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Longitudinally stiffened web purlins under shear and bending moment
Thin–Walled Structures 148 (2020) 106616
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Longitudinally stiffened web purlins under shear and bending moment Juliana Maria Mazzeti Silva *, Maximiliano Malite Department of Structural Engineering, S~ ao Carlos School of Engineering, University of S~ ao Paulo, Av. Trabalhador S~ ao-Carlense, 400, 13566-590, S~ ao Carlos, SP, Brazil
A R T I C L E I N F O
A B S T R A C T
Keywords: Cold-formed sections High strength steel Longitudinal web stiffeners Shear buckling. Combined bending and shear Direct strength method
This paper aims to better understand the influence of longitudinal web stiffeners in the structural behavior of cold-formed steel purlins. A set of specimens with stiffened web and plain web cross-sections were tested on predominantly shear, combined bending and shear, and bending only. Numerical analyses carried out in CUFSM and ANSYS investigated the buckling behavior of the sections, allowing the determination of elastic critical stresses and the calculation of the resistance for the tested specimens by Direct Strength Method (DSM). Nu merical results showed that the intermediate web stiffeners improve the buckling strength in loading cases of pure bending and pure shear. Experimental and numerical data were used to evaluate the interaction between bending moment and shear force, and showed that the circular interaction diagram was conservative and the results were closer to the trilinear interaction diagram. The comparison between experimental data and DSM predictions demonstrated that DSM provides accurate estimates for shear and flexural strengths of purlins with stiffened web cross-section.
1. Introduction Cold-formed steel (CFS) members are widely used in civil engineer ing. One of the most common CFS applications are as purlins in roof systems, usually with C or Z-sections. Between these two cross-sections, Z-section is distinguished by the possibility of overlapping using different flanges widths, which provides continuity and double thickness at the internal supports. CFS members find great applicability in roof systems worldwide due to advantages such as ease of fabrication, flex ibility and high strength-to-weight ratio. Over the last few years, the growing demand for even more economical and versatile structures has led to the development of members with complex geometry cross-sections. The manufacture of high strength steel (with yield stress up to 550 MPa) results in reduction of thicknesses and consequently sections with more folds and stiffeners [1]. Intermediate stiffeners are commonly inserted into the web of CFS purlins to improve the structural performance. In cross-sections with high web slenderness ratios, web stiffeners may weaken local buckling by reducing the impact of bending moment and/or shear force. One example of stiffened web cross-section is ZAE-section (Fig. 1), developed in Brazil by Modular Building Systems [2]. ZAE-section presents Z-shape with 90� lips and two longitudinal intermediate stiffeners in the web. These longitudinal stiffeners are located near the flanges, allowing the connection through the web with the bracing and other components of
roof systems. The failure strength of CFS purlins with stiffened web can be pre dicted by Direct Strength Method (DSM) according to AISI S100-16 [3]. The elastic critical moment can be determined by Finite Strip Method (FSM) or Generalized Beam Theory (GBT) easily in freeware software such as CUFSM [4] and GBTul [5]. However, the calculation of shear strength for complex geometry sections is not simple. The major reason is that CUFSM and GBTul cannot perform elastic buckling analyses under pure shear stresses. In addition, the classic solution by Theory of Elastic Stability assumes the web as an isolated flat plate, simply sup ported at the four edges and subjected to pure shear. Thus, the effect of the shear flow over the cross-section and the influence of flanges, lips and intermediate stiffeners are not accounted for. The structural behavior of stiffened web purlins is poorly under stood, especially in situations of shear and combined bending and shear as occur at the internal supports in continuous systems. Most of the re searches in literature focus on purlins of plain web (without any web stiffeners), mainly with channel section. LaBoube and Yu [6] conducted a pioneering study in CFS channels members on combined bending and shear. The study was an experimental investigation of simply supported beams with short spans and loading introduced directly into the web by bolted connections at the mid-span. Pham and Hancock [7] developed a similar research analyzing cases of predominantly shear and combined bending and shear, and included some tests of bending only based on
* Corresponding author. E-mail address: [email protected] (J.M.M. Silva). https://doi.org/10.1016/j.tws.2020.106616 Received 30 May 2019; Received in revised form 3 December 2019; Accepted 13 January 2020 Available online 27 January 2020 0263-8231/© 2020 Elsevier Ltd. All rights reserved.
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purlins with plain web Z-section and with stiffened web ZAE-section under shear and bending moment. The main goal was to investigate the influence of intermediate web stiffeners on the structural behavior of purlins. An experimental analysis was carried out considering cases of predominantly shear, combined bending and shear, and bending only. Numerical analyses using finite strip software CUFSM [4] and finite element software ANSYS [12] were performed to obtain elastic critical stresses and buckling modes. Experimental and numerical data were compared to DSM predictions and used to evaluate the interaction be tween bending moment and shear force. 2. Experimental analysis 2.1. Test rig and specimens Fig. 1. ZAE-profile.
Twenty-two laboratory tests were conducted in a set of CFS purlins with plain web and with stiffened web [13]. Three different series of tests were performed: predominantly shear, combined bending and shear, and bending only. Predominantly shear tests and combined bending and shear tests were performed in specimens with both stiff ened and unstiffened web cross-sections for comparison purposes. Bending only tests were carried out just for specimens with web stiff eners because previous results for plain web specimens had already shown good agreement with DSM predictions [14,15]. The design of the test rig was based on works developed by LaBoube
four-point loading arrangement. Recently, the same authors experi mentally and numerically investigated the structural behavior of CFS channels with longitudinally stiffened webs under predominantly shear [1], combined bending and shear [8], and pure bending [9]. Pham et al. [10,11] also studied the effect of Tension Field Action (TFA) in increasing the shear strength of members with and without web stiffeners. This paper presents an experimental and numerical study of CFS
Fig. 2. Predominantly shear and combined bending and shear tests setup. 2
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and Yu [6] and Pham and Hancock [7]. Predominantly shear tests and combined bending and shear tests (Fig. 2) consisted of a simply sup ported beam with concentrated load at mid-span. The difference be tween them was the shear span-to-height ratio, a/h, equals one for predominantly shear tests, and equals two or three for combined bending and shear tests. Bending only tests (Fig. 3) consisted of the well-known four-point loading scheme, in which the central region has uniform bending moment and null shear force. It is worth mentioning that h parameter is the height of the flat portion of the web. For stiffened web ZAE-section, the h parameter was defined as if the web was plain (intermediate web stiffeners were disregarded). The inside bend radii of the sections were assumed to be equal 4.75 mm at the corners. All tests were performed using a servo-controlled hydraulic jack and a reaction frame system. In order to prevent lateral buckling, the purlins were tested in pairs with bottom flanges facing inwards. A gap was left between the two purlins of each specimen, to ensure experimental as sembly. At the supports and loading points, rigid profiles attached to cleats were placed and connected to the purlins to loading transfer and
lateral bracing, aiming to simulate the similar conditions of a real structure. Hence, the loading was applied directly into the web. The purlins and cleats were connected by M16 high tensile bolts preten sioned up to 200 N.m. To better define the boundary conditions, filler plates were put between the purlins and cleats. Tests with and without distortion restraint were carried out. The distortion restraint was given by 50�50�3.00 mm equal angle steel straps, connecting both top and bottom flanges of the two tested purlins in each specimen. The purpose of these straps was preventing distor tional buckling and forcing local buckling to occur [7]. In predominantly shear tests and in combined bending and shear tests, the straps were put in all spans, at both sides of the loading point. In bending only tests, only the central region had straps. The number of straps was variable and dependent of the span length. Self-tapping screws connected straps and flanges. Fig. 4(a) shows an example of specimen with straps. In predominantly shear tests and in combined bending and shear tests, steel rectangular bars were connected by bolts to the straps attached to the top and bottom flanges of the purlins at the supports
Fig. 3. Bending only tests setup. 3
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Fig. 4. Combined bending and shear test with straps and lateral flange restraints: (a) side view and (b) detail of lateral flanges restraint at the support.
region. These rectangular bars, shown in Fig. 4(b), provided lateral re straint to the purlins’ flanges and contributed to lateral bracing at the supports. For lateral bracing in bending only tests, the regions between the supports and loading points had steel decks connected to the top flanges of both purlins by self-tapping screws, as shown in Fig. 5. CFS members of Z-section and ZAE-section were tested. Both crosssections present 90� lips and slightly different flange widths to provide the fitting of two purlins inversely placed, combining top flange with bottom flange. ZAE-section also has two longitudinal intermediate stiffeners in the web, near the flanges. Fig. 6 shows cross-sections ge ometry and definitions, and Table 1 summarizes the average dimensions of the specimens measured by scanner. The Z and ZAE-sections with the same cross-section height are equivalent, and the differences between lip lengths and flanges widths are because some adjustments have to be made in the cross-sections dimensions in order to minimize losses during the steel coils slitting in the production of the purlins. However, since these differences were small, they had no major implications on the results. Labels identified the test specimens according to the type of section, cross-section height, series of test and presence or lack of straps. For instance, the label ZAE 245-1.S defines the specimen as follows:
Fig. 6. Definitions of cross-section dimensions: (a) Z-section and (b) ZAE-section.
- ZAE: stiffened web section (alternatively, Z indicates plain web section); - 245: cross-section height in millimeters; - 1: value of ratio a/h, which is 1 for predominantly shear tests, and 2 or 3 for combined bending and shear tests (alternatively, M indicates bending only tests); - S: straps (alternatively, N means no straps).
different values of vertical bolt spacing were adopted taking into ac count the cross-section geometry and construction features. Fig. 7 shows the connection configurations utilized in the purlins, where it can be noticed that at the same cross-section height the bolts are much closer to each other in ZAE-section than in Z-section. The reason is that longitu dinal stiffeners do not allow the connection by bolts near the flanges, thus they have to be concentrated in the central region of the web for ZAE-section specimens.
The boundary conditions in the tests aimed to reproduce the condi tions used in practice for purlins. For each cross-section height and type,
2.2. Material properties The purlins’ material is ZAR-450 steel, with minimum nominal yield stress of 450 MPa, and with a standard zinc-coating thickness of approximately 0.025 mm on each side of the steel plate. Tensile coupon tests according to ASTM A370:2017 [16] determined the material properties of the test specimens. For each thickness, three steel coupons were extracted from a random specimen in the flat surfaces of the top flange (CP 1), web (CP 2) and bottom flange (CP 3). Table 2 summarizes the values of yield stress (fy) and ultimate stress (fu). The adopted values for Young’s modulus and Poisson’s ratio of steel were 200,000 MPa and 0.3, respectively. 3. Numerical analysis Numerical analyses were carried out to investigate the buckling behavior of CFS purlins under bending moment or shear force. The finite strip method (FSM) and the finite element method (FEM) were used to
Fig. 5. Steel decks at the support region in a bending only test. 4
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Table 1 Cross-section dimensions of the specimens. Specimen
H (mm)
B1 (mm)
B2 (mm)
D1 (mm)
D2 (mm)
A (mm)
B (mm)
C (mm)
E (mm)
F (mm)
G (mm)
A1 (mm)
A2 (mm)
t (mm)
ZAE ZAE ZAE ZAE ZAE ZAE ZAE ZAE N
245-1.S 245-2.S 245-3.S 245-1.N 245-2.N 245-3.N 245-M.S 245-M.
247 247 247 246.5 246.5 246.5 246.5 246.5
69 69 69 69 69 69 69 69
53 53 53 55 55 55 55 55
20 20 20 22 22 22 22 22
21 21 21 21 21 21 21 21
49 49 49 48 48 48 48 48
31 31 31 29 29 29 29 29
13 13 13 14 14 14 14 14
60 60 60 63 63 63 63 63
37 37 37 36 36 36 36 36
15 15 15 13 13 13 13 13
7.1 7.1 7.1 6.7 6.7 6.7 6.7 6.7
9.4 9.4 9.4 10.1 10.1 10.1 10.1 10.1
1.25 1.25 1.25 1.25 1.25 1.25 1.25 1.25
ZAE ZAE ZAE ZAE ZAE ZAE ZAE ZAE N
340-1.S 340-2.S 340-3.S 340-1.N 340-2.N 340-3.N 340-M.S 340-M.
