Duerr, M.

Saal, H.

English

Missouri University of Science and Technology (Missouri S&T): Scholars' Mine

Missouri University of Science and Technology Scholars' Mine International Specialty Conference on Cold-Formed Steel Structures (2006) - 18th International Specialty Conference on Cold-Formed Steel Structures Oct 26th, 12:00 AM Investigation of the Shear Stiffness of Profiled Steel Sheeting Diaphragms M. Duerr H. Saal Follow this and additional works at: https://scholarsmine.mst.edu/isccss  Part of the Structural Engineering Commons Recommended Citation Duerr, M. and Saal, H., "Investigation of the Shear Stiffness of Profiled Steel Sheeting Diaphragms" (2006). International Specialty Conference on Cold-Formed Steel Structures. 4. https://scholarsmine.mst.edu/isccss/18iccfss/18iccfss-session7/4 This Article - Conference proceedings is brought to you for free and open access by Scholars' Mine. It has been accepted for inclusion in International Specialty Conference on Cold-Formed Steel Structures by an authorized administrator of Scholars' Mine. This work is protected by U. S. Copyright Law. Unauthorized use including reproduction for redistribution requires the permission of the copyright holder. For more information, please contact scholarsmine@mst.edu. 549 Eighteenth International Specialty Conference on Cold-Formed Steel Structures Orlando, Florida, U.S.A, October 26 & 27, 2006   Investigation of the shear stiffness of profiled steel sheeting diaphragms with only two edges fastened  Duerr, M.1 and Saal, H.2 Abstract The shear stiffness of profiled steel sheeting diaphragms can be used to stabilize members and structures as well. The available design rules ac-counting for this effect suppose that all four edges of the shear diaphragm are fastened. As the shear connectors, which are used to fasten the sheeting to the parallel members, cause additional work, costs and effort, the dia-phragm is - according to its uniaxial load bearing behaviour for transverse loading – often fastened only at the two edges, which are normal to the span of the profiles.  The effect of free edges parallel to the span is investigated numerically and verified experimentally for the diaphragm under shear loading. This nonlin-ear investigation includes the flexibility of the fasteners which connect the sheeting to the supporting structure. The result of the investigations is a formula for calculating the shear stiffness of profiled steel sheeting dia-phragms with only two edges fastened.   1. Introduction The ultimate load of slender beams can be increased by using the diaphragm effect of covering trapezoidal sheeting attached to the compression flange. The profile is fixed in the lateral direction at the location where the sheeting  is attached when the shear stiffness S of the sheeting meets the following requirement according to [1]:  22222 7025.0hhlEIGIlEIS zT ⋅⎟⎟⎠⎞⎜⎜⎝⎛ ++≥ ππω  (1)                                                           1 Scientific Assistant, Versuchsanstalt für Stahl, Holz und Steine, Universität Karlsruhe (TH), D-76128 Karlsruhe, Germany 2 Full Professor, idem. 550 A restriction is given by [2], which allows the application of (1) only if all four edges of the diaphragm are fastened. Actually often only the two edges, which are normal to the span of the profiles, are fastened. This is because fastening along the longitudinal edges requires high constructive effort or increased use of material, e.g. by using special shear connectors. The effect of free longitudinal edges was investigated numerically and experimentally such that the shear stiffness and strength of trapezoidal sheeting with this type of fastening can also be taken into account for increasing the lateral torsional buckling load.  2. Testing of diaphragms   The experiments with the trapezoidal sheeting diaphragms were performed with the shear frame shown in figure 1.    