Introduction

The design of structural steel connections requires evaluation of many limit states, consideration of many behavioral effects, and adherence to many requirements. The AISC Specification, AISC Manual, and other references describe methods of design used in US practice. Currently, the most widely used methods rely predominantly on calculations that can be performed by hand. However, advances in computer hardware and software enable a different type of design that relies on nonlinear structural analysis. 

The use of nonlinear analysis in design can be advantageous for complex or unique connections, where the assumptions of traditional calculations are unproven. Yet, the same limit states, design considerations, and design requirements apply. Good connection design comes from engineers who know these design criteria and how their tools capture them. 

This document is intended to be a detailed, but not exhaustive, listing of limit states, design considerations, and design requirements relevant to structural steel design and a description of how they are considered in traditional calculations and in IDEA StatiCa using the component-based finite element method. 

This document is constantly being updated as verification and investigation exercises are still in process. 

The content in this article references the 2022 AISC Specification and the 16^th Edition AISC Manual. 

Limit States

Weld Rupture

The AISC Specification includes provisions for groove welds, fillet welds, and plug and slot welds. Of these, complete joint penetration (CJP) groove welds and fillet welds are the only types that can currently be defined in IDEA StatiCa.

CJP groove welds and butt welds in IDEA StatiCa, are modeled by directly connecting the components using multi-point constraints. The multi-point constraints introduce no flexibility. Also, the strength of these welds is not checked since the strength of the CJP groove welds is controlled by the base metal.

Fillet welds are also modeled using multi-point constraints and an equivalent weld shell element that approximates the elastoplastic behavior of the weld. The forces in these shell elements are extracted and used as required strengths for comparison to available strengths computed according to the AISC Specification.

The available strength of welds is defined in AISC Specification Section J2.4. For fillet welds, the nominal strength is the product of the nominal stress of the weld metal, F_nw, the effective area of the weld, A_we, and a directional strength increase factor, k_ds. AISC Specification Table J2.5 sets F_nw = 0.6F_EXX and references AISC Specification Section J2.2a for the definition of A_we. For each weld segment, A_we is taken as the throat thickness times the length of the weld segment. The reductions to the effective length for long welds in AISC Specification Section J2.2b are not applied; however, the effects of long welds are captured explicitly as described in the entry on Deformation Compatibility in Long Connections.

The directional strength increase factor is defined in AISC Specification Section J2.4. When strain compatibility of the various weld elements is considered (as is the case in IDEA StatiCa because the stiffness of the welds and connecting elements are explicitly modeled), k_ds is a function of the angle between the line of action of the required force and the weld longitudinal axis. IDEA StatiCa determines the line of action from the internal forces in the equivalent weld shell element and computes k_ds and the nominal strength for each weld segment.

To illustrate the effect of the directional strength increase, consider the welded specimens tested experimentally by Miazga and Kennedy (1989). The specimens had loading angles of 0, 15, 30, 45, 60, 75, and 90 degrees as shown in the figure below where the units are millimeters. The plates were fabricated with CAN3-G40.21-M8 grade 300W steel. The exterior plates had a measured yield strength of 52.8 ksi. The interior plates had a measured yield strength of 50.2 ksi. E48014 electrodes with a nominal strength of F_EXX = 70 ksi were used.

The maximum permitted applied loads were determined for each specimen in IDEA StatiCa using models with measured plate material properties, nominal filler metal properties, and including resistance factors. The maximum permitted applied loads were normalized by the total length of weld in the connection and are shown in the figure below. Also shown are the design strength according to the AISC Specification (including the directional strength increase factor and resistance factor) and the experimental strength.

The angle of loading measured from the weld longitudinal axis for each specimen, as output by IDEA StatiCa in the weld results, is listed in the table below.

The IDEA StatiCa and AISC Specification strengths are both much less than the experimental strengths. There are several reasons the experimental strengths are higher: they do not include resistance factors, the actual filler metal strength is likely more than the nominal strength, and the actual failure area of the weld is likely greater than assumed in design calculations.

The strengths from IDEA StatiCa are slightly less than those according to the AISC Specification, but both show an increase with loading angle. Furthermore, the geometric angle of the specimen differs from the angle of loading measured from the weld longitudinal axis as output by IDEA StatiCa. These differences occur because welds are divided into short segments when modeled in IDEA StatiCa. Unlike traditional calculations where the demands along the length of the weld are assumed to be uniform, the weld segments experience different demands based on the stiffness of the weld and connecting elements. The angle output by IDEA StatiCa is for the weld segment that has the greatest utilization ratio. Often this is a segment at the end of a weld. For these specimens, the aggregate effect of the non-uniform demands is a slight reduction in strength.

A special case applies for fillet welds to the ends of rectangular HSS loaded in tension where k_ds = 1.0. In IDEA StatiCa, the directional strength increase factor is not used for fillet welds to the ends of rectangular HSS, regardless of loading.

AISC Specification Section J2.4 also defines the strength of the base metal. For fillet welds, AISC Specification Table J2.5 references AISC Specification Section J4 for base metal checks. Base metal strength checks are described in greater detail in the entry on Weld Base Metal Strength.

Weld Base Metal Strength

In welded connections, the strength of the connecting elements adjacent to the weld is called base metal strength. In many cases, potential limit states can be identified, and the available strength of the base metal can be computed using the provisions of AISC Specification Section J4. The evaluation of these limit states in IDEA StatiCa is described in entries on the individual limit states, including Tensile Yielding, Tensile Rupture, Shear Yielding and Rupture, and Block Shear Rupture.

However, in some connections, potential limit states adjacent to the weld are hard to identify, and the available strength of the base metal cannot be directly computed by hand. For these cases, the AISC Manual provides Equations 9-6 and 9-7 for the minimum base metal thickness that matches the weld with some assumptions. This equation is not evaluated in IDEA StatiCa since potential base metal limit states need not be identified a prior, and strength is assessed with the 5% plastic strain limit. However, engineers may still use the limit to size welds and connecting elements.

IDEA StatiCa provides an option to check base metal capacity at the fusion face. This check can be enabled in the “Code setup” window. This check is not commonly performed in US practice and is generally not necessary when the filler metal is appropriately matched to the base metal. The commentary on AISC Specification Section J2.4 states that tests have demonstrated that the stress on the fusion area is not critical in determining the shear strength of fillet welds.

Bolt Shear and Tensile Rupture

The available strength of bolts subject to tension or shear is defined in AISC Specification Section J3.7. The available strength of bolts subject to combined tension and shear is defined in AISC Specification Section J3.8. IDEA StatiCa uses these provisions directly to compute available strengths, compared to required strengths determined from nonlinear analysis. As specified, the required tensile strength determined from nonlinear analysis includes tension resulting from prying action.

A footnote in AISC Specification Table J3.2 requires that the nominal shear stress, F_nv, of A307 bolts be reduced when the grip of a bolt is greater than five times its diameter. This reduction is not implemented in IDEA StatiCa. Therefore, the nominal shear stress of long A307 bolts needs to be manually adjusted in the materials tab.

Bearing and Tearout at Bolt Holes

The strength of bolts in shear can be limited by bearing or tearout at the bolt holes. It is sometimes common practice to evaluate bearing and tearout separate from bolt shear rupture. However, bolt groups can fail with some bolts rupturing and others tearing out. A user note in AISC Specification Section J3.7 states “The effective strength of an individual fastener may be taken as the lesser of the fastener shear strength per Section J3.7 or the bearing or tearout strength at the bolt hole per Section J3.11. The strength of the bolt group is taken as the sum of the effective strengths of the individual fasteners.”

IDEA StatiCa evaluates the strength of each bolt individually, with required strengths determined from the nonlinear analysis and available strengths calculated using the provisions of the AISC Specification. This evaluation complies with the user note in AISC Specification Section J3.7. However, IDEA StatiCa does not simply sum up the effective strengths of the individual fasteners. The approach taken by IDEA StatiCa can result in a conservative underestimation of strength.

Consider the three-bolt connection shown below. The connection is short and the stiffness of the three bolts is equal because the load-deformation response for bolts in IDEA StatiCa does not depend on edge distance, therefore the applied load is shared approximately equally among the bolts. The strength of the bolt with the 1 in. edge distance is controlled by tearout. IDEA StatiCa indicates failure when the first bolt reaches 100% utilization. Since the bolt with 1 in. edge distance has the lowest available strength (ϕr_n = ϕ1.2dtF_u = 17.4 kips), it reaches 100% utilization first. The other bolts are stronger (ϕr_n = 35.8 kips, AISC Manual Table 7-1) but they do not reach 100% utilization, resulting in a connection strength of 52.5 kips. By traditional calculations, each bolt is assumed to reach its effective strength, resulting in a connection strength of 89.0 kips, 70% greater than the strength from IDEA StatiCa.

Three-bolt bolted connection

Three-bolt bolted connection with 57.5 kips of applied load

Two sets of equations are provided in AISC Specification Section J3.11a, one when deformation at the bolt hole at service load is a design consideration and one when deformation at the bolt hole at service load is not a design consideration. The choice of whether deformation at the bolt hole at service load is a design consideration can be made in the “Code setup” window.