342 342 342 342 342 342 342 342
98 98 98 96 96 96 96 96
82 82 82 81 81 81 81 81
17 17 17 23 23 23 23 23
21 21 21 21 21 21 21 21
56 56 56 58 58 58 58 58
35 35 35 36 36 36 36 36
17 17 17 16 16 16 16 16
54 54 54 57 57 57 57 57
30 30 30 33 33 33 33 33
13 13 13 14 14 14 14 14
11.6 11.6 11.6 10.8 10.8 10.8 10.8 10.8
14 14 14 10.5 10.5 10.5 10.5 10.5
1.75 1.75 1.75 1.75 1.75 1.75 1.75 1.75
247 247 247 246.5 246.5 246.5
67 67 67 71 71 71
55 55 55 51 51 51
25 25 25 26 26 26
25 25 25 22 22 22
– – – – – –
– – – – – –
– – – – – –
– – – – – –
– – – – – –
– – – – – –
– – – – – –
– – – – – –
1.25 1.25 1.25 1.25 1.25 1.25
Z 245-1.S Z 245-2.S Z 245-3.S Z 245-1.N Z 245-2.N Z 245-3.N
NOTE: t is the total thickness of steel with zinc coating.
Fig. 7. Vertical bolt spacing: (a) Z-section with 245 mm height, (b) ZAE-section with 245 mm height and (c) ZAE-section with 340 mm height.
perform elastic buckling analyses and evaluate the elastic critical buckling stresses and buckling modes in pure bending and pure shear.
3.2. Shear buckling The FEM computer program ANSYS v.17.0 [12] was used in the shear buckling analysis because FSM freeware programs such as CUFSM and GBTul do not perform elastic buckling analysis of sections under pure shear stresses. The modeling of shear buckling was based on the researches by Pham et al. [17,18]. A finite element model of CFS members with stiffened and unstiffened web was generated to determine elastic critical stresses and buckling modes in pure shear loading. The ANSYS routine was the setting of geometry and creation of a mechanical model in which the boundary conditions and loading were applied. Initially, a linear static analysis was performed to obtain a stress field that was then used in the shear buckling analysis (eigenvalue buckling analysis).
3.1. Buckling due to bending moment To assess the elastic critical buckling stresses and buckling modes due to bending moment, the FSM computer program CUFSM v.4.05 [4] was employed in the elastic buckling analysis of CFS members with and without stiffened web. Both ZAE and Z-sections were modeled with the dimensions from the experimental analysis and steel thickness without coating. Bending stresses were applied in the cross-sections considering a moment about the axis perpendicular to the web (restrained bending).
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the web’s area) was used to create the non-uniform shear stress distri bution according to Eq. (1), which provided the shear stress values in each strip. For the sake of simplification, V1 was evaluated as if the web was plain even for stiffened web cross-sections.
Table 2 Coupon test results. Thickness (mm)
Coupon position – CP
fy (MPa)
fu (MPa)
1.25
1 2 3
495.38 481.51 481.89
575.73 582.09 576.53
486.26 7.90 0.0162
578.12 3.46 0.0060
508.17 504.18 518.99
594.87 593.85 603.05
510.44 7.66 0.0150
597.26 5.05 0.0084
Average (μ) Standard deviation (σ) Coefficient of variation (σ/μ) 1.75
1 2 3 Average (μ) Standard deviation (σ) Coefficient of variation (σ/μ)
τi ¼
Fi;iþ1 ¼ ðτiþ1
Z
Xi
Yi
1, 2, 5, 6, 9 and 10 3, 4, 7 and 8 11 and 12 13, 14, 15, 16, 17, 18, 19 and 20
0 0 0 0
0 1 0 0
1 0 0 0
0 0 0 1
0 0 0 0
(2)
4. Results and discussion 4.1. Experimental results Table 4 summarizes the experimental results, including experimental force (FT), shear span length (a), shear capacity (VT) and bending moment capacity (MT). As one specimen has two purlins, the loading in each purlin is half of the experimental force. According to Table 4, in predominantly shear tests and in combined bending and shear tests the experimental force for specimens without straps was lower than for specimens with straps. The lowest values of experimental force in tests without straps occurred due to the rotation of the top flanges at the supports (Fig. 12). This happened mostly for stiffened web ZAE-section specimens due to the connection configura tion, with bolts far from the flanges and close to each other, concen trated at the middle of the web because of the longitudinal stiffeners. In plain web Z-section specimens, bolts are far from each other and near the flanges, which provided lateral restraint and avoided the flanges rotation. In the tests with distortion restraint, the straps and flange re straint bars prevented the rotation at the supports. Table 4 shows that tests with stiffened web specimens had experi mental force values lower than tests with plain web specimens. This appears to be an inconsistency because longitudinal stiffeners are sup posed to increase the strength. However, it happened because of the different connection configurations used in the tests and shown in Fig. 7, which was proven by additional tests. These additional tests (Z 245-1.S (a) and Z 245-1.N(a) in Table 4) were carried out in two Z-section specimens using the same connection configuration of ZAE-section, with bolts far from flanges and close to each other. Tests with and without straps were performed considering only the ratio a/h equals one, in which case the rotation at the supports was more pronounced. The re sults showed that the flanges rotation at the supports region occurred in these further tests, and the experimental force decreased from 125.83 kN to 98.47 kN for the specimen with straps, and from 99.64 kN to 64.09 kN for the specimen without straps. Hence, the further results for unstiffened web specimens approximated the previous results for
Table 3 Boundary conditions. Y
τi Þat
where Fi,iþ1 ¼ longitudinal force applied between strips i and i þ 1; τi ¼ stress in strip i; τiþ1 ¼ stress in strip i þ 1; a ¼ length; and t ¼ thickness. Elastic shear buckling analyses were performed with stress fields from linear static analyses carried out previously. Material and geo metric nonlinearities were not considered, thus the results refer to ideal structures in elastic phase. Although there are many possible buckling modes for each cross-section, only the one associated with the lowest stress value is relevant.
3.2.2. Loading and elastic shear buckling analysis Shear loads in the FEM model was simulated by means of pure shear forces directly applied on all edges of the two end cross-sections. In order to consider the effect of shear flow varying around the crosssection, the geometry was divided into strips and shear forces were applied on their edges. The strips’ widths were defined according to the maximum element size adopted for the mesh, so that each strip was composed by one or two finite element rows. Since the shear forces applied on the strips had variable magnitudes and generated uniform shear stresses in each one of them, the stress distribution in the nu merical model approximated the non-uniform shear stress distribution that occurs in a cross-section with a shear force parallel to the web passing by the shear center, as can be seen in Fig. 9. Initially, the webs of Z and ZAE-sections were assumed subjected to a uniform and unitary shear stress, as depicted in Fig. 10. The V1 shear force related to this shear stress distribution (unitary stress multiplied by
X
(1)
where τi ¼ shear stress in strip i; V1 ¼ shear force that generates uniform and unitary stress in the web; Szi ¼ static moment of strip i about Z-axis; t ¼ thickness; and Iz ¼ moment of inertia of the section about Z-axis. Eq. (1) calculates the stresses in each strip. To turn these stresses into forces actually applied in the numerical model, they were multiplied by the area of the corresponding strip. Applying average and uniform values of stress in each strip leads to a stress difference between adjacent strips, as shown in Fig. 11. Therefore, longitudinal forces corresponding to this stress difference were applied between strips to ensure the balance in the model. These forces were evaluated according to Eq. (2).
3.2.1. Geometry and boundary conditions ZAE and Z-sections in FEM model approximated the real crosssections, without any discretization of the cold-formed corners at the vertices and intermediate web stiffeners. As was done in the elastic buckling analysis due to bending moment, the cross-sections were modeled with the dimensions from the experimental analysis and real thickness, discounting the zinc coating thickness, which has no struc tural role. The finite element SHELL 181 from ANSYS library was used in the model. This element has four nodes and six degrees of freedom at each node, three translations and three rotations, associated with the coor dinate axes X, Y and Z. To provide an adequate refinement level, a quadrilateral mesh with maximum element size of 10 mm was generated. Table 3 and Fig. 8 show the boundary conditions in the finite element model. All edges of the end cross-sections were simply supported, including the intermediate web stiffeners of ZAE-section, in which local axes were created and displacements were restrained orthogonally to the plate (Fig. 8(c)). Although this simply supported boundary condition is idealized and rarely occurs in practice, it is very important to better understand the theoretical behavior of CFS members under pure shear [17]. A node at the middle of the web at one end section was also restrained in axial direction (X-axis), to avoid rigid body motion.
Edges
V1 Szi tIz
NOTE: “0” denotes free and “1” denotes fixed. 6
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Fig. 8. Boundary conditions: (a) Z-profile, (b) ZAE-profile and (c) details of the intermediate web stiffeners of ZAE-profile.
Fig. 11. Unbalanced stresses between adjacent strips.
Fig. 9. Non-uniform shear stress distribution in the numerical model.
Consequently, there was an increase of the purlins capacity in these tests because TFA improves the shear resistance due to the post-buckling strength reserve. Due to the short span length, predominantly shear tests had large influence of concentrated loads introduced into the web by bolts. For stiffened web specimens with straps, local shear buckling at the web was observed experimentally (Fig. 13(a)). For unstiffened web specimens, the failure mode was a combination of shear buckling and web crippling (Fig. 13(b)), which was also reported by Pham and Hancock [11]. The reason for the occurrence of web crippling is the lack of bolts at the middle of the web. In combined bending and shear tests with straps, the failure occurred by local buckling at the web for ZAE-section specimens, as shown in Fig. 14(a) and (b). For Z-section specimens, a combination of local buckling and web deformation at the loading point was observed in the tests, as depicted in Fig. 14(c) and (d). Bending only tests had experimental results very close for specimens with and without straps. Hence, the straps were not able of properly avoid the distortion and the failure occurred due to distortional buckling in all tests, as shown in Fig. 15.
Fig. 10. Uniform and unitary shear stress distribution in the web.
stiffened web specimens. The highest experimental force values in tests with plain web Zsection specimens also can be justified by Tension Field Action (TFA). Besides the analyzed Z cross-section has high web slenderness ratio, the connection configuration with bolts near the flanges worked such as transverse stiffeners, which allowed the development of TFA.
4.2. Numerical results 4.2.1. Buckling due to bending moment Elastic buckling analyses of ZAE and Z-sections subjected to bending moment were carried out in CUFSM [4]. Fig. 16 shows an example of 7
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ZAE-section with 245 mm height and 1.25 mm thickness (ZAE 245�1.25) and its equivalent plain web Z-section (Z 245�1.25). It is important to notice that the elastic critical stresses shown in the graphics are the maximum stress values in the cross-section. As can be seen in Fig. 16, the longitudinal web stiffeners significantly increase the local buckling strength. Due to the longitudinal stiffeners, the critical stresses at local buckling rise from 150.94 MPa in Z-section to 392.19 MPa in ZAE-section for compression in the top flange and from 136.54 MPa in Z-section to 553,87 MPa in ZAE-section for compression in the bottom flange, which represents increases of over 150% and 300%, respectively. Nevertheless, the stiffeners have no considerable effect in the distortional and lateral-torsional buckling. For instance, the critical stresses at distortional buckling in the case of compression in the top flange are 239.59 MPa for Z-section and 250.85 MPa for ZAEsection, and in the case of compression in the bottom flange, the cor responding values are 246.24 MPa and 259.56 MPa. This represents a difference between the critical distortional buckling stresses of only about 5%. For both loading cases analyzed in Fig. 16, the elastic critical stresses for the stiffened web ZAE-section are higher at local buckling than at distortional buckling, which does not happen for the plain web Z-sec tion. The highest values of elastic critical stresses at local buckling are a great advantage of stiffened web purlins, because the combination of bending and shear is usually computed with only the flexural strength at local buckling. Thus, using the same cross-section height and practically the same amount of material (reducing the flanges and lips widths), it is possible to increase the combined bending and shear strength by means of longitudinal web stiffeners.
