Behavior in Multi-axial Stress States

Concrete

3.6 Behavior in Multi-axial Stress States

Concrete in real structures may be subjected to complex stress states resulting from restrained volume change and applied loads. For example, simple beams under transverse loading experience stresses due to combined flexure and shear, as well as stresses from bearing at the supports and from bond between longitudinal reinforcement and concrete. In columns and shear walls these actions are further complicated by the addition of axial loads. Later chapters will consider these subjects in detail. This chapter describes how concrete as a material responds to multi-axial stress and strain states.

3.6.1 Plain Concrete in Biaxial Stress State

The biaxial stress state is defined in Figure 3.11 by principal stresses f₁, f₂, and f₃, where f₁ and f₂ are positive in compression and f₃ = 0. Kupfer et al.

(1969) report tests in which loading began in an unstressed state, and then the stresses f₁ and f₂ were increased proportionally until failure. The continuous curve in Figure 3.11 shows the strength envelope for various ratios f₁/f₂. The envelope was relatively insensitive to concrete compressive strength for the range tested [2700 to 8400 psi (19 to 58 MPa)]. Point a corresponds to a uniaxial compression test along axis 1.

Point b corresponds to a uniaxial tension test along axis 1. Point c demonstrates how a small tensile stress in the 2 direction causes significant reduction in the compressive stress at failure in the 1 direction. Under biaxial compression with f₁ = f₂ (point d), the strength increase over the uniaxial compressive strength f_c′ is about 15%, whereas the maximum strength increase is about 25% for f₁⁄f₂ ≈ 2 (point e). Failure at points d and e is due to tensile splitting failure in the unrestrained 3 direction. To achieve greater strength, it is necessary to restrain dilation in the 3 direction. Section 3.6.3 covers behavior under triaxial loading.

FIGURE 3.11 Concrete strength envelope under biaxial loading. (After Kupfer et al., 1969, courtesy of American Concrete Institute.)

Figure 3.12 plots relations between strain and stress ratio f₁⁄f_c′. The continuous curve in Figure 3.12a is for uniaxial compressive loading.

Poisson’s ratio, v = ε₂/ε₁ = ε₃/ε₁ can be scaled from the data for uniaxial loading. Typical values are ν = 0.15 to 0.20. As compressive stress approaches f_c′, the rate of transverse straining increases rapidly. As shown in Figure 3.12b, the volumetric strain (∆V⁄/V = ε₁ + ε₂ + ε₃) increases beyond this point, which is a sign of more extensive micro-cracking just before failure. Similar results are shown for biaxial compression with f₁/f₂

= 1/1 and 1/0.52. The stress at which the minimum volume is achieved is sometimes referred to as the critical stress. Observed values of the critical stress usually occur in the range of 0.75f_c′ to 1.0f_c′, the value apparently depending on details of the load and measurement apparatus, with typical values commonly taken as 0.85f_c′ to 0.9f_c′.

FIGURE 3.12 Biaxial compressive loading: (a) stress–strain relations; (b) volumetric strain. (After Kupfer et al., 1969, courtesy of American Concrete Institute.)

3.6.2 Reinforced Concrete in Biaxial Loading

In a reinforced concrete section, concrete may be subjected to complex strain fields including tensile strains that cause concrete cracking. To study this behavior, reinforced concrete panels have been tested under in-plane shear and normal stresses (Vecchio and Collins, 1986; Hsu, 1993). A principal finding of these studies is that the compressive stress capacity of concrete reduces as it is subjected to increasing transverse tensile strain.

Figure 3.13 shows results from the modified compression field theory (Vecchio and Collins, 1986). According to this model, the concrete principal compressive strength f₁ is a function not only of the principal compressive strain ε₁ but also of the principal tensile strain ε₂ (Figure 3.13a). The relation between maximum principal compressive stress capacity f_c1max and the principal tensile strain ε₂ (defined negative for tension) is shown in Figure 3.13b. This results in a family of concrete compressive stress–strain relationships for different values of ε₂ as shown in Figure 3.13d.

FIGURE 3.13 Biaxial response of reinforced concrete under in-plane loading: (a) stress–strain relationship for cracked concrete in compression; (b) relation between maximum compressive stress and transverse tensile strain; (c) correlation of test data for cracked concrete in compression; (d) three-dimensional representation of compressive stress–strain relationship. (After Vechio and Collins, 1986, courtesy of American Concrete Institute.)

In Figure 3.13a, the stress–strain relationship is

in which

and ε₂ is negative in tension.

The softened truss model (Hsu, 1993) proposes a softening coefficient in the form

Whereas the modified compression field theory applies the softening coefficient only to the stress, the softened truss model also applies the softening coefficient to the strain, such that the strain corresponding to peak stress f_c1max decreases with increasing (more negative) principal tensile strain.

The parabolic form of Eq. (3.8) is adopted from traditional stress–strain models for concrete, and is applicable for longitudinal strains up to approximately 1.75ε₀. The softening coefficient of Eq. (3.9) or (3.10) will be useful in explaining observed shear strength of structural concrete members (see Chapter 7).

3.6.3 Plain Concrete in Triaxial Stress State

Section 3.6.1 shows that the maximum compressive strength under biaxial loading is only moderately higher than the uniaxial compressive strength, and there is little effect on strain capacity. Only a moderate effect is observed because the concrete is unconfined in the 3 direction, such that failure in that direction is not restrained. Tests show that if we can confine the concrete in all directions by applying external compressive stress, the behavior of concrete can be dramatically changed.

Figure 3.14 plots stress–strain curves obtained from concrete cylinders uniformly confined by external stress in the 2 and 3 directions. The tests were conducted by applying an external hydrostatic pressure to concrete cylinders sealed in a rubber membrane, and then loading in the longitudinal direction. Note that both the stress and the strain capacities are significantly increased by confinement.

FIGURE 3.14 Stress–strain relationships for normalweight concrete confined by hydrostatic pressure and then loaded in axial compression. (After Richart et al., 1928, courtesy of the University of Illinois at Urbana–Champaign Archives.)

Richart et al. (1928) proposed that the axial strength of confined concrete could be represented by

in which f₃ is the smallest principal compressive stress (positive in compression). The test data show that the value of k is slightly higher for low confinement stress than for higher confinement stress. Richart et al.

(1928) recommended a single value of k = 4.1. Subsequent comparisons with larger data sets demonstrate that Eq. (3.11) with k = 4.1 adequately models the data trend over a large range of confinement stresses (Figure 3.15).

FIGURE 3.15 Comparison of measured confined concrete strength versus Eq. (3.11). (After Hobbs et al., 1977.)