Different equations are also provided in AISC Specification Section J3.11a for long-slotted holes when the slot is perpendicular to the direction of force. Slotted holes can be defined in IDEA StatiCa using the plate editor. The bearing and tearout equations in the AISC Specification for long-slotted holes are used for all slotted holes in IDEA StatiCa, regardless of slot length.

AISC Specification Section J3.11b requires the use of the bearing provisions of Section J7 for bolts or rods that pass completely through an unstiffened box member or hollow structural section (HSS). This provision is not implemented in IDEA StatiCa and bearing is evaluated in such connections as though they are regular bolted connections with all plies making firm contact. A warning is provided in the report if the bolt grip length is larger than the sum of the thicknesses of connected plates. 

When evaluating tearout, IDEA StatiCa determines the clear distance, in the direction of the force, between the edge of the hole and the edge of the adjacent hole or edge of the material, l_c, using the direction of force for each bolt from the nonlinear analysis. This feature is particularly helpful for eccentrically loaded bolt groups where the direction of force varies from bolt to bolt. The tearout limit state was investigated for bracket plate connections in this article and for single plate shear connections in this article.

Bearing (Local Compressive Yielding)

AISC Specification Section J7 defines the available strength for the limit state of bearing (local compressive yielding). These provisions apply to specific cases of contact between steel components but are not implemented in IDEA StatiCa.

For finished surfaces and ends of fitted bearing stiffeners, while the contact bearing pressure is not checked against the limit prescribed in the AISC Specification, stresses in contacts can be plotted and yielding of the steel components often provides a more controlling limit since the allowable bearing pressure exceeds the yield strength.

IDEA StatiCa evaluates the bearing strength of bolts or rods that pass completely through an unstiffened box or HSS member as though they are regular bolted connections with all plies making firm contact and not using the provisions of AISC Specification Section J7. A warning is provided in the report if the bolt grip length is larger than the sum of the thicknesses of connected plates. See also Bearing and Tearout at Bolt Holes.

Expansion rollers and rockers cannot be modeled in IDEA StatiCa. Pins were introduced to IDEA StatiCa in version 24.0 and are currently only available for design by Eurocode.

Slip

Connections are required to be designed as slip-critical when they are subject to fatigue load with reversal of loading direction, when they use oversized holes, when slip at the faying surfaces would be detrimental to the performance of the structure, and for other reasons. Available strength for the limit state of slip is defined in AISC Specification Section J3.9 with additional provisions in Section J3.10 for combined tension and shear in slip-critical connections. IDEA StatiCa uses these provisions directly to compute available strengths which are compared to required strengths determined from nonlinear analysis.

The slip coefficient, μ, is defined in the code setup. The factor for fillers, h_f, is determined automatically.

Differences between IDEA StatiCa and hand calculations can occur because of the reduction factor for tension, k_sc, defined in AISC Specification Section J3.10. IDEA StatiCa uses the tension in the bolt from the nonlinear analysis to calculate k_sc, even if the tension in the bolt was not caused by applied tension that reduces the net clamping force. For example, in an extended end-plate moment connection with a slip-critical connection between the end plate and the column flange (such as shown below), moment in the beam causes tension in the bolts in IDEA StatiCa. Physically, any loss of clamping force near the bolts on the tension side of the beam due to moment will be offset by an increase in clamping force near the bolts on the compression side of the beam. In hand calculations, the factor k_sc would not be used for this connection (unless the beam has a net tension force). But since IDEA StatiCa evaluates bolts individually, k_sc is conservatively applied to the bolts on the tension side of the beam reducing the overall slip strength of the connection. Incidental tension in a predominantly shear-loaded connection and tension from prying action are also conservatively included when calculating k_sc in IDEA StatiCa. 

AISC Specification Section J3.9 requires that slip-critical connections be designed for the limit states of bearing-type connections in addition to slip. IDEA StatiCa does not check bolt rupture, bearing, or tearout for bolts designated to transfer force through friction. Additionally, slip-critical connections are modeled differently than bearing-type connections in IDEA StatiCa. In slip-critical connections, the forces are transferred from one plate to another over a larger area more representative of force transfer through friction. The larger spread of transfer forces can lead to increased strength of connecting elements for limit states such as block shear rupture. For most connections, slip strength is less than the strength for the limit states of bearing-type connections. However, engineers should be aware of these limitations and address them in design. It is recommended that slip-critical connections be analyzed twice in IDEA StatiCa: once as a slip-critical connection (i.e., with shear force transfer type set to “Friction”) and again as a bearing type connection (i.e., with shear force transfer type set to “Bearing – tension/shear interaction”) to ensure all limit states are evaluated appropriately.

Tensile Yielding

Tensile yielding is among the most fundamental limit states in structural steel design. The nominal strength for tensile yield is defined in AISC Specification (2022) Section D2 for tension members and Section J4.1 for connecting elements as the specified minimum yield stress, F_y, times the gross area, A_g. Despite the simplicity of this equation, it is not used to assess strength in IDEA StatiCa. Members and connecting elements are modeled in IDEA StatiCa with shell elements that are assigned a nonlinear stress-strain relationship consisting of a linear elastic region and a linear plastic region. Shell elements can experience stress along multiple axes and the stress-strain relationships account for this. If subject to uniaxial stress, the stiffness in the elastic range is the modulus of elasticity, E, the stiffness in the plastic range is one-thousandth of the modulus of elasticity, E/1000, and the transition between elastic and plastic occurs at a stress of F_y times a resistance factor of 0.9 for LRFD or divided by a safety factor of 1.67 for ASD.

Instead of limiting the required strength to be no more than the available strength (e.g., R_u ≤ ϕR_n), IDEA StatiCa limits plastic strain to 5%. While this is a fundamentally different assessment, resulting strengths for tensile yield of the gross section of a member or component from the two approaches will never differ by a large amount. Minor differences can arise for two reasons: 1) the small increase in stress after yielding in IDEA StatiCa and 2) small differences in cross-sectional area.

A small post-yield stiffness (one-thousandth of the elastic stiffness) is used in IDEA StatiCa to avoid the computational difficulties that would arise with zero post-yield stiffness. At the 5% plastic strain limit, this results in approximately 0.05×E/1000 = 0.05×(29,000 ksi)/1000 = 1.45 ksi stress above the yielding stress. For ASTM A992 steel with a F_y of 50 ksi and using LRFD, tensile yield initiates in IDEA StatiCa at 0.9×50 ksi = 45 ksi. The extra 1.45 ksi of stress accumulated post-yield can lead to an approximately 3% increase in strength.

Structural steel members are modeled with shell elements in IDEA StatiCa resulting in some simplifications of the physical geometry. The shell elements only represent rectangular components thus fillets are neglected. Additionally, since shell elements are connected at nodes located at the center of thickness, there is some overlap at joints of cross-sectional elements. The figure below shows the simplifications for a wide flange shape. The simplifications cause small differences in cross-sectional area which can affect tensile yield strength. For a W14x159, the cross-sectional area listed in AISC Manual Table 1-1 is 46.7 in.². The cross-sectional area when modeled as in IDEA StatiCa is 2b_ft_f+(d-t_f)t_w = 2(15.6 in.)(1.19 in) + (15.0 in. – 1.19 in.)(0.745 in.) = 47.4 in.², where the cross-sectional dimensions were also determined from AISC Manual Table 1-1. This is a 1.5% difference.

The overall effect of these minor differences can be observed in a simple IDEA StatiCa model of a splice connection between two W14x159 (ASTM A992) steel shapes. The splice is butt welded (e.g., CJP) and loaded in tension. Per the AISC Specification (2022), the design strength of the wide flange tension member is 0.9×(50 ksi)×(46.7 in.²) = 2,100 kips. The maximum load that can be applied to the connection in IDEA StatiCa (version 22.1) is 2,180 kips, 4% greater than the design strength as calculated per the AISC Specification. The distribution of plastic strain in the connection shows that the full cross-section has yielded.

Tensile Rupture

Provisions for the limit state of tensile rupture are in AISC Specification Chapter D. These provisions are referenced in AISC Specification Section J4.1 for connecting elements. The nominal strength for tensile rupture is calculated as the tensile strength of the material, F_u, times the effective net area, A_e. The effective net area accounts for material removed, including bolt holes, and the effect of shear lag through the shear lag factor, U, defined in AISC Specification Table D3.1. A resistance factor of ϕ = 0.75 is applied to the nominal strength to determine the design strength.

The limit state of tensile rupture is not directly evaluated in IDEA StatiCa. It is captured by limiting the amount of plastic strain any component can experience. The default plastic strain limit in IDEA StatiCa is 5%. Neither F_u nor the resistance factor of ϕ = 0.75 are used in IDEA StatiCa. IDEA StatiCa uses a bilinear stress-strain relationship in which yield occurs at the yield stress of the steel, F_y, times a reduction factor equal to 0.9 by default (the user can adjust this factor). After yield, the stiffness of steel is only one thousandth of the modulus of elasticity. This post yield stiffness is included for numerical stability and does not provide any significant strain hardening. Additionally, IDEA StatiCa does not use the shear lag factors of AISC Specification Table D3.1. Instead, shear lag is modeled explicitly.

