470 S.K. Clubley and R.Y. Xiao Table 1: Summary of typical tested specimens. Plate Thickness (mm) Failure Load (KN) Maximum Slip (mm) Failure Mode 12 900 1.95 Brittle 10 870 4.70 Ductile 8 850 3.40 Ductile 6 820 5.63 Ductile 5 770 6.00 Ductile 3. NON-LINEAR NUMERICAL MODELLING A comprehensive numerical programme was conducted on the tested specimens. The initial stages of numerical modelling have been concentrated on the definition of the geometric model, associated constraints and the constituent material properties. Subsequent selection of finite element types in the computer program will determine the mathematical model applied to achieve solution convergence. This in turn will govern the level of physical behavioural model accuracy. Both material and geometric non-linearity were considered. The data obtained from the laboratory is limited and does not detail behaviour at varying geometric specification. A wide range of parameters was selected to examine their influence on the shear strength of Bi-Steel plates. Numerical analysis has enabled all geometric and material properties to be varied. From this research, conclusions about Bi-Steel behaviour and corresponding theoretical modelling are suggested. Figure 4: Meshed geometric model prior to solution. Geometric non-linearity will occur if the model is allowed to experience large strain displacement. Following initial geometric construction the model requires a mesh of nodes and elements as shown in Figure 4. Symmetry has been used to provide efficient computing. Meshing parameters and subsequent specification can prove detrimental to the success of the solution. The level of mesh refinement will largely govern solution accuracy and the cost of associated computation time. Bi- Steel is a complicated three-dimensional problem and requires careful use of isoparametric elements to construct curved surfaces. Within the Ansys program Solid45 and Solid65 were chosen to represent steel and concrete entities respectively. Solid45 has eight nodes each with three degrees of Testing and Modellling of Bi-Steel Plate Subject to Push-Out Loading 471 freedom; design is primarily that of three-dimension consideration. Rate dependent and independent behaviour is permitted through load application with tolerance for stress stiffening, large deflections and large strain capability. Tetrahedral shape definition is to be avoided in favour of prismatic assignment. Solid65 is capable of cracking in three orthogonal directions in tension, crushing in compression, plastic deformation and creep. The element is defined by eight nodes each having three degrees of freedom. All models were computed on Silicon Graphics Octane machines and solved on a mainframe Silicon Graphics Power Challenge 2000 tandem CPU array. 3.1 Modelling without use of a contact element. Initial computing simulation without a gap element between steel and concrete materials involves the use of entity merge commands in areas considered to be key load paths. The merge will represent full interaction at steel and concrete interface. The increase in recorded data is representative of over stiffening. For this reason it was considered appropriate to invoke area merge commands between contact surfaces of the shear connector and concrete only. Entity interface at these two points would ensure load path transference between the concrete core and steel skin across the whole plate length. The standard non-contact modelling may be considered as the benchmark for subsequent gap element as shown in Figure 5. Loading is assumed uniform over area, eccentricity is zero and an arbitrary friction constant is defined. The material specification used was typical of the material used in the laboratory tests. Figure 5: Standard contact Bi-Steel model (concrete omitted for clarity). Initial data analysis indicated the presence of over stiffness and the subsequent tendency to produce failure loads generally in excess of what could be reasonably expected under laboratory conditions. Considering the models individually, the most accurate models are those with reduced plate thickness, which typically experience ductile deformation. Large plate sizes appear accurate during early to mid stages of load application. All of the aforementioned modelling errors are a result of the unrealistically high surface interaction invoked at the shear connector. The result summary graph, Figure 6, indicates a key relationship between plate and connector spacing which was also later noted with increased accuracy in the contact element model study. It would appear that the greatest governing factor defining failure strength is plate spacing. Connector pitch adjusts failure load marginally, producing a group of lines clustered and collecting in a similar fashion around a locus. A change in the mode of failure appears between plate spacing one hundred and two hundred millimetres, indicating the possibility of plastic hinge formation and the corresponding relocation at varying model geometry. 