Abstract. 1 Introduction

Transcription

1 Ultimate deformation capacity of reinforced concrete slabs under blast load J.C.A.M. van Doormaal, J. Weeheijm TNO PrinsMaurits Laboratory, P.O. Box 45, 2280 AA Rijswijk, The Netherlands Abstract In this paper a test method to determine the deformation capacity and the resistance-deformation curve of blast-loaded slabs is described. This method was developed at TNO-PML. The method has been used to determine the ultimate deformation capacity of some simply supported reinforced concrete slabs in order to increase the knowledge of this parameter. The influence of the following parameters on the dynamic deformation capacity has been studied: the thickness of the slab, the amount of bending reinforcement and the diameter of the bending reinforcement. Preliminary results have been obtained. 1 Introduction Reinforced concrete is often used for structures that must be able to resist extreme transient loadings, such as explosions and impact. The strategy in designing such a structure is to let the structure absorb the offered energy through plastic deformation, i.e. bending deformation. Therefore, it is essential to have knowledge of the dynamic deformation capacity of reinforced concrete. The deformation capacity together with the resistance against deformation quantifies the energy absorption capacity of the structure. The deformation limits that are prescribed in design manuals, such as TM % have been observed to be conservative in many cases. A certain degree of conservatism is of course inevitable for design rules which must be generally applicable. But here, the conservatism is mainly due to a gap in the knowledge of the dynamic deformation capacity of reinforced concrete. More knowledge of the dynamic deformation capacity is necessary for more economical designs or for realistic estimates of the protection level of a structure.

2 384 Structures Under Shock And Impact In order to increase the insight into the deformation capacity of reinforced concrete slabs under a blast load, a test method has been developed at TNO- PML. It has been applied in an experimental programme. 2 Test method The principle of the test method is based on a theoretical view on the deformation process using the single-degree-of-freedom approach (see Biggs^). The mechanics of the SDOF system prescribe that at any time the sum of the inertia forces (/ = M^a) and the resistance force (R) must be in balance with the applied load (P): P = I + R (1) This implies that the inertia force must suddenly increase when the resistance falls away suddenly; or, which is the same, that the acceleration must suddenly increase. This is shown in Figure 1 for a slab with an idealised linear elasticpure plastic resistance against bending. Acceleration a=p/me\ Time Figure 1: Acceleration of centre of an ideal slab fi At time f = fo, a triangular blast load reaches the slab and is reflected. Due to the peak load P, the slab experiences an acceleration of P/Mg in the centre, where Mg is the representative mass for the bending motion. The load decreases linearly with time until it reaches zero at time f/. Simultaneously, the slab deflects and builds up a resistance against this bending. Because of equilibrium of forces, the acceleration must decrease. At time t = f/, the slab fails. The resistance falls away. This manifests itself in a sudden increase in the acceleration, because the inertia forces must take over from the bending resistance. There is more to be observed in Figure 1. There is a kink in the curve at time t = tp. This kink corresponds with the transition from elastic to plastic bending behaviour. Since the resistance of the slab does not increase anymore when it has reached the plastic phase, the acceleration decreases less rapidly. Two distinguishing marks in the acceleration signal are defined, which characterise respectively the transition from elastic to plastic behaviour and the failure of the slab. These marks would be good and objective criteria to

