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Resistance of slender austenitic stainless steel I- girders subjected to patch loading
Structures 20 (2019) 924–934
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Resistance of slender austenitic stainless steel I- girders subjected to patch loading
T
Carlos Gracianoa, , Nelson Loaizaa, Euro Casanovab ⁎
a b
Universidad Nacional de Colombia, Facultad de Minas Sede Medellín, Departamento de Ingeniería Civil, A.A. 75267 Medellín, Colombia Universidad del Bío-Bío, Departamento Ingeniería Civil y Ambiental, Avenida Collao 1202, Concepción, Código Postal 4051381, Concepción, Chile
ARTICLE INFO
ABSTRACT
Keywords: Patch loading Resistance Resistance function Stainless steel Ultimate strength
This paper presents a numerical investigation on the patch loading resistance of slender austenitic stainless steel plate girders. Current design provisions for the resistance to patch loading of stainless steel girders are based on the plastic collapse mechanism observed in experimental and numerical studies conducted for carbon steel girders, disregarding the strain hardening capacities of stainless steel. At present, strength-curves approaches are used within European standards to deal with stability problems in steel plated structural elements. In this regard, three parameters require special attention: the yield load for plastic resistance, the resistance function depending on element slenderness, and the elastic buckling load. In this paper, an experimental dataset is firstly collected from the literature for comparative analysis. Subsequently, an extensive parametric study is conducted through nonlinear finite element analyses covering a wide range of slender stainless steel I-girder sections. Then, a resistance function is calibrated throughout a statistical evaluation of experimental and numerical results. Finally, the results show significant improvements in the predicted patch loading resistances of slender stainless steel I-girders.
Notation a bf E Fcr Fu Fy fyf fyw hw kF Leff ly MR MS m n s ss tf tw ⁎
length of web panel width of flange Young's modulus critical buckling load ultimate resistance calculated with the finite element model yield resistance flange yield strength web yield strength depth of web panel buckling coefficient effective length for resistance effective loaded length bending resistance applied bending moment mean value nonlinear material parameter standard deviation length of patch load flange thickness web thickness
v αF0 γM1 ϵ F0 F
ν σ σ0.2 σ1.0 σu χF
coefficient of variation v[=s/m] imperfection factor partial safety factor strain plateau length slenderness parameter Poisson's ratio stress stress at 0.2% strain stress at 1.0% strain ultimate stress resistance function
1. Introduction In recent years, an increasing awareness towards sustainability and low-cost maintenance has led to the improvement and development of material alloys used as construction materials. In this regard, stainless steel alloys play an important role in the design of structural elements in corrosive environments, and to improve their structural performance under fire conditions. Current design provisions for the resistance to patch loading of
Corresponding author at: Universidad Nacional de Colombia, Facultad de Minas, Departamento de Ingeniería Civil, A.A. 75267 Medellín, Colombia. E-mail address: [email protected] (C. Graciano).
https://doi.org/10.1016/j.istruc.2019.07.008 Received 28 May 2019; Received in revised form 18 July 2019; Accepted 20 July 2019 2352-0124/ © 2019 Institution of Structural Engineers. Published by Elsevier Ltd. All rights reserved.
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F
bf
ss y
tw
hw x
tf a Fig. 1. Plate girder subjected to patch loading (Notation).
stainless steel plate girders [1] are derived from those of carbon steel plate girders [2]. The provisions in the EC3 Part 1–5 [2] are based upon a plastic collapse mechanism proposed by Lagerqvist and Johansson [3] that considers an idealized elastic perfectly plastic material behavior, which is typical for carbon steel. Stainless steel exhibits nonlinear material strain-stress curves with significant strain hardening that should be considered in the strength calculations for economical and efficient structural designs [4–6]. Fig. 1 illustrates a typical setup for a plate girder subjected to patch loading. In 2003, Unosson et al. [7] conducted an experimental investigation on the resistance of austenitic stainless steel plate girders subjected to patch loading. Extending the slenderness of the girders, Unosson et al. [8] also performed a numerical study to investigate the influence of combined patch loading and bending, as well as the influence of the girder slenderness. The results showed that the predicted resistances calculated with a preliminary version of the EC3 Part 1.4 [9] were conservative. Recently, dos Santos et al. [10] studied experimentally and numerically the patch loading resistance of stocky welded plate girders manufactured of stainless steel. A comparison between experimental and numerical resistances with theoretical predictions was conducted, the results showed that the latter were generally rather conservative, particularly for girders with decreasing web slenderness. These conservative results for the stockier sections were also attributed to neglecting the pronounced strain hardening exhibited by stainless steel members with stocky webs. These behaviors have also been observed for stocky stainless steel cross sections under other loading configurations and addressed by means of the deformation based Continuous Strength Method (CSM) [4–6]. The CSM was originally developed for stainless steel and carbon steel materials, and is a deformation-based design framework that allows for the beneficial influence of strain hardening [4]. In recent years, the plate buckling rules in the EC3 Part 1–5 [2] are under review for harmonization of various buckling phenomena in plated structures [11]. Müller [12] introduced a proposal harmonizing the resistance function shape χF for all verifications of structural compressed members in the EC3 Part 1–5 [2]. This formulation comprises some dimensionless parameters, which can be calibrated in order to achieve a desired level of safety. After statistical evaluations of the resistance function [12], these parameters have been calibrated for carbon steel plate girders subjected to patch loading [13–15], leading to improvements in the predicted resistances. This paper aims at investigating the resistance to patch loading of slender austenitic stainless steel plate girders. At first, a dataset of experimental results is collected from the literature for comparative analysis. Secondly, an extensive parametric study is conducted through nonlinear finite element analyses in order to expand the available database. Thereafter, a statistical evaluation of the results is performed in order to calibrate a resistance function for the patch loading resistance
of slender stainless steel plate girders. Next, a new safety factor is determined according to the guidelines provided in international design codes. Finally, the results show a significant improvement in the predicted resistances. 2. Resistance models in European standards 2.1. Resistance according to the EC3 Part1.4 [1] - stainless steel In accordance with EC3 Part1.4 [1], the patch loading resistance for stainless steel plate girders without intermediate stiffener FRd is calculated by
FRd = f yw Leff tw /
(1)
M1
where fyw is the yield strength of the web, and Leff is the effective length for resistance to transverse loads
Leff =
F
(2)
ly
The reduction factor χF due to local buckling is F
=
0.5
1.0
(3)
F
and the effective loaded length ly is given by
l y = ss + 2t f (1 +
m1 + m2 )
a
(4)
where m1 and m2 are dimensionless parameters
m1 = f yf bf /f yw tw m2 = 0.02(h w /t f )2
(5)
In Eq. (4), m2 should be taken as zero if F < 0.5. The slenderness parameter F is obtained from F
=
Fy (6)
Fcr
Fy is the yield resistance
Fy = fyw tw l y
(7)
The buckling load Fcr is
Fcr = 0.9kF Et w3 / h w kF = 6 + 2
hw a
(8)
2
(9)
2.2. Resistance function update for carbon steel girders After consensus within the TWG 8.3 [15], any modification in the resistance function should be based on the proposal presented by Müller [12]. This proposal harmonizes the resistance function shape for all 925
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girders tested by dos Santos et al. [10]. All the experiments were conducted using austenitic stainless steel of three different grades: Grade EN 1.4301 for tests 1 to 5, Grade EN 1.4404 for tests 6 to 10, and Grade EN 1.4571 for tests 11 to 21. It must be mentioned that, the investigation conducted herein considers girders subjected to only patch loading, therefore tests with combined bending moment and patch loading are discarded. The interaction with bending is taken into account when the applied moment MS is 50% greater than the bending resistance MR of the girder. To verify this, MS and MR are calculated following the procedure described in EC3 Part 1–4 [1], based on the effective width concept. Therefore, from the test reported in Table 2 only 6 out of the 21 experiments are taken into account: 5 from Unosson et al. [7] and only 1 (IOF-H152L150-SS30) from dos Santos et al. [10].
Table 1 αF0, F 0 and γM1 for different resistance models. Proposal
αF0
Müller [12] Davaine [13] Gozzi [14] Chacón et al. [15]
0.34 0.21 0.50 1.00 0.75
F0
γM1
0.8 0.8 0.6 0.5 0.5
– 1.1 1.0 1.0 1.1
verifications of structural compressed members within the EC3 Part 1–5 [2]. A general form of this proposal is presented as follows F
=
1.0 2
+
1
(10)
F
4. Nonlinear finite element model
where
=
1 [1 + 2
F0 ( F
F0
)+
F]
Geometrically and materially nonlinear with imperfection analyses (GMNIA) are performed to evaluate the influence of various geometrical parameters on the postbuckling behavior of the slender stainless steel girders subjected to patch loading. To achieve that, a nonlinear finite element model is developed using ANSYS [17]. The web panel, flanges and vertical stiffeners of the girder are modeled using the fournode shell element S181 [17] with six degrees of freedom at each node. Material nonlinearities are considered in the stainless steel model. In this case, a multilinear stress-strain curve was defined using the Eq. (12) of true stress-strain curves with strain hardening provided in the EC3 Part 1.4-Annex C.2 [1].