Table 4 Experimental results. FT ðkNÞ 4
Specimen
aðmmÞ
FT ðkNÞ
ZAE ZAE ZAE ZAE ZAE ZAE ZAE ZAE
245-1.S 245-2.S 245-3.S 245-1.N 245-2.N 245-3.N 245-M.S 245-M.N
233 466 699 233 466 699 1200 1200
95.76 76.70 61.88 67.74 51.27 38.26 33.03 34.57
23.94 19.18 15.47 16.94 12.82 9.56 8.26 8.64
5.58 8.94 10.81 3.95 5.97 6.69 9.91 10.37
ZAE ZAE ZAE ZAE ZAE ZAE ZAE ZAE
340-1.S 340-2.S 340-3.S 340-1.N 340-2.N 340-3.N 340-M.S 340-M.N
327 654 981 327 654 981 1650 1650
185.70 143.72 120.02 150.74 79.20 54.01 63.85 61.56
46.43 35.93 30.00 37.69 19.80 13.50 15.96 15.39
15.18 23.50 29.43 12.32 12.95 13.25 26.34 25.39
Z 245-1.S Z 245-2.S Z 245-3.S Z 245-1.N Z 245-2.N Z 245-3.N
233 466 699 233 466 699
125.83 86.79 56.70 99.64 54.27 42.30
31.46 21.70 14.17 24.91 13.57 10.58
7.33 10.11 9.91 5.80 6.32 7.39
Z 245-1.S(a)a Z 245-1.N(a)a
233 233
98.47 64.09
24.62 16.02
5.74 3.73
VT ¼
MT ¼
FT a ðkN:mÞ 4
a Z 245-1.S(a) and Z 245-1.N(a) are additional tests performed in Z specimens with the same connection setup of ZAE specimens.
4.2.2. Shear buckling Shear buckling analysis using ANSYS [12] determined the elastic critical stresses and buckling modes on pure shear loading. Fig. 17 de picts the elastic critical stresses according to the length for a stiffened web section (ZAE 245�1.25) and its equivalent unstiffened section (Z 245�1.25). The comparison between the results shows that the longi tudinal web stiffeners enhance shear strength by increasing the magni tude of elastic shear buckling stresses in the length range from about 100 mm to 2500 mm. For example, at 200 mm length the critical shear stresses are 53.53 MPa for Z-section and 99.37 MPa for ZAE-section, which represents an increase of about 85% due to the longitudinal web stiffeners. Conversely, below 100 mm and above 2500 mm the buckling stresses are almost the same for Z and ZAE-sections, and therefore the longitudinal stiffeners have no significant effect in the critical shear stresses. Fig. 17 also shows the buckling modes at some lengths for members under pure shear stresses. In plain web section, the local buckling in the web occurs for lengths up to 2500 mm. In stiffened web section, the local buckling mode appears for lengths up to about 500 mm; from 500 mm to 1500 mm, a combination of local and distortional buckling is observed; and, as the length increases, the distortional buckling becomes more relevant. For both analyzed cross-sections, a flexural-torsional mode can
Fig. 12. Rotation of the top flanges at the supports in a combined bending and shear test without straps (ZAE 340-2.N).
results for equivalent cross-sections, considering compression in the top flange (Fig. 16(a)) and compression in the bottom flange (Fig. 16(b)). These two loading cases were evaluated because in continuous and sleeved multi-span systems the purlins can be used in both positions. The analyzed cross-sections are the same of the tests: stiffened web
Fig. 13. Failure modes in predominantly shear tests with straps: (a) ZAE 245-1.S and (b) Z 245-1.S. 8
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Fig. 14. Failure modes in combined bending and shear tests with straps: (a) ZAE 245-2.S, (b) ZAE 245-3.S, (c) Z 245-2.S and (d) Z 245-3.S.
Fig. 15. Distortional buckling in bending only tests with and without straps: (a) ZAE 245-M.S and (b) ZAE 245-M.N.
Fig. 16. Elastic buckling analyses of a stiffened web cross-section (ZAE 245�1.25) and its equivalent plain web cross-section (Z 245�1.25) considering a moment about the axis perpendicular to the web: (a) compression in the top flange and (b) compression in the bottom flange.
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Fig. 17. Elastic shear buckling analysis for a stiffened web section (ZAE 245�1.25) and its equivalent plain web section (Z 245�1.25).
be identified at long lengths such as 5000 mm. Hancock and Pham [19] also reported the occurrence of this buckling mode in elastic buckling analysis of CFS channels. However, as the authors stated, the flexural-torsional buckling due to shear is unlikely because pure shear loading could not occur at long lengths. Table 5 presents a comparison between the critical shear stresses from ANSYS and from Theory of Elastic Stability. The stiffened web section (ZAE 245�1.25) and its equivalent unstiffened web section (Z 245�1.25) were analyzed in ANSYS using the numerical model described previously and the cross-section dimensions from the experi mental program. The length was considered as the shear spans, defined
by the distance between bolts rows at the supports and loading points, according to the ratio a/h. The critical stresses obtained from eigenvalue buckling analyses in ANSYS for stiffened web members (τcr_ANSYS_ZAE) and plain web members (τcr_ANSYS_Z) were compared to each other and to the values resulting from Theory of Elastic Stability for the plain web members (τcr_Z). In the classic solution by Theory of Elastic Stability, the critical shear stress is evaluated considering the web as an isolated flat plate, simply supported at the four edges and under pure shear stresses. The elastic critical stress and shear buckling coefficient are given by Timoshenko and Gere [20] as presented in Eq. (3) and Eq. (4).
τcr ¼ kv Table 5 Comparison between elastic critical shear stresses from ANSYS and from Theory of Elastic Stability for stiffened web section (ZAE 245�1.25) and unstiffened web section (Z 245�1.25). a= h
τcr
1 2 3
91.95 68.35 55.60
ANSYS ZAE ðMPaÞ
τcr
ANSYS Z ðMPaÞ
46.12 33.83 31.66
12ð1
kv ¼ 5:34 þ
π2 E ν2 Þðh=tÞ2
(3)
4:00
(4)
ða=hÞ2
where τcr ¼ critical shear stress, kv ¼ shear buckling coefficient, E ¼ Young’s modulus, ν ¼ Poison’s ratio, h ¼ height of flat portion of the web, t ¼ thickness and a ¼ length. According to Table 5, the critical shear stresses for the analyzed stiffened web members can be up to twice the critical stresses for its
τcr Z ðMPaÞ 44.78 30.40 27.73
10
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equivalent plain web members. Therefore, longitudinal stiffeners really increase the buckling strength in a significant way. Moreover, for unstiffened web members, the critical stresses from ANSYS are very close to the critical stresses from Theory of Elastic Stability. The maximum difference between these values is about 14%. These results suggest that the influence of flanges and lips in shear buckling is small and negligible. The effect of the shear flow varying around the crosssection also can be ignored. Hence, the simplifications of the classic solution by Theory of Elastic Stability are acceptable for plain web members.
- Local buckling strength: (7)
� Mnl ¼ 1
(8)
Combined bending and shear is very important in the design of stiffened web sections due to the effect of the bending moment in the shear strength. Whereas longitudinal web stiffeners significantly in crease the shear strength, the bending moment capacity is not increased at the same level [1]. Thus, the interaction between bending moment and shear force must be considered. The combination of bending and shear was evaluated with experimental and numerical data, using cir cular and trilinear interaction curves, as depicted by Eq. (5) and Eq. (6), respectively. � �2 � �2 MT VT þ ¼1 (5) Mn Vn
where MT ¼ bending moment capacity, VT ¼ shear capacity, Mn ¼ nominal flexural strength and Vn ¼ nominal shear strength. Table 6 presents the nominal flexural strength and nominal shear strength for the tested purlins, both evaluated according to Direct Strength Method (DSM) in AISI S100-16 [3]. DSM rules for flexure and for shear are shown in Eq. (7) to Eq. (15). The determination of nominal flexural strength by DSM used the critical elastic buckling moments calculated with the critical stresses from elastic buckling analysis in CUFSM. In the determination of nominal shear strength, the critical stresses from ANSYS were employed to obtain the elastic shear buckling force. The shear strength results for plain web Z-section specimens include TFA, experimentally observed in these tests.
Mnl ðkN:mÞ
Mnd ðkN:mÞ
Vn ðkNÞ
Vn
245-1.S 245-2.S 245-3.S 245-1.N 245-2.N 245-3.N 245-M.S 245-M.N
12.88 12.88 12.88 13.46 13.46 13.46 13.46 13.46
9.62 9.62 9.62 10.08 10.08 10.08 10.08 10.08
25.87 19.23 15.64 26.13 19.37 16.03 14.90 14.90
– – – – – – – –
ZAE ZAE ZAE ZAE ZAE ZAE ZAE ZAE
340-1.S 340-2.S 340-3.S 340-1.N 340-2.N 340-3.N 340-M.S 340-M.N
35.40 35.40 35.40 35.77 35.77 35.77 35.77 35.77
24.05 24.05 24.05 26.09 26.09 26.09 26.09 26.09
43.55 32.38 28.23 42.91 31.11 27.29 26.17 26.17
– – – – – – – –
9.54 9.54 9.54 9.56 9.56 9.56
10.33 10.33 10.33 10.23 10.23 10.23
12.98 9.52 8.91 12.98 9.49 8.90
36.44 32.48 31.69 36.40 32.40 31.64
TFA
0:22
� �0:5 �� �0:5 Mcrd Mcrd My for λd > 0:673 My My
(9) (10)
Vn ¼ Vy for λv � 0:815
(11)
pffiffiffiffiffiffiffiffiffiffiffi Vn ¼ 0:815 Vcr Vy for 0:815 < λv � 1:227
(12)
Vn ¼ Vcr for λv > 1:227
(13)
- Shear strength with TFA: Vn ¼ Vy for λv � 0:776
(14)
� Vn ¼ 1
(15)
0:15
� �0:4 �� �0:4 Vcr Vcr Vy for λv > 0:776 Vy Vy
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi where λv ¼ Vy =Vcr , Vn ¼ nominal shear strength, Vy ¼ yield shear force of cross-section and Vcr ¼ elastic shear buckling force. The yield shear force of cross-section is given by Vy ¼ 0:6Aw fy , where Aw ¼ web’s area and fy ¼ yield stress. For all the analyzed sections, the values of Mnl and Mnd shown in Table 6 were obtained by DSM using elastic critical stresses from CUFSM, although it assumes the calculations for pure bending cases. This is applicable even for specimens with predominantly shear and combined bending and shear because the gradient of bending moment does not significantly affect the results for local and distortional buck ling modes. The effect of the bending moment gradient is especially important only for global buckling mode (lateral-torsional buckling), which was not evaluated in this research because the specimens had adequate lateral bracing. Fig. 18 shows bending and shear interaction in predominantly shear tests and in combined bending and shear tests with stiffened web ZAEsection purlins. For specimens with straps (Fig. 18(a)), the interaction was evaluated with the flexural strength at local buckling, because straps provided distortion restraint and forced the purlins to fail in local buckling. For specimens without straps (Fig. 18(b)), the flexural strength at distortional buckling was used in the interaction expressions, because numerical analyses showed that its corresponding critical stresses were lower than the critical stresses at local buckling. On the one hand, most of the results for tests with straps got above circular and trilinear interaction curves. On the other hand, most of the results for tests without straps got below both the interaction curves due to premature failure caused by rotation at the supports. Fig. 19 presents the interaction between bending moment and shear force in tests involving plain web Z-section purlins with and without
Table 6 DSM results for nominal flexural strength and nominal shear strength. ZAE ZAE ZAE ZAE ZAE ZAE ZAE ZAE
�0:4 �� �0:4 Mcrl Mne for λl > 0:776 Mne
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where λl ¼ Mne =Mcrl , λd ¼ My =Mcrd , Mnl ¼ nominal flexural strength at local buckling, Mnd ¼ nominal flexural strength at distortional buckling, Mne ¼ critical elastic lateral-torsional buckling moment, My ¼ member yield moment, Mcrl ¼ critical elastic local buckling moment and Mcrd ¼ critical elastic distortional buckling moment. - Shear strength without TFA:
(6)
Specimen
Mcrl Mne
Mnd ¼ My for λd � 0:673 � Mnd ¼ 1
� � � � MT VT 0:6 þ ¼ 1:3 Mn Vn
� 0:15
- Distortional buckling strength:
4.3. Combined bending and shear
Z 245-1.S Z 245-2.S Z 245-3.S Z 245-1.N Z 245-2.N Z 245-3.N
Mnl ¼ Mne for λl � 0:776
ðkNÞ
11
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Fig. 18. Bending and shear interaction for stiffened web purlins in predominantly shear and combined bending and shear tests: (a) specimens with straps and (b) specimens without straps.
