Available online at www.sciencedirect.com
ScienceDirect Procedia Engineering 172 (2017) 1102 – 1109
Modern Building Materials, Structures and Techniques, MBMST 2016
Numerical investigation on web crippling behaviour of cold-formed C-section beam with vertical stiffeners Gintaras Šakalysa*, Alfonsas Daniūnasb a, b
Department of Steel and Timber Structures, Vilnius Gediminas Technical University, Saulėtekio al. 11, Vilnius LT-10223, Lithuania
Abstract Present paper investigates an efficiency of application of vertical stiffeners in the web of cold-formed C-section beams under local concentrated loading. The stiffeners would be cold-formed – an I-section cut is made on the web and the cut edges are folded into the inner side of the profile. A lipped channel beam with a length of 1 m was selected to investigate the efficiency of vertical web stiffeners. As a validation of rational stiffener parameters requires an extensive experimental program that is time-consuming and prohibitively expensive, thus, the numerical modelling using finite element method (FEM) was adopted. The numerical modelling was performed according to the standardized methodology of American Iron and Steel Institute (AISI) covering four loading and boundary conditions: 1) interior-one-flange (IOF); 2) interior-two-flange (ITF); 3) end-one-flange (EOF); and 4) end-two-flange (ETF). The influence of stiffeners geometry on the ultimate stiffened web crippling strength was assessed using the data from the numerical modelling of one hundred specimens. The obtained results were compared to the ultimate web crippling capacity of unstiffened beams. The relationships of load-vertical web displacement and load-horizontal web displacement were derived. The analysis of the obtained results formulated the directions for a further research. © by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license © 2017 2016Published The Authors. Published by Elsevier Ltd. (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the organizing committee of MBMST 2016. Peer-review under responsibility of the organizing committee of MBMST 2016 Keywords: Web crippling; Beams; Cold-formed steel; Finite element analysis; Stiffeners; Load-capacity.
1. Introduction Nowadays, a developed manufacturing technology of cold-formed thin-walled profiles is a simple and less energyconsuming process. Due to the rapid mass production, easy transportation, simple mounting, light weight and high
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1877-7058 © 2017 Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license
(http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the organizing committee of MBMST 2016
doi:10.1016/j.proeng.2017.02.171
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load-bearing capacity, the cold-formed thin-walled elements became widespread in the construction industry all over the world. A wide selection from a variety of profiles enables them to be used as the load-bearing walls, floor and roof structural elements. However, the buckling failure is characteristic for the cold-formed profiles, making the structural application and the design of such elements a complicated task. The modification and calibration of the calculation methods assessing the buckling effect require a comprehensive experimental and theoretical research. A local failure of the web due to the concentrated loading, support reaction or the combination of the latter two is one of the most frequent concerns in the design of thin-walled bending members. Previous investigations by Hetrakul and Yu [1] revealed that the ultimate web crippling strength depends on the six key parameters: 1) the thickness of the web t; 2) the tilt angle of the web θ; 3) the yield tensile strength fy; 4) the slenderness of the web h/t; 5) the ratio between the bearing length and the thickness of the web n/t; and 6) the ratio between the inside bend radius and the thickness of the web r/t. Over the past 40 years, a significant amount of the research has been performed to propose the analytical methods for buckling calculations based on the relationships between the governing parameters and the ultimate web crippling strength [2], [3]. Nowadays, the software based on the finite element method allows performing a variety of experiments using virtual, though, accurate and precise models and validating the calculation techniques [6]. The study performed by Macdonald et al. [9] revealed that the experimentally obtained and numerically calculated results are quite similar. The latter observation allows using the numerical modelling as an accurate tool for the investigation of deformation behaviour of the thin-walled C-section bending members. In recent years, an extensive research has turned towards the development of C-section elements [7], [8]. The enhanced shapes of the profiles are being created by incorporating the longitudinal stiffeners into the web of the crosssection. However, a more efficient alternative for the reduction of web slenderness would be achieved applying the vertical stiffeners, though there were no research investigating the application of vertical stiffeners. The present study investigates the influence of geometry parameters of vertical stiffeners on the load-carrying capacity of the web under the four different loading conditions, as described in Figure 2. The numerical modelling was performed according to the standardized methodology of American Iron and Steel Institute (AISI) [5]. A lipped channel beam with a length of 1 m was selected to investigate the behaviour of the wall (Fig. 1a). The vertical stiffeners were formed in a standard C-section bending element making an I-section cut on the web and folding the cut edges into the inner side of the profile as shown in Figure 1b and c. All forms of the analysed stiffeners are presented in Figure 3. In total, one hundred numerical modelling experiments were performed. The influence of stiffeners geometry on the ultimate stiffened web crippling strength was obtained. The latter results were compared to the ultimate web crippling capacity of unstiffened beams. The relationships of load-vertical web displacement and load-horizontal web displacement were determined. The analysis of the obtained results sets the directions for a further research. Nomenclature EOF IOF ETF ITF PFEA Ps,FEA E fy
