difference between the results of Lee’s modified equation and FEM are about 11% and 30% for non-compact and slender LWPs, respectively. Also, by increasing the temperature to 600� C, especially for slender plates, AISC and Lee’s equations yield values that are more nonconservative for USS such that the difference between the results of FEM and Lee reaches 40%. However, this difference for AISC equation is about 18%. According to Fig. 11, both modified AISC and Lee equations are more accurate for compact web panels at higher temperature of 800� C. However, these equations generally lead to the unsafe condition for noncompact and compact plates. The difference between the results of modified AISC equation and FEM are about 10% and 15% for noncompact and slender LWPs, respectively. That is while; these differ ences for Lee’s modified equation reaches almost 18% and 35%, respectively. Comparison the USS results at 400 and 600� C with those of 800� C shows the different shear behavior of the plates at high temper atures compared to the mid-range temperatures. The cause of this discrepancy is examined in a step-by-step method in Ref. [22]. In 2014, Garlock and Glassman [22] numerically evaluated the Basler–Thürli mann’s theoretical equation [7] at elevated temperatures, which serves
3.3. Ultimate shear strength at elevated temperatures
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi � ky;T :kE;T : φv 0:6σyw;20 h tw
¼
for
(6b)
where Rλ,T is the shear strength reduction factor due to high slenderness at elevated temperature, and λw,T-AASHTO represents the NWS parameter at a specific temperature (T) which is calculated based on the AASHTO ESB coefficient (KAASHTO).
According to this figure, both AISC and Lee modified design equations are more accurate in compact plates with plastic shear buckling. How ever, by increasing the NWS parameter, the difference between Vu-AISC, Vu-Lee and Vu-FEM is increased. The maximum difference between the results of Lee’s equation and FEM for compact, non-compact, and slender web plates is about 1.5%, 15%, and 19%, respectively. On the other hand, this difference between the results of AISC prediction and FEM for compact, non-compact, and slender web plates reaches 1.1%, 23%, and 28%, respectively. It is important to note that this article ex amines the USS of LWPs without considering the effects of residual stress and large initial geometrical imperfections. In addition, AISC pre dictions considering the effects of residual stress and large initial geometrical imperfection are close to those obtained using FEM, espe cially in compact and non-compact web panels. Therefore, certain reduction factors must be applied to AISC predictions to take the effect of mentioned parameters on the shear strength degradation into account.
Vu;T ¼
AASHTO
C � 0:3 0:1 < C < 0:3 C � 0:1
(5b)
(5c)
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Journal of Building Engineering 29 (2020) 101070
Table 4 Ultimate shear strength at 400� C. Model
D/tw
λw,400-AASHTO
λw,400-AISC
Vu-FEM (1)
Vu-AISC (2)
Vu-Lee (3)
Eq (7) (4)
Ratio (2/1)
Ratio (3/1)
Ratio (4/1)
PG3-1 PG3-2 PG3-3 PG3-4 PG3-5 PG3-6
125 100 83.33 71.42 62.50 50
2.456 1.965 1.637 1.404 1.228 0.982
2.456 1.965 1.637 1.404 1.228 0.982
571.07 885.36 1328.23 1829.07 2381.62 3658.47
598.99 943.17 1368.62 1877.06 2470.24 3499.20
736.48 1120.52 1568.77 2139.29 2636.98 3480.00
546.06 888.34 1313.63 1876.14 2407.14 3396.61
1.049 1.065 1.030 1.026 1.037 0.956
1.290 1.266 1.181 1.170 1.107 0.951
0.96 1.00 0.99 1.03 1.01 0.93
PG4-1 PG4-2 PG4-3 PG4-4 PG4-5 PG4-6
125 100 83.33 71.42 62.50 50
2.588 2.070 1.725 1.479 1.294 1.035
2.504 2.003 1.669 1.431 1.252 1.002
560.83 868.78 1303.09 1797.33 2741.76 3444.49
587.55 925.16 1342.47 1841.20 2423.06 3499.20
689.78 1078.45 1496.08 2023.85 2559.53 3480.00
503.50 841.71 1233.30 1747.33 2300.15 3343.86
1.048 1.065 1.030 1.024 0.884 1.016
1.230 1.241 1.148 1.126 0.934 1.010
0.90 0.97 0.95 0.97 0.98 0.97
PG5-1 PG5-2 PG5-3 PG5-4 PG5-5 PG5-6
125 100 83.33 71.42 62.50 50
2.588 2.070 1.725 1.479 1.294 1.035
2.504 2.003 1.669 1.431 1.252 1.002
555.22 864.91 1301.48 1795.23 2338.42 3427.82
587.55 925.16 1342.47 1841.20 2423.06 3499.20
689.78 1078.45 1496.08 2023.85 2559.53 3480.00
503.50 841.71 1233.30 1747.33 2300.15 3343.86
1.058 1.070 1.031 1.026 1.036 1.021
1.242 1.247 1.150 1.127 1.095 1.015
0.91 0.97 0.95 0.97 0.98 0.98
PG6-1 PG6-2 PG6-3 PG6-4 PG6-5 PG6-6
125 100 83.33 71.42 62.50 50
2.588 2.070 1.725 1.479 1.294 1.035
2.504 2.003 1.669 1.431 1.252 1.002
565.59 865.57 1303.97 1797.25 2337.15 3525.11
587.55 925.16 1342.47 1841.20 2423.06 3499.20
689.78 1078.45 1496.08 2023.85 2559.53 3480.00
503.50 841.71 1233.30 1747.33 2300.15 3343.86
1.039 1.069 1.030 1.024 1.037 0.993
1.220 1.246 1.147 1.126 1.095 0.987
0.89 0.97 0.95 0.97 0.98 0.95
Note: 1 inch ¼ 25.4 mm, 1 kip ¼ 4.45 kN, 1 psi ¼ 6.895 kPa. Table 5 Ultimate shear strength at 600� C. Model
D/tw
λw,600-AASHTO
λw,600-AISC
Vu-FEM (1)
Vu-AISC (2)
Vu-Lee (3)
Eq (7) (4)
Ratio (2/1)
Ratio (3/1)
Ratio (4/1)
PG3-1 PG3-2 PG3-3 PG3-4 PG3-5 PG3-6
125 100 83.33 71.42 62.50 50
2.530 2.024 1.687 1.446 1.265 1.012
2.530 2.024 1.687 1.446 1.265 1.012
236.40 389.99 580.75 817.59 1079.73 1711.53
273.27 430.30 624.40 856.36 1126.99 1644.62
333.27 515.12 717.41 973.85 1218.40 1635.60
244.91 404.76 595.40 846.47 1102.32 1582.23
1.156 1.103 1.075 1.047 1.044 0.961
1.410 1.321 1.235 1.191 1.128 0.956
1.04 1.04 1.03 1.04 1.02 0.92
PG4-1 PG4-2 PG4-3 PG4-4 PG4-5 PG4-6
125 100 83.33 71.42 62.50 50
2.666 2.133 1.777 1.523 1.333 1.066
2.580 2.064 1.720 1.474 1.290 1.032
275.73 381.32 565.63 798.08 1256.78 1598.13
268.05 422.08 612.47 840.01 1105.46 1644.62
312.96 496.49 685.22 922.72 1183.06 1635.60
226.41 384.06 559.84 789.58 1053.73 1557.65
0.972 1.107 1.083 1.053 0.880 1.029
1.135 1.302 1.211 1.156 0.941 1.023
0.98 1.01 0.99 0.99 1.00 0.97
PG5-1 PG5-2 PG5-3 PG5-4 PG5-5 PG5-6
125 100 83.33 71.42 62.50 50
2.666 2.133 1.777 1.523 1.333 1.066
2.580 2.064 1.720 1.474 1.290 1.032
288.18 380.85 566.54 799.22 1058.06 1594.36
268.05 422.08 612.47 840.01 1105.46 1644.62
312.96 496.49 685.22 922.72 1183.06 1635.60
226.41 384.06 559.84 789.58 1053.73 1557.65
0.930 1.108 1.081 1.051 1.045 1.032
1.086 1.304 1.209 1.155 1.118 1.026
0.98 1.01 0.99 0.99 1.00 0.98
PG6-1 PG6-2 PG6-3 PG6-4 PG6-5 PG6-6
125 100 83.33 71.42 62.50 50
2.666 2.133 1.777 1.523 1.333 1.066
2.580 2.064 1.720 1.474 1.290 1.032
242.85 382.13 569.92 802.70 1057.84 1682.60
268.05 422.08 612.47 840.01 1105.46 1644.62
312.96 496.49 685.22 922.72 1183.06 1635.60
226.41 384.06 559.84 789.58 1053.73 1557.65
1.104 1.105 1.075 1.046 1.045 0.977
1.289 1.299 1.202 1.150 1.118 0.972
0.99 1.01 0.98 0.98 1.00 0.93
Note: 1 inch ¼ 25.4 mm, 1 kip ¼ 4.45 kN, 1 psi ¼ 6.895 kPa.
as the basis of the AISC and AASHTO Specifications. In this research, the Basler–Thürlimann theoretical equation for predicting the USS of plates was evaluated by replacing web panel yield stress (σyw;20 ) with the stress at the fire condition, σyw,T ¼ kyσyw,20. According to results, it was rec ommended that for 20� C � T � 800� C, σyw,20 be substituted with kyσyw, � 20 and for T > 800 C, σyw,20 be substituted with kpσyw,20. Based on this research and the results of the present paper, the difference in shear behavior of plates at 800� C with their behavior at temperatures less than 800� C will be justified. However, by comparing the results in Figs. 9–11, it is observed that all data at high temperatures follow the same general trend. In addition, it can be mentioned that the difference between the
results of Eqs. (5) and (6) and those obtained from FEM is because the above equations are not basically provided for high temperatures. Therefore, it is necessary to propose a new equation to achieve the USS of LWPs under pure shear by taking into account strength degradation caused by high temperatures. 3.3.3. The proposed shear design equation In this section, FEM results are used to develop an equation for predicting the USS of LWPs at elevated temperatures. This equation is a function of NWS parameter at high temperatures, λw-AASHTO, based on the AASHTO ESB coefficient [37]. In addition, the effects of large 9
G.H. Pourmoosavi et al.
