DAMAGE AND ENERGY DISSIPATION IN CONCRETE UNDER CYCLIC UNIAXIAL COMPRESSIVE LOADING IN QUASI STATIC TESTS

Transcription

1 DAMAGE AND ENERGY DISSIPATION IN CONCRETE UNDER CYCLIC UNIAXIAL COMPRESSIVE LOADING IN QUASI STATIC TESTS E. Schwabach, E. Raue, H. G. Timmler Bauhaus University Weimar, Germany 1. Introduction Concrete is a construction material, that s nonhomogeneous because of its mix design, its manufacturing process and its structure formed during curing. In general concrete is modeled as an homogeneous, ideal material. This is the aspect in the macro level of the concrete. A closer inspection of the structural changes or structural damage respectively, leads at least to a view into the submacroscopic level or meso level of concrete. In that level the concrete is considered as a twocomponent material. For that the coarse concrete aggregate (size ~> 4 mm) is embedded in an homogeneous matrix consisting primarily of the cement paste with its pore system and the embedded fine gravel grading. Even before loading, concrete contains some flaws or pore cracks or rather micro cracks due to the hardening process. In compression, the fracture mechanical properties of concrete are governed by those of its two components, cement paste and aggregate, and the composite behavior in the contact zone between aggregate and matrix. The age dependent properties (increase in strength and elastic modulus etc.) are primarily attributable to changes in the matrix component caused by changes in its hydration state. The strength of concrete is to a large extent dependent upon that of its cement paste fraction. In various research papers [1, 2, 3, 4, 5], the structural changes occurring during the loading of concrete have been described with the compaction of particles in the mortar matrix (plastic deformation) and the propagation, growth and union of micro cracks between aggregate and matrix and in the matrix itself (quasi-plastic deformation). Sporadically there are cracks through the coarse concrete aggregate too. The change in initial elastic modulus (progressive reduction in stiffness) after several unloading and reloading events is due to fact, that micro cracks can t close completely after unloading events [6, 7; cf. Fig. 1]. Sinha, Gerstle und Tulin 1964 [6] Bahn und Hsu 1998 [7] Fig. 1: Cyclic compression tests out of reference papers The progressive formation and union of micro cracks to larger cracks produces the fracture mechanism ultimately. Plain concrete usually fails in a brittle manner, caused by multiple fracture surfaces. It s plausible to assume that cracking of some sort and significance occurs at all stress levels and that damage is progressive. A quantitative evaluation of damage occurring in concrete during loading and unloading depending on the applied stress level is useful. Especially for the serviceability limit state a quantitative assessment of the damage is more important in the pre peak range than in the post peak range of the stress strain curve. Two methods of doing this are described in this paper. In one, the change in stiffness of the concrete, as measured by the elastic modulus for reloading to a particular stress, is used as a measure of damage. In the other, damage is assessed on an energy basis. 2. Elementary experimental analysis The behavior of concrete is dependent upon its load history. The nonlinear nature of the stress strain relationship of plain concrete subjected to a series of cycles of uniaxial compressive loading (increasing strain) and unloading (decreasing strain) is shown in Fig. 1. In general the envelope curve for a concrete subjected to cyclic axial compression coincides well with the stress strain curve under monotonically increasing strain, especially in the ascending part. Moreover the cyclic test delivers information about the structural changes occurring com-

