Punching shear resistance of reinforced concrete slabs under pneumatic loads

Transcription

1 Punching shear resistance of reinforced concrete slabs under pneumatic loads João Pedro Infante Gonçalves Instituto Superior Técnico, Lisboa, Portugal October 2016 Abstract The purpose of this work is to evaluate the punching shear resistance of reinforced concrete slabs under pneumatic loads and compare the results of a test on a slab under a pneumatic load with the results of a test on a slab supported by concrete column under concentrated loads, to evaluate if the punching shear failure is similar in both cases. The results for PR1 test were obtained from research by Vaz Rodrigues and Muttoni [1] and the results for PG-10 test from Guandalini [2]. First, some analysis methods for the punching shear problem were described. After that, a description of the results of the experimental tests is presented: PR1 test for the slab under a pneumatic load and PG-10 test for the slab supported by a concrete column. This description includes the materials that were used, the loads that were applied and the results of the tests. Then, the results of both tests are compared with the NP EN [3] and with the ultimate flexural load obtained using yield line method. Next it was developed a linear finite elements model [4] for both load cases, pneumatic load and column load, and distribution of shear force along the control perimeter was evaluated. Afterwards it was investigated the slab behaviour with non-linear elements to better understand the influence of yielding and cracking on the distribution of shear force along the control perimeters, in both pneumatic and column cases. Also some examples using the the non-linear model were used to validate the simulations in the computational model. Finally the results of the linear and non-linear models are compared and conclusions are drawn. Keywords: Punching shear, pneumatic loads, modelling, slab, reinforced concrete 1 Introduction 2 Literature review The simplest analysis method regarding punching shear is the shear stress on a critical perimeter [5]. It compares the shear stress on a critical section with the maximum shear stress. While analysing the influence of some variables in the punching shear problem, it is possible to see that are differences in the considered models. The beam analogy is rather successful at predicting the flexural behaviour of a slab, while failing to correctly predict the punching shear failure mode. One of the best models designed is the strut and tie model [5], that provides a load path for shear forces after cracking and explains the role of flexural reinforcement in determining shear strength. This model considers inclined compressions struts that carry shear force to steel ties. Based on this model, a bond model was developed and it considered that the compression struts are in fact curved arches [6]. This model combines the shear stress theory and the truss model, and its results match the experimental data. 3 Results description This chapter consists in the description of the punching shear failure mechanism and description on the results of both PR1 test and PG-10 test. 1

2 3.1 Ultimate limit state of punching shear The punching shear failure of a slab is a brittle and sudden failure, mainly conditioned by the concrete resistance and more common on slabs under loads applied on small areas. When one of the columns of a floor slab collapses, the pillars right next to it receive extra load and may collapse too, since they weren't designed for that extra load, leading to a progressive collapse of the structure. In order to prevent that from happening, it is possible to reinforce the slab with stirrups next to the area where the failure occurs or with a thickening of the slab around the column. However, in some cases, like a bridge, it is not possible to predict where the vehicular loads will be more intense and it is impractical to reinforce the whole structure due to construction problems. 3.2 PR1 test This test consists of a punching shear test, on a slab with a central support that is a flat jack made of a copper sheet envelope with water inside. The flat jack has a nominal surface of 0,156 m 2 and a nominal diameter = 446 mm. The loads were applied using four hydraulic jacks below the strong floor and the resultant force was measured using four load cells between the jacks and the strong floor [1]. The flat jack simulates a road load, which causes contact pressures on the surface of the slab that are approximately constant until the failure of the slab Dimensions and detail The slab is 3 m x 3 m x 0,25 m and is supported in 8 points by jacks. The reinforcement at the top layer is and at the bottom layer is, which means a reinforcement ratio of 0,33%. The slab doesn't have any vertical shear reinforcement. The average effective depth between both directions at the top layer is 210 mm and the concrete cover is 30 mm Materials and actions The concrete and reinforcement were tested before the slab was constructed and when the failure happened, the properties were measured again. In Table 1 are the concrete properties and in Table 2 are the reinforcement properties. Table 1: Concrete properties f ck [MPa] f ctm [MPa] E c [GPa] 35,17 2,23 31,84 Table 2: Reinforcement properties f yk [MPa] f t [MPa] The slab was subjected to 9 load stages, where the load increased at each stage. After the load stage #6, the slab was unloaded and after 5 days was reloaded. The failure load is 599,1 kn and it happened for a rotation of 51,5 mrad, which is the average of the rotations on the north side of the slab and the south side of the slab. The rotations were measured with the inclinometers. 3.3 PG-10 test This test belongs to a series of 10 trials within the Stefano Guandalini doctoral thesis in which the author seeks to develop a new physical model to determine the punching shear resistance of reinforced concrete slabs Dimensions and detail The slab has the same dimensions as the PR1 test slab, 3 m x 3 m x 0,25 m, but the central support is a reinforced concrete square column, monolithically connected to the slab, with 0,26 m x 0,26 m. It also has the same reinforcement layout, at the top layer and at the bottom layer, with a reinforcement ratio of 0,33%. In this type of test, the contact pressures are not uniform and tend to increase near the column edges as the deflection of the slab increases Materials and actions The properties of the concrete and the properties of the reinforcement are in Tables 3 and 4, respectively: Table 3: Concrete properties f ck [MPa] f ctm [MPa] E c [GPa] 28,5 2,20 29,5 2