Journal of Building Engineering 29 (2020) 101070
Table 6 Ultimate shear strength at 800� C. Model
D/tw
λw,800-AASHTO
λw,800-AISC
Vu-FEM (1)
Vu-AISC (2)
Vu-Lee (3)
Eq (7) (4)
Ratio (2/1)
Ratio (3/1)
Ratio (4/1)
PG3-1 PG3-2 PG3-3 PG3-4 PG3-5 PG3-6
125 100 83.33 71.42 62.50 50
2.272 1.817 1.515 1.298 1.136 0.909
2.272 1.817 1.515 1.298 1.136 0.909
66.86 104.00 153.00 219.10 272.00 382.90
71.23 112.17 162.76 223.23 293.77 384.91
90.14 131.14 186.19 256.96 303.66 382.80
68.42 106.43 159.60 230.68 283.76 382.47
1.065 1.079 1.064 1.019 1.080 1.005
1.348 1.261 1.217 1.173 1.116 1.000
1.023 1.023 1.043 1.053 1.043 0.999
PG4-1 PG4-2 PG4-3 PG4-4 PG4-5 PG4-6
125 100 83.33 71.42 62.50 50
2.394 1.915 1.596 1.368 1.197 0.957
2.316 1.853 1.544 1.323 1.158 0.926
66.02 95.89 146.06 210.00 262.37 400.06
69.87 110.02 159.65 218.96 288.16 384.91
83.83 125.73 176.84 242.11 294.45 382.80
62.64 100.45 149.23 213.98 270.88 376.53
1.058 1.147 1.093 1.043 1.098 0.962
1.270 1.311 1.211 1.153 1.122 0.957
0.949 1.048 1.022 1.019 1.032 0.941
PG5-1 PG5-2 PG5-3 PG5-4 PG5-5 PG5-6
125 100 83.33 71.42 62.50 50
2.394 1.915 1.596 1.368 1.197 0.957
2.316 1.853 1.544 1.323 1.158 0.926
71.00 107.35 160.00 228.00 303.00 390.43
69.87 110.02 159.65 218.96 288.16 384.91
83.83 125.73 176.84 242.11 294.45 382.80
62.64 100.45 149.23 213.98 270.88 376.53
0.984 1.025 0.998 0.960 0.951 0.986
1.181 1.171 1.105 1.062 0.972 0.980
0.882 0.936 0.933 0.939 0.894 0.964
PG6-1 PG6-2 PG6-3 PG6-4 PG6-5 PG6-6
125 100 83.33 71.42 62.50 50
2.394 1.915 1.596 1.368 1.197 0.957
2.316 1.853 1.544 1.323 1.158 0.926
70.07 105.57 163.00 233.08 290.000 396.86
69.87 110.02 159.65 218.96 288.16 384.91
83.83 125.73 176.84 242.11 294.45 382.80
62.64 100.45 149.23 213.98 270.88 376.53
0.997 1.042 0.979 0.939 0.994 0.970
1.196 1.191 1.085 1.039 1.015 0.965
0.894 0.952 0.916 0.918 0.934 0.949
Note: 1 inch ¼ 25.4 mm, 1 kip ¼ 4.45 kN, 1 psi ¼ 6.895 kPa.
slenderness values, Rλ,T, high temperature, RT, and initial geometrical imperfection, RImp, are considered. Firstly, the effect of large slenderness values is considered based on the equation proposed by Lee et al. [6], Eq. (6b). Next, to derive a reduction factor as a function of NWS parameter, RT, for predicting the USS reduction of a perfect LWP due to higher temperatures, a regression analysis and data fitting are performed. The correlation between the FEM results and those predicted by the 1/RT curve is shown in Figs. 9–11. Finally, for determining the USS reduction factor due to initial distortions, the proposed equation by Ghadami and Broujerdian [26] is used. Based on these assumptions, the final equation is obtained as Eq. (7) where Imp is the initial geometric imperfection magnitude applied to the perfect finite element model, and λw-Lee is the NWS parameter, which is calculated using the ESB coefficient value of Lee et al. [39] in Eqs. (2f) to (2g). The ratio of predicted values based on the proposed equation to the ones obtained by FEM, Vu-FEM, with respect to the NWS parameters of λw,400-AISC, λw,600-AISC and λw,800-AISC, are reported in Tables 4–6 According to the all results, the maximum and minimum difference between the results of the proposed equation with those of FEM results are about þ5.3% and 11.8%, respectively. In addition, the correlation factors, R2, of the mentioned equation at 400� C, 600� C and 800� C are obtained as 0.999, 0.998, and 0.995, respectively. Therefore, there is a good conformity between the USS determined from Eq. (7) and FEM analysis for LWPs at elevated temperatures.
Fig. 9. The ratio of Vu-AISC and Vu-Lee to the FEM results vs. λw-AISC at 400� C.
4. Conclusion In the present work, elastic shear buckling and ultimate shear strength of long steel web panels (LWPs) with a/D between 3.0 and 6.0 subjected to pure shear were investigated at both ambient and elevated temperatures by the finite element method. Ninety-six plate girders with compact, non-compact and slender web plates were numerically analyzed and the obtained results were compared with those of AISC 360-16 [38] and Lee’s modified equation [6]. Next, finite element re sults were used to develop an equation for predicting the ultimate shear strength of long steel plate girder web panels by considering the strength degradation caused by high slenderness, high temperature and initial geometrical imperfections.
Fig. 10. The ratio of Vu-AISC and Vu-Lee to the FEM results vs. λw-AISC at 600� C.
10
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Journal of Building Engineering 29 (2020) 101070
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) 263–270, https://doi.org/10.1016/J.JCSR.2015.04.025. [2] A. Ghadami, V. Broujerdian, Flexure–shear interaction in hybrid steel I-girders at ambient and elevated temperatures, Adv. Struct. Eng. 22 (2019) 1501–1516, 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) 508–518, 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) 412–420, 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) 1631–1643, https://doi.org/10.1016/j. jcsr.2009.04.001. [6] S.C. Lee, D.S. Lee, C.H. Yoo, Ultimate shear strength of long web panels, J. Constr. Steel Res. 64 (2008) 1357–1365, https://doi.org/10.1016/J.JCSR.2008.01.023. [7] K. Basler, Strength of plate girders in shear, J. Struct. Div. ASCE. 87 (7) (1961) 151–181. [8] D.M. Porter, K.C. Rockey, H.R. Evans, The collapse behavior of plate girders loaded in shear, J. Struct. Eng. 53 (1975) 313–325. [9] T. Hoglund, Simply Supported Long Thin Plate I-Girders without Web Stiffeners Subjected to Distributed Transverse Load, IABSE Colloq. Des. Plate Box Girders Ultimate Strength, 1971, pp. 85–97. London. [10] T. Hoglund, Design of thin plate I girders in shear and bending, R. Inst Technol Bull. 94 (1973). [11] S.C. Lee, C.H. Yoo, D.Y. Yoon, Behavior of intermediate transverse stiffeners attached on web panels, J. Struct. Eng. 128 (2002) 337–345. [12] S.C. Lee, C.H. Yoo, D.Y. Yoon, New design rule for intermediate transverse stiffeners attached on web panels, J. Struct. Eng. 129 (2003) 1607–1614. [13] S.C. Lee, C.H. Yoo, Strength of plate girder web panels under pure shear, J. Struct. Eng. 124 (1998) 184–194, https://doi.org/10.1061/(ASCE)0733-9445(1998)124: 2(184). [14] C.H. Yoo, S.C. Lee, Mechanics of web panel postbuckling behavior in shear, J. Struct. Eng. 132 (2006) 1580–1589, https://doi.org/10.1061/(ASCE)0733-9445 (2006)132:10(1580). [15] A. Reis, N. Lopes, P. Vila Real, Ultimate shear strength of steel plate girders at normal and fire conditions, Thin-Walled Struct. 137 (2019) 318–330, https://doi. org/10.1016/J.TWS.2019.01.013. [16] V. Vimonsatit, K.H. Tan, S.K. Ting, Shear strength of plate girder web panel at elevated temperature, J. Constr. Steel Res. 63 (2007) 1442–1451, https://doi.org/ 10.1016/J.JCSR.2007.01.002. [17] B. British Standards Institution, Structural Use of Steelwork in Building: Code of Practice for Design-Rolled and Welded Sections, BSI, 1990. [18] K. Tan, Z. Qian, Experimental behaviour of a thermally restrained plate girder loaded in shear at elevated temperature, J. Constr. Steel Res. 64 (2007) 596–606, https://doi.org/10.1016/j.jcsr.2007.12.008. [19] Z. Qian, K. Tan, Deflection behaviour of plate girders loaded in shear at elevated temperatures, J. Constr. Steel Res. 65 (4) (2009) 991–1000. [20] E. Aziz, V. Kodur, An approach for evaluating the residual strength of fire exposed bridge girders, J. Constr. Steel Res. 88 (2013) 34–42. [21] M. Salminen, M. Heinisuo, Numerical analysis of thin steel plates loaded in shear at non-uniform elevated temperatures, J. Constr. Steel Res. 97 (2014) 105–113, https://doi.org/10.1016/j.jcsr.2014.02.002. [22] 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) 395–406, https://doi.org/10.1016/j.jcsr.2014.05.021. [23] 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. 105–112. [24] V.K.R. Kodur, M.Z. Naser, Effect of shear on fire response of steel beams, J. Constr. Steel Res. 97 (2014) 48–58, https://doi.org/10.1016/j.jcsr.2014.01.018. [25] 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) 91–103. [26] A. Ghadami, V. Broujerdian, Shear behavior of steel plate girders considering variations in geometrical properties, J. Constr. Steel Res. 153 (2019) 567–577, https://doi.org/10.1016/j.jcsr.2018.11.009. [27] G. Quan, S. Huang, I. Burgess, An analytical approach to modelling shear panels in steel beams at elevated temperatures, Eng. Struct. 85 (2015) 73–82. [28] A. Reis, N. Lopes, P. Real, Shear-bending interaction in steel plate girders subjected to elevated temperatures, Thin-Walled Struct. 104 (2016) 34–43, https://doi.org/ 10.1016/j.tws.2016.03.005. [29] EN 1993-1-5, Eurocode3: Design of Steel Structures - Part 1-5: Plated Structural Elements, European Committee for Standardisation (CEN), 2006. [30] EN 1993-1-2, Eurocode3: Design of Steel Structures - Part 1-2: General Rules Structural Fire Design, European Committee for Standardisation (CEN), 2005. [31] 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) 1–12, https:// doi.org/10.1016/j.jcsr.2015.10.001. [32] A. Reis, N. Lopes, E. Real, P. Real, Numerical modelling of steel plate girders at normal and elevated temperatures, Fire Saf. J. 86 (2016) 1–15, https://doi.org/ 10.1016/j.firesaf.2016.08.005.
