PRIMARY AND SECONDARY REINFORCEMENTS IN CORBELS UNDER COMBINED ACTION OF VERTICAL AND HORIZONTAL LOADINGS

Transcription

1 16th International Conference on Composite Structures ICCS 16 A. J. M. Ferreira (Editor) FEUP, Porto, 2011 PRIMARY AND SECONDARY REINFORCEMENTS IN CORBELS UNDER COMBINED ACTION OF VERTICAL AND HORIZONTAL LOADINGS Mehdi Rezaei 1, S.A.Osman 2, N.E. Shanmugam 3 Department of Civil & Structural Engineering, Faculty of Engineering & Built Environment National University of Malaysia (UKM) 1 rezaei_mrc@yahoo.com, 2 saminah@eng.ukm.my, 3 shan@eng.ukm.my Keywords: Corbel, Combination of Horizontal and Vertical Loading Summary. This paper is concerned with the effects of primary and secondary reinforcements in corbels under combined action of horizontal and vertical loadings. Strut-and-Tie models and Cantilever Beam method recommended by PCI (Pre-stress/Precast Concrete Institute) are considered. The two methods are assessed and validated. The results showed that Cantilever Beam method is suitable if 20% or less of the vertical load is involved as the horizontal force. It was also found that Strut-and-Tie method provides more primary and secondary reinforcements compared to the Cantilever Beam method. 1

2 1 INTRODUCTION Extensive experimental programs have been undertaken over the last decades to establish the behavior of reinforced concrete non-flexural members, such as deep beams, nibs, corbels, beam-column joints. Experimental studies have shown that non-flexural members display an increased shear capacity relative to flexural members. Based on the results obtained from the studies a number of empirical models were developed for design of corbels. The main limitations of the empirical approaches are a limited range of shear span to depth ratios for which design equations apply and, an inability to explain precisely the mechanics of corbels behavior. Finite element studies incorporating non-linear materials models have been used in combination with experimental programs to determine the mechanics of non-flexural members including corbels. 16 life-size corbels were tested by Yong et al. [1994] to investigate the effects of horizontal force, reinforcement ratio, and shear-span-to-depth ratio. A specially designed test setup was used to induce a horizontal load equal to 20% of the vertical load. The primary steel yielded before failure almost in all cases. The truss analogy model was found to provide relatively accurate strength predictions compared to the American Concrete Institute s procedure. Six full-scale reinforced concrete corbel specimens were tested by Mohamed Almeer [2004], to study the influence of steel and polypropylene fibers, headed bars, and horizontal loading. The experimental values of ultimate load capacity were compared with predictions using simplified and refined Strut-and-Tie models; the refined model was found to provide better prediction. The results showed that the addition of horizontal force (20% of vertical) to the test specimen resulted in ultimate load carrying capacity dropping by almost 20%. Addition of this force increased the flexibility of the test specimens. Studies have been carried out on high-strength reinforced concrete corbels by researchers [Stephan, 1997; Mohamed et al., 1997; Fattuhi, 1994, 1995] with artificial substances such as steel fiber, plastic mesh, and etc. Only vertical load was considered in these studies. Comparison between the experimental and calculated strengths of corbels showed that the two values obtained in each case are in satisfactory agreement. It is clear from what has been observed that the studies on the effects of horizontal force on strength of reinforced concrete corbels are inconclusive. One reason why the researchers avoided the study with the application of horizontal and vertical forces simultaneously could be because of the practical problems that may arise in realizing that type of loading in the tests. In order to fill the gap in the results for the combined action of horizontal and vertical loading on corbels, the current study aims to provide some information in this regard. In this paper corbels are designed with different ratios of primary and secondary reinforcements and they are analyzed by two methods viz. Strut-and-Tie model and Cantilever Beam model, recommended by PCI. Finite element software package, LUSAS version14.1, was employed in the investigations. 2

3 2 FINITE ELEMENT ANALYSIS OF CORBELS 2.1 Finite Element Modeling and Its Accuracy Finite element software package LUSAS was used to analyze corbels in this study. A typical meshed corbel adopted in the analysis is shown in Fig. 1. It is necessary to establish the accuracy of the finite element modeling. Therefore, seven corbels tested by Stephan [1996] were considered in this study. All specimens were modeled and analyzed using LUSAS and the results obtained are shown in Fig. 2 in which the experimental ultimate load is plotted against the corresponding values obtained from the finite element analysis for all the seven specimens. It is clear from the figure that finite element results are close to the corresponding experimental values, maximum deviation being 15%. The difference could be attributed to the assumptions used in the modeling since some of the material properties and dimensions are not given in the paper. Since LUSAS was found to predict the results with acceptable accuracy the software package was used for further analyses. Fig. 1 A Half Meshed Corbel, Using LUSAS Fig. 2 Comparison between Finite Element Results and Test Results 3

