textbild PDF MATERIAL PROPERTIES OF HSC UNDER TRIAXIAL COMPRESSION Andreas Rogge Lehrstuhl für Massivbau, Technische Universität München, Germany The paper presents standard triaxial tests on high str[.]
MATERIAL PROPERTIES OF HSC UNDER TRIAXIAL COMPRESSION Andreas Rogge Lehrstuhl für Massivbau, Technische Universität München, Germany The paper presents standard triaxial tests on high strength concrete with a mean cylinder compressive strength of 90 MPa The resulting maximum stresses are evaluated by ultimate strength criteria Stress-strain relationships under lateral pressure for normal and high-strength concrete are presented and compared with respect to the observed failure process The results show a good agreement with theoretical considerations INTRODUCTION Multiaxial stress states often determine the behaviour of structures such as panels, confined columns or elements loaded over a limited area For practical design these effects are usually considered qualitatively of by use of global factors C to be applied as fc,3ax = fc,1ax + C σlateral Due to the different microstructure of high strength concrete resulting in a more brittle stressstrain behaviour the regulations used for normal strength concrete may not be maintained without further investigations A series of about 100 load tests on concrete under various compressive stress states are performed to validate these factors for normal strength concrete and to extend and modify their application to high strength concrete Additionally the numerical, three-dimensional calculation of complex structures is a state-ofthe-art Therefore the lateral and longitudinal strains of concrete under triaxial pressure are measured during the tests Stress-strain relationships for concrete under triaxial stress states are formulated as constitutive laws which can be implemented into finite element programs Within the research project presented in this paper the behaviour of high strength concrete under longitudinal and rotationally symmetric lateral loading is examined using triaxial pressure vessels The investigated stress states are especially representative for members under restricted lateral deformations e.g confined concrete columns or anchorages of tendons The project continues the extensive work at the Technische Universität München in the field of multiaxial testing of concrete during the last decades [1 to 5] Compared to previous standard triaxial tests [6] the experimental techniques could be improved by modernized test controlling which enables various deformation- or stress-controlled load paths Thus the post-peak behaviour of concrete under different loading conditions can be examined EXPERIMENTAL PROGRAM All triaxial tests are performed using cylindrical specimens h/d = 40/15 cm According to previous investigations the chosen h/d-value of 2.7 is sufficient to minimize the lateral constraint of the loading plates [6], [7] The ratio of diameter and maximum size of aggregates (dg = 16 mm) provides a homogenous stress distribution over the cross-section Dependent on their dimension, shape, strength and curing conditions the boundary zones of moulded specimens may have a deformational behaviour, which differs up to 50% from the core concrete [8], [9] To avoid this influence the cylinders are drilled from a massive concrete block The possible disadvantage of drilled or sawn specimens resulting from flat aggregate particles spalling off the surface was not observed during the tests The tests are performed with three different concrete grades B 45, B 75 and B 95 according to the German standard DIN 1045 [10] and the annex to DIN 1045 for high-strength concrete [11] The values used for classification are related to the 5%-quantiles of the cube compressive strength The concrete mixtures and the results of the conformity tests after 28 days are given in table The curing of the specimens is performed in accordance with ISO 2736 Table Concrete mixtures and material properties Component unit B 45 B 75 B 95 Sand 0/4 mm [kg/m3] 899 839 832 Gravel 4/8 mm [kg/m3] 180 466 462 Gravel 8/16 mm [kg/m3] 719 560 554 Cement CEM I 42,5 [kg/m3] 360 430 430 Silica slurry 50:50 M.-% [kg/m3] - - 52 Water (additional) [kg/m3] 198 137 107 Superplasticizer [kg/m3] - 15 15 [-] 0.55 0.35 0.32 Mean cylinder compressive strength [MPa] 40.0 73.0 92.1 Young's modulus [MPa] 31000 40000 43000 Water/binder-ratio The construction of the triaxial cells is shown in fig schematically The lateral compressive stress is generated by hydraulic pressure To avoid fluid penetration into the specimens the cylinders are sealed by a rubber tube during the test The longitudinal stress is applied by a servo-controlled uniaxial testing machine with a maximum load of MN A specimen fully prepared for built-in is shown in fig To facilitate the built-in of the cylinders they are connected with the loading plates by three screws on each end The main number of tests is performed using a deformation-controlled rise of the longitudinal stress σ3 by 0.005 mm/s The lateral stress σ1 is raised proportionally to the measured longitudinal stress σ1 = kprop σ3 Beyond peak-stress the lateral stress is kept constant The investigated load paths are principally shown in fig Due to the construction of the cells the lateral stress is limited to σ1 = -30 MPa The longitudinal stress is limited to σ3 = -215 MPa Related to concrete columns these boundary values represent a stress state of about 11 Vol.