Fig. 11. The ratio of Vu-AISC and Vu-Lee to the FEM results vs. λw-AISC at 800� C.
The results are briefly discussed as follows: 1. According to linear analysis, the elastic shear buckling strength of AISC 360-16 [38] is more conservative than FEM results such that the difference is about 40% for all web panel slenderness ranges. However, when using ESB coefficient of Lee et al. [39], which con siders the effects of the flange-to-web stiffness ratio, tf/tw, the maximum difference is about 2%. Therefore, the equation proposed by Lee et al. is more reliable for design purposes. 2. According to nonlinear analysis at ambient temperatures, Lee’s modified design equation [6] is more accurate in predicting the ul timate shear strength of compact long web panels with plastic shear buckling without considering the effect of residual stress and large initial geometrical imperfections. However, the difference between these two results increases with increasing the slenderness param eter. This difference for compact, non-compact, and slender web plates is about 1.5%, 15%, and 19%, respectively. 3. According to nonlinear analysis at ambient temperatures, AISC 36016 [38] provides an exact equation for predicting the ultimate shear strength of compact long web panels regardless of the effects of re sidual stress and large initial geometrical imperfections. The differ ence between the results of AISC 360-16 and those of FEM increases with increasing slenderness parameter. This difference is equal to 1.1%, 23%, and 28% for, respectively, compact, non-compact, and slender web panels. 4. According to nonlinear analysis at elevated temperatures, Lee’s modified design equation [6] to fire situation is more accurate for compact plates. However, this equation approaches non-conservative results for higher slenderness such that the differ ence reaches almost 40%. 5. By investigating the numerical analysis of long web panels at elevated temperatures, a new design equation was proposed to pre dict the ultimate shear strength of long web panels by considering the strength degradation caused by high slenderness, high temper ature and initial geometrical imperfections. 6. Further experimental study is recommended to verify the proposed shear design equation for long web panels with a/D between 3 and 6. Appendix A. Supplementary data Supplementary data related to this article can be found at https://do i.org/10.1016/j.jobe.2019.101070.
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[33] 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) 136–143, https://doi.org/10.1016/j.jcsr.2016.01.004. [34] M.M. Alinia, A. Gheitasi, M. Shakiba, Postbuckling and ultimate state of stresses in steel plate girders, Thin-Walled Struct. 49 (4) (2011) 455–464, https://doi.org/ 10.1016/j.tws.2010.12.008. [35] 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) 1017–1031, https://doi.org/10.1016/j.tws.2011.03.007. [36] A.J. Daley, D. Brad Davis, D.W. White, Shear strength of unstiffened steel I-section members, J. Struct. Eng. 143 (2017), https://doi.org/10.1061/(ASCE)ST.1943541X.0001639, 04016190.
[37] AASHTO, in: Bridge Design Specifications, American Association of State Highway and Transportation Officials, Washington, DC, 2014. [38] ANSI/AISC 360-16, Specification for Structural Steel Buildings, American. Inst. Steel Constr., 2016, pp. 1–612. [39] S.C. Lee, J.S. Davidson, C.H. Yoo, Shear buckling coefficients of plate girder web panels, Comput. Struct. 59 (1996) 789–795, https://doi.org/10.1016/0045-7949 (95)00325-8. [40] M. Amani, M.M. Alinia, M. Fadakar, Imperfection sensitivity of slender/stocky metal plates, Thin-Walled Struct. 73 (2013) 207–215, https://doi.org/10.1016/J. TWS.2013.08.010. [41] Abaqus 6.14, ABAQUS Analysis User’s Manual, ABAQUS Inc., 2016.
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Journal of Building Engineering journal homepage: http://www.elsevier.com/locate/jobe
Numerical investigation on ultimate shear strength of long steel plate girder web panels at high temperatures G.H. 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
Keywords: Fire condition Shear strength Long web panel Plate slenderness Finite element method
In this article, elastic shear buckling and ultimate shear strength of long steel plate girder web panels subjected to pure shear loading are investigated at both ambient and elevated temperatures by the finite element method (FEM). Ninety-six plate girders with compact, non-compact and slender long web panels are numerically analyzed and compared. From linear eigenvalue analysis, it is shown that AISC 360-16 predictions are more conservative than FEM results such that the difference is about 40% for all ranges of web panel slenderness. According to nonlinear analysis results at ambient temperature, there must be reduction factors applied to AISC 360-16 non-conservative predictions for considering the strength degradation caused by large slenderness values and effects due to initial geometrical imperfection. Furthermore, it is observed that the adopted equation of AISC 360-16 for fire situations yields values that are more non-conservative such that the difference reaches almost 18%. In this regard, the FEM results are used to develop a new equation for predicting the ultimate shear strength of long web panels by taking into account strength degradation caused by large slenderness values, high tem perature and initial geometrical imperfections.
1. Introduction 1.1. Background Steel plate I-girders have extensively been employed in the con struction of bridges and buildings. A conventional method of acquiring remarkable strength-to-weight ratios is to consider deep slender webs for I-girders, although shear buckling is a threat to slender web plates [1–3]. 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 thick web plates. Simultaneous geometrical instability and material yielding are notable phenomena in non-compact web plates [5]. There has been limited research on the postbuckling strength behavior of long web panels [6]. Basler [7] and Porter et al. [8] inde pendently developed a method to estimate the postbuckling reserve shear strength due to tension field action in steel plate girders. The
proposed methods, deal with web panels with aspect ratios of a/D � 3, and they are fundamentally based on the assumption that in order to the web panel to develop a complete tension field there must be anchor systems such as rigid flanges and adjacent panels [6]. where, a is the transverse stiffeners’ distance relative to each other, and D is the web €oglund [9] experimentally investigated the plate depth. In 1971, Ho shear behavior of long web panels and developed a theory to consider their shear strength [10]. According to test results, plate girders without intermediate transverse stiffeners showed substantial postbuckling strengths. Despite the former assumptions of existing classical tension field theories, Lee and Yoo [11–14] based on the analytical and exper imental studies found that even a simply supported panel without the flanges and adjacent panels are able to develop the postbuckling strength that is quantitatively equivalent to results which are observed in tests. Recently, these authors have emphasized the presence of post buckling strength in the long web panels. Although the postbuckling strength of plate girders with long web panels can be significant, how ever, it is not considered in many steel structures design codes due to lack of a comprehensive study [6]. AISC 360-10 limited the ultimate shear strength of slender web panels to their elastic shear buckling
* Corresponding author. E-mail address: [email protected] (S.A. Mousavi Ghasemi). https://doi.org/10.1016/j.jobe.2019.101070 Received 14 July 2019; Received in revised form 12 November 2019; Accepted 14 November 2019 Available online 19 December 2019 2352-7102/© 2019 Published by Elsevier Ltd.
