Three-Dimensional Benchmark Problems for Design by Advanced Analysis: Impact of Twist

Ziemian, Ronald D.; Batista Abreu, Jean C.; Denavit, Mark D.; Denavit, Tsu-Jung L.

doi:10.1061/(ASCE)ST.1943-541X.0002224

Open access

Technical Papers

Oct 5, 2018

Three-Dimensional Benchmark Problems for Design by Advanced Analysis: Impact of Twist

Authors: Ronald D. Ziemian, M.ASCE [email protected], Jean C. Batista Abreu [email protected], Mark D. Denavit, M.ASCE [email protected], and Tsu-Jung L. Denavit [email protected]Author Affiliations

Publication: Journal of Structural Engineering

Volume 144, Issue 12

https://doi.org/10.1061/(ASCE)ST.1943-541X.0002224

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Abstract

Design codes for steel structures continue to provide additional opportunities for the use of more advanced methods of nonlinear analysis. As the complexity of the analysis increases, it becomes more important for the designer to validate the capabilities of their analysis software and, just as importantly, their ability to properly utilize it. Many benchmark problems can be found in the literature; however, few include three-dimensional behavior and fewer include I-shaped sections, which are torsionally flexible and where warping restraint can be significant. This paper introduces a new set of benchmark problems that contribute to a much needed database of examples in which the accurate modeling of three-dimensional or spatial behavior is essential. A secondary goal of the paper is to illustrate new steel design provisions and the type of comparisons that were part of their development.

Introduction

A growing number of specifications worldwide are providing designers with opportunities to take advantage of the ever-increasing capabilities of nonlinear analysis software. One such example is the AISC specification for structural steel buildings. The current AISC specification (AISC 2016b) defines several different methods and tiers of design for stability, each of which satisfies a set of general requirements and varies by which effects are included directly in the analysis. This paper focuses on variations of the direct analysis method of design.

The basic tier of the direct analysis method is defined in Chapter C of the AISC specification (AISC 2016b). In this method, required strengths are calculated from a second-order elastic analysis that includes consideration of system (but not member) imperfections and reductions to the stiffness of the structure. The benefit of using this method is that, for the limit state of flexural buckling, the available strength can be computed considering only the unbraced length of the column, without the need to compute an effective length factor.

More recently developed variations of the direct analysis method are defined within Appendix 1 of the AISC specification (AISC 2016b). These tiers represent logical extensions of the basic direct analysis method and provide the opportunity for engineers to utilize more advanced methods of analyses in their design.

The first of these design methods requires the use of a second-order elastic analysis that is more rigorous than the traditional

P - Δ

and

P - δ

methods expected for application of the basic direct analysis method. Traditional

P - Δ

and

P - δ

analyses are defined as methods that account for the effects of axial force on flexure, such as the use of moment amplification factors, but differ from more rigorous analysis methods, which directly ensure that equilibrium is satisfied on the deformed geometry of the system. In addition to the requirements from the basic direct analysis method of considering system sway imperfections and reductions to stiffness, this method further requires the consideration of member out-of-straightness imperfections and a more robust handling of torsional deformations. In exchange for these more stringent requirements, the compression strength of members may be taken as their cross-sectional strength; in other words, member and system instabilities will be detected by the analysis and hence do not need to be checked by code-specific equations. This new approach can be particularly useful for design situations in which the definitions of the unbraced lengths are not readily apparent. Such cases include arches and unbraced Vierendeel trusses, in which the member may be curved or the axial force varies significantly along the unbraced length.

The second of the design methods that appear in Appendix 1 of the AISC specification (AISC 2016b) requires the use of an inelastic analysis, thereby allowing the engineer to explicitly take advantage of the potential redistribution of member and connection forces and moments that can occur as a result of localized yielding. This method is not the focus of this paper, but analyses will be presented that are in general accordance with this method as a point of comparison to the elastic analysis results.

Given that the preceding descriptions are only intended to highlight the direct analysis method and the spectrum of requirements placed on this design method as a function of the method of analysis being employed, the AISC specification (AISC 2016b) should be consulted for complete details. For design methods such as these that rely heavily on rigorous second-order analyses, it is essential for the engineer to confirm the capabilities of the analysis software and, just as importantly, their ability to use it. One approach to such a validation includes comparisons with published benchmark problems. Numerous two-dimensional benchmark problems can be found in the literature. While some three-dimensional benchmark problems exist (Teh 2004), few are available for torsionally flexible I-shaped sections. The purpose of this paper is to build upon prior work (Ziemian and Batista Abreu 2016, 2018) to provide results of four benchmark problems in which significant twists develop primarily as a result of second-order effects. These three-dimensional validation studies, which focus on beams and beam-columns, are intended to represent only a few of the array of benchmark problems that an engineer should exercise before using advanced methods of analysis to design steel structures.

Effect of Twist on Strength

Consider the beam-column shown in Fig. 1 and assume that it is only subjected to a uniformly distributed load,

w_{y}

, that tracks the direction of gravity and therefore always remains vertical. For now, it is further assumed that no other forces are applied (i.e.,

P = 0

,

w_{x} = 0

,

M_{x} = 0

, and

M_{y} = 0

). Under this loading, a geometrically perfect structure would bend only in plane with no out-of-plane deflections or twist. With an initial out-of-plane imperfection,

δ_{x o}

, however, the member will bend in plane and out of plane and also twist under the given load. Based on equilibrium, the bending moments at midspan are

M_{u x} = w_{y} L^{2} / 8

about the original (undeformed)

x

-axis and

M_{u y} = 0

in the direction of the original (undeformed)

y

-axis. However, the local coordinates are rotated by an amount equal to the twist of the member,

θ_{z}

, and thus a transformation [Eq. (1)] is necessary to identify the major-axis bending moment,

M_{u x^{'}}

, and the minor-axis bending moment,

M_{u y^{'}}

M_{u x^{'}} = M_{u x} \cos θ_{z} - M_{u y} \sin θ_{z} M_{u y^{'}} = M_{u x} \sin θ_{z} + M_{u y} \cos θ_{z}

(1)

Fig. 1. General description of member investigated.

To gain further insight, the moments from Eq. (1) will be substituted into the governing interaction equation [Eq. (2)] from the AISC specification (AISC 2016b), which in the specification is identified as Eq. (H1-1). The authors contend that the moments oriented along the local cross-section axes (i.e.,

M_{u x^{'}}

and

M_{u y^{'}}

) are the moments that would be used in any interaction equation design checks and that the interaction ratio be computed on a cross section by cross section basis along the length of the member. Some designers conservatively combine the maximum demands within the member span, regardless of whether or not they occur at the same location

I_{H11} \leq 1.0

(2a)

I_{H11} = {\begin{cases} \frac{P_{u}}{ϕ P_{n}} + \frac{8}{9} (\frac{M_{u x^{'}}}{ϕ M_{n x}} + \frac{M_{u y^{'}}}{ϕ M_{n y}}) & when \frac{P_{u}}{ϕ P_{n}} \geq 0.2 \\ \frac{P_{u}}{2 ϕ P_{n}} + (\frac{M_{u x^{'}}}{ϕ M_{n x}} + \frac{M_{u y^{'}}}{ϕ M_{n y}}) & when \frac{P_{u}}{ϕ P_{n}} < 0.2 \end{cases}

(2b)

Given that

P_{u} = 0

and assuming

θ_{z}

is small (i.e.,

\cos θ_{z} = 1

and

\sin θ_{z} = θ_{z}

), the strength limitation is given by Eq. (3)

\frac{w_{y} L^{2} / 8}{ϕ M_{n x}} + \frac{(w_{y} L^{2} / 8) θ_{z}}{ϕ M_{n y}} \leq 1.0

(3)

By rearranging terms, the strength limitation can be converted to an inherent limitation on the twist [Eq. (4)]

θ_{z} \leq [{(\frac{w_{y} L^{2} / 8}{ϕ M_{n x}})}^{- 1} - 1] / \frac{M_{n x}}{M_{n y}}

(4)

The maximum twist from Eq. (4) is shown in Fig. 2 for several ratios of major axis to minor axis available strength. Several observations can be made from these curves. First, in order to maximize the utilization of the major-axis bending capacity of this member, the amount of twist resulting from the initial out-of-plane member imperfection (or some other perturbation, such as a small amount of out-of-plane eccentricity in the applied load) will need to be limited. Also of note is that the limitation on twist is more stringent for members with large ratios of major axis to minor axis available strength. Such large ratios are not unusual and such sections are typically deemed efficient for members designed primarily to resist major-axis bending.