End-one-flange Interior-one-flange End-two-flange Interior-two-flange Unstiffened web crippling ultimate capacity obtained by using finite element analysis Stiffened web crippling ultimate capacity obtained by using finite element analysis Elastic modulus Tensile yield stress
fu fy t h hs b d n r k
Ultimate tensile stress Tensile yield stress Web thickness Section height Stiffener height Opening width Stiffener width Bearing length Inside bend radius friction coefficient
2. Finite element model description A finite element modelling was performed using the software of ANSYS 16.0. All specimens were modelled accounting for the symmetry condition, i.e. setting the plane of the symmetry along the paired C-section beams. Axes Z, Y and X are the longitudinal, vertical and perpendicular to the plane of the web axes of the beam, respectively. A
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designation of constraint degrees of freedom UX, UY, UZ and RX, RY, RZ indicates the constraint displacements and rotations with respect to global coordinate axes X, Y, and Z, respectively.
Fig. 1. (a) C-section; (b) form of cut; (c) folding direction.
2.1. Finite element type, mesh size, and steel properties All the beams were modelled using SHELL181 finite elements. It is a four-node, plane element with six degrees of freedom at each node: displacements in the X, Y, and Z directions, and rotations about the X, Y, and Z axes [4]. SHELL181 is suitable for a large strain nonlinear analysis of thin shell structures. Support, load transferring, and tie plates were modelled using volume finite elements. The maximum size of the finite element was set to 3 mm around the stiffeners and at the contact zones. A mesh size of the rest of the beam was 6×6 mm. The curves in the bending zone were divided into two parts. The following material properties were used for the SHELL181 element: the steel grade S350GD; the yield tensile strength fy = 350 N/mm2; the ultimate tensile strength fu = 420 N/mm2; the Young's modulus of elasticity E = 2.1×105 N/mm2; and the Poisson's ratio ν = 0.3. The same steel grade is used in the manufacturing of the real cold-formed structural elements. The material behavior was defined by elastic-perfectly plastic stress-strain curve. A residual stresses arising during the formation process of the cross-section were neglected in the further analysis. The steel grade S235 was adopted for modelling the support, load transferring, and tie plates assuming a linear elastic material behavior. 2.2. Nonlinear analysis An effect of large strains, nonlinear material behavior, nonlinear behavior of contact zones near the support and load transferring plates were assessed determining the nonlinear failure behavior of the specimens subjected to the external loading. As the analysis on the geometry parameters were performed at the theoretical level, the crosssectional geometric imperfections and loading eccentricities were neglected. A Newton-Raphson method was adopted for the calculation of the equilibrium equations. 2.3. Loading In order to determine the specimen behavior after the loss of load-carrying capacity, the acting load was modelled as a 15 mm displacement. The displacement was set on the top of the load transferring plate towards the direction of
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Y axis making all other displacements of the surface constraint. The ultimate web crippling capacity was determined from the relationship of displacement and support reactions. 2.4. ETF loading and boundary conditions (Fig. 2a) At the end of the specimen, where the load was applied, the specimen was supported on the plates with a width n = 60 mm and the thickness of 10 mm. A bottom surface of the support plate was fixed, i.e., all six degrees of freedom were constraint. A top surface of the load transferring plate with the same dimensions as the support plate had the UX and UZ displacements and RX, RY, RZ rotations constraint. Both plates were rigidly connected to the flanges of the beam at the contact area of the circle having a diameter D = 24 mm. Such a contact simulated a washer having a center point in the intersection of the axial lines of the beam flanges and plates. The remaining area between the plates and the beam was described by a friction contact with the friction coefficient k = 0.2. Another end of the beam was supported on the same-size plate with a rigid connection of its bottom surface. A contact area between the latter plate and the beam was described by a friction coefficient k = 0.2. In order to prevent out of the plane displacement of the unloaded end of the beam, the top flanges were bonded together using a plate with a low bending stiffness (crosssectional dimensions of 40×1 mm). 2.5. ITF loading and boundary conditions (Fig. 2b) The specimen was supported on the centrally modelled support plate with a dimensions of 60×10 mm. A bottom surface of the support plate was fixed, i.e., all six degrees of freedom were constraint. A load transferring plate was modelled on the top flange at the same position symmetrical to the axis Z. The contacts between the plates and the beam were the same. Both plates were rigidly connected to the flanges of the beam at the contact area of the circle having a diameter D = 24 mm. The rest of the contact area was described by a friction contact with the friction coefficient k = 0.2. The flanges of the beam were bonded with a slender tie plates having a cross-sectional dimensions of 40×1 mm.