2 bined during compressive loading. The change of stiffness in the reloading stress strain path and unloading plastic (residual) and quasi plastic deformations are the major parameters that control general behavior of concrete under cyclic compressive loading. For investigating the pre peak range of the stress strain curve it s favorable to perform the loading pattern under load control. To receive detailed information of the structural behavior up to failure, the applied load level should be gradually increased by small, constant load steps (~ 5 % cal F u ) [8]. The principle of the interaction between cyclic quasi static loading and structural response of the concrete is shown in Fig. 2. consist of the recovered elastic deformation el and the reversible short-term creep deformation cr,2 3. stress (σ = F/A) 1 cr,1 2 el E c pl, 1 + cr, F Force (F) Force (F) cr,2 3 el F 4 3 strain () 4 = pl 2 3 cr, time (t) 4 3 deformation () Fig. 2: Load cycle and response to several cyclic loading The structural response is greatly affected by the loading rate and loading last term. It s commonly thought that when loading periods last over 5 seconds, the measured strain has already been affected by creep, the so called long term deformation of concrete [9]. In practice, the experiment can not mostly be achieved successfully in such a short time. It leads to the result that the short term (instantaneous) and long term (time dependent) deformation are always mixed. Fig. 3 shows a typical short term stress strain curve of concrete in case of loading, loading last term and full unloading, in which some important deformation characteristics are exhibited. The deformation on loading: 1 = el + pl, 1 + cr, 1 = 1 (1) contains elastic strain el (recoverable deformation), plastic strain pl, 1 (irrecoverable deformation) and short term creep deformation cr, 1. During unloading, the deformation: 2 3 = el + cr,2 3 (2) Fig. 3: Deformation characteristics during a full loading event Defining cr, 1 = cr,2 3, the plastic deformation not due to short term creep may be written as: pl, 1 = = pl,d (3) In case of a constant loading rate and constant loading last and unloading last term respectively, cr,3 4 is the reversible component cr,el of the time dependent creep deformation cr,1 2. The plastic and quasi plastic deformation after total unloading last term consisting of irreversible short term creep deformation cr,pl or structural changes respectively caused by plasticity and micro cracking is: pl = 1 + cr, cr,3 4 = pl,d + cr,pl (4) The plastic proportion cr,pl, due to irreversible short term creep then can be expressed as follows: cr,pl = cr,1 2 cr,3 4 (5) To receive a quantitative assessment of damage occurring in concrete during loading it s reasonable to use an energy basis, because it completely represents the interaction between imposed force or load and structural re-

3 sponse of the concrete. Similar to Spooner and Dougill [3] the total energy or work done during a loading unloading cycle: 1 E = E d + E cr + E el = σ1( ) d + σ ( 2 1) (6) load level as well as unloading to near zero load level (Fig. 5) is very useful. The registered structural changes on the next higher load level then is only due to the applied load increase [8]. F n can be separated into three components (Fig. 4). The energy dissipated in damage or rather structural changes (micro cracking and plasticity) is represented by: 1 5 E d = σ1 ( ) d + σ ( 5 1) σ2 ( ) d (7) 4 I.e. there is a directly relation between structural changes and parallel dissipated energy. E cr corresponding to the energy dissipated in creep, and E el representing the recovered elastic component of the total energy E. F 2 F 1 time (t) Fig. 5: Loading pattern for investigations of the energy dissipation [8] To assess the structural damage effected of forces it s useful to separate the energy dissipating mechanisms or the energy E d respectively dissipated by the two mechanisms. An approximate solution for that problem is described in the next section. stress (σ) Energy dissipated in damage (E d ) Energy dissipated in creep (E cr ) 3. Quantitative assessment of the structural changes The total energy dissipated in structural damage E d is composed of the energy dissipated in micro-cracking E d,cr and the energy dissipated in plastic behavior E d,pl : σ 1 () σ 2 () 4 3 Elastic energy (E el ) strain () Fig. 4: Division of the total energy involved during a loading unloading cycle The experimental results for or E respectively are greatly affected by the loading rate and loading last term, e.g., 1 doesn t reach the same value as that when the loading rate is lower or cr,1 2 reaches a higher value as that when the loading last term lasts shorter. So for carry out an experiment it s very important to have a constant, equal loading and unloading rate (~ 1/24 cal F u [kn/s]) and constant, equal loading and unloading last terms (3 15 min). To achieve a steady damage state on a load level (stable equilibrium), a several repeated loading up to that E d = E d,cr (σ, 1 ) + E d,pl (σ, pl ) (8) The progressive reduction in stiffness after unloading and reloading, indicated by the reduced initial slope of the reloading curve, corresponds to structural damage due to micro cracking. Thus the variation of an initial secant modulus E c (cf. Fig. 3) provides an assessment of the relative damage during loading. The change in initial secant modulus E c on the same load or rather stress level can be written as (cf. Fig. 6): k Ec = 1 1i,2 / 1i,1 resp. (9a) k Ec = 1 1i,3 / 1i,2 between two following stress levels as: (9b) k Ec = 1 E c(i+1),1 / E ci,3 with: (1) E cj,1,2,3 = σ j / 1j,1,2,3 (11)