Also, the stresses that develop in connection regions are rarely purely uniaxial. IDEA StatiCa uses the von Mises yield criterion to identify when yielding occurs under these complex stress states which can lead to an apparent increase in strength. To illustrate this effect, consider the simple splice connection shown in the figure below. The strength of the central plate near the bolts controls the strength of this connection. Based on hand calculation procedures, one may expect that the strength that IDEA StatiCa will determine would be the stress at which yielding occurs times the net area (shown by a red dotted line in the figure). For this connection, the net area is (1/2 in.)×(8 in. – 2d_h) = 2.875 in.², where the diameter of the hole, d_h, equals 1-1/8 in. (note that IDEA StatiCa does not include the 1/16 in. for damage described in AISC Specification Section B4.3b, see the entry on Net Area Determination for additional information - ADD ANCHOR). For LRFD, the stress at which yielding occurs in IDEA StatiCa is 0.9F_y and there is minimal strain hardening (see the entry on Tensile Yielding for additional information). For the A36 material used in this example, yielding will occur at 0.9(36 ksi) = 32.4 ksi. Therefore, one may expect that the strength of this connection in IDEA StatiCa would be (2.875 in.²)×(32.4 ksi) = 93.1 kips. However, because the stress is not purely uniaxial at the net section, the other components of stress effectively increase the yield stress normal to the net area, and 5% plastic strain is not achieved until an applied load of 111.7 kips.

Taken individually, the differences between traditional calculations and IDEA StatiCa result in lower strengths in IDEA StatiCa (only using F_y and not F_u), higher strengths in IDEA StatiCa (using a material strength reduction factor of 0.9 instead of ϕ = 0.75), and different strengths depending on the specific connection (explicitly modeling shear lag instead of using the shear lag factor, U). Taken together, the differences usually, but not always, result in equal or lower strength from IDEA StatiCa than from traditional calculations.

The limit state of tensile rupture was investigated in this study through comparison to hundreds of experimental results. The results show that IDEA StatiCa is generally conservative, especially at the nominal strength level, but there are some cases where the available strength from IDEA StatiCa is greater than that calculated according to the AISC Specification. Using measured material and geometric properties without resistance factors applied, the strength from IDEA StatiCa was less than or equal to the experimentally observed strength for all but 12 specimens out of 529 (9 of which were fabricated with high-strength steel, F_y = 122.8 ksi) and less than or equal to the expected tensile rupture strength computed using design equations for all but 30 specimens out of 529. Using nominal material and geometric properties with resistance factors applied, the strength from IDEA StatiCa was found to be greater than the strength computed according to the AISC Specification for some connections without physical counterparts, especially plate tension members with relatively short welds, and rectangular HSS tension members. Given that the experimental data for these cases is limited, work is ongoing to determine if the differences are the result of unconservatism in IDEA StatiCa or conservatism in the AISC Specification equations.

Compressive Yielding and Buckling

The available strength of affected elements of members and connecting elements in compression is defined in AISC Specification Section J4.4. When the slenderness ratio, L_c/r, is less than or equal to 25 compressive yielding applies and the nominal strength is computed as the product of the specified minimum yield stress and the gross area (i.e., P_n = F_yA_g). Like for Tensile Yielding, the limit state of compressive yielding is evaluated in IDEA StatiCa with the 5% plastic strain limit.

When the slenderness ratio, L_c/r, is greater than 25, the provisions of AISC Specification Chapter E apply. Limit states in AISC Specification Chapter E include flexural buckling, torsional buckling, and flexural-torsional buckling. The nonlinear analysis performed in IDEA StatiCa is nonlinear because it includes effects such as yielding and contact. The analysis typically does not consider geometric nonlinearities such as P-Δ effects (geometric nonlinearities are considered when HSS are used as bearing members).

Engineers must also perform a linear buckling analysis to detect buckling. A linear buckling analysis can determine the elastic buckling load, expressed as a ratio of the applied load. While providing useful information that can guide design, the linear buckling analysis does not consider potential yielding that can reduce stiffness and the buckling load (i.e., inelastic buckling), nor does it consider the effects of initial geometric imperfections. Because of these limitations, to use IDEA StatiCa, the connection needs to be stocky enough that neither elastic buckling nor inelastic buckling will occur. The elastic buckling load ratio provides a convenient measure of stockiness (or slenderness).

Consider the slenderness ratio limit in AISC Specification Section J4.4 of L_c/r ≤ 25 to assume compressive yielding. A slenderness ratio of L_c/r = 25 corresponds to an elastic critical stress F_e = π²E/(L_c/r)² = π²(29,000 ksi)/(25)² = 458 ksi. For A36 steel, this corresponds to 14 times the factored yield stress for LRFD and 21 times the factored yield stress for ASD. For grade 50 steel, the elastic critical stress corresponds to 10 times the factored yield stress for LRFD and 15 times the factored yield stress for ASD. Accordingly, the elastic buckling load ratio should be kept greater than these ratios to avoid cases where inelastic buckling could control.

The appropriate limit on the elastic buckling load ratio varies based on the connection configuration. For plate buckling, the limit is much lower. Based on the limiting width-to-thickness limits in AISC Specification Table B4.1a, the elastic critical buckling load ratio should be kept to no less than 3 for LRFD and 4.5 for ASD . An evaluation of bracket plates identified elastic critical buckling load ratio limits of 4 for LRFD and 6 for ASD. The use of a critical buckling load ratio limit of 3 has been evaluated for bearing stiffeners (report coming soon), coped beams, and beam-over-column connections.

Elements of connections that are slender enough for inelastic buckling to occur still have strength, potentially enough strength for a given application. However, without the capability to accurately quantify inelastic buckling strength in IDEA StatiCa these cases must be avoided.

Shear Yielding and Rupture

The available strength of affected elements of members and connecting elements in shear is defined in AISC Specification Section J4.2. This section describes two limit states: shear yielding and shear rupture. For both limit states, IDEA StatiCa does not compute the available strength per the AISC Specification, but rather relies on the 5% plastic strain limit to evaluate if the connection is sufficiently strong.

In tension, the stress-strain relationship used in IDEA StatiCa is linear up to yield, with a stiffness equal to the modulus of elasticity, then linear thereafter, with a stiffness equal to one-thousandth of the modulus of elasticity. Yield in tension occurs at the specified minimum yield stress of the steel, F_y, times 0.9 for LRFD or divided by 1.67 for ASD. IDEA StatiCa uses the von Mises yield criterion to determine when yielding begins under multi-axial states of stress. According to the von Mises yield criterion, material subject to pure shear will yield when the shear stress equals the yield stress divided by the square root of 3. The inverse of the square root of 3 is approximately equal to 0.577 which is approximately equal to the 0.6 factor applied to the shear strength equations in the AISC Specification. This difference, or similar differences when the element is not strictly in pure shear, can lead to differences between IDEA StatiCa and traditional calculations. The small amount of strain hardening can also lead to differences as described in the entry on Tensile Yielding.

Differences can also arise because in AISC Specification Section J4.2, the resistance factor for shear yielding is defined as 1.00 and the safety factor for shear yielding is defined as 1.50. IDEA StatiCa does not utilize these factors and instead reduces the yield point by a factor of 0.9 for LRFD or by dividing by 1.67 for ASD based on the typical resistance factor and safety factor for yielding.

Other differences exist for the limit state of shear rupture. As described for the limit state of Tensile Rupture, IDEA StatiCa does not utilize the tensile strength of the steel, F_u, nor the resistance factor or safety factor for shear rupture. Again, the yield point in tension is taken as 0.9F_y for LRFD and F_y/1.67 for ASD. The result of these differences depends on the ratio of the material strengths. Also in bolted connections, the net area subjected to shear typically passes through the centerlines of the bolts. The distribution of plastic strains at the limit point in IDEA StatiCa can be different, as was seen for single plate shear connections in this article.

As an example of the combined result of the differences between the AISC Specification equations and IDEA StatiCa, consider the two beam splice connections shown in the figures below. For both, two W27×94 beams made of A992 steel are connected by splice plates on either side of the web. The splice plates are 3/8 in. thick and made of A36 steel.

The welded connection is controlled by shear yielding of the splice plates. The design strength for the plates is ϕR_n = ϕ0.6F_yA_gv = (1.0)0.6(36 ksi)(2 × 3/8 in. × 16 in.) = 259 kips. In IDEA StatiCa, the splice plates reach a plastic strain of 5% when subject to a shear load of 236 kips. The difference in strengths is mostly due to the use of ϕ = 1.0 in the AISC Specification equations and a reduction of 0.9 on the yield stress in IDEA StatiCa.

The bolted connection is controlled by shear rupture of the splice plates. The design strength for the plates is ϕR_n = 210 kips. In IDEA StatiCa, the splice plates reach a plastic strain of 5% when subject to a shear load of 213 kips, nearly the same as the design strength according to the AISC Specification, indicating that the differences counteract each other and result in a safe design.