Figure 1: shear frame in horizontal projection  The displacements of the frame were measured with two displacement transducers v1 and v2 parallel to the direction of the applied load and at the two supports of the frame with two transducers v3 and v4 normal to this direction. The displacement due to shear is  )(2 4321 vvbavvvs −−+=  (2)  With the shear displacement vs the shear stiffness S is obtained from   svbaFTS Δ⋅⋅Δ=ΔΔ= γ  (3) with  551  bFT Δ=Δ   increment of shear flow T (4) avsΔ=Δγ  increment of diaphragm angle  (5) The experiments were performed with profiles LS 5/35/1035 with tN=1.0mm and steel grade S320GD+Z as shown in figure 2.   Figure 2: nominal dimensions of trapezoidal sheet LS 5/35/1035  The span of the profiles was parallel to the short span of the frame. They were connected in every trough to the frame with self-drilling screws with tapping screw thread type EJOT JT2-12-5.5x35 V16. This type of screws was  also used at the longitudinal edge with spacing of eR=150mm if the four edges of the diaphragm were fastened. Self-drilling screws with tapping screw thread type EJOT JT2-4.8x19 were used as seem fasteners with spac-ing eL=150mm.  Some of the fasteners were tested separately in small scale tests according to [3] to determine their flexibility when subjected to a shear force.  Table 1 provides additional description and results of the diaphragm tests which were performed in the laboratories of the Versuchsanstalt für Stahl, Holz und Steine.   diaphragm test 1 diaphragm test 2 type of fastening four sides fastened two sides fastened dimensions a – length of diaphragm b – width of diaphragm  3000mm 6831mm  3000mm 6624mm characteristic loads  85kN: plastic deformations 100kN: obvious profile distortion    166kN: failure of fastener  55kN: plastic deformations 70kN: obvious profile dis-tortion 105kN:buckling of rib at longitudinal edge 121kN: failure of fastener  shear stiffness S 7350kN/m 6185kN/m Table 1: description and results of diaphragm tests  3. Numerical Analysis  Numerical analyses were performed with the Finite Element program AN-SYS to investigate the shear stiffness of diaphragms with only two edges 552 fastened. The FE-model of the trapezoidal sheeting under shear loading is shown in figure 3.    Figure 3: FE-model of trapezoidal sheeting and supporting structure  The fasteners were modeled by nonlinear springs with an implemented and approximated load-deformation-behavior according to figure 4. This graph resulted from the small scale tests according to [3].   Figure 4: approximated load-deformation-graph for fasteners  For verification of the FE-model profiles with different cross-sections and different thickness were numerically analyzed for four-sided fastening. The results of this analysis were compared to [4] which is based on [5]. As [5] is based on the theory of elasticity for this comparison neither for the spring elements nor for the supporting structure any flexibilities were allowed. 553 Table 2 compares the shear stiffness obtained from the numerical analysis with the stiffness according to [4].  shear stiffness S [kN/m] profile width of diaphragm b [m] thickness[mm] FEM [4] difference [%] 35/207 3.0 0.75 1.00 1.25 4215 8621 15540 4366 8831 14926 4 2 4 105/345 6.0 0.75 1.00 1.25 807 1727 3039 794 1669 2941 2 3 3 135/310 6.0 0.75 1.00 1.25 1053 2198 3829 1103 2306 4036 5 5 5 Table 2: comparison of FE-calculated shear stiffness S with [4]  The small differences shown in table 2 for the results obtained by the two methods demonstrate that the FE-model is suitable for numerical analyses of shear diaphragms.   In order to verify the FE-model with the performed tests it was extended to include the nonlinear load-deformation-relation of the fasteners shown in figure 4, the  mechanical properties of the trapezoidal sheeting and the stiff-ness of the supporting structure. The comparison of experimental and nu-merical shear displacement vs for a diaphragm with fastening along four edges is shown in figure 5.   Figure 5: Load-deformation-relations obtained from the test and from the FE-analysis for the diaphragm with fastening along four edges  554 The experimental shear stiffness evaluated for Tcrit=3.7kN/m according to [4] deviates from the FEM-based value at the same load level  by 8%. The difference of the experimental and the FEM-based value of the ultimate load is 11%. Both the experimental and FEM-based ultimate load was limited by the maximum forces of the fasteners located in the corners of the dia-phragm.   