Advanced concrete technology6 strength and failure of concrete under short term, cyclic and sustained loading

Advanced concrete technology6 strength and failure of concrete under short term, cyclic and sustained loading Advanced concrete technology6 strength and failure of concrete under short term, cyclic and sustained loading Advanced concrete technology6 strength and failure of concrete under short term, cyclic and sustained loading Advanced concrete technology6 strength and failure of concrete under short term, cyclic and sustained loading Advanced concrete technology6 strength and failure of concrete under short term, cyclic and sustained loading Advanced concrete technology6 strength and failure of concrete under short term, cyclic and sustained loading Advanced concrete technology6 strength and failure of concrete under short term, cyclic and sustained loading Advanced concrete technology6 strength and failure of concrete under short term, cyclic and sustained loading

Strength and failure of concrete under shortterm, cyclic and sustained loading John Newman 6.1.1 The structure of concrete Concrete is a multiphase material containing cement paste (unhydrated and hydrated compounds), fluids, aggregates, discontinuities, etc The overall mechanical and physical properties of such a composite system depend on the volume fractions and properties of the various constituents and the mechanisms of interaction, whether mechanical, physical or chemical, between the separate phases 6.1.2 Stresses and strains At a location in an element of material the generalized stress (strain) state in one, two or three dimensions comprising direct and shear stresses (strains) can be decomposed geometrically to a system of mutually perpendicular principal stresses (strains) el, ~2, ~3 6/4 Strength and failure of concrete under short-term, cyclic and sustained loading (1~1, 1~2, F_.3) acting orthogonal to the principal planes on which the shear stresses (strains) are zero 6.1.3 Deformation and failure theories ~::~,~::::~: ~: :~ :~:~:::~: ~ ::::::: :: :~::::::::::: ::: :::: ~::: :::~ ~: ::~::::::::::: ::::::::::::::::::::: Since the eighteenth century many theories and models have been proposed to explain or predict the deformation, fracture and failure of composite systems These are categorized in Table 6.1 Table 6.1 Categories of theories and models for the behaviour of composite materials Category Theory/model Remarks 'Classical' theories Maximum principal stress or strain Maximum shear stress Maximum strain energy of distortion Maximum octahedral shear stress Internal friction theory Mohr theory, etc Mathematical models Fundamental theory Structural models 'Mixture' laws Rheological models Comprising elements for elasticity, plasticity and viscosity Statistical models Distributions of properties of elements Physical models Simulations of real material (Griffith theory, finite element models, etc.) It is beyond the scope of this chapter to discuss all of these in detail but the following is a summary of the advantages and disadvantages of the various approaches, paticularly with regard to their use for concrete Category These predict failure when a particular function of stress or strain reaches a critical value and have limited application to concrete Category Such models are based on fundamental theories of physics and mechanics and allow the evaluation of stresses and strains within composite materials and for different geometrical arrangements of homogeneous materials Inglis in 1913 considered an elliptical crack in an ideal elastic solid under uniform uniaxial tension applied at 90 ° to the major axis of the crack For a major axis of 2b and a minor axis of 2c the radius of the crack tip is b2/c and the maximum stress at the crack tip is o(1 + 2c/b) where o is the stress applied to the boundary of the solid The relationship between the radius of the crack tip (nondimensionalized) and the intensification of stress at the crack tip (1 + 2c/b) is shown in Figure 6.1 (Inglis, 1913) Figure 6.1 demonstrates the large stresses developed around sharp cracks which is relevant to the cracks, flaws and voids present in concrete The model developed by Goodier in 1933 predicts a stress intensification of x the applied stress around a spherical Strength and failure of concrete under short-term, cyclic and sustained loading Stress intensification factor 400 300 200 100 0.00001 0.0001 0.001 0.01 0.1 Radius of crack tip - log scale 100 10 Figure 6.1 Relationship between radius of crack tip and stress at crack tip using Inglis's solution inclusion in an elastic material (as predicted for the Inglis model with b = c) and this is relevant to aggregate particles in hardened cement paste Category Assuming concrete to be a two-phase material (matrix and aggregate) then its stiffness (elastic modulus Ec) can be calculated using models in which the matrix phase (Em) and aggregate phase (Ea) are arranged in various configurations and proportions All models described below assume that all phases are elastic and the simplest are the Dantu upper and lower bound models (Dantu, 1958) which give the highest and lowest values for Ec The upper bound model, in which both phases experience the same strain