Yielding Under Combined Actions

Members and connecting elements are often subject to multiple actions simultaneously, including axial force, bending moment, shear, and torsion. AISC Specification Section J4 does not provide specific requirements for connecting elements subject to combined actions. However, AISC Manual Part 9 describes several approaches for assessing connecting elements subject to combined actions. One approach is to superimpose stresses calculated based on elastic beam theory and use a first yield criterion. Another approach is to use interaction equations that approximate the limit of plastic strength. One such equation that applies to rectangular members under in-plane loading is AISC Manual Equation 9-1.

\[ \frac{M_r}{M_c} + \left ( \frac{P_r}{P_c} \right )^2 + \left ( \frac{V_r}{V_c} \right )^4 \le 1.0 \]

where M_r, P_r, and V_r are the required flexural, axial, and shear strengths, respectively; and M_c, P_c, and V_c are the available flexural, axial, and shear strengths, respectively.

Dowswell (2015) presented a more general equation for rectangular members under in-plane and out-of-plate loading.

\[ \left ( \frac{P_r}{P_c} \right )^2 + \left ( \frac{T_r}{T_c} \right )^2 + \left ( \frac{V_r}{V_c} \right )^4  + \left ( \left ( \frac{M_{rx}}{M_{cx}} \right )^{1.7} + \left ( \frac{M_{ry}}{M_{cy}} \right )^{1.7} \right )^{0.59} \le 1.0 \]

where T_r, M_rx, and M_ry are the required torsional, major-axis flexural, and minor-axis flexural strengths, respectively; and T_c, M_cx, and M_cy are available torsional, major-axis flexural, and minor-axis flexural strengths, respectively.

In IDEA StatiCa, connecting elements are modeled with shell finite elements that are assigned a multi-axial plasticity material model that uses the von Mises yield criterion (use of the von Mises yield criterion is also described in AISC Manual Part 9). As load is applied in the model, the individual shell elements experience general states of stress which are evaluated using the criterion to determine if yielding has occurred. If yielding occurs, the stiffness of the material is reduced to 1/1000 of the initial stiffness and the analysis continues.

To illustrate the differences between strengths computed using interaction equations and IDEA StatiCa, consider the connection shown below. The middle “test” plate has a thickness of 1 in., height of 6 in., a length of 10 in., and is made of A36 steel. Both the connecting plates and hollow section members were selected to be strong and stiff. Analyses were performed subjecting the test plate to biaxial loading, consisting of axial tension and bending moment about the major- and minor-axes to determine maximum permitted applied loads (i.e., the loads which cause 5% plastic strain in the test plate). For these analyses, the geometric nonlinear (GMNA) option was turned off in the code setup. Also, the maximal size of elements was changed to 0.25 in. and the minimal size of elements was changed to 0.10 in. to create a finer mesh and capture the stress distribution more accurately.

Results of the IDEA StatiCa analyses are shown in the figure below. Interaction diagrams based on the Dowswell (2015) equation are also shown in the figure. Available strengths used for the calculated interaction diagrams are P_c = ϕP_n = 194.4 kips, M_cx = ϕM_nx = 24.3 kip-ft, and M_cy = ϕM_ny = 4.05 kip-ft. Differences are seen between the IDEA StatiCa results and those from the interaction equation, including when only one action is applied. The causes of the differences under a single action are described in the entries on flexural yielding and tensile yielding. The differences between IDEA StatiCa and the approximate equation for combined actions are greater, but the IDEA StatiCa results show clear interaction effects.

Block Shear Rupture

Block shear rupture is a combined tension and shear failure in which a block of material is torn away from a member or connecting element. The available strength for the limit state of block shear rupture is defined in AISC Specification Section J4.3. As with the limit state of tensile rupture, the limit state of block shear rupture is not directly evaluated in IDEA StatiCa. It is captured by limiting the amount of plastic strain any component can experience to a maximum of 5% (the user can change this limit). Key differences between the traditional calculations and IDEA StatiCa result from the stress-strain relationship used in IDEA StatiCa. Only minimal post-yield hardening is included (i.e., stresses do not reach F_u), and the yield stress is reduced by 0.9 for LRFD (i.e., not ϕ = 0.75 as specified for block shear rupture).

A comparison between traditional calculations and IDEA StatiCa for the limit state of block shear rupture in bolted connections is presented in this article. The results of comparison show that strength from IDEA StatiCa can be greater than that according to the AISC Specification for some cases, especially if the ratio of tensile strength to yield strength (F_u/F_y) is relatively low. However, researchers have identified that the provisions of the AISC Specification can be conservative in comparison to experimental results. The block shear rupture strength from IDEA StatiCa was found to be accurate or conservative in comparison to the Canadian standard (CSA S16) and an alternative design equation proposed by researchers.

Strength for the limit state of block shear rupture in IDEA StatiCa can vary based on the shear force transfer type of the bolts. In IDEA StatiCa, forces are transferred from one plate to another over a larger area for slip-critical connections than for bearing-type connections. The larger spread of transfer forces, while physically representative of load transfer by friction, can lead to different block shear rupture failure paths and increased strength. For most connections, slip strength is less than block shear rupture strength. However, since slip-critical connections are required to be designed for the limit states of bearing-type connections in addition to slip (AISC Specification Section J3.9), it is recommended that slip-critical connections be analyzed twice in IDEA StatiCa: once as a slip-critical connection (i.e., with shear force transfer type set to “Friction”) and again as a bearing type connection (i.e., with shear force transfer type set to “Bearing – tension/shear interaction”). 

To illustrate this effect, consider the connection shown below between a W14x99 (A992) tension member and two plates. The connection is made with (4) 1 in. diameter A490 bolts in standard holes and Class B surfaces. The design strength of this connection for the limit state of slip is \(\phi R_n = 289\textrm{ kips}\), however, block shear rupture controls the strength of the connection with a design strength of \(\phi R_n = 148 \textrm{ kips}\). When modeled in IDEA StatiCa and the shear force transfer type of the bolts is set to “Friction”, applied loads of up to 263 kips can be applied before the utilization of the bolts reaches 100%. The difference between this strength and the 289-kip design strength for the limit state of slip is because tension in the bolts develops in the model and is conservatively treated as an applied tension in IDEA StatiCa. At 263 kips of applied tension and using “Friction” bolts, the plastic strain in the web is 3.5%, below the 5% limit. When the shear force transfer type for the bolts is set to “Bearing – tension/shear interaction” the maximum applied load decreases to 183 kips with plastic strain in the web controlling. The difference between this strength and the 148-kip design strength for the limit state of block shear rupture is predominantly conservatism in the AISC Specification equation for block shear rupture as described in this article. According to the Canadian standard (CSA S16), the design strength of this connection for the limit state of block shear rupture is 181 kips, approximately equal to the strength from IDEA StatiCa. The figure below shows the plastic strain in the web at the maximum applied load for each shear force transfer type. The distributions of plastic strain are clearly different and demonstrate the larger spread of transfer forces for “Friction” bolts in IDEA StatiCa. Additional discussion can be found in the entry on Slip.

Flexural Yielding

The nominal strength for flexural yielding is defined in AISC Specification (2022) Chapter F for flexural members and Section J4.5 for connecting elements. The nominal strength for the limit state of flexural yielding is generally taken as the specified minimum yield stress, F_y, times the plastic section modulus, Z. In IDEA StatiCa, instead of limiting the required strength to be no more than the available strength (e.g., M_u ≤ ϕM_n), members and connecting elements are modeled with shell elements that are assigned a nonlinear stress-strain relationship consisting of a linear elastic region and a linear plastic region, and the plastic strain is limited to 5%.

Modeling members and connecting elements as shell elements results in some simplifications of the physical geometry. For example, shell elements only represent rectangular components thus, fillets are neglected. Additionally, since shell elements are connected at nodes located at the center of thickness there is some overlap at joints of cross-sectional elements. The figure below shows the simplifications for a wide flange shape.

Wide flange shape as modeled in IDEA StatiCa

For a W24x176, the plastic section modulus about the major axis (x axis) listed in the AISC Steel Construction Manual (2023) Table 1-1 is 511 in.³. The plastic section modulus about the major axis of the cross-section formed by the shell elements (with cross-sectional dimensions determined from AISC Manual Table 1-1) is calculated as follows:

\[\frac{t_w(d-t_f)^2}{4}+2b_f t_f \left ( \frac{d-t_f}{2} \right ) = \frac{0.75 \textrm{ in.}(25.2 \textrm{ in.}-1.34\textrm{ in.})^2}{4}+2(12.9\textrm{ in.}) (1.34\textrm{ in.}) \left ( \frac{25.2\textrm{ in.}-1.34\textrm{ in.}}{2} \right ) = 519.2 \textrm{ in.}^3\]

This is 1.6% greater than the plastic section modulus listed in the AISC Manual table.

The stress distribution at the plastic strain limit in IDEA StatiCa will also be different than the idealized stress distribution used to compute M_p. Unlike the idealized stress distribution, the stresses will be lower than F_y near the neutral axis since the plastic strain limit will be reached at a finite curvature. Also, the stresses will be greater than F_y at the extreme fibers of the cross section because a small amount of post-yield hardening is assumed in the stress-strain relationship in IDEA StatiCa.