3 Table 4: Reinforcement properties f yk [MPa] f t [MPa] The failure load is 540 kn and it happened for a rotation of 22,3 mrad. 4. Comparison with EC2 and ultimate flexural load In this chapter, the test results of PR1 test and PG-10 test are compared to the NP EN [3] and with the ultimate flexural load. 4.1 PR1 test Assessment of resistance to punching shear Using the equation (1) the punching shear stress can be determined: (1) where is the monolithic ratio, is the control perimeter of the central support, is the average effective depth between both directions at the top layer and is the maximum load applied during the test. In Table 5 are the values needed to calculate the punching shear stress of the PR1 test slab: Table 5: Punching shear stress of the slab [m] [m] [kn] [kn/m 2 ] 1 0,21 4,04 614,5 724,3 Concerning the punching shear strength of the slab according to the NP EN [3], it is obtained using equation (2): (2) where, with d in mm, is the reinforcement ratio and is the compression stress of the concrete. In Table 6 are all the values needed to determine the punching shear strength of the slab: Table 6: Punching shear strength of the slab [mm] [%] [MPa] [kn/m 2 ] 1 0, ,9759 0,33 35, Comparing the two values, it follows that the punching shear stress acting on the slab is smaller than the punching shear strength of the slab according to the NP EN [3] Determination of ultimate flexural load In order to compare the maximum load applied to the slab with the flexural failure load, it is necessary to determine the maximum resistant moment of the superior reinforcement, which is, that corresponds to 6,87 cm 2 /m. Using the equations (3), (4) and (5), it is possible to determine the maximum resistant moment. (3) (4) (5) where is the mechanic ratio of reinforcement, is the width where the reinforcement is distributed, is the average effective depth between both directions at the top layer, f cd is the design value of the compression strength of the concrete, f yd is the design value of the steel yield strength, is the reduced bending moment of the section and M Rd is the maximum resistant moment of the section. 3