is shown in Figure 6.2(a) For this arrangement, and assuming zero Poisson's ratio for the constituents, then Ec = (EmVm + EaVa) where Va and Vp are the volume fractions of the matrix and aggregate respectively ~ lll I i III N ~ A/ 1A1&1 &1 1A1A A N ~AAAA&AAA IIIIIIIIII (a) (b) ~ ] (c) (d) Matrix Aggregate Figure 6.2 Two-phase models for concrete The Dantu lower bound model, in which both phases experience the same stress, is shown in Figure 6.2(b) Again, assuming zero Poisson's ratio for the constituents and the same notation then 1/Ec = (Vm/Em + Va/Ea) Hansen (1968) suggested that the upper bound model is more relevant to hard aggregate particles in soft paste matrix with the lower bound model being more relevant for soft particles in a hard matrix He also considered that, due to strain disturbances around aggregate particles in concrete, the actual E-values lie between upper and lower bounds 6/5 6/6 Strength and failure of concrete under short-term, cyclic and sustained loading The Hirsch-Dougill model combined the upper and lower bound models as shown in Figure 6.2(c) Assuming zero Poisson's ratio for the constituents, for a volume proportion of lower bound model of x then 1/Ec = x[1/(VrnEm + VaEa)] + (1 - x)[Vm/Em + Va[Ea] However, both the lower bound and Hirsch/Dougill models predict Ec = for Ea = which is clearly incorrect To overcome this problem Counto (1964) proposed the model in Figure 6.2(d) For this model 1/Ec = (1 - ~/ Va)/Em + 1/{[(1 - ~/ Va)/~J Va]Em + Ea} Assuming for a paste with a water/cement ratio of 0.45 that Ep = 12.5 kN/mm and E a = 50 kN/mm the relationships between Ec and Va for the various models are shown in Figure 6.3 50 I ~'45 E 40 Z 35 v (D 30 25 Upper bound ~ I :~','" ,,,'" f.~ ~- ,,- ,"" -" Z~-":.'"'' "'~'15 '~;:";1"~'~'~" \ ,~( :~'~- IEP at 28 daY~l5-~0 for w/c = 0.451 0.0 Hirsch x = 0.7 ~ "o 20 o / 5"/ /.;;/ "N ~ Counto Lowerbound I I I 1.0 0.2 0.4 0.6 0.8 Volume fraction of aggregate Figure 6.3 Relationships between elastic modulus of concrete and volume fraction of aggregate for various models assuming Ep = 12.5 and Ea = 50 kN/mm Category These are based on the use of combinations of elements modelling stiffness (elastic springs), plasticity (yield stress) and viscosity (damper) For example, the combination of elements shown in Figure 6.4 will give the stress/strain relationship shown but only if the relevant constants for the various elements are assumed (Teeni and Staples, 1969) (n ~ Yield o~ ~Â0 ~ (~1 CO Strain $2 I starts to break starts to br4 e ~ ' i e l d s 1] tt (~ = yield stress (plastic) S = stiffness (elastic) 1] = viscosity (viscous) Strain Figure 6.4 Rheological model for the stress/strain relationship of concrete • Strength and failure of concrete under short-term, cyclic and sustained loading Category These are based on the statistical distribution of element properties and are of limited use for concrete since, as for Category models, assumptions must be made before they can be applied Category There is a large discrepancy between the theoretical strength of a brittle material (as calculated from the bonding forces between atoms) and its observed fracture strength The theoretical estimate gives values of 10x to 1000x those determined experimentally In 1920 Griffith (1920) proposed that this difference could be explained by the presence of microscopic flaws or cracks that always exist under normal conditions at the surface and within the interior of a body of material, with each crack tip acting as a stress-raiser, as discussed in Category models above During crack propagation there is a release of elastic strain energy (i.e some of the energy stored as the material is elastically deformed) and new free surfaces are created which increases the surface energy of the system A preexisting crack can propagate when the elastic strain energy released during crack extension is equal to, or greater than, the surface energy of the newly formed crack surface Using Inglis solution (Inglis, 1913) this indicates that, for a single elliptical crack in a thin plate subjected to uniform uniaxial tension, the critical stress for crack propagation is: Ocrit = (2TwE/rr, c) °5 where 7w = energy required to form a fracture surface E = elastic modulus 2c = crack length However, the various parameters required by the Griffith criterion to determine the critical stress needed for crack propagation in real materials are difficult to evaluate For a heterogeneous material such as concrete the task is impossible since (a) many cracks of different sizes, shapes and orientations either pre-exist or are formed under load and (b) the solid particles of aggregate etc act both as crack arrestors and stress intensifiers Nevertheless, the theory is useful as an aid to understanding the fracture and failure process and other models have been developed from the Griffith approach One such model has been developed by Hoek (1965) for rocks and by Newman (1973a) for concrete Consider a thin plate of concrete or another brittle material with a pre-existing 'closed' crack aligned at an arbitrary angle and apply a uniform uniaxial (compressive) stress to the plate (Figure 6.5) When the stress is gradually increased to a certain stress level the stress