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Journal of Building Engineering 29 (2020) 101070
strength. Currently, AISC 360–16, has improved the shear design equation for slender web panels. This development indicates that past research has emphasized existing the post buckling shear strength of the long web panels. In other words, the ultimate shear strength of slender web panels is not limited to their elastic shear strength. Given the newness of the topic, to gain a better understanding of the shear char acteristics of unstiffened web plates, the following review is provided with the aim of studies at elevated temperatures.
buckling, either at elevated or normal temperatures. The exactness of the model was verified using FEM. The shear-bending behavior of ho mogenous I-girders when exposed to fire with the aid of numerical methods was the subject of another study by Reis et al. [28]. They showed that the EC3 [29] recommendation for the safety assessment of the studied problem was suitable for design purposes if stress-strain reduction factors of Ref. [30] were directly applied at high tempera tures. For steel plate girders, the same researchers also provided novel design criteria so as to estimate the USS when fire occurs [31]. Using a numerical approach, Reis et al. [32] examined the effect of several pa rameters 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. [33] 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 was between 19% and 29% of approaches based on the explicit flange model. They also showed that the buckling coefficients and conse quently critical buckling loads decreased with increasing aspect ratio.
1.2. Literature review There have been extensive studies on the shear behavior of steel plates at ambient temperatures. Fire design is another important aspect that should be addressed separately [15]. 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. [16] to estimate the failure loads when a mixture of axial and shear or pure shear stresses were 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 became the dominant failure mode at such temperatures. Noteworthy is that the authors extended the ambient temperature relations proposed in the BS 5950: Part 1 standard [17] so as to study axial stress and material conditions at elevated temperatures. In an empirical study under fire condition, Tan and Qian [18] investigated the behavior of twelve simply-supported steel plate girders with thermal axial restraints. It was observed that the ultimate shear strength (USS) decreased significantly under a thermal restrain effect. This influence appeared to be more significant with higher thermal restraint and with greater web plate slenderness. 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 [19]. A recent study by Aziz and Kodur [20] on steel bridge girders employed a nonlinear finite element method (FEM) to estimate their residual strength, indicating that the highest fire temperature (along with the corresponding steel temperature) is the most influential parameter. In another study on thin steel plates, Salminen [21] benefited from FEM to propose an analytical method capable of esti mating the shear resistance at uneven high temperatures. The shear buckling characteristics of plate girder webs are among other 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 [22]. Scandella et al. [23] showed that the failure of steel plate girders as a result of shear web buckling when exposed to fire is a probable phe nomenon 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 eval uation of shear capacity such as the temperature-dependent degradation of components [24,25]. 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 web slenderness parameter which was inca pable of considering the impact of boundary conditions on the web panel’s shear behavior [1–3,26]. In a study on beams of Class 1, Quan et al. [27] developed a theoretical model for analyzing the shear
1.3. Elastic shear strength of long web panels at ambient temperatures Many studies have investigated the elastic shear buckling (ESB) behavior of plates [1,3,26,34,35]. Based on these researches, the ESB strength of web panels at ambient temperature, Vcr,20, subjected to pure shear loading, is obtained using the classical plate buckling equation as: Vcr;20 ¼ K
π2 E20 t3w 12 ð1 υ2 Þ D
(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 [13,22] and is found as � 5:34 Kss ¼ 4 þ 2 for ϕ ¼ a D < 1 (2a) ϕ Kss ¼ 5:34 þ
Ksf ¼
4 for ϕ ¼ a ϕ2
5:34 2:31 þ 8:39 ϕ þ ϕ ϕ2
Ksf ¼ 8:98 þ
5:61 ϕ2
� D�1
3:44 for ϕ ¼ a
� 1:99 for ϕ ¼ a D�1 ϕ3
(2b) � D<1
(2c) (2d)
where Kss is the ESB coefficient of plates with simply-supported condi tions along all four edges, Ksf is the ESB coefficient of plates with two opposite simply-supported edges and two other fixed edges, ϕ ¼ a/D is the web panel aspect ratio, and a is the transverse stiffeners’ distance relative to each other. In Basler’s model [13,22,36], 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, AASHTO [37] (Article 6.10.9.3.2–7) and AISC 360-16 [38] (Article G2-5) 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 [39]. In these cases, Lee et al. [39] proposed Eqs. (2f) to (2g) to calculate the ESB coefficient. For unstiffened long web panels (LWPs) with a/D > 3, the ESB strength is obtained assuming the ESB coefficient KAISC ¼ 5.34 and KAASHTO ¼ 5 in accordance with AISC 360-16 [38] (Article G2-5) and AASHTO [37] (Article 6.10.9.2), respectively. It must be noted that, according to AASHTO [37] (Article 6.10.9.1), interior web panels of I-shaped members (1) without a lon gitudinal stiffener and with a transverse stiffener spacing not exceeding 2
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Journal of Building Engineering 29 (2020) 101070
3D, or (2) with one or more longitudinal stiffeners and with a transverse stiffener spacing not exceeding 1.5D, shall be considered as stiffened web panels. Otherwise, the panel shall be considered unstiffened. � 5 KAASHTO ¼ KAISC ¼ 5 þ 2 for φ ¼ a D � 3 (2e) ϕ KLee ¼ Kss þ
4 Ksf 5
Kss
KLee ¼ Kss þ
4 Ksf 5
Kss
�
� 2 2 3
� 1
�
tf tw
�� for 0:5 �
tf <2 tw
tf �2 tw
for
1.5. Aims and scopes According to the above literature review and to the best of our knowledge, there is a lack of comprehensive research on the shear behavior of long web panels (LWPs) with a/D between 3.0 and 6.0 at both ambient and elevated temperatures, where a/D is the web panel aspect ratio, a is the transverse stiffeners’ distance relative to each other, and D is the web plate depth. In addition, it was found that the LWP shear design relationships of AISC 360-16 [38] and Lee [6] do not take fire effects into consideration. Moreover, most researchers have dis cussed their results based on web plate slenderness ratio parameter, D/tw, which is unable to take the effect of web plate boundary conditions and material properties into account [1,3,26]. Therefore, in this article, firstly a new effective parameter, namely non-dimensional web plate slenderness (NWS), is introduced. This parameter is employed in a nonlinear finite element analysis to investigate the elastic shear buckling (ESB) and ultimate shear strength (USS) of LWPs at both ambient and elevated temperatures. Then, to examine the predictions of AISC 360-16 [38] and Lee et al. [6] in case of fire, 96 plate girders with compact, non-compact and slender LWPs are numerically analyzed and compared. To this end, the web panel shear design relationships mentioned in AISC 360-16 and those of Lee et al. [6] are modified to be used in fire con ditions. This is achieved by direct utilization of steel stress-strain reduction factors in EN1993-1-2 [30] at elevated temperatures. Finally, a new equation is proposed to obtain the USS of LWPs under pure shear by considering the strength degradation caused by large slenderness values, high temperatures and initial geometrical imperfection.
(2f) (2g)
1.4. Ultimate shear strength of LWPs at ambient temperatures 1.4.1. AISC design equation According to AISC specifications (Chapter G) [38], the design ulti mate shear strength (USS) of unstiffened web plates at ambient tem peratures (20� C–150� C), Vu-AISC,20 ¼ φvVn-AISC,20 is found as: � Vu AISC;20 ¼ φv 0:6σ yw;20 h tw for λw;20 � 1:1 (3a) Vu
AISC;20
λw;20
¼ φv 0:6σ yw;20 h tw
AISC ¼
D tw
�
�
1:1
λw;20
� for λw;20 > 1:1
AISC
(3b)
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
σ yw;20
(3c)
KAISC E20
where Vn-AISC,20 is the nominal shear strength at 20� C, φv ¼ 0.9 repre sents the shear resistance factor, σyw,20 is the yield stress of web plate at 20� C, and h signifies the overall depth of girder. Also, λw,20-AISC repre sents the non-dimensional web slenderness (NWS) parameter at 20� C which is calculated based on the AISC 360-16 ESB coefficient (KAISC). One should note that based on the NWS parameter, web panels can be classified as slender, non-compact, and compact plates [1–3,5,26,40]. Accordingly, plates with λw,20 � 1.1 and λw,20 > 1.37 are considered as compact and slender web panels, respectively. In addition, plates with 1.1 < λw,20 � 1.37 are considered as non-compact web panels with in elastic shear buckling failure mode.
2. Materials and methods 2.1. Specifications of steel plate girder models In order to investigate the shear behavior of LWPs under pure shear at both ambient and elevated temperatures, 96 models with slender (S), non-compact (NC) and compact (C) webs at four temperatures of 20� C, 400� C, 600� C and 800� C were constructed using the FEM software package, ABAQUS. Selecting a wide range of thickness and NWS pa rameters provides the possibility for studying the elastic/plastic buck ling behavior of slender and stocky 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 – 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, PG3-2 represents the PG3 girder with a girder length of a ¼ 3000mm, an aspect ratio of a/D ¼ 3, and slender web plate with thickness of tw ¼ 10mm. As another example, PG4-6 represents the PG4 girder with an aspect ratio of 4, and compact web plate with thickness of tw ¼ 20mm. 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 [2,6]. In addition, to eliminate the lateral torsional buckling phenomenon, the entire mid-nodes of flanges are constrained in the y direction [1,2,26].