Fig. 19. Bending and shear interaction for plain web purlins in predominantly shear and combined bending and shear tests: (a) shear strength by DSM without TFA and (b) shear strength by DSM with TFA.
straps. The interaction expressions considered the flexural strength at local buckling because all tested specimens failed in this way. Moreover, shear strength used in the interaction expressions were computed by DSM without TFA (Fig. 19(a)) and with TFA (Fig. 19(b)). Fig. 19(a) shows that using DSM without TFA all of the test points lie much above the interaction curves. This conservative trend is because the shear strength evaluated without TFA is underestimated, since the experi mental tests have indicated an increase of the ultimate capacity of the purlins due to TFA. Fig. 19(b) shows that using DSM with TFA all of the results for specimens without straps lie below circular interaction curve and most of the results for specimens with straps lie above trilinear interaction curve. The only exception is the result of the additional test Z 245-1.S(a), which got below both interaction curves although this specimen had straps. The explanation is that the connections with bolts far from flanges did not provide suitable lateral restraint and hindered
the complete development of TFA in this case. The same reasoning can be applied to the specimens without straps, in which the lack of straps and the consequent lack of lateral restraint prevented the full mobili zation of TFA. In bending only tests, the experimental results showed that failure occurred by distortional buckling. For that reason, the interaction ex pressions were evaluated considering the flexural strength at distor tional buckling. As can be seen in Fig. 20, the results overcame the circular interaction curve and reached out the trilinear interaction curve. The outcomes also suggest there was no significant difference between tests with and without straps. It is important to emphasize that in the practice of cold-formed steel structures the purlins typically have greater span-to-depth ratios. Consequently, the influence of the bending moment is much more pro nounced than the influence of the shear force, and the effect of TFA is 12
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shear with and without TFA is presented in Fig. 22. Only the results for predominantly shear tests with straps and for combined bending and shear tests with straps were considered due to the premature failure observed in tests without straps. The shear yield load (Vy) was used to pffiffiffiffiffiffiffiffiffiffiffiffiffiffi � calculate the shear slenderness λv ¼ Vy =Vcr and the nondimesional � shear capacity VT =Vy . Fig. 22 shows that all of the experimental data for stiffened web specimens are very close to the DSM curve for shear without TFA. Therefore, DSM provides accurate predictions for the shear strength of these cross-sections and the model generated in ANSYS gives adequate values of elastic critical shear stresses. However, for plain web specimens, the experimental results lie above DSM curve without TFA and approximate to the DSM curve with TFA, but lie below it. The reason for this fact is that a full TFA were not developed in the tests, probably because of the connection configuration with only two bolts. The lack of bolts over the full depth of the web at the supports and loading points did not allow an adequate level of transverse restraint, affecting the complete development of TFA in tests with plain web specimens. The influence of the connection setup in the development of TFA was corroborated by Z 245-1.S(a) specimen, where the bolts were put closer together as for ZAE-sections and the consequent reduction of lateral restraint caused the test point to drop, coming close to the DSM curve without TFA.
Fig. 20. Bending and shear interaction in bending only tests.
minimized. For these reasons, the design of longitudinally stiffened web purlins can be done with the flexural strength at distortional buckling and the best interaction diagram to be utilized is the trilinear.
5. Conclusions This paper has presented experimental and numerical analyses per formed in CFS purlins with plain web Z-section and stiffened web ZAEsection to better understand the effect of longitudinal web stiffeners in the structural behavior. Twenty-two specimens were tested in predom inantly shear, combined bending and shear, and bending only. The re sults from experimental analysis showed that in predominantly shear tests and in combined bending and shear tests the failure occurred with local shear buckling in the web for specimens with straps. For specimens without straps, the lack of lateral restraint due to the connection configuration caused a premature failure characterized by rotation of the top flanges at the supports. In bending only tests, the ultimate ca pacities for purlins with and without straps were very close, because the straps did not prevent the distortion and the failure occurred by distortional buckling in all tests. Moreover, the experimental program also showed that, against all expectations, the specimens with stiffened web cross-section presented experimental force values lower than the specimens with plain web cross-section because of the development of TFA in these tests and not in those. Numerical analyses carried out in CUFSM and ANSYS investigated the buckling behavior of the cross-sections. The numerical results indi cated that the web stiffeners enhance the buckling strength in both loading cases of pure bending and pure shear. In buckling due to bending moment, the longitudinal stiffeners increase the critical stresses at local buckling but have no considerable effect in distortional and lateral-torsional buckling modes. In shear buckling, the stiffeners also lead to an increase of the elastic critical stresses, and buckling modes are mostly local for unstiffened web members and a combination of local and distortional for stiffened web members. The strength of the tested specimens was determined by DSM using the elastic critical stresses from numerical analyses. Experimental and numerical data were used to evaluate the interaction between bending moment and shear force. These results were plotted against the inter action diagrams and showed that the circular interaction diagram was conservative in most cases and the experimental data were closer to the trilinear interaction diagram. The experimental data were also compared to DSM predictions and demonstrated good agreement. The only exception was the case of the specimens with plain web crosssection, in which the lack of transverse restraint due to screwed con nections with few bolts hindered the full development of TFA, so the experimental results lied above DSM curve without TFA but below DSM
4.4. Comparison between experimental results and DSM predictions The appropriateness of DSM in the design of the stiffened and unstiffened cross-sections was analyzed by plotting the experimental results against the DSM curves. Fig. 21 presents the comparison between the bending moment capacity in bending only tests and the DSM curves of local and distortional buckling. For the tested specimens, the yield moment (My) was used to compute the distortional slenderness ðλd ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi My =Mcrd Þ and the nondimensional bending capacity ðMT =My Þ. The use pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi � of distortional slenderness instead of local slenderness λl ¼ My =Mcrl with the experimental data is because the laboratory tests and numerical analyses showed that these specimens failed at distortional buckling. As can be seen in Fig. 21, the experimental results for bending only tests practically match the DSM curve of distortional buckling, indicating that DSM predictions were quite accurate. The comparison between the shear capacity and the DSM curves for
Fig. 21. Comparison between bending moment capacity and DSM curves of local and distortional buckling. 13
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Thin-Walled Structures 148 (2020) 106616
Fig. 22. Comparison between shear capacity and DSM curves with and without TFA.
curve with TFA. This research remains ongoing with prediction of new experimental tests and parametric numerical analyses with full and detailed FEM models including post-buckling behavior, so that the re sults in Figs. 18–22 could be more robust.
[6] R.A. LaBoube, W.W. Yu, Cold-formed steel web elements under combined bending and shear, in: Proceedings of the 4th International Specialty Conference on ColdFormed Steel Structures, 1978, pp. 219–251. [7] C.H. Pham, G.J. Hancock, Experimental investigation of high-strength cold-formed C-sections in combined bending and shear, J Struct Eng Am Soc Civil Eng 136 (7) (2010) 866–878. [8] C.H. Pham, L.A. Bruneau, G.J. Hancock, Experimental study of longitudinally stiffened web channels subjected to combined bending and shear, J Struct Eng Am Soc Civil Eng 141 (11) (2015), 04015018. [9] C.H. Pham, G.J. Hancock, Experimental investigation and direct strength design of high-strength complex C-sections in pure bending, J Struct Eng Am Soc Civil Eng 139 (11) (2013) 1842–1852. [10] C.H. Pham, G.J. Hancock, Tension field action for cold-formed sections in shear, J. Constr. Steel Res. 72 (2012) 168–178. [11] C.H. Pham, D. Zelenkin, G.J. Hancock, Effect of flange restraints on shear Tension Field Action in cold-formed C-sections, J. Constr. Steel Res. 129 (2017) 42–53. [12] Swanson Analysis System Inc, ANSYS Help, 2016, version 17.0. [13] J.M.M. Silva, Cold-formed Steel Purlins with Stiffened Web: Emphasis on Shear and Combined Bending and Shear (In Portuguese), Thesis (M. Sc. in Structural Engineering), S~ ao Carlos School of Engineering, University of S~ ao Paulo. S~ ao Carlos, S~ ao Paulo, Brazil, 2018, pp. 1–144. [14] C. Yu, B.W. Schafer, Local buckling tests on cold-formed steel beams, J. Struct. Eng. 129 (12) (2003) 1596–1606. [15] C. Yu, B.W. Schafer, Distortional buckling tests on cold-formed steel beams, J. Struct. Eng. 132 (4) (2006) 515–528. [16] American Society for Testing and Material (ASTM), Standard test method and definitions for mechanical testing of steel products, ASTM 370 (2017). [17] S.H. Pham, C.H. Pham, G.J. Hancock, Numerical Simulation of Cold-Formed Channel Sections with Intermediate Web Stiffeners Undergoing Pure Shear, Research Report R930, School of Civil Engineering, University of Sydney, 2012. [18] S.H. Pham, C.H. Pham, G.J. Hancock, Direct strength method of design for shear including sections with longitudinal web stiffeners, Thin-Walled Struct. 81 (2014) 19–28. [19] G.J. Hancock, C.H. Pham, A Signature Curve for Cold-Formed Channel Sections in Pure Shear, Research Report R919, School of Civil Engineering, University of Sydney, 2011. [20] S.P. Timoshenko, J.M. Gere, Theory of Elastic Stability, McGraw-Hill Book Co. Inc., New York, NY, 1961.
CRediT authorship contribution statement Juliana Maria Mazzeti Silva: Conceptualization, Methodology, Validation, Investigation, Data curation, Writing - original draft, Visu alization. Maximiliano Malite: Conceptualization, Methodology, Writing - review & editing, Supervision. Acknowledgements ~o de Aperfeiçoa This study was financed in part by the Coordenaça mento de Pessoal de Nível Superior – Brasil (CAPES) - Finance Code 001. The authors would like to thank Modular Building Systems for donating the specimens for the experimental analysis. References [1] C.H. Pham, G.J. Hancock, Numerical investigation of longitudinally stiffened web channels predominantly in shear, Thin-Walled Struct. 86 (2015) 47–55. [2] Modular Building Systems (in Portuguese). http://www.modularsc.com.br/, 2012. (Accessed 2 September 2019). [3] American Iron and Steel Institute (AISI), North American Specification for the Design of Cold-Formed Structural Members, AISI S100-16, Washington, DC, 2016. � any, Buckling analysis of cold-formed steel members using [4] B.W. Schafer, S. Ad� CUFSM: conventional and constrained finite strip methods, in: Proceedings of the 18th International Specialty Conference on Cold-Formed Steel Structures, 2006, pp. 39–54. [5] R. Bebiano, P. Pina, N. Silvestre, D. Camotim, GBTUL – a GBT-based code for thinwalled member analysis, in: Proceedings of the 5th International Conference on Thin-Walled Structures, 2008, pp. 1173–1180.