The overall effect of these minor differences can be observed in a simple splice connection between two W24x176 (ASTM A992) steel shapes. The splice is butt welded (e.g., CJP) and loaded in major axis bending. The design strength of the wide flange according to the AISC Specification (2022) with resistance factor, ϕ = 0.9, is 0.9 × 50 ksi × 511 in.³ = 1916.3 kip-ft. The maximum moment that can be applied to the connection in IDEA StatiCa (version 23.0) is 2000.7 kip-ft., 4.4% greater than the design strength as calculated per the AISC Specification. The plastic strain distribution at the limit is shown in the figure below. As expected, the top and bottom flanges have yielded, but web at the neutral axis remains elastic.

Plastic strain distribution for a W24x176 flexural member at the 5% plastic strain limit

The relationship between applied moment and maximum plastic strain is shown in the figure below. The design flexural strength calculated using the plastic section modulus from the AISC Manual is shown as ϕM_p (Manual). The design flexural strength calculated using the plastic section modulus calculated as shown above based on the representation of the section in IDEA StatiCa is shown as ϕM_p (IDEA).

Applied moment vs plastic strain for a W24x176 flexural member

For a wide flange beam, most of the flexural resistance is captured by the in-plane behavior of the shell elements. The out-of-plane behavior of the shell elements can be evaluated through an investigation of plate bending.

For a plate (ASTM A36, F_y = 36 ksi) of width, b = 10 in. and thickness, t = 0.5 in., the plastic section modulus for out-of-plane bending is calculated as Z = bt²/4 = 0.625 in.³, and the design strength, ϕM_p, with resistance factor, ϕ = 0.9 is calculated as 0.9 x 36 ksi x 0.625 in.³ = 20.25 kip-in. The geometric simplifications described above for a wide flange section don’t apply to a simple rectangular plate but differences in the stress distribution remain. The maximum moment that can be applied to the plate in IDEA StatiCa (version 23.0) is 19.66 kip-in., 2.9% less than the design strength as calculated per the AISC Specification. The plastic strain distribution for the plate loaded in minor axis bending and a plot of applied moment vs plastic strain are presented in the figures below.

Plastic strain distribution for plate out-of-plane bending at the 5% plastic strain limit

Applied moment vs plastic strain for a plate loaded in minor axis bending

Flexural Rupture

Flexural rupture is among the limit states identified for affected elements of members and connecting elements in flexure in AISC Specification Section J4.5. Flexural rupture can occur when a moment is applied to a cross-section with material removed, such as bolt holes. AISC Specification Chapter J does not define the available strength for the limit state of flexural rupture. AISC Specification Section F13.1 addresses flexural rupture for members with bolt holes in the tension flange, and guidance is provided for flexural rupture of affected and connecting elements in Part 9 of the AISC Manual. Specifically, AISC Manual Equation 9-8 defines the nominal strength for flexural rupture as the product of the specified minimum tensile strength and the net plastic section modulus of the affected or connecting element. The AISC Manual further defines the resistance factor as \(\phi=0.75\) and safety factor as \(\Omega = 2.00\) for flexural rupture.

As with the limit state of tensile rupture, IDEA StatiCa does not evaluate strength equations for flexural rupture. Instead, the limit state of flexural rupture is evaluated using the plastic strain limit. Thus, like for tensile rupture, differences arise because the stress strain relationship used in IDEA StatiCa has minimal strain hardening past yield whereas the design equation uses the tensile strength of the material and because IDEA StatiCa reduces the stress at yielding by a factor of 0.9 (for LRFD) whereas a resistance factor of 0.75 is used for flexural rupture. Additional differences, specific to flexural rupture, result from the use of a plastic section modulus in the design equation, which assumes a uniform stress in either tension or compression. In IDEA StatiCa, stresses are an analysis result and not necessarily uniform.

To examine the net effect of these differences, consider the splice plates tested by Mohr and Murray (2008). They tested 14 specimens altogether; the six tests of the first series with three different bolt patterns are investigated here. The plates were installed between two W27x84 beams. The whole assembly was loaded in four-point bending, subjecting the plate to pure flexure. The dimensions of the largest plates, those with 7 bolts in each vertical row, are shown below. Tests were also performed with 5 and 3 bolts in each vertical row with similar dimensions. The measured yield strength of the plates was F_y = 49.5 ksi, the measured tensile strength of the plates was F_u = 72.1 ksi, and the measured thickness of the plates was t = 0.370 in.

The design strength, \(\phi M_n\), of the plates was computed according to the AISC Specification for the limit state of flexural yielding and the AISC Manual for the limit state of flexural rupture. Measured material and geometric properties were used in these calculations and resistance factors were applied. IDEA StatiCa models of the three connections were also built using measured material and geometric properties of the plates. Resistance factors remained as their default values. Properties of the beams and bolts were increased from nominal values to ensure the failure mode matched that of the experiment. The maximum permitted applied moment from IDEA StatiCa, M_IDEA, was determined iteratively. The results of these calculations are shown in the figure below along with the experimental strength, M_exp. The experimental strength was taken as the average of the reported strengths for the two specimens of each bolt pattern. The moments in the figure are for each plate noting that there were two plates for each specimen, one on each side of the beams.

In the physical experiments, all the specimens failed by flexural rupture. Flexural rupture also controls the moment strength of the plates since \(\phi M_{n,rupture} < \phi M_{n,yield}\). IDEA StatiCa, however, does not clearly distinguish between these two limit states; both are evaluated using the 5% plastic strain limit. The plastic strain in the plates at the maximum permitted applied load is shown for the cases with 7 and 3 bolts in each vertical row below.

The maximum permitted applied moment from IDEA StatiCa, M_IDEA, is approximately 5% greater than \(\phi M_{n,rupture}\) for these cases, a slightly unconservative result in comparison to the AISC Manual equation. However, M_IDEA is approximately 20% lower than M_exp for these cases. While it is expected that M_IDEA is less than M_exp since no reduction factor was applied to the experimental results, the difference indicates that there is a safety margin.

Concrete Crushing

At column bases bearing stresses are developed on concrete footings and foundations. AISC Specification (2022) Section J8 provides an equation for the strength of the concrete for the limit state of concrete crushing that is identical to the equivalent provisions in ACI 318 (ACI 2019). The strength depends on the area of steel bearing on a concrete support, the geometry of the concrete support, and the specified compressive strength of concrete.

IDEA StatiCa uses these provisions to evaluate concrete crushing. However, some differences between IDEA StatiCa and traditional hand calculations in the evaluation of concrete crushing arise because of differences in the underlying analysis approach. In hand calculations it is common to assume that the bearing stress is unform over the contact area. In IDEA StatiCa, the stiffness of the concrete footing, the stiffness of the column base, and contact are explicitly modeled resulting in a more physically realistic, non-unform distribution of bearing stress. The bearing area in IDEA StatiCa is computed as the area of steel that is in contact with the concrete and with a bearing stress greater than a cut-off value (the stress cut-off is defined as a ratio to the peak bearing stress with the ratio selectable in the code setup). This can result in a relatively complex shape for the bearing area as shown in the figure below. Nonetheless, the total bearing force, bearing area, and geometrically similar area in the concrete support are computed for use in the code equation.

Three-dimensional view (left) and plan view (right) of stress in concrete at the steel-concrete interface of a concentrically loaded base plate connection. The boundary of the bearing area (A₁ in AISC Specification Section J8) is shown as a solid black line in the plan view. Note the irregular shape that follows the stress contours and the anchor rod holes. The concrete supporting surface (A₂ in AISC Specification Section J8) is shown as the hatched region of the plan view and is similarly irregular.

Additional information can be found in these articles:

• https://www.ideastatica.com/support-center/general-theoretical-background#Structural_model_of_a_concrete_block
• https://www.ideastatica.com/support-center/check-of-components-according-to-aisc
• https://www.ideastatica.com/support-center/check-of-concrete-blocks-according-to-aisc
• https://www.ideastatica.com/support-center/base-plate-connections-aisc

Flange Local Bending

Flange local bending is among the limit states that apply to concentrated forces applied normal to the flange of wide-flange sections and similar built-up shapes. It applies only to tensile concentrated forces. The nominal strength for the limit state of flange local bending is defined in AISC Specification (2022) Section J10.1.

As described in the commentary on Section J10.1, the limit state of flange local bending was originally intended to prevent weld fracture which could occur prematurely because of uneven demands due to flange deformation. However, more recent tests have shown that weld fracture does not occur when flange local bending strength is exceeded, but rather that the flange local bending strength represents a lower bound at which flange deformation could lead to premature flange local buckling or be detrimental to other aspects of member performance. The commentary further notes that while flange deformations can also occur under compressive forces, the AISC Specification does not require flange local bending be checked for compressive forces because it is customary to perform the check for tensile forces only.

As shown in the figure above, both the uneven stress distribution and flange deformations are explicitly modeled in IDEA StatiCa. Each weld segment is checked independently for strength. Cases like shown in the figure above were examined in the calibration and subsequent validation and verification of the weld model in IDEA StatiCa. However, for shapes other than HSS, local flange deformations are not checked against a limit, their effect on member performance is not evaluated, and their magnitude cannot be directly obtained from the model. As a result, the limit state of flange local bending is not evaluated in IDEA StatiCa. Where the flange local bending controls traditional calculations, significantly higher strengths can be obtained from IDEA StatiCa. Where flange deformations are a concern, it is recommended to evaluate the limit state outside of IDEA StatiCa.

