8th International Symposium on Steel Structures, November 5-7, 2015, Jeju, Korea

Comprehensive Stability Design of Steel Members and Systems via Inelastic Buckling Analysis

Donald W. White1*, Woo Yong Jeong2, and Oğuzhan Toğay 3

1*School of Civil and Environmental Engineering, Georgia Institute of Technology, Atlanta, GA, USA. dwhite@ce.gatech.edu (corresponding author)

2 School of Civil and Environmental Engineering, Georgia Institute of Technology, Atlanta, GA, USA. wyjeong77@gmail.com

3 School of Civil and Environmental Engineering, Georgia Institute of Technology, Atlanta, GA, USA. oguzhantogay@gatech.edu

Abstract This paper presents a comprehensive approach for the design of structural steel members and systems via the use of buckling analysis combined with appropriate column, beam and beam-column inelastic stiffness reduction factors. The stiffness reduction factors are derived from the ANSI/AISC 360 Specification column, beam and beam-column strength provisions. The resulting procedure provides a rigorous check of member design resistances accounting for continuity effects across braced points, as well as lateral and/or rotational restraint from other framing including a wide range of types and configurations of stability bracing. No separate checking of the corresponding underlying Specification member design resistance equations is required. In addition, no calculation of design strength factors, such as effective length (K) factors and moment gradient and/or load height (Cb) factors, is necessary. The buckling analysis model directly captures the fundamental mechanical responses associated with the design strength factors. For the design of beam- columns and frames, the approach may be used with the AISC Direct Analysis Method (the DM) or the Effective Length Method (the ELM). The DM or ELM requirements are satisfied in the second-order elastic analysis calculation of the required member strengths (i.e., the internal member forces), accounting for pre-buckling load-displacement effects. The buckling analysis captures the member design resistances. This approach provides a particularly powerful mechanism for the design of frames utilizing general stepped and/or tapered I-section members. Keywords: Buckling Analysis, Inelastic Stiffness Reduction Factors, Stability Design 1. Introduction Within the context of the Effective Length Method of design (the ELM), engineers have often calculated ine-lastic buckling effective length (K) factors to achieve a more accurate and economical design of columns. This process involves the determination of a stiffness reduction factor,, which captures the loss of rigidity of the column due to the spread of yielding, including initial residual stress effects, as a function of the magnitude of the col-umn axial force. Several different tau factor equations are used in practice, but there is only one that fully captures the implicit inelastic stiffness reduction associated with the AISC column strength curve (AISC 2010 and 2015). This tau factor typically is referred to as a. What many engineers do not realize is that the ELM does not actually require the calculation of K factors at all. The column theoretical buckling load can be computed directly and used in determining the design resistance rather than being obtained implicitly via the use of K. Furthermore, if the stiffness reduction 0.9 x 0.877 x a is incorporated in a buckling analysis of the member or structural system, the calculations may be set up such that, if the member or structure buckles at a given multiple of the required design load, Pu in LRFD, this load Pu is equal to the design axial resistance cPn. If the load multiplier required to reach the buckling load is greater than 1.0, with the column stiffnesses calculated based on 0.9 x 0.877 x a, the member or structure satisfies the AISC Specification column strength requirements without the need for further

checking. The column strength requirements are inher-ently included in the buckling calculations.

The above approach can be applied not only to account for column end rotational restraint from supports or other structural framing. In addition, it can be employed to rig-orously evaluate the column strength given the modeled stiffness of various types and combinations of stability bracing. Furthermore, since the bracing stiffness require-ments of the AISC Specification Appendix 6 are based on multiplying the ideal bracing stiffness, which is the brac-ing stiffness necessary to achieve a column buckling strength equal to the required column axial load, by 2/ = 2/0.75 in LRFD, a buckling analysis that incorporates the column a factor(s) can be used as a rigorous stiffness check of the column stability bracing. Even more power-ful is that the above approach can be extended to the bracing and member design of beams and beam-columns.

This paper first reviews the development and proper use of the column stiffness reduction factor, a. It then focuses on the extension of column buckling analysis procedures based on a to the rigorous assessment of general I-section beam and beam-column strength limit states via buckling analysis and appropriate inelastic stiffness reduction factors. 2. Column Buckling Analysis using the AISC Inelastic Stiffness Reduction Factor a The column inelastic stiffness reduction factor a is the most appropriate of various stiffness reduction estimates

