Concrete Multiaxial Behaviour under Static Loading

Biaxial tensile strength tests of concrete are not standardised. The tests may be performed on plates, on cylinders, as well as on cubes. Nevertheless, they all are laborious and not trivial [192]. Practically, the biaxial tensile strength of concrete is equal to the uniaxial tensile strength [114]. A statistical evaluation conducted in [107] and based on published results reveals an increase of tensile strength of approximately 5% in relation to f_ct for a stress ratio ofσ_c2⇑σ_c1 =0.25 and a decrease of tensile strength of 2% forσ_c2⇑σ_c1 =1.00.

Schröder [170], however, reports that the biaxial tensile strength byσ_c2⇑σ_c1 = 1.00 is reduced with growing values of the uniaxial compression strengthf_c. Also the deformation capacity of concrete specimens under biaxial tension has been observed to decrease with increasingσ_c2⇑σ_c1-rates. Though, the number of performed tests does not permit an overall generalisation.

Similar to the concrete mechanical behaviour under uniaxial tension, the response behaviour under biaxial tension depends on the test facility and the specimen properties [170]. Likewise, a major number of parameters like aggregate size, water-cement-ratio, concrete strength, strain velocity, etc. have been found to influence the biaxial tensile behaviour significantly. It is, thus, difficult to determine a proper value for the biaxial tension strength of the concrete. Considering that the concrete behaviour under tension exhibits a great variability due e.g. the environmental conditions, the post-treatment, and the present residual stresses, it seems appropriate to assume a constant strength under biaxial tension states.

Kupfer [105] performed biaxial load tests on concrete slabs for different stress configura-tions: Tension-tension, compression-compression, and compression-tension. Through the use of brush bearing platens he was able to apply uniform stress conditions and to avoid, on the same time, strain restrain effects on the load application zones of the concrete specimens. For compression-compression stress states withσ_c2⇑σ_c1 = −1.00⇑ −1.00 he noticed an increase of the compression strength f_cby a factor of 16%. After the formation of numerable microcracks the specimen failure occurred owing to an approximately 30^○ diagonal macrocrack. A similar behaviour was observed for compression-tension stress states as long asσ_c1≤1⇑15⋃︀σ_c2⋃︀(σ_c1≥0≥σ_c2), otherwise the main cracks were perpendicu-lar to the tension stress. However, the achieved compression strength f_cwas considerably

3.2 Concrete 55

σ₁

σ₂

σ /σ = -1/-1_{1 2}

σ₁

σ /σ = -1/0.052_{1 2} σ₁

σ /σ = 1/0.54_{1 2}

σ₂

σ₂

σ₁

σ /σ = -1/0.103_{1 2} σ₂

σ₁

σ /σ = 1/1_{1 2}

σ₂

Figure 3.16:Failure modes of tested slabs under biaxial stress states, redrawn from [105]

reduced depending on the value of the transverse tensile stress. Additionally, the defor-mation behaviour of concrete was also extremely affected under compression-tension stress states since the presence of tensile stresses leads to a brittleness rise. The maximal strength increase of 27% was found by a ratioσ_c2⇑σ_c1 = −1.00⇑ −0.50 [104]. The measured σ_c−_c−curves (Fig. 3.18) contain information about the modified concrete ductility in dependence of the biaxial stress state; also the Young’s modulus of elasticityE_cand the elastic Poisson’s ratioµ_cmay be obtained. Kupfer’s results show, like in uniaxial concrete tests, a decreasing ductility with growing concrete strength fc [104].

The mechanical concrete behaviour under triaxial stress states depends on the applied stress ratio. Also the concrete composition and the porosity degree play an important role [192]. The inner structure, however, experiences in a more or less degree tension stresses similarly to specimens under unixial stresses. According to [157], tested concrete cylinders under triaxial compression present a positive lateral strain owing to the Poisson’s ratio if the transverse compression loads are not high. As a result, the specimen failure in this case is a consequence of transverse tension stresses between matrix and aggregates, exactly as under unixial compression and exhibiting almost the same phases of tightening and linear-elastic course, then a phase of stable cracking, a phase of volume dilatancy closed to the curve peak, and finally a softening post-peak phase [192]. Are the lateral compression loads high, then a more elevate concrete ductility and a greater distribution of microcracks over the specimen height may be noticed. Under nearly hydrostatic stress states (σ_c3 ≈ σ_c2 ≈ σ_c1 ≤ 0), practically no failure is observed. After the concrete inner pores collapse due to compression, a material hardening without any material softening may be registered [157]. Only very elevate load values produce a failure: Compact test specimens are crushed while slender test cylinders experience a diagonal fracture surface