472 S.K. Clubley and R.Y. Xiao Figure 6: Plate spacing, connector spacing and plate thickness influence on shear strength. From the regression analysis in Figure 6 the following equations define failure load as a function of plate spacing. In Figure 6 the regression trendline is indicated in bold. 9 6mm steel plates 9 8mm steel plates F = 0.02D 2 - 10D + 390 (1) F = 0.009D 2 - 4.6D + 77 (2) Where: F = Failure load per shear connector (KN) D = Plate spacing (mm) 3.2 Modelling with a contact element. Following the benchmark analysis of the standard model the next stage of accuracy progression is the introduction of a gap element. Upon initial consideration of software documentation and element libraries it was decided that Contac52 would represent the interface between steel and concrete surfaces and provide a more realistic load path for transmission of normal and shear force. Contac52 allows numerous parameter definitions, most importantly values for normal and tangential stiffness, Kn & Ks. The gap element may be judged analogous to a spring assembly. Consequently, incorrect stiffness definition will produce either unrealistic elastic 'bounce' during load application or too large an inertia to movement caused by shear action. The introduction of tangential stiffness, Ks, is a key step regarding the modelling of energy dissipation during shear action, characteristic of a classical push-out test. The ability to successfully transfer energy and maintain the desired load path propagation is defined by modelling accuracy of physical chemical bonds and corresponding friction force generated. Upon examination of the relative slip versus load graph produced by numerical analysis shown in Figure 7, it has been noted that large geometry models slip considerable less in longitudinal direction UX than smaller specification models. Behaviour appears not subject to proportionality with respect to their global size. This performance is curious due to the expected deflection of large span beams according to classical structural Testing and Modellling of Bi-Steel Plate Subject to Push-Out Loading 473 mechanics. A possible explanation for this may be due to the fact that shear connectors can be divided into two categories, either rigid or flexible. The corresponding classification provides for alternate failure mechanisms. Rigid connectors tend to exhibit higher stress concentrations in the concrete surrounding them resulting in crushing. Flexible connectors are generally more consistent in failure behaviour. Therefore, smaller size equals increased rigidity, which implies in the case of Bi-Steel, extended crushing appears locally to shear connectors. Relative slip is possible through and past crushed concrete zone with applied force not transferred to steel plates fully due to reduced surface area. Consequently, relative slip of concrete versus steel increases. In contradiction, large span shear connector equals increased flexibility producing reduced concentrated local crushing. Traction force and corresponding friction force increases, which implies passage through crushed matrix reduces. Figure 7: Relative slip versus load comparison for varied spacing of 6mm thick plate. When plate spacing is increased to three hundred millimetres or greater, all regions of maximum stress intensity are generated at weld interface with considerable local problems of concrete crushing. In addition, the nearest load path is the adjacent steel plate resulting in large UZ wave displacement particularly evident at large connector spacing. The concrete core has now ceased to become an effective load path. Subsequently, smaller plate sizes are capable of displaying ductile deformation before ultimate load as indicated at the weld interface. Therefore, the curve reduction noted is smaller, but large plate thickness inhibit ductile behaviour due to increased local stiffness which in turn promotes earlier failure at the weld interface with regression of a plastic hinge into the shear connector. Connector spacing of one hundred millimetres or less is very unlikely to deform in a UZ wave shape even for small plate thickness at high loads. This factor allows the whole plate surface area to remain in contact with the concrete surface with retention of the chemical interface bond. Increased shear action is necessary to remove the resistance to shear force. 