4 The Table 7 presents all values necessary to calculate the maximum resistant moment of the slab: Table 7: Maximum resistant moment of the slab [cm 2 /m] [m] [m] [MPa] [MPa] [knm/m] 6,87 1,00 0,21 35, , , ,05 According to Guandalini [2], the ultimate flexural load can be calculated through the method of rupture lines. In his work, Guandalini deduces the equation that can be used to determine the ultimate flexural load (6): where B is the width of the slab, b is the width of the support, b Q is the equivalent radius and m u is the maximum resistant moment. In Table 8 are presented all the values necessary to calculate the ultimate flexural load: Table 8:Ultimate flexural load B [m] b [m] [m] [m -1 ] [knm/m] [kn] 3 0,446 1,38 8, ,05 700,12 Comparing both values it follows that the maximum load applied to the slab is lower than the ultimate flexural load. In Table 9 are presented all the values compared for the PR1 test slab: Table 9: Comparison between, and [kn] [kn] [kn] 614,5 682,96 700, PG-10 test Assessment of resistance to punching shear In Table 9 are all the values needed to determine the punching shear stress of the slab and in Table 10 all the values necessary to calculate the punching shear strength of the slab according to the NP EN [3]: Table 9: Punching shear stress of the slab [m] [m] [kn] [kn/m 2 ] 1 0,21 3, ,8 Table 10: Punching shear strength of the slab [mm] [%] [MPa] [kn/m 2 ] 1 0, ,9759 0,33 28,5 750,74 For the PG-10 test slab, the punching shear stress of the slab is lower than the punching shear strength of the slab Determination of ultimate flexural load The calculations for the maximum resistant moment are shown in Table 11 and the values necessary to determine the ultimate flexural load are shown in Table 12: Table 11: Maximum resistant moment of the slab [cm 2 /m] [m] [m] [MPa] [MPa] [knm/m] 6,87 1,00 0,21 28, ,0662 0, ,90 (6) 4

5 Table 12: Ultimate flexural load B [m] b [m] [m] [m -1 ] [knm/m] [kn] 3 0,260 1,38 8, ,90 648,87 In this case, the maximum load applied on the slab is smaller than the ultimate flexural load In Table 13 is shown the comparison between, and : Table 13: Comparison between, and [kn] [kn] [kn] ,17 648, Comparison between PR1 test and PG-10 test In order to be able to compare the results of two tests with different characteristics, the following equation (7) was used: (7) where is the maximum load applied to the slab, d is the average effective depth between both directions at the top layer, u 1 is the control perimeter of the central support and f ck is the characteristic value of the compression strength of the concrete. The Table 14 shows the comparison of the results of both tests using the equation (7): Table 14: Comparison between PR1 and PG-10 tests [m] [m] [MPa] [kn] PR1 0,21 4,04 35,17 614,5 0,2211 PG-10 0,21 3,68 28,50 540,0 0,2288 From this comparison, it follows that the maximum load applied to each test slab is very similar, when the differences in the materials or how the load is applied are accounted. 5 Linear modelling of slab behaviour using a finite element program Using the finite elements program [4], the slabs from each test were analysed with more detail. 5.1 Slab calculation model The slabs were modelled using Shell elements with thickness 0,25 m. In both cases, the material properties adopted were those of the PG-10 test, which are shown in Table 15: Table 15: Material properties for the computational model [MPa] [MPa] [GPa] [MPa] [MPa] [MPa] 28, , The control perimeter is at a distance of d/2, which is 0,105 m. 5.2 Considered loads in slab analysis Column load The column works as an elastic support and the concentrated loads are located on the jacks. The resultant of these forces added with the self weight of the slab totalizes V = 596,25 kn Pneumatic load In this case the supports are now located in the jacks and a constant pressure is applied on the central support. This pressure results from the self weight of the structure and the failure load of the slab, in a total of V = 596,7 kn, and translates into a pressure P = 8826,92 kn/m Description of results In the next figure, it shown that the shear distribution is similar for both cases, despite the shear being higher in the column load case, near the corners of the column. The shear distribution for 5

6 the pneumatic load case is nearly uniform. The reaction of the slab for the PR1 test is the exact same as for the PG-10 test. Figure 1: Shear comparison for both cases 6 Fundamentals of the analytical model The Modified Darwin-Pecknold reinforced concrete material model [7] describes a nonlinear two-dimensional model that can account for the interaction between bending and shear in shear wall structures. It considers nonlinear effects such as the cracking of concrete and, when combined with steel, the yield of the reinforcement. It does not consider the tensile strength of concrete. 6.1 Uniaxial concrete material behaviour This model considers a trilinear approximation when a user defined stress-strain curve is specified, in order to simplify the nonlinear model. 6.2 Initial Elastic Stress-Strain Relationship Until the material yields or cracks, it has an elastic linear relationship with the initial value of Young's modulus E 0 and Poisson's ratio ν. 6.3 Post-yield or cracked material behaviour After yield or cracking, the Young's modulus varies and Poisson's ratio is overlooked. 6.4 Strength reduction under perpendicular tensile strength The shear stresses on concrete can lead to cracking in one direction and compression in the other. The compression strength of concrete depends on the magnitude of the tensile strain in the perpendicular direction [7]. In these cases, the effective compression strength of concrete can be significantly smaller than the original compression strength [8]. When that happens, a reduction factor is applied to the stress-strain curve and it is reduced. However, the initial Young's modulus remains the same. 6.5 Important numerical considerations When compared to the directional material models, the modified Darwin-Pecknold model has a higher degree of nonlinearity. Due to this, some refinement of the mesh may be necessary, but shouldn't be over-refined. Using the Poisson's ratio equal to zero may improve convergence in some cases. 7 Validation of the nonlinear model In the following chapter, the materials nonlinear effects will be considered and as such, it is necessary to validate the non linear model before analysing the slabs of PR1 and PG-10 tests. 7.1 Material properties In the next figures are shown the stress-strain curve for the concrete and steel adopted for the nonlinear model. 6