56 Chapter 3 Material Behaviour

95% confidence interval mean A 5%-/ 95%-quantile A

Figure 3.17:(a) Statistical data analysis of biaxial tension behaviour of concrete, redrawn from [107]; (b) Biaxial concrete strength in theσ_c1−σ_c2-plane, redrawn from [104]

of about 25...35° and hence a shear failure. Furthermore, the concrete strength f_c was identified as an influencing parameter since lower values of f_ccause the test specimens to suffer a crush failure rather than a shearing. Under multiaxial stress states the failure mode is not unambiguous [157] and in many cases it is defined based on the formation of macrocracks, leading to a prominent increase of transverse strains and to a volume dilatation. The transition between a brittle and ductile concrete mechanical behaviour is known assoftening-hardening transition[192]. Torrenti [192] classifies the concrete failure in form of growing microcracking together with a material collapse, material softening, and porosity increase as Mode I. On the other hand, Mode II corresponds to a failure in form

σ /σ_{1 2}

Figure 3.18:(a) strain relationships of concrete under biaxial compression; (b) Stress-strain relationships of concrete under combined compression and tension, redrawn from [105]

3.2 Concrete 57

Figure 3.19:Stress-strain relationships of concrete under biaxial tension, redrawn from [105]

of microcracking combined with shear bands.

- !₃σ - !₂σ

Figure 3.20: (a) Fracture stress surface of concrete with Haigh-Westeergard coordinates, redrawn from [157]; (b) Deviator plane (left) and Rendulic plane (right) of concrete, redrawn from [185]

Rogge [157] classifies the different available calculation models of the mechanical behaviour of concrete into those according to the classical failure theory, where materials are considered isotropic, and those which base on fracture mechanics. In general, the mechanical behaviour of concrete under monotonic static loading is characterised by a nonlinear curve behaviour due to existing and growing microcracks resulting from increasing loads. However, un- and reloading processes are substantially more complicate to describe and calculate. One of the main difficulties is the mathematical modeling of the memory capacity of concrete [157] since it is evident that properties of the experienced load history like deformation values may decisively affect the actual mechanical answer (see Ch.3.2.4).

The general Mohr-Coulomb fracture-stress-hypothesis corresponds to the classical failure theories and is frequently employed owing to its simple mathematical background – a linear equation [114] – with an acceptable degree of accuracy. Shear failure is an usual failure mode of concrete. Though, the resulting shear stressesτ_cat the fracture surface

58 Chapter 3 Material Behaviour

are strongly governed by the corresponding active normalσ_c stresses. Therefore, the general calculation model of Mohr-Coulomb, similar to the behaviour of soil, describes the ultimate shear stressτ_cas a function of normal stresses and of internal cohesion. According to [114], the Mohr-Coulomb failure model is inadequate for a numerical implementation due to the discontinuous curve progress.

⋃︀τ_c⋃︀ =c−σ_ctanϕc. (3.20)

Equation (3.20) leads to an uniaxial tension strength of

ft=ccotϕc. (3.21)

A stress state with 0<σ_c< f_tprovides an alternative equation for the uniaxial tension strength with

ft=ccotϕc= 1

2(σ_c1+σ_c3) +1 2

σ_c1−σ_c3

sinϕc . (3.22)

Reshaping equation3.22leads to an expression of the general Mohr-Coulomb yield criterion under principal stresses:

Y=σ_c1(1+sinϕc) −σ_c3(1−sinϕc) −2ccosϕc=0. (3.23) Withσ_c1=0 andσ_c3= −f_c, one obtains for the uniaxial compression strength:

f_c= 2ccosϕ_c

1−sinϕ_c, (3.24)

and withσ_c1= f_tandσ_c3=0 for the uniaxial tension strength:

f_t= 2ccosϕc

1+sinϕc. (3.25)

According to [157], the calculation model of Mohr-Coulomb is valid as long as the applied stress are not excessively high. Marti [114] neglects the strength increase under biaxial compression-compression but seeks to appropriately model the reduced concrete strength under compression-tension. For the inclination of the failure curve in theσ_c−τ_c -plane he suggests tanϕc = 3⇑4 (ϕc ≈ 37^○), a test-based value which is appropriate for not too high hydrostatic compression values but which leads to an unacceptable tension strength. Inserting tanϕ_c = 3⇑4 into equation (3.24) one obtains c = 0.25f_c. This value together with tanϕc =3⇑4 in equation3.25leads to f_t =0.25f_c. However, tests deliver a

3.2 Concrete 59

value of f_t=0.10f_c[114], [178]. For this reason, a modification turns mandatory which is known as the modified Coulomb yield criterion [112] with ft= fct.