1.4.2. Lee’s modified design equation Lee et al. [6] investigated the validity of an arbitrary limit imposed by Basler on the maximum aspect ratio of a transversely stiffened web panel and developed a new equation for predicting the USS of web panels with high aspect ratios (3 � a/D � 6) at ambient temperatures. According to this research, the USS of an LWP at 20� C, Vu-Lee,20, without considering the effects of residual stress, and large initial geometrical imperfections, is calculated as: � Vu Lee;20 ¼ Rλ;20 0:58σ yw;20 D tw ð0:6C þ 0:4Þ (4a) 8 for < Rλ;20 ¼ 1 for Rλ;20 ¼ 1:35C þ 0:6 : Rλ;20 ¼ 5:62C þ 0:145 for 8 C¼1 > > > > > 1 :12 >
D AASHTO ¼ tw
C � 0:3 0:1 < C < 0:3 C � 0:1
for
λw;T
AASHTO
< 1:12
for
1:12 � λw;T
AASHTO
for
λw;T
> 1:40
AASHTO
(4b)
� 1:40
(4c)
2.2. Material properties
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
σyw;20
KAASHTO E20
(4d)
In this article, the stress-strain reduction factors provided in EN 1993-1-2 [30] specific to steel for elevated temperatures are employed to consider fire effects. In fact, the design criteria at ambient tempera tures are modified using the reduction factors such that they could be exploited for fire conditions [2,22,28]. At ambient temperature, yield
where Rλ,20 is the shear strength reduction factor due to high slenderness at 20� C, and λw,20-AASHTO represents the NWS parameter at 20� C which is calculated based on the AASHTO ESB coefficient (KAASHTO). 3
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Journal of Building Engineering 29 (2020) 101070
Table 1 Specifications of the FEM models (1 inch ¼ 25.4 mm). Model
a/D
Temperature (� C)
tw (mm)
tf (mm)
D/tw
λw,20-AASHTO
λw,20-AISC
Web plate classificationa
PG3-1 PG3-2 PG3-3 PG3-4 PG3-5 PG3-6
3
20, 400, 600, 800
8 10 12 14 16 20
16 20 24 28 32 40
125 100 83.33 71.42 62.50 50
2.055 1.644 1.370 1.174 1.027 0.822
2.055 1.644 1.370 1.174 1.027 0.822
S S NC NC C C
PG4-1 PG4-2 PG4-3 PG4-4 PG4-5 PG4-6
4
20, 400, 600, 800
8 10 12 14 16 20
16 20 24 28 32 40
125 100 83.33 71.42 62.50 50
2.165 1.732 1.443 1.237 1.083 0.866
2.095 1.676 1.397 1.197 1.048 0.838
S S NC NC C C
PG5-1 PG5-2 PG5-3 PG5-4 PG5-5 PG5-6
5
20, 400, 600, 800
8 10 12 14 16 20
16 20 24 28 32 40
125 100 83.33 71.42 62.50 50
2.165 1.732 1.443 1.237 1.083 0.866
2.095 1.676 1.397 1.197 1.048 0.838
S S NC NC C C
PG6-1 PG6-2 PG6-3 PG6-4 PG6-5 PG6-6
6
20, 400, 600, 800
8 10 12 14 16 20
16 20 24 28 32 40
125 100 83.33 71.42 62.50 50
2.165 1.732 1.443 1.237 1.083 0.866
2.095 1.676 1.397 1.197 1.048 0.838
S S NC NC C C
a
S: Slender; NC: Non-compact; C: Compact.
Fig. 2. Stress-strain reduction factors of steel at high temperatures.
Fig. 1. Boundary condition and pure shear loading of models in the FEM analysis.
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 diagrams of Eurocode reduction factors related to yield stress (σy), Young’s modulus (E) and proportional limit stress (σ p), ky,T, kE,T and kp,T, respectively with respect to temperature, where the superscript T refers to high temper ature values. 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.
Fig. 3. Material properties of FEM models at ambient and elevated temperatures.
2.3. Numerical modeling and validation 2.3.1. Numerical modeling The numerical modeling and analysis are performed using ABAQUS 4
G.H. Pourmoosavi et al.
Journal of Building Engineering 29 (2020) 101070
[41]. 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 LWPs. In this method, if the model is defined as a perfect plate without initial geometrical imperfections, the buckling phenome non may not be observed and the system yields under a specific load [1, 26]. 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 [6]. 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 PG3-1 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 0.42%.
Fig. 5. Comparing the experimental and numerical analysis results.
3.1. Elastic shear buckling strength In this section, the ESB strength of models is discussed at both ambient and elevated temperatures. The ESB strength of each model is obtained by linear eigenvalue analysis using FEM. For example, Fig. 6 shows the first mode shapes of PG3-1, PG4-1, PG5-1, and PG6-1 models at 20� C. The ESB strength of the finite element (Vcr-FEM), AISC (Vcr-AISC), and Lee et al. (Vcr-Lee) models are shown in Table 2. According to this table, AISC prediction is more conservative than FEM results such that the difference is about 40% for all ranges of web panel slenderness. On the other hand, the ESB coefficient of Lee et al. which considers the effects of flange-to-web stiffness ratio, tf/tw, on the ESB strength of girders, leads to more accurate results with a maximum difference of 2%. It must be noted that similar results with a similar error percentage should be obtained at high temperatures.
2.3.2. Verification of numerical modeling process Definitely the best way to corroborate any numerical analysis is to compare the results with experimental ones [26,31,32]. In 2007, Vimonsatit et al. [16] 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 verify of the nu merical modeling process the test model TG3 is 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. [16]. 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 dif ference 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.
3.2. Ultimate shear strength at ambient temperatures In this section, the USS of LWPs is discussed at ambient temperatures. The USS strength of each model is obtained by the nonlinear Riks analysis using FEM. For example, Fig. 7 shows the failure mode shapes corresponding to the PG3-1, PG4-1, PG5-1, and PG6-1 models at 20� C. The USS strength of the finite element (Vu-FEM), AISC (Vu-AISC), and Lee et al. (Vu-Lee) models is shown in Table 3. The ratio of predicted values based on AISC (Eq. (3)) and Lee et al. [6] (Eq. (4)) to the ones obtained by the FEM model, Vu-FEM, with respect to the NWS parameter, λw,20-AISC, is plotted in Fig. 8. � (7a) Vu;T proposed ¼ Rλ;T RT RImp ky;T 0:58σyw;20 D tw ð0:6C þ 0:4Þ
3. Results and discussion In this section, the results are investigated in three parts. In the first part, the ESB strength of the FEM models is discussed. In the second part, the USS of models is investigated at ambient temperature. Finally, the third part is devoted to the USS of LWPs considering the reduction factors due to large web panel slenderness (Rλ,T), high temperatures (RT), and initial geometric imperfections (RImp).
8 for < Rλ;T ¼ 1 for Rλ;T ¼ 1:35C þ 0:6 : Rλ;T ¼ 5:62C þ 0:145 for 8 > < RT ¼ 1 > : RT ¼
1 1:03 � ðλw;T
0:3 AASHTO Þ
C � 0:3 0:1 < C < 0:3 C � 0:1
(7b)
for
λw;T
AASHTO
< 0:9
for
λw;T
AASHTO
� 0:9
� �0:05 8 tw > > � ð1:15 0:3λw;Lee Þ � 1 > RImp ¼ 1:03 � < Imp � �0:05 > > > : RImp ¼ 1:03 � tw � ð0:34 þ 0:3λw;Lee Þ � 1 Imp
(7c)
for
λw;Lee < 1:37
for
λw;Lee � 1:37 (7d)
Fig. 4. Convergence study and mesh sensitivity in the finite element models. 5
Journal of Building Engineering 29 (2020) 101070
G.H. Pourmoosavi et al.
Fig. 6. First mode shapes of PG3-1, PG4-1, PG5-1, and PG6-1 models at 20� C. Table 2 Elastic shear buckling strength at ambient temperature (20� C). a/D
Model
D/tw
λw,20-AASHTO
λw,20-AISC
Vcr-FEM (1)
Vcr-AISC (2)
Vcr-Lee (3)
Ratio (2/1)
Ratio (3/1)
3
PG3-1 PG3-2 PG3-3 PG3-4 PG3-5 PG3-6
125 100 83.33 71.42 62.50 50
2.055 1.644 1.370 1.174 1.027 0.822
2.055 1.644 1.370 1.174 1.027 0.822
816.00 1591.80 2746.60 4354.50 6488.40 12620.0
513.66 1003.23 1733.59 2752.87 4109.25 8025.87
812.59 1587.10 2742.50 4355.00 6500.75 12696.78
0.629 0.630 0.631 0.632 0.633 0.636
0.996 0.997 0.999 1.000 1.002 1.006
4
PG4-1 PG4-2 PG4-3 PG4-4 PG4-5 PG4-6
125 100 83.33 71.42 62.50 50
2.165 1.732 1.443 1.237 1.083 0.866
2.095 1.676 1.397 1.197 1.048 0.838
795.85 1552.40 2678.70 4246.80 6327.70 12370.0
494.22 965.27 1667.99 2648.71 3953.76 7722.19
791.96 1546.79 2672.85 4244.39 6335.64 12374.30
0.621 0.622 0.623 0.624 0.625 0.624
0.995 0.996 0.998 0.999 1.001 1.000
5
PG5-1 PG5-2 PG5-3 PG5-4 PG5-5 PG5-6
125 100 83.33 71.42 62.50 50
2.165 1.732 1.443 1.237 1.083 0.866
2.095 1.676 1.397 1.197 1.048 0.838
786.82 1534.80 2648.30 4198.50 6255.90 12168.0
494.22 965.27 1667.99 2648.71 3953.76 7722.19
782.05 1527.45 2639.43 4191.31 6256.42 12219.57
0.628 0.629 0.630 0.631 0.632 0.635
0.994 0.995 0.997 0.998 1.000 1.004
6
PG6-1 PG6-2 PG6-3 PG6-4 PG6-5 PG6-6
125 100 83.33 71.42 62.50 50
2.165 1.732 1.443 1.237 1.083 0.866
2.095 1.676 1.397 1.197 1.048 0.838
787.64 1539.90 2657.10 4212.50 6276.80 12209.0
494.22 965.27 1667.99 2648.71 3953.76 7722.19
776.64 1516.87 2621.15 4162.29 6213.10 12134.97
0.627 0.627 0.628 0.629 0.630 0.632
0.986 0.985 0.986 0.988 0.990 0.994
Note: 1 inch ¼ 25.4 mm, 1 kip ¼ 4.45 kN, 1 psi ¼ 6.895 kPa.