14
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Full length article
Longitudinally stiffened web purlins under shear and bending moment Juliana Maria Mazzeti Silva *, Maximiliano Malite Department of Structural Engineering, S~ ao Carlos School of Engineering, University of S~ ao Paulo, Av. Trabalhador S~ ao-Carlense, 400, 13566-590, S~ ao Carlos, SP, Brazil
A R T I C L E I N F O
A B S T R A C T
Keywords: Cold-formed sections High strength steel Longitudinal web stiffeners Shear buckling. Combined bending and shear Direct strength method
This paper aims to better understand the influence of longitudinal web stiffeners in the structural behavior of cold-formed steel purlins. A set of specimens with stiffened web and plain web cross-sections were tested on predominantly shear, combined bending and shear, and bending only. Numerical analyses carried out in CUFSM and ANSYS investigated the buckling behavior of the sections, allowing the determination of elastic critical stresses and the calculation of the resistance for the tested specimens by Direct Strength Method (DSM). Nu merical results showed that the intermediate web stiffeners improve the buckling strength in loading cases of pure bending and pure shear. Experimental and numerical data were used to evaluate the interaction between bending moment and shear force, and showed that the circular interaction diagram was conservative and the results were closer to the trilinear interaction diagram. The comparison between experimental data and DSM predictions demonstrated that DSM provides accurate estimates for shear and flexural strengths of purlins with stiffened web cross-section.
1. Introduction Cold-formed steel (CFS) members are widely used in civil engineer ing. One of the most common CFS applications are as purlins in roof systems, usually with C or Z-sections. Between these two cross-sections, Z-section is distinguished by the possibility of overlapping using different flanges widths, which provides continuity and double thickness at the internal supports. CFS members find great applicability in roof systems worldwide due to advantages such as ease of fabrication, flex ibility and high strength-to-weight ratio. Over the last few years, the growing demand for even more economical and versatile structures has led to the development of members with complex geometry cross-sections. The manufacture of high strength steel (with yield stress up to 550 MPa) results in reduction of thicknesses and consequently sections with more folds and stiffeners [1]. Intermediate stiffeners are commonly inserted into the web of CFS purlins to improve the structural performance. In cross-sections with high web slenderness ratios, web stiffeners may weaken local buckling by reducing the impact of bending moment and/or shear force. One example of stiffened web cross-section is ZAE-section (Fig. 1), developed in Brazil by Modular Building Systems [2]. ZAE-section presents Z-shape with 90� lips and two longitudinal intermediate stiffeners in the web. These longitudinal stiffeners are located near the flanges, allowing the connection through the web with the bracing and other components of
roof systems. The failure strength of CFS purlins with stiffened web can be pre dicted by Direct Strength Method (DSM) according to AISI S100-16 [3]. The elastic critical moment can be determined by Finite Strip Method (FSM) or Generalized Beam Theory (GBT) easily in freeware software such as CUFSM [4] and GBTul [5]. However, the calculation of shear strength for complex geometry sections is not simple. The major reason is that CUFSM and GBTul cannot perform elastic buckling analyses under pure shear stresses. In addition, the classic solution by Theory of Elastic Stability assumes the web as an isolated flat plate, simply sup ported at the four edges and subjected to pure shear. Thus, the effect of the shear flow over the cross-section and the influence of flanges, lips and intermediate stiffeners are not accounted for. The structural behavior of stiffened web purlins is poorly under stood, especially in situations of shear and combined bending and shear as occur at the internal supports in continuous systems. Most of the re searches in literature focus on purlins of plain web (without any web stiffeners), mainly with channel section. LaBoube and Yu [6] conducted a pioneering study in CFS channels members on combined bending and shear. The study was an experimental investigation of simply supported beams with short spans and loading introduced directly into the web by bolted connections at the mid-span. Pham and Hancock [7] developed a similar research analyzing cases of predominantly shear and combined bending and shear, and included some tests of bending only based on
* Corresponding author. E-mail address: [email protected] (J.M.M. Silva). https://doi.org/10.1016/j.tws.2020.106616 Received 30 May 2019; Received in revised form 3 December 2019; Accepted 13 January 2020 Available online 27 January 2020 0263-8231/© 2020 Elsevier Ltd. All rights reserved.
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purlins with plain web Z-section and with stiffened web ZAE-section under shear and bending moment. The main goal was to investigate the influence of intermediate web stiffeners on the structural behavior of purlins. An experimental analysis was carried out considering cases of predominantly shear, combined bending and shear, and bending only. Numerical analyses using finite strip software CUFSM [4] and finite element software ANSYS [12] were performed to obtain elastic critical stresses and buckling modes. Experimental and numerical data were compared to DSM predictions and used to evaluate the interaction be tween bending moment and shear force. 2. Experimental analysis 2.1. Test rig and specimens Fig. 1. ZAE-profile.
Twenty-two laboratory tests were conducted in a set of CFS purlins with plain web and with stiffened web [13]. Three different series of tests were performed: predominantly shear, combined bending and shear, and bending only. Predominantly shear tests and combined bending and shear tests were performed in specimens with both stiff ened and unstiffened web cross-sections for comparison purposes. Bending only tests were carried out just for specimens with web stiff eners because previous results for plain web specimens had already shown good agreement with DSM predictions [14,15]. The design of the test rig was based on works developed by LaBoube
four-point loading arrangement. Recently, the same authors experi mentally and numerically investigated the structural behavior of CFS channels with longitudinally stiffened webs under predominantly shear [1], combined bending and shear [8], and pure bending [9]. Pham et al. [10,11] also studied the effect of Tension Field Action (TFA) in increasing the shear strength of members with and without web stiffeners. This paper presents an experimental and numerical study of CFS
Fig. 2. Predominantly shear and combined bending and shear tests setup. 2
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and Yu [6] and Pham and Hancock [7]. Predominantly shear tests and combined bending and shear tests (Fig. 2) consisted of a simply sup ported beam with concentrated load at mid-span. The difference be tween them was the shear span-to-height ratio, a/h, equals one for predominantly shear tests, and equals two or three for combined bending and shear tests. Bending only tests (Fig. 3) consisted of the well-known four-point loading scheme, in which the central region has uniform bending moment and null shear force. It is worth mentioning that h parameter is the height of the flat portion of the web. For stiffened web ZAE-section, the h parameter was defined as if the web was plain (intermediate web stiffeners were disregarded). The inside bend radii of the sections were assumed to be equal 4.75 mm at the corners. All tests were performed using a servo-controlled hydraulic jack and a reaction frame system. In order to prevent lateral buckling, the purlins were tested in pairs with bottom flanges facing inwards. A gap was left between the two purlins of each specimen, to ensure experimental as sembly. At the supports and loading points, rigid profiles attached to cleats were placed and connected to the purlins to loading transfer and
lateral bracing, aiming to simulate the similar conditions of a real structure. Hence, the loading was applied directly into the web. The purlins and cleats were connected by M16 high tensile bolts preten sioned up to 200 N.m. To better define the boundary conditions, filler plates were put between the purlins and cleats. Tests with and without distortion restraint were carried out. The distortion restraint was given by 50�50�3.00 mm equal angle steel straps, connecting both top and bottom flanges of the two tested purlins in each specimen. The purpose of these straps was preventing distor tional buckling and forcing local buckling to occur [7]. In predominantly shear tests and in combined bending and shear tests, the straps were put in all spans, at both sides of the loading point. In bending only tests, only the central region had straps. The number of straps was variable and dependent of the span length. Self-tapping screws connected straps and flanges. Fig. 4(a) shows an example of specimen with straps. In predominantly shear tests and in combined bending and shear tests, steel rectangular bars were connected by bolts to the straps attached to the top and bottom flanges of the purlins at the supports
Fig. 3. Bending only tests setup. 3
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Fig. 4. Combined bending and shear test with straps and lateral flange restraints: (a) side view and (b) detail of lateral flanges restraint at the support.
region. These rectangular bars, shown in Fig. 4(b), provided lateral re straint to the purlins’ flanges and contributed to lateral bracing at the supports. For lateral bracing in bending only tests, the regions between the supports and loading points had steel decks connected to the top flanges of both purlins by self-tapping screws, as shown in Fig. 5. CFS members of Z-section and ZAE-section were tested. Both crosssections present 90� lips and slightly different flange widths to provide the fitting of two purlins inversely placed, combining top flange with bottom flange. ZAE-section also has two longitudinal intermediate stiffeners in the web, near the flanges. Fig. 6 shows cross-sections ge ometry and definitions, and Table 1 summarizes the average dimensions of the specimens measured by scanner. The Z and ZAE-sections with the same cross-section height are equivalent, and the differences between lip lengths and flanges widths are because some adjustments have to be made in the cross-sections dimensions in order to minimize losses during the steel coils slitting in the production of the purlins. However, since these differences were small, they had no major implications on the results. Labels identified the test specimens according to the type of section, cross-section height, series of test and presence or lack of straps. For instance, the label ZAE 245-1.S defines the specimen as follows:
Fig. 6. Definitions of cross-section dimensions: (a) Z-section and (b) ZAE-section.
- ZAE: stiffened web section (alternatively, Z indicates plain web section); - 245: cross-section height in millimeters; - 1: value of ratio a/h, which is 1 for predominantly shear tests, and 2 or 3 for combined bending and shear tests (alternatively, M indicates bending only tests); - S: straps (alternatively, N means no straps).
different values of vertical bolt spacing were adopted taking into ac count the cross-section geometry and construction features. Fig. 7 shows the connection configurations utilized in the purlins, where it can be noticed that at the same cross-section height the bolts are much closer to each other in ZAE-section than in Z-section. The reason is that longitu dinal stiffeners do not allow the connection by bolts near the flanges, thus they have to be concentrated in the central region of the web for ZAE-section specimens.
The boundary conditions in the tests aimed to reproduce the condi tions used in practice for purlins. For each cross-section height and type,
2.2. Material properties The purlins’ material is ZAR-450 steel, with minimum nominal yield stress of 450 MPa, and with a standard zinc-coating thickness of approximately 0.025 mm on each side of the steel plate. Tensile coupon tests according to ASTM A370:2017 [16] determined the material properties of the test specimens. For each thickness, three steel coupons were extracted from a random specimen in the flat surfaces of the top flange (CP 1), web (CP 2) and bottom flange (CP 3). Table 2 summarizes the values of yield stress (fy) and ultimate stress (fu). The adopted values for Young’s modulus and Poisson’s ratio of steel were 200,000 MPa and 0.3, respectively. 3. Numerical analysis Numerical analyses were carried out to investigate the buckling behavior of CFS purlins under bending moment or shear force. The finite strip method (FSM) and the finite element method (FEM) were used to
Fig. 5. Steel decks at the support region in a bending only test. 4
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Table 1 Cross-section dimensions of the specimens. Specimen
H (mm)
B1 (mm)
B2 (mm)
D1 (mm)
D2 (mm)
A (mm)
B (mm)
C (mm)
E (mm)
F (mm)
G (mm)
A1 (mm)
A2 (mm)
t (mm)
ZAE ZAE ZAE ZAE ZAE ZAE ZAE ZAE N
245-1.S 245-2.S 245-3.S 245-1.N 245-2.N 245-3.N 245-M.S 245-M.
247 247 247 246.5 246.5 246.5 246.5 246.5
69 69 69 69 69 69 69 69
53 53 53 55 55 55 55 55
20 20 20 22 22 22 22 22
21 21 21 21 21 21 21 21
49 49 49 48 48 48 48 48
31 31 31 29 29 29 29 29
13 13 13 14 14 14 14 14
60 60 60 63 63 63 63 63
37 37 37 36 36 36 36 36
15 15 15 13 13 13 13 13
7.1 7.1 7.1 6.7 6.7 6.7 6.7 6.7
9.4 9.4 9.4 10.1 10.1 10.1 10.1 10.1
1.25 1.25 1.25 1.25 1.25 1.25 1.25 1.25
ZAE ZAE ZAE ZAE ZAE ZAE ZAE ZAE N
340-1.S 340-2.S 340-3.S 340-1.N 340-2.N 340-3.N 340-M.S 340-M.