(11)
In Eq. (11), the imperfection factor αF0 and the plateau length F 0 , need calibration in order to achieve a desired level of safety. Chacón et al. [15] performed a statistical evaluation of a new proposal for the patch loading resistance of carbon steel plate girders. Accordingly, the study concluded that a greater plateau length F 0 leads to a less conservative formulation, whereas a greater imperfection factor αF0 gives safer resistance predictions. Table 1 presents values for imperfection factors αF0, the plateau lengths F 0 and partial safety factors γM1 available for various patch loading resistance models. For each proposal, αF0and F 0 were attained after calibrating Eqs. (10) and (11), while γM1 was determined employing a standard evaluation procedure provided in EN 1990 [16].
n
=
E
+ 0.002
(12)
02
To validate the numerical model, computed ultimate resistances Fu are compared against experimental loads Fexp obtained by Unosson [7]. Fig. 2 shows the material stress-strain curves corresponding to each specimen tested in [7]. The whole I-girders are modeled, initial geometrical imperfections are also considered within this analysis. All nodes in the web plate are modified resembling the first eigenmode of a plate girder subjected to patch loading. Hence, initial curvatures in both transverse and longitudinal directions of the web plate are considered. In spite of the fact
3. Previous experimental studies As mentioned in the introduction, experimental studies have been conducted to evaluate the resistance of stainless steel girders subjected to patch loading. In this section, a brief description of experimental results found in the literature is addressed. Table 2 summarizes dimensions and material properties of 21 available experimental tests, of which, 5 correspond to girders tested by Unosson et al. [7], and 16 Table 2 Test studies of stainless steel girders subjected to patch loading. #
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
Test
Pli 4301:1 Pli 4301:2 Pli 4301:3 Pli 4301:4 Pli 4301:5 IOF-H150-L150-SS60 IOF-H150-L200-SS60 IOF-H150-L300-SS60 IOF-H150-L400-SS60 IOF-H150-L450-SS60 IOF-H152-L150-SS30 IOF-H152-L300-SS30 IOF-H152-L450-SS30 IOF-H152-L600-SS30 IOF-H152-L750-SS30 IOF-H102-L300-SS5 IOF-H102-L300-SS7.5 IOF-H102-L300-SS10 IOF-H102-L300-SS12.5 IOF-H102-L300-SS15 IOF-H102-L300-SS20
a
hw
tw
bf
tf
ss
σ02w
σ02f
Ew
Ef
Fexp
[mm]
[mm]
[mm]
[mm]
[mm]
[mm]
[MPa]
[MPa]
[GPa]
[GPa]
[kN]
998 996 1397 1623 1682 150 200 301 402 452 150 301 451 598 751 299 299 299 298 298 299
238 238 316 438 401 150 150 150 150 150 152 153 152 152 152 101 102 101 101 102 101
4.10 4.10 4.10 4.10 8.80 6.95 6.87 6.88 6.81 6.87 6.20 6.18 6.22 6.18 6.13 4.89 4.98 4.92 4.94 4.99 4.91
118.5 119.9 121.0 121.2 120.4 75.8 75.8 75.8 75.7 75.7 160.0 159.0 159.6 159.6 159.8 67.9 67.9 67.8 67.9 67.8 67.8
11.7 11.9 11.9 11.9 12.0 9.9 9.8 9.8 9.8 9.8 8.7 8.8 8.7 8.9 8.7 5.1 5.2 5.2 5.2 5.1 5.1
40.0 80.0 40.0 40.0 40.0 60.0 60.0 60.0 60.0 60.0 30.0 30.0 30.0 30.0 30.0 5.0 7.5 10.0 12.5 15.0 20.0
297 297 297 297 245 274 274 274 274 274 272 272 272 272 272 222 222 222 222 222 222
285 285 285 285 285 267 267 267 267 267 227 227 227 227 227 222 222 222 222 222 222
200 200 200 200 200 197 197 197 197 197 191 191 191 191 191 187 187 187 187 187 187
200 200 200 200 200 197 197 197 197 197 205 205 205 205 205 187 187 187 187 187 187
Tests 1–5 come from Unosson [7]; Tests 6–21 come from dos Santos et al. [10]. 926
176.0 196.0 168.0 169.0 478.0 424.4 393.1 368.6 342.2 340.0 340.0 322.2 301.1 296.7 275.0 126.7 132.3 121.8 143.2 130.8 142.5
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Fig. 4. Convergence study of the finite element mesh.
Fig. 2. Stress-strain curves of the specimens tested in [7].
Overall, the numerical model accurately captures the nonlinear responses of the slender stainless steel girders subjected to patch loading. Table 3 shows a comparison between the experimental resistances Fexp and those computed with the numerical model Fu, a good agreement is found with an average difference mΔ = 6.83%, a standard deviation sΔ = 1.33 and coefficient of variation equal to vΔ = 0.19. 5. Parametric study 5.1. General According to the review performed in Section 3, only 6 tests satisfied the patch loading case. In order to expand this database, an extensive parametric study is carried out. Table 4 presents the fixed geometrical and material properties used herein, it must be pointed out that these material properties were the same for all the components (web and flanges) of the plate girder. Moreover, a maximum geometrical web imperfection of w0 = min(hw/100, tw) was included in the numerical model, this value meets the recommendations described in EC3 Part 1.4 - Annex C.2 [1], where the rules design for FE simulations/ modeling should be taken from EC3 Part 1.5 - Annex C [2]. These imperfections are large enough to attain a lower bound limit for design [22]. Table 5 summarizes the values of the geometric variables employed in the parametric study. For each combination of variables presented in Table 5, the ultimate resistance of the stainless steel girder was computed, therefore a total of 7020 numerical simulations were performed. These dimensions are similar to the girder proportions tested by Unosson [7]. It is worth quoting that, this section aims at investigating the effect of girder dimensions, covering a wide range of slenderness, on the patch loading resistance of slender stainless steel I-girders. Therefore, based on the effective width concept [1], the cross-section for all modeled girders classifies as slender Class 4 cross-sections.
Fig. 3. Adopted finite element mesh.
that residual stress patterns have been defined for welded austenitic and ferritic stainless steel sections [18], residual stresses were not accounted for in the analysis since these have a diminished impact on patch loading resistance [19–21]. For stress states involving the full cross-section of the elements, such as bending and compression, residual stresses can detriment the ultimate strength [22], but this is barely the case for patch loading in which the stress state is rather localized. As shown in Fig. 3, two boundary conditions are applied to the model, first, for the support ends, simple support conditions are defined, i.e. displacements in x and y directions were restricted in the nodes along support A, and the nodes on support B were free to move longitudinally (x-direction). Next, the load is applied on the nodes located over an equivalent length ss in the upper flange, in which the displacements in the x and z directions, and all rotations are restricted, allowing only the vertical displacement in the y direction. Moreover, the nonlinear response in the postbuckling region is accurately captured using the Riks method [23], an arc-length based incremental method. This method allows the capture of the structural response considering the material nonlinearities and unstable collapse of the structure. After conducting a convergence study (see Fig. 4) using as a basis the geometry of the second specimen (Pli 4301:2) tested by Unosson [7], a finite element mesh with 4352 elements was chosen as shown in Fig. 3, i.e. a mesh with 2112 elements in the web panel, 880 elements in each flange, and 240 elements in each transverse stiffener. Similarly to [24], a maximum geometric imperfection of w0 = 0.003hw was used for the validation of the numerical model. Fig. 5 presents a comparison between experimental and numerical load-deflection curves corresponding to the five specimens tested in [7] (see Table 2). For all the specimens, the numerical model describes accurately the gradual loss of resistance in the postbuckling region.
5.2. Effect of the panel aspect ratio a/hw Fig. 6 shows the computed resistances Fu in terms of the panel aspect ratio a/hw for various patch loading lengths ss/hw and web slenderness hw/tw. It can be seen that, the ultimate resistance decreases proportionally with the panel aspect ratio. When the patch loading support is largely distributed over the loaded flange, the ultimate resistance presents a nonlinear reduction as the panel aspect ratio is increased, while for short patch loading lengths the decrease is almost linear. Moreover, for large values of patch loading lengths Fu decays considerably when a/hw is increased. As seen in Fig. 6(f), for ss/hw = 0.50 and hw/tw = 200, the ultimate resistance is reduced from Fu = 72.1 to 60.2 kN (a 16.5% of decrease), for an increasing panel aspect ratio from a/hw = 1 to 2, respectively. On the other hand, for a short loading length ss/hw = 0.10 and hw/tw = 200 (Fig. 6(f)), Fu is only reduced from 46.9 to 43.4 kN (a 7.5% of decrease) once a/hw is increased from 1 to 2, 927
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Fig. 5. Experimental and numerical load-deflection curves comparison. Table 3 Comparison between experimental Fexp and computed resistances Fu. #
Test
1 2 3 4 5
Pli Pli Pli Pli Pli
4301:1 4301:2 4301:3 4301:4 4301:5
Table 5 Varied geometrical dimensions in the numerical database.