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Journal of Building Engineering 29 (2020) 101070
Fig. 7. Failure mode shapes of PG3-1, PG4-1, PG5-1, and PG6-1 models at 20� C. Table 3 Ultimate shear strength at ambient temperature (20� C). a/D
Model
D/tw
λw,20-ASHTO
λw,20-
Vu-FEM (1)<
Vu-AISC (2)
Vu-Lee (3)
Ratio (2/1)
Ratio (3/1)
3
PG3-1 PG3-2 PG3-3 PG3-4 PG3-5 PG3-6
125 100 83.33 71.42 62.50 50
2.055 1.644 1.370 1.174 1.027 0.822
2.055 1.644 1.370 1.174 1.027 0.822
977.57 1555.19 2077.87 2432.45 2785.40 3485.95
715.93 1127.31 1635.81 2243.51 2757.89 3499.20
867.31 1302.46 1859.39 2368.44 2784.00 3480.00
0.732 0.725 0.787 0.922 0.990 1.004
0.887 0.837 0.895 0.974 0.999 0.998
4
PG4-1 PG4-2 PG4-3 PG4-4 PG4-5 PG4-6
125 100 83.33 71.42 62.50 50
2.165 1.732 1.443 1.237 1.083 0.866
2.095 1.676 1.397 1.197 1.048 0.838
953.71 1523.96 2074.67 2431.23 2788.36 3535.08
702.25 1105.77 1604.56 2200.66 2757.89 3499.20
836.54 1242.36 1779.31 2297.56 2784.00 3480.00
0.736 0.726 0.773 0.905 0.989 0.990
0.877 0.815 0.858 0.945 0.998 0.984
5
PG5-1 PG5-2 PG5-3 PG5-4 PG5-5 PG5-6
125 100 83.33 71.42 62.50 50
2.165 1.732 1.443 1.237 1.083 0.866
2.095 1.676 1.397 1.197 1.048 0.838
941.55 1510.79 2072.31 2430.98 2783.09 3474.01
702.25 1105.77 1604.56 2200.66 2757.89 3499.20
836.54 1242.36 1779.31 2297.56 2784.00 3480.00
0.746 0.732 0.774 0.905 0.991 1.007
0.888 0.822 0.859 0.945 1.000 1.002
6
PG6-1 PG6-2 PG6-3 PG6-4 PG6-5 PG6-6
125 100 83.33 71.42 62.50 50
2.165 1.732 1.443 1.237 1.083 0.866
2.095 1.676 1.397 1.197 1.048 0.838
938.75 1507.15 2073.13 2428.85 2777.06 3470.02
702.25 1105.77 1604.56 2200.66 2757.89 3499.20
836.54 1242.36 1779.31 2297.56 2784.00 3480.00
0.748 0.734 0.774 0.906 0.993 1.008
0.891 0.824 0.858 0.946 1.002 1.003
AISC
Note: 1 inch ¼ 25.4 mm, 1 kip ¼ 4.45 kN, 1 psi ¼ 6.895 kPa.
7
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Journal of Building Engineering 29 (2020) 101070
where Vn,T is the nominal shear strength, λw,T is the NWS parameter at a constant temperature of T, λw,20 is the NWS parameter at 20� C and Vp,20 ¼ 0.6σyw,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 λw, T � 1.1 and λw,T > 1.37 are considered as compact and slender plates, respectively. Also, web panels with 1.1 < λw,T � 1.37 are considered to be non-compact plates with inelastic shear buckling failure mode. On the other hand, Lee’s modified design equation for unstiffened long web plates at elevated temperatures, Vu-Lee,T, shall be determined as � (6a) Vu Lee;T ¼ Rλ;T ky;T 0:58σyw;20 D tw ð0:6C þ 0:4Þ 8 for < Rλ;T ¼ 1 Rλ;T ¼ 1:35C þ 0:6 for : Rλ;T ¼ 5:62C þ 0:145 for
Fig. 8. The ratio of Vu-AISC and Vu-Lee to the FEM results vs. λw-AISC at 20� C.
8 C¼1 > > > > > 1 :12 >
λw;T
for
λw;T
AASHTO
< 1:12
for
1:12 � λw;T
AASHTO
for
λw;T
> 1:40
sffiffiffiffiffiffiffi ky;T � λw;20 AASHTO ¼ kE;T
AASHTO
� 1:40
8 C¼1 > > > > > 1 :12 >
(7e)
λw;T
(7f)
AASHTO
3.3.1. Evaluation method Part 1–5 of EC3 [29] defines no specific criterion for USS verification under fire conditions. In this article, the stress-strain reduction factors provided in EN 1993-1-2 [30] 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 [2,22,28]. Based on the direct application of the reduction factors of EC3 for steel stress-strain rela tionship, the modified design shear strength of AISC for unstiffened web plates at elevated temperatures, Vu,T ¼ φvVn,T, shall be determined as: � Vu;T ¼ ky;T : φv 0:6σ yw;20 h tw for λw;T � 1:1 (5a)
λw;T
AISC ¼
sffiffiffiffiffiffiffi ky;T : λw;20 kE;T
AISC
�
1:1
λw;20
AISC
� for λw;T > 1:1
λw;T
AASHTO
< 1:12
for
1:12 � λw;T
AASHTO
for
λw;T
> 1:40
sffiffiffiffiffiffiffi ky;T : λw;20 kE;T
AASHTO
AASHTO
� 1:40
(6c)
(6d)
3.3.2. Ultimate shear strength In this section, the USS of LWPs is discussed at elevated temperatures (400� C, 600� C and 800� C). The USS of the finite element (Vu-FEM), AISC (Vu-AISC), and Lee et al. (Vu-Lee) models are reported in Tables 4–6. The ratio of predicted values based on AISC (Eq. (5)) and Lee et al. (Eq. (6)) to the ones obtained by FEM, Vu-FEM, with respect to the NWS parame ters of λw,400-AISC, λw,600-AISC, and λw,800-AISC are plotted in Figs. 9–11, respectively. According to Figs. 9 and 10, both AISC and Lee equations are more accurate and in the safe region for compact plates. However, these equations lead to the unsafe condition of Ratio >1 for higher slenderness values. The difference between the USS obtained via ABAQUS and the results of modified AISC equation (Eq. (5)) at 400� C reaches almost 4% and 7% for non-compact and slender LWPs, respectively. However, this difference between the results of Lee’s modified equation and FEM are about 11% and 30% for non-compact and slender LWPs, respectively. Also, by increasing the temperature to 600� C, especially for slender plates, AISC and Lee’s equations yield values that are more nonconservative for USS such that the difference between the results of FEM and Lee reaches 40%. However, this difference for AISC equation is about 18%. According to Fig. 11, both modified AISC and Lee equations are more accurate for compact web panels at higher temperature of 800� C. However, these equations generally lead to the unsafe condition for noncompact and compact plates. The difference between the results of modified AISC equation and FEM are about 10% and 15% for noncompact and slender LWPs, respectively. That is while; these differ ences for Lee’s modified equation reaches almost 18% and 35%, respectively. Comparison the USS results at 400 and 600� C with those of 800� C shows the different shear behavior of the plates at high temper atures compared to the mid-range temperatures. The cause of this discrepancy is examined in a step-by-step method in Ref. [22]. In 2014, Garlock and Glassman [22] numerically evaluated the Basler–Thürli mann’s theoretical equation [7] at elevated temperatures, which serves
3.3. Ultimate shear strength at elevated temperatures
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi � ky;T :kE;T : φv 0:6σyw;20 h tw
¼
for
(6b)
where Rλ,T is the shear strength reduction factor due to high slenderness at elevated temperature, and λw,T-AASHTO represents the NWS parameter at a specific temperature (T) which is calculated based on the AASHTO ESB coefficient (KAASHTO).