342 342 342 342 342 342 342 342
98 98 98 96 96 96 96 96
82 82 82 81 81 81 81 81
17 17 17 23 23 23 23 23
21 21 21 21 21 21 21 21
56 56 56 58 58 58 58 58
35 35 35 36 36 36 36 36
17 17 17 16 16 16 16 16
54 54 54 57 57 57 57 57
30 30 30 33 33 33 33 33
13 13 13 14 14 14 14 14
11.6 11.6 11.6 10.8 10.8 10.8 10.8 10.8
14 14 14 10.5 10.5 10.5 10.5 10.5
1.75 1.75 1.75 1.75 1.75 1.75 1.75 1.75
247 247 247 246.5 246.5 246.5
67 67 67 71 71 71
55 55 55 51 51 51
25 25 25 26 26 26
25 25 25 22 22 22
– – – – – –
– – – – – –
– – – – – –
– – – – – –
– – – – – –
– – – – – –
– – – – – –
– – – – – –
1.25 1.25 1.25 1.25 1.25 1.25
Z 245-1.S Z 245-2.S Z 245-3.S Z 245-1.N Z 245-2.N Z 245-3.N
NOTE: t is the total thickness of steel with zinc coating.
Fig. 7. Vertical bolt spacing: (a) Z-section with 245 mm height, (b) ZAE-section with 245 mm height and (c) ZAE-section with 340 mm height.
perform elastic buckling analyses and evaluate the elastic critical buckling stresses and buckling modes in pure bending and pure shear.
3.2. Shear buckling The FEM computer program ANSYS v.17.0 [12] was used in the shear buckling analysis because FSM freeware programs such as CUFSM and GBTul do not perform elastic buckling analysis of sections under pure shear stresses. The modeling of shear buckling was based on the researches by Pham et al. [17,18]. A finite element model of CFS members with stiffened and unstiffened web was generated to determine elastic critical stresses and buckling modes in pure shear loading. The ANSYS routine was the setting of geometry and creation of a mechanical model in which the boundary conditions and loading were applied. Initially, a linear static analysis was performed to obtain a stress field that was then used in the shear buckling analysis (eigenvalue buckling analysis).
3.1. Buckling due to bending moment To assess the elastic critical buckling stresses and buckling modes due to bending moment, the FSM computer program CUFSM v.4.05 [4] was employed in the elastic buckling analysis of CFS members with and without stiffened web. Both ZAE and Z-sections were modeled with the dimensions from the experimental analysis and steel thickness without coating. Bending stresses were applied in the cross-sections considering a moment about the axis perpendicular to the web (restrained bending).
5
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the web’s area) was used to create the non-uniform shear stress distri bution according to Eq. (1), which provided the shear stress values in each strip. For the sake of simplification, V1 was evaluated as if the web was plain even for stiffened web cross-sections.
Table 2 Coupon test results. Thickness (mm)
Coupon position – CP
fy (MPa)
fu (MPa)
1.25
1 2 3
495.38 481.51 481.89
575.73 582.09 576.53
486.26 7.90 0.0162
578.12 3.46 0.0060
508.17 504.18 518.99
594.87 593.85 603.05
510.44 7.66 0.0150
597.26 5.05 0.0084
Average (μ) Standard deviation (σ) Coefficient of variation (σ/μ) 1.75
1 2 3 Average (μ) Standard deviation (σ) Coefficient of variation (σ/μ)
τi ¼
Fi;iþ1 ¼ ðτiþ1
Z
Xi
Yi
1, 2, 5, 6, 9 and 10 3, 4, 7 and 8 11 and 12 13, 14, 15, 16, 17, 18, 19 and 20
0 0 0 0
0 1 0 0
1 0 0 0
0 0 0 1
0 0 0 0
(2)
4. Results and discussion 4.1. Experimental results Table 4 summarizes the experimental results, including experimental force (FT), shear span length (a), shear capacity (VT) and bending moment capacity (MT). As one specimen has two purlins, the loading in each purlin is half of the experimental force. According to Table 4, in predominantly shear tests and in combined bending and shear tests the experimental force for specimens without straps was lower than for specimens with straps. The lowest values of experimental force in tests without straps occurred due to the rotation of the top flanges at the supports (Fig. 12). This happened mostly for stiffened web ZAE-section specimens due to the connection configura tion, with bolts far from the flanges and close to each other, concen trated at the middle of the web because of the longitudinal stiffeners. In plain web Z-section specimens, bolts are far from each other and near the flanges, which provided lateral restraint and avoided the flanges rotation. In the tests with distortion restraint, the straps and flange re straint bars prevented the rotation at the supports. Table 4 shows that tests with stiffened web specimens had experi mental force values lower than tests with plain web specimens. This appears to be an inconsistency because longitudinal stiffeners are sup posed to increase the strength. However, it happened because of the different connection configurations used in the tests and shown in Fig. 7, which was proven by additional tests. These additional tests (Z 245-1.S (a) and Z 245-1.N(a) in Table 4) were carried out in two Z-section specimens using the same connection configuration of ZAE-section, with bolts far from flanges and close to each other. Tests with and without straps were performed considering only the ratio a/h equals one, in which case the rotation at the supports was more pronounced. The re sults showed that the flanges rotation at the supports region occurred in these further tests, and the experimental force decreased from 125.83 kN to 98.47 kN for the specimen with straps, and from 99.64 kN to 64.09 kN for the specimen without straps. Hence, the further results for unstiffened web specimens approximated the previous results for
Table 3 Boundary conditions. Y
τi Þat
where Fi,iþ1 ¼ longitudinal force applied between strips i and i þ 1; τi ¼ stress in strip i; τiþ1 ¼ stress in strip i þ 1; a ¼ length; and t ¼ thickness. Elastic shear buckling analyses were performed with stress fields from linear static analyses carried out previously. Material and geo metric nonlinearities were not considered, thus the results refer to ideal structures in elastic phase. Although there are many possible buckling modes for each cross-section, only the one associated with the lowest stress value is relevant.
3.2.2. Loading and elastic shear buckling analysis Shear loads in the FEM model was simulated by means of pure shear forces directly applied on all edges of the two end cross-sections. In order to consider the effect of shear flow varying around the crosssection, the geometry was divided into strips and shear forces were applied on their edges. The strips’ widths were defined according to the maximum element size adopted for the mesh, so that each strip was composed by one or two finite element rows. Since the shear forces applied on the strips had variable magnitudes and generated uniform shear stresses in each one of them, the stress distribution in the nu merical model approximated the non-uniform shear stress distribution that occurs in a cross-section with a shear force parallel to the web passing by the shear center, as can be seen in Fig. 9. Initially, the webs of Z and ZAE-sections were assumed subjected to a uniform and unitary shear stress, as depicted in Fig. 10. The V1 shear force related to this shear stress distribution (unitary stress multiplied by
X
(1)
where τi ¼ shear stress in strip i; V1 ¼ shear force that generates uniform and unitary stress in the web; Szi ¼ static moment of strip i about Z-axis; t ¼ thickness; and Iz ¼ moment of inertia of the section about Z-axis. Eq. (1) calculates the stresses in each strip. To turn these stresses into forces actually applied in the numerical model, they were multiplied by the area of the corresponding strip. Applying average and uniform values of stress in each strip leads to a stress difference between adjacent strips, as shown in Fig. 11. Therefore, longitudinal forces corresponding to this stress difference were applied between strips to ensure the balance in the model. These forces were evaluated according to Eq. (2).
3.2.1. Geometry and boundary conditions ZAE and Z-sections in FEM model approximated the real crosssections, without any discretization of the cold-formed corners at the vertices and intermediate web stiffeners. As was done in the elastic buckling analysis due to bending moment, the cross-sections were modeled with the dimensions from the experimental analysis and real thickness, discounting the zinc coating thickness, which has no struc tural role. The finite element SHELL 181 from ANSYS library was used in the model. This element has four nodes and six degrees of freedom at each node, three translations and three rotations, associated with the coor dinate axes X, Y and Z. To provide an adequate refinement level, a quadrilateral mesh with maximum element size of 10 mm was generated. Table 3 and Fig. 8 show the boundary conditions in the finite element model. All edges of the end cross-sections were simply supported, including the intermediate web stiffeners of ZAE-section, in which local axes were created and displacements were restrained orthogonally to the plate (Fig. 8(c)). Although this simply supported boundary condition is idealized and rarely occurs in practice, it is very important to better understand the theoretical behavior of CFS members under pure shear [17]. A node at the middle of the web at one end section was also restrained in axial direction (X-axis), to avoid rigid body motion.
Edges
V1 Szi tIz
NOTE: “0” denotes free and “1” denotes fixed. 6
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Fig. 8. Boundary conditions: (a) Z-profile, (b) ZAE-profile and (c) details of the intermediate web stiffeners of ZAE-profile.
Fig. 11. Unbalanced stresses between adjacent strips.
Fig. 9. Non-uniform shear stress distribution in the numerical model.
Consequently, there was an increase of the purlins capacity in these tests because TFA improves the shear resistance due to the post-buckling strength reserve. Due to the short span length, predominantly shear tests had large influence of concentrated loads introduced into the web by bolts. For stiffened web specimens with straps, local shear buckling at the web was observed experimentally (Fig. 13(a)). For unstiffened web specimens, the failure mode was a combination of shear buckling and web crippling (Fig. 13(b)), which was also reported by Pham and Hancock [11]. The reason for the occurrence of web crippling is the lack of bolts at the middle of the web. In combined bending and shear tests with straps, the failure occurred by local buckling at the web for ZAE-section specimens, as shown in Fig. 14(a) and (b). For Z-section specimens, a combination of local buckling and web deformation at the loading point was observed in the tests, as depicted in Fig. 14(c) and (d). Bending only tests had experimental results very close for specimens with and without straps. Hence, the straps were not able of properly avoid the distortion and the failure occurred due to distortional buckling in all tests, as shown in Fig. 15.
Fig. 10. Uniform and unitary shear stress distribution in the web.
stiffened web specimens. The highest experimental force values in tests with plain web Zsection specimens also can be justified by Tension Field Action (TFA). Besides the analyzed Z cross-section has high web slenderness ratio, the connection configuration with bolts near the flanges worked such as transverse stiffeners, which allowed the development of TFA.
4.2. Numerical results 4.2.1. Buckling due to bending moment Elastic buckling analyses of ZAE and Z-sections subjected to bending moment were carried out in CUFSM [4]. Fig. 16 shows an example of 7
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ZAE-section with 245 mm height and 1.25 mm thickness (ZAE 245�1.25) and its equivalent plain web Z-section (Z 245�1.25). It is important to notice that the elastic critical stresses shown in the graphics are the maximum stress values in the cross-section. As can be seen in Fig. 16, the longitudinal web stiffeners significantly increase the local buckling strength. Due to the longitudinal stiffeners, the critical stresses at local buckling rise from 150.94 MPa in Z-section to 392.19 MPa in ZAE-section for compression in the top flange and from 136.54 MPa in Z-section to 553,87 MPa in ZAE-section for compression in the bottom flange, which represents increases of over 150% and 300%, respectively. Nevertheless, the stiffeners have no considerable effect in the distortional and lateral-torsional buckling. For instance, the critical stresses at distortional buckling in the case of compression in the top flange are 239.59 MPa for Z-section and 250.85 MPa for ZAEsection, and in the case of compression in the bottom flange, the cor responding values are 246.24 MPa and 259.56 MPa. This represents a difference between the critical distortional buckling stresses of only about 5%. For both loading cases analyzed in Fig. 16, the elastic critical stresses for the stiffened web ZAE-section are higher at local buckling than at distortional buckling, which does not happen for the plain web Z-sec tion. The highest values of elastic critical stresses at local buckling are a great advantage of stiffened web purlins, because the combination of bending and shear is usually computed with only the flexural strength at local buckling. Thus, using the same cross-section height and practically the same amount of material (reducing the flanges and lips widths), it is possible to increase the combined bending and shear strength by means of longitudinal web stiffeners.