Fig. 2. Upper bounds on permitted twist angle, $θ_{z}$ .

In general, the reason for this behavior is that assessing equilibrium on the deformed shape (as is physically accurate) requires that a portion of the major-axis demands be resolved into minor-axis demands and vice versa as the section twists. This transformation can have a significant impact on the design interaction equation checks, particularly when the member has a relatively small minor-axis available strength. These observations are general and apply to any method of design. Traditional methods of design account for some of these effects within the calculation of available strength for the limit state of lateral-torsional buckling. Newer advanced methods address the issue more completely and explicitly with specific requirements for the handling of torsional deformations in the analysis.

Benchmark Problem Details

Four benchmark problems based on the member shown in Fig. 1 are presented in this work, all of which have the same geometric and material properties. The member was comprised of an I-shaped wide flange

W 460 \times 97

(

W 18 \times 65

) cross-section profile with a span and unbraced length of

L = 6.1 m

(240 in.). The cross-sectional properties of this shape are listed in Table 1. The material properties were as follows: modulus of elasticity

E = 200 GPa

(29,000 ksi); shear modulus of elasticity

G = 77 GPa

(11,154 ksi); Poisson’s ratio

ν = 0.3

; and yield stress

F_{y} = 345 MPa

(50 ksi). However, all of the elastic analyses presented in this work utilized stiffness reductions (i.e., both moduli were reduced by 0.8) and all of the inelastic analyses presented in this work utilized both stiffness and strength reductions (i.e., both moduli were reduced by 0.9 and

0.9 F_{y}

was used as the yield stress in the analyses).

Table 1. Cross-sectional properties

Parameter	Value
Depth of section, $d$	467 mm (18.4 in.)
Width of flange, $b_{f}$	193 mm (7.59 in.)
Thickness of web, $t_{w}$	11.4 mm (0.450 in.)
Thickness of flange, $t_{f}$	19.1 mm (0.750 in.)
Cross-sectional area, $A$	$12,300 {mm}^{2}$ ( $19.1 {in.}^{2}$ )
Moment of inertia about the major axis, $I_{x}$	$445 \times 10^{6} {mm}^{4}$ ( $1,070 {in.}^{4}$ )
Moment of inertia about the minor axis, $I_{y}$	$22.8 \times 10^{6} {mm}^{4}$ ( $54.8 {in.}^{4}$ )
Torsional constant, $J$	$1,140 \times 10^{3} {mm}^{4}$ ( $2.73 {in.}^{4}$ )
Warping constant, $C_{w}$	$1,140 \times 10^{9} {mm}^{6}$ ( $4,240 {in.}^{6}$ )

All members were simply supported (pin support at one end and roller support at the other end) with longitudinal rotation or twist at the member ends restrained. Member ends were assumed to be warping free with cross-flange bending not resisted at the supports. The axial, flexural, and uniformly distributed loads were assumed to be factored and were applied proportionally. All applied loads remained aligned to the original global coordinate system throughout each analysis even after deformations occurred. Thus, for example, a uniformly distributed gravity load (

w_{y}

in Fig. 1) was applied in the vertical direction throughout the loading history.

The design strength of the member was computed using the provisions of the AISC specification (AISC 2016b). The selected wide-flange section was compact; thus the strengths were not affected by local buckling. The cross-sectional axial strength was

ϕ P_{n s} = 3,825 kN

(860 kips). Given the unbraced length and support conditions, the compressive strength was

ϕ P_{n} = 952 kN

(214 kips). The cross-sectional major-axis flexural strength was

ϕ M_{n x} = 676.2 kN - m

(5,985 kip-in.). The member major-axis flexural strength depends on the lateral-torsional buckling modification factor,

C_{b}

, which was computed based on the shape of the moment diagram. For the case of uniform moment distribution (

C_{b} = 1

), the major-axis flexural strength was

ϕ M_{n x} = 380.9 kN - m

(3,371 kip-in.). For the case of uniformly distributed load and associated parabolic moment distribution (

C_{b} = 1.14

), the major-axis flexural strength was

ϕ M_{n x} = 434.2 kN - m

(3,843 kip-in.). The minor-axis flexural strength was

ϕ M_{n y} = 114.4 kN - m

(1,013 kip-in.). With these values, the major axis to minor axis available strength ratios were

M_{n x} / M_{n y} = 3.33

with

C_{b} = 1.0

and

M_{n x} / M_{n y} = 3.79

with

C_{b} = 1.14

. Neither of these strengths nor the governing interaction equation [Eq. (2)] account for the normal stresses produced by cross-flange bending due to warping. The AISC specification does not address how to account for such stresses in the calculation of available strength; however, methods have been proposed and have been implemented in other specifications (White and Grubb 2005).

Analysis Details

Several commercial and research programs were used to produce the analysis results presented in this paper. In general, all of these programs are known to possess rigorous three-dimensional geometric nonlinear (second-order) analysis capabilities; equilibrium on the deformed shape is ensured. For each of the benchmark problems, results are presented for a variety of analysis types. Given that most current structural engineering designs are completed using analyses that employ elastic line or framework elements, the focus of this paper was to establish such benchmark results. Results from four types of elastic beam analyses are presented, labeled (1), (2a), (2b), and (2c). Additionally, elastic shell analysis results labeled (S) are presented, as well as select results from an inelastic shell analysis labeled (NS). A comparison of the characteristics of the different elastic analysis types is presented in Table 2 and each is described in detail in the following sections.

Table 2. Details of analysis types

Characteristic	Elastic analysis type
Characteristic	(1)	(2a)	(2b)	(2c)	(S)
Finite-element type	Beam	Beam	Beam	Beam	Shell
Includes stiffness reduction (i.e., $0.8 E$ )	Yes	Yes	Yes	Yes	Yes
Includes initial geometric imperfection	No	No	Yes^a	Yes^a	Yes^a
Includes geometric nonlinearity	No	Yes	Yes	Yes	Yes
Geometric nonlinearity includes impact of twist	N/A	No	Yes	Yes	Yes
Includes warping resistance	No	No	No	Yes	Yes
Includes shear deformations	No	No	No	No	Yes
Utilizes Euler-Bernoulli beam theory (i.e., plane sections remain plane)	Yes	Yes	Yes	No	No
Includes material nonlinearity	No	No	No	No	No
Compressive strength used in interaction equation	$ϕ P_{n}$	$ϕ P_{n}$	$ϕ P_{n s}$	$ϕ P_{n s}$	$ϕ P_{n s}$
Analysis software used for tabulated results	Hand calculations	FE++	FE++	ANSYS or Abaqus^b	Abaqus

a

Initial geometric imperfections are not defined and not included for Problems 1 and 3.

b

Abaqus used for Problem 4 and for the determination of applied load ratio at

I_{H 11} = 1

for all problems.

