To account for the complex behavior of steel structures, both geometric and material nonlinearities should generally be considered. Traditionally, any limit state that is not included in the structural analysis must be accounted for using appropriate design checks. For example, a linear (first-order) analysis does not capture member buckling; thus, a corresponding member buckling check is required. Advanced analysis reduced the number of required design checks by incorporating various limit states into the analysis itself.

Ideally, all design checks can be eliminated and the capacity of a structural system defined simply by the peak load factor reached during the analysis. However, although frame and member instability (i.e., P-Δ and P-δ) effects are generally accounted for in an advanced analysis, more complex phenomena such as local buckling (and other cross-section level buckling modes) cannot be captured using beam elements and are thus typically excluded from the analysis and considered approximately through cross-section classification and corresponding capacity checks. The design approach proposed herein departs from this practice and employs the CSM strain limits to mimic the effects of local buckling in beam elements. It is hence possible to use the same advanced nonlinear analysis with beam elements for structures comprising cross-sections of all four classes, since the strain limits control the degree of local buckling, spread of plasticity, and plastic redistribution in a rational manner. This is in contrast to the step-wise treatment in current design codes, whereby full global plastic redistribution is allowed in the case of Class 1 cross-sections, while no redistribution is permitted for other classes of cross-section.

The CSM is a deformation-based design approach that relates the cross-section slenderness to the deformation capacity [11]. Instead of classifying cross-sections into discrete groups, the CSM base curve defines the peak strain that a cross-section can endure. Utilizing this strain limit, integration of the corresponding stress distribution obtained from the adopted material model (see Section 2) over the area of the cross-section yields the cross-section capacity. Including the CSM strain limits in advanced analysis enables the CSM cross-section capacity to be computed directly, since numerical integration is performed at each load increment of the analysis.

The CSM base curve, shown in Fig. 6, provides a continuous relationship between the cross-section slenderness λ¯p and its deformation capacity. The latter is defined in normalized form as ε_csm/ε_y, where ε_csm is the maximum compressive strain a cross-section can sustain and ε_y is the yield strain. The base curve is split into two parts: Eq. (7) applies to non-slender sections with λ¯p≤0.68, which are referred to as Class 1 to 3 cross-sections in EN 1993-1-1, and Eq. (8) applies to slender cross-sections with λ¯p>0.68, which are referred to as Class 4 cross-sections in EN 1993-1-1. Note that in Eq. (7), two upper bounds have been placed on the predicted CSM strain ratio ε_csm/ε_y; the first limit Ω defines the level of plastic deformation that is deemed tolerable at ultimate limit state on a given project with a recommended value of 15, while the second limit of C₁ε_u, where C₁ is a material coefficient described in Section 2 and ε_u is the material ultimate strain which may be determined using Eq. (6), defines a “cut-off” strain to avoid over-predictions of material strength. Also note that both Eqs. (7) and (8) pass through the identified transition point, that is, (0.68, 1) in Fig. 6, thus ensuring compatibility between the CSM base curves for non-slender and slender cross-sections.

The accuracy of the CSM strain limits used in advanced analysis depends on the ability to accurately determine the cross-section slenderness. The cross-section slenderness is a dimensionless parameter that quantifies susceptibility to local instability and is defined by Eq. (9), where f_y is the yield stress and σ_cr is the elastic critical buckling stress.

Various methods are available to calculate the elastic critical buckling stress. Standard plate theory, utilizing buckling coefficients k such as those presented in EN 1993-1-5 [20], is simple to use yet ignores any element interaction by assuming simply supported boundary conditions along the adjoined plate edges. The elastic critical buckling stress can then be calculated using Eq. (10), where t and b are the relevant plate thickness and width, v is the Poisson's ratio; the plate element with the lowest buckling stress is then used to define the cross-section slenderness through Eq. (9).

Alternatively, approximate expressions calibrated against numerical results [21,22] may be used to determine the elastic critical buckling stress of the full cross-section. Plate interaction is implicitly accounted for in these expressions, which increases their accuracy. The expressions in Ref. [21] only cover pure compression and pure major/minor axis bending, whereas in structural frames, cross-sections are typically subjected to a combination of compression and bending; the expressions developed in Ref. [22] address this limitation and cover compression, bending and combined compression plus bending. Numerical tools, such as the constrained and unconstrained finite strip method (CUFSM) [23], can also be used to determine the elastic critical buckling stress of the full cross-section; CUFSM is employed in the present study.

Experiments [24,25] have shown that the maximum in-plane bending resistance of a member is greater under a moment gradient than under uniform bending. The increase in capacity arises from the restraint afforded to the critical section (i.e., that is under maximum bending moment) by the lower stressed adjacent regions, thus delaying the onset of local buckling in both the elastic and plastic regimes. A moment gradient is of course present in most practical applications, yet EN 1993-1-1 and the CSM base curve do not consider the effects of local moment gradients on the local stability of cross-sections.

It is proposed to exploit the beneficial influence of moment gradients on local stability by applying the CSM strain limit to a strain extracted from the advanced analysis that is averaged over a finite member length, as opposed to considering simply the most highly stressed (or strained) cross-section, as is customary. Since local buckling requires a finite length of member over which to develop—that is, the local buckling half-wavelength L_b—this represents a suitable distance over which to average the strains. This approach is in line with the observations of Lay and Galambos [26], who demonstrated experimentally that inelastic local buckling commences only when yielding has extended to a finite length of member related to the local buckling half-wavelength. By applying the CSM strain limit to an averaged strain, it is therefore possible to directly account for the magnitude of the local moment gradient; under uniform moment, the averaging approach predicts the cross-section capacity, while for increasing moment gradients, the peak moment capacity increases accordingly. The strain-averaging approach also reduces the sensitivity of the strength predictions to the mesh density employed in the FE model.