Fexp
Fu
Δ
Variable
[kN]
[kN]
%
Value
176.0 196.0 168.0 169.0 478.0
186.8 208.0 183.4 179.3 446.2
6.14 6.12 9.17 6.09 6.65
Table 4 Geometric and material properties of the plate girder. bf [mm]
E [GPa]
120
200
υ 0.30
σ0.2 [MPa]
n
σ1.0 [MPa]
σu [MPa]
285
9
334
611
hw [mm]
a/hw
hw/tw
ss/hw
tf/tw
400 600
1.00 1.25 1.50 1.75 2.00
75 100 125 150 175 200
0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50
1.00 1.25 1.50 1.75 2.00 2.25 2.50 2.75 3.00 3.25 3.50 3.75 4.00
increment when the web slenderness is reduced. As expected, girders with low web slenderness ratios (thicker webs) show a better performance for patch loading. This could be observed in Fig. 7(d) where for any value of flange-to-web thickness ratio tf/tw the resistance of a thicker web (hw/tw = 75) subjected to a wide loading length of ss/ hw = 0.40 is around 5.6 times the resistance of a slender girder (hw/ tw = 200). It is interesting to notice that the resistance is also influenced by the thickness ratio tf/tw. The results show that increasing flange thickness results in an enhanced resistance, this can be appreciated in Fig. 7(a) for
respectively. 5.3. Effect of the web slenderness hw/tw Furthermore, the resistance Fu is also affected by the slenderness of the web panel hw/tw. Fig. 7 illustrates the behavior of Fu in terms of hw/ tw for various flange-to-web thickness ratios tf/tw and patch loading lengths ss/hw. In this case, the ultimate resistance presents a nonlinear 928
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Fig. 6. Ultimate resistance Fu in terms of the panel aspect ratio a/hw (hw = 400 mm, tf/tw = 2).
Fig. 7. Ultimate resistance Fu versus the web slenderness hw/tw (hw = 400 mm, a/hw = 1.50).
929
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Fig. 8. Ultimate resistance Fu as function of the loading length ss/hw (tf/tw = 2, a/hw = 1.50).
short lengths ss/hw = 0.10 and thicker webs (hw/tw = 75), where Fu rises from 201.3 to 319.6 kN (an increment of 58.8%), when the thickness ratio is goes from tf/tw = 1 to 4. This also occurs for slender webs (hw/tw = 200) and wide lengths ss/hw = 0.40, in which the resistance increases 56.2% for an increasing ratio tf/tw going from 1 to 4.
stainless steel girders in Section 5. In this calibration, the imperfection factor αF0 and the plateau length F 0 are adjusted in order to achieve a desired level of safety. Following the same procedure employed by Chacón et al. [15] a sensitivity analysis is performed. The examined variable within this evaluation is the ratio between the resistance obtained with the numerical model Fu and the resistance FRd⁎ predicted with Eq. (3) (without the partial safety factor) using the new resistance functions given in Eqs. (10) and (11). Table 6 summarizes basic statistics corresponding to each evaluated case of αF0 and F 0 . As observed in Table 6, for high values of the plateau length F 0 the formulation tends to overestimate the ultimate resistance. This can be appreciated in the statistics, where the percentage of values lower than Fu/FRd⁎ = 1 increases considerably with F 0 , while the mean, maximum and minimum values of Fu/FRd⁎ decrease. On the other hand, standard deviations and coefficients of variation slightly rise for increasing plateau lengths. Concerning the imperfection factor αF0, the statistical evaluation shows that the theoretical model slightly underestimates the ultimate resistance when αF0 is high, i.e. the formulation is more conservative. The results show that the standard deviation for the ratio Fu/FRd⁎ increases proportionally with αF0. Overall, Table 6 shows that for high F 0 and low αF0 values (bottom left corner) the prediction of ultimate resistance is less conservative, while for low F 0 and high αF0 values (top right corner) there is an overestimation of the ultimate resistance. Based on these observations, the combination of values that best fit the available results occurs when plateau length is F 0 = 0.1 and the imperfection factor is αF0 = 0.4. The goodness of the fit is demonstrated by observing the statistics of the ratio Fu/FRd⁎, in which their corresponding values are acceptable (max = 1.52; min = 0.99), with a low standard deviation of s = 0.09.
5.4. Effect of the patch loading length ss/hw Fig. 8 depicts the influence of the patch loading length-to-web height ratio ss/hw on the resistance for two web heights hw, and various web height-to-web thickness ratios hw/tw. As observed, the ultimate resistance rises with ss/hw, in this case the increase is almost linear. For any value of hw and hw/tw, the numerical results show that the resistance of a girder subjected to ss/hw = 0.10 is approximately 1.37 to 1.47 times the resistance of a girder with ss/hw = 0.50. In addition to the influence of the patch loading length, the variation of the ultimate resistance is also related to the height of the girder. As seen in Fig. 8, for any value of ss/hw a significant increment of around 110% is achieved by increasing hw from 400 to 600 mm. 6. Statistical evaluation 6.1. Calibration of the resistance function χF As mentioned earlier, it has been demonstrated that ultimate resistance predictions for carbon steel girders subject to patch loading improve when the resistance function χF is calculated using Eq. (10) [15]. Hence, in this section a statistical calibration of the resistance function is conducted using the numerical database obtained for slender 930
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Table 6 Statistics for the ratio between computed and predicted resistances Fu/FRd⁎ for slender austenitic stainless steel girders. λF0
αF0 0.20
0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90
0.40
0.60
min
max
m
s
v
% <1
min
max
m
s
v
% <1
min
max
m
s
v
% <1
min
max
m
s
v
% <1
0.83 0.81 0.80 0.79 0.76 0.74 0.70 0.67 0.62
1.28 1.27 1.26 1.25 1.24 1.23 1.22 1.21 1.20
1.09 1.08 1.06 1.04 1.03 1.01 0.99 0.97 0.95
0.07 0.08 0.08 0.08 0.09 0.09 0.10 0.11 0.12
0.07 0.07 0.08 0.08 0.09 0.09 0.10 0.11 0.12
11.00 17.05 24.91 32.56 39.23 45.21 50.44 55.06 60.71
0.99 0.97 0.94 0.90 0.85 0.80 0.73 0.67 0.62
1.52 1.50 1.48 1.47 1.45 1.43 1.41 1.40 1.38
1.30 1.27 1.25 1.22 1.19 1.16 1.13 1.09 1.06
0.09 0.09 0.10 0.10 0.11 0.12 0.13 0.14 0.16
0.07 0.07 0.08 0.09 0.09 0.10 0.12 0.13 0.15
0.07 0.40 0.87 1.68 3.33 12.95 20.90 25.64 35.31
1.13 1.10 1.04 0.98 0.91 0.84 0.74 0.67 0.62
1.75 1.72 1.69 1.67 1.64 1.62 1.59 1.57 1.54
1.49 1.45 1.41 1.38 1.34 1.29 1.25 1.21 1.16
0.11 0.11 0.12 0.13 0.14 0.15 0.16 0.18 0.20
0.07 0.08 0.08 0.09 0.10 0.11 0.13 0.15 0.17
0.00 0.00 0.00 0.04 0.41 1.65 6.37 18.13 21.88
1.27 1.20 1.13 1.05 0.97 0.87 0.74 0.67 0.62
1.96 1.93 1.90 1.87 1.83 1.80 1.77 1.73 1.70
1.67 1.62 1.57 1.52 1.47 1.42 1.37 1.31 1.25
0.12 0.13 0.14 0.15 0.16 0.18 0.19 0.21 0.24
0.07 0.08 0.09 0.10 0.11 0.12 0.14 0.16 0.19
0.00 0.00 0.00 0.00 0.04 0.60 2.49 10.04 18.69
6.2. Partial safety factor γM1
V =
In the previous section, the resistance function χF was calibrated. In order to guarantee the required safety level of the patch loading resistance model, the next step is to determine the safety factor γM1, following the guidelines provided in the EN 1990 - Annex D [16]. This factor considers the uncertainties of the geometrical and material properties related to the resistance model. The steps to calculate the partial safety factor are described as follows
with i
rt = grt (X )
s2 =
where grt is the resistance function of the numerical model and X is the array of values of the basic variables.
_
With the purpose of comparing the deviation of the predicted results, the least squares best-fit to the slope (b) is calculated
=
rei b rti
1 n
n i
(18)
i=1
n
1 n
1
(
)2
i
(19)
i=1
Following the recommendations of Davaine [13], an additional coefficient that considers the deviations between the numerical model and the experimental results was included in this calibration. The procedure to obtain the coefficient of variation was the same as the one followed to calculate the coefficient of variation Vδ. In this case, the value obtained was VFEM = 0.0763.