According to this figure, both AISC and Lee modified design equations are more accurate in compact plates with plastic shear buckling. How ever, by increasing the NWS parameter, the difference between Vu-AISC, Vu-Lee and Vu-FEM is increased. The maximum difference between the results of Lee’s equation and FEM for compact, non-compact, and slender web plates is about 1.5%, 15%, and 19%, respectively. On the other hand, this difference between the results of AISC prediction and FEM for compact, non-compact, and slender web plates reaches 1.1%, 23%, and 28%, respectively. It is important to note that this article ex amines the USS of LWPs without considering the effects of residual stress and large initial geometrical imperfections. In addition, AISC pre dictions considering the effects of residual stress and large initial geometrical imperfection are close to those obtained using FEM, espe cially in compact and non-compact web panels. Therefore, certain reduction factors must be applied to AISC predictions to take the effect of mentioned parameters on the shear strength degradation into account.
Vu;T ¼
AASHTO
C � 0:3 0:1 < C < 0:3 C � 0:1
(5b)
(5c)
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Journal of Building Engineering 29 (2020) 101070
Table 4 Ultimate shear strength at 400� C. Model
D/tw
λw,400-AASHTO
λw,400-AISC
Vu-FEM (1)
Vu-AISC (2)
Vu-Lee (3)
Eq (7) (4)
Ratio (2/1)
Ratio (3/1)
Ratio (4/1)
PG3-1 PG3-2 PG3-3 PG3-4 PG3-5 PG3-6
125 100 83.33 71.42 62.50 50
2.456 1.965 1.637 1.404 1.228 0.982
2.456 1.965 1.637 1.404 1.228 0.982
571.07 885.36 1328.23 1829.07 2381.62 3658.47
598.99 943.17 1368.62 1877.06 2470.24 3499.20
736.48 1120.52 1568.77 2139.29 2636.98 3480.00
546.06 888.34 1313.63 1876.14 2407.14 3396.61
1.049 1.065 1.030 1.026 1.037 0.956
1.290 1.266 1.181 1.170 1.107 0.951
0.96 1.00 0.99 1.03 1.01 0.93
PG4-1 PG4-2 PG4-3 PG4-4 PG4-5 PG4-6
125 100 83.33 71.42 62.50 50
2.588 2.070 1.725 1.479 1.294 1.035
2.504 2.003 1.669 1.431 1.252 1.002
560.83 868.78 1303.09 1797.33 2741.76 3444.49
587.55 925.16 1342.47 1841.20 2423.06 3499.20
689.78 1078.45 1496.08 2023.85 2559.53 3480.00
503.50 841.71 1233.30 1747.33 2300.15 3343.86
1.048 1.065 1.030 1.024 0.884 1.016
1.230 1.241 1.148 1.126 0.934 1.010
0.90 0.97 0.95 0.97 0.98 0.97
PG5-1 PG5-2 PG5-3 PG5-4 PG5-5 PG5-6
125 100 83.33 71.42 62.50 50
2.588 2.070 1.725 1.479 1.294 1.035
2.504 2.003 1.669 1.431 1.252 1.002
555.22 864.91 1301.48 1795.23 2338.42 3427.82
587.55 925.16 1342.47 1841.20 2423.06 3499.20
689.78 1078.45 1496.08 2023.85 2559.53 3480.00
503.50 841.71 1233.30 1747.33 2300.15 3343.86
1.058 1.070 1.031 1.026 1.036 1.021
1.242 1.247 1.150 1.127 1.095 1.015
0.91 0.97 0.95 0.97 0.98 0.98
PG6-1 PG6-2 PG6-3 PG6-4 PG6-5 PG6-6
125 100 83.33 71.42 62.50 50
2.588 2.070 1.725 1.479 1.294 1.035
2.504 2.003 1.669 1.431 1.252 1.002
565.59 865.57 1303.97 1797.25 2337.15 3525.11
587.55 925.16 1342.47 1841.20 2423.06 3499.20
689.78 1078.45 1496.08 2023.85 2559.53 3480.00
503.50 841.71 1233.30 1747.33 2300.15 3343.86
1.039 1.069 1.030 1.024 1.037 0.993
1.220 1.246 1.147 1.126 1.095 0.987
0.89 0.97 0.95 0.97 0.98 0.95
Note: 1 inch ¼ 25.4 mm, 1 kip ¼ 4.45 kN, 1 psi ¼ 6.895 kPa. Table 5 Ultimate shear strength at 600� C. Model
D/tw
λw,600-AASHTO
λw,600-AISC
Vu-FEM (1)
Vu-AISC (2)
Vu-Lee (3)
Eq (7) (4)
Ratio (2/1)
Ratio (3/1)
Ratio (4/1)
PG3-1 PG3-2 PG3-3 PG3-4 PG3-5 PG3-6
125 100 83.33 71.42 62.50 50
2.530 2.024 1.687 1.446 1.265 1.012
2.530 2.024 1.687 1.446 1.265 1.012
236.40 389.99 580.75 817.59 1079.73 1711.53
273.27 430.30 624.40 856.36 1126.99 1644.62
333.27 515.12 717.41 973.85 1218.40 1635.60
244.91 404.76 595.40 846.47 1102.32 1582.23
1.156 1.103 1.075 1.047 1.044 0.961
1.410 1.321 1.235 1.191 1.128 0.956
1.04 1.04 1.03 1.04 1.02 0.92
PG4-1 PG4-2 PG4-3 PG4-4 PG4-5 PG4-6
125 100 83.33 71.42 62.50 50
2.666 2.133 1.777 1.523 1.333 1.066
2.580 2.064 1.720 1.474 1.290 1.032
275.73 381.32 565.63 798.08 1256.78 1598.13
268.05 422.08 612.47 840.01 1105.46 1644.62
312.96 496.49 685.22 922.72 1183.06 1635.60
226.41 384.06 559.84 789.58 1053.73 1557.65
0.972 1.107 1.083 1.053 0.880 1.029
1.135 1.302 1.211 1.156 0.941 1.023
0.98 1.01 0.99 0.99 1.00 0.97
PG5-1 PG5-2 PG5-3 PG5-4 PG5-5 PG5-6
125 100 83.33 71.42 62.50 50
2.666 2.133 1.777 1.523 1.333 1.066
2.580 2.064 1.720 1.474 1.290 1.032
288.18 380.85 566.54 799.22 1058.06 1594.36
268.05 422.08 612.47 840.01 1105.46 1644.62
312.96 496.49 685.22 922.72 1183.06 1635.60
226.41 384.06 559.84 789.58 1053.73 1557.65
0.930 1.108 1.081 1.051 1.045 1.032
1.086 1.304 1.209 1.155 1.118 1.026
0.98 1.01 0.99 0.99 1.00 0.98
PG6-1 PG6-2 PG6-3 PG6-4 PG6-5 PG6-6
125 100 83.33 71.42 62.50 50
2.666 2.133 1.777 1.523 1.333 1.066
2.580 2.064 1.720 1.474 1.290 1.032
242.85 382.13 569.92 802.70 1057.84 1682.60
268.05 422.08 612.47 840.01 1105.46 1644.62
312.96 496.49 685.22 922.72 1183.06 1635.60
226.41 384.06 559.84 789.58 1053.73 1557.65
1.104 1.105 1.075 1.046 1.045 0.977
1.289 1.299 1.202 1.150 1.118 0.972
0.99 1.01 0.98 0.98 1.00 0.93
Note: 1 inch ¼ 25.4 mm, 1 kip ¼ 4.45 kN, 1 psi ¼ 6.895 kPa.