The test with only two edges of the diaphragm fastened was investigated with the same FE-model with the only difference that the fastening at the longitudinal edges was removed. The comparison of experimental and nu-merical shear displacement vs for a diaphragm with fastening along two edges is shown in figure 6.   Figure 6: Load-deformation-relation resulting from experimental testing and FEM for fastening along two edges  In this case the experimental shear stiffness evaluated for Tcrit=3.7kN/m according to [4] deviates from the FEM-based value at the same load level  by 13%. The difference of the experimental and the FEM-based value of the ultimate load is 1%. Both the experimental and FEM-based ultimate load was limited by the maximum forces of the fasteners located in the corners of the diaphragm. Figure 5 and figure 6 show the overall close agreement of numerical and experimental results for the load-displacement-curve.   4. Parametric study with the verified Finite-Element-model   With the verified FE-model a parametric study was performed for six differ-ent types of trapezoidal sheeting with profile height ranging from 30mm to 153mm. The study included the profile LS 5/35/1035 with its effective ge-555 ometry and material properties. The characteristic values of the other inves-tigated profiles types are listed in table 3.       type of profile 30/221 40/183 59/225 105/345 153/280 thickness tK [mm] 0.96 0.96 0.96 0.96 0.96 profile height [mm] 30 40 59 105 153 width top flange [mm] 120 119 140 210 119 width bottom flange [mm] 40 40 35 40 40 pitch of corrugations [mm] 221 183 225 345 280       length of diaphragm [mm] 3000 3000 3000 5000 5000 Tcrit according to [4] [kN/m] 4.1 3.7 3.0 1.8 2.5 S according to [4] [kN/m] 11389 5640 2940 1378 1898 Table 3: characteristic values of investigated profile types   Diaphragms with different a/b-ratios were numerically investigated with four sides fastened and with two sides fastened. This investigation reflected the nonlinear behavior of the fasteners as shown in figure 4 and bi-linear material properties with E=210000N/mm² and fy=320N/mm²  for the trape-zoidal profiles. Figure 7 shows the shear stiffness S for diaphragms fastened along four edges with a/b-ratios ranging from 0.25 to 1.75.    Figure 7: variation of shear stiffness with different a/b-ratios for dia-phragms with fastening along four edges  Figure 7 shows that the shear stiffness decreases with increasing a/b-ratio. This influence, which amounts for the profile 30/221 to 12%, is due to the flexibility of the fasteners and the interaction of longitudinal and transverse 556 forces in the corners of the diaphragm. The numerical analysis shows that with rigid fasteners the shear stiffness is independent of the a/b- ratio.  Removing the fastening along the longitudinal edges increased the influence of the ratio a/b on  the shear stiffness S significantly as figure 8 shows.    Figure 8: variation of shear stiffness with different a/b-ratios for dia-phragms with fastening along two edges  The obvious loss in shear stiffness with increasing a/b-ratio for diaphragms with fastening along two edges is explained by figure 9, which shows the distribution of the forces of the fasteners along the edge.   Figure 9: forces of the fasteners along edge for a/b=0.5   It is obvious that the fasteners close to the corners of the diaphragm are mainly exposed to forces Fx resulting from the free longitudinal edge. As the number of fasteners decreases with smaller width b of the diaphragm the forces Fx per fastener will increase significantly and cause larger deforma-557 tions. This is proven by figure 10 which shows the changes connected with an increase of a/b form 0.5 to 1.0. Figure 10 also shows, that with this in-crease the force of the fastener increases more for profiles with small profile height than for higher ones.    