474 S.K. Clubley and R.Y. Xiao 4. MATHEMATICAL MODELLING OF PLATE DEFLECTED SHAPE A mathematical model is required to support the Bi-Steel design process. This must compare favourably with previous experimental data and in addition correlate with numerical modelling already conducted. The solution to the problem is sought with the application of the Laplace equation. 4.1 Formulation of deflected shape by the Laplace equation. Consider that plate displacement surface between shear connectors is governed by the Laplace Equation 3. This assumption comes from the plate measurements recorded in the laboratory tests. Testing and Modellling of Bi-Steel Plate Subject to Push-Out Loading Hence the UZ deformation shape of the plate is described by: ~ u(x, y) 4A I 1 1 =~ e -y sinx +-e -3y sin3x + ~r 3 (5) 475 4.2 Validation against the test data. The value of the interaction displacement is defined by the UZ displacement at the first shear connector, weld perimeter. Currently, this is obtained from numerical modelling analysis in the absence of concise 'real' world physical data. It was found that the equation was very accurate at the considered point of contraflexure but accuracy reduced moderately at peak/trough values. Typically accuracy error moved between 0.1% and 40% for displacement predictions at midpoint and peak/troughs respectively. Quantitatively, the discrepancy in each case is only several hundredths of a millimetre. Measurement this small would be extremely difficult to record consistently in the laboratory during specimen loading. Table 2 indicates the accuracy of Equation 5. Table 2: UZ deformation shape error at midpoint between shear connectors. Plate Size 6 8 10 12 14 16 (mm) Numerical 2.77 2.89 5.23 1.94 0.83 0.34 Modelling (ram) Equation (5) 2.59 2.99 5.00 1.89 0.82 0.36 Answer (ram) % Error 6.94 3.34 4.60 2.65 1.22 5.56 Difference It is shown that greatest error is achieved consistently on plate sizes that promote ductile wave displacements of the Bi-Steel plate between the shear connectors. However, co-ordinates of peak/trough displacement at one third, two thirds distance between the shear connectors display a consistent error difference of approximately 30% regardless of the steel plate thickness. Further evaluation indicates that higher load conditions produce increased displacement stability, while lower load predictions become more difficult to make accurately. Generally, the Laplace equation was consistently more accurate at the one-third point than at two-thirds distance between shear connector one and two. Error difference experienced between the two positions was typically in the region of 20%. 5. CONCLUSIONS The following conclusions can be drawn based on the testing, numerical modelling and mathematical analysis for Bi-Steel plates. 1. From the recent testing it has revealed that Bi-Steel rods and plates have significant shear strength. The shear strength is greatly affected by several important parameters. These include plate spacing, rod spacing and rod diameter. 2. From load-deformation relationships it can be seen that Bi-Steel plates have high ductility and deformation capacity. For very thick plates (> 14mm), the failure can be brittle if Bi-Steel rod numbers are small. The failure will be initiated by the shear failure of local welds. 476 S.K. Clubley and R.Y. Xiao 3. Graphical plots from numerical analysis show plate thickness and plate spacing govern stress distribution at local weld perimeter. Rod spacing will largely determine the out of plane UZ plate deformation shape. 4. Preliminary design formulae for shear strength of Bi-Steel plate have been proposed. These include the consideration of plate spacing and rod diameter. A general equation is being developed for design purposes. 5. The deformation shape of Bi-Steel plates has been established through the derivation of the Laplace equation. Validation against the test results has proved Equation (5) is accurate. This will be very useful for the serviceability design of Bi-Steel plate. Further testing, numerical simulation and design procedures are being conducted. These research results will be published in stages according to the plan. Acknowledgements: This research was jointly funded by British Steel Plc and a CASE studentship from the Engineering and Physical Science Research Council (EPSRC). References: 1. MOY S. S. J., XIAO R. Y. and LILLESTONE D. Tests for British Steel on the shear strength of the studs used in the Bi-Steel system. University of Southampton - Department of Civil & Environmental Engineering, 199.8, May. 2. OEHLERS D. J. and SVED G. Composite beams with limited slip capacity shear connectors. Journal of Structural Engineering, 1995, Volume 121, June, 932-938. 