Note that flexural yielding of flanges in bolted connections is considered a separate limit state. In traditional calculations, the available strength is typically determined using yield line theory as described by Dowswell (2011) for general connections or Eatherton and Murray (2023) for end-plate moment connections. IDEA StatiCa captures this limit state through explicit modeling of the flange as shown in the figure below.

Web Local Yielding

Web local yielding is among the limit states that apply to concentrated forces applied normal to the flange of wide-flange sections and similar built-up shapes. The nominal strength equations for web local yielding in AISC Specification Section J10.2 are based on yield of the web over a length equal to the length of bearing plus an assumed spread of the force through the flange. While yielding of the web is modeled explicitly in IDEA StatiCa, several features of the design equations are not. The equations assume a stress gradient of 2.5:1 through the flange and the fillet of rolled shapes. In IDEA StatiCa, the flange is modeled with shell elements and the fillet is neglected, thus the spread of forces depends largely on the constraints between the flange and web. There are two separate equations in AISC Specification Section J10.2 for web local yielding depending on the distance of the force away from the member ends. In IDEA StatiCa, reduction in strength due to proximity to the member end is captured by directly modeling the member. A resistance factor of ϕ = 1.00 and safety factor of Ω = 1.50 apply to the limit state of web local yielding. IDEA StatiCa does not utilize these factors and instead reduces the yield point by a factor of 0.9 for LRFD or by dividing by 1.67 for ASD based on the typical resistance factor and safety factor for yielding.

The overall effect of these differences has been investigated for beam-over-column connections in this article and generic concentrated forces in this report.

Web Compression Buckling

Web compression buckling is among the limit states that apply to concentrated forces applied normal to the flange of wide-flange sections and similar built-up shapes. It applies when a pair of forces compress the web from both flanges at the same location along the length of the member. AISC Specification Section J10.5 provides an equation for the nominal strength for web compression buckling. The equation is based on the elastic buckling strength of a simply supported plate subjected to equal and opposite concentrated forces.

In IDEA StatiCa, design for web compression buckling can be accomplished by ensuring the elastic critical buckling load is sufficiently large (see discussion in the entry on Compressive Yielding and Buckling). Through comparisons to geometric and material nonlinear analysis with imperfections included (GMNIA) an elastic critical buckling load ratio of 3 was determined to be an appropriate lower limit.

Web Panel-Zone Shear Yielding

The available strength for the limit state of panel-zone shear yielding of wide-flange and similar built-up shapes is defined in AISC Specification Section J10.6. Four different equations are provided in this section for the nominal strength. One pair of equations is provided for when the effect of inelastic panel-zone deformation on frame stability is not accounted for in the analysis and another pair for when it is. The first pair of equations limits panel-zone behavior to the elastic range. The second pair of equations provides greater strength; however, plastic deformation of the panel-zone is necessary to achieve greater strength. The additional deformations can significantly increase overall frame deformations and second-order effects. If the potential for inelastic panel-zone deformation is not accounted for in the calculation of member and connection required strengths, then AISC Specification Section J10.6 requires panel-zone behavior to be limited to the elastic range.

In IDEA StatiCa, panel-zone shear yielding is modeled explicitly with nonlinear shell elements and is limited by a plastic strain limit. The limit state of panel-zone shear yielding was explored for extended end plate moment connections in this article and for bolted flange plate moment connections in this article. Using the default plastic strain limit of 5%, the strength from IDEA StatiCa exceeds that from the AISC Specification for when the effect of inelastic panel-zone deformation on frame stability is not accounted for in the analysis. However, reducing the plastic strain limit to a small value (e.g., 0.1%) in IDEA StatiCa enforces essentially elastic behavior and results in strengths that are accurate in comparison to the AISC Specification equations for when the effect of inelastic panel-zone deformation on frame stability is not accounted for in the analysis.

Engineers should know if the effect of inelastic panel-zone deformation on frame stability was accounted for in the analysis to determine required strengths (i.e., not the IDEA StatiCa analysis). And, if it was not, they should limit panel-zone behavior to be essentially elastic.

Connections to HSS Members

AISC Specification (2022) Chapter K includes additional requirements, beyond those of Chapter J, that apply to connections to HSS members and box sections that behave like HSS members. Chapter K is organized by connection type and the requirements are often accompanied by limits of applicability. However, Chapter K does not prohibit the use of connections of other configurations or those outside the limits of applicability.

The limit states described in the tables of Chapter K are evaluated in IDEA StatiCa by explicit modeling and the 5% plastic strain limit. The effects of parameters defined in Section K1, including the effective width for connections to rectangular HSS to account for uneven stress distributions, the chord-stress interaction parameter, and end distance are also modeled explicitly. To increase accuracy, geometric nonlinearity is included in the model by default when a hollow cross-section is used as a bearing member.

The commentary on Chapter K states “When inelastic finite element analysis is used, peak strains in the thick shell (T × T × T) elements should not exceed 0.02/T at the nominal capacity, where T is the thickness in inches.” Neglecting the difference between strain and plastic strain, the limiting value of this recommendation is greater than the 5% used by IDEA StatiCa when the thickness is less than 0.4 in. While the strain limit in the commentary recommendation is more restrictive than the default limit in IDEA StatiCa for thicker tubes, the 5% plastic strain limit is more widely recognized as an acceptable limit for strength design including by the Steel Tube Institute.

Chapter K is based on strength limit states only. As a result, large deformations can occur in connections that meet the requirements of Chapter K. Nonetheless, local out-of-plane deformation of HSS members is checked in IDEA StatiCa against a limit of 3% of the smallest transverse dimension of the cross-section (i.e., diameter or width) based on the requirements of other standards.

As the provisions of Chapter K are based largely on international research and the work of international committees, verifications to other standards are generally informative to US practice. Several verification studies for connections to HSS members are available on the IDEA StatiCa website, including for connections between rectangular hollow sections, circular hollow sections, plate and rectangular hollow sections, and plate and circular hollow sections.

Design Considerations and Requirements

Design Basis

Design for strength according to the AISC Specification is performed with either the provisions for load and resistance factor design (LRFD) or the provisions for allowable strength design (ASD). While these two approaches have different required strengths and different available strengths, the nominal strengths are the same and final designs should be similar if not the same.

Required strengths are greater for LRFD than for ASD due to the greater load factors in the LRFD load combinations. Differences in required strengths can also arise when required strengths are computed using nonlinear analysis and the level of nonlinearity depends on the level of loading. To compensate for this in design for stability the AISC Specification requires that all load-dependent effects be calculated at a level of loading corresponding to LRFD load combinations or 1.6 times ASD load combinations. IDEA StatiCa follows a different approach. In IDEA StatiCa, the yield stress for shell elements is taken as 0.9F_y for LRFD and F_y/1.67 for ASD, with the 0.9 and 1.67 corresponding to the typical resistance factor and safety factor for yielding limit states. For most cases, this results in maximum permitted applied loads that are 1.5 times greater for LRFD than they are for ASD, consistent with the provisions of the AISC Specification. However, the modulus of elasticity is not reduced in IDEA StatiCa for either LRFD or ASD. Therefore, the ratio of stiffness to strength differs between approaches resulting in some consequences in design. For buckling, the limiting elastic buckling load ratio differs between LRFD and ASD. Also, where the stiffness of a connection impacts its strength, e.g., long welded connections, the ratio of maximum permitted applied load between LRFD and ASD can deviate from 1.5. Most of the validation studies comparing IDEA StatiCa to the AISC Specification were performed for LRFD.

IDEA StatiCa implements provisions for ASD as defined in the 2022 AISC Specification. The provisions in the 2022 AISC Specification for ASD differ from those in historic standards such as the 1989 AISC Specification which is included in the 9^th edition AISC Manual (commonly referred to as the “green book”). The historic provisions for ASD focused on elastic behavior and had more differences with LRFD. The current provisions for ASD are more consistent with LRFD, including common nominal strength calculations.

Structural Steel Materials

AISC Specification Section A3.1 includes requirements for structural steel materials. In this section, Table A3.1 lists specific materials that have a history of satisfactory performance and are considered to perform as anticipated in the provisions of the AISC Specification. The listed materials include those for rolled shapes with yield stress of up to 80 ksi and plates with yield stress of up to 100 ksi. Materials other than those listed in Table A3.1 are permitted when their use is determined to be acceptable by the engineer of record. Many factors can affect the suitability of materials, including intended use, strength properties in transverse directions, ductility, and weldability.

Given the broad verification of IDEA StatiCa to the provisions of the AISC Specification, the materials listed in Table A3.1 can also be considered to perform as anticipated in the software. Use of materials not listed in Table A3.1 is not prohibited but remains subject to the judgement of the engineer of record. The commentary on AISC Specification Section A3.1 includes discussion on the factors that impact the suitability of materials and guidance for assessing suitability.