(14)
and the error term δi is determined for each value of numerical resistance rei and theoretical resistance rti i
(17)
- Thereafter, the coefficients of variations Vxi for the basic variables (geometry and material) were defined. In this case, based on the statistical evaluation of data collected from the steel producers and manufacturers of stainless steel sections conducted by Baddoo and Francis [25], the coefficients used for the material and geometry were: Vx1 = 0.105 for austenitic steel material and Vx2 = 0.0214 for I-section stainless steel member subjected to compression.
- Next, numerical resistances re and theoretical resistances rt are compared (see Fig. 9)
re rt rt2
(16)
1
Additionally, a normality test should be carried out. To this purpose, a Kolmogorov-Smirnoff test was performed to verify the normality of the distribution of the errors.
(13)
_
exp(s 2)
= ln( i ) =
- First, a design model is developed using the following expression
b=
0.80
(15)
- Then, the final coefficient of variation is computed in order to take into account all variations in one parameter
- Subsequently, the coefficient of variation of the error is computed as
Vr2 = (V 2 + 1)
j
(Vxi2 + 1)
i=1
2 1 + VFEM
(20)
-Furthermore, the design and characteristic resistances (rd, rk) are calculated using the following expressions
rd = bgrt ( Xm) e (
k Q 0.5Q2)
rk = bgrt ( Xm ) e (
k Q 0.5Q2)
_
_
(21) (22)
where Xm is the mean array of values of the basic variables and Q is _
computed as follows
Q=
ln(Vr2 ) + 1
(23)
with k∞ equal to 3.04 and 1.64 for design and characteristic resistance [16], respectively. The partial safety factor is obtained dividing rk by rd
Fig. 9. Numerical vs. theoretical resistances. 931
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Table 7 αF0, F 0 and γM1 obtained from the statistical evaluation. αF0 0.40
M1
=
F0
γM1
0.10
1.05
rk rd
(24)
and latter corrected with the purpose of adjusting for better statistical variations M1
=
M1
rn rk
(25)
with rn as the nominal resistance, calculated employing Eq. (26) with Xn known as the array of nominal values of the basic variables
_
rn = bgrt ( Xn ) e (
k Q 0.5Q 2)
_
(26)
After following the procedure presented above, the partial safety factor obtained for the numerical database was γM1∗ = 1.05. This result shows correspondence and reliability with the partial safety factors provided in the EC3 Part 1.4 [1], where the resistance models for stainless steel cross sections under local buckling employ partial factors less or equal than 1.10. 6.3. Comparison of the revised formulation with the numerical results A comparison between numerical results derived from the parametric study and ultimate resistances computed using Eq. (1), in combination to the newly calibrated resistance function for stainless steel girders developed in Section 6.1, is carried out. Table 7 summarizes the imperfection factor αF0 and the plateau length F 0 corresponding to the calibration of the resistance function, along with the partial safety factor γM1 obtained for stainless steel girders subjected to patch loading. On the one hand, Fig. 10 displays the ratio between resistances computed numerically Fu and resistances FRd calculated using the current provisions of the EC3 Part 1.4 [1] versus various geometrical relationships. On the other hand, Fig. 11 shows the same relationship, but FRd is calculated using the resistance function in Eq. (10) and considering the values reported in Table 7 (with the already computed partial safety factor). The basic statistics for the ratio Fu/FRd corresponding to both ultimate resistance approaches are presented in Table 8. As observed in Fig. 10, there is a significant scatter in the ratio Fu/ FRd (max = 2.18; min = 1.02), when the ultimate resistance is predicted with the current provisions in the EC3 Part1.4 [1]. As seen in Fig. 10(a) and (c), the results indicate that there is no influence of the panel aspect ratio a/hw and the patch loading length ss/hw on the ultimate load ratio Fu/FRd. However, the results also show that the ratio Fu/ FRd rises for increasing values of web slenderness hw/tw and thickness ratio tf/tw (Fig. 10(b) and (d)). Additionally, it is interesting to observe that when the ultimate resistance is computed using the modified proposal (see Fig. 11) the scatter in the load ratio Fu/FRd is significantly reduced. This is clearly observed with the statistics for the ratio Fu/FRd in Table 8 with a mean ratio m = 1.36 and a standard deviation s = 0.09. Once the dimensionless parameters of the resistance function have been calibrated, the resistance curve is evaluated. To this purpose, Fig. 12 presents a comparison of the numerical results Fu/Fy and the resistance curves of the current EC3 Part1.4 formulation [1] and the resistance curve calibrated herein. As expected, there is an increase of Fu/Fy when the slenderness parameter is reduced. It is also observed that, the resistance curve of the EC3 Part1.4 [1] which is plotted using Eq. (3) and originally developed for carbon steel girders underestimates the resistance of the stainless steel girders
Fig. 10. Fu/FRd in terms of various geometrical parameters (FRd calculated according EC3: 1–4 provisions [1]).
subjected to patch loading. While, Eq. (10) corresponding to the resistance curve calibrated in Section 6.1 presents an acceptable lower bound for the resistance values obtained numerically. 932
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Table 8 Statistical values of Fu/FRd. Approach
min
max
m
s
v
EC3 Part1.4 [1] Revised
1.02 1.04
2.18 1.59
1.61 1.36
0.19 0.09
0.12 0.07
Fig. 12. Fu/Fy versus the slenderness parameter.
7. Conclusions This paper presents a comprehensive evaluation of the resistance of slender austenitic stainless steel I- girders subjected to patch loading. To this purpose, an extensive parametrical evaluation with more than 7000 numerical simulations was carried out, in order to present some design recommendations of the effect of several geometrical parameters on the ultimate resistance. Based on these results, a revised formulation to predict the ultimate resistance is discussed. From the analyses conducted herein, the following conclusions are drawn:
• The ultimate resistance of slender austenitic stainless steel plate •
• •
girders enhances significantly when the patch loading length-to-web height ratio ss/hw is incremented, and also when the web panel height is increased. The results have demonstrated that is possible to improve the ultimate resistance predictions when the non dimensionless parameters used to calculate the resistance function are calibrated for stainless steel girders subjected to patch loading. Moreover, the results also show the reliability of the resistance model, this is established after obtaining a partial safety factor within the limits established in the EC3 Part1.4 [1] for stainless steel cross sections under local buckling. Only slender stainless steel plate girders have been analyzed herein, therefore an additional work should be performed for girders with low web slenderness ratios. Finally, the resistance model was validated using a limited number of experimental results available in the literature, therefore more tests are necessary to fully validate it.
Declaration of Competing Interest All authors have participated in (a) conception and design, or analysis and interpretation of the data; (b) drafting the article or revising it critically for important intellectual content; and (c) approval of the final version. This manuscript has not been submitted to, nor is under review at, another journal or other publishing venue. The authors have no affiliation with any organization with a direct or indirect financial interest in the subject matter discussed in the manuscript.
Fig. 11. Fu/FRd in terms of various geometrical parameters (FRd calculated with modified proposal).