Table 4 Experimental results. FT ðkNÞ 4
Specimen
aðmmÞ
FT ðkNÞ
ZAE ZAE ZAE ZAE ZAE ZAE ZAE ZAE
245-1.S 245-2.S 245-3.S 245-1.N 245-2.N 245-3.N 245-M.S 245-M.N
233 466 699 233 466 699 1200 1200
95.76 76.70 61.88 67.74 51.27 38.26 33.03 34.57
23.94 19.18 15.47 16.94 12.82 9.56 8.26 8.64
5.58 8.94 10.81 3.95 5.97 6.69 9.91 10.37
ZAE ZAE ZAE ZAE ZAE ZAE ZAE ZAE
340-1.S 340-2.S 340-3.S 340-1.N 340-2.N 340-3.N 340-M.S 340-M.N
327 654 981 327 654 981 1650 1650
185.70 143.72 120.02 150.74 79.20 54.01 63.85 61.56
46.43 35.93 30.00 37.69 19.80 13.50 15.96 15.39
15.18 23.50 29.43 12.32 12.95 13.25 26.34 25.39
Z 245-1.S Z 245-2.S Z 245-3.S Z 245-1.N Z 245-2.N Z 245-3.N
233 466 699 233 466 699
125.83 86.79 56.70 99.64 54.27 42.30
31.46 21.70 14.17 24.91 13.57 10.58
7.33 10.11 9.91 5.80 6.32 7.39
Z 245-1.S(a)a Z 245-1.N(a)a
233 233
98.47 64.09
24.62 16.02
5.74 3.73
VT ¼
MT ¼
FT a ðkN:mÞ 4
a Z 245-1.S(a) and Z 245-1.N(a) are additional tests performed in Z specimens with the same connection setup of ZAE specimens.
4.2.2. Shear buckling Shear buckling analysis using ANSYS [12] determined the elastic critical stresses and buckling modes on pure shear loading. Fig. 17 de picts the elastic critical stresses according to the length for a stiffened web section (ZAE 245�1.25) and its equivalent unstiffened section (Z 245�1.25). The comparison between the results shows that the longi tudinal web stiffeners enhance shear strength by increasing the magni tude of elastic shear buckling stresses in the length range from about 100 mm to 2500 mm. For example, at 200 mm length the critical shear stresses are 53.53 MPa for Z-section and 99.37 MPa for ZAE-section, which represents an increase of about 85% due to the longitudinal web stiffeners. Conversely, below 100 mm and above 2500 mm the buckling stresses are almost the same for Z and ZAE-sections, and therefore the longitudinal stiffeners have no significant effect in the critical shear stresses. Fig. 17 also shows the buckling modes at some lengths for members under pure shear stresses. In plain web section, the local buckling in the web occurs for lengths up to 2500 mm. In stiffened web section, the local buckling mode appears for lengths up to about 500 mm; from 500 mm to 1500 mm, a combination of local and distortional buckling is observed; and, as the length increases, the distortional buckling becomes more relevant. For both analyzed cross-sections, a flexural-torsional mode can
Fig. 12. Rotation of the top flanges at the supports in a combined bending and shear test without straps (ZAE 340-2.N).
results for equivalent cross-sections, considering compression in the top flange (Fig. 16(a)) and compression in the bottom flange (Fig. 16(b)). These two loading cases were evaluated because in continuous and sleeved multi-span systems the purlins can be used in both positions. The analyzed cross-sections are the same of the tests: stiffened web
Fig. 13. Failure modes in predominantly shear tests with straps: (a) ZAE 245-1.S and (b) Z 245-1.S. 8
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Fig. 14. Failure modes in combined bending and shear tests with straps: (a) ZAE 245-2.S, (b) ZAE 245-3.S, (c) Z 245-2.S and (d) Z 245-3.S.
Fig. 15. Distortional buckling in bending only tests with and without straps: (a) ZAE 245-M.S and (b) ZAE 245-M.N.
Fig. 16. Elastic buckling analyses of a stiffened web cross-section (ZAE 245�1.25) and its equivalent plain web cross-section (Z 245�1.25) considering a moment about the axis perpendicular to the web: (a) compression in the top flange and (b) compression in the bottom flange.
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Fig. 17. Elastic shear buckling analysis for a stiffened web section (ZAE 245�1.25) and its equivalent plain web section (Z 245�1.25).
be identified at long lengths such as 5000 mm. Hancock and Pham [19] also reported the occurrence of this buckling mode in elastic buckling analysis of CFS channels. However, as the authors stated, the flexural-torsional buckling due to shear is unlikely because pure shear loading could not occur at long lengths. Table 5 presents a comparison between the critical shear stresses from ANSYS and from Theory of Elastic Stability. The stiffened web section (ZAE 245�1.25) and its equivalent unstiffened web section (Z 245�1.25) were analyzed in ANSYS using the numerical model described previously and the cross-section dimensions from the experi mental program. The length was considered as the shear spans, defined
by the distance between bolts rows at the supports and loading points, according to the ratio a/h. The critical stresses obtained from eigenvalue buckling analyses in ANSYS for stiffened web members (τcr_ANSYS_ZAE) and plain web members (τcr_ANSYS_Z) were compared to each other and to the values resulting from Theory of Elastic Stability for the plain web members (τcr_Z). In the classic solution by Theory of Elastic Stability, the critical shear stress is evaluated considering the web as an isolated flat plate, simply supported at the four edges and under pure shear stresses. The elastic critical stress and shear buckling coefficient are given by Timoshenko and Gere [20] as presented in Eq. (3) and Eq. (4).
τcr ¼ kv Table 5 Comparison between elastic critical shear stresses from ANSYS and from Theory of Elastic Stability for stiffened web section (ZAE 245�1.25) and unstiffened web section (Z 245�1.25). a= h
τcr
1 2 3
91.95 68.35 55.60
ANSYS ZAE ðMPaÞ
τcr
ANSYS Z ðMPaÞ
46.12 33.83 31.66
12ð1
kv ¼ 5:34 þ
π2 E ν2 Þðh=tÞ2
(3)
4:00
(4)
ða=hÞ2
where τcr ¼ critical shear stress, kv ¼ shear buckling coefficient, E ¼ Young’s modulus, ν ¼ Poison’s ratio, h ¼ height of flat portion of the web, t ¼ thickness and a ¼ length. According to Table 5, the critical shear stresses for the analyzed stiffened web members can be up to twice the critical stresses for its
τcr Z ðMPaÞ 44.78 30.40 27.73
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equivalent plain web members. Therefore, longitudinal stiffeners really increase the buckling strength in a significant way. Moreover, for unstiffened web members, the critical stresses from ANSYS are very close to the critical stresses from Theory of Elastic Stability. The maximum difference between these values is about 14%. These results suggest that the influence of flanges and lips in shear buckling is small and negligible. The effect of the shear flow varying around the crosssection also can be ignored. Hence, the simplifications of the classic solution by Theory of Elastic Stability are acceptable for plain web members.
- Local buckling strength: (7)
� Mnl ¼ 1
(8)
Combined bending and shear is very important in the design of stiffened web sections due to the effect of the bending moment in the shear strength. Whereas longitudinal web stiffeners significantly in crease the shear strength, the bending moment capacity is not increased at the same level [1]. Thus, the interaction between bending moment and shear force must be considered. The combination of bending and shear was evaluated with experimental and numerical data, using cir cular and trilinear interaction curves, as depicted by Eq. (5) and Eq. (6), respectively. � �2 � �2 MT VT þ ¼1 (5) Mn Vn
where MT ¼ bending moment capacity, VT ¼ shear capacity, Mn ¼ nominal flexural strength and Vn ¼ nominal shear strength. Table 6 presents the nominal flexural strength and nominal shear strength for the tested purlins, both evaluated according to Direct Strength Method (DSM) in AISI S100-16 [3]. DSM rules for flexure and for shear are shown in Eq. (7) to Eq. (15). The determination of nominal flexural strength by DSM used the critical elastic buckling moments calculated with the critical stresses from elastic buckling analysis in CUFSM. In the determination of nominal shear strength, the critical stresses from ANSYS were employed to obtain the elastic shear buckling force. The shear strength results for plain web Z-section specimens include TFA, experimentally observed in these tests.
Mnl ðkN:mÞ
Mnd ðkN:mÞ
Vn ðkNÞ
Vn
245-1.S 245-2.S 245-3.S 245-1.N 245-2.N 245-3.N 245-M.S 245-M.N
12.88 12.88 12.88 13.46 13.46 13.46 13.46 13.46
9.62 9.62 9.62 10.08 10.08 10.08 10.08 10.08
25.87 19.23 15.64 26.13 19.37 16.03 14.90 14.90
– – – – – – – –
ZAE ZAE ZAE ZAE ZAE ZAE ZAE ZAE
340-1.S 340-2.S 340-3.S 340-1.N 340-2.N 340-3.N 340-M.S 340-M.N
35.40 35.40 35.40 35.77 35.77 35.77 35.77 35.77
24.05 24.05 24.05 26.09 26.09 26.09 26.09 26.09
43.55 32.38 28.23 42.91 31.11 27.29 26.17 26.17
– – – – – – – –
9.54 9.54 9.54 9.56 9.56 9.56
10.33 10.33 10.33 10.23 10.23 10.23
12.98 9.52 8.91 12.98 9.49 8.90
36.44 32.48 31.69 36.40 32.40 31.64
TFA
0:22
� �0:5 �� �0:5 Mcrd Mcrd My for λd > 0:673 My My
(9) (10)
Vn ¼ Vy for λv � 0:815
(11)
pffiffiffiffiffiffiffiffiffiffiffi Vn ¼ 0:815 Vcr Vy for 0:815 < λv � 1:227
(12)
Vn ¼ Vcr for λv > 1:227
(13)
- Shear strength with TFA: Vn ¼ Vy for λv � 0:776
(14)
� Vn ¼ 1
(15)
0:15
� �0:4 �� �0:4 Vcr Vcr Vy for λv > 0:776 Vy Vy
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi where λv ¼ Vy =Vcr , Vn ¼ nominal shear strength, Vy ¼ yield shear force of cross-section and Vcr ¼ elastic shear buckling force. The yield shear force of cross-section is given by Vy ¼ 0:6Aw fy , where Aw ¼ web’s area and fy ¼ yield stress. For all the analyzed sections, the values of Mnl and Mnd shown in Table 6 were obtained by DSM using elastic critical stresses from CUFSM, although it assumes the calculations for pure bending cases. This is applicable even for specimens with predominantly shear and combined bending and shear because the gradient of bending moment does not significantly affect the results for local and distortional buck ling modes. The effect of the bending moment gradient is especially important only for global buckling mode (lateral-torsional buckling), which was not evaluated in this research because the specimens had adequate lateral bracing. Fig. 18 shows bending and shear interaction in predominantly shear tests and in combined bending and shear tests with stiffened web ZAEsection purlins. For specimens with straps (Fig. 18(a)), the interaction was evaluated with the flexural strength at local buckling, because straps provided distortion restraint and forced the purlins to fail in local buckling. For specimens without straps (Fig. 18(b)), the flexural strength at distortional buckling was used in the interaction expressions, because numerical analyses showed that its corresponding critical stresses were lower than the critical stresses at local buckling. On the one hand, most of the results for tests with straps got above circular and trilinear interaction curves. On the other hand, most of the results for tests without straps got below both the interaction curves due to premature failure caused by rotation at the supports. Fig. 19 presents the interaction between bending moment and shear force in tests involving plain web Z-section purlins with and without
Table 6 DSM results for nominal flexural strength and nominal shear strength. ZAE ZAE ZAE ZAE ZAE ZAE ZAE ZAE
�0:4 �� �0:4 Mcrl Mne for λl > 0:776 Mne
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where λl ¼ Mne =Mcrl , λd ¼ My =Mcrd , Mnl ¼ nominal flexural strength at local buckling, Mnd ¼ nominal flexural strength at distortional buckling, Mne ¼ critical elastic lateral-torsional buckling moment, My ¼ member yield moment, Mcrl ¼ critical elastic local buckling moment and Mcrd ¼ critical elastic distortional buckling moment. - Shear strength without TFA:
(6)
Specimen
Mcrl Mne
Mnd ¼ My for λd � 0:673 � Mnd ¼ 1
� � � � MT VT 0:6 þ ¼ 1:3 Mn Vn
� 0:15
- Distortional buckling strength:
4.3. Combined bending and shear
Z 245-1.S Z 245-2.S Z 245-3.S Z 245-1.N Z 245-2.N Z 245-3.N
Mnl ¼ Mne for λl � 0:776
ðkNÞ
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Fig. 18. Bending and shear interaction for stiffened web purlins in predominantly shear and combined bending and shear tests: (a) specimens with straps and (b) specimens without straps.