All of the problems were created and analyzed using imperial units (i.e., kips and inches). The presented results were obtained by converting moments and deflections from imperial units to SI units. Additionally, only the magnitude (absolute value) of each result is presented, regardless of the sign of the result from the analysis software.

Elastic Beam Analyses

Four different types of elastic beam analyses were performed for each problem in this work. The analysis types differed in their treatment of geometric nonlinearities, torsional resistance, and initial geometric imperfections (Table 2).

The first analysis type, labeled (1), captured only first-order linear elastic behavior on a structure with no imperfections. This analysis type was included for comparative purposes and was not representative of any particular design methodology, although a stiffness reduction of 0.8 was included per the AISC specification requirements for the direct analysis method. The axial load within the member for all analyses presented in this work was less than the threshold at which the additional AISC flexural stiffness reduction factor,

τ_{b}

, would take on a value less than unity. While nearly any analysis software can perform analyses at this level, the results presented were obtained through hand calculations.

The second analysis type, labeled (2a), was representative of the method typically employed within the direct analysis method of design defined within Chapter C of the AISC specification (AISC 2016b). In this case, and specific to the problems presented in this paper, a traditional second-order (

P - δ

) analysis was employed in which the presence of axial force amplified deflections and moments about the separate principal axes of the cross section. In an effort to employ as few software packages as possible, this was accomplished by employing a rigorous second-order elastic analysis with all torsional deformations restrained (twist at each node along the length of the member was prevented). The tabulated results were obtained using the program FE++ accessed through the MASTAN2 (Version 3.5.5) graphical user interface; however, other software (e.g., OpenSees ) and SAP2000 ) provide nearly identical results. Initial geometric imperfections were not modeled and a reduction factor of 0.8 was applied to all stiffnesses.

The third analysis type, labeled (2b), was representative of the variation of the direct analysis method of design which utilizes advanced elastic analysis and is defined within Appendix 1 of the AISC specification (AISC 2016b). In these analyses, torsional deformations were included but using software that only accounts for the continuous flow of shear stresses on the section, commonly called uniform or St. Venant torsional resistance. It is permitted within the methodology to neglect stiffness related to warping restraint because doing so is typically conservative. An engineer may choose to neglect the additional torsional strength and stiffness provided by warping restraint because it is not common for analysis software to include these effects in beam elements. The tabulated results for this analysis type were also obtained using the program FE++. In accordance with the design methodology, the initial geometric imperfections were modeled where defined and a reduction factor of 0.8 was applied to all stiffnesses.

The fourth analysis type, labeled (2c), was representative of the same design methodology as (2b), but considered warping deformations along the length of the member within the analysis. The tabulated results for this analysis type were obtained with either ANSYS or Abaqus, with similar results between the two programs and when compared with other software with similar capabilities. [e.g., ADINA and MASTAN2 (version 3.5.5)]. In accordance with the design methodology, the initial geometric imperfections were modeled where defined and a reduction factor of 0.8 was applied to all stiffnesses.

In each of these cases, analysis results were reported from models in which the cross-sectional properties were provided; in other words, the general or integrated properties, such as the cross-sectional area and moment of inertia, were defined as input. Although not reported here, nearly identical results were obtained when the dimensions of the sections were provided as input, with the software computing the integrated section properties. Shear deformations were not included in any of the analyses that employed beam elements. Based on the results of a mesh refinement study presented subsequently, 40 line elements along the length of the beam were used in each analysis. The primary reason for this refinement was that the distributed load, where defined, was applied as point or concentrated loads at each node (element end) based on the tributary length.

When computing the interaction ratio [Eq. (2b)], the compressive strength was taken as that for the limit state of flexural buckling (

ϕ P_{n}

) for analysis types (1) and (2a) and the cross-sectional strength (

ϕ P_{n s}

) for analysis types (2b), (2c), and (S), which is described in the following section. The major-axis moment strength was taken as that for the limit state of lateral-torsional buckling for all analysis types. Despite the requirement to model imperfections and the advanced handling of torsional deformations, the available moment strength for the design by elastic analysis method described in Appendix 1 of the AISC specification (AISC 2016b) is not different than that for the more basic methods of design. The fact that cross-sectional strength cannot be used generally within this method somewhat diminishes its attractiveness (e.g., while the unbraced length need not be defined for flexural buckling, it still needs to be defined for lateral-torsional buckling). Future design provisions may allow for a method of design for structural steel members based solely on cross-sectional strength, perhaps by modifying the stiffness reduction. The stiffness reduction employed by the direct analysis method may be too simple to accurately capture inelastic lateral-torsional buckling resistance.

Elastic Shell Analysis

The fifth analysis type, labeled (S), utilized shell elements instead of beam elements. As noted in the previous section, analysis programs that consider warping restraint in beam elements are not common. This restraint, however, will naturally be included when modeling the member with shell elements. While modeling an entire structure with shell elements is impractical, it is feasible for special investigations of substructures or individual members. The tabulated results for this analysis type were obtained using Abaqus.

Models for this analysis type were built with 4-node shell elements with reduced integration. Noting the symmetry of the geometry, loading, and boundary conditions, only half of the span was modeled as shown in Fig. 3. Based on a mesh refinement study described subsequently, 16 elements were used along the width of each flange, 34 elements were used along the height of the web, and 240 of such element groups were used along the length of the half-span. The resulting elements were approximately square with a side dimension of approximately 12.7 mm (0.5 in.). To avoid material overlap at the junction of the web and flange, the shell elements were oriented such that the nodes were placed at the inside faces of the flanges and at the centroid of the web (Fig. 4). The fillets at the junction of the web and flange were neglected in these analyses. Distributed loads were applied as point loads located at the centroid of the web (except in Problem 4, where load height effects were investigated) and spaced a distance

L / 40

along the length of the member to mimic the load application used for the beam elements.

Fig. 3. Isometric view of the shell element model: (a) undeformed; and (b) deformed.

Fig. 4. Supported end boundary conditions for the shell element model.

Boundary conditions at the midspan end (plane of symmetry) were defined to enforce the symmetry. Specifically, all nodes at the midspan were constrained against translation in the longitudinal (

z

) direction (i.e.,

δ_{z} = 0

, thereby providing a fully restrained warping condition) and against rotation about the lateral (

x

) and vertical (

y

) axes (i.e.,

θ_{x} = θ_{y} = 0

). Boundary conditions at the supported end were defined to mimic a roller support with a warping free condition (Fig. 4). Three rigid body constraints were defined, one for each of the top flange, web, and bottom flange. Duplicate nodes were defined at the junction between the web and the flanges in order that no nodes were part of more than one rigid body constraint. The three translational degrees of freedom for each pair of duplicate nodes were tied to each other using stiff springs, allowing each of the flange rigid bodies to rotate independently of the web rigid body (i.e., plane sections do not remain plane). The node at the centroid of the web was constrained against translation in the lateral (

x

) and vertical (

y

) directions (i.e.,

δ_{x} = δ_{y} = 0

) and against rotation about the twist (

z

) axis (i.e.,

θ_{z} = 0

).