Prying Action

In bolted connections, contact between connecting elements can increase tensile forces beyond those due to applied loads only. This phenomenon is known as prying action and only occurs in connections with tensile bolt forces. The contact that increases bolt forces occurs because of deformation of the connecting element. Thus, prying action is a design consideration for both bolts and connecting elements.

The relative stiffness and strength of the bolts and the connecting elements control the behavior. If the connecting elements are stiff relative to the bolts, then the connecting elements will deform without bending back and making contact and no prying action will occur. In this case, the strength of the bolts will control the design. If the connecting elements are weak relative to the bolts, then the connecting elements will yield and impart prying forces on the bolts, but also limiting the force in the bolts. In this case, the strength of the connecting elements will control the design. In between, the strength of the bolts and the connecting elements simultaneously control the design.

Guidance for considering prying action in design is provided in Part 9 of the AISC Manual. The equations presented in the AISC Manual were developed for the common cases of a tee and back-to-back angles and validated against experimental data. IDEA StatiCa explicitly models the stiffness and strength of bolts and connecting elements, including contact, thus prying action is naturally captured by the analysis regardless of the specific configuration. A comparison between the AISC Manual equations and IDEA StatiCa results has been performed for T-stub connections. A similar comparison to the design approach for prying action recommended in the Guide to Design Criteria for Bolted and Riveted Joints (Kulak et al. 1987) has also been performed. Prying action is featured in other verification examples including for bracing connections and extended end-plate moment connections.

Deformation Compatibility in Long Connections

In long end-loaded connections, the difference in elongation between the connected elements is greatest at the ends of the connection. As a result, the stress in bolts and welds in long end-loaded connections is not uniform. Since it is common in traditional calculations to assume uniform stress, the AISC Specification includes reductions to the length of long end-loaded welds and to the nominal shear stress of bolts. AISC Specification Section J2.2b defines the effective length of end-loaded fillet welds including reductions when the length of the weld exceeds 100 times the weld size. The values of nominal shear stress in AISC Specification Table J3.2 include a 10% reduction to account for length effects and an additional reduction is required for end-loaded connections with a fastener pattern length greater than 38 in.

IDEA StatiCa does not implement these reductions directly. Rather, the underlying behavior that motivates these reductions is modeled explicitly. IDEA StatiCa models the stiffness of bolts, welds, and connecting elements, thus the non-uniform distribution of stress in bolts and welds arises naturally. With the strength of bolts and weld segments assessed individually, the resulting connection strength is comparable to that from traditional calculations. A detailed comparison between IDEA StatiCa and determined from traditional calculations for long end-loaded connections is presented in this article.

Deformation Compatibility in Eccentrically Loaded Bolt and Weld Groups

Bolts and welds in eccentrically loaded groups are subject to direct shear plus additional shear from the induced moment. The resulting stress in the bolts or welds varies in both magnitude and direction from bolt to bolt and from weld segment to weld segment. As described in Parts 7 and 8 of the AISC Manual, engineers may use the instantaneous center of rotation method or the elastic method to analyze eccentrically loaded bolt or weld groups. Calculations using the instantaneous center of rotation method are typically performed using tabulated values provided in the AISC Manual.

In IDEA StatiCa, the required strength of bolts and weld segments is determined from results of the nonlinear analysis. Each bolt and weld segment is modeled individually and equilibrium is enforced. Available strengths are determined according to the AISC Specification.

The instantaneous center of rotation method is also based on nonlinear analysis, but there are key differences between the nonlinear analyses of the instantaneous center of rotation method and IDEA StatiCa. In the instantaneous center of rotation method, the connecting elements are assumed to be rigid, which is not the case for IDEA StatiCa. The force-deformation response of bolts and welds also differs between the two methods. The force-deformation response used in IDEA StatiCa for bolts and welds is bilinear and is described in the theoretical background.

The differences generally result in similar or lower strengths from IDEA StatiCa as shown in this article on bracket plate connections. Comparisons between traditional calculations and IDEA StatiCa for eccentrically loaded bolt groups are also made in this article on single-plate shear connections.

Bolts in Combination with Welds

Accurate strength prediction is more difficult when bolts and welds share load on a common faying surface. The lower ductility of welds in comparison to bolts can lead to brittle fracture before the full strength of the bolt is realized. AISC Specification Section J1.8 allows bolts and welds to be considered as sharing load only in certain circumstances.

According to Section J1.8, bolts can be considered as sharing load with welds only in the design of shear connections on a common faying surface where strain compatibility between the bolts and welds is considered. The section also describes a case with pretensioned high-strength bolts and longitudinal fillet welds where the nominal strength is permitted to be determined as the nominal slip strength plus the nominal weld strength. The bolts and welds must each support a specified proportion of the load and a resistance factor of ϕ = 0.75 or safety factor of Ω = 2.00 applies to the combined joint.

Strength checks for bolts and welds are independent in IDEA StatiCa with no special handling of when bolts and welds share load. Given the explicit modeling of the stiffness of bolts, welds, members, and connecting elements, strain compatibility is always considered in IDEA StatiCa. When bolts and welds share load, the required strength of each is based on their relative stiffness and the available strength is computed as usual. This is true even for tension connections; therefore, it is recommended to not model bolts and welds as sharing load for tension connections and instead only rely on one or the other.

To illustrate the differences between the method provided in AISC Specification Section J1.8 and IDEA StatiCa, consider the connection between plates subjected to tension shown below.

According to the AISC Specification, when the connection is designed as slip-critical, the design strength of the bolts alone is ϕR_n = 133 kips (R_n = 133 kips). The design strength of the welds alone is ϕR_n = 290 kips (R_n = 386 kips). When combining bolts and welds, the total connection strength is ϕR_n = 0.75 (133 + 386) = 389 kips since all requirements in Section J1.8 to allow the summation of bolt and weld strengths are met.

In IDEA StatiCa, the maximum permitted applied tension is 126 kips when the bolts alone are modeled and 277 kips when the welds alone are modeled. The difference between the bolt strength in IDEA StatiCa and the 133-kips design strength is because tension in the bolts develops in the model and is conservatively treated as an applied tension in IDEA StatiCa (see entry on Slip). The difference between the weld strength in IDEA StatiCa and the 277-kips design strength is because of non-uniform demands along the length of the weld in IDEA StatiCa. When both bolts and welds are modeled, the maximum permitted applied tension is 394 kips, with both the bolts and welds showing 100% utilization. This value is closely comparable to the AISC Specification strength of 389 kips.

If the bolts are assumed to be bearing type, the design strength for the bolts according to the AISC Specification is ϕR_n = 245 kips. While the AISC Specification permits bolts to be considered as sharing load with welds in shear connections, it does not provide a method to assess the strength when the bolts do not meet the requirements of a slip-critical connection. Therefore, it would be common to assess the strength of this connection as that of the welds alone or ϕR_n = 290 kips.

In IDEA StatiCa, when the bolts are modeled as bearing bolts and the welds are not modeled, the maximum permitted applied tension matches the AISC Specification design strength of 245 kips. When the bolts are modeled as bearing bolts and the welds are modeled, the maximum permitted applied tension is 311 kips with weld strength being the controlling limit. This strength is only 12% more than the strength of the welds alone according to IDEA StatiCa. The minor increase in strength with the addition of bearing bolts is because the bolts are less stiff than the welds and thus do not attract much load before the welds reach 100% utilization.  

Effect of Hole Size

AISC Specification (2022) Section J3.3 describes the use of standard, oversize, short-slotted, and long-slotted holes for bolts in structural steel connections. Standard holes are the default in IDEA StatiCa. Oversize holes can be achieved by editing the hole diameter in the bolt assembly. Slotted holes can be defined for plates in the plate editor.

Hole size affects several aspects of behavior, and some design requirements are based on hole size.

• The material removed for bolt holes affects net area. This effect is addressed explicitly in IDEA StatiCa through the definition of the shell element model for members and connecting elements. However, the additional 1/16 in. for damage required by AISC Specification Section B4.3b is not automatically implemented (see Net Area Determination)
• Hole size affects the clear distance used to determine tearout strength. This effect is addressed explicitly in IDEA StatiCa by computing the clear distance based on the geometry of the connected material and the direction of the force in the individual bolt.
• Oversize holes are not permitted in bearing-type connections. IDEA StatiCa does not check this requirement and will allow the use of bearing shear force transfer with oversize.
• The resistance factor for limit state of slip depends on hole type. IDEA StatiCa does not automatically adjust the resistance factor for hole type. The resistance factor can be set manually in the Code setup.

Hole size can affect the load-deformation response of the bolt. The bolt load-deformation model used in IDEA StatiCa does not depend on hole size, but the shear transfer is assumed to be zero in the long direction of slotted holes.

Mill Underrun

Variation in member length can result in meaningful differences to the dimensions used in connection design. In several calculations in the AISC Design Examples, a 1/4 in. tolerance is subtracted from a length to account for possible mill underrun. IDEA StatiCa does not account for possible mill underrun automatically, but possible mill underrun can be considered by manually defining the connection with the assumed underrun.