7 V [kn] σ (kn/m 2 ) σ (kn/m 2 ) Betão ,00E-03 0,00E+00 2,00E ε (m/m) 4,00E-03 Aço ,2-0, ,1 0, ε (m/m) Figure 2: Stress-strain curve of the concrete Figure 3: Stress-strain curve of the steel 8 Nonlinear modelling of slab behaviour using a finite element program The nonlinear model's purpose is to bridge the gap between the results of the linear model for the different load cases, pneumatic load and column load. 8.1 Nonlinear effects on a reinforced concrete slab The nonlinear model has in consideration the nonlinear effects of the reinforced concrete slab such as concrete cracking and yielding of the reinforcement Concrete cracking When a reinforced concrete element is loaded, it presents an elastic linear behaviour. Then it reaches the cracking moment and the section begins to crack. After that, the bending stiffness begins to decrease and at each new crack there is an increase in deformation Yielding of the reinforcement The reinforcement gives a greater resistance to the reinforced concrete as well as a capacity to deform after it reaches the cracking moment, with a ductile behaviour until failure [9]. When the section reaches the yielding moment, the reinforcement yields. After this point, the structure continues to deform due to the ductile behaviour until it reaches the tensile strength of the steel and the reinforced concrete element reaches failure. 8.2 Slab calculation model In order to be able to compare the results of the nonlinear model to the linear model, the yielding stresses are the same as in the linear model. Both slabs of the nonlinear model have the same mesh. Since the purpose is to refine the mesh to obtain better results, a mesh simplification was considered. 8.3 Pneumatic load Considered loads in slab analysis The load cases considered were increases of 50 kn until 600 kn Results The load-displacement relationship for the nonlinear model is very close to the same curve from the PR1 test results, despite having a higher rotation in failure ,0 20,0 40,0 60,0 80,0 100,0 ΨN-S [mrad] Ensaio PR1 Modelo de cálculo NL V.flex Figure 4: Comparison between the nonlinear model results and the PR1 test results 7

8 V [kn] Slab deflection during the test As it is expected, the deflection of the slab increases as the load applied to the slab increases. Near the failure load, the slab deflection varies at a lower rate that when the load applied is lower Stresses in the upper reinforcement As the load applied to the slab increases, the stresses along the control perimeter increase as well. When the load is 500 kn, the control perimeter and a section of nearby elements reach the value of the yielding stress of the reinforcement. 8.4 Column load Considered loads in slab analysis The load cases considered were increases of 50 kn until 600 kn Results This time, the results of the computational model are very similar to the results of the experimental test PG-10. The rotation of the slab obtained from the model for the failure load was 24,38 mrad and the rotation of the slab from the experimental test results was 22,3 mrad, meaning it is a good approximation Modelo de cálculo NL Ensaio PG-10 V,flex 0 0,00 5,00 10,00 15,00 20,00 25,00 30,00 ΨN-S [mrad] Figure 5: Comparison between the nonlinear model results and the PG-10 test results 8.5 Results description The results of the nonlinear model show that there is an increase of the shear in the linear region of the column and the shear distribution for both cases is more similar than the distributions for the linear model. Figure 6: Shear comparison for both cases 9 Results analysis In this chapter is presented a final evaluation of the results. 9.1 Modelling options Comparing both cases, it is possible to verify in Table 18 that the experimental loads of both tests are lower than the punching shear resistance of the slab, calculated from the NP EN [3]. 8

