Strength under multiaxial loading

So far in this chapter our discussions on compressive strength have been concerned with the effects of uniaxial loading, i.e. where s1 (or sx) is finite, and the orthogonal stresses s2 (or sy) and s3 (or sz) are both zero. In many, perhaps most, structural situations concrete will be subject to a multiaxial stress state (i.e. s2 and/or s3 as well as s1 are finite).

This can result in considerable modifications to the failure stresses, primarily by influencing the cracking pattern.

A typical failure envelope under biaxial stress (i.e.

s3 = 0) is shown in Fig. 21.18, in which the applied stresses, s1 and s2, are plotted non-dimensionally as proportions of the uniaxial compressive strength, sc. Firstly, it can be seen that concretes of different

strengths behave very similarly when plotted on this basis. Not surprisingly, the lowest strengths in each case are obtained in the tension–tension quadrant.

The effect of combined tension and compression is to reduce considerably the compressive stress needed for failure even if the tensile stress is significantly less than the uniaxial tensile strength. The cracking pattern over most of this region (Type 1 in Fig. 21.18) is a single tensile crack, indicating that the failure criterion is one of maximum tensile strain, with the tensile stress enhancing the lateral tensile strain from the compressive stress. In the region of near uniaxial compressive stress, i.e. close to the compres-sive stress axes, the cracking pattern (Type 2) is essentially the same as that in the central region of the cylinder shown in Fig. 21.1b, i.e. the cracks form all around the specimen approximately parallel to the compressive load. In the compression–compression quadrant, the cracking pattern (Type 3) becomes more regular, with the cracks forming in the plane of the applied loads, splitting the specimen into slabs.

Under equal biaxial compressive stresses, the failure stress is somewhat larger than the uniaxial strength.

Both Type 2 and Type 3 crack patterns also indicate a limiting tensile strain failure criterion, in the direc-tion perpendicular to the compressive stress(es).

With triaxial stresses, if all three stresses are compressive then the lateral stresses (s2 and s3) act in opposition to the lateral tensile strain produced by s1. This in effect confines the specimen, and results in increased values of s1 being required for failure, as illustrated in Fig. 21.19 for the case of

(a) (b)

Fig. 21.17 The effect of sustained compressive and tensile loading on the stress–strain relationship for concrete:

(a) compressive loading; (b) tensile loading (after Rusch, 1960; Domone, 1974).

uniform confining stress (i.e. s2 = s3); the axial strength (s1ult) can be related to the lateral stress by the expression:

s1ult = sc + Ks2 (or s3) (21.9) where K has been found to vary between about 2 and 4.5.

In describing strength tests in Section 21.1.1, we said that when a compressive stress is applied to a specimen by the steel platen of a test machine, the lateral (Poisson effect) strains induce restraint forces

in the concrete near the platen owing to the mismatch in elastic modulus between the concrete and the steel. This is therefore a particular case of triaxial stress, and the cause of the higher strength of cubes compared to longer specimens such as cylinders.

References

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De Rooij MR, Bijen JMJM and Frens G (1998). Introduc-tion of syneresis in cement paste. Proceedings of Inter-national Rilem Conference on the Interfacial Transition Zone in Cementitious Composites, Israel (eds Katz A, Bentur A, Alexander M and Arliguie G). E&FN Spon, London, pp. 59–67.

Domone PL (1974). Uniaxial tensile creep and failure of concrete. Magazine of Concrete Research, 26 (No. 88), 144–152.

FIP/CEB (1990). State-of-the-art report on high strength concrete, Thomas Telford, London.

Foulger D (2008). Simplification of Technical Systems.

Diploma in Advanced Concrete Technology project report, Institute of Concrete Technology, Camberley, UK.

Fig. 21.18 Failure envelopes and typical fracture patterns for concrete under biaxial stress s1 and s2 relative to uniaxial stress sc (after Kupfer et al., 1969; Vile, 1965).

0 20 40 60 80 100

0 2 4 6 8

σ2 or σ3 (MPa)

σ1 (MPa)

σ2= σ3

1

Fig. 21.19 The effect of lateral confining stress (s2, s3) on the axial compressive strength (s1) of concretes of two different strengths (from FIP/CEB, 1990).

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Maso JC (ed.) (1992). Proceedings of International RILEM Conference on Interfaces in Cementitious Composites, Toulouse, October. E&FN Spon, London, p. 315.

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Portland Cement Association (1968). Design and Con-trol of Concrete Mixes, 11th edition, Stokie, lllinois, USA.

Rusch H (1960). Researches toward a general flexural theory for structural concrete. Proceedings of the American Concrete Institute, 57 (No. 7), 1–28.

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Mix design is the process of selecting the proportions of cement, water, fine and coarse aggregates and, if they are to be used, additions and admixtures to pro­

duce an economical concrete mix with the required fresh and hardened properties. It is often, perhaps justifiably, referred to as ‘mix proportioning’ rather than ‘mix design’. The cement and other binder con­

stituents are usually the most expensive component(s), and ‘economical’ usually means keeping its/their content as low as possible, without, of course, compromising the resulting properties. There may be other advantages, such as reduced heat of hydration (Chapter 19), drying shrinkage or creep (Chapter 20).