Fig. 2. Loading and boundary conditions (a) ETF; (b) ITF; (c) IOF; (d) EOF.
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2.6. EOF and IOF loading and boundary conditions (Fig. 2c, d) The specimen was supported on the ends of the modelled support plates with a dimensions of 60×10 mm. A bottom surface of the plates had the UX and UY displacements and RY and RZ rotations constraint. In order to obtain a web failure at the ends (EOF) of the beam, the out of the plane displacement of the web was constraint at the area along the width of load transferring plate. As for the web failure at the load transferring zone (IOF), the out of the plane displacement of the web was constraint at the ends of the beam along the width of support plates. A load transferring plate having a dimensions of 60×10 mm was modelled at the center of the beam with a top surface constraint in the directions UX and UZ. The contacts between the plates and the beam were the same. Both plates were rigidly connected to the flanges of the beam at the contact area of the circle having a diameter D = 24 mm. The rest of the contact area was described by a friction contact with the friction coefficient k = 0.2. The top flanges of the beam were bonded with a slender tie plates having a cross-sectional dimensions of 40×1 mm. 3. Parametric study In order to determine the efficiency of stiffeners, a parametric study consisting of numerical modelling results of 96 beams with stiffeners under four loading and boundary conditions as well as four specimens without stiffeners were
Fig. 3. (a) variety of stiffeners; (b) cross-section of stiffeners with b=20 mm; (c) cross-section of stiffeners with b=16 mm; (d) cross-section of stiffeners with b=12 mm; (e) cross-section of stiffeners with b=8 mm.
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performed. All the numerical modelling experiments were conducted with a single type of C20010 beam by varying the opening width of the stiffened zone (8 mm ≤ b ≤ 20 mm) and the height of the stiffener (60 mm ≤ h s ≤ 160 mm) as shown in Figure 3. In total, 24 stiffeners with various geometry were modelled and tested under four different loading and boundary conditions. A single stiffener was modelled at every zone expected to a failure for each of the specimen. Depending on the loading conditions, the stiffeners were localized at the center of acting load or support reaction. A central horizontal axis of the stiffeners coincided with the longitudinal axis of the specimens. Table 1. Web crippling strength results. Test number ETF-1 ETF-2 ETF-3 ETF-4 ETF-5 ETF-6 ETF-7 ETF-8 ETF-9 ETF-10 ETF-11 ETF-12 ETF-13 ETF-14 ETF-15 ETF-16 ETF-17 ETF-18 ETF-19 ETF-20 ETF-21 ETF-22 ETF-23 ETF-24 IOF-1 IOF-2 IOF-3 IOF-4 IOF-5 IOF-6 IOF-7 IOF-8 IOF-9 IOF-10 IOF-11 IOF-12 IOF-13 IOF-14 IOF-15 IOF-16 IOF-17 IOF-18 IOF-19 IOF-20 IOF-21 IOF-22 IOF-23 IOF-24
b (mm) 20 20 20 20 20 20 16 16 16 16 16 16 12 12 12 12 12 12 8 8 8 8 8 8 20 20 20 20 20 20 16 16 16 16 16 16 12 12 12 12 12 12 8 8 8 8 8 8
hs/h 0,3 0,4 0,5 0,6 0,7 0,8 0,3 0,4 0,5 0,6 0,7 0,8 0,3 0,4 0,5 0,6 0,7 0,8 0,3 0,4 0,5 0,6 0,7 0,8 0,3 0,4 0,5 0,6 0,7 0,8 0,3 0,4 0,5 0,6 0,7 0,8 0,3 0,4 0,5 0,6 0,7 0,8 0,3 0,4 0,5 0,6 0,7 0,8