In these shell analyses, the major- and minor-axis bending moments were computed by integration via a free-body section cut. This approach gave moments oriented to the global coordinate axes, which were then resolved to the deformed member axes using Eq. (1). The displacements and twist (i.e.,

δ_{y}

,

δ_{x}

, and

θ_{z}

) were obtained from the node at the centroid of the cross section at midspan.

Inelastic Shell Analysis

The sixth analysis type, labeled (NS), utilized shell elements and included geometric and material nonlinearity. It is generally representative of the design by advanced inelastic analysis method defined within Appendix 1 of the AISC specification (AISC 2016b). This analysis type utilized a model that was the same as that for the elastic shell analyses apart from a different material constitutive model and the explicit inclusion of residual stresses (in contrast, the elastic analyses implicitly included the effect of residual stress with the 0.8 stiffness reduction on the material modulus).

The constitutive relation used for these analyses employs the von Mises yield criterion, associated plastic flow, and isotropic hardening. The modulus of elasticity and yield stress were reduced by a factor of 0.9 per the AISC specification (AISC 2016b). A hardening ratio of

E / 1,000 = 0.2 GPa

(29 ksi) was assumed. The engineering stress and strain relation was converted to true stress and strain as required for input to Abaqus (Fig. 5).

Fig. 5. Constitutive relation for inelastic shell element model.

Residual stresses implemented in the model were based on the Lehigh residual stress pattern (Galambos and Ketter 1959) with a maximum compressive residual stress of

f_{r c} = 0.3 F_{y} = 103 MPa

(15 ksi) at the flange tips. Based on equilibrium, the maximum tensile residual stress at the web was determined from Eq. (5) as

f_{r t} = 44.3 MPa

(6.42 ksi)

f_{r t} = f_{r c} \frac{b_{f} t_{f}}{b_{f} t_{f} + t_{w} (d - 2 t_{f})}

(5)

Residual stresses were defined directly for each element as an initial condition (prestressing). Although the idealized residual stress pattern varies linearly across the flange, it was discretized in the analysis model such that each element has a uniform residual stress (Fig. 6). It should be noted that the Lehigh residual stress pattern, while simple and commonly used for all types of wide-flange sections, was not developed for beam-type sections such as the one examined in this work, and other residual stress patterns may be more representative.

Fig. 6. Residual stress pattern for inelastic shell element model.

While the elastic analyses were performed employing a load control solution scheme, the inelastic analyses were performed using an arc length method in order that the analysis could be run into the postpeak range and the limit point could be readily identified.

Benchmark Problems

Problem 1: Beam Subjected to Biaxial Bending and Zero Axial Force

Most structural engineers and design codes associate second-order effects with additional moments produced by gravity loads acting through lateral displacements of stories (

P - Δ

sway moments) and lateral deformations of members (

P - δ

member moments)—this is termed as a traditional second-order analysis and was described previously. This first benchmark problem provided numerical results for a situation in which no axial force existed but there were still significant differences between results obtained when ensuring equilibrium on the deformed shape versus undeformed shape. In other words, second-order effects can still be significant even in the absence of axial force.

This problem consisted of a single member with properties and support conditions defined previously and with reference to Fig. 1, and with parameters provided in Table 3. Applied loading at each end included only the equal and opposite end moments of

M_{x} = 323.7 kN - m

(2,865 kip-in.) and

M_{y} = 17.17 kN - m

(152 kip-in.). The axial force and uniformly distributed loads shown in Fig. 1 were taken as zero (i.e.,

P = 0

,

w_{x} = 0

, and

w_{y} = 0

) and no initial imperfections were included within any of the analyses (i.e.,

δ_{x o} = 0

). Although the AISC specification (AISC 2016b) requires modeling of member imperfections for design by advanced analysis, these provisions have been purposely disregarded in order to provide a benchmark problem with a geometry that is relatively easy to create. For the shell analyses,

M_{x}

was applied at the node at the centroid of the web and

M_{y}

was applied in two equal halves at the nodes at the centroid of each the top and bottom flanges.

Table 3. Parameters for problems

Parameter^a	Benchmark problem number
Parameter^a	1	2	3	4
$w_{x}$	0	0	Varies^b	0
$w_{y}$	0	Varies^c	Varies^b	Varies^d^,^e
$M_{x}$	323.7 kN-m (2,865 kip-in.)	0	0	0
$M_{y}$	17.17 kN-m (152 kip-in.)	0	0	0
$P$	0	Varies^c	Varies^b	Varies^e
$δ_{x o}$	0	Eq. (6)	0	Eq. (6)

a

See Fig. 1 for parameter definitions.

b

See Table 6.

c

See Table 5.

d

Load applied at top of top flange (i.e.,

d / 2

above beam centerline).

e

See Table 7.

A mesh refinement study was performed for this problem to select the discretization used for all problems. Several analyses of Types (2b), (2c), and (S) were performed using different mesh densities. Twist results from the analyses are shown in Fig. 7. The study indicated that multiple elements along the length of the beam are necessary to achieve accurate results. In typical design practice, the use of one or a few elements along the length of a member is common. In contrast, Fig. 7 shows that 10 elements along the length are required to provide nearly convergent results. However, noting the goals of this work and that these models are small and can be analyzed quickly regardless, 40 elements along the length was selected as the mesh density for beam analyses in this work. Similarly, a relatively fine mesh is required for accurate results from the shell analyses. In the mesh refinement study for the shell models, the aspect ratio of the elements was held constant at about 1 in order for the total number of elements used in the analyses to be proportional to the square of the number of elements along the flange. Given these results, 16 elements along the flange width was selected as the mesh density for the shell analyses.

Comprehensive results from each analysis type are provided in Table 4. The format of the tabulated results is the same for all problems and will be described in detail here. The results include primary values obtained at the midspan of the member under the given loads:

M_{u x^{'}}

and

M_{u y^{'}}

, the major- and minor-axis moments about a coordinate system aligned with the deformed (rotated) section;

δ_{y}

and

δ_{x}

, the vertical (

y

) and lateral (

x

) displacements; and the longitudinal twist

θ_{z}

. Also included is the ratio from the governing interaction equation Eq. (H1-1) from the AISC specification (AISC 2016b),

I_{H 11}

, which is computed using Eq. (2b) with all strength terms computed in accordance with the specification. In the determination of this ratio, the major-axis flexural strength was taken, for this problem, as

ϕ M_{n x} = 380.9 kN - m

(3,371 kip-in.) with the appropriate moment gradient factor (

C_{b} = 1

) for a uniform moment distribution. The compressive strength was taken as either the member compressive strength including length effects,

ϕ P_{n}

, or the cross-sectional compressive strength,

ϕ P_{n s}

, depending on the analysis type employed (Table 2). For simplicity, no reduction in strength was made to account for the normal stresses produced by cross-flange bending, which is pertinent when computing the interaction ratio for cases in which the analysis includes warping restraint.