3 Structures Under Shock And Impact 385 determine the specified moments. So, measuring simultaneously the acceleration and the deflection of a slab, would be sufficient to find the maximum elastic deflection and the ultimate deflection. The acceleration signal specifies at which two points in time the deflection must be read. The method has been extended, because the possibility that the concept would not work was thought realistic. A different structural behaviour than assumed in the concept might ruin the application of the theory. The structure might lose its resistance gradually instead of suddenly. Then there will not be a jump in the acceleration. Likewise, the transition from elastic to plastic behaviour is generally more gradual with an intermediate elastic-plastic phase. It will not manifest itself by a kink in the acceleration signal. The concept has been extended such that the complete resistancedeformation curve can be determined. Eqn (1) can be rewritten as R = P-M,a (2) All the parameters on the right-hand side of eqn (2) are known or can be measured. The load and the acceleration can be measured. The effective mass Mg follows from the deformation shape and the mass of the tested slab. So, it is possible to find the resistance as a function of time. Next, by relating this resistance to the deflection the resistance-deformation curve can be obtained. This extension improves the benefits of the test method and makes it more robust. The opportunity to follow the complete resistance-deformation behaviour, makes the initially stated requirement of a leap in the acceleration redundant, because it can be observed when the resistance has reduced to zero. Moreover, much more information on the slab behaviour is obtained with only a little more effort, such as the maximum resistance and the absorbed energy. 3 Experimental programme The method was used in an experimental programme to learn more about the dynamic deformation capacity of reinforced concrete. Simply supported oneway slabs were tested in the 2 m diameter blast simulator at TNO-PML. The slabs were loaded by a shock wave above their capacity and they failed in bending. Parameters that were varied in the tests are the thickness of the slabs, the amount of reinforcement and the diameter of the reinforcement. The slabs had a length of 1.2 m, a support length of 1.1 m and a width of 0.85 m. The slabs were symmetrically reinforced (FeB 500). No shear reinforcement was used, because it was not expected that the slabs would fail due to shear. The reinforcement in the transverse direction was the same for all slabs, 6 mm bars at an in-between distance of 150 mm. The other data on the slabs are given in Table 1. During the test, the load, the acceleration and the deformation were measured simultaneously. The acceleration and the deformation were measured at four different positions, so that the distribution over the slab is known. The points of measurement, which are all on the unloaded side of the slab, are given

4 386 Structures Under Shock And Impact in Figure 2. The load was not measured on the slab itself, but just underneath it on the mask that surrounds the slab and closes the blast simulator. This is sufficient because of the uniform distribution of the pressure over the crosssection of the blast simulator. Table 1: Thickness [m] Reinforcement ratio [%] Diameter of rods [mm] In-between distance [mm] Number of rods Cube strength concrete [MPa] E-modulus of concr. [GPa] Number of slabs Tested slabs slab 1.1 slab slab slab Al A2 j A3 A4 * * m-f * + * - - Dl D2j D3 D ', I g _.. _ 190» 1 Figure 2: Points of measurement ; 4 Application of test method To show how the test method can be applied and how it meets its requirements, an example is given here. A slab of type 2.1 was chosen for this example. The chosen slab was loaded by a shock wave with a peak pressure of 160 kpa and a duration of approximately 90 ms. This load caused the slab to fail in the midsection. The compression zone was crushed and the tensile reinforcement was broken. Figure 3 shows the accelerations measured in locations A2 and A3. It must be remarked that these accelerations are the sum of the contribution of a rigid motion and that of a bending motion. A rigid motion is imposed on the slab by the blast simulator, which also moves under the shock load.

5 Structures Under Shock And Impact Figure 3: Acceleration in A2 and A3 i * * ' ' r Time [m/s] The acceleration signals show that the test concept works for the simply supported slabs of reinforced concrete. Even the rigid motion does not obscure the distinguishing mark. A sudden increase is clearly visible at t«36.8 ms. This is the moment of failure. The leap cannot be attributed to the rigid motion, which should gradually damp and not show a discontinuity. A kink, which should point out the change from elastic to plastic behaviour is not visible; that mark is probably not present because there is no sudden change in behaviour. In order to obtain the resistance-deformation curve of the bending behaviour of the tested slab, the contribution of the rigid motion in the measured signals must first be eliminated. This can be achieved by splitting up the displacement and the acceleration into three parts: the contribution of the rigid motion, dr or a^ the contribution of elastic bending, d^ or &<,,; and the contribution of plastic bending, dpi or a/,/. The elastic and plastic variables at each location are related to each other by respectively the elastic and the plastic deformation shapes 0. Mathematically this is described by dw = <4 + ^W + dp/w = ^ + ^,/M-4/(0)4-0^W-^(0) (3) for the displacement, where x is the distance from the centre of the slab. For the acceleration, a similar relation can be written down. For each measuring point, such a relation can be formulated. Since there are four measuring points, this will result in a set of four equations with three unknown quantities (three independent equations for the acceleration). This set of equations can be solved with the method of least squares if the elastic and plastic deformation shapes are known. It was observed that the global deformation can well be described with the theoretical (= static) deformation shapes, which are given by