for the AISC design assessment of steel columns via buckling analysis. This is because a is derived inherently from the AISC column strength curve. Therefore, when configured properly with a buckling analysis, the internal axial force in the column(s) is equal to cPn at incipient buckling of the analysis model. The a factor accounts implicitly for residual stress and initial geometric imper-fection effects, as well as the traditional higher margin of safety specified by AISC for slender columns. This factor is not the most appropriate inelastic stiffness reduction for a second-order load-deflection analysis, such as an anal-ysis conducted to satisfy the requirements of the Direct Analysis Method of design (the DM). The separate b factor has been adopted by AISC for use with the Direct Analysis Method. The b factor is intended to account predominantly just for nominal residual stress effects. If used with a second-order load- deflection analysis per the DM, the a factor gives falsely high internal forces and correspondingly low strength predictions. This is because the engineer would effectively be double-counting geo-metric imperfection effects in the DM if a were used, since geometric imperfections are modeled explicitly as part of the DM calculation of the internal forces. It is recommended that a can be used, along with an eigen-value buckling analysis, to provide a rigorous assessment of the member resistances, given the internal forces ob-tained from the above DM second-order load-deflection analysis. The use of a in a buckling analysis allows the engineer to obtain a rigorous prediction of column strengths per the AISC column strength equations, ac-counting for continuity effects across braced points and lateral and/or rotational restraint from other framing, including general stability bracing. With this approach, no separate checking of the corresponding underlying Spec-ification member resistance equations is needed. The mechanical responses associated with effective length (K) factors and moment gradient and load height (Cb) factors are captured rigorously without the difficulty of deter-mining these factors. Refined estimates of the member strengths are obtained without the modeling of detailed member out-of-straightness imperfections.

The a factor has been used extensively for the calcu-lation of column inelastic effective lengths for use in the AISC Effective Length Method of design (the ELM). However, as has been well documented throughout the literature, one does not have to actually calculate column effective lengths to use the ELM. The ELM can be em-ployed with an explicit buckling analysis to determine the column internal axial force at theoretical buckling, Pe, or the corresponding column axial stress Fe (in essence a “direct” buckling analysis, but the word “direct” is being avoided here to avoid confusion with the DM). This type of application of the ELM, including the application of a, is documented in detail in (ASCE 1997).

An expression for a can be derived as follows. The derivation is shown only in the context of AISC LRFD to keep the development succinct and clear.

Generally, one can write the AISC factored column design resistance as

0.9 (0.877) 0.9 (0.877)c n e a eP P P (1)

where 0.9 is the resistance factor used by the AISC Specification for column axial compression, 0.877 is a factor applied generally to the elastic column buckling resistance in the AISC Specification to obtain the nominal column elastic buckling resistance (accounting for geo-metric imperfection and partial yielding effects for col-umns that fail by theoretical elastic buckling, as well as an implicit increased margin of safety for slender columns), and a is the column inelastic stiffness reduction factor. The column inelastic buckling load, considering just a and not considering the additional 0.9 and 0.877 factors, may be written as

e a eP P (2)

where Pe is the theoretical column elastic buckling load.

For nonslender element columns with 4

9e

y

P

P , or

0.390 ,c n

c y

P

P

the AISC column inelastic strength

equation may be written as

0.658a y

e

P

Pc n

c y

P

P

(3)

Upon recognizing, from Eq. (1) that

0.877

ne

PP (4)

this relationship may be substituted into Eq. (2), giving

0.877

0.658a y

n

P

Pc n

c y

P

P

(5)

Using this form of the column inelastic strength equation, one can take the natural logarithm of both of its sides and then solve for a as follows:

0.877

ln ln 0.658a y

n

P

Pc n

c y

P

P

(6)

ln 0.877 ln 0.658yc na

c y n

PP

P P

(7)

2.724 lnc n c na

c y c y

P P

P P

(8)

(The c factor is included in both the numerator and de-nominator of the fractions in Eq. (8) to facilitate the next step of the development shown below.) The final Eq. (8) has been used widely for column inelastic buckling cal-culations in the context of the AISC Specification. This equation can be applied most clearly by substituting an internal axial force Pu for cPn, such that a can be

thought of conceptually as an effective reduction on the member flexural rigidity (EI) at a given level of axial load.

As such, the a equation becomes, for 0.390u

c y

P

P

:

2.724 lnu ua

c y c y

P P

P P

(9a)

This equation is valid only for column buckling within the

inelastic range. For 0.390 ,u

c y

P

P

elastic buckling

controls and 1a (9b)

The above a expressions can be employed with buckling analysis capabilities, such as those provided by the pro-grams Mastan (Ziemian 2014) and SABRE2 (White et al. 2015), to explicitly (or “directly”) calculate the maximum column strength for any axially loaded problem.

The most streamlined application of a with a buckling analysis to determine column strength is as follows: 1. Construct an overall buckling analysis model for the

problem at hand. (This can be done easily for basic structural problems using programs such as Mastan and SABRE2.)

2. Apply the desired factored loads from a given LRFD load combination to the above model. These applied loads produce the column internal axial forces Pu.