4 The energy dissipated in internal micro cracking can be approximately estimated as shown in Fig. 6. Thus the energy dissipated in the first cycle (primary cycle) of a stress level σ i can be expressed as: E d,cri,1 = ½ σ i ( 1i,2 1i,1 ) (12) It is noticeable that this approach provides a quantitative view of the damage occurring during loading caused by the individual mechanisms micro cracking and plasticity. The next section deals with the evaluation of experimental results of concrete subjected to cyclic uniaxial compressive loading using the described techniques. 4. Analysis of cyclic loading tests stress σ i+1 σ i E d,cri,1 E d,cr(i+1),1 The following test evaluation is restricted to the recorded data of a test series performed on three plain concrete cylinders undertaken in connection with [8]. The dimension of every specimen is given in Table 1. Table 1: Dimensions of the normal concrete cylinders Specimen Mean diameter d m [cm] Height h [cm] E ci,(i+1),1,2,3 Z 1 9,71 29,75 1i,1 strain Z 2 9,66 29,8 1i,2 Z 3 9,68 29,85 1i,3 1(i+1),1 1(i+1),2 Fig. 6: Energy dissipated in micro cracking The value pl represents the plastic and quasi plastic deformation in the mortar matrix in the form of compaction and residual cracking effects. The energy dissipated for that can be approximated as shown in Fig. 7. For the primary cycle of the stress level σ i it s found to be: E d,pli,1 = ½ (σ i + σ i 1 ) pli,1 (13) stress σ i+1 σ i σ i 1 E d,pli,1 pli,1 E d,pli,2 E d,pli,3 E d,pl(i+1),1 pli,2 pli,3 pl(i+1), E d,pl(i+1),2 pl(i+1),2 strain Fig. 7: Energy dissipated in plastic and quasi plastic effects The concrete mix proportions are constant for all three test specimens. The compressive strength was tested at age 128 days on six 15 mm cubes of the same batch. The mean value of the peak stress (characteristic strength), converted to 15/3 mm cylinders, was f cm = 47 N/mm². Considering EC 2, the concrete may be classified to strength class C 35/45 (f cm = 43 N/mm², E cm = 335 N/mm²). The concrete specimens were tested under several cyclic uniaxial compressive loading and unloading as shown in Fig. 5. All tests were performed under load control up to failure of the specimens to study the pre peak range of the stress strain curve. The tests were carried out by a servo-controlled material testing system with a hydraulic capacity of 63t. To measure the deformation during testing, in the middle range three strain gauges were applied round the concrete cylinder. Considering the slenderness ratio of the test specimens (h/d m 3) in this range an uniaxial stress distribution is acceptable. During the test a basic load of 25 kn remained to be applied. The loading and unloading rate used in the tests is 1,5 kn per second. The loading and unloading last term was constant 3 min. Two repetitions of load cycle on every load level were performed. The differential load or force respectively between the load steps was constant 25 kn.

5 Fig. 8 illustrates the relation between force and longitudinal compressive strain for specimen Z 1, that s similar for all specimens. Due to carrying out the tests under force control, the plain concrete cylinders failed in a brittle manner. Force F [kn] strain [µm/m] Fig 8: Force (longitudinal compressive) strain curve (Z 1) that calculations for all specimens is shown in Fig. 11. In the current case the discrepancy of the total energy dissipated up to failure in comparison to Fig. 9 is on average 2,62 %. 5,5 5, 4,5 4, 3,5 3, 2,5 2, sum epl,1 1,5 e1,1 1,,5 e1,2, Σ pl,1 ; 1,1 ; 1,2 [µm/m] Fig. 1: Deformation characteristics up to failure (Z 1) 8 Using the recorded data, the total energy of structural changes or damage respectively E d (see Fig. 4) was incremental analyzed for every primary cycle of the loading pattern. The development of E d for every specimen depending on the applied stress level up to failure is shown in Fig. 9. E d,cr & E d,pl [kn/cm² µm/m] Ed,pl(Z-1) Ed,pl(Z-2) Ed,pl(Z-3) Ed,cr(Z-1) Ed,cr(Z-2) Ed,cr(Z-3) E d [kn/cm² µm/m] Z-1 Z-2 Z-3,,5 1, 1,5 2, 2,5 3, 3,5 4, 4,5 5, 5,5 Fig. 9: Total energy of structural damage up to failure Fig. 1 illustrates the deformation 1 [cf. Fig 3, Eq. (1)] of every primary loading curve ( 1,1 ), following loading curve ( 1,2 ) and the sum of the plastic deformation pl after every primary cycle ( pl,1 ) related to the applied stress level up to failure for specimen Z 1 (similar for all specimens). Using Eq. (12) and Eq. (13), the proportions of the individual dissipative mechanisms in the total absorbed energy E d can be approximately determined. The result of Fig. 11:,,5 1, 1,5 2, 2,5 3, 3,5 4, 4,5 5, 5,5 Proportions of the energy dissipated in damage up to failure Fig. 11 distinctly shows, that until failure of the specimens in essence energy is dissipated by mechanisms due to plastic effects (E d,pl ). The proportion of E d dissipated by micro crack formation and growth (E d,cr ) is with an average value of 17,3 % comparatively small, but not negligible. Because it usually indicates the failure or rather fracture mechanism of plain concrete subjected to compressive loads. The secant stiffness or secant modulus E c [Eq. (11)] respectively for every primary cycle depending on the applied stress level up to failure for every specimen is illustrated in Fig. 12. The initial value of E c on average correlates relatively well with the normative value (E cm = 335 kn/cm²) corresponding to EC 2, despite another definition and different slenderness ratio of the test specimens. However, the nearly linear degradation in stiffness with increasing stress level is more important. In the current

