MODELING SLAB-COLUMN CONNECTIONS REINFORCED WITH GFRP UNDER LOCALIZED IMPACT

Transcription

1 MODELING SLAB-COLUMN CONNECTIONS REINFORCED WITH GFRP UNDER LOCALIZED IMPACT QI ZHANG and AMGAD HUSSEIN Faculty of Engineering, Memorial University of Newfoundland St. John s, Newfoundland, Canada, A1B 3X5 and ABSTRACT: In recent years there has been increased interest in the use of glass fibrereinforced polymers (GFRP) as reinforcement for concrete slabs because of its corrosion resistance. There are major differences between static and short-duration dynamic loads because the local effects of the impact are often over before the structure can globally react to the effects of impact. In this study, a finite element (FE) model is constructed for FRP-reinforced concrete slabs under static loading and short duration impact including blast waves (soft impact) and falling rocky block (hard impact). Based on the transient analysis, the response of the slab under localized impact is investigated. A good agreement with the test data is obtained for the static analysis using this FE model. Consequently, the FE model is applied under blast wave impact. In such case, the maximum amplitude of displacement and reaction force is proportional to the overpressure. The periods of oscillated displacement are similar to the period of the first mode in the modal analysis. However, the displacements due to high overpressure exhibit more disorderly behaviour than those with low overpressure, and the reaction force with high-pressure decays much slower than the others. The duration period of the rocky block impact is much longer than that of the blast wave. The reaction force with an impact of 5 MPa reached 577 kn, which is 2.3 times higher than the ultimate loading in the static analysis; the displacement spectrum and the reaction force spectrum show the significant difference from those of the lower impact or shorter duration pulse. Keywords: GFRP, localized impact, concrete slabs, modeling, finite element method. INTRODUCTION In recent years there has been increased interest in the use of glass fibre-reinforced polymer bars (GFRP) for concrete slabs. As one of the new promising technologies in construction, GFRP material solves the durability problem due to corrosion of steel reinforcement. There are major differences between static and short-duration dynamic loads. Typically, static loads do not produce inertia effects in the structural response, are time independent, and are assumed to act on the structure for long periods of time. Comparably, short-duration dynamic loads, such as induced by explosions, impact or debris impact, are non-oscillatory pulse loads. The structural response under short duration dynamic effects could be significantly different than static loading cases. This is due to the local effects of the impact which are over before the structure can globally react to the effects of impact. With the remarkable development of the computation technology, numerical simulations play an increasingly significant role. In this study, the finite element program ANSYS 1 was used to construct finite element (FE) model under the short duration impacts such as blast waves. Based on the transient analysis, the response of concrete 1

2 slabs reinforced with GFRP under localized impact was investigated. Different parameters were defined in the finite element model such as load function: rise time, peak load, duration, and shapes of the loading and unloading branches of the function. FE STATIC MODEL CONSTRUCTION AND VERIFICATION Modeling of Slab-Column Connections In 1982, the American Society of Civil Engineers presented an extensive review of the underlying theory and the application of the finite element method to the linear and nonlinear analysis of reinforced concrete structures in the state of-the-art report 2. It was stated that the results from the FE analysis significantly relies on the stress-strain relationship of the materials, failure criteria chosen, simulation of the crack of concrete and the interaction of the reinforcement and concrete. Because of the complexity in short- and long-term behaviour of the constituent materials of the reinforced concrete, the ANSYS finite element program 1, operating on a Windows 2 system, introduces a three-dimension element, Solid65, which is capable of simulating cracking and crushing. The model is also capable of simulating the interaction between the two constituents, concrete and reinforcement. Thus, it can be used to describe the behaviour of the composite reinforced concrete material. Although the Solid65 can describe the reinforcing bars, an additional element, Link8, was used in this study to investigate the stress along the reinforcement. The GFRP reinforcing bar is assumed to carry stress along its axis only and have the perfect bond between the concrete (Solid65 element) and the GFRP bar (Link8 element). In the static analysis, the stress-strain relations of concrete are modified to represent the presence of a crack. A plane of weakness in a direction normal to the crack face and a shear transfer coefficient t are introduced in the Solid65 element. The shear strength reduction for those subsequent loads, which induce shear across the crack surface, is considered by defining the value of t. This is important to accurately predict the loading after cracking, especially when calculating the strength of concrete member dominated by shear, such as slabs and shear walls. Based on a preliminary study, the t is defined as.2 in this study. The density ( ) and the Poisson s ratio ( ) of concrete were assumed as 23 kg/mm 3 and.2. The shear transfer coefficient for a closed crack, c, is widely accepted within.9 to 1.. In the current study, c was taken as.9. The ultimate uniaxial compressive strength (f c ) was used based on the experimental results 3. The Canadian Standard Association 4 defines the elastic modulus and the ultimate uniaxial tensile strength (f r ) as follows: ' ( fc ) γ Ec = = 284MPa 23 r '.6 c f = f = MPa (2) (1) The constitutive model for concrete is defined using the uniaxial compression stressstrain behaviour of concrete. An elastic-perfectly plastic model with concrete strain crushing limit was applied in this model. The GFRP reinforcement in the experimental study 3 is the V-ROD * bar as produced by Pultrall. The tension tests 3 of the bars showed the linear elastic behaviour of the bar 2