Contact and Friction

Steel cannot physically pass through steel, yet this is the default behavior in finite element analyses. Contact surfaces need to be defined to prevent material from overlapping when deformed. Surface-to-surface contact is defined automatically with bolt group operations. Surface-to-surface contact may be defined with the "Bolt group/contact” operation. Edge-to-edge contact or edge-to-surface contact can be defined with the “General weld or contact” operation.

Not all potential contact surfaces are defined automatically by IDEA StatiCa. Therefore, it is important for the user to have a good understanding of the intended behavior of the connection and to examine the deformed shape to confirm the connection is modeled and behaving as intended.

Contact bearing can be an efficient means of force transfer in a connection if the connection has been detailed and the surfaces are carefully prepared such that bearing will exist (Muir 2015). Since special detailing is necessary to ensure contact bearing is effective, edge-to-edge and edge-to-surface contact is not defined automatically in IDEA StatiCa but can be defined manually using the “General weld or contact” operation. Bolted column splices are an example where the definition of edge-to-edge contact between the members will reduce the demands in the bolts resulting in a more efficient connection. Use of contact bearing can also be efficient in conjunction with welds between columns and base plates. Welds, by default, are not defined with contact and thus also checked for compressive forces. Combining weld and contact operations can enable the use of smaller welds. Welds are stiff and will attract load even when combined with contact but demands from compressive forces will rarely exceed the capacity, even if the size of the weld is reduced.

Friction at steel-to-steel contact surfaces is conservatively neglected in IDEA StatiCa, except for bolts designated as transferring shear forces through friction (i.e., slip-critical bolts). Only considering friction when pretensioned bolts provide the clamping force is typical for traditional calculations also. However, some differences in results between IDEA StatiCa and traditional calculations can occur because of friction. For example, AISC Specification Section J3.10 defines a reduction factor to be applied to the slip strength when a slip-critical connection is subject to combined tension and shear. The reduction factor is based on the tension load applied to the connection. IDEA StatiCa has no way of quantifying how much of a tension load in a bolt is due to applied load versus other sources such as prying action. If prying action induces tension into a slip-critical bolt, then the slip strength will be reduced in IDEA StatiCa. The slip strength per traditional calculations would not be reduced. A detailed investigation of this difference is described for T-stub connections in this article.

Net Area Determination

AISC Specification (2022) Section B4.3b requires that the width of a bolt hole be taken as 1/16 in. greater than the nominal dimension of the hole when computing the net area in tension or shear. Implementation of this requirement reduces the net area to account for possible damage around a bolt hole during drilling or punching operations. This requirement affects limit states such as net section tensile rupture and block shear rupture but does not affect the limit state of tearout at the bolt holes.

In IDEA StatiCa, the default bolt assemblies have a hole diameter equal to the nominal hole dimension. Therefore, while the 1/16 in. can be added to the bolt hole diameter manually by editing the bolt assembly, this requirement is not automatically addressed in IDEA StatiCa. If the hole diameter of the bolt assembly is increased, the increased diameter will apply for all aspects of the analysis, including evaluation of tearout. Additional discussion on how hole size affects results in IDEA StatiCa can be found in the entry on the Effect of Hole Size.

AISC Specification (2022) Section B4.3b also includes provisions for the determination of net area when a chain of bolt holes extends across a part in any diagonal or zigzag line. For these cases, the net width of the part is obtained by deducting from the gross width the sum of the diameters (including the 1/16 in. for damage) of all holes in the chain, and adding, for each gage space in the chain, the quantity s²/4g, where

g = transverse center-to-center spacing (gage) between fastener gage lines

s = longitudinal center-to-center spacing (pitch) of any two consecutive bolt holes

The resulting net width is different from the length of the failure surface (i.e., the red dashed line in the figure below) and accounts for the combination of tension and shear along the inclined plane. Since IDEA StatiCa does not explicitly calculate the net area, the net width provisions are not implemented in the software. However, the potential for failure along a diagonal or zigzag line of bolts, including the interaction of tension and shear along the inclined plane, is captured explicitly by modeling the connected elements.

The effect of staggering bolt lines can be observed in a simple splice connection. A test plate is bolted between two reaction plates and loaded in tension. The thickness of the test plate is 1/2 in. and the thickness of each reaction plate is 3/8 in. All plates are 6 in. wide and conform to ASTM A572 Gr 50 (F_y = 50 ksi, F_u = 65 ksi). The connection has (6) 7/8 in. diameter A325 bolts in two staggered lines. The spacing between bolts in a line is 3 in., the gage, g, is 3 in., and the edge distance is 1.5 in. The amount of stagger between the two lines of bolts is measured by the dimension s.

A three-dimensional view of the connection with s = 1.5 in. is shown in the figure below.

Analyses were performed for connections with dimension s varying from zero (i.e., no stagger) to 3 in. in increments of 0.5 in. The strength per the AISC Specification was calculated using the provisions of Section B4.3b. The limit state of tensile rupture along the zigzag line shown as a red dashed line in the figure above controlled for all cases. The strength per IDEA StatiCa was determined iteratively using stress-strain analysis by adjusting the applied load input to a value that the program deems safe but if increased by a small amount (0.1 kip) the program would deem unsafe. The 5% plastic strain limit controlled for all cases. The results of the analyses are shown in the figure below.

The AISC Specification results show a clear trend of increasing strength with dimension s. The IDEA StatiCa results show less sensitivity to the dimension s and the strength is greater than that from the AISC Specification for all but the s = 3 in. case. However, the expected zigzag failure pattern is captured by the model as demonstrated in the figure below which shows the plastic strain in the test plate at the maximum permitted applied load.

Fillet Weld Size Requirements

AISC Specification (2022) Section J2.2b includes limitations for fillet welds.

Items (a)-(c) of Section J2.2b specify geometric limitations on the size and minimum length of fillet welds. These limitations are checked during the calculation if the “Detailing” option is checked in the “Code setup”. The specific limitations checked are described in this article. A weld will not pass the code check due to a detailing error if any limitations are not satisfied. Dimensions near or at the limit may not be evaluated as expected due to numerical precision or rounding.

Item (d) of Section J2.2b specifies the effective length of fillet welds including reductions for long end-loaded fillet welds. IDEA StatiCa does not compute the effective length for fillet welds and thus does not use these provision's direction, but the effect of non-uniform stress distribution on the strength of end-loaded fillet welds is captured through explicit modeling of the stiffness of the weld and connected material. See this article for a detailed investigation of this provision.

Items (e)-(i) of Section J2.2b specify limitations that are not checked by IDEA StatiCa and, if applicable, must be evaluated by the engineer separately.

Design Wall Thickness for HSS

AISC Specification (2022) Section B4.2 requires that the thickness of walls be taken as the design wall thickness, t, in strength calculations for hollow structural sections (HSS). The design wall thickness is equal to the nominal thickness, t_nom, for box sections and HSS produced according to ASTM A1065/A1065M or ASTM A1085/A1085M. The design wall thickness is equal to 0.93 times the nominal wall thickness (i.e., t = 0.93t_nom) for other standards approved for use by the Specification, including ASTM A500/A500M. ASTM A500 Gr. C is the preferred material specification in the United States for rectangular and round HSS (Tavarez 2022).

IDEA StatiCa does not automatically adjust the wall thickness of HSS cross-sections based on the material. Therefore, the user needs to be aware of this requirement and ensure that the proper thickness is assigned.

When defining the cross-section in IDEA StatiCa, predefined cross-sections in the category labeled “HSS (AISC 15.0 - A1085, A1065)” have wall thickness equal to the nominal wall thickness and those in the category labeled “HSS (AISC 15.0 - A500, A501, A618, A847)” have wall thickness equal to 0.93 times the nominal wall thickness.

References

AISC (2022), Specification for Structural Steel Buildings, American Institute of Steel Construction, Chicago, IL.

AISC (2023), Steel Construction Manual, 16^th Edition, American Institute of Steel Construction, Chicago, IL.

Dowswell, B. (2011). “A Yield Line Component Method for Bolted Flange Connections.” Engineering Journal, AISC, 48(2nd Quarter), 93–116.

Dowswell, B. (2015). “Plastic Strength of Connection Elements.” AISC Engineering Journal, 52(1st Quarter), 47–66.

Eatherton, M. R., and Murray, T. M. (2023). End-Plate Moment Connections. Design Guide 39, American Institute of Steel Construction, Chicago, Illinois.

Kulak, G. L., Fisher, J. W., and Struik, J. H. A. (1987). Guide to Design Criteria for Bolted and Riveted Joints, Second Edition. John Wiley & Sons, Inc.

Miazga, G. S., and D. L. Kennedy. (1989), “Behaviour of fillet welds as a function of the angle of loading,” Canadian Journal of Civil Engineering, 16 (4): 583–599.

Muir, L. (2015), “Bear It and Grin” Modern Steel Construction, AISC. (December).

Mohr, B. A., and Murray, T. M. (2008). “Bending Strength of Steel Bracket and Splice Plates.” Engineering Journal, AISC, 45(2), 97–106.

Tavarez, J. (2022), “Are You Properly Specifying Materials?” Modern Steel Construction, AISC. (June), 16-22.