6 case the E modulus reduces average 22,2 % up to failure of the specimens, i.e. on an average 1,48 % per load cycle (present 15 6,25 % of the ultimate load). E c [kn/cm²] Z-1 34 Z Z ,,5 1, 1,5 2, 2,5 3, 3,5 4, 4,5 5, 5,5 Fig. 12: Development of the secant stiffness E c up to failure 5. Conclusions A technique is described which gives a quantitative assessment of the damage mechanisms occurring during the cyclic compressive loading of plain concrete. Damage is indicated by the change in initial E modulus and of energy dissipated in damage. Most attention being given to the energy method. Test evaluations show that the processes associated with damage are continuously effective during increasing applied stress level. A small share of the energy dissipated in damage is due to effects related to micro cracking. Up to the failure basically energy is dissipated by effects due to the plastic behavior of the concrete. The initial E modulus or rather stiffness of the investigated specimens gradually falls up to failure. Thus it is no constant as supposed in standards. A caused damage on a certain stress level effects an degradation of the initial stiffness on reloading. This reduction in stiffness is basically caused by micro cracking. This paper is restricted to concrete behavior in uniaxial compression. Generalization to other systems of loads is possible and may be useful. 3. Spooner, D.C.; Dougill, J. W.: A quantitative assessment of Damage sustained in concrete during compressive loading, Magazine of Concrete Research, 27(1975), No. 92, p Spooner, D.C.; Pomeroy, C.D.; Dougill, J. W.: Damage and energy dissipation in cement pastes in compression. Magazine of Concrete Research, 28(1976), No. 94, p Wischers, G.: Aufnahme und Auswirkungen von Druckbeanspruchungen auf Beton. beton 28 (1978) No. 2, p und No. 3, p Sinha, B.P.; Gerstle, K.H.; Tulin, L.G.: Stress Strain Relations for Concrete Under Cyclic Loading. ACI Materials Journal, Vol. 61, Febr. 1964, p Bahn, B.Y.; Hsu, C.T.T.: Stress Strain Behavior of Concrete under Cyclic Loading. ACI Materials Journal, V.95, No.2, March April 1998, p Bolle, G.: Zur Bewertung des Belastungsgrades biegebeanspruchter Stahlbetonkonstruktionen anhand von Last Verformungs Informationen. Dissertation 1999, Bauhaus Universität Weimar. 9. Hui, Rongyan, Huang, Guoxing and Yi, Bingre: The Creep of Concrete (in chinese) Peking: Railway Press, 1998 Įteikta Enrico SCHWABACH. Dipl.-Ing. Department of Rein-forced Concrete Structures. Bauhaus University Weimar, Marienstraße 13, D Weimar, Germany.... Erich RAUE. Prof. Dr.-Ing. habil. Department. of Reinforced Concrete Structures. Bauhaus University Weimar, Marienstraße 13, D Weimar, Germany.... Hans Georg TIMMLER. Dr.-Ing. Department. of Reinforced Concrete Structures. Bauhaus University Weimar, Marienstraße 13, D Weimar, Germany. References 1. Spooner, D.C.: The stress-strain relationship for hardened cement pastes in compression. Mag. Of Concr. Res. 24 (1972) No. 79, p Spooner, D.C.; Pomeroy, C.D.: Energy dissipating processes in the compression of cement paste and concrete. Cem. And Concr. Res. 3 (1973) No. 4, p