6 388 Structures Under Shock And Impact - 'rh-lr'*' (4a) Lj is the support length of the slab. So, these shape functions can be used to split up the measured signals and eliminate the contribution of the rigid motion. It must be remarked that the real bending behaviour of the slab is different than the global behaviour. High modes are present in the response of the slab. These high modes cause deviations in the calculated (elastic and plastic) bending deflections. So, the results must be interpreted carefully. From the sum of the two bending deflections, the ultimate deflection can be read at the point of time that has been indicated by the acceleration signal as the moment of failure. In this phase, the first mode is dominant and a good estimate of the deformation capacity is obtained. The resistance can be obtained by subtracting the inertia forces from the load. This last parameter is equal to the product of the pressure and the loaded area of the slab. The total inertia force is equal to the sum of each part of the acceleration multiplied by the corresponding effective mass. Having found the resistance as a function of time, it can be coupled to the deflection in the centre of the slab and the resistance-deformation curve is obtained. The result for the sample slab is shown in Figure 4. (4b) [kn] [m] Figure 4: Resistance-deformation curve The curve may look a little chaotic, and there is the remarkable aspect of an initially negative deflection. Both aspects are due to the presence of high frequency modes in the response of the slab, whereas the results are treated as if the slab responded in the first mode. The high frequency modes are most

7 dominant in the initial phase of the deformation process. This manifests itself in the lacking behind of the centre of the slab. Consequently, initially a negative deflection is found in the centre (see Figure 5, which shows the shapes of the slab at several points in time). In later stages, the first mode is dominant. The high frequency modes cause a vibration around the first mode response. The global behaviour of the slab can be well recognised. Only the elastic phase is disturbed too much by the high frequency modes. The maximum resistance and the ultimate deflection of the slab can easily be read from the curve. The energy that has been absorbed by the slab can be found by calculating the area underneath the resistance-deformation curve. 5.50E X-coordinate along slab [m] -2.00E E-02J 3.00E E E E-02 Figure 5: Deformed shape of slab 5. Discussion of test results With the above-described test method, the maximum resistance, the ultimate deflection and the absorbed energy has been determined for each slab. The average results are summarised in Table 2. Instead of the ultimate deflection d^ the maximum support rotation 0% is given. These two parameters are directly related to each other by tan(0^) = 2d» /L,. Slab type Table 2: Summary of test results Max. resistance Max. support rotation [kn] n / Absorbed energy [kj] 1.3/ The results, in 0^, show only a small amount of scatter (within ± 7%) for each slab type, except for slab type 1.1, which failed in two different ways; that is the reason for the two values in the table. In the following discussion, the value of 5.7 will be used, because it corresponds with the same failure mode as that of the other slabs.

8 390 Structures Under Shock And Impact The maximum resistance that is found is no point for discussion. The maximum resistance almost corresponds with what was expected based on TM '. Comparison of the maximum support rotation of the different types of slabs is more interesting and shows how the deformation capacity was qualitatively influenced by the test parameters. Comparing the result of slab type 1.1 with that of slab type 1.2 shows that the reinforcement ratio is only of minor importance for the deformation capacity. The difference between slab types 2.1 and 2.2 shows that the diameter of the reinforcement has a strong influence on the deformation capacity of the slab. The slab with the thicker rods could sustain a much larger deformation before it failed. Comparison of the results of slabs 1.1 and 2.1 shows that a thicker slab has a smaller deformation capacity. Can we understand these observations? The reduced deformation capacity of the thicker slab 2.1 compared to slab 1.1 can easily be explained by the larger strains and therefore the larger stresses in thicker slabs for a certain bending deformation. The strains and stresses are larger in the concrete as well as in the reinforcement. A similar explanation can be used for the irrelevancy of the reinforcement ratio. At a certain deformation, the strains and stresses in the two slabs are the same. The strong influence of the diameter on the deformation capacity cannot be explained that simply. In equally thick slabs, the strains and stresses at a certain bending deformation are of the same order. Yet, it has been observed that the slabs fail at a completely different deformation. To explain this phenomenon, it is necessary to know how the slab failed. It is believed that failure of the slab started in the compression zone. Here the concrete was crushed. At the moment of crushing, the load is transferred to the reinforcement. The deformation process then goes on. The problem however with small bars under a compressive load is that they can buckle, especially when with ongoing deformation, more concrete in the compressive zone crushes and spalls. And that is what has occurred in our opinion. The tested slabs failed due to buckling of the compression reinforcement. This assumption is supported by the observation that some bars had twisted. Buckling being the cause of failure explains why the slabs with the thicker reinforcement bars had a much larger deformation capacity. The equation for the resistance against buckling, ow*/^, is given by where E is the Young's modulus of the bar material, 0 is the diameter of the bar and i is the buckling length. The compressive stress in the bars is the same for all slabs, namely equal to the yield stress. This follows from the equations of balance. From eqn (5) it then follows that the free length of the bars (the amount of spalled concrete)

