Experimental data on, and mathematical description of, failure surfaces

The use of non-linear computer-based methods for the analysis of concrete structures sub-jected to complex stress states requires that both the strength and the deformational prop-erties of concrete should be expressed in a suitable form. The deformational propprop-erties have been the subject of the previous section in which a mathematical description of the stress–strain behaviour of the material under generalised stress was outlined. The present section complements the above constitutive properties and is concerned with the cal description of the strength properties of concrete (Kotsovos 1979). Such a mathemati-cal description is considered essential, since most of the strength criteria proposed to date for use in practical structural design (e.g., Hannant 1974, Kotsovos and Newman 1977, Hobbs et al. 1977, Lowe 1978, Newman and Newman 1978) have not been expressed in a suitable form for computer applications. Furthermore, certain criteria (Hannant 1974) have been formulated in such a way that the convexity principle (Drucker 1967) does not hold, whereas the formulation of others has been based on the over-simplified assumptions that the effect of the intermediate principal stress on the strength properties is negligible

Predicted

Figure 3.18 (Continued) Stress–strain relationships for various concretes under stress states σ1 = σ2 > σ3

(triaxial ‘extension’): (a) fc = 31.7 MPa; (b) fc = 46.9 MPa; (c) fc = 62.1 MPa.

40 50

30

20

10

0

00.52 1.0 Predicted relationships

f_c = 31.7 N/mm²

ε₃

ε₂ =ε₃ ε₃

ε₂ ε₁

ε₁ ε₁ = ε₂

σ1 (Compressive) (N/mm2)

Strain × 10³ Compressive Tensile

–4 –3 –2 –1 0 1 2 3 4

σ₂/σ₁ =

Figure 3.19 Stress–strain relationships for under biaxial (and uniaxial) compression for a typical concrete (with fc = 31.7 MPa).

–8 –6 –4

ε₂ = ε₃ ε₁

–2 0 2 4 6 8

70 Predicted relationships Popovics (other empirical

relationships) f_c = 70 N/mm²

60 N/mm² 40 N/mm²

32 N/mm² 25 N/mm² Barnard

(experimental values) 60

50 40 30 20 10 0 σ1 (N/mm2)

Strain (mm/m) Compressive Tensile

Figure 3.20 Stress–strain relationships for various concretes under uniaxial compression.

(Hobbs et al. 1977) or that concrete behaves elastically up to a limiting principal tensile strain which defines ultimate strength and is regarded as a material constant (Lowe 1978).

The derivation of mathematical expressions given here is based on an analysis of strength data obtained in the course of investigations of the behaviour of concrete under multi-axial stress states carried out at Imperial College and described elsewhere (Newman 1973,

040 50

2Axial stress (N/mm) 100

150 Axial and lateral strain × 10³

Compressive Tensile

Figure 3.21 Typical scatter on (triaxial) constitutive material data for a given concrete (with f_c = 46.9 MPa) tested under various levels of maximum confining pressure. (a) Variation of axial and lateral strains with total axial stress for triaxial ‘compression’, triaxial ‘extension’ and C–C–T tests; (b) variation of volumetric strain with total axial stress for triaxial ‘compression’, triaxial ‘extension’ and C–C–T tests; (c) variation of lateral strain with axial strain for triaxial ‘compression’, triaxial ‘extension’ and C–C–T tests; (d) variation of lateral strain with axial strain for triaxial ‘extension’ and C–C–T tests.

(Continued)

Kotsovos 1974, 1979, Newman and Newman 1978). The testing techniques used to obtain these data (see Section 2.1.1 and Newman 1974) have been validated by comparing them with those obtained in the international cooperative programme of research into the effect of testing techniques and apparatus on the behaviour of concrete under biaxial and triaxial stress states referred to previously (Gerstle et al. 1978, 1980).

50

Axial strain × 103 Axial strain × 103

20

Lateral strain × 10³ Compressive Lateral compressive strain × 10³

Compressive

Figure 3.21 (Continued) Typical scatter on (triaxial) constitutive material data for a given concrete (with fc = 46.9 MPa) tested under various levels of maximum confining pressure. (a) Variation of axial and lateral strains with total axial stress for triaxial ‘compression’, triaxial ‘extension’ and C–C–T tests; (b) variation of volumetric strain with total axial stress for triaxial ‘compression’, triaxial ‘extension’ and C–C–T tests; (c) variation of lateral strain with axial strain for triaxial

‘compression’, triaxial ‘extension’ and C–C–T tests; (d) variation of lateral strain with axial strain for triaxial ‘extension’ and C–C–T tests.

As in the case of the constitutive relations, a mathematical description of the strength envelope of concrete, which is governed by combinations of maximum stresses that define a given failure criterion, is most readily formulated in terms of hydrostatic and deviatoric components acting on the octahedral plane. Therefore, it is convenient to define the stress space by the orthogonal coordinate system (σ1, σ2, σ3) of principal stresses. (The convention that compressive stresses are positive will be adopted.) Then, viewing the coordinate system and the octahedral (or deviatoric) plane from the hydrostatic axis, which intersects this plane at right angles, it is easy to see that the stress space can be divided into six regions, within which the following conditions are satisfied:

region 1: σ1 ≥ σ2 ≥ σ3 (3.47)

These regions are shown clearly in Figure 3.22.