3 and the elastic modulus (E s ) was found to be equal to 42, MPa. The measured ultimate stress (f y ) was 63 MPa. Poisson s ratio ( ) is assumed to be.3. In this study, one quarter of the slab was modelled due to the two axes of symmetry. The boundary condition of these two edges along the symmetry lines was defined based on the symmetry of displacement. In the finite element of concrete structures, it is important to select an appropriate mesh size to meet the requirement of accuracy and computation speed. The mesh size used at a distance of 2.5 times the effective depth from the edge of load (column) was chosen as 6 mm. The other mesh size was defined as the global mesh size of 12 mm. The symmetry boundary condition and the layered mesh are illustrated in the Fig. 1. Fig.1 The FE model Newton-Raphson equilibrium iterations were used for nonlinear analysis. A displacement controlled incremental loading was applied through a column stub. This was used to simulate the actual loading used in the experimental program. Small initial load step was used for detecting the first crack in the connections. Then, automatic time stepping was used to control the load step sizes. Line search and the predictor-corrector methods were also used in the nonlinear analysis for accelerating the convergence. The failure of the connection was defined when the solution for a small displacement increment did not converge. Consequently, the finite element model was constructed following the above-mentioned assumptions and considerations. The details of model geometry are shown in the Fig.1. Verification of the Finite Element Model with the Static Analysis The constructed finite element model was used to simulate slab GS1 that was tested in the experimental program 3 conducted at Memorial University. The FE results are compared to the measured ultimate load and the associated deflection at the centre of the tested slabs. The slab was reinforced with GFRP bars. The square slab had a side dimension equal to 192 mm and a thickness of 15 mm. The effective depth of the slab was 1 mm. The reinforcement ratio was 1.18%. A concrete cover of 5 mm was used. 3

4 The connection can be regarded as an elastic plate before the crack forms. At that stage, the finite element results were compared to the results of an elastic plate analysis of the slabs 5. The results of both methods are shown in the Fig. 2. As expected, the deflection is linear, and the finite element model shows excellent correlation with the hand calculation before the cracks form in the model. The finite element model in the non-linear range was verified by comparing the predicted response of the model with the experimental results. The results are shown in Fig. 3. The results show reasonable agreement between the FE predictions and the test results. Hence, the proposed FE model was applied to analyze the behaviour of the connections under localized impact. Loading (KN) Calculated_GS1 ANSYS_GS Deflection (mm) Fig. 2: Finite Element Model Verification in the Elastic Stage Loading (kn) d GS1 ANSYS_GS Deflection (mm) Fig. 3 Load-defelction graph of GS1 FE MODEL UNDER LOCALIZED IMPACT Hard Impact and Soft Impact Krauthammer 6 carried out an extensive investigation on the behaviour of slabs under impact. The basic theory 7 on impact is briefly outlined. Consider a mass, M, impacting a structure with resistance, R(u), the equation of dynamic equilibrium, based on Newton s second law, is: ( ) M u + R u = (3) The structure has a mass, and there is an impact resistance between the mass and the structure. The equation of equilibrium becomes: ( ) M u R u u (4) = ( ) ( ) M u R u u R u (5) = where M 1, ü 1, u 1 are the mass, acceleration and displacement of the impacting body (impactor), respectively. M 2, ü 2 and u 2 are the mass, acceleration and displacement of the impacted structure, respectively. R 1 and R 2 are the impact and structural resistance, respectively. This system of equations describes the case of "hard impact" where the equations of dynamic equilibrium for the structure and impacting body are coupled. In the case of hard impact, the impactor's kinetic energy is transformed into deformation energy 4