as the basis of the AISC and AASHTO Specifications. In this research, the Basler–Thürlimann theoretical equation for predicting the USS of plates was evaluated by replacing web panel yield stress (σyw;20 ) with the stress at the fire condition, σyw,T ¼ kyσyw,20. According to results, it was rec ommended that for 20� C � T � 800� C, σyw,20 be substituted with kyσyw, � 20 and for T > 800 C, σyw,20 be substituted with kpσyw,20. Based on this research and the results of the present paper, the difference in shear behavior of plates at 800� C with their behavior at temperatures less than 800� C will be justified. However, by comparing the results in Figs. 9–11, it is observed that all data at high temperatures follow the same general trend. In addition, it can be mentioned that the difference between the
results of Eqs. (5) and (6) and those obtained from FEM is because the above equations are not basically provided for high temperatures. Therefore, it is necessary to propose a new equation to achieve the USS of LWPs under pure shear by taking into account strength degradation caused by high temperatures. 3.3.3. The proposed shear design equation In this section, FEM results are used to develop an equation for predicting the USS of LWPs at elevated temperatures. This equation is a function of NWS parameter at high temperatures, λw-AASHTO, based on the AASHTO ESB coefficient [37]. In addition, the effects of large 9
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Journal of Building Engineering 29 (2020) 101070
Table 6 Ultimate shear strength at 800� C. Model
D/tw
λw,800-AASHTO
λw,800-AISC
Vu-FEM (1)
Vu-AISC (2)
Vu-Lee (3)
Eq (7) (4)
Ratio (2/1)
Ratio (3/1)
Ratio (4/1)
PG3-1 PG3-2 PG3-3 PG3-4 PG3-5 PG3-6
125 100 83.33 71.42 62.50 50
2.272 1.817 1.515 1.298 1.136 0.909
2.272 1.817 1.515 1.298 1.136 0.909
66.86 104.00 153.00 219.10 272.00 382.90
71.23 112.17 162.76 223.23 293.77 384.91
90.14 131.14 186.19 256.96 303.66 382.80
68.42 106.43 159.60 230.68 283.76 382.47
1.065 1.079 1.064 1.019 1.080 1.005
1.348 1.261 1.217 1.173 1.116 1.000
1.023 1.023 1.043 1.053 1.043 0.999
PG4-1 PG4-2 PG4-3 PG4-4 PG4-5 PG4-6
125 100 83.33 71.42 62.50 50
2.394 1.915 1.596 1.368 1.197 0.957
2.316 1.853 1.544 1.323 1.158 0.926
66.02 95.89 146.06 210.00 262.37 400.06
69.87 110.02 159.65 218.96 288.16 384.91
83.83 125.73 176.84 242.11 294.45 382.80
62.64 100.45 149.23 213.98 270.88 376.53
1.058 1.147 1.093 1.043 1.098 0.962
1.270 1.311 1.211 1.153 1.122 0.957
0.949 1.048 1.022 1.019 1.032 0.941
PG5-1 PG5-2 PG5-3 PG5-4 PG5-5 PG5-6
125 100 83.33 71.42 62.50 50
2.394 1.915 1.596 1.368 1.197 0.957
2.316 1.853 1.544 1.323 1.158 0.926
71.00 107.35 160.00 228.00 303.00 390.43
69.87 110.02 159.65 218.96 288.16 384.91
83.83 125.73 176.84 242.11 294.45 382.80
62.64 100.45 149.23 213.98 270.88 376.53
0.984 1.025 0.998 0.960 0.951 0.986
1.181 1.171 1.105 1.062 0.972 0.980
0.882 0.936 0.933 0.939 0.894 0.964
PG6-1 PG6-2 PG6-3 PG6-4 PG6-5 PG6-6
125 100 83.33 71.42 62.50 50
2.394 1.915 1.596 1.368 1.197 0.957
2.316 1.853 1.544 1.323 1.158 0.926
70.07 105.57 163.00 233.08 290.000 396.86
69.87 110.02 159.65 218.96 288.16 384.91
83.83 125.73 176.84 242.11 294.45 382.80
62.64 100.45 149.23 213.98 270.88 376.53
0.997 1.042 0.979 0.939 0.994 0.970
1.196 1.191 1.085 1.039 1.015 0.965
0.894 0.952 0.916 0.918 0.934 0.949
Note: 1 inch ¼ 25.4 mm, 1 kip ¼ 4.45 kN, 1 psi ¼ 6.895 kPa.
slenderness values, Rλ,T, high temperature, RT, and initial geometrical imperfection, RImp, are considered. Firstly, the effect of large slenderness values is considered based on the equation proposed by Lee et al. [6], Eq. (6b). Next, to derive a reduction factor as a function of NWS parameter, RT, for predicting the USS reduction of a perfect LWP due to higher temperatures, a regression analysis and data fitting are performed. The correlation between the FEM results and those predicted by the 1/RT curve is shown in Figs. 9–11. Finally, for determining the USS reduction factor due to initial distortions, the proposed equation by Ghadami and Broujerdian [26] is used. Based on these assumptions, the final equation is obtained as Eq. (7) where Imp is the initial geometric imperfection magnitude applied to the perfect finite element model, and λw-Lee is the NWS parameter, which is calculated using the ESB coefficient value of Lee et al. [39] in Eqs. (2f) to (2g). The ratio of predicted values based on the proposed equation to the ones obtained by FEM, Vu-FEM, with respect to the NWS parameters of λw,400-AISC, λw,600-AISC and λw,800-AISC, are reported in Tables 4–6 According to the all results, the maximum and minimum difference between the results of the proposed equation with those of FEM results are about þ5.3% and 11.8%, respectively. In addition, the correlation factors, R2, of the mentioned equation at 400� C, 600� C and 800� C are obtained as 0.999, 0.998, and 0.995, respectively. Therefore, there is a good conformity between the USS determined from Eq. (7) and FEM analysis for LWPs at elevated temperatures.
Fig. 9. The ratio of Vu-AISC and Vu-Lee to the FEM results vs. λw-AISC at 400� C.
4. Conclusion In the present work, elastic shear buckling and ultimate shear strength of long steel web panels (LWPs) with a/D between 3.0 and 6.0 subjected to pure shear were investigated at both ambient and elevated temperatures by the finite element method. Ninety-six plate girders with compact, non-compact and slender web plates were numerically analyzed and the obtained results were compared with those of AISC 360-16 [38] and Lee’s modified equation [6]. Next, finite element re sults were used to develop an equation for predicting the ultimate shear strength of long steel plate girder web panels by considering the strength degradation caused by high slenderness, high temperature and initial geometrical imperfections.
Fig. 10. The ratio of Vu-AISC and Vu-Lee to the FEM results vs. λw-AISC at 600� C.
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Journal of Building Engineering 29 (2020) 101070
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) 263–270, https://doi.org/10.1016/J.JCSR.2015.04.025. [2] A. Ghadami, V. Broujerdian, Flexure–shear interaction in hybrid steel I-girders at ambient and elevated temperatures, Adv. Struct. Eng. 22 (2019) 1501–1516, 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) 508–518, 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) 412–420, 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) 1631–1643, https://doi.org/10.1016/j. jcsr.2009.04.001. [6] S.C. Lee, D.S. Lee, C.H. Yoo, Ultimate shear strength of long web panels, J. Constr. Steel Res. 64 (2008) 1357–1365, https://doi.org/10.1016/J.JCSR.2008.01.023. [7] K. Basler, Strength of plate girders in shear, J. Struct. Div. ASCE. 87 (7) (1961) 151–181. [8] D.M. Porter, K.C. Rockey, H.R. Evans, The collapse behavior of plate girders loaded in shear, J. Struct. Eng. 53 (1975) 313–325. [9] T. Hoglund, Simply Supported Long Thin Plate I-Girders without Web Stiffeners Subjected to Distributed Transverse Load, IABSE Colloq. Des. Plate Box Girders Ultimate Strength, 1971, pp. 85–97. London. [10] T. Hoglund, Design of thin plate I girders in shear and bending, R. Inst Technol Bull. 94 (1973). [11] S.C. Lee, C.H. Yoo, D.Y. Yoon, Behavior of intermediate transverse stiffeners attached on web panels, J. Struct. Eng. 128 (2002) 337–345. [12] S.C. Lee, C.H. Yoo, D.Y. Yoon, New design rule for intermediate transverse stiffeners attached on web panels, J. Struct. Eng. 129 (2003) 1607–1614. [13] S.C. 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Fig. 11. The ratio of Vu-AISC and Vu-Lee to the FEM results vs. λw-AISC at 800� C.
The results are briefly discussed as follows: 1. According to linear analysis, the elastic shear buckling strength of AISC 360-16 [38] is more conservative than FEM results such that the difference is about 40% for all web panel slenderness ranges. However, when using ESB coefficient of Lee et al. [39], which con siders the effects of the flange-to-web stiffness ratio, tf/tw, the maximum difference is about 2%. Therefore, the equation proposed by Lee et al. is more reliable for design purposes. 2. According to nonlinear analysis at ambient temperatures, Lee’s modified design equation [6] is more accurate in predicting the ul timate shear strength of compact long web panels with plastic shear buckling without considering the effect of residual stress and large initial geometrical imperfections. However, the difference between these two results increases with increasing the slenderness param eter. This difference for compact, non-compact, and slender web plates is about 1.5%, 15%, and 19%, respectively. 3. According to nonlinear analysis at ambient temperatures, AISC 36016 [38] provides an exact equation for predicting the ultimate shear strength of compact long web panels regardless of the effects of re sidual stress and large initial geometrical imperfections. The differ ence between the results of AISC 360-16 and those of FEM increases with increasing slenderness parameter. This difference is equal to 1.1%, 23%, and 28% for, respectively, compact, non-compact, and slender web panels. 4. According to nonlinear analysis at elevated temperatures, Lee’s modified design equation [6] to fire situation is more accurate for compact plates. However, this equation approaches non-conservative results for higher slenderness such that the differ ence reaches almost 40%. 5. By investigating the numerical analysis of long web panels at elevated temperatures, a new design equation was proposed to pre dict the ultimate shear strength of long web panels by considering the strength degradation caused by high slenderness, high temper ature and initial geometrical imperfections. 6. Further experimental study is recommended to verify the proposed shear design equation for long web panels with a/D between 3 and 6. Appendix A. Supplementary data Supplementary data related to this article can be found at https://do i.org/10.1016/j.jobe.2019.101070.
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[37] AASHTO, in: Bridge Design Specifications, American Association of State Highway and Transportation Officials, Washington, DC, 2014. [38] ANSI/AISC 360-16, Specification for Structural Steel Buildings, American. Inst. Steel Constr., 2016, pp. 1–612. [39] S.C. Lee, J.S. Davidson, C.H. Yoo, Shear buckling coefficients of plate girder web panels, Comput. Struct. 59 (1996) 789–795, https://doi.org/10.1016/0045-7949 (95)00325-8. [40] M. Amani, M.M. Alinia, M. Fadakar, Imperfection sensitivity of slender/stocky metal plates, Thin-Walled Struct. 73 (2013) 207–215, https://doi.org/10.1016/J. TWS.2013.08.010. [41] Abaqus 6.14, ABAQUS Analysis User’s Manual, ABAQUS Inc., 2016.
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