References [1] EN 1993-1-4 Eurocode 3. Design of steel structures - part 1–4: general rules – supplementary rules for stainless steels CEN. Brussels: European Committee for
933
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C. Graciano, et al. Standardization; 2006. [2] EN 1993-1-5. Eurocode 3: design of steel structures – part 1–5: plated structural elements. Brussels: European Committee for Standardization (CEN); 2006. [3] Lagerqvist O, Johansson B. Resistance of I-girders to concentrated loads. J Constr Steel Res 1996;39(2):87–119. [4] Afshan S, Gardner L. The continuous strength method for structural stainless steel design. Thin-Walled Struct 2013;68:42–9. [5] Ahmed S, Ashraf M, Anwar-Us-Saadat M. The continuous strength method for slender stainless steel cross-sections. Thin-Walled Struct 2016;107:362–76. [6] Zhao O, Afshan S, Gardner L. Structural response and continuous strength method design of slender stainless steel cross-sections. Eng Struct 2017;140:14–25. [7] Unosson E. Patch loading of stainless steel girders: experiments and finite analyses Licentiate thesis Division of steel Structures, Luleå University of Technology; 2003. [12, February 2003]. [8] Unosson E, Olsson A. Stainless steel girders: Resistance to concentrated loads and shear. In stainless steel in structures: International experts seminar 20/05/2003-20/ 05/2003. Stainless steel in structures: International experts seminar 20th may 2003, Ascot, UK. Proceedings. 2003. p. 123–30. [9] prEN 1993-1-4. Eurocode 3: design of steel structures - part 1–4: general rules – supplementary rules for stainless steels CEN. Brussels: European Committee for Standardization; 2002. [10] dos Santos GB, Gardner L, Kucukler M. Experimental and numerical study of stainless steel I-sections under concentrated internal one-flange and internal twoflange loading. Eng Struct 2018;175:355–70. [11] Johansson B, Veljkovic M. Review of plate buckling rules in EN 1993-1-5. Steel Constr 2009;2(4):228–34. [12] Müller C. Zum Nachweis ebener Tragwerke aus Stahl gegen seitliches Ausweichen Doctoral Thesis Shaker Verlag: RWTH Aachen, Lehrstuhl für Stahlbau; 2003. No. 47. [In German]. [13] Davaine L. Formulation de la résistance au lancement d'une âme métallique de pont
[14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25]
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raidie longitudinalement Doctoral Thesis D05–05 France: INSA de Rennes; 2005. [In French]. Gozzi J. Patch loading resistance of plated girders – ultimate and service ability limit state Doctoral Thesis Luleå University of Technology, Division of Steel Structures; 2007. 2007:30, Luleå, Sweden. Chacón R, Mirambell E, Kuhlmann U, Braun B. Statistical evaluation of the new resistance model for steel plate girders subjected to patch loading. Steel Constr 2012;5(1):10–5. EN 1990. Eurocode – basis of structural design. 2002. ANSYS. Ansys release 19 elements reference. 2018. Yuan HX, Wang YQ, Shi YJ, Gardner L. Residual stress distributions in welded stainless steel sections. Thin-Walled Struct 2014;79:38–51. Granath P. Behavior of slender plate girders subjected to patch loading. J Constr Steel Res 1997;42:1–19. Chacon R, Mirambell E, Real E. Influence of designer-assumed initial conditions on the numerical modelling of steel plate girders subjected to patch loading. ThinWalled Struct 2009;47:391–402. Chacon R, Serrat M, Real E. The influence of structural imperfections on the resistance of plate girders to patch loading. Thin-Walled Struct 2012;53:15–25. Couto C, Real PV. Numerical investigation on the influence of imperfections in the local buckling of thin-walled I-shaped sections. Thin-Walled Struct 2019;135. (89–08). Riks E. An incremental approach to the solution of snapping and buckling problems. Int J Solids Struct 1979;15:529–51. Unosson E, Olsson A, Lagerqvist O. A numerical study of the resistance of stainless steel girders subjected to concentrated forces. Proc. Nordic steel const conf 18–20th June 2001 Finland, Helsinki. 2001. p. 799–806. Baddoo NR, Francis P. Development of design rules in the AISC design guide for structural stainless steel. Thin-Walled Struct 2014;83:200–8.
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Structures journal homepage: www.elsevier.com/locate/structures
Resistance of slender austenitic stainless steel I- girders subjected to patch loading
T
Carlos Gracianoa, , Nelson Loaizaa, Euro Casanovab ⁎
a b
Universidad Nacional de Colombia, Facultad de Minas Sede Medellín, Departamento de Ingeniería Civil, A.A. 75267 Medellín, Colombia Universidad del Bío-Bío, Departamento Ingeniería Civil y Ambiental, Avenida Collao 1202, Concepción, Código Postal 4051381, Concepción, Chile
ARTICLE INFO
ABSTRACT
Keywords: Patch loading Resistance Resistance function Stainless steel Ultimate strength
This paper presents a numerical investigation on the patch loading resistance of slender austenitic stainless steel plate girders. Current design provisions for the resistance to patch loading of stainless steel girders are based on the plastic collapse mechanism observed in experimental and numerical studies conducted for carbon steel girders, disregarding the strain hardening capacities of stainless steel. At present, strength-curves approaches are used within European standards to deal with stability problems in steel plated structural elements. In this regard, three parameters require special attention: the yield load for plastic resistance, the resistance function depending on element slenderness, and the elastic buckling load. In this paper, an experimental dataset is firstly collected from the literature for comparative analysis. Subsequently, an extensive parametric study is conducted through nonlinear finite element analyses covering a wide range of slender stainless steel I-girder sections. Then, a resistance function is calibrated throughout a statistical evaluation of experimental and numerical results. Finally, the results show significant improvements in the predicted patch loading resistances of slender stainless steel I-girders.
Notation a bf E Fcr Fu Fy fyf fyw hw kF Leff ly MR MS m n s ss tf tw ⁎
length of web panel width of flange Young's modulus critical buckling load ultimate resistance calculated with the finite element model yield resistance flange yield strength web yield strength depth of web panel buckling coefficient effective length for resistance effective loaded length bending resistance applied bending moment mean value nonlinear material parameter standard deviation length of patch load flange thickness web thickness
v αF0 γM1 ϵ F0 F
ν σ σ0.2 σ1.0 σu χF
coefficient of variation v[=s/m] imperfection factor partial safety factor strain plateau length slenderness parameter Poisson's ratio stress stress at 0.2% strain stress at 1.0% strain ultimate stress resistance function
1. Introduction In recent years, an increasing awareness towards sustainability and low-cost maintenance has led to the improvement and development of material alloys used as construction materials. In this regard, stainless steel alloys play an important role in the design of structural elements in corrosive environments, and to improve their structural performance under fire conditions. Current design provisions for the resistance to patch loading of
Corresponding author at: Universidad Nacional de Colombia, Facultad de Minas, Departamento de Ingeniería Civil, A.A. 75267 Medellín, Colombia. E-mail address: [email protected] (C. Graciano).
https://doi.org/10.1016/j.istruc.2019.07.008 Received 28 May 2019; Received in revised form 18 July 2019; Accepted 20 July 2019 2352-0124/ © 2019 Institution of Structural Engineers. Published by Elsevier Ltd. All rights reserved.
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F
bf
ss y
tw
hw x
tf a Fig. 1. Plate girder subjected to patch loading (Notation).
stainless steel plate girders [1] are derived from those of carbon steel plate girders [2]. The provisions in the EC3 Part 1–5 [2] are based upon a plastic collapse mechanism proposed by Lagerqvist and Johansson [3] that considers an idealized elastic perfectly plastic material behavior, which is typical for carbon steel. Stainless steel exhibits nonlinear material strain-stress curves with significant strain hardening that should be considered in the strength calculations for economical and efficient structural designs [4–6]. Fig. 1 illustrates a typical setup for a plate girder subjected to patch loading. In 2003, Unosson et al. [7] conducted an experimental investigation on the resistance of austenitic stainless steel plate girders subjected to patch loading. Extending the slenderness of the girders, Unosson et al. [8] also performed a numerical study to investigate the influence of combined patch loading and bending, as well as the influence of the girder slenderness. The results showed that the predicted resistances calculated with a preliminary version of the EC3 Part 1.4 [9] were conservative. Recently, dos Santos et al. [10] studied experimentally and numerically the patch loading resistance of stocky welded plate girders manufactured of stainless steel. A comparison between experimental and numerical resistances with theoretical predictions was conducted, the results showed that the latter were generally rather conservative, particularly for girders with decreasing web slenderness. These conservative results for the stockier sections were also attributed to neglecting the pronounced strain hardening exhibited by stainless steel members with stocky webs. These behaviors have also been observed for stocky stainless steel cross sections under other loading configurations and addressed by means of the deformation based Continuous Strength Method (CSM) [4–6]. The CSM was originally developed for stainless steel and carbon steel materials, and is a deformation-based design framework that allows for the beneficial influence of strain hardening [4]. In recent years, the plate buckling rules in the EC3 Part 1–5 [2] are under review for harmonization of various buckling phenomena in plated structures [11]. Müller [12] introduced a proposal harmonizing the resistance function shape χF for all verifications of structural compressed members in the EC3 Part 1–5 [2]. This formulation comprises some dimensionless parameters, which can be calibrated in order to achieve a desired level of safety. After statistical evaluations of the resistance function [12], these parameters have been calibrated for carbon steel plate girders subjected to patch loading [13–15], leading to improvements in the predicted resistances. This paper aims at investigating the resistance to patch loading of slender austenitic stainless steel plate girders. At first, a dataset of experimental results is collected from the literature for comparative analysis. Secondly, an extensive parametric study is conducted through nonlinear finite element analyses in order to expand the available database. Thereafter, a statistical evaluation of the results is performed in order to calibrate a resistance function for the patch loading resistance
of slender stainless steel plate girders. Next, a new safety factor is determined according to the guidelines provided in international design codes. Finally, the results show a significant improvement in the predicted resistances. 2. Resistance models in European standards 2.1. Resistance according to the EC3 Part1.4 [1] - stainless steel In accordance with EC3 Part1.4 [1], the patch loading resistance for stainless steel plate girders without intermediate stiffener FRd is calculated by
FRd = f yw Leff tw /
(1)
M1
where fyw is the yield strength of the web, and Leff is the effective length for resistance to transverse loads
Leff =
F
(2)
ly
The reduction factor χF due to local buckling is F
=
0.5
1.0
(3)
F
and the effective loaded length ly is given by
l y = ss + 2t f (1 +
m1 + m2 )
a
(4)
where m1 and m2 are dimensionless parameters
m1 = f yf bf /f yw tw m2 = 0.02(h w /t f )2
(5)
In Eq. (4), m2 should be taken as zero if F < 0.5. The slenderness parameter F is obtained from F
=
Fy (6)
Fcr
Fy is the yield resistance
Fy = fyw tw l y
(7)
The buckling load Fcr is
Fcr = 0.9kF Et w3 / h w kF = 6 + 2
hw a
(8)
2
(9)
2.2. Resistance function update for carbon steel girders After consensus within the TWG 8.3 [15], any modification in the resistance function should be based on the proposal presented by Müller [12]. This proposal harmonizes the resistance function shape for all 925
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girders tested by dos Santos et al. [10]. All the experiments were conducted using austenitic stainless steel of three different grades: Grade EN 1.4301 for tests 1 to 5, Grade EN 1.4404 for tests 6 to 10, and Grade EN 1.4571 for tests 11 to 21. It must be mentioned that, the investigation conducted herein considers girders subjected to only patch loading, therefore tests with combined bending moment and patch loading are discarded. The interaction with bending is taken into account when the applied moment MS is 50% greater than the bending resistance MR of the girder. To verify this, MS and MR are calculated following the procedure described in EC3 Part 1–4 [1], based on the effective width concept. Therefore, from the test reported in Table 2 only 6 out of the 21 experiments are taken into account: 5 from Unosson et al. [7] and only 1 (IOF-H152L150-SS30) from dos Santos et al. [10].