9 Structures Under Shock And Impact 391 can be much larger in the slabs with the thicker reinforcement rods. Therefore, slab type 2.2 failed first at a much larger deformation. With the hypothesis that buckling is the cause of failure, the results can be studied quantitatively. Eqn (5) describes mathematically when buckling occurs, i.e. at the moment that the free length of the compression bars exceeds the buckling length I. So, failure starts when over a length t the concrete has crushed and spalled. Crushing of concrete starts when a certain compressive strain, ^.cm, is exceeded. Since the same concrete quality is used for all slabs, this critical compressive strain is the same for all slabs. With the use of the elastic deformation shape and with the assumption of a linear distribution of strains over a cross-section, the following relation can be derived for the maximum compressive strain in the cross-section at location jc (6) Here T is the thickness of the slab and K is the quotient between the maximum tensile strain and the compressive strain. By substituting x = 1/2 and e^m for in eqn (6), a relation is found for the ultimate deflection of the slab: d,. - AT H2I- 27 (7) From this relation it can be read that the ultimate deflection is inversely proportional to the thickness of the slab and proportional to the square of the buckling length t. This parameter in its turn is proportional to the diameter of the reinforcement, as is shown by eqn (5). So the ultimate deflection should be proportional to fift. Figure 6 shows that the test results conform to this theoretical hypothesis. The results of the tests almost lie on a straight line Figure 6: Maximum support rotation as a function of

10 392 Structures Under Shock And Impact The observation that the ultimate deflection is proportional to $IT cannot be used for all slabs. It is only valid for slabs that will fail in the compression zone. Furthermore, not all possible parameters of influence are studied. In order to increase the applicability of the present results, it should be investigated and defined when slabs fail in compression, and the influence of other parameters should also be taken into account. 7 Conclusions A test method has been described with which the resistance-deflection curve of a reinforced concrete slab can be determined. It is shown that the test method works well. Especially the behaviour at failure can be determined very well. With the test method, several slabs were tested in order to find their maximum deformation capacity. Analysis and comparison of all results led to the following conclusions: failure was initiated by crushing in the compression zone; the deformation capacity decreases with the thickness of the slab; the reinforcement ratio has a slight influence on the deformation capacity; the diameter of the reinforcement has a significant influence on the deformation capacity; the deformation capacity seems to be related linearly to the parameter <f?/t. So, this parameter (Figure 6) might provide a simple method to estimate the deformation capacity of a slab. Since this relationship is only based on a small number of tests on an even smaller number of slab types, the validity of the prediction method is limited. It must be stated that extension of the present results to slabs that do not fall within the category of tested slabs (other sizes, other concrete quality, other kind of steel) is not allowed. Other slabs might fail in another mode (for instance due to failure of the tensile reinforcement), then the present observations are no longer valid. The present results however are a good starting point for further investigation into the deformation capacity of reinforced concrete. Acknowledgement Acknowledgement is made to the Dutch Minister/ of Defense for funding this research and for permission to publish this paper. We also thank Ir. D. Boon from DGW&T/CD/TB for his contribution in the discussion of the results. References 1. US Department of the Army, the Navy and the Air Force, TM 5-] 300, Structures to resist the Effects of Accidental Explosions, November J.M. Biggs, Introduction to Structural Dynamics, McGraw-Hill Inc., 1964.