3. KALFAS C., PAVLIDIS P. and GALOUSSIS E. Inelastic behaviour of shear connection by a method based on FEM. Journal of Constructional Steel Research, 1997, Volume 44, No. 1-2, 107- 114. 4. UY B. and BRADFORD M. A. Local buckling of thin steel plates in composite construction: Experiment and theory. Proceedings of the Institution of Civil Engineers - Structures and Buildings, 1995, Volume 110, November, 426-440. 5. UY B. and BRADFORD M. A. Elastic local buckling of steel plates in composite steel - concrete members. Journal of Engineering Structures, 1996, Volume 18, No. 3, 193-200. 6. SCHUURMAN R. G. and STARK J. W. B. Longitudinal shear resistance of composite slabs. Proceedings of the Engineering Foundation Conference, 1997, 89-103. 7. AN L. and CEDERWALL K. Push-out tests on studs in high strength and normal strength concrete. Journal of Constructional Steel Research, 1996, Volume 36, No.l, 15-29. 8. OEHLERS D. J. and JOHNSON R. P. The strength of stud shear connections in composite beams. The Structural Engineer, 1987, Volume 65B, No.2, June, 44-48. 9. ANWAR HOSSAIN K. M. and WRIGHT H. D. Performance of profiled concrete shear panels. Journal of Structural Engineering- ASCE, 1998, Volume 124, 368-381. 10. KEMP A. R. and TRINCHERO P. E. Horizontal shear failures around connectors used with steel decking. Proceedings of the Engineering Foundation Conference, 1997, 104-118. 11. CLUBLEY S. K., XIAO R. Y. and MOY S. S. J., Computational structural analysis and testing of Bi-Steel plates- Six and Twelve month progress report for British Steel. University of Southampton - Department of Civil & Environmental Engineering, 1999, June, 145 & 270 pages. RECTANGULAR TWO-WAY RC SLABS BONDED WITH A STEEL PLATE J. W. Zhangl, J.G. Teng 2 and Y.L. Wong 2 1 Department of Structural Engineering, Southeast University, Nanjing, China. 2 Department of Civil and Structural Engineering The Hong Kong Polytechnic University, Hong Kong, China. ABSTRACT External bonding of steel plates has been widely used for retrofitting RC structures. Many studies have been carried out on RC beams bonded with steel plates, but little research exists on two-way RC slabs strengthened using this technique. This paper is therefore concerned with the strength of rectangular two-way RC slabs bonded with steel plates subject to a central patch load. Experimental results on square RC slabs bonded with square steel plates are first summarized. A yield line analysis of rectangular two-way plated RC slabs is then presented based on experimental observations of the formation of yield lines. Finally, a design procedure based on the yield line analysis is proposed for practical use, which incorporates an empirical modification factor based on the experimental results. KEYWORDS: Slabs, Steel Plates, Yield Line Analysis, Strengthening, Bonding, Adhesive INTRODUCTION Among the many strengthening techniques available, the method of plate bonding has been an attractive one in recent years, due to its simplicity and speed of application and minimum increases in structural self-weight and size. Steel plates and fibre-reinforced plastic (FRP) plates have both been used in plate bonding, depending on the requirement of a particular situation. Steel plates have been used very widely to strengthen RC beams and also slabs. Two recent cases of plate bonding to slabs are reported in Civil Engineers Australia (1995) and Godfrey and Sharkey (1996), and both used steel plates. Although a great deal of research has been carried out in recent years on this method of strengthening for RC beams, the only previous study on two-way slabs is that by Erki and Hefferman (1995) which reported some tests on small two-way slabs bonded with FRP sheets to enhance the punching shear failure load. This paper is therefore concerned with the strength of rectangular two-way RC slabs bonded with steel plates subject to a central patch load. High patch loads requiting strengthening of structures often arise in practice. Examples include local loads due to the installation of a piece of heavy equipment and column loads on floor slabs due to the removal or addition of columns. Experimental results on square 477 478 J.W. Zhang et al. RC slabs bonded with square steel plates are first summarized. A yield line analysis of rectangular two- way plated RC slabs is then presented based on experimental observations of the formation of yield lines. Finally, a design procedure based on the yield line analysis is proposed for practical use, which incorporates an empirical modification factor based on the experimental