Table 1 αF0, F 0 and γM1 for different resistance models. Proposal
αF0
Müller [12] Davaine [13] Gozzi [14] Chacón et al. [15]
0.34 0.21 0.50 1.00 0.75
F0
γM1
0.8 0.8 0.6 0.5 0.5
– 1.1 1.0 1.0 1.1
verifications of structural compressed members within the EC3 Part 1–5 [2]. A general form of this proposal is presented as follows F
=
1.0 2
+
1
(10)
F
4. Nonlinear finite element model
where
=
1 [1 + 2
F0 ( F
F0
)+
F]
Geometrically and materially nonlinear with imperfection analyses (GMNIA) are performed to evaluate the influence of various geometrical parameters on the postbuckling behavior of the slender stainless steel girders subjected to patch loading. To achieve that, a nonlinear finite element model is developed using ANSYS [17]. The web panel, flanges and vertical stiffeners of the girder are modeled using the fournode shell element S181 [17] with six degrees of freedom at each node. Material nonlinearities are considered in the stainless steel model. In this case, a multilinear stress-strain curve was defined using the Eq. (12) of true stress-strain curves with strain hardening provided in the EC3 Part 1.4-Annex C.2 [1].
(11)
In Eq. (11), the imperfection factor αF0 and the plateau length F 0 , need calibration in order to achieve a desired level of safety. Chacón et al. [15] performed a statistical evaluation of a new proposal for the patch loading resistance of carbon steel plate girders. Accordingly, the study concluded that a greater plateau length F 0 leads to a less conservative formulation, whereas a greater imperfection factor αF0 gives safer resistance predictions. Table 1 presents values for imperfection factors αF0, the plateau lengths F 0 and partial safety factors γM1 available for various patch loading resistance models. For each proposal, αF0and F 0 were attained after calibrating Eqs. (10) and (11), while γM1 was determined employing a standard evaluation procedure provided in EN 1990 [16].
n
=
E
+ 0.002
(12)
02
To validate the numerical model, computed ultimate resistances Fu are compared against experimental loads Fexp obtained by Unosson [7]. Fig. 2 shows the material stress-strain curves corresponding to each specimen tested in [7]. The whole I-girders are modeled, initial geometrical imperfections are also considered within this analysis. All nodes in the web plate are modified resembling the first eigenmode of a plate girder subjected to patch loading. Hence, initial curvatures in both transverse and longitudinal directions of the web plate are considered. In spite of the fact
3. Previous experimental studies As mentioned in the introduction, experimental studies have been conducted to evaluate the resistance of stainless steel girders subjected to patch loading. In this section, a brief description of experimental results found in the literature is addressed. Table 2 summarizes dimensions and material properties of 21 available experimental tests, of which, 5 correspond to girders tested by Unosson et al. [7], and 16 Table 2 Test studies of stainless steel girders subjected to patch loading. #
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
Test
Pli 4301:1 Pli 4301:2 Pli 4301:3 Pli 4301:4 Pli 4301:5 IOF-H150-L150-SS60 IOF-H150-L200-SS60 IOF-H150-L300-SS60 IOF-H150-L400-SS60 IOF-H150-L450-SS60 IOF-H152-L150-SS30 IOF-H152-L300-SS30 IOF-H152-L450-SS30 IOF-H152-L600-SS30 IOF-H152-L750-SS30 IOF-H102-L300-SS5 IOF-H102-L300-SS7.5 IOF-H102-L300-SS10 IOF-H102-L300-SS12.5 IOF-H102-L300-SS15 IOF-H102-L300-SS20
a
hw
tw
bf
tf
ss
σ02w
σ02f
Ew
Ef
Fexp
[mm]
[mm]
[mm]
[mm]
[mm]
[mm]
[MPa]
[MPa]
[GPa]
[GPa]
[kN]
998 996 1397 1623 1682 150 200 301 402 452 150 301 451 598 751 299 299 299 298 298 299
238 238 316 438 401 150 150 150 150 150 152 153 152 152 152 101 102 101 101 102 101
4.10 4.10 4.10 4.10 8.80 6.95 6.87 6.88 6.81 6.87 6.20 6.18 6.22 6.18 6.13 4.89 4.98 4.92 4.94 4.99 4.91
118.5 119.9 121.0 121.2 120.4 75.8 75.8 75.8 75.7 75.7 160.0 159.0 159.6 159.6 159.8 67.9 67.9 67.8 67.9 67.8 67.8
11.7 11.9 11.9 11.9 12.0 9.9 9.8 9.8 9.8 9.8 8.7 8.8 8.7 8.9 8.7 5.1 5.2 5.2 5.2 5.1 5.1
40.0 80.0 40.0 40.0 40.0 60.0 60.0 60.0 60.0 60.0 30.0 30.0 30.0 30.0 30.0 5.0 7.5 10.0 12.5 15.0 20.0
297 297 297 297 245 274 274 274 274 274 272 272 272 272 272 222 222 222 222 222 222
285 285 285 285 285 267 267 267 267 267 227 227 227 227 227 222 222 222 222 222 222
200 200 200 200 200 197 197 197 197 197 191 191 191 191 191 187 187 187 187 187 187
200 200 200 200 200 197 197 197 197 197 205 205 205 205 205 187 187 187 187 187 187
Tests 1–5 come from Unosson [7]; Tests 6–21 come from dos Santos et al. [10]. 926
176.0 196.0 168.0 169.0 478.0 424.4 393.1 368.6 342.2 340.0 340.0 322.2 301.1 296.7 275.0 126.7 132.3 121.8 143.2 130.8 142.5
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Fig. 4. Convergence study of the finite element mesh.
Fig. 2. Stress-strain curves of the specimens tested in [7].
Overall, the numerical model accurately captures the nonlinear responses of the slender stainless steel girders subjected to patch loading. Table 3 shows a comparison between the experimental resistances Fexp and those computed with the numerical model Fu, a good agreement is found with an average difference mΔ = 6.83%, a standard deviation sΔ = 1.33 and coefficient of variation equal to vΔ = 0.19. 5. Parametric study 5.1. General According to the review performed in Section 3, only 6 tests satisfied the patch loading case. In order to expand this database, an extensive parametric study is carried out. Table 4 presents the fixed geometrical and material properties used herein, it must be pointed out that these material properties were the same for all the components (web and flanges) of the plate girder. Moreover, a maximum geometrical web imperfection of w0 = min(hw/100, tw) was included in the numerical model, this value meets the recommendations described in EC3 Part 1.4 - Annex C.2 [1], where the rules design for FE simulations/ modeling should be taken from EC3 Part 1.5 - Annex C [2]. These imperfections are large enough to attain a lower bound limit for design [22]. Table 5 summarizes the values of the geometric variables employed in the parametric study. For each combination of variables presented in Table 5, the ultimate resistance of the stainless steel girder was computed, therefore a total of 7020 numerical simulations were performed. These dimensions are similar to the girder proportions tested by Unosson [7]. It is worth quoting that, this section aims at investigating the effect of girder dimensions, covering a wide range of slenderness, on the patch loading resistance of slender stainless steel I-girders. Therefore, based on the effective width concept [1], the cross-section for all modeled girders classifies as slender Class 4 cross-sections.