9 Table 18: Punching shear stresses and the punching shear strength obtained from EC2 in kn/m 2 Test EC2 Pneumatic (PR1) 724,30 805,00 Column (PG-10) 698,80 750,74 Table 19: Punching shear load and the ultimate flexural load in kn Test Nonlinear Ultimate flexural load Pneumatic 614,50 600,00 700,12 Column 540,00 597,79 646, Calculations models Pneumatic load When analyzing the linear model, the distribution of shear along the support control perimeter was nearly uniform, with lower shear in the linear region of the support relatively to the edges of the support. Considering the nonlinear model, the shear distribution maintained relatively constant, but in this case, the shear is higher around the edges of the support and lower in the linear region of the support. When comparing these two models, both of them converge to the failure load of the slab, regardless of how the load is applied. Figure 6: Shear comparison for the pneumatic load Column load When the nonlinear effects are considered in the model, there is a decrease of the shear in the corners of the column and an increase in the linear region of the column. The failure load for each of the models, linear and nonlinear, is V = 596,69 kn and V = 597,79 kn, respectively. Despite the shear distribution around the control perimeter being different, both models converge to the failure load of the slab. It is possible to realize that the consideration of the nonlinear effects affects the shear distribution in this case, especially near the corners of column because it is a discontinuity zone where there is a higher concentration of shear. Yet the value of the failure load doesn't change. The use of a nonlinear model allows a better evaluation of the test results by considering the materials nonlinear effects. Figure 7: Shear comparison for the column load 9

10 10 Conclusions This thesis allowed the evaluation of the punching shear resistance of reinforced concrete slabs under pneumatic loads and also allowed the comparison with slabs supported by concrete columns under concentrated loads. The comparison of the experimental test results with the regulations enabled a better understanding of how these regulations can be applied. The use of finite elements model allowed a more detailed analysis of the test results. When the slab was modelled using nonlinear elements, some effects that were discarded in the linear model were considered. The pneumatic load case had similar shear distributions for both the linear and the nonlinear models, but for the column load case the shear distribution had some differences considering the models used. In conclusion, after all the analysis, the models that were studied are very similar despite the load application mode. However, the uniform shear distribution of the slab under pneumatic loads is less critical that the distribution of the slab under column load, because in the latter, the stresses accumulate near the edges of the reinforced concrete column. In short the load application mode does not seem to have a very significant effect and therefore it seems correct that this influence is not reflected into any specific parameter considered at a modern construction codes. References [1] VAZ RODRIGUES, Rui - Shear Strength of Reinforced Concrete Bridge Deck Slabs. Lausanne : Tese de doutoramento [2] GUANDALINI, Stefano - Poinçonnement symétrique des dalles en béton armé. Lausanne : Tese de doutoramento [3] Eurocódigo 2 - Projeto de estruturas de betão, Parte 1-1: Regras gerais e regras para edifícios, NP EN , Comité Europeu de Normalização, Bruxelas, Bélgica, 2010, 225 pp. [4] Manual do Software SAP2000 Ultimate V Csi Berkeley [5] ALEXANDER, S. and SIMMONDS, S. - Shear-Moment Transfer in Slab-Column connections, Structural Engineering Report No. 141, University of Alberta, Edmonton, Alberta, 1986, 95 pp. [6] ALEXANDER, S. and SIMMONDS, S. - Bond Model for Concentric Punching Shear, ACI Structural Journal, V. 89, No. 3, 1992, pp [7] DARWIN, D. and PECKNOLD, D.A.W. - Modified Darwin-Pecknold 2-D reinforced concrete material model. : 2015 [8] VECCHIO, F.J. and COLLINS, M.P. - The Modified Compression-Field Theory for Reinforced Concrete Elements Subjected to Shear. ACI Journal, Paper No , Março-Abril 1986 [9] CÂMARA, José Noronha da et al - Estruturas de Betão I, Folhas de apoio às aulas. 2014/