Transverse Elements-to-HSS: Reliability of the Plate Local Yielding Limit State

by Jeffrey A. Packer and Ian G. Kennedy
Department of Civil & Mineral Engineering, University of Toronto, Ontario, Canada
August 2026

A transverse element, welded to the face of a rectangular HSS member, may arise in a transverse plate-to-HSS connection (Figure 1(a)) or an HSS-to-HSS gapped or overlapped K-connection (Figure 1(b)).  In the latter, the “transverse element” is the wall of a branch member that is transverse to the HSS member on which it sits. Thus, in Figure 1(b), wall i of the overlapping branch is the transverse element that sits on transverse wall j of the overlapped branch.

The design of transverse plate-to-HSS welded connections is covered by Chapters J and K of the AISC 360 Specification (AISC, 2022), with elaboration in Part 9 of the AISC Manual (AISC, 2023). The pertinent limit states are, however, more clearly displayed in AISC Design Guide 24 (DG24) (Packer and Olson, 2024), and are reproduced in Table 1. Limit states 4, 5 and 6, in that table, apply when the ratio of the width of the branch (plate) is approximately matched to the width of the chord, or β ~ 1.0. Usually the plate is sized to fit on the chord flat connecting face (β < 1.0) and then, for limit states 1, 2 and 3, it has been shown that limit state 3 (local yielding of the branch plate) theoretically governs over limit states 1 and 2, according to the simple nominal strength formulas (Packer, 2015; Packer and Olson, 2024). The local yielding limit state for transverse plates (or plate-like elements), highlighted in Table 1, is expressed by Eq. (K1-1) in the AISC Specification (AISC, 2022) in the format of a transverse element effective width (B_e). The importance of this transverse element local yielding limit state is such that it appears in the limit states applied to T-, Y-, Cross and gapped/overlapped K-connections involving rectangular HSS (Packer and Olson 2024).   

The purpose of this article is to delve into the background to this plate local yielding limit state, and to investigate the reliability of the assigned available strength by using contemporary best-practice structural reliability methods. As one can see in Table 1, the limit state nominal strength formula in question has a LRFD resistance factor (ϕ) of 0.95, whereas – being a yielding element in tension or compression – one might expect a resistance factor of 0.90 per AISC Specification Section J4.1 or Section J4.4 (AISC, 2022).

Background and Available Data

A series of transverse plate-to-HSS connection tests was performed in The Netherlands more than 45 years ago (Wardenier et al., 1981), using low-strength, hot-formed hollow sections. That research produced plate effective width expressions for B_e and B_ep, Eqs. (K1-1) and (K1-2) respectively in the AISC 360 Specification, with one difference – the constant in the equations was 13.5, rather than 10 (which can be seen in equations for limit states 2 and 3 in Table 1). Over time, this constant of 13.5 has been rationalized downwards, to 11.7 for Grade 50 HSS (Davies and Packer, 1982), and lower still in order to provide sufficient “safety”. In the first edition of CIDECT Design Guide No. 3 on rectangular HSS connections (Packer et al., 1992) this constant was 10, and has remained at this value to the most-recent technical publication on this topic (CEN, 2024), with an implied resistance factor of 1.0. The available database for transverse plate-to-rectangular HSS connections is shown in Table 2, which is augmented through the inclusion of additional experiments and numerical studies (Lu et al., 1993).

Table 3 provides the results of the experimental/numerical tests (T_nx) for the specimens listed in Table 2. To investigate the reliability of the plate local yielding limit state expression, specimens that failed in this manner must be identified. Neither Wardenier et al. (1981) nor Lu et al. (1993) explicitly state the failure mode of individual specimens, though recorded experimental observations can be used to identify instances of weld failure or tube tearing. The characterization of other specimens is done by analyzing the force-displacement behavior of each test. If the transverse plate element fails by local yielding, the experimental data will show a distinct peak load, after which the force decreases. Conversely, specimens experiencing chord plastification (limit state 1 in Table 1) will not exhibit a peak load, as large displacements will be supported before an eventual strength plateau is reached. The ascertained failure modes are corroborated by the values of β in Table 2; chord plastification may govern when β is moderate to low (i.e., less than 0.75), whereas plate local yielding often governs when β is approximately equal to the width of the chord flat connecting face (which is often the prevalent case, see Figure 1(a)).