The transformation of the orthogonal coordinate system (σ1, σ2, σ3) into the cylindrical coordinate system (z = (3)^1/2σo, r = (3)^1/2τo, ϑ) has been outlined in Appendix A. Accordingly, z is related to the hydrostatic stress that coincides with the space diagonal σ1 = σ2 = σ3, while the radius r is similarly related to the magnitude of the deviatoric stress component, the rotational variable ϑ defining the latter’s orientation on the octahedral plane. (Clearly, the hydrostatic and deviatoric stresses are obtained by contracting the (z, r) coordinates by a constant factor of (3)^1/2.) With these preliminaries, the strength envelope may be described by reference to both coordinate systems, and this is shown in Figure 3.22. The resulting ultimate-strength variation obeys the convexity principle usually associated with failure surfaces (Drucker 1967), and is open in compression since concrete can sustain increasing values of deviatoric stress for increasing hydrostatic compressive stress levels, that is, cross sections of the strength envelope (perpendicular to the z axis) become larger as σo increases.

If isotropic material behaviour is assumed, the ultimate-strength surface possesses a six-fold symmetry about the space diagonal σ1 = σ2 = σ3. Therefore, it follows that only one-sixth of the closed curve defining the failure boundary on a deviatoric plane (Figure 3.22b) is required for its definition. Now, experimental data are readily obtainable for τoe and τoc (the factor (3)^1/2 will henceforth be dropped, that is, the deviatoric plane will be shrunk to the curve τou rather than (3)^1/2τou). These values correspond to axisymmetric stress states easily imposed in a tri-axial test. Thus, τoe (for ϑ = 0°) is obtained by setting σ1 = σ2 > σ3 (triaxial ‘extension’) while τoc (for ϑ = 60°) follows by setting σ1 > σ2 = σ3 (triaxial ‘compression’). In this way, τoe and τoc

values can be determined for various levels of hydrostatic stress σo. For each σo, the value of τou for any ϑ intermediate between 0° and 60° may be interpolated between the values of τou at 0° and 60° by means of the following expression (Willam and Warnke 1974):

τ τ τ τ ϑ τ τ τ

This expression describes a smooth convex curve with tangents perpendicular to the directions of τoe and τoc at ϑ = 0° and 60°, respectively (see Figure 3.22). Therefore, it fol-lows that a full description of the strength surface may be established once the variations of τoe and τoc with σo are determined.

Figure 3.23 shows such variations of τoe and τoc. These combinations of octahedral stresses (σo, τo) at the ultimate-strength level, which appear normalised with respect to the uniaxial cylinder compressive strength f_c were obtained from triaxial tests carried out at Imperial College on a wide range of concretes (with f_c varying approximately between 15 and 65 MPa) subjected to the axisymmetric stress states σ1 > σ2 = σ3 > 0 (triaxial ‘compres-sion’), σ1 = σ2 > σ3 > 0 (triaxial ‘extension’) and σ1 = σ2 > 0 > σ3 (triaxial ‘tension’ C–C–T).

Full details of these tests can be found elsewhere (Kotsovos and Newman 1977, Gerstle et al. 1978). Figure 3.23 indicates that, for the portion of the stress space investigated, the ultimate-strength envelopes are essentially independent of f_c, that is, the type of concrete.

Furthermore, since the stress-path effects on ultimate strength have been found small enough to be regarded as insignificant for practical purposes (see Section 2.3.3.3), the two envelopes

σ₁ σ₁ >σ₂ > σ₃

σ₂ >σ₁ > σ₃

σ₂ >σ₃ > σ₁ σ₃ >σ₂ > σ₁ σ₃ >σ₁ > σ₂

σ₃ σ₂

σ₁ >σ₃ > σ₂

σ₃ σ₁

σ₂ = σ₃ σ₂

√(3)τ_oe

√(3)τ_oc

Deviatoric plane

√(3)τ_ou

√(3)τ_oe

r = √(3)τ_ou

√(3)τ_oc

z = √(3)σ₀ (σ₁ = σ₂ = σ₃) 60°

θ

Deviatoric plane (a)

(b)

θ = 60°

θ = 0° θ

Figure 3.22 Schematic representation of the ultimate-strength surface: (a) general view in stress space; (b) typical cross section of the strength envelope with a deviatoric plane (i.e., a plane of constant σo, viewed along the axis σ1 = σ2 = σ3).

of Figure 3.23 is considered to describe adequately the strength of most ordinary concretes likely to be encountered in practice when these are subjected to axisymmetric stress states.