and strain intensification near the crack tips causes small cracks to initiate to stabilize the system This stage of the fracture process (Stage I) has been termed 'stable fracture initiation' Further increases in stress cause these cracks to propagate in a direction essentially parallel to the direction of the applied stress but when the stress is maintained constant propagation ceases This stage (Stage II) has been termed 'stable fracture propagation' and ends at a critical stress level after which propagation continues even when the stress is maintained constant This latter stage (Stage III) has been termed 'unstable fracture propagation' and ends with failure of the material This process has been confirmed experimentally by Hoek and from finite element modelling by Newman (unpublished report) 6/7 618 Strength and failure of concrete under short-term, cyclic and sustained loading Crack propagation I Original I flaw/crack I I Stable I I crack I I initiati°n I i | Figure 6.5 Initiation and propagation from a single crack in a brittle material Summary Most concretes can be simplified as two-phase materials in which stiffer and stronger particles are embedded in a softer and weaker matrix For such materials experience has shown that most of the above theories and models not adequately explain or predict behaviour but can help in the understanding of the stresses induced within a composite material under load The influence of the cracks and flaws within concrete can best be explained by Griffith-type physical models and that of aggregate by Goodier-type mathematical models In view of these complexities, normal practice is to test concrete and fit the results to relationships which have been derived on the basis of a knowledge of fundamental material behaviour but which can be used by engineers 6.1.4 Deformation of concrete .~ :~:::::: :.:,: ,:::~.:::~ , ~ ::.:.,:::::: :~:: ~ : ~:::~:::::.:.: ~:::: :: :~:::::: :~::::: :::::::: Stress-strain relationships for a typical concrete subjected to short-term monotonic uniaxial compressive loading to ultimate and beyond are shown in Figure 6.6 To allow the measurement of post-ultimate behaviour the load application was controlled to produce a constant rate of axial deformation (see below) Axial and lateral strains were measured using electrical resistance strain gauges (ERSG) and the overall axial deformation of the 25 ] a ~ Lateral strain ~ l ~ ~ (ERSGs) I I Ultimate ~ I /-/I '~ Axialstrain (platen-platen) 101]/ L A(XE~;G:i)nl X I I I I I 2000 4000 6000 8000 10000 Strain (microstrain) Figure 6.6 Typical relationships between stress and strain for concrete under uniaxial compression - 6000 - 4000 - 2000 Strength and failure of concrete under short-term, cyclic and sustained loading specimen using linearly variable displacement transducers (LVDT) measuring platen/ platen deformation Although the axial and lateral relationships appear reasonably linear up to about 4060 per cent of ultimate strength, they are not strictly linear Thus, unlike steel (see Figure 6.7), concrete has no readily identifiable elastic limit and, for simplicity, engineers resort to tangent and secant 'elastic' moduli (E values, defined in the next section) for design purposes **," ¢.)(~ _~ " x~ X ° Axial strain Figure 6.7 Comparison between stress-strain relationships for steel and concrete (not to scale) Up to a stress of about 60 per cent of ultimate the lateral tensile strain (in a direction orthogonal to the applied compressive stress) is a near-constant proportion of the axial compressive strain The constant of proportionality is termed Poisson's ratio (v) and has a value of between 0.15 and 0.20 depending on the mix and its constituents Above this stress the tensile strain increases at a much faster rate than the compressive strain and v increases to above 0.5 (the value for rubber) Since the material is discontinuous at these stress levels the concept of Poisson's ratio is not valid To add to this complexity, the deformations are not reversible (i.e are inelastic) and are time-dependent 6.1.5 Modulus of elasticity (E-value) ~ ~,~ ~ ~:~ ~ ~ ~ ~,~:~ ~ ~~ ~, ~ ~:~ ~ ~:~: ~:~:::~ :~:::~::,~: :,~ ~ .:~:.~::~:.~:: :~ ~.~ ~:~ : :::~ :: ::~ E-value is the ratio between stress and strain but, as discussed in section 6.1.4, the stressstrain relationship for concrete is non-linear and the material is not strictly elastic Thus, the concept is not strictly applicable so for structural design and assessment three types of E-value are used, namely secant modulus, tangent modulus and initial tangent modulus (Figure 6.8) The secant and tangent moduli can be determined from the stress-strain relationship from a short-term static test in which a specimen is loaded in uniaxial compression A procedure for determining secant modulus is described in BS 1881: Part 121 in which a 150 dia.