Fig. 3. Adopted finite element mesh.
that residual stress patterns have been defined for welded austenitic and ferritic stainless steel sections [18], residual stresses were not accounted for in the analysis since these have a diminished impact on patch loading resistance [19–21]. For stress states involving the full cross-section of the elements, such as bending and compression, residual stresses can detriment the ultimate strength [22], but this is barely the case for patch loading in which the stress state is rather localized. As shown in Fig. 3, two boundary conditions are applied to the model, first, for the support ends, simple support conditions are defined, i.e. displacements in x and y directions were restricted in the nodes along support A, and the nodes on support B were free to move longitudinally (x-direction). Next, the load is applied on the nodes located over an equivalent length ss in the upper flange, in which the displacements in the x and z directions, and all rotations are restricted, allowing only the vertical displacement in the y direction. Moreover, the nonlinear response in the postbuckling region is accurately captured using the Riks method [23], an arc-length based incremental method. This method allows the capture of the structural response considering the material nonlinearities and unstable collapse of the structure. After conducting a convergence study (see Fig. 4) using as a basis the geometry of the second specimen (Pli 4301:2) tested by Unosson [7], a finite element mesh with 4352 elements was chosen as shown in Fig. 3, i.e. a mesh with 2112 elements in the web panel, 880 elements in each flange, and 240 elements in each transverse stiffener. Similarly to [24], a maximum geometric imperfection of w0 = 0.003hw was used for the validation of the numerical model. Fig. 5 presents a comparison between experimental and numerical load-deflection curves corresponding to the five specimens tested in [7] (see Table 2). For all the specimens, the numerical model describes accurately the gradual loss of resistance in the postbuckling region.
5.2. Effect of the panel aspect ratio a/hw Fig. 6 shows the computed resistances Fu in terms of the panel aspect ratio a/hw for various patch loading lengths ss/hw and web slenderness hw/tw. It can be seen that, the ultimate resistance decreases proportionally with the panel aspect ratio. When the patch loading support is largely distributed over the loaded flange, the ultimate resistance presents a nonlinear reduction as the panel aspect ratio is increased, while for short patch loading lengths the decrease is almost linear. Moreover, for large values of patch loading lengths Fu decays considerably when a/hw is increased. As seen in Fig. 6(f), for ss/hw = 0.50 and hw/tw = 200, the ultimate resistance is reduced from Fu = 72.1 to 60.2 kN (a 16.5% of decrease), for an increasing panel aspect ratio from a/hw = 1 to 2, respectively. On the other hand, for a short loading length ss/hw = 0.10 and hw/tw = 200 (Fig. 6(f)), Fu is only reduced from 46.9 to 43.4 kN (a 7.5% of decrease) once a/hw is increased from 1 to 2, 927
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Fig. 5. Experimental and numerical load-deflection curves comparison. Table 3 Comparison between experimental Fexp and computed resistances Fu. #
Test
1 2 3 4 5
Pli Pli Pli Pli Pli
4301:1 4301:2 4301:3 4301:4 4301:5
Table 5 Varied geometrical dimensions in the numerical database.
Fexp
Fu
Δ
Variable
[kN]
[kN]
%
Value
176.0 196.0 168.0 169.0 478.0
186.8 208.0 183.4 179.3 446.2
6.14 6.12 9.17 6.09 6.65
Table 4 Geometric and material properties of the plate girder. bf [mm]
E [GPa]
120
200
υ 0.30
σ0.2 [MPa]
n
σ1.0 [MPa]
σu [MPa]
285
9
334
611
hw [mm]
a/hw
hw/tw
ss/hw
tf/tw
400 600
1.00 1.25 1.50 1.75 2.00
75 100 125 150 175 200
0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50
1.00 1.25 1.50 1.75 2.00 2.25 2.50 2.75 3.00 3.25 3.50 3.75 4.00
increment when the web slenderness is reduced. As expected, girders with low web slenderness ratios (thicker webs) show a better performance for patch loading. This could be observed in Fig. 7(d) where for any value of flange-to-web thickness ratio tf/tw the resistance of a thicker web (hw/tw = 75) subjected to a wide loading length of ss/ hw = 0.40 is around 5.6 times the resistance of a slender girder (hw/ tw = 200). It is interesting to notice that the resistance is also influenced by the thickness ratio tf/tw. The results show that increasing flange thickness results in an enhanced resistance, this can be appreciated in Fig. 7(a) for
respectively. 5.3. Effect of the web slenderness hw/tw Furthermore, the resistance Fu is also affected by the slenderness of the web panel hw/tw. Fig. 7 illustrates the behavior of Fu in terms of hw/ tw for various flange-to-web thickness ratios tf/tw and patch loading lengths ss/hw. In this case, the ultimate resistance presents a nonlinear 928
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Fig. 6. Ultimate resistance Fu in terms of the panel aspect ratio a/hw (hw = 400 mm, tf/tw = 2).
Fig. 7. Ultimate resistance Fu versus the web slenderness hw/tw (hw = 400 mm, a/hw = 1.50).
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Fig. 8. Ultimate resistance Fu as function of the loading length ss/hw (tf/tw = 2, a/hw = 1.50).
short lengths ss/hw = 0.10 and thicker webs (hw/tw = 75), where Fu rises from 201.3 to 319.6 kN (an increment of 58.8%), when the thickness ratio is goes from tf/tw = 1 to 4. This also occurs for slender webs (hw/tw = 200) and wide lengths ss/hw = 0.40, in which the resistance increases 56.2% for an increasing ratio tf/tw going from 1 to 4.
stainless steel girders in Section 5. In this calibration, the imperfection factor αF0 and the plateau length F 0 are adjusted in order to achieve a desired level of safety. Following the same procedure employed by Chacón et al. [15] a sensitivity analysis is performed. The examined variable within this evaluation is the ratio between the resistance obtained with the numerical model Fu and the resistance FRd⁎ predicted with Eq. (3) (without the partial safety factor) using the new resistance functions given in Eqs. (10) and (11). Table 6 summarizes basic statistics corresponding to each evaluated case of αF0 and F 0 . As observed in Table 6, for high values of the plateau length F 0 the formulation tends to overestimate the ultimate resistance. This can be appreciated in the statistics, where the percentage of values lower than Fu/FRd⁎ = 1 increases considerably with F 0 , while the mean, maximum and minimum values of Fu/FRd⁎ decrease. On the other hand, standard deviations and coefficients of variation slightly rise for increasing plateau lengths. Concerning the imperfection factor αF0, the statistical evaluation shows that the theoretical model slightly underestimates the ultimate resistance when αF0 is high, i.e. the formulation is more conservative. The results show that the standard deviation for the ratio Fu/FRd⁎ increases proportionally with αF0. Overall, Table 6 shows that for high F 0 and low αF0 values (bottom left corner) the prediction of ultimate resistance is less conservative, while for low F 0 and high αF0 values (top right corner) there is an overestimation of the ultimate resistance. Based on these observations, the combination of values that best fit the available results occurs when plateau length is F 0 = 0.1 and the imperfection factor is αF0 = 0.4. The goodness of the fit is demonstrated by observing the statistics of the ratio Fu/FRd⁎, in which their corresponding values are acceptable (max = 1.52; min = 0.99), with a low standard deviation of s = 0.09.
5.4. Effect of the patch loading length ss/hw Fig. 8 depicts the influence of the patch loading length-to-web height ratio ss/hw on the resistance for two web heights hw, and various web height-to-web thickness ratios hw/tw. As observed, the ultimate resistance rises with ss/hw, in this case the increase is almost linear. For any value of hw and hw/tw, the numerical results show that the resistance of a girder subjected to ss/hw = 0.10 is approximately 1.37 to 1.47 times the resistance of a girder with ss/hw = 0.50. In addition to the influence of the patch loading length, the variation of the ultimate resistance is also related to the height of the girder. As seen in Fig. 8, for any value of ss/hw a significant increment of around 110% is achieved by increasing hw from 400 to 600 mm. 6. Statistical evaluation 6.1. Calibration of the resistance function χF As mentioned earlier, it has been demonstrated that ultimate resistance predictions for carbon steel girders subject to patch loading improve when the resistance function χF is calculated using Eq. (10) [15]. Hence, in this section a statistical calibration of the resistance function is conducted using the numerical database obtained for slender 930
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Table 6 Statistics for the ratio between computed and predicted resistances Fu/FRd⁎ for slender austenitic stainless steel girders. λF0
αF0 0.20
0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90
0.40
0.60
min
max
m
s
v
% <1
min
max
m
s
v
% <1
min
max
m
s
v
% <1
min
max
m
s
v
% <1
0.83 0.81 0.80 0.79 0.76 0.74 0.70 0.67 0.62
1.28 1.27 1.26 1.25 1.24 1.23 1.22 1.21 1.20
1.09 1.08 1.06 1.04 1.03 1.01 0.99 0.97 0.95
0.07 0.08 0.08 0.08 0.09 0.09 0.10 0.11 0.12
0.07 0.07 0.08 0.08 0.09 0.09 0.10 0.11 0.12
11.00 17.05 24.91 32.56 39.23 45.21 50.44 55.06 60.71
0.99 0.97 0.94 0.90 0.85 0.80 0.73 0.67 0.62
1.52 1.50 1.48 1.47 1.45 1.43 1.41 1.40 1.38
1.30 1.27 1.25 1.22 1.19 1.16 1.13 1.09 1.06
0.09 0.09 0.10 0.10 0.11 0.12 0.13 0.14 0.16
0.07 0.07 0.08 0.09 0.09 0.10 0.12 0.13 0.15
0.07 0.40 0.87 1.68 3.33 12.95 20.90 25.64 35.31
1.13 1.10 1.04 0.98 0.91 0.84 0.74 0.67 0.62
1.75 1.72 1.69 1.67 1.64 1.62 1.59 1.57 1.54
1.49 1.45 1.41 1.38 1.34 1.29 1.25 1.21 1.16
0.11 0.11 0.12 0.13 0.14 0.15 0.16 0.18 0.20
0.07 0.08 0.08 0.09 0.10 0.11 0.13 0.15 0.17
0.00 0.00 0.00 0.04 0.41 1.65 6.37 18.13 21.88
1.27 1.20 1.13 1.05 0.97 0.87 0.74 0.67 0.62
1.96 1.93 1.90 1.87 1.83 1.80 1.77 1.73 1.70
1.67 1.62 1.57 1.52 1.47 1.42 1.37 1.31 1.25
0.12 0.13 0.14 0.15 0.16 0.18 0.19 0.21 0.24
0.07 0.08 0.09 0.10 0.11 0.12 0.14 0.16 0.19
0.00 0.00 0.00 0.00 0.04 0.60 2.49 10.04 18.69
6.2. Partial safety factor γM1
V =
In the previous section, the resistance function χF was calibrated. In order to guarantee the required safety level of the patch loading resistance model, the next step is to determine the safety factor γM1, following the guidelines provided in the EN 1990 - Annex D [16]. This factor considers the uncertainties of the geometrical and material properties related to the resistance model. The steps to calculate the partial safety factor are described as follows
with i
rt = grt (X )
s2 =
where grt is the resistance function of the numerical model and X is the array of values of the basic variables.