-% confining reinforcement Herein a tensile stress of the reinforcement reaching the tensile strength (ft = 550 MPa) of the common German type of reinforcing steel is assumed Figure Triaxial cell Figure Specimen ready for built-in Figure Investigated load paths The longitudinal and lateral deformations are measured inside the cells by inductive strain gauges, which are able to resist the maximum fluid pressure of 300 bar without any significant deviations from linearity The lateral strain is measured in the middle of the cylinder between two opposite surface points using a surrounding steel frame (fig 4) The longitudinal strain is measured along two opposite surface lines with gauge lengths of 150 mm Figure Arrangement of lateral strain gauge ULTIMATE STRENGTH SURFACE General Three-dimensional stress states are usually described using the principal stresses σ1, σ2, σ3 or the stress invariants I1, J2, J3 [12] However, for the evaluation of the present test results the Haigh-Westergaard coordinate system is used due to the possible geometrical interpretation The meaning of this coordinates in the principal stress space is shown in fig for a typical ultimate strength surface of concrete σ1 +σ + σ 3 ξ = ⋅σ m σm = ρ = (σ − σ m )2 + (σ − σ m )2 + (σ − σ m )2 cosθ = ⋅ (σ − σ m ) − (σ − σ m ) − (σ − σ m ) ⋅ρ Figure Haigh-Westergaard coordinate system ξ ρ θ projected distance from the origin on the hydrostatic axis deviation on the hydrostatic plane perpendicular to the hydrostatic axis angle with the projection of σ1 on the hydrostatic plane The ultimate strength surface is usually described by the shape of the meridians (θ = const.) and the deviatoric planes (ξ = const.) Most of the modern criteria use parabolic formulations for the meridians The section of the deviatoric planes with the meridians changes from a triangular shape for low hydrostatic stresses to an almost circular shape for very high triaxial stresses Usually the threefold symmetric deviatoric plane is defined geometrically by the ξ-dependent ratio 0.5 < ρt /ρc < 1.0 of two special meridians: - compressive meridian ρc: - tensile meridian ρt: σ1 = σ2 ≥ σ3 σ1 ≥ σ2 = σ3 An often used ultimate strength criterion was formulated by Hsieh, Ting and Chen [13]: ρ ρ ξ F (ξ , ρ ,ϑ ) = a ⋅ + ( b1 ⋅ cosϑ + b2 ) ⋅ + c ⋅ fc fc fc ≤ (1) The factors a, b, c represent the quadratic equation for the meridians of the surface; the trigonometric equation for factor b describes the varying shape of the deviatoric planes The equation is made dimensionless by dividing with the concrete compressive strength fc Nevertheless, the fitting factors a, b1, b2 and c are still dependent on the compressive strength of the investigated concrete This criterion is almost equivalent to the well-known criterion of Ottosen [14] with only slight differences in the description of the deviatoric planes Due to the nature of standard triaxial tests using cylindrical specimens the possible stress states are limited to the compressive meridian of the so-called Rendulic-plane (σ1 = σ2) with θ = 60° = const Therefore eq (1) is reduced to a simple parabolic equation in ξ and ρ Evaluation The maximum stresses obtained from the tests on B 95 are summarized in fig using the stress coordinates of the Rendulic-plane σ3 and σ1 The intersection of the compressive meridian with the hydrostatic axis representing the triaxial tensile strength was assumed to be equal to the uniaxial tensile strength fctm = 5.0 MPa The other two parameters were determined by the least squares method in ξ, ρ-coordinates: ρ ρ ξ + 7.0 + 10.2 F (ξ , ρ , ϑ ) = 19 f c,1ax f c ,1ax f c ,1ax ≤ (2) The value of fctm was taken from splitting tensile tests on cylinders h/d = 30/15 cm using the transformation fctm = 0.9 fctm,sp Since the values of a, b, c are heavily influenced by the choice of fctm further evaluations will use the tensile strength found in direct tension tests The relation factor fc was set to the uniaxial strength fc,1ax = 85 MPa found in the triaxial cells on cylinders h/d = 40/15 cm Figure Ultimate strength results for B 95 The formulation of a complete ultimate strength surface according to eq (1) requires at least one value on the tensile meridian, e.g the biaxial compressive strength Since this value can not be obtained by the present test equipment an evaluation including b1 and b2 of eq (1) is omitted at this stage of the project The investigated range of stress states with σ1 = σ2 and ≥ σ1 ≥ -30 MPa can also be described by a linear relationship acc to eq (3) with sufficient correlation of the test results Both relationships are compared in fig fc,3ax = fc,1ax + 6.0 ⋅ σ1,min (3) STRESS-STRAIN RELATIONSHIP Concrete grade B 95 A typical stress-strain relationship of a high-strength concrete B 95 is shown in fig Up to a stress level of 85% of the maximum longitudinal stress the stress increases approximately linear The measured strains ε1 (lateral) and ε3 (longitudinal) correspond in good agreement to the values calculated with the deformation parameters found in standard concrete tests: ε3 = σ − ν σ1 Ecm where: ε1 = (1 − ν ) σ1 − ν σ Ecm Ecm = 43000 MPa ν = 0.18 (4) Young's modulus Poisson's ratio Figure Stress-strain diagram for concrete B 95 Shortly after the beginning of cracking the maximum longitudinal stress is reached, the stress level decreases rapidly The measured strains in the post-peak region indicate the failure mode of the cylindrical specimen While gauges and show a significant rise, gauge even shows a slight reduction of the longitudinal strain This behaviour is typical for a shear band failure mode The inclined damage zone is located over a very limited height of the specimen The regions above and beneath the shear band show an elastic relaxation due to the reduction of the longitudinal stress (gauge 1) The measured deformations of gauges and indicate the relative displacement of the failure wedges along the shear band The combination of both reveals the inclination of the shear band α = 55 65° towards the vertical axis This angle was confirmed after the removal of the specimen from the triaxial cell Concrete grade B 45 Compared to the high strength concrete the stress-strain diagram of a normal strength concrete B 45 (fig 8) shows an earlier formation of microcracks and therefore an increasing curvature of the curve beginning at about 50% of the maximum stress level The cracks have no dominant direction and are located over the whole height of the specimen except the laterally constrained areas directly beneath the loading plates The failure eventually occurs on an inclined shear band which is formed by the combination of existing cracks Figure Stress-strain diagram for concrete B 45 CONCLUSIONS For the investigated range of stress states the rise of the longitudinal compressive strength under lateral pressure can be formulated by a linear relationship of the following type: fc,3ax = fc,1ax + C ⋅ σ1,min (5) The factor C = 6.0 obtained from the triaxial tests on high strength concrete B 95 is certainly higher than the factor C = 4.8 found for normal strength concrete B 45 This increase results from the improved bond between aggregate and cement matrix leading to a more homogenous stress distribution in the cross-section However, the improved bond also leads to a more brittle behaviour due to the delayed development of microcracks The steeper ascent of the stress-strain curve produces a higher amount of elastically stored energy in the specimen to be released at peak level The observed failure process of the high strength concrete cylinders demonstrate that the released energy can not be redistributed within the specimen The failure occurs shortly after the development of the shear band at ultimate stress level without any significant ductility in the post-peak region REFERENCES 10 11 12 13 14 Scholz, U et al.: Versuche zum Verhalten von Beton unter dreiachsiger Kurzzeitbeanspruchung Berlin: Beuth, 1995 (DAfStb Heft 447) Lanig, N.; Stöckl, S.; Kupfer, H.: Versuche zum Kriechen und zur Restfestigkeit von Beton bei mehrachsiger Beanspruchung Berlin: Beuth, 1991 (DAfStb Heft 420) Linse, D.: Lösung versuchstechnischer Fragen bei der Ermittlung des Festigkeits- und Verformungsverhaltens von Beton unter dreiachsiger Beanspruchung Berlin: Ernst & Sohn, 1978 (DAfStb Heft 292) Menne, B.: Zur Traglast der ausmittig gedrückten Stahlbetonstütze mit Umschnürungsbewehrung In: Berlin: Ernst & Sohn, 1977 (DAfStb Heft 285) Kupfer, H.: Das Verhalten des Betons unter mehrachsiger Kurzzeitbelastung unter besonderer Berücksichtigung der zweiachsigen Beanspruchung In: Berlin: Ernst & Sohn, 1973 (DAfStb Heft 229) Lanig, N.: Langzeitverhalten von Beton bei mehrachsiger Beanspruchung München: Diss., 1988 van Mier, J.: Strain-Softening of Concrete under Multiaxial Loading Conditions Eindhoven (Netherlands): Diss., 1984 Stöckl, S.: Das unterschiedliche Verformungsverhalten der Rand- und Kernzonen von Beton Berlin: Ernst & Sohn, 1966 (DAfStb Heft 185) Schickert, G.: Schwellenwerte beim Betondruckversuch Berlin: Ernst & Sohn, 1980 (DAfStb Heft 312) DIN 1045: Beton und Stahlbeton: Bemessung und Ausführung Beuth: Juli 1988 Deutscher Ausschuß für Stahlbeton (Hrsg.): Richtlinie für hochfesten Beton: Ergänzung zu DIN 1045/07.88 für die Festigkeitsklassen B 65 bis B 115 (August 1995) Chen, W.: Plasticity in Reinforced Concrete New York: McGraw-Hill, 1982 Hsieh, S.; Ting, E.; Chen, W.: A plastic-fracture model for concrete In: Chen, W.; Ting, E (Edit.): Fracture in Concrete New York: 1980 (Proceedings of the ASCE National Convention 1980, Hollywood, Florida) Ottosen, N.: A failure criterion for concrete In: Proceedings of the American Society of Civil Engineers, Journal of The Engineering Mechanics Division (1977) 525-535 ... principally shown in fig Due to the construction of the cells the lateral stress is limited to σ1 = -30 MPa The longitudinal stress is limited to σ3 = -215 MPa Related to concrete columns these... The construction of the triaxial cells is shown in fig schematically The lateral compressive stress is generated by hydraulic pressure To avoid fluid penetration into the specimens the cylinders... differences in the description of the deviatoric planes Due to the nature of standard triaxial tests using cylindrical specimens the possible stress states are limited to the compressive meridian of the