σ1

Figure 3.21:Modified Coulomb yield criterion of concrete, redrawn from [114]

Further calculation models are discussed in [157] and [185]. Specially the model devel-oped by Willam and Warnke [201] is considered appropriate for a numerical implementa-tion with the finite element method since it presents no geometrical discontinuities but continuously differentiable surfaces. However, both models are quiet complicate to deal with due to the many parameters which have to be derived.

Willam and Warnke [201] developed a simplified 3-parameter model with a conical failure surface and a non-circular base section in the principal stress state. The model leads to an acceptable agreement with test data in the lower compression regime. For higher compression values it shows however, in accordance to the authors, appreciable errors. A further refinement results in a 5-parameter model, where principal meridians are described as parabolas instead of straight lines, connected by an ellipsoidal surface.

In the deviatoric plane, the failure curve is defined as ellipse in a section betweenθ_c=0^○ and θ_c = 60^○ and then mirrored to the remaining sections. It is rather triangular for low compression values while it becomes more circular for higher compression values.

For the sake of a stable numerical implementation, special attention was given to the smoothness criterion (continuous surfaces and varying tangents) as well to convexity (no reflexion points). In sum, the model after [201] follows an "elastic, perfectly plastic material formulation in compressions and an elastic, perfectly brittle behaviour under tension".

The mathematical definition of the failure surface of concrete under multiaxial stress states may be expressed by means of a three-dimensional stress space with the principal

60 Chapter 3 Material Behaviour

30/15 40/15

Figure 3.22:Observed failure mechanisms at encased concrete cylinders withh⇑D= 30⇑15 cm and at a drilled concrete cylinder withh⇑D=40⇑15 cm, redrawn from [157]

stressesσ_c1,σ_c2,σ_c3, or using the octahedron stressesσ_c0,τ_c0with the Lode angleθ_c, or also the Haigh-Westergaard cylindrical coordinatesξc,ρ_c,θ_c, whereξcdescribes the hydrostatic axis,ρ_cthe deviatoric plane perpendicular toξ_candθ_ccorresponds to the Lode angle with

ξc= 1

⌋︂3σ_cI= 1

⌋︂3(σ_c1+σ_c2+σ_c3), (3.26)

ρ_c=

⌈︂2s_cII= 1

⌋︂3(︀(σ_c1−σ_c2)²+ (σ_c2−σ_c3)²+ (σ_c3−σ_c1)²⌋︀

12, (3.27)

cosθ_c= 2σ_c1−σ_c2−σ_c3

⌋︂6ρ_c . (3.28)

The failure surface of the 5-parameter model after [201] is described using the Haigh-Westergaard cylindrical coordinates

Y(ξc,ρ_c,θ_c) = 1 rc(ξc,ρ_c,θ_c)

ρ_c

⌋︂5fc, (3.29)

with the radius functionr_cin the deviatoric plane

r_c(ξ_c,θ_c) =

2r2(r²₂−r²₁)cosθ_c+r2(2r₁−r2)

⌉︂

4(r²₂−r²₁)cos²θ_c+5r₁−4r₁r2

4(r²₂−r²₁)cos²θ_c+ (r₂−2r₁)²

. (3.30) The principal meridians are parabolas whose parameters are to be quantified based on

3.2 Concrete 61 Rogge [157] combines the models of Willam and Warnke [201]: Ellipses describe the deviatoric plane while the meridians follow the concept of Ottosen. The failure surface is defined in sections, what at first increases the number of parameters to be determined.

According to the author, the model exhibits a better adaptability for numerical purposes.

Speck [185] analyses the material behaviour of high-strength concrete under multiaxial stress states. The interesting results published in [185] reflect that the multiaxial stress state has a significant lower influence on the mechanical performance of high-strength concrete compared to normal-strength concrete. Accordingly, the biaxial strength is practically equal to the uniaxial strength fc. However, through the application of fibres, the high-strength concrete attains a similar ductility as normal-strength concrete.

Axial compression [MPa]

Figure 3.23:Measured stress-strain-curves of concrete cylinders under axial compressionσ_c1 and transverse compressionσ_c3=kpropσ_c1, redrawn from [157]

Im Dokument FATIGUE OF THE TENSION-STIFFENING EFFECT IN REINFORCED CONCRETE (Seite 72-79)