4 2.2 Details of The Corbels Detailed dimensions of the corbels and the reinforcements are shown in Fig. 3. For all corbels considered in the analyses the overall dimensions were kept the same. The corbel width, 254 mm was the same as the width of the column supporting the corbels, the overall corbel depth and its effective depth being 406 mm and 376 mm, respectively. The depth at the free edge of the corbel was 203 mm. The cross section of the column supporting the corbel was 305 mm 254 mm with total length taken as 965 mm for the purpose of analysis. The column was reinforced with four 16-mm diameter bars located at the corners and tied by 9 mm diameter ties spaced at 216 mm center to center along the column length. The primary and secondary reinforcements designed as per the code requirements are summarized in Table 1. Secondary reinforcement usually includes two or more steel bars. The total cross-sectional area for secondary reinforcement is generally divided into a number of steel bars spread equally. In this study, two secondary reinforcements were used. In all corbels, the materials used were the same having properties such as compressive strength of concrete equal to 30 MPa, the yield stress of steel equal to 420 MPa, Young s modulus of concrete and steel equal to N/mm 2 and N/mm 2, respectively, and Poisson s ratio of 0.2 and 0.3 for concrete and steel, respectively. There are two methods, Cantilever Beam method and Strut-and-Tie method, proposed in the PCI recommendations to determine the primary and secondary reinforcements for corbels. For the corbel dimensions chosen, area of reinforcements has been calculated by the two methods for different combinations of vertical and horizontal loadings the details of which are shown in Table 1. In the table, the corbel specimens with different combinations of horizontal and vertical loadings are identified as H/V0.0, H/V0.1 etc. indicating the horizontal to vertical loading ratios of 0.0 and 0.1 and so on. Both primary and secondary reinforcement areas calculated by the two methods are presented in the table. In Strut-and-Tie method, however, neither primary nor secondary reinforcement values are given for corbels H/V0.8, H/V0.9 and H/V1.0. In these corbels, the quadratic expression derived for the determination of reinforcement areas by truss analogy method results in larger column width, not covered in the present study. In the finite element analyses of the corbels the top and the bottom ends of the column were assumed clamped with the load applied incrementally on a bearing pad, 100mm 254mm 15mm, made of steel as shown in the figure. Taking advantage of symmetry in geometry, loading and support conditions only a half of the specimen was modeled for the analyses. Mesh size of mm and distribution of mesh similar to the one shown in Fig. 1 were chosen based on convergence studies carried out to determine the optimal mesh that gives a relatively accurate solution and one that takes low computational time. Results are presented in the form of plots, horizontal to vertical load ratios against reinforcement areas and, vertical and horizontal loads against the respective displacements. 4

5 Fig. 3 Reinforcement Detailing (Dimensions in mm) 3 RESULTS AND DISCUSSION Figs. 4 and 5 show the variation of primary and secondary reinforcements, respectively, with horizontal to vertical load ratios. In each of the two figures, the variations as obtained by the two methods, Cantilever Beam method and Strut-and-Tie method are compared. It can be seen from Fig. 4 that Strut-and-Tie method provides larger cross sectional area for primary reinforcements. The difference in the area of primary reinforcements gradually increases with the load ratios. For example, the area of reinforcement predicted by the Cantilever Beam method and Strut-and-Tie method for load ratio of 0.0 is 272 mm 2 and 355 mm 2, respectively with a difference of 83 mm 2 whilst the corresponding values for load ratio of 0.7 are 752 mm 2 and 949 mm 2 with a difference of 197 mm 2. Variations in secondary reinforcements with load ratio are totally different, as predicted by the two methods. It can be seen from Fig. 5 that the Cantilever Beam method provides almost same area of secondary reinforcement irrespective of the load ratios but on the other hand the area of secondary reinforcement provided by Strutand-Tie method increases linearly with load ratio. Strut-and-Tie method theoretically does not provide any secondary reinforcement when horizontal force is not involved. A minimum amount of horizontal stirrup reinforcement must, however, be provided to avoid diagonal tension failure. Therefore, the ACI 318 recommends designing corbels for a minimum of 0.2 times vertical force applied in the horizontal direction. It is clear from Fig. 5 that Strut-and- Tie method gives secondary reinforcement quite close to that by Cantilever Beam method for a load ratio of around

6 Designation Table 1: Primary and Secondary Reinforcements Vertical Load (kn) Horizontal Load (kn) Beam theory Truss Analogy A prim A sec A prim A sec H/V H/V H/V H/V H/V H/V H/V H/V H/V H/V H/V Fig. 4 Cross Sectional Areas of Primary Reinforcement versus Ratios of Horizontal to Vertical Force 6