_
With the purpose of comparing the deviation of the predicted results, the least squares best-fit to the slope (b) is calculated
=
rei b rti
1 n
n i
(18)
i=1
n
1 n
1
(
)2
i
(19)
i=1
Following the recommendations of Davaine [13], an additional coefficient that considers the deviations between the numerical model and the experimental results was included in this calibration. The procedure to obtain the coefficient of variation was the same as the one followed to calculate the coefficient of variation Vδ. In this case, the value obtained was VFEM = 0.0763.
(14)
and the error term δi is determined for each value of numerical resistance rei and theoretical resistance rti i
(17)
- Thereafter, the coefficients of variations Vxi for the basic variables (geometry and material) were defined. In this case, based on the statistical evaluation of data collected from the steel producers and manufacturers of stainless steel sections conducted by Baddoo and Francis [25], the coefficients used for the material and geometry were: Vx1 = 0.105 for austenitic steel material and Vx2 = 0.0214 for I-section stainless steel member subjected to compression.
- Next, numerical resistances re and theoretical resistances rt are compared (see Fig. 9)
re rt rt2
(16)
1
Additionally, a normality test should be carried out. To this purpose, a Kolmogorov-Smirnoff test was performed to verify the normality of the distribution of the errors.
(13)
_
exp(s 2)
= ln( i ) =
- First, a design model is developed using the following expression
b=
0.80
(15)
- Then, the final coefficient of variation is computed in order to take into account all variations in one parameter
- Subsequently, the coefficient of variation of the error is computed as
Vr2 = (V 2 + 1)
j
(Vxi2 + 1)
i=1
2 1 + VFEM
(20)
-Furthermore, the design and characteristic resistances (rd, rk) are calculated using the following expressions
rd = bgrt ( Xm) e (
k Q 0.5Q2)
rk = bgrt ( Xm ) e (
k Q 0.5Q2)
_
_
(21) (22)
where Xm is the mean array of values of the basic variables and Q is _
computed as follows
Q=
ln(Vr2 ) + 1
(23)
with k∞ equal to 3.04 and 1.64 for design and characteristic resistance [16], respectively. The partial safety factor is obtained dividing rk by rd
Fig. 9. Numerical vs. theoretical resistances. 931
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Table 7 αF0, F 0 and γM1 obtained from the statistical evaluation. αF0 0.40
M1
=
F0
γM1
0.10
1.05
rk rd
(24)
and latter corrected with the purpose of adjusting for better statistical variations M1
=
M1
rn rk
(25)
with rn as the nominal resistance, calculated employing Eq. (26) with Xn known as the array of nominal values of the basic variables
_
rn = bgrt ( Xn ) e (
k Q 0.5Q 2)
_
(26)
After following the procedure presented above, the partial safety factor obtained for the numerical database was γM1∗ = 1.05. This result shows correspondence and reliability with the partial safety factors provided in the EC3 Part 1.4 [1], where the resistance models for stainless steel cross sections under local buckling employ partial factors less or equal than 1.10. 6.3. Comparison of the revised formulation with the numerical results A comparison between numerical results derived from the parametric study and ultimate resistances computed using Eq. (1), in combination to the newly calibrated resistance function for stainless steel girders developed in Section 6.1, is carried out. Table 7 summarizes the imperfection factor αF0 and the plateau length F 0 corresponding to the calibration of the resistance function, along with the partial safety factor γM1 obtained for stainless steel girders subjected to patch loading. On the one hand, Fig. 10 displays the ratio between resistances computed numerically Fu and resistances FRd calculated using the current provisions of the EC3 Part 1.4 [1] versus various geometrical relationships. On the other hand, Fig. 11 shows the same relationship, but FRd is calculated using the resistance function in Eq. (10) and considering the values reported in Table 7 (with the already computed partial safety factor). The basic statistics for the ratio Fu/FRd corresponding to both ultimate resistance approaches are presented in Table 8. As observed in Fig. 10, there is a significant scatter in the ratio Fu/ FRd (max = 2.18; min = 1.02), when the ultimate resistance is predicted with the current provisions in the EC3 Part1.4 [1]. As seen in Fig. 10(a) and (c), the results indicate that there is no influence of the panel aspect ratio a/hw and the patch loading length ss/hw on the ultimate load ratio Fu/FRd. However, the results also show that the ratio Fu/ FRd rises for increasing values of web slenderness hw/tw and thickness ratio tf/tw (Fig. 10(b) and (d)). Additionally, it is interesting to observe that when the ultimate resistance is computed using the modified proposal (see Fig. 11) the scatter in the load ratio Fu/FRd is significantly reduced. This is clearly observed with the statistics for the ratio Fu/FRd in Table 8 with a mean ratio m = 1.36 and a standard deviation s = 0.09. Once the dimensionless parameters of the resistance function have been calibrated, the resistance curve is evaluated. To this purpose, Fig. 12 presents a comparison of the numerical results Fu/Fy and the resistance curves of the current EC3 Part1.4 formulation [1] and the resistance curve calibrated herein. As expected, there is an increase of Fu/Fy when the slenderness parameter is reduced. It is also observed that, the resistance curve of the EC3 Part1.4 [1] which is plotted using Eq. (3) and originally developed for carbon steel girders underestimates the resistance of the stainless steel girders
Fig. 10. Fu/FRd in terms of various geometrical parameters (FRd calculated according EC3: 1–4 provisions [1]).
subjected to patch loading. While, Eq. (10) corresponding to the resistance curve calibrated in Section 6.1 presents an acceptable lower bound for the resistance values obtained numerically. 932
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Table 8 Statistical values of Fu/FRd. Approach
min
max
m
s
v
EC3 Part1.4 [1] Revised
1.02 1.04
2.18 1.59
1.61 1.36
0.19 0.09
0.12 0.07
Fig. 12. Fu/Fy versus the slenderness parameter.
7. Conclusions This paper presents a comprehensive evaluation of the resistance of slender austenitic stainless steel I- girders subjected to patch loading. To this purpose, an extensive parametrical evaluation with more than 7000 numerical simulations was carried out, in order to present some design recommendations of the effect of several geometrical parameters on the ultimate resistance. Based on these results, a revised formulation to predict the ultimate resistance is discussed. From the analyses conducted herein, the following conclusions are drawn:
• The ultimate resistance of slender austenitic stainless steel plate •
• •
girders enhances significantly when the patch loading length-to-web height ratio ss/hw is incremented, and also when the web panel height is increased. The results have demonstrated that is possible to improve the ultimate resistance predictions when the non dimensionless parameters used to calculate the resistance function are calibrated for stainless steel girders subjected to patch loading. Moreover, the results also show the reliability of the resistance model, this is established after obtaining a partial safety factor within the limits established in the EC3 Part1.4 [1] for stainless steel cross sections under local buckling. Only slender stainless steel plate girders have been analyzed herein, therefore an additional work should be performed for girders with low web slenderness ratios. Finally, the resistance model was validated using a limited number of experimental results available in the literature, therefore more tests are necessary to fully validate it.
Declaration of Competing Interest All authors have participated in (a) conception and design, or analysis and interpretation of the data; (b) drafting the article or revising it critically for important intellectual content; and (c) approval of the final version. This manuscript has not been submitted to, nor is under review at, another journal or other publishing venue. The authors have no affiliation with any organization with a direct or indirect financial interest in the subject matter discussed in the manuscript.
Fig. 11. Fu/FRd in terms of various geometrical parameters (FRd calculated with modified proposal).
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