Reliability Analysis

The reliability of the local yielding limit state expression in Table 1 can be investigated by calculating resistance factors using practical reliability assessment methods. Several methods are provided in different technical standards, though different underlying assumptions and simplifications often make it unclear which method should be used. Therefore, this article provides example calculations for three common methods. Based on the results of these analyses, the calculated resistance factors can be compared against the values that are currently recommended for design by DG24 (Packer and Olson, 2024) and the AISC Specification (2022). A comparison of the accuracy of different reliability assessment methods can also be made, to inform an optimised approach to calculating resistance factors that balances complexity and accuracy.

In LRFD, resistance factors are calculated to achieve a predetermined probability of failure. This is accomplished by using target reliability indices (β⁺) as inputs within reliability assessment methods. Often, resistance factors will be calculated using a value of β⁺ and then rounded downwards to a convenient number (e.g., 0.93 rounded to 0.90). This approach simplifies design and produces conservative solutions. Therefore, resistance factors used in design can be assigned any value less than or equal to the calculated magnitude, with a practical upper limit of ϕ = 1.00 for ultimate limit states. For the case of local yielding of transverse plate-to-HSS welded connections, a target reliability index of β⁺ = 3.0 is used, as given by ASCE 7 (2022) for ductile failure modes.

Separation Factor Approach

The first method under investigation is known as the separation factor approach (SFA) and is recommended by ASCE 7 (2022). This approach has been used extensively to produce resistance factors since the origin of LRFD, though various technical standards provide different values of the separation factor (α_R). ASCE 7 (2022) suggests a value of 0.7, however the most used definition of α_R is 0.55 (Galambos and Ravindra, 1973). The SFA also requires the resistance bias coefficient (ρ_R) and coefficient of variation (CoV) (V_R), which account for the difference between the actual and nominal resistance. These parameters capture the variation in material and geometric variables, as well as the professional factor, which considers the variable accuracy of the nominal strength expressions. The resistance bias coefficient can be calculated as the product of the material, geometric, and professional factors, and the resistance CoV as the square root of the sum of the squares of the corresponding CoVs. Representative material and geometric statistics can be produced from experimental databases or found in the literature, whereas the professional factor and CoV are taken as the mean value and the CoV of the ratio of the experimental (or numerical) to nominal strengths. For the case of local yielding of transverse plate-to-HSS welded connections, the professional factor statistics are ρ_P = 1.57 and V_P = 0.15 and the material and geometric statistics are taken from the literature for ASTM A500 Grade B/C rectangular HSS and ASTM A572 Grade 50 plate: ρ_M = 1.28, V_M = 0.09, ρ_G = 0.89, and V_G = 0.021 (Schmidt and Bartlett, 2002; Liu et al., 2007; Liu, 2016). The resistance factor can then be calculated using the SFA, which gives a value of ϕ = 1.34.

AISC 360-27 Appendix 9

An alternative reliability assessment method is presented in Appendix 9 of the forthcoming (2027) edition of the AISC Specification. This method is based on a similar approach provided in AISI S100 (2020), though statistical parameters including the calibration coefficient (C_ϕ = 1.48) and the CoV of the load effect (V_Q = 0.19) have been recalibrated for structural steel, as opposed to cold-formed (or light-gauge) steel. The 2027 Specification Appendix 9 method also uses material, geometric, and professional bias coefficients and CoVs, though values for these statistics are provided in Appendix 9, depending on the type of component and failure mode. For local yielding of transverse plate-to-HSS, statistics are taken for the case of “yielding in the connection elements”, which gives the following parameters: ρ_M = 1.16, V_M = 0.10, ρ_G = 1.00, and V_G = 0.05 (AISC, 2025). This method also includes a sample size correction factor, to account for the inherent uncertainty when the database is small. Because only 10 experimental/numerical tests exist for the local yielding limit state (Table 3), it is expected that the sample size correction factor will have a non-negligible impact. The resistance factor can be calculated using the 2027 Specification Appendix 9 method, which gives a value of ϕ = 1.15.