PFEA (kN) 2,70 2,70 2,70 2,70 2,70 2,70 2,70 2,70 2,70 2,70 2,70 2,70 2,70 2,70 2,70 2,70 2,70 2,70 2,70 2,70 2,70 2,70 2,70 2,70 6,92 6,92 6,92 6,92 6,92 6,92 6,92 6,92 6,92 6,92 6,92 6,92 6,92 6,92 6,92 6,92 6,92 6,92 6,92 6,92 6,92 6,92 6,92 6,92
PS,FEA (kN) 2,72 2,87 3,15 3,59 4,20 4,95 2,73 2,83 3,12 3,54 4,14 4,95 2,75 2,87 3,14 3,53 4,11 4,89 2,76 2,88 3,08 3,36 3,76 4,10 6,93 6,93 6,92 6,90 6,53 6,39 6,93 6,92 6,92 6,84 6,57 6,48 6,93 6,92 6,90 6,83 6,60 6,56 6,92 6,92 6,90 6,86 6,61 6,50
PS,FEA/PFEA 1,009 1,065 1,169 1,332 1,557 1,835 1,013 1,051 1,156 1,314 1,534 1,834 1,021 1,066 1,163 1,309 1,523 1,812 1,025 1,069 1,143 1,247 1,392 1,519 1,001 1,001 1,000 0,996 0,943 0,924 1,001 1,001 0,999 0,988 0,949 0,936 1,001 1,000 0,998 0,988 0,954 0,947 1,000 1,000 0,998 0,992 0,955 0,939
Test number ITF-1 ITF-2 ITF-3 ITF-4 ITF-5 ITF-6 ITF-7 ITF-8 ITF-9 ITF-10 ITF-11 ITF-12 ITF-13 ITF-14 ITF-15 ITF-16 ITF-17 ITF-18 ITF-19 ITF-20 ITF-21 ITF-22 ITF-23 ITF-24 EOF-1 EOF-2 EOF-3 EOF-4 EOF-5 EOF-6 EOF-7 EOF-8 EOF-9 EOF-10 EOF-11 EOF-12 EOF-13 EOF-14 EOF-15 EOF-16 EOF-17 EOF-18 EOF-19 EOF-20 EOF-21 EOF-22 EOF-23 EOF-24
b (mm)
hs/h
20 20 20 20 20 20 16 16 16 16 16 16 12 12 12 12 12 12 8 8 8 8 8 8 20 20 20 20 20 20 16 16 16 16 16 16 12 12 12 12 12 12 8 8 8 8 8 8
0,3 0,4 0,5 0,6 0,7 0,8 0,3 0,4 0,5 0,6 0,7 0,8 0,3 0,4 0,5 0,6 0,7 0,8 0,3 0,4 0,5 0,6 0,7 0,8 0,3 0,4 0,5 0,6 0,7 0,8 0,3 0,4 0,5 0,6 0,7 0,8 0,3 0,4 0,5 0,6 0,7 0,8 0,3 0,4 0,5 0,6 0,7 0,8
PFEA (kN) 6,45 6,45 6,45 6,45 6,45 6,45 6,45 6,45 6,45 6,45 6,45 6,45 6,45 6,45 6,45 6,45 6,45 6,45 6,45 6,45 6,45 6,45 6,45 6,45 3,38 3,38 3,38 3,38 3,38 3,38 3,38 3,38 3,38 3,38 3,38 3,38 3,38 3,38 3,38 3,38 3,38 3,38 3,38 3,38 3,38 3,38 3,38 3,38
PS,FEA (kN) 6,26 6,11 5,95 5,82 5,77 6,01 6,29 6,14 5,99 5,86 5,80 6,02 6,31 6,17 6,02 5,89 5,81 6,01 6,40 6,25 5,82 5,33 4,95 4,98 3,43 3,52 3,67 3,84 4,07 4,41 3,45 3,55 3,71 3,89 4,18 4,58 3,48 3,59 3,75 3,94 4,23 4,66 3,63 3,76 3,94 4,20 4,54 4,61
PS,FEA/PFEA 0,971 0,946 0,923 0,902 0,894 0,931 0,975 0,951 0,928 0,908 0,899 0,934 0,978 0,957 0,934 0,913 0,900 0,932 0,992 0,969 0,902 0,825 0,768 0,773 1,014 1,040 1,084 1,137 1,204 1,305 1,022 1,052 1,097 1,152 1,237 1,355 1,030 1,062 1,108 1,164 1,252 1,378 1,073 1,111 1,167 1,242 1,343 1,364
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4. Results, conclusions and future work The influence of vertical stiffeners on the ultimate stiffened web crippling strength was analyzed using the data from the numerical modelling of one hundred specimens. The obtained results are presented in Table 1 and Figure 4. As can be observed in Figure 4, the application of stiffeners is not efficient under the loading and boundary conditions of ITF and IOF, though the contrary results are evident under the loading and boundary conditions of ETF and EOF. With the increasing height of the stiffeners, the highest efficiency is achieved under the ETF loading and boundary condition. This is explained by the fact that under such a condition the unstiffened web of the beam buckles through eits entire height at relatively low loading intensity. Under the EOF loading and boundary condition, the efficiency also increases with the increasing height of the stiffener, though the increment is not so high. This is due to the fact
Fig. 4. Web crippling ultimate capacity dependence of stiffeners geometric parameters.