Table 4. Results for Problem 1

Parameter	Analysis type	Value
$M_{u x^{'}}$ [kN-m (kip-in.)]	(1)	323.7 (2,865)
	(2a)	323.7 (2,865)
	(2b)	200.4 (1,774)
	(2c)	304.2 (2,692)
	(S)	300.0 (2,656)
$M_{u y^{'}}$ [kN-m (kip-in.)]	(1)	17.2 (152)
	(2a)	17.2 (152)
	(2b)	254.8 (2,255)
	(2c)	112.1 (992)
	(S)	122.7 (1,086)
$δ_{y}$ [mm (in.)]	(1)	21.1 (0.831)
	(2a)	21.1 (0.831)
	(2b)	197.9 (7.791)
	(2c)	50.0 (1.970)
	(S)	55.7 (2.192)
$δ_{x}$ [mm (in.)]	(1)	21.9 (0.861)
	(2a)	21.9 (0.861)
	(2b)	194.7 (7.666)
	(2c)	111.5 (4.390)
	(S)	120.2 (4.733)
$θ_{z}$ [rad (degrees)]	(1)	0.0000 (0.00)
	(2a)	0.0000 (0.00)
	(2b)	0.8523 (48.83)
	(2c)	0.3000 (17.19)
	(S)	0.3352 (19.21)
$I_{H 11}$	(1)	1.00
	(2a)	1.00
	(2b)	2.75
	(2c)	1.78
	(S)	1.86
ALR at $I_{H 11} = 1$	(1)	1.00
	(2a)	1.00
	(2b)	0.70
	(2c)	0.81
	(S)	0.79
${ALR}_{ult}$	(NS)	0.85

Each of the results described previously was obtained at an applied load ratio

(ALR) = 1.0

, which corresponds to 100% of the described loads being applied. Additional results are presented for cases with applied load ratios not equal to 1. In these cases, ratios between the applied loads are equal but the magnitude is scaled by the value of ALR. The result ALR at

I_{H 11} = 1

is the applied load ratio at which results when input to Eq. (2b) yield a value of unity: in other words, the maximum load that can be applied according to the design methodology (i.e., without violating the specification’s interaction equation). Finally, the value

{ALR}_{ult}

is given for the inelastic shell analyses. This value is the applied load ratio at the limit point, or the maximum load that can be applied according to the analysis model.

Given that there is no axial force, most engineers and design codes would assume that a first-order elastic analysis is adequate. Under such an analysis [Analysis Type (1) in Table 4], the beam deflected vertically in plane and horizontally out of plane, but no twist was computed along its length. With no twist, the resulting midspan moments were equal to the applied end moments. Further, the specific values defined for the applied end moments were purposely selected such that the beam would be assessed as adequate (

I_{H 11} = 1

) in meeting the requirements of the governing interaction equation [Eq. (2)]. For reference, the major- and minor-axis demands equalled 85% and 15% of the member’s available major- and minor-axis moment capacities, respectively. Given that the interaction equation results in a value of unity for

ALR = 1.0

, the member proportion could be defined as optimal.

Nearly identical results were obtained for Analysis Type (2a) where a second-order elastic analysis was used, but the effects of twist were neglected. However, rigorous second-order analyses that account for twist and thereby assess equilibrium on the deformed shape provided significantly different results. Load path results from the different analysis types are shown in Fig. 8. As can be seen, for Analysis Type (1), the path taken by the analysis was linear and at

ALR = 1

the resulting moments lay directly on the interaction strength. The paths for the other analysis types were distinctly nonlinear, with more rapidly increasing minor-axis moment as the member twists. For Analysis Type (2b), without the beneficial effects of warping restraint, the calculated twist was the largest among the analysis types, and thus more of the applied major-axis moment

M_{x}

was resolved into minor-axis moment, thereby providing interaction equation results that indicate significant overstress (i.e.,

I_{H 11} = 2.75

). Much of the nonlinearity and minor-axis moment was produced in the latter portion of the analysis, as indicated by the fact that

ALR = 0.70

at

I_{H 11} = 1

. In other words, while the interaction equation results indicated a 175% overstress, only a 30% reduction in the applied loads is required to bring the member into compliance.

As expected, Analysis Types (2c) and (S) produced similar results, and since the analyses included the beneficial effects of warping restraint there was less twist, and therefore the minor-axis moments and interaction equation results were also lower. The inelastic shell analysis (NS) followed a similar path to Analysis Types (2c) and (S) prior to its limit point. The limit point from the inelastic analysis occurred at

{ALR}_{ult} = 0.85

. This result compares well with results from Analysis Types (2c) and (S), which predicted limit points at

ALR = 0.81

and 0.79, respectively. As expected, and because of the neglect of warping restraint, Analysis Type (2b) predicted an earlier limit point at

ALR = 0.70

. Analysis Types (1) and (2a) both showed some unconservative error since they both predicted a limit point of

ALR = 1.0

. Generally, a maximum of 5% unconservative error is permitted for beam-column design methodologies (ASCE 1997).

Problem 2: Beam-Column Subjected to an In-Plane Distributed Load

The second problem used the same geometry, member properties, and support conditions as defined previously and shown in Fig. 1. The applied loads included a range of combinations of in-plane uniformly distributed load,

w_{y}

, and axial force,

P

, such that each combination was near the ultimate strength of the member. The out-of-plane distributed load and applied end moments were taken as

w_{x} = 0

,

M_{x} = 0

, and

M_{y} = 0

.

Appendix 1 of the AISC specification (AISC 2016b) states that for design by advanced analysis, initial member imperfections shall be directly modeled and that the magnitude of the imperfections shall be the maximum amount considered in the design. Strict adherence to these rules can be challenging for the simple problems considered in this work, and more so for complex practical cases with multiple members. Many forms of imperfection exist (e.g., for this problem one could expect initial offsets in the lateral and vertical directions as well as initial twist of the member), but not all will significantly affect the results. A simplified approach to initial imperfections was adopted in this work: only the most salient imperfections were modeled and only when necessary to cause a perturbation that initiates member twist. Thus, none of the problems presented in this work were strictly in adherence to the provisions for modeling initial imperfections in the AISC specification. It is expected that engineers may make similar simplifications in practice.

The initial geometric imperfections were defined as a half sine wave with maximum magnitude of

L / 1,000

at midspan in the lateral (

x

) direction (Fig. 1). The magnitude of the initial geometric imperfections was selected as the maximum tolerance on the out-of-straightness given in the AISC code of standard practice (AISC 2016a). As stated previously, no initial geometric imperfections were defined in the vertical (

y

) in-plane direction or as twist. For both the beam and shell analyses, the initial geometric imperfections were applied by shifting all nodes in the lateral (

x

) direction based on their position in the longitudinal (

z

) direction per Eq. (6)

δ_{x o} (z) = \frac{L}{1,000} \sin (\frac{π z}{L})

(6)

Reported lateral deflections (

δ_{x}

) are from the imperfect structure and do not include the initial imperfections.

The analysis results are presented in Table 5, which is formatted in the same way as Table 4. For the determination of the interaction strength ratio, the major-axis flexural strength was determined according to the AISC specification (AISC 2016b) with

ϕ M_{n x} = 434.2 kN - m

(3,843 kip-in.), which includes a moment gradient factor of

C_{b} = 1.14

representative of a uniformly distributed load. While the moment diagram for the geometric nonlinear analyses is not exactly parabolic, the variation is not severe enough to justify a different value of

C_{b}

. This research is the basis for a version of Table 5 that appears in the commentary to the AISC (2016b) specification.

