Nonlinearity and strength in 1D site response analyses: a simple constitutive approach

Abstract

When performing seismic site response analyses in the design practice, one of the main issues consists in finding a compromise between the limited availability of site-specific experimental data to characterise the cyclic soil behaviour and the necessity of a proper constitutive representation of such behaviour, including nonlinearity, hysteresis and strength. This paper presents a new constitutive approach for 1D nonlinear soil models, capable of describing such features with a simple and easy to calibrate constitutive equation, requiring the definition of few parameters, all derived from conventional field and laboratory data. The main element of novelty in the proposed approach consists in recognizing that the soil shear strength is a key ingredient not only to limit the maximum shear stress that the soil can experience at large strains, but also to describe the dependence of the nonlinear soil properties on mean effective stress and plasticity index, as observed in the medium strain range. Based on a thorough comparison with laboratory data, it is shown that the proposed model provides a very good description of the soil behaviour, in the whole strain range. Its application in site response analyses is further validated on centrifuge data and verified against other nonlinear constitutive soil models. Finally, using a well-characterised site in Italy as reference, it is demonstrated that an erroneous prediction of the soil shear strength can lead to a misinterpretation of the amplification phenomena within the soil deposit, together with a possible gross underestimation of the actual surface acceleration.

Appendices

Appendix 1

Nonlinear models including the soil shear strength as a constitutive ingredient should predict, from a phenomenological point of view, the maximum horizontal shear stress (τ_lim) that can develop under the vertical propagation of shear waves, compatibly with the assumed failure criterion and the imposed stress/strain boundary conditions.

Starting from an initial isotropic stress state (σ_v0′ = σ_h0′ = p₀′), as in standard laboratory RC and TS tests, and assuming drained conditions and a purely frictional failure criterion, the maximum horizontal shear stress at yield is given by (Fig. 21a):

Fig. 21

Mohr circles and stress–strain conditions imposed to a soil element under shear loading: a initial isotropic stress state and pure frictional failure criterion; b initial geostatic stress state and cohesive-frictional failure criterion; c initial geostatic stress state and purely cohesive failure criterion

$$\tau_{vh,y} = p'_{0} \sin \phi '$$

(14)

If σ_v′ and σ_h′ cannot evolve, due to the imposed stress boundary condition (p′ = p₀′), then τ_lim = τ_vh,y. In this case, τ_lim coincides with the radius of the Mohr circle at failure.

More in general, starting from an initial geostatic stress state (σ_h0′ = K₀σ_v0′) and assuming a cohesive-frictional failure criterion (Fig. 21b), the maximum horizontal shear stress at yield is given by Hardin and Drnevich (1972b):

$$\tau_{vh,y} = \sqrt {\left[ {\left( {\frac{{\sigma '_{v0} + \sigma '_{h0} }}{2}} \right)\sin \phi ' + c'\cos \phi '} \right]^{2} - \left( {\frac{{\sigma '_{v0} - \sigma '_{h0} }}{2}} \right)^{2} }$$

(15)

According to Eq. (15), τ_vh,y depends both on the distance of the Mohr circle at rest from the axis origin and on its radius. If the horizontal axial strain is restrained, due to the imposed kinematic condition (ε_h = 0), then the horizontal normal stress can evolve after the onset of yielding, with σ_h′ → σ_v0′, and the failure condition is attained when σ_h′ = σ_v0′, corresponding to which:

$$\tau_{lim} = \tau_{vh,f} = \sigma '_{v0} \sin \phi ' + c'\cos \phi '$$

(16)

The same considerations hold for the case of a purely cohesive failure criterion (c = c_u, ϕ = ϕ_u = 0), corresponding to which (Fig. 21c):

$$\tau_{vh,y} = \sqrt {c_{u}^{2} - \left( {\frac{{\sigma_{v0} - \sigma_{h0} }}{2}} \right)^{2} }$$

(17)

and

$$\tau_{lim} = \tau_{vh,f} = c_{u}$$

(18)

Different definitions of τ_lim have been proposed so far in the scientific literature, as detailed in Table 4. For cohesive-frictional soils, the equation used by Régnier et al. (2016) is not consistent with the assumed failure criterion and an initial geostatic stress state, while Hardin and Drnevich (1972b) and Wichtmann and Triantafyllidis (2013) make use of Eq. (15) at yield. The equation used by Yee et al. (2013) and Groholski et al. (2016), instead, is derived under the assumption that soil failure occurs on the horizontal plane, which can be achieved only if the horizontal stress satisfies the condition:

Table 4 Definitions of τ_lim proposed in the literature for cohesive-frictional and purely cohesive failure criteria

$$\sigma_{h} = \sigma_{v0} \left( {\frac{2}{{\cos^{2} \phi '}} - 1} \right)$$

(19)

For purely cohesive soils, Afacan et al. (2014) make use of Eq. (18).

Appendix 2

The nonlinear soil model proposed by Phillips and Hashash (2009) (referred to as the PH model in this work) is defined by a standard hyperbolic function for the backbone (Matasovic and Vucetic 1993):

$${{\varPhi }}\left( \gamma \right) = \frac{{G_{0} \gamma }}{{1 + \beta \left( {\frac{\gamma }{{\gamma_{ref} }}} \right)^{s} }}$$

(20)

where β > 0 and s > 0 are model constants. The reference shear strain, which describes the dependence of the G(γ)/G₀ curve on p′, is defined as:

$$\gamma_{ref} = P_{4} \left( {\frac{{\sigma '_{v} }}{{\sigma_{ref} }}} \right)^{{P_{5} }}$$

(21)

where σ_v′ is the vertical effective stress, σ_ref is a reference stress and P₄ and P₅ are model parameters (Hashash and Park 2001). Equation (20) implies that, for γ → ∞, the function τ(γ) converges to a finite value (= G₀γ_ref/β) only in the case s = 1, while tending to either infinity or zero, for s < 1 and s > 1 respectively.

The unloading–reloading rule is given by Eq. (9), where:

$${{\varPsi }}\left( \gamma \right) = \frac{{G_{0} \gamma }}{{1 + \beta \left( {\frac{{\gamma_{m} }}{{\gamma_{ref} }}} \right)^{s} }}$$

(22)

and

$$\alpha \left( {\gamma_{m} } \right) = P_{1} - P_{2} \left[ {1 - \frac{{G\left( {\gamma_{m} } \right)}}{{G_{0} }}} \right]^{{P_{3} }}$$

(23)

in which P₁, P₂ and P₃ are additional model constants, controlling the D(γ)curve.

Table 5 reports, for all the soils analysed in this study, the values of the model parameters obtained from the best-fit of the experimental data.

Table 5 PH model: values of the model constants for the soils analysed in this study