Fig. 5. Stress distribution in deformed specimen (a) ETF-22; (b) EOF-24; (c) ITF-23; (d) IOF-8.
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that the buckling of the unstiffened web takes place not over its entire height but at the bottom part only. Once the critical slenderness of the stiffener is reached, the width d of the stiffener has no influence on the load carrying capacity of the web and web failure occurs not over the height of the stiffener. However, having a small width d of the stiffener, it becomes too slender to resist buckling and the failure occurs in the stiffener itself (Fig. 5a, b, and c.). The latter effect is observed in the specimens ETF-22, ETF-23, ETF-24, ITF-22, ITF-23, ITF-24, and EOF24. Due to the right angles at the ends of the stiffener, a pronounced stress concentration was observed (Fig. 5). Before the loss of load-capacity of the web, a plastic deformations were formed in these zones. The analysis of stress distribution under the ITF loading and boundary conditions revealed that an area of high stresses is formed in the middle of the stiffener at the longitudinal axis of the beam. Following to the observed stress distribution it can be concluded that during the formation of the buckling wave along the Z axis the opening of the stiffener facilitates the out of plane deformation. This explains the ineffectiveness of the stiffeners under the ITF loading and boundary conditions. The same tendency is observed under the IOF loading and boundary conditions at the high height of stiffeners. In the following studies, the critical slenderness of stiffeners will be determined corresponding to the failure of the web without buckling of the stiffener. An apparent efficiency of the use of stiffeners under the EOF and ETF loading and boundary conditions requires offering a calculation methodology for the practical applications. A further research will contribute to the investigation of smoother forms of the stiffeners preventing stress concentrations. It will also cover the efficiency analysis of vertical stiffeners made without an openings. References [1] N. Hetrakul, W. W. Yu, Structural Behavior of Beam Webs Subjected to Web Crippling and a Combination of Web Crippling and Bending, Civil Engineering Study 78-4, Cold-Formed Steel Series, Final Report, University of Missouri-Rolla, Rolla, Missouri, USA, 1978. [2] G. D. Ratliff , Interaction of Concentrated Loads and Bending in C-shaped Beams, Proceedings of the Third Specialty Conference on ColdFormed Steel Structures, University of Missouri-Rolla, 1975. [3] B. Young, G. J. Hancock, Experimental Investigation of Cold-Formed Channels Subjected to Combined Bending and Web Crippling, Proceedings of the 15th International Specialty Conference on Cold-Formed Steel Structures, University of Missouri-Rolla, 2000. [4] ANSYS Inc., Ansys mechanical APDL element reference, Canonsburg, USA, 2013. [5] American Iron and Steel Institute, TS-9-05 Standard test method for determining the web crippling strength of cold-formed steel beams, USA, 2008. [6] M. Macdonald, M. A. Heiyantuduwa, A design rule for web crippling of cold-formed steel lipped channel beams based on nonlinear FEA, ThinWalled Structures 53 (2012) 123–130. [7] L. Sundararajah, P. Keerthan, M. Mahendran, Web crippling behaviour of C-section beams with web ribs, in: 23rd Australasian Conference on the Mechanics of Structures and Materials, Byron Bay, Australia, 9-12 December 2014, pp. 553-558. [8] Y. Chen, X. Chen, Ch. Wang, Experimental and finite element analysis research on cold-formed steel lipped channel beams under web crippling, Thin-Walled Structures 87 (2015) 41–52. [9] M. Macdonald, M. A. Heiyantuduwa, M. Kotelko, J. Rhodes, Web crippling behaviour of thin-walled lipped channel beams, Thin-Walled Structures 49 (2011) 682–690. [10] D. Dubina, V. Ungureanu, R. Landolfo, Design of Cold-formed Steel Structures, Eurocode 3: Design of Steel Structures, Part 1-3: Design of Cold-formed Steel Structures, ECCS, 2012 [11] W. W. Yu, COLD-FORMED STEEL DESIGN third edition, John Wiley & Sons, USA, 2000.