5 in both the impactor and the structure. Furthermore, if the impactor is assumed to be rigid and is arrested by the structure, its kinetic energy is transformed into deformation energy in the structure and the penetration takes place. Generally, the displacement of the impacting mass is much larger than the structural displacement (i.e., u 1 >> u 2 ), and therefore, Equation (4) can be rewritten as: ( ) M u R u (6) = Equation (6) can be solved together with the Newton s second law: 1 ( ) ( ) Then, Equation (5) can be rewritten as follows: R t = F t (7) ( ) ( ) ( ) M u + R u = R u = F t (8) This case, where u 1 >> u 2, permits one to uncouple Equations (4) and (5), and it is defined as "soft impact". The impact forcing function, F(t), can be calculated from Equation (7) by assuming that the responding structure is rigid (i.e., u 2 = ), and then to compute the response of the deforming structure from Equation (8). The explosive waves acting on structures are close to the "soft impact" response. In the event of an explosion, stored energy is released very rapidly in the form of an audible blast, thermal radiation and expanding shock waves. When a shock wave strikes the structure, a reflection of the shock wave takes place; in addition, a stress wave will propagate through the structure. Damping Coefficient in the FE Model In dynamic equilibrium, the equations of motion for multi-degree of freedom systems are given by: [ M ] u + [ C] u + [ K ] u = F ( t) (9) where u is the relative displacement, u is the relative acceleration and u is the relative velocity, [M], [C] and [K] are the mass, damping, and stiffness matrix, respectively. F(t) represents the impact force or transient force. Since the mass matrix and stiffness are determined in the FE model based on the physical description of structures, the damping matrix, which is the unique property in the dynamic analysis, is presented as follows. Energy dissipation in the form of damping is commonly idealized in linear elastic dynamic analysis as viscous or velocity proportional. In reality, damping forces may be proportional to the velocity or to some power of velocity. The most effective means of deriving a suitable damping matrix is to assume appropriate values of modal damping ratios for all significant modes of vibration of the structure and then compute a damping matrix based on these damping ratios 8. In this study a Raleigh type mass and stiffness proportional damping of the following form is used. [ C] α [ M ] β [ K ] = + (1) where and are the constants derived by assuming suitable damping ratios for two modes of vibration. Using normal coordinate transformation of the equations of motion, the damping ratio of the n th mode is: 5

6 α ωn λn = + β 2ω 2 n where n is the circular frequency of the n th mode. For mass dependent damping n is inversely proportional to the frequency such that higher modes have little damping. On the contrary, stiffness proportional damping is proportional to the frequency of the structure and results in higher damping for higher modes thus decreasing the contribution of higher modes to the response of the structure Moreover, the equation of motion in the normal mode can be expressed as: ( ) * * * m q + c q + k q = f t where q represents the normal coordinate and (11) (12) c * = 2ξ n ω n m * c is defined as: where n is the critical damping ratio in the n th normal mode. For concrete element with light reinforcement and a few cracks, n is generally considered as a constant of 2% in all the normal modes. Before cracks form, concrete elements exhibit linear elastic behaviour, which indicates the unchanged stiffness. Therefore, in this study, the damping matrix is assumed to be only dependent on the mass matrix. According to the modal analysis used, the multiplier could be determined by a simple calculation based on the period of the first mode (Table 1): Thus, the mass matrix multiplier, * * * c = αm = 2ξ n ω n m 1 * (13) (14) 2π α = 2ξ nωn = 2 ξn ( ) = 2.2 = 1.24 (15) T.222, for damping is assumed as 1.5 in this study. Table 1 Modal periods and frequencies for model of slab-column connections Mode Frequency (cycles/s) Period (s) Mode Frequency (cycles/s) Period (s) Mode Frequency (cycles/s) Period (s)