Table 5. Results for Problem 2

Parameter	Analysis type	Loading condition
Parameter	Analysis type	I	II	III	IV
$P$ [kN (kips)]	—	0.0 (0.0)	333.6 (75.0)	556.0 (125.0)	778.4 (175.0)
$w_{y}$ [ $kN / m (kip / ft)$ ]	—	58.4 (4.0)	43.8 (3.0)	29.2 (2.0)	14.6 (1.0)
$M_{u x^{'}}$ [kN-m (kip-in.)]	(1)	271.2 (2,400)	203.4 (1,800)	135.6 (1,200)	67.8 (600)
	(2a)	271.2 (2,400)	207.1 (1,833)	139.8 (1,237)	70.7 (626)
	(2b)	269.6 (2,386)	206.3 (1,826)	139.5 (1,235)	70.5 (624)
	(2c)	271.1 (2,399)	207.0 (1,832)	139.8 (1,237)	70.7 (626)
	(S)	271.0 (2,399)	207.1 (1,833)	139.9 (1,238)	70.7 (626)
$M_{u y^{'}}$ [kN-m (kip-in.)]	(1)	0.0 (0)	0.0 (0)	0.0 (0)	0.0 (0)
	(2a)	0.0 (0)	0.0 (0)	0.0 (0)	0.0 (0)
	(2b)	29.2 (258)	26.4 (234)	21.7 (192)	34.9 (309)
	(2c)	6.3 (56)	11.8 (104)	15.8 (140)	32.1 (284)
	(S)	7.9 (70)	14.1 (124)	17.7 (157)	34.5 (306)
$δ_{y}$ [mm (in.)]	(1)	14.7 (0.580)	11.1 (0.435)	7.4 (0.290)	3.7 (0.145)
	(2a)	14.7 (0.580)	11.3 (0.443)	7.6 (0.299)	3.8 (0.151)
	(2b)	17.6 (0.694)	13.3 (0.524)	8.7 (0.342)	5.1 (0.201)
	(2c)	15.0 (0.589)	11.7 (0.460)	8.1 (0.318)	4.7 (0.186)
	(S)	16.0 (0.629)	12.6 (0.495)	8.7 (0.344)	5.2 (0.205)
$δ_{x}$ [mm (in.)]	(1)	0.0 (0.000)	0.0 (0.000)	0.0 (0.000)	0.0 (0.000)
	(2a)	0.0 (0.000)	0.0 (0.000)	0.0 (0.000)	0.0 (0.000)
	(2b)	24.6 (0.967)	24.2 (0.951)	21.2 (0.833)	35.5 (1.397)
	(2c)	5.4 (0.214)	11.0 (0.435)	15.6 (0.616)	32.8 (1.292)
	(S)	6.9 (0.271)	13.3 (0.523)	17.6 (0.692)	35.5 (1.397)
$θ_{z} [rad (degrees)]$	(1)	0.0000 (0.00)	0.0000 (0.00)	0.0000 (0.00)	0.0000 (0.00)
	(2a)	0.0000 (0.00)	0.0000 (0.00)	0.0000 (0.00)	0.0000 (0.00)
	(2b)	0.1078 (6.19)	0.0790 (4.53)	0.0471 (2.70)	0.0358 (2.05)
	(2c)	0.0233 (1.33)	0.0290 (1.66)	0.0266 (1.52)	0.0260 (1.49)
	(S)	0.0292 (1.67)	0.0368 (2.11)	0.0327 (1.87)	0.0316 (1.81)
$I_{H 11}$	(1)	0.62	0.77	0.86	0.96
	(2a)	0.62	0.77	0.87	0.96
	(2b)	0.87	0.75	0.58	0.62
	(2c)	0.68	0.62	0.53	0.60
	(S)	0.69	0.64	0.55	0.62
ALR at $I_{H 11} = 1$	(1)	1.60	1.30	1.16	1.05
	(2a)	1.61	1.29	1.15	1.04
	(2b)	1.03	1.07	1.13	1.09
	(2c)	1.28	1.26	1.25	1.11
	(S)	1.23	1.17	1.20	1.08
${ALR}_{ult}$	(NS)	1.33	1.23	1.23	1.16

As a means of quantifying the effect of neglecting shear deformations in the beam analyses, additional analyses were performed that were similar to Analysis Type (2c) but with shear deformations included. The shear area in the major-axis direction was taken as

d t_{w}

and the shear area in the minor-axis direction was taken as

5 b_{f} t_{f} / 3

(CSI 2017). In these analyses, the deformations (i.e.,

δ_{y}

,

δ_{x}

, and

θ_{z}

) were as much as 5% greater than those presented in Table 5 and the moments (i.e.,

M_{u x^{'}}

and

M_{u y^{'}}

) were different by as much as 2%.

Similar to the previous problem, only minor differences exist between Analysis Types (1) and (2a). For both of these analysis types, the initial geometric imperfections were not included and thus the results only show in-plane behavior and thus the geometric nonlinearity in Analysis Type (2a) is limited to

P - δ

effects (moment magnification due to axial force). For the other analysis types, which include the initial geometric imperfections, full three-dimensional behavior is observed with twist, out-of-plane deflections, and minor-axis moments. Load path results for Analysis Type (2b) are shown in Fig. 9.

Fig. 9. Problem 2 results, Analysis Type (2b).

When comparing

{ALR}_{ult}

from the inelastic shell analyses to ALR at

I_{H 11} = 1

from the elastic analyses, the method of design by advanced elastic analyses (2c) and (S) accurately captures the strength of the member with only minor unconservative errors in some cases. The neglect of warping resistance is again shown to be a conservative assumption, consistently leading to lower strengths. Analysis Type (2a) shows unconservative errors for Loading Conditions I and II with no or lower axial compression and conservative error for Loading Conditions III and IV with higher axial compression.

Problem 3: Beam-Column Subjected to In-Plane and Out-of-Plane Distributed Loads

The third benchmark problem was similar to Problem 2 except that an out-of-plane uniformly distributed load,

w_{x}

, was applied along with the in-plane distributed load

w_{y}

and axial force

P

. Initial member out-of-straightness imperfection was not included. Although the AISC specification (AISC 2016b) requires modeling of member imperfections for design by advanced analysis, these provisions have been purposely disregarded in order to provide a benchmark problem with a geometry that is relatively easy to create. Neglecting initial geometric imperfections can have significant effect on specific results (e.g., moments and deformations), especially in the presence of higher axial loads. However, the effect on the calculated strength of the system is typically less. For this problem, the value of ALR at

I_{H 11} = 1

would only be 3% to 5% lower for Analysis Type (2b) if initial imperfections were included as they were in Problem 2.

The analysis results for this problem are presented in Table 6, which employs a format similar to Table 4. For the determination of the interaction strength ratio, the major-axis flexural strength was taken as

ϕ M_{n x} = 434.2 kN - m

(3,843 kip-in.), as it was in Problem 2.