In this section, the proposed advanced analysis is applied to a series of individual members, as well as continuous beams and a frame, in order to illustrate how the CSM strain limits are able to mimic local buckling, control the level of inelastic redistribution of forces and moments, and capture the beneficial effects of moment gradients. For comparison, the capacity predictions according to EN 1993-1-1 are also shown. In all cases, calibrated shell FE models featuring local geometric imperfections were employed to provide benchmark results. The shell models were developed in Abaqus [27] and solved using the modified Riks method [28]. The quad-linear material model described in Section 2 was adopted in the simulations.

The normalized bending capacity at the critical cross-section of a series of beams with cross-sections of different local slenderness analyzed under four-point bending and as a cantilever is shown in Fig. 7, where M_max is the maximum bending moment, M_pl is the plastic moment capacity, and M_el is the elastic moment capacity. It may be seen that application of the CSM strain limits to the beam element models yields safe-sided predictions of the shell FE model results across the range of cross-section slenderness. According to EN 1993-1-1, there is an artificial “step” in bending capacity from Class 2 to 3 cross-sections, which can be eliminated using the modified elastic–plastic bending capacity [29] to account for partial yielding in Class 3 cross-sections. For Class 4 cross-sections, the strain limits predict similar capacities to Eurocode 3 but avoid the cumbersome calculation procedures associated with the effective width method. The shell FE models capture an increase in bending capacity of 5%–10% across the range of cross-section slenderness from the uniform moment case to the cantilever case, due to the beneficial influence of the moment gradient. Similarly, the proposed advanced analysis using beam elements accurately predicts an increase of 2%–7% due to the moment gradient, which is accounted for through the strain-averaging approach. On the other hand, EN 1993-1-1 does not capture this effect and does not distinguish between the two cases, but predicts a single bending capacity for each cross-section.

At an isolated member level, there is no scope to allow for moment redistribution. In indeterminate structures, however, such as continuous beams, the benefits of the proposed method can become even more significant. In addition to the local cross-section strength and the effects of moment gradients, the collapse load of an indeterminate system depends on the level of inelastic force and moment redistribution. Fig. 8 shows the ultimate load factor at collapse, α_u, normalized by the load factor at which the first yield occurs, α_el, for shell FE models of a series of continuous beams under point loads P multiplied by load factor α, along with the capacity predictions from the proposed method of design by advanced analysis and Eurocode 3. EN 1993-1-1 only permits the use of plastic analysis methods, which are able to capture the redistribution of forces and moments as the material yields, for Class 1 cross-sections. On the other hand, Class 2 cross-sections must be analyzed elastically and thus benefit from no redistribution, even though some yielding occurs before their plastic moment capacity is reached. The resulting steps in resistance predictions clearly do not reflect the shell FE results and are an overly simplistic representation of reality. The use of a consistent advanced analysis, in which the CSM strain limits define the cross-section dependent level of permissible redistribution, is shown to provide significantly more accurate capacity prediction. The average capacity prediction for the 46 analyzed continuous beams according to EN 1993-1-1 is 0.810 of the shell FE results, with a coefficient of variation (COV) of 5.3%. In comparison, the proposed method of design by advanced analysis predicts a mean capacity of 0.924 of the shell FE results, with a COV of 4.9%.

Considering structural behavior at the frame level, a slightly modified version of “Test frame 2,” as reported by Avery and Mahendran [30], is used to illustrate the full benefits of the proposed design method. The single bay portal frame comprised a single Class 3 cross-section throughout. Compared with the original test frame, the material model was adjusted to the quad-linear material model described earlier with an elastic modulus E = 200 000 MPa, a yield stress f_y = 360 MPa, and an ultimate stress f_u = 510 MPa, matching the measured flange material properties of “Test frame 2.” The global sway imperfection was based on the recommendations of EN 1993-1-1, while the member imperfections were taken as l/1000 in the most unfavorable direction, where l is the member length; both P-Δ and P-δ effects are therefore directly accounted for in the advanced analysis.

The load versus normalized strain (averaged over L_b from the critical cross-section) responses of the shell and beam FE models of the single bay portal frame are shown in Fig. 9. The measured test collapse load of the frame was P = 135.0 kN. In the shell FE model, some yielding is observed at the base of the right column prior to local buckling and ultimately collapse of the frame at a load of P = 133.9 kN. In contrast, the beam FE model cannot capture the local buckling at the base of the column and reaches a global sway failure at a peak load of P = 156.0 kN when allowing for unlimited strains—clearly an unsafe situation, since the shell FE collapse load is over-predicted by 16%.

To ensure safe capacity predictions, suitable cross-section and member design checks are required by EN 1993-1-1 following an elastic analysis of the structure; failure is predicted at a load of P = 108.9 kN. This is conservative by 20% compared with the shell FE model, since no account or benefit is taken from the spread of plasticity. Allowing for partial plastification in Class 3 cross-sections using a linear transition from the plastic to the elastic bending moment capacity through the elastic-plastic moment M_ep [29] increases the failure load prediction to P = 116.9 kN.

The proposed method of design by advanced analysis defines failure of the frame as the load level at which the CSM strain limit of 1.35ε_y for a modeled cross-section with λ¯p = 0.626 is reached (which occurs first at the base of the right-hand column). This occurs at a load level of P = 127.0 kN, which is only 6% shy of the shell FE collapse load. It should be emphasized again that the strains considered in the averaging procedure are the extreme fiber compressive flange strains of all elements fully contained within L_b. The described frame example illustrates both the ease of application of the proposed approach and the enhancements in design efficiency that can be achieved.