Briefing: Factored material properties and limit state loads—unlikely extreme or impossible pretense Free

A limit state is typically defined in design codes as ‘any limiting condition beyond which the construction ceases to fulfil its intended function’.

Limit state design (LSD) is a formal way of stating the design criteria in a performance-based way. The LSD process recognises there are many ways a construction may fail to satisfy its intended purpose, e.g. strength, deflection and stability.

Probabilistic design methods give more consistency in the safety of structures than working stress methods. Thus partial factors and LSD structural codes were developed as a convenient way to carry out routine probabilistic-based structural design.

Failure is described by a performance function G = f(x₁, x₂, …, x_i) where x_i is the collection of parameters. Values of G < 0 define failure and values of G ≥ 0 indicate satisfactory performance. The limit state is therefore G = 0. The probability of failure, P_f, can be explicitly calculated as P_f = p(G < 0).

A simple two-parameter system is: G = R − S, where S is the load on, and R is the strength of a construction. Both R and S have some variability represented by their respective probability density functions. The design task is to make R large enough so that the P_f = p(G < 0) = p(R < S) is acceptably small. The design equation becomes

where S_k and R_k are characteristic values, and γ_S and ϕ_R are partial factors. The characteristic values may be chosen as any representative value of the parameters. It may be expressed in terms of the mean values, $S¯$ and $R¯$⁠, as $Sk=δSS¯$ and $Rk=δRR¯$⁠.

Table 1 gives the statistically calculated load and resistance factors for two different choices of the characteristic values, for the design criterion of P_f = 0·00135 (coefficients of variation, V_R = 0·25 and V_S = 0·2).

In the first case the characteristic values are chosen as the mean values of the parameter. In this case the design equation becomes: 1·112S_k ≤ 0·263R_k. For the second case the characteristic values are chosen as ‘unlikely’ values. The design equation becomes: 0·837S_k ≤ 0·447R_k. These equations are equivalent and yield identical design results.

For the second case, γ_S—the load factor—is less than unity. That is, the limit state design load is less than the characteristic design load. If the second equation is multiplied by a factor of 2 the load factor is 1·67 and the resistance factor is 0·89, which are more familiar values.

The previous example demonstrates that limit state design load (the ultimate load) has no physical meaning; in fact it may even be physically impossible for that load to be applied. The limit state load and factored material strength are mathematical concepts used to simplify a complex probabilistic analysis.

The Australian loading code (AS1170) specifies γ_S = 1·25 for liquid pressure, where the density of liquid is well defined and its depth cannot be exceeded. The probability distribution of this type of load is very narrow. The ultimate load—equal to 1·25 times the hydrostatic load—is physically impossible. So what then does this mean? The answer lies in the probabilistic nature of the partial factor limit state method. For the simple case (G = R − S) the variation in P_f with load factor is shown in Table 2. A load factor as high as 1·25 is required to ensure the probability of failure is acceptable (P_f < 0·03%).

In comparison, the same structure loaded with self-weight only (load factor = 1·25) has a P_f approximately equal to 0·005%. To have the same probability of failure, the load factor for the liquid pressure would be 1·35.

The partial factors for load and resistance (and material parameters) are interdependent and a result of statistical analysis. Clearly the ultimate design load has no physical interpretation. Likewise factored soil strength parameters are not intended to represent a realistic extreme possibility and are not a physically possible ‘very severe’ design situation. The factored soil strength parameters may in fact be physically impossible (as for water in the example above). Designers like to put a meaningful physical interpretation on the ultimate design load or capacity. We like to imagine some unlikely extreme situation. Unfortunately in geotechnical engineering such interpretation is not possible and often very confusing—particularly with respect to retaining walls.

Another source of confusion in the application of LSD is the use of analysis models. A designer considering the strength limit state of a structure or the overturning limit state of a retaining wall will typically apply a physical interpretation to these limiting conditions. For example, collapse of the structure or toppling of the wall. In doing so the designer is attracted to a model that realistically describes the physical behaviour at the perceived physical limit. However, the limit state is a mathematical concept defined by the performance function G. Hence the appropriate model for analysis is implied by this definition.

For example, the strength limit state for structural design is often defined by the performance function G = σ_yield − σ. For this definition an elastic analysis is the appropriate model. A designer interpreting the strength limit state as one in which the structure is verging on collapse is confused because ductile plastic behaviour and stress redistribution are not included in the model. Similarly for the stability (overturning) limit state design of a gravity retaining wall the analysis model need not accurately describe the perceived physical limiting situation. If we imagine a physical interpretation, the usual analysis method of examining moment equilibrium about the toe is unrealistic since it implies an infinite bearing pressure at the toe. However this does not mean it is an unsuitable model for assessing the overturning limit state.

Limit state design codes were introduced into structural design to incorporate probabilistic techniques in routine design. Designers were comfortable with LSD because a meaningful physical interpretation was possible. Unfortunately in geotechnical design trying to attach a physical meaning generally leads to confusion. This paper explains why. It briefly outlines the fundamental basis underlying the derivation and purpose of partial factors. In summary, the following points must be understood to avoid confusion in the discussion and application of partial factor based LSD methods.

• The ultimate load and factored material strengths are mathematical concepts. They have no physical meaning.

• Design or factored material parameters do not represent an extreme design situation. They may be physically impossible.

• Any value of a design parameter within its probability distribution may be chosen as the characteristic value. No particular value or definition is implied by the LSD method. Design codes and standards usually state a definition for use within the standard on which the specified partial factors are dependent.

• The limit state design analysis models need not describe the real behaviour at the perceived equivalent physical limit condition. Analysis models need only be consistent with the definition of the limit state.