AISC 360-27 Appendix 9

The first-order reliability method (FORM) is another approach that has been used extensively in North America and is generally considered to be the most accurate closed-form reliability assessment method. This is a result of the FORM considering the variation in both the resistance and load variables, the latter of which is ignored in the SFA and simplified in the AISC 2027 Specification Appendix 9 method. An approximate FORM can be performed by assuming a governing load combination, which, for steel structures, is typically 1.2D + 1.6L from ASCE 7 (2022) (Ellingwood et al., 1980). Like the SFA, the FORM uses the bias coefficient and CoV of the resistance to account for material, geometric, and professional factor variation, so statistical parameters are identical to this approach. The calculation of the CoV of the loads (V_Q) and the resistance factor require bias coefficients and CoVs for the dead (D) and live (L) loads, which are taken as: ρ_D = 1.05, V_D = 0.10, ρ_L = 1.00, and V_L = 0.25 (Ellingwood and Culver, 1977; Ellingwood et al., 1980). Because the ratio of the live-to-dead loads (L/D) can vary, a conservative approach is to calculate resistance factors across the range of 0 ≤ L/D ≤ 3 and select the minimum value. However, it is important to note that the principal dead load combination (1.4D) governs when L/D = 0. The FORM can be used to calculate resistance factors for incremental values of L/D, which are listed in Table 4. From Table 4, a representative value of ϕ = 1.13 is selected.

Monte Carlo Simulation

In contrast to the previous closed-form reliability assessment methods, numerical approaches can be used to determine resistance factors with a high degree of accuracy, though they are often overly complex and time consuming for routine usage. Monte Carlo simulations (MCS) have been used historically for reliability analyses, as they can converge to the “exact” resistance factor. MCS operate by randomly sampling probabilistic distributions of all input parameters to calculate a sample resistance and load effect. When this is repeated many times over a range of representative connections, the probability of failure can be determined. Thus, MCS produce a distribution of the reliability index (β⁺) over the range 0 ≤ L/D ≤ 3 for each trial resistance factor under assessment. The exact solution is the trial resistance factor that exceeds the target reliability index over the full L/D range. For the case of local yielding of transverse plate-to-HSS welded connections, the likely exact resistance factor was determined to be ϕ = 1.13 by preliminary approximate MCS. This is shown as the purple curve in Figure 2, plotted alongside the results for resistance factors of ϕ = 0.90, 0.95, and 1.00. The dip in reliability at L/D = 0.1 is produced due to the switch between the principal live and dead load combinations, which is seen to produce the minimum reliability for this case.

Conclusion

Figure 2 shows that the exact resistance factor of ϕ = 1.13 is sufficient to meet the target reliability index of β⁺ = 3.0, and that ϕ = 0.90, 0.95, and 1.00 all produce excessive reliability. Therefore, the use of a resistance factor of ϕ = 0.95 for the local yielding limit state of Table 1 provides adequate structural reliability. The primary reason for the magnitude of the exact resistance factor being much greater than typical values is the professional factor, which has a bias coefficient of ρ_P = 1.57 and a CoV of V_P = 0.15. These values indicate an overly conservative correlation between the nominal expression and the experimental/numerical test results. Given this poor correlation and the small experimental database, an opportunity for further research exists in the conduction of additional experimental/numerical tests and the derivation of a more-accurate nominal strength expression.

While the SFA, the 2027 AISC Specification Appendix 9, and FORM all predict resistance factors greater than 1.00, their results can be analyzed to determine the relative accuracies of each method. In comparison to the exact solution, the best prediction was provided by the FORM, followed by the 2027 AISC Specification Appendix 9 method, which was slightly unconservative. This is likely the result of using the statistical parameters given by Appendix 9 over the values from the literature used in the FORM. This is confirmed by recalculating the resistance factor using the material and geometric statistics form the literature, which gives a value of ϕ = 1.13, equal to that of the FORM and the MCS. Conversely, the resistance factor produced by the SFA was unconservative and would produce unsafe solutions if used. This analysis shows that the FORM should be used to calculate resistance factors, though the 2027 AISC Specification Appendix 9 method can also provide accurate resistance factors if input statistics are fully representative of the limit state.

References

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