results. It should be remarked that if a single steel plate is too big for convenient handling in construction, a number of orthogonally placed steel strips may be used instead to achieve the same amount of external reinforcement. The work presented here is equally applicable to such slabs. EXPERIMENTS ON SQUARE SLABS Experimental Results A total of five square RC slabs bonded with square steel plates were tested by the authors (Zhang et al., 1999). Only a brief summary of the experimental results in relation to the yield line analysis to be TABLE 1 PROPERTIES OF MATERIALS Materials Concrete Rebar Type 1800x 1800x70 (mm) Mild steel d76.5,@ 150mm centres, in both directions, average concrete cover for the two directions =16.5mm Steel plate Mild steel Adhesive ET epoxy resin *Assumed values Elastic Compressive Yield Ultimate modulus strength stress tensile stress (N/mm 2 ) (Nlmm z ) (N/mm z ) (N/mm 2 ) - 26.4 - - 200000* 200000* 5960 340 431 - 335 417 94 - 11 TABLE 2 EXPERIMENTAL RESULTS AND COMPARISON WITH YIELD LINE ANALYSIS Specimen SB1 (control) Dimensions of steel plate (in mm) and plate-to-slab area ratio (%) No plate Initial cracking loadPcr (kY) and relative increase Ycr against SB1 (%) 21 (0.00) Experimental ultimate load Pe (kN) and relative increase Ye against SBI(%) 55.0 (0.00) Theoretical ultimate load Pu (1~) 45.2 SB2 500•215 (8.65) 40 (90.5) 67.5 (22.7) 56.0 SB3 500•215 (8.65) 40 (90.5) 65.0 (18.2) 56.0 SB4 850•215 (25.0) 60 (186) 85.0 (54.5) 75.7 SB5 1400• 1400• 1 100 (376) 165 (200) 200 (67.8) Pu ee 0.82 0.83 0.86 0.92 1.21 presented in this paper is given here. The slabs all had the same dimensions of 1800 mm x 1800 mm x 70 mm. They were simply supported with a span of 1700 mm between supports and subject to a Rectangular Two-Way RC Slabs Bonded with a Steel Plate 479 central patch load over an area of 150 mm x 150 mm. Details of the material properties and dimensions of the bonded steel plates are given in Tables 1 and 2 respectively. Table 2 also gives the initial cracking loads and ultimate loads of the test slabs. These experiments showed that bonding of steel plates to the soffit of slabs can greatly increase both the cracking load and the ultimate load of two-way RC slabs. Debonding of the steel plate from the slab is unlikely as in all four tests on plated slabs, no debonding failure was found. This is contrary to the case of plated beams. In this sense, the plate bonding method is more suited for slabs than for beams. Failure of the plated slabs was by the formation of yield lines and the failure mode was ductile. The final cracking patterns and hence yield line patterns of slabs SB2, SB3 and SB4 are similar. Most of the cracks were on the soffit of the slab. At the edges of the steel plate, an abrupt change in stiffness and strength occurs. As a result, main cracks occurred around the plate perimeter. In addition, in each zone between the corner of the steel plate and that of the slab, there were four or five main diagonal cracks. The final cracking patterns of slab SB4 are shown in Figure 1. Slab SB5 which was bonded with a large steel plate had a different yield line pattern. Figure 1 Final Cracking Patterns of Slab SB4 Figure 2 Triangular Yield Line Pattern for a Square Plated Slab TABLE 3 ACCURACY OF THE TRIANGULAR YIELD LINE PATTERN Slabs k 2 ( k 3 ) SB1 0.0882 SB2 0.294 SB3 0.294 SB4 0.5 SB5 k 1 0.267 0.207 0.207 0.146 Pe (kN) 55 67.5 65 85 Pu (kN) 45.2 56.0 56.0 75.7 Error (%) -17.8 -17.0 -13.8 -10.9 0.824 0.052 165 200 21.2 Yield Line Analysis Based on the experimentally observed yield line pattems, yield line analyses (Johansen, 1962; Jones and Wood, 1967; Kong and Evans, 1987) were carried out for the test slabs (Zhang et al., 1999). In particular, three of four different yield line pattems explored were suitable for slabs with cracking patterns similar to those shown in Figure 1, with the fourth being suitable for slab SB5. The differences in these three yield line patterns lie in the assumption of yield line patterns in the zones between the comers of the steel plate and the slab. . No. 1-2 , 10 7- 114. 4. UY B. and BRADFORD M. A. Local buckling of thin steel plates in composite construction: Experiment and theory. Proceedings of the Institution of Civil Engineers - Structures. and minimum increases in structural self-weight and size. Steel plates and fibre-reinforced plastic (FRP) plates have both been used in plate bonding, depending on the requirement of a particular. Structural Engineering- ASCE, 1998, Volume 124, 36 8-3 81. 10. KEMP A. R. and TRINCHERO P. E. Horizontal shear failures around connectors used with steel decking. Proceedings of the Engineering Foundation