× 300 mm specimen is loaded to 33 per cent of ultimate stress and the slope of stress/strain relationship measured after conditioning to near linearity by progressively loading and unloading The initial tangent modulus can be determined using ultrasonic (BS 1881: Part 203) and dynamic (BS 1881: Part 209) methods 6/9 6/10 Strengthand failure of concreteundershort-term,cyclicand sustainedloading J Initial ~ ° tangent ^~': ~ modulus " ~ / ~ I/ ~ / / ,'."1" ,% ~ / / , / / / Strain Figure 6.8 Diagram of stress-strain relationship for concrete under uniaxial compression The E-value of concrete is influenced generally by the same factors as strength and a relationship between strength and E-value for normal density concrete (BS DD ENV 11) is: Ec = 9.5(fck + 8)0.33 where fck = characteristic cylinder strength (MPa) For lightweight concrete E-value varies with both strength and density and the following relationship can be used (ACI 209R-92): E c = 4.3 x 10-5 × p 1.s × fcyi5 The above relationships are approximate only and the E-value should be determined experimentally where its use is important 6.1.6 Poisson's ratio ~ ~ ::~ ~ ~ 0.2 0.3 with all stresses compressive (triaxial 'compression') 0.2 0.3 > 0.1 with all stresses compressive (triaxial 'extension') ' - 0"3 > 0"1 with 0"2 and 0"3 compressive and 0"1 tensile (compression/compression/ tension) A hydrostatic cell pressure (confining pressure) was first applied and then the axial (deviator) stress either increased or decreased to failure Figure 6.29 shows the stress paths and the form of failure envelopes in stress space Also shown are typical forms of stress-strain relationships exhibited from the start of deviator loading It is usual to describe all tests in which the axial stress is increased in compression to failure as 'compression' tests, since failure occurs in the compression mode Tests in which the axial stress is decreased to failure are termed 'extension' tests with failure occurring in the tensile mode even if the resultant axial stress is compressive A special case of this latter type of test is when the specimen is suitably shaped, usually by 'waisting', such that the confining pressure can induce a tensile stress in the axial direction This test is termed a compression/compression/tension (C/C/T) test 6/29 6/30 Strength and failure of concrete under short-term, cyclic and sustained loading ~1/~c 12 10 I Laterall ' /~1 / X i 6'5 ~ 6'~ ~ _' j ~ ~ x i a l ,extenSiOn o3/(~c Figure 6.29 Typical failure envelopes and stress-strain relationships for concrete under triaxial compression and extension Stress-strain relationships Figure 6.30 shows the variation of axial and lateral strains for a concrete mix Each relationship covers the entire loading regime, including hydrostatic as well as deviatoric, for a particular maximum level of confining pressure applied 500 C-'" Confining pressure 138.3 MPa cU Q 59.0 M 250 22.5 MPa 6.8 -100 000 " MPaf~ ~ MPa (not visible) -50 000 Total axial and lateral strains (E2 = %) (MPa) 50 000 Figure 6.30 Stress-strain relationships for a concrete under triaxial 'compression' with various confining pressures Figure 6.30 shows that, for a given confining pressure the axial stress-axial strain relationship for 'compression' tests gradually increases in curvature with increasing stress The corresponding axial stress-lateral strain relationship, however, exhibits marked increases Strength and failure of concrete under short-term, cyclic and sustained loading in curvature only as ultimate failure is approached This phenomenon can be attributed to the process of cracking and the alignment of the cracks for this type of loading Similar behaviour is observed under 'extension' and C/C/T stress states Concrete also exhibits a transition from relatively 'brittle' to more 'ductile' behaviour as the maximum confining pressure is increased At low confining pressures, the slope of both the axial and lateral stress-strain curves reverses distinctly at ultimate At higher confining pressures, this reversal is less obvious Stage I behaviour As discussed previously, studies of the fracture mechanisms of brittle materials (e.g rocks and concrete) indicate that such materials fail by crack extension in a direction parallel to the maximum principal compressive stress, or in other words, orthogonal to the maximum principal tensile stress For uniaxial and biaxial compression and tension this behaviour can be easily identified, as it can for triaxial 'extension' tests or 'compression' tests However, for triaxial 'compression' at high confining pressures the mode of failure changes from 'brittle' to 'ductile' as evidenced by the stress-strain relationships and no clear fracture mechanism is visible since the cracks are extremely small and localized As in the case of uniaxial and biaxial stress states an indication of the level at which cracking becomes significant can be identified by examining the relationships between the strains in the directions of maximum and minimum principal stress, since such cracking will influence the strain in the direction of the minimum compressive stress more than in the direction of the maximum compressive stress Therefore, as the cracks extend, the ratio of the strain in the direction of minimum compressive stress to the strain in the direction of maximum compressive stress should increase numerically A typical variation of this relationship for a concrete under triaxial 'compression' is shown in Figure 6.31 Only a small portion of the relationship is shown in each case up to and just Confining MPa 138.3 MPa /22.5 pressure MPa -10 000 MPa ,-'E" ( •-~ b X