7 Fig. 5 Cross Sectional Areas of Secondary Reinforcement versus Ratios of Horizontal to Vertical Force It has been noted in the ultimate load analyses of the corbels that a crack initiates first at the re-entrant corner and then propagates along the column-corbel interface. At the inner edge of the bearing plate a second crack is formed propagating much faster than the first one. While the first crack continues to propagate along the column face the second crack progresses towards the junction of the column and the sloping face of the corbel. The second crack, which becomes the primary or major crack, eventually runs between the inner edge of the bearing plate and the column-corbel junction at the sloping face and results in the failure of the corbel. Ten reinforced concrete corbels, listed in Table 1, designed by Cantilever Beam method were analyzed by the finite element method and the results are presented in the form of load-displacement plots as shown in Figs. 6 and 7. Vertical or horizontal displacements plotted on the horizontal axis correspond to those measured under the load. Fig. 6 shows the load-deflection curves for vertical loads and the corresponding plots for horizontal loads are given in Fig. 7. In each of these figures curves are plotted showing the variation of deflection with vertical load or horizontal load for different values of load ratios. It can be seen from the figures that the load deflection relation is nonlinear even from the early stages of loading. The horizontal and vertical load interaction is reflected in the two sets of curves. In the presence of vertical load the gradual increase in horizontal load results in gradual drop in vertical load capacity. In all cases the load tends to increase and stop further increase when the ultimate load condition is reached. The results for ultimate loads are summarized in Table 2. The results show that the Cantilever Beam method is appropriately able to predicate the behavior 7

8 of corbels without horizontal load, on the contrary it grossly overestimates the load-carrying capacity of corbels when horizontal load is involved, even though, for the corbel involving 20% of the vertical force is still satisfactory. It is clear from the results that Cantilever Beam method is overestimated when greater than 20% of vertical load is involved as horizontal load. It should be noted that the horizontal to vertical force ratio is limited to 1.0 by the PCI (Nu/Vu 1). Unlike vertical load-deflection curves, in horizontal load-deflection curves the ductility enhances. Also, the rate change of the ductility of the horizontal load-deflection curves is considerably greater than the vertical load-deflection curves as well as toughness of the horizontal load-deflection curves. Fig. 6 Vertical Load-Deflection Curves Fig. 7 Horizontal Load-Deflection Curves 8

9 Designation Table 2 Results for ultimate loads Ultimate Vertical load (kn) Ultimate Horizontal load (kn) H/V H/V H/V H/V H/V H/V H/V H/V H/V H/V H/V CONCLUSIONS Finite element method has been employed to analyze corbels subjected to combined action of vertical and horizontal loads. Two different methods, Cantilever Beam Method and Strutand-Tie method proposed by the PCI have been used to determine secondary and primary reinforcements and the analyses carried out to determine the ultimate load carrying capacity. From the results obtained following conclusions are drawn: 1. The amount of primary and secondary reinforcements obtained by using the Strut-and-Tie method is more than the corresponding amount provided by the Cantilever Beam method. 2. As per the Strut-and-Tie method, the cross sectional area of secondary reinforcement is considerably larger in the presence of horizontal load. 3. Cantilever Beam method is suitable when 20% or less of vertical load is involved. 4. As expected, the ultimate vertical load carrying capacity drops considerably in the presence of horizontal load. REFERENCES [1] Yook-Kong Yong, and P. Blaguru, Behavior of Reinforced High-Strength- Concrete Corbels, American Society of Civil Engineers (ASCE), Vol.120, No.4, pp (1982). [2] Nijad I. Fattuhi. Reinforced Corbel Made With High-Strength Concrete and Various Secondary Reinforcement, ACI Structural Journal, Vol.91, No.3, pp (1994). [3] S.J. Foster, R.E. Powell, and H.S. Selim, Performance of High Strength Corbels, ACI Structural Journal 93, pp (1997). 9

10 [4] Nijad I. Fattuhi,Strength of FRC Corbels in Flexure, Journal of Structural Engineering, Vol. 120, No. 2,, pp (1994). [5] PCI Design Handbook (6 th Edition), Precast and Prestressed Concrete Institute, Chapter 6, 46 pp (2004). [6] Mohamed Almeer, Effects of Fibers and Headed Bars on the Response of Concrete Corbels, MS Thesis, Department of Civil Engineering and Applied Mechanics, McGill University (2004). [7] Moahmed S., and Houssam T., Design of High Preformance Concrete Corbels, Journal of Materials and Structures, Vol.31, November, pp (1998). [8] ACI Committee 318, Buildding Code Requiremtns for reinforced Concrete (ACI 318), American Concrete Institute. 10