Table 6. Results for Problem 3

Parameter	Analysis type	Loading condition
Parameter	Analysis type	I	II	III	IV
$P$ [kN (kips)]	—	0.0 (0.0)	222.4 (50.0)	444.8 (100.0)	667.2 (150.0)
$w_{y}$ [ $kN / m$ ( $kip / ft$ )]	—	58.4 (4.0)	43.8 (3.0)	29.2 (2.0)	14.6 (1.0)
$w_{x}$ [ $kN / m$ ( $kip / ft$ )]	—	5.8 (0.4)	4.4 (0.3)	2.9 (0.2)	1.5 (0.1)
$M_{u x^{'}}$ [kN-m (kip-in.)]	(1)	271.2 (2,400)	203.4 (1,800)	135.6 (1,200)	67.8 (600)
	(2a)	271.2 (2,400)	205.9 (1,822)	138.9 (1,229)	70.3 (622)
	(2b)	248.3 (2,198)	199.6 (1,767)	137.4 (1,216)	70.1 (620)
	(2c)	267.0 (2,363)	203.5 (1,802)	138.2 (1,223)	70.2 (621)
	(S)	265.7 (2,352)	203.4 (1,800)	138.1 (1,223)	70.3 (622)
$M_{u y^{'}}$ [kN-m (kip-in.)]	(1)	27.1 (240)	20.3 (180)	13.6 (120)	6.8 (60)
	(2a)	27.1 (240)	26.6 (235)	25.4 (225)	22.1 (196)
	(2b)	111.6 (988)	66.4 (588)	41.8 (370)	26.8 (237)
	(2c)	54.2 (480)	43.9 (389)	34.9 (309)	25.5 (226)
	(S)	60.0 (531)	47.7 (422)	36.9 (327)	26.4 (234)
$δ_{y}$ [mm (in.)]	(1)	14.7 (0.580)	11.1 (0.435)	7.4 (0.290)	3.7 (0.145)
	(2a)	14.7 (0.580)	11.2 (0.440)	7.5 (0.297)	3.8 (0.150)
	(2b)	42.4 (1.670)	19.6 (0.773)	10.0 (0.394)	4.3 (0.171)
	(2c)	19.2 (0.755)	13.7 (0.539)	8.8 (0.345)	4.2 (0.165)
	(S)	21.6 (0.850)	2.1 (0.597)	9.5 (0.375)	4.5 (0.177)
$δ_{x}$ [mm (in.)]	(1)	28.8 (1.133)	21.6 (0.849)	14.4 (0.566)	7.2 (0.283)
	(2a)	28.7 (1.130)	27.9 (1.100)	26.7 (1.050)	23.1 (0.909)
	(2b)	97.0 (3.820)	62.7 (2.470)	41.7 (1.640)	27.4 (1.080)
	(2c)	52.1 (2.050)	43.5 (1.712)	35.6 (1.400)	26.4 (1.040)
	(S)	57.7 (2.271)	47.5 (1.869)	37.8 (1.488)	27.5 (1.083)
$θ_{z} [rad (degrees)]$	(1)	0.0000 (0.00)	0.0000 (0.00)	0.0000 (0.00)	0.0000 (0.00)
	(2a)	0.0000 (0.00)	0.0000 (0.00)	0.0000 (0.00)	0.0000 (0.00)
	(2b)	0.3230 (18.51)	0.1580 (9.05)	0.0700 (4.01)	0.0232 (1.33)
	(2c)	0.1010 (5.79)	0.0682 (3.91)	0.0404 (2.31)	0.0165 (0.95)
	(S)	0.1225 (7.02)	0.0826 (4.73)	0.0482 (2.76)	0.0195 (1.12)
$I_{H 11}$	(1)	0.86	0.81	0.85	0.89
	(2a)	0.86	0.86	0.95	1.02
	(2b)	1.55	1.07	0.74	0.48
	(2c)	1.09	0.88	0.68	0.47
	(S)	1.14	0.91	0.70	0.48
ALR at $I_{H 11} = 1$	(1)	1.16	1.24	1.18	1.12
	(2a)	1.16	1.15	1.05	0.99
	(2b)	0.84	0.97	1.12	1.20
	(2c)	0.92	1.05	1.18	1.22
	(S)	0.93	1.05	1.16	1.20
${ALR}_{ult}$	(NS)	0.98	1.07	1.17	1.25

Problem 4: Beam-Column Subjected to an In-Plane Distributed Load Applied at Top Flange

In the previous problems, the transverse distributed loads were applied at the centroid of the cross section. This is generally what is assumed in equations for the computation of available flexural strength, such as in the AISC specification (AISC 2016b). Loads applied above the centroid can induce a destabilizing tipping effect that can increase the twist and reduce the strength of the member. One way of accounting for these load-height effects is with a specially calibrated lateral-torsional buckling modification factor, or moment gradient factor,

C_{b}

, as suggested by Helwig et al. (1997). However, given the requirement of accurate modeling of twist within the advanced elastic analysis method of design, it is possible that load-height effects can be accounted for directly within the rigorous second-order analysis. This example problem explores that possibility while also providing an additional set of results that may be used for validation purposes.

This benchmark problem is the same as Problem 2 except that the load was applied at the center of the top face of the top flange. The load height was simulated in the beam analyses by defining rigid posts modeled as stiff beam elements at each node along the length of the member, with each oriented in the positive vertical (

y

) direction with a length of half the depth of the section. The point loads representing

w_{y}

were applied to the nodes at the top of these posts. A similar approach was used for the shell element models, but the posts were connected to the node at the junction of the web and top flange and their length was equal to the thickness of the flange because the nominal node location was at the bottom of the top flange (Fig. 4). This idealized method of applying the load does not provide any tipping restraint, which exists in many practical situations and can negate load-height effects (Helwig et al. 1997).

The analysis results for this problem are presented in Table 7, which is similar in format to Table 4. For the determination of the interaction strength ratio, the major-axis flexural strength was again taken as

ϕ M_{n x} = 434.2 kN - m

(3,843 kip-in.). With the load-height effect modeled within the analyses, it was not included in the calculation of this design strength; in other words, the moment gradient factor of

C_{b} = 1.14

continued to be employed.

Table 7. Results for Problem 4

Parameter	Analysis type	Loading condition
Parameter	Analysis type	I	II	III	IV
$P$ [kN (kips)]	—	0.0 (0.0)	333.6 (75.0)	556.0 (125.0)	778.4 (175.0)
$w_{y}$ [ $kN / m$ ( $kip / ft$ )]	—	58.4 (4.0)	43.8 (3.0)	29.2 (2.0)	14.6 (1.0)
$M_{u x^{'}}$ [kN-m (kip-in.)]	(1)	271.2 (2,400)	203.4 (1,800)	135.6 (1,200)	67.8 (600)
	(2a)	271.4 (2,402)	207.2 (1,834)	139.8 (1,237)	70.8 (626)
	(2b)	121.9 (1,079)	138.3 (1,224)	137.6 (1,218)	70.4 (624)
	(2c)	269.1 (2,382)	206.1 (1,824)	139.4 (1,234)	70.5 (624)
	(S)	259.0 (2,293)	205.3 (1,817)	139.8 (1,237)	70.7 (626)
$M_{u y^{'}}$ [kN-m (kip-in.)]	(1)	0.0 (0)	0.0 (0)	0.0 (0)	0.0 (0)
	(2a)	0.0 (0)	0.0 (0)	0.0 (0)	0.0 (0)
	(2b)	242.2 (2,144)	215.7 (1,909)	51.7 (457)	38.5 (341)
	(2c)	32.0 (283)	28.4 (252)	20.9 (185)	33.5 (297)
	(S)	88.9 (787)	44.8 (396)	25.4 (225)	36.9 (327)
$δ_{y}$ [mm (in.)]	(1)	14.7 (0.580)	11.1 (0.435)	7.4 (0.290)	3.7 (0.145)
	(2a)	14.7 (0.580)	11.2 (0.443)	7.6 (0.299)	3.8 (0.151)
	(2b)	199.5 (7.854)	142.8 (5.623)	14.8 (0.581)	5.7 (0.223)
	(2c)	27.5 (1.084)	26.2 (1.032)	20.5 (0.805)	4.9 (0.194)
	(S)	25.5 (1.004)	17.8 (0.699)	9.6 (0.379)	5.6 (0.219)
$δ_{x}$ [mm (in.)]	(1)	0.0 (0.000)	0.0 (0.000)	0.0 (0.000)	0.0 (0.000)
	(2a)	0.0 (0.000)	0.0 (0.000)	0.0 (0.000)	0.0 (0.000)
	(2b)	117.5 (4.627)	149.6 (5.891)	49.0 (1.929)	39.0 (1.535)
	(2c)	18.3 (0.720)	13.7 (0.538)	8.6 (0.339)	34.2 (1.345)
	(S)	46.3 (1.824)	40.8 (1.606)	24.8 (0.977)	37.8 (1.488)
$θ_{z}$ [rad (degrees)]	(1)	0.0000 (0.00)	0.0000 (0.00)	0.0000 (0.00)	0.0000 (0.00)
	(2a)	0.0000 (0.00)	0.0000 (0.00)	0.0000 (0.00)	0.0000 (0.00)
	(2b)	1.1060 (63.37)	0.7976 (45.70)	0.1495 (8.57)	0.0473 (2.71)
	(2c)	0.1186 (6.80)	0.0854 (4.90)	0.0440 (2.52)	0.0314 (1.80)
	(S)	0.2059 (11.80)	0.1403 (8.04)	0.0587 (3.36)	0.0392 (2.25)
$I_{H 11}$	(1)	0.62	0.77	0.86	0.96
	(2a)	0.63	0.77	0.87	0.96
	(2b)	2.40	2.25	0.84	0.65
	(2c)	0.90	0.77	0.58	0.61
	(S)	1.37	0.91	0.62	0.63
ALR at $I_{H 11} = 1$	(1)	1.60	1.30	1.16	1.05
	(2a)	1.60	1.29	1.15	1.04
	(2b)	0.74	0.86	1.02	1.07
	(2c)	1.02	0.97	1.12	1.09
	(S)	0.98	1.02	1.09	1.07
${ALR}_{ult}$	(NS)	1.03	1.06	1.15	1.14