7 Impact Force Time History The pressure time history of a blast wave can be idealized as shown in Fig. 4. The illustration is an idealization of an explosion in free air. The pressure time history is divided into a positive and a negative phase. In the positive phase, maximum overpressure rises instantaneously and decays to atmospheric pressure in a short time. For the negative phase, the maximum negative pressure has much lower amplitude than the maximum overpressure. The duration of the negative phase is much longer compared to the positive duration. The positive phase is more interesting in studies of blast wave effects on concrete buildings because of its high amplitude of the overpressure and the Fig. 4 Pressure time history of a blast concentrated impulse. Therefore, in this study, the completed blast waves were applied in the FE simulation as well as only the positive part was taken into consideration in the different duration periods. FE Modeling of Blast Wave Impact 6 2MPa Pressure A Blast wave impact simulation was 4 5MPa Pressure conducted using the model constructed and verified under static loads. The modified pressure-time history in Corley 9 was 2 adopted in this study (Fig.5). The connection was analyzed under two load conditions. The two pulses were applied on the column stub (Fig.1). Since the column -2 stub is short and strong enough so that the impact can be considered to be imposed on Time (ms) the slab. The same approach was applied to investigate the falling block impact. The Fig. 5 Pressure-time history (Corley) positive pulse had a rise time of.22 seconds and the duration period of.7 seconds. The negative pulse, one tenth of positive pulse with duration period of.63 seconds, was also considered in this study. This pulse produced an impact force to the connection. The vertical displacement of the centre of the slab (Fig.6) and the reaction force (Fig.7) were investigated. Pressure (MPa) 7

8 .9.2 Displacement (mm) MPa overpressure -.6 Displacement (mm) MPa overpressure -.2 Fig. 6 Vertical displacement with an overpressure 15 Reactio Force(KN) MPa overpressure Reaction Force(KN) MPa overpressure Fig. 7 Reaction Force with an overpressure The maximum amplitude of displacement and reaction force is proportional to the overpressure. Compared with the response of slab-column connection under static loading (Fig.3), the reaction force could reach the higher value than the ultimate loading in static loading. The displacement is much smaller than the corresponding displacement under lower loading. In addition, the periods of oscillated displacement are similar to the period of the first mode (Table 1); however, the displacement with high overpressure exhibits much more disorderly than those with low overpressure, and the reaction force due to high pressure decays much slower than the other. Besides the absolute amplitude of impact force, which is larger than the ultimate loading under static loading, both phenomena show the harm of the high overpressure on the structure. 8

9 Displacement (mm) s duration pulse -.3 Displacement (mm) s duration pulse Displacement (mm) s duration pulse -.5 Displacement (mm) s duration pulse -.6 Fig. 8 Vertical displacement with different duration pulse Furthermore, the duration period of blast wave is another significant parameter that affects the response of a structure. This response was examined under different duration periods. The results are shown in the Fig. 8 and Fig. 9. The negative pulse is much smaller than the positive pulse. The impact force with an overpressure, 2 MPa, therefore, only included the positive pulse. A rise time of the pulse is 1 6 seconds and the duration periods are in range of.5s, which is smaller than the period of the major modes, to.2 seconds, similar to the period of the first mode (Table 1). The shape of their displacement spectrums is similar, but it becomes smoother when the duration period increases. It is interesting to note that the maximum displacement amplitude with.1 seconds duration period is larger than that with.2 seconds duration periods. The exception is case of the pulse with duration period,.5, which is smaller than the period of the major modes. The corresponding maximum amplitude of reaction forces reached 28 kn that is similar to that with overpressure of 5 MPa. This demonstrates that the increase of the duration period induces a more serious effect on structures. 9