This lack of influence of loading history for both stress–strain relations and the ultimate-strength envelope was discussed in Section 2.3.3.2, where it was argued that the unsystem-atic variability of the relevant data is larger than the scatter due to path dependency. Figure 3.24 shows the justification for such an argument in the case of failure data: it is evident that the scatter of ultimate stresses for concrete of a given f_c for different loading paths is smaller than the scatter of ultimate values for concretes of different f_c, but following a given loading path (Kotsovos 1984). A similar justification for adopting the OUFP level as the failure limit (as opposed to the slightly higher maximum sustained stress level – see Section 2.3.3.1) may be seen by reference to Figure 3.25, which shows that the unsystematic variation of the maxi-mum stress level for various concretes far exceeds the deviation between OUFP and US level for a given concrete (Kotsovos 1984).

A mathematical description of the two strength envelopes in Figure 3.23 may be obtained by fitting curves to the experimental data. Such an approach leads to the following expressions:

τoc/fc = 0 944. (σo/fc +0 05. )^{0 724.} (3.54) τoe/fc = 0 633. (σo/fc +0 05. )^{0 857.} (3.55) Equations 3.54 and 3.55 represent two open-ended convex envelopes the slopes of which tend to become equal to that of the space diagonal as σo tends to infinity. These expressions, together with Equation 3.53, define an ultimate-strength surface which conforms with gen-erally accepted shape requirements such as six-fold symmetry and convexity with respect to the space diagonal, open-ended shape which tends to become cylindrical as σo tends to infin-ity and so forth (Franklin 1970). A three-dimensional (3-D) view of this ultimate-strength surface is shown in Figure 3.26.

σ1 = σ² > σ³

Figure 3.23 Combinations of octahedral stresses at ultimate strength for concrete under triaxial ‘compres-sion’ and triaxial ‘exten‘compres-sion’.

5

4

A

B Path 2

Path 1 σ_a

σ_c

σc σ_a

σa =σc

2 2

1

σ^a =σ^c 1

3

2 Stress

paths

1

00 1 2

A B

√(2)σ_c/f_c σa /fc

3

Figure 3.24 A, unsystematic variability of ‘failure’ data obtained from tests using stress path 1 for concretes with fc varying between approximately 15 and 65 MPa; B, stress-path effect on ‘failure’ data for a typical concrete (with fc = 31.7 MPa).

5

4

A B

‘Maximum stress’ level

‘Failure’ level σ_a

σ_c

σ^a =σ^c 3

2

1

00 1 2

A B

√(2)σ_c/f_c σa /fc

3

Figure 3.25 A, unsystematic variability of ‘maximum stress’ level exhibited by concretes with fc varying between approximately 15 and 65 MPa; B, deviation of ‘failure’ level from ‘maximum stress’

level for a typical concrete (with fc = 47 MPa).

It will be noticed that the validity of expressions (3.54) and (3.55) is limited by the constraint that tensile hydrostatic stress states cannot exceed 5% of the uniaxial cylinder compressive strength f_c. This leads to consideration of the question of what experimental data there are for states of stress in which at least two of the principal stresses are tensile, and how the model describes the failure envelopes under such conditions. Experimental evidence of this type is very scarce and, moreover, is invariably associated with large scatter. Under such circumstances, the model smoothly extrapolates the C–C–T portion of the failure surface into regions where more than one principal stress is tensile. A typical cross section of the failure envelope is shown in Figure 3.27 (corresponding to the axisymmetric case) (Kotsovos and Pavlovic 1986), and the result is a smooth surface in the ‘tension’ region which provides a conservative estimate to a parameter that is subject to a very large degree of unsystematic variability, and which, further-more, represents a small absolute order of magnitude (relative to other stress values) in the stress space. (It is also important to recall the well-known fact that the testing of brittle materials in tension is usually more problematic than the determination of their compressive properties; this was stressed already by Föppl [Timoshenko 1953] and is still largely relevant today.)

On the basis of expressions (3.53) through (3.55), checks may be carried out to ascertain whether a state of stress lies inside or outside the failure envelope. The actual procedure consists of the following steps.

• The octahedral stresses and the rotational variable (σo, τo, ϑ) are calculated either from the principal stresses (σ1, σ2, σ3) – computed previously on the basis of the global stresses σij, that is, (σx, σy, σz, τxy, τxz, τyz) – or directly from the first, second and third stress invariants expressed in terms of σij (see Appendix A).

σ₁ = σ₂ =σ₃ σ₂

σ₃ σ₁

Figure 3.26 Three-dimensional view of the predicted ultimate-strength surface.

• The ultimate deviatoric stresses at ϑ = 0° and 60° (i.e., τoe and τoc, respectively) are calculated for the existing state of hydrostatic stress σo.

• The ultimate deviatoric stress τou for the existing rotational angle ϑ is calculated on the basis of the interpolation formula defined by (τoe, τoc and ϑ).

• The values of τo and τou are compared; if τo > τou, the state of stress lies outside the failure envelope.

In document Finite Element Analysis (Page 111-120)