A comparison between the strength results from Table 5 for Problem 2 and Table 7 for Problem 4 is presented in Fig. 10. In this figure, the values of ALR are ALR at

I_{H 11} = 1

for the elastic analyses and

{ALR}_{ult}

for the inelastic analyses. As expected, the inelastic shell analyses, which are most closely representative of the physical problem, indicate a reduction in strength when the load is applied at the top flange. This reduction varies with the magnitude of the distributed load. Also as expected, only negligible load-height effects were noted using Analysis Types (1) and (2a) because the effects of twist were neglected in these analyses. This indicates that for the standard direct analysis method, reductions in the calculated design strength, such as those proposed by Helwig et al. (1997), are necessary. However, for Analysis Types (2b), (2c), and (S), there was an increase in twist and reduction in strength on the order of what was observed in the inelastic shell analyses. This indicates that load-height effects can be represented accurately within the analyses and it may not be necessary to include adjustments for these effects within the design strength (i.e., using a modified

C_{b}

). However, a more complete study would be necessary to make any general conclusions.

Fig. 10. Comparison of results from Problems 2 and 4.

Conclusions

Four benchmark problems have been presented in this paper, each using the same laterally unbraced I-shaped member subject to various conditions of in-plane and out-of-plane loading effects as well as applied axial force. These benchmark problems are meant to be valuable for validating the proper use of advanced methods of nonlinear analysis for situations in which the accurate modeling of spatial behavior is important. Just as importantly, the problems highlight important effects associated with twist that are relevant as the design profession moves toward employing three-dimensional nonlinear analysis, specifically:

•

For situations in which the major-axis flexure demands of an I-shaped member approach its capacity, only a small amount of twist needs to occur before such major-axis demands are resolved into significant and potentially troublesome minor-axis demands.

•

Second-order, or geometrically nonlinear, analysis is required to assess such demands, even for cases in which there is no axial force.

•

For the cases investigated in this work, new methods of design which require advanced elastic analyses that explicitly model twist provide strength results that are in close agreement with results from advanced inelastic analyses.

•

While it is common to account for only uniform (St. Venant) torsional resistance, the use of analysis programs that account for both uniform and nonuniform (warping) torsional resistance results in more accurate and smaller deflections, twist, and minor-axis bending moment demands. However, the additional normal stresses produced by the corresponding cross-flange bending when warping is included should be considered in the evaluation of available strength.

Furthermore, this work has identified several aspects of the design process for which engineers may appreciate more clarity and recommendations. With this in mind, additional research may be necessary to address these gaps in knowledge, and the following topics could help guide this future work:

•

Importance of including member initial geometric imperfections, lateral and/or torsional, within an analysis model.

•

Significance of warping stresses in the evaluation of the available strength of members.

•

Interaction equations based on moment demands that are resolved to a coordinate system aligned with the deformed shape of the member.

•

Need for interaction equation checks made at a cross section by cross section basis along the length of the member.

•

Stiffness reductions necessary to provide for inelastic lateral-torsional buckling within an elastic design methodology.

•

Further identification of criteria to be used to characterize structural systems in which more advanced methods of nonlinear analysis are warranted.

References

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Information & Authors

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Published In

Journal of Structural Engineering

Volume 144 • Issue 12 • December 2018

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This work is made available under the terms of the Creative Commons Attribution 4.0 International license, http://creativecommons.org/licenses/by/4.0/.

History

Received: Sep 18, 2017

Accepted: Jun 18, 2018

Published online: Oct 5, 2018

Published in print: Dec 1, 2018

Discussion open until: Mar 5, 2019

Authors

Affiliations

Ronald D. Ziemian, M.ASCE [email protected]

Professor, Dept. of Civil and Environmental Engineering, Bucknell Univ., Lewisburg, PA 17837 (corresponding author). Email: [email protected]

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Jean C. Batista Abreu [email protected]

Assistant Professor, Dept. of Engineering and Physics, Elizabethtown College, Elizabethtown, PA 17022. Email: [email protected]

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Mark D. Denavit, M.ASCE [email protected]

Assistant Professor, Dept. of Civil and Environmental Engineering, Univ. of Tennessee, Knoxville, TN 37996. Email: [email protected]

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Tsu-Jung L. Denavit [email protected]

Research Assistant Professor, Dept. of Civil and Environmental Engineering, Univ. of Tennessee, Knoxville, TN 37996. Email: [email protected]

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Three-Dimensional Benchmark Problems for Design by Advanced Analysis: Impact of Twist

Abstract

Introduction

Effect of Twist on Strength

Benchmark Problem Details

Analysis Details

Elastic Beam Analyses

Elastic Shell Analysis

Inelastic Shell Analysis

Benchmark Problems

Problem 1: Beam Subjected to Biaxial Bending and Zero Axial Force

Problem 2: Beam-Column Subjected to an In-Plane Distributed Load

Problem 3: Beam-Column Subjected to In-Plane and Out-of-Plane Distributed Loads

Problem 4: Beam-Column Subjected to an In-Plane Distributed Load Applied at Top Flange

Conclusions

References

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Abstract

Introduction

Effect of Twist on Strength

Benchmark Problem Details

Analysis Details

Elastic Beam Analyses

Elastic Shell Analysis

Inelastic Shell Analysis

Benchmark Problems

Problem 1: Beam Subjected to Biaxial Bending and Zero Axial Force

Problem 2: Beam-Column Subjected to an In-Plane Distributed Load

Problem 3: Beam-Column Subjected to In-Plane and Out-of-Plane Distributed Loads

Problem 4: Beam-Column Subjected to an In-Plane Distributed Load Applied at Top Flange

Conclusions

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