Full-range stress–strain curves for stainless steel alloys

Abstract

The paper develops an expression for the stress–strain curves for stainless steel alloys which is valid over the full strain range. The expression is useful for the design and numerical modelling of stainless steel members and elements which reach stresses beyond the 0.2% proof stress in their ultimate limit state. In this stress range, current stress–strain curves based on the Ramberg–Osgood expression become seriously inaccurate principally because they are extrapolations of curve fits to stresses lower than the 0.2% proof stress. The extrapolation becomes particularly inaccurate for alloys with pronounced strain hardening.

The paper also develops expressions for determining the ultimate tensile strength (σ_u) and strain (ϵ_u) for given values of the Ramberg–Osgood parameters (E₀, σ_0.2, n). The expressions are compared with a wide range of experimental data and shown to be reasonably accurate for all structural classes of stainless steel alloys. Based on the expressions for σ_u and ϵ_u, it is possible to construct the entire stress–strain curve from the Ramberg–Osgood parameters (E₀, σ_0.2, n).

Introduction

Stainless steel alloys have low proportionality limits and extended strain-hardening capability. The pronounced yield plateau familiar from structural steel is nonexistent and so an equivalent yield stress is used in structural design, usually chosen as a suitable proof stress. The nonlinear stress–strain behaviour is acknowledged in the American [1], Australian [2] and South African [3] standards for cold-formed stainless steel structures which define the stress-strain curve in terms of the Ramberg–Osgood expression [4],

$$\text{ε=}\text{σ}\text{E}{}_0\text{+p}\text{σ}\text{σ}{}_{\mathrm{<mtext>p</mtext>}}{}^{\mathrm{n}}\text{.}$$

Eq. (1) was originally developed for aluminium alloys but has proven suitable for other nonlinear metals including stainless steel alloys. It involves the initial Young’s modulus (E₀), the proof stress (σ_p) corresponding to the plastic strain p, and a parameter (n) which determines the sharpness of the knee of the stress–strain curve. In the design of aluminium and stainless steel structures, it has become industry practice to use the 0.2% proof stress (σ_0.2) as the equivalent yield stress. For this proof stress, the stress–strain relationship takes the form,

$$\text{…}$$

$$\text{ε=}\text{σ}\text{E}{}_0\text{+0.002}\text{σ}\text{σ}{}_{0.2}{}^{\mathrm{n}}\text{.}$$

It has also become standard practice to determine the parameter (n) using the 0.01% and 0.2% proof stresses which leads to the following expression,

$$\text{n=}\text{ln(}\text{20}\text{)}\text{ln(}\text{σ}{}_{0.2}\text{/}\text{σ}{}_{0.01}\text{)}\text{.}$$

Eq. (3) ensures that the Ramberg–Osgood approximation matches exactly the measured stress–strain curve at the 0.01% and 0.2% proof stresses. It generally provides close approximations to measured stress–strain curves for stresses up to the 0.2% proof stress.

In concentrically loaded columns, the strains are small when reaching the ultimate load for all practical ranges of length. It is therefore possible to base the design on the Ramberg-Osgood curve and achieve close agreement with experimental strengths, e.g. see [5], [6]. This result was used [7] to develop a direct relationship between the column strength and the parameters n and e, where e is the nondimensional proof stress,

$$\text{e=}\text{σ}{}_{0.2}\text{E}{}_0\text{.}$$

However, structural components which undergo significant straining before reaching their ultimate capacity, such as plates in compression or shear, compact beams failing by in-plane bending and tension members, may develop stresses beyond the 0.2% proof stress and strains well in excess of the 0.2% total strain,

$$\text{ε}{}_{0.2}\text{=}\text{σ}{}_{0.2}\text{E}{}_0\text{+0.002.}$$

When the strains exceed the 0.2% total strain (ϵ_0.2), the Ramberg–Osgood curve obtained on the basis of the 0.01% and 0.2% proof stresses may become seriously inaccurate, tending to produce too high stresses, as shown in Fig. 1. This particularly applies to alloys with low values of n. In this strain range, it is necessary to use a refined expression for the stress–strain curve with wider applicability range. This paper aims to develop such an expression within the following constraints:

• Current values of n, such as those given in the American, Australian and South African standards for stainless steel structures, shall remain applicable. This implies that the stress–strain curve can be accurately determined using the Ramberg–Osgood expression for stresses up to the 0.2% proof stress.

• In the stress range between the 0.2% proof stress and the ultimate tensile strength (σ_u), the stress–strain curve shall be defined in terms of a minimum of additional parameters.

These constraints ensure simplicity in the stress–strain curve formulation. It will be demonstrated that it is possible to obtain agreement with measured stress–strain curves within tolerances that would be deemed acceptable for a wide range of applications.