Figure 10: increase of maximum forces of edge fasteners and displacements with a/b increasing from 0.5 to 1.0  Figure 9 and 10 demonstrate the non-uniform distribution of the shear forces for diaphragms with only two edges fastened. This deviates from the uni-form distribution supposed in [5] for a diaphragm fastened along four edges. The shear forces are concentrated in the corners of the diaphragm with fas-tening only along two edges.   5. Evaluation of numerical results and approximated calculation  Figure 11 shows the ratio of the two-sided shear stiffness (see figure 8) to four-sided shear stiffness (see figure 7) for the investigated profile types subject to the a/b- ratio of the diaphragm.  558  Figure 11: ratio of shear stiffness with fastening along two edges to that with fastening along four edges for different profiles and a/b-ratios  For a diaphragm with free edges parallel to the span and with b→∞ it is obvious that the shear stiffness S is the same as for a diaphragm fastened along four edges. This is explained by the decreasing influence of the free longitudinal edge with increasing width of the diaphragm. The loss of stiff-ness is mainly controlled by the shear stiffness of the diaphragm with fasten-ing along four edges.  The graphs of figure 11 can be approximated by using straight lines with  ba⋅−= γβ 1  (6) Figure 12 shows the factor γ, which is easily taken from figure 11 depending on the shear stiffness of the diaphragm with fastening along four edges.   Figure 12: variation of factor γ with shear stiffness of diaphragm with fas-tening along four edges  559 Thus the shear stiffness S2-sided of diaphragms with two sides fastened can be calculated depending on the shear stiffness S4-sided by means of factor γ ac-cording to figure 12 or approximated by  ⎟⎠⎞⎜⎝⎛⋅+= −900033.017.0 4 sidedSγ  (7) to  sidedsided SbaS 42 )1( ⋅⋅−=− γ  (8) Equation (8) applies for trapezoidal sheeting according to [4] with profile heights ranging from 35mm to 175mm and a/b-ratios of the diaphragm less than 1.75 if fasteners are at least as stiff as described by figure 4.      6. Conclusions The shear stiffness of diaphragms fastened only along two edges can easily be derived from that of diaphragms fastened along all four edges by the application of a knock-down-factor. This factor depends on the a/b-ratio and the shear stiffness of the diaphragm with all four edges fastened.  This results from numerical investigations which were verified both by comparison with [3] and with experimental results.  The equation which is given to calculate the shear stiffness for diaphragms with only two edges fastened applies for trapezoidal sheeting according to [4] with profile heights ranging from 35mm to 175mm and a/b-ratios of the diaphragm less than 1.75 if the fasteners are at least as stiff as described by figure 4. The knock-down-factor applies for the stiffness under the design load according to [4]. The load-displacement behavior of the diaphragms which was numerically found and experimentally confirmed is linear up to this load-level for both types of fastenings. The ultimate loads of the dia-phragms are much higher than these design loads.   7. References [1] DIN 18800 Teil 2, Stahlbauten – Stabilitätsfälle, Knicken von Stäben und Stabwerken. Berlin, November 1990  [2] Beuth-Kommentare Stahlbauten, Kommentar zu DIN 18800-1, bis -4, Ausgabe 1998-05 [3] Entwurf zu DIN 18807 Teil 4,: Trapezprofile im Hochbau – Verbindungen zwischen Stahltrapezprofilen und Bauteilen aus Stahl – Ermittlung der Tragfähigkeit durch Versuche. Berlin, Dezember 1999 560 [4] DIN 18807 Teil 1, Trapezprofile im Hochbau – Stahltrapezprofile – Allgemeine Anforderungen, Ermittlung der Tragfähigkeitswerte durch Berechnung. Berlin, Juni 1987  [5] Strehl, C.: Berechnung regelmäßig periodisch aufgebauter Faltwerksquerschnitte unter Schubbelastung am Beispiel des Trapezblechs, Dissertation, Darmstadt, 1976   

Investigation of the Shear Stiffness of Profiled Steel Sheeting Diaphragms with Only Two Edges Fastened

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Investigation of the Shear Stiffness of Profiled Steel Sheeting Diaphragms

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