10 8 3 Reaction Force (KN) s duration pulse -12 Reaction Force (KN) s duration pulse -3 Reaction Force(KN) s duration pulse -3 Reaction Force (KN) s duration pulse -3 Fig. 9 Reaction Force with different duration pulse FE Modeling of Falling Rocky Block Impact The impact pulse-time history of the rocky block impact is similar to that of the blast wave. However, its duration period is much longer, which is illustrated in the typical force-time history in the rocky block impact experiment 8. Also, because of the rigidity of rock, the penetration will take place if rocky block impact the concrete structure with high kinetic energy. Due to the high complexity of this incident, this response was not considered in the current study. The pressure-time history used in the FE simulation is shown in the Fig. 1. The related responses are given in the Fig. 11. The reaction force with an impact of 5 MPa reached 577 kn, which is 2.3 times Time (ms) Fig. 1 Rocky impact pressure-time history higher than the ultimate loading under the static analysis. The displacement spectrum and the reaction force spectrum show the significant difference from that of lower impact or shorter duration pulse. This feature represents the impact force reaching the limit of the capacity of the structure. Pressure (MPa) MPa Pressure 5MPa Pressure 1

11 On the other hand, the periods of oscillated displacement under the impact of 5MPa, around.6 seconds, is much larger than the period of the first mode in the modal analysis. This can be accounted for the appearance of too many cracks, which impairs the slab stiffness dramatically. The different behaviour of the structure under 5 MPa impact was further analysed. The high strain,.34, and the full development of crack indicated that the structure would be failed soon. Displacement(mm) MPa impact -.6 Reaction Force (KN) MPa impact -2 Displacement (mm) MPa impact -3. Reaction Force (KN) MPa impact -3 Fig. 11 Displacement / Reaction Force with an impact of 2 MPa / 5 MPa SUMMARY AND CONCLUSION (1) A full-size slab-column connection model, with proper boundary conditions, was constructed using ANSYS. A static analysis was conducted first to verify the accuracy of the model. The concrete constitutive model included the elastic-perfectly plastic model, crack condition and crush limit. GFRP reinforcement was defined as a linear elastic material. A layered mesh and an appropriate shear transfer coefficient were used. Good agreement with the test data was obtained. (2) Under blast wave impact, the maximum amplitude of displacement and reaction force is proportional to the overpressure. The periods of oscillated displacement are similar to the period of the first mode in the modal analysis; however, the displacement due to high overpressure exhibits much more disorderly than those with 11

12 low overpressure, and the reaction force with high pressure decays much slower than the others. (3) When the blast wave with different duration period was investigated, the increase of the duration period induced a more serious effect on the connection. In addition, the maximum displacement amplitude with.1 seconds duration period is larger than that with.2 seconds duration periods. (4) The duration period of the rocky block impact is much longer than that of the blast wave. The reaction force with an impact of 5 MPa reached 577 kn, which is 2.3 times higher than the ultimate loading in the static analysis; the displacement spectrum and the reaction force spectrum show the significant difference from those of the lower impact or shorter duration pulse. This feature represents the impact force is reaching the limit of the capacity of the structure. REFERENCES 1. ANSYS, ANSYS User s Manual Revision 5.5, ANSYS, Inc., Canonsburg, Pennsylvania, ASCE Task Committee on Finite Element Analysis of Reinforced Concrete Structures. State-of-the-Art Report on Finite Element Analysis of Reinforced Concrete, ASCE Special Publications, Rashid, M., The Behaviour of Slabs Reinforced with GFRP, master thesis, Faculty of Engineering, Memorial University of Newfoundland, St. John s, Canada, CSA. Design of concrete structures. Standard CSA-A , Canadian Standards Association, Rexdale, Ont, S. Timoshenko and S. Woinowsky-Krieger, Theory of Plates and Shells. MCGraw- Hill, Inc., New York, Krauthammer, T., Structural Concrete Slabs Under Impulsive Loads, Fortifikatorisk Notat Nr 211/93, Norwegian Defence Construction Service, Krauthammer, T., and Zineddin. M., "Structural Concrete Slabs under Localized Impact," Proc. 9th International Symposium on Interaction of the Effects of Munitions with Structures, Berlin, Germany, 3-7 May Clough, R.W. and J. Penzien, "Dynamics of Structures," Second Edition, McGraw-Hill Book Company, Corley, W. G. and Oesterle, W. G., Dynamic Analysis to Determine Source of Blast Damage, Abnormal Loading on Structures edited by K. S. Virdi, R. S. Mathews, J. L. Clarke and F. K. Garas. Published in 2 by E & FN Spon, 11 New Fetter lane, London EC4P 4EE, UK, 2. 12