Fig. 19. Bending and shear interaction for plain web purlins in predominantly shear and combined bending and shear tests: (a) shear strength by DSM without TFA and (b) shear strength by DSM with TFA.
straps. The interaction expressions considered the flexural strength at local buckling because all tested specimens failed in this way. Moreover, shear strength used in the interaction expressions were computed by DSM without TFA (Fig. 19(a)) and with TFA (Fig. 19(b)). Fig. 19(a) shows that using DSM without TFA all of the test points lie much above the interaction curves. This conservative trend is because the shear strength evaluated without TFA is underestimated, since the experi mental tests have indicated an increase of the ultimate capacity of the purlins due to TFA. Fig. 19(b) shows that using DSM with TFA all of the results for specimens without straps lie below circular interaction curve and most of the results for specimens with straps lie above trilinear interaction curve. The only exception is the result of the additional test Z 245-1.S(a), which got below both interaction curves although this specimen had straps. The explanation is that the connections with bolts far from flanges did not provide suitable lateral restraint and hindered
the complete development of TFA in this case. The same reasoning can be applied to the specimens without straps, in which the lack of straps and the consequent lack of lateral restraint prevented the full mobili zation of TFA. In bending only tests, the experimental results showed that failure occurred by distortional buckling. For that reason, the interaction ex pressions were evaluated considering the flexural strength at distor tional buckling. As can be seen in Fig. 20, the results overcame the circular interaction curve and reached out the trilinear interaction curve. The outcomes also suggest there was no significant difference between tests with and without straps. It is important to emphasize that in the practice of cold-formed steel structures the purlins typically have greater span-to-depth ratios. Consequently, the influence of the bending moment is much more pro nounced than the influence of the shear force, and the effect of TFA is 12
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shear with and without TFA is presented in Fig. 22. Only the results for predominantly shear tests with straps and for combined bending and shear tests with straps were considered due to the premature failure observed in tests without straps. The shear yield load (Vy) was used to pffiffiffiffiffiffiffiffiffiffiffiffiffiffi � calculate the shear slenderness λv ¼ Vy =Vcr and the nondimesional � shear capacity VT =Vy . Fig. 22 shows that all of the experimental data for stiffened web specimens are very close to the DSM curve for shear without TFA. Therefore, DSM provides accurate predictions for the shear strength of these cross-sections and the model generated in ANSYS gives adequate values of elastic critical shear stresses. However, for plain web specimens, the experimental results lie above DSM curve without TFA and approximate to the DSM curve with TFA, but lie below it. The reason for this fact is that a full TFA were not developed in the tests, probably because of the connection configuration with only two bolts. The lack of bolts over the full depth of the web at the supports and loading points did not allow an adequate level of transverse restraint, affecting the complete development of TFA in tests with plain web specimens. The influence of the connection setup in the development of TFA was corroborated by Z 245-1.S(a) specimen, where the bolts were put closer together as for ZAE-sections and the consequent reduction of lateral restraint caused the test point to drop, coming close to the DSM curve without TFA.
Fig. 20. Bending and shear interaction in bending only tests.
minimized. For these reasons, the design of longitudinally stiffened web purlins can be done with the flexural strength at distortional buckling and the best interaction diagram to be utilized is the trilinear.
5. Conclusions This paper has presented experimental and numerical analyses per formed in CFS purlins with plain web Z-section and stiffened web ZAEsection to better understand the effect of longitudinal web stiffeners in the structural behavior. Twenty-two specimens were tested in predom inantly shear, combined bending and shear, and bending only. The re sults from experimental analysis showed that in predominantly shear tests and in combined bending and shear tests the failure occurred with local shear buckling in the web for specimens with straps. For specimens without straps, the lack of lateral restraint due to the connection configuration caused a premature failure characterized by rotation of the top flanges at the supports. In bending only tests, the ultimate ca pacities for purlins with and without straps were very close, because the straps did not prevent the distortion and the failure occurred by distortional buckling in all tests. Moreover, the experimental program also showed that, against all expectations, the specimens with stiffened web cross-section presented experimental force values lower than the specimens with plain web cross-section because of the development of TFA in these tests and not in those. Numerical analyses carried out in CUFSM and ANSYS investigated the buckling behavior of the cross-sections. The numerical results indi cated that the web stiffeners enhance the buckling strength in both loading cases of pure bending and pure shear. In buckling due to bending moment, the longitudinal stiffeners increase the critical stresses at local buckling but have no considerable effect in distortional and lateral-torsional buckling modes. In shear buckling, the stiffeners also lead to an increase of the elastic critical stresses, and buckling modes are mostly local for unstiffened web members and a combination of local and distortional for stiffened web members. The strength of the tested specimens was determined by DSM using the elastic critical stresses from numerical analyses. Experimental and numerical data were used to evaluate the interaction between bending moment and shear force. These results were plotted against the inter action diagrams and showed that the circular interaction diagram was conservative in most cases and the experimental data were closer to the trilinear interaction diagram. The experimental data were also compared to DSM predictions and demonstrated good agreement. The only exception was the case of the specimens with plain web crosssection, in which the lack of transverse restraint due to screwed con nections with few bolts hindered the full development of TFA, so the experimental results lied above DSM curve without TFA but below DSM
4.4. Comparison between experimental results and DSM predictions The appropriateness of DSM in the design of the stiffened and unstiffened cross-sections was analyzed by plotting the experimental results against the DSM curves. Fig. 21 presents the comparison between the bending moment capacity in bending only tests and the DSM curves of local and distortional buckling. For the tested specimens, the yield moment (My) was used to compute the distortional slenderness ðλd ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi My =Mcrd Þ and the nondimensional bending capacity ðMT =My Þ. The use pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi � of distortional slenderness instead of local slenderness λl ¼ My =Mcrl with the experimental data is because the laboratory tests and numerical analyses showed that these specimens failed at distortional buckling. As can be seen in Fig. 21, the experimental results for bending only tests practically match the DSM curve of distortional buckling, indicating that DSM predictions were quite accurate. The comparison between the shear capacity and the DSM curves for
Fig. 21. Comparison between bending moment capacity and DSM curves of local and distortional buckling. 13
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Fig. 22. Comparison between shear capacity and DSM curves with and without TFA.
curve with TFA. This research remains ongoing with prediction of new experimental tests and parametric numerical analyses with full and detailed FEM models including post-buckling behavior, so that the re sults in Figs. 18–22 could be more robust.
[6] R.A. LaBoube, W.W. Yu, Cold-formed steel web elements under combined bending and shear, in: Proceedings of the 4th International Specialty Conference on ColdFormed Steel Structures, 1978, pp. 219–251. [7] C.H. Pham, G.J. Hancock, Experimental investigation of high-strength cold-formed C-sections in combined bending and shear, J Struct Eng Am Soc Civil Eng 136 (7) (2010) 866–878. [8] C.H. Pham, L.A. Bruneau, G.J. Hancock, Experimental study of longitudinally stiffened web channels subjected to combined bending and shear, J Struct Eng Am Soc Civil Eng 141 (11) (2015), 04015018. [9] C.H. Pham, G.J. Hancock, Experimental investigation and direct strength design of high-strength complex C-sections in pure bending, J Struct Eng Am Soc Civil Eng 139 (11) (2013) 1842–1852. [10] C.H. Pham, G.J. Hancock, Tension field action for cold-formed sections in shear, J. Constr. Steel Res. 72 (2012) 168–178. [11] C.H. Pham, D. Zelenkin, G.J. Hancock, Effect of flange restraints on shear Tension Field Action in cold-formed C-sections, J. Constr. Steel Res. 129 (2017) 42–53. [12] Swanson Analysis System Inc, ANSYS Help, 2016, version 17.0. [13] J.M.M. Silva, Cold-formed Steel Purlins with Stiffened Web: Emphasis on Shear and Combined Bending and Shear (In Portuguese), Thesis (M. Sc. in Structural Engineering), S~ ao Carlos School of Engineering, University of S~ ao Paulo. S~ ao Carlos, S~ ao Paulo, Brazil, 2018, pp. 1–144. [14] C. Yu, B.W. Schafer, Local buckling tests on cold-formed steel beams, J. Struct. Eng. 129 (12) (2003) 1596–1606. [15] C. Yu, B.W. Schafer, Distortional buckling tests on cold-formed steel beams, J. Struct. Eng. 132 (4) (2006) 515–528. [16] American Society for Testing and Material (ASTM), Standard test method and definitions for mechanical testing of steel products, ASTM 370 (2017). [17] S.H. Pham, C.H. Pham, G.J. Hancock, Numerical Simulation of Cold-Formed Channel Sections with Intermediate Web Stiffeners Undergoing Pure Shear, Research Report R930, School of Civil Engineering, University of Sydney, 2012. [18] S.H. Pham, C.H. Pham, G.J. Hancock, Direct strength method of design for shear including sections with longitudinal web stiffeners, Thin-Walled Struct. 81 (2014) 19–28. [19] G.J. Hancock, C.H. Pham, A Signature Curve for Cold-Formed Channel Sections in Pure Shear, Research Report R919, School of Civil Engineering, University of Sydney, 2011. [20] S.P. Timoshenko, J.M. Gere, Theory of Elastic Stability, McGraw-Hill Book Co. Inc., New York, NY, 1961.
CRediT authorship contribution statement Juliana Maria Mazzeti Silva: Conceptualization, Methodology, Validation, Investigation, Data curation, Writing - original draft, Visu alization. Maximiliano Malite: Conceptualization, Methodology, Writing - review & editing, Supervision. Acknowledgements ~o de Aperfeiçoa This study was financed in part by the Coordenaça mento de Pessoal de Nível Superior – Brasil (CAPES) - Finance Code 001. The authors would like to thank Modular Building Systems for donating the specimens for the experimental analysis. References [1] C.H. Pham, G.J. Hancock, Numerical investigation of longitudinally stiffened web channels predominantly in shear, Thin-Walled Struct. 86 (2015) 47–55. [2] Modular Building Systems (in Portuguese). http://www.modularsc.com.br/, 2012. (Accessed 2 September 2019). [3] American Iron and Steel Institute (AISI), North American Specification for the Design of Cold-Formed Structural Members, AISI S100-16, Washington, DC, 2016. � any, Buckling analysis of cold-formed steel members using [4] B.W. Schafer, S. Ad� CUFSM: conventional and constrained finite strip methods, in: Proceedings of the 18th International Specialty Conference on Cold-Formed Steel Structures, 2006, pp. 39–54. [5] R. Bebiano, P. Pina, N. Silvestre, D. Camotim, GBTUL – a GBT-based code for thinwalled member analysis, in: Proceedings of the 5th International Conference on Thin-Walled Structures, 2008, pp. 1173–1180.
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