3. Reduce the elastic modulus of the structural members, E, by 0.9 x 0.877 = 0.7893.

4. Reduce the member moments of inertia by a, based on Pu, using Eqs. (9) with = 1. (Alternately, this step and step 3 may be replaced by a single step where ei-ther the elastic modulus E or the moment of inertia I is reduced by the net stiffness reduction factor, SRF = 0.9 x 0.877 x a.)

5. Solve for the inelastic buckling load of the above model. Vary the applied loads by the common applied load scale factor , calculate the a values based on the scaled load levels (using the corresponding internal loads Pu), and solve for the multiple of the current loading, , at which the system buckles. Iterate on these calculations until = 1, indicating that the sys-tem buckles at the load level specified at the start of the buckling analysis. The corresponding internal axial forcesPu in the model at incipient buckling are then “directly” equal to the column axial capacities cPn. This is a very powerful approach to obtain the most

accurate column design axial strengths, accounting for the inelastic characteristics of the different members at the strength limit state, continuity effects across braced points with adjacent member lengths and various other member end and intermediate restraints.

Traditionally, the ELM has been used with a basic version of this approach to determine the influence of end rotational restraint on columns. The member restraints need not be limited to just column end rotational restraints though. When applied to column buckling problems, the

above procedure gives an accurate estimate of the column strength accounting for the restraint offered by the bracing system. The engineer simply needs to include the lateral (and more generally, also torsional) stiffnesses provided by the bracing system in the buckling analysis.

In fact, if desired, rather than solving for the column buckling load for a given set of bracing stiffnesses, one can consider a given LRFD applied factored loading Pu (with = 1) and then solve for the required bracing stiff-nesses necessary to develop the critical buckling strength equal to this factored load level. These bracing stiffnesses are commonly referred to as the ideal bracing stiffness values, βi, corresponding to a given desired load level Pu. (The required bracing stiffness is then taken as 2i/, with = 0.75, per the AISC (2010 and 2015) Appendix 6.)

Some engineers have suggested that 0.8b, the general stiffness reduction used in the AISC Direct Analysis Method, should be used for all problems, including cal-culation of column inelastic buckling loads. This is cer-tainly possible, but such an approach misses the clear advantage of having a buckling analysis procedure that rigorously determines the value of cPn accounting for all end and intermediate restraint effects. This issue can be understood by comparing the net stiffness reduction factors (SRFs) 0.9 x 0.877 x a and 0.8b as shown in Fig. 1. The SRF of 0.8b generally does not give an accurate estimate of the column strength cPn when used in a buckling analysis calculation. It does perform reasonably well at giving an appropriate but less rigorous estimate of the column strengths if used as part of a second-order analysis in which appropriate geometric imperfections are included per the requirements of the DM.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 0.2 0.4 0.6 0.8 1

Net SR

F

Pu/Py

0.9 x 0.877 x a0.8 x b

Figure 1. Comparison of the net column stiffness reduc-

tion factors (SRFs) 0.9 x 0.877 x a and 0.8b 3. Net Column Stiffness Reduction Factor (SRF) for Columns with Slender Cross-Section Elements The draft 2016 AISC Specification (AISC 2015) has adopted a unified effective width approach to characterize the axial resistance of members having slender cross- section elements under uniform axial compression. Using

this approach, the member axial resistance is expressed simply as

0.9c n cr eP F A (10)

where Fcr is the column critical stress determined using the member gross cross-section properties and Ae is the cross-section effective area obtained by summing the effective widths times the thicknesses for all the cross- section elements. In many practical situations, the most economical welded I-section members have slender webs. Beam-type rolled wide flange sections, i.e., sections that have a depth-to-flange width d/bf greater than about 1.7, also often have slender webs. Therefore, it is important to define how the column inelastic SRFs should be deter-mined for these cases. Stated succinctly, the column net SRF is given by

0.9 x 0.877 x ea

g

ASRF

A (11)

for these cross-section types, where a is calculated from

Eqs. (9), but using the ratio u

c ye

P

P

instead of u

c y

P

P

, and

0.9c ye y eP F A (12)

In addition, the plate effective widths should be calculated based on the axial stress f = Pu /Ae. As such, since Ae is dependent on f, while f is also dependent on Ae, Ae and f generally must be solved for iteratively. These iterations are reasonably fast and are simple to handle numerically. For manual calculation, f may be taken conservatively as Fy. For columns with simply-supported end conditions, the above calculations produce the same result as the streamlined unified effective width equations in the draft 2016 AISC Specification (AISC 2015). For columns with general end and intermediate restraints, the above calcu-lations account for the idealized relative stiffnesses in the structural system with the same rigor as the more basic method for nonslender element members explained in Section 2. 4. Basic Column Inelastic Buckling Example Figure 2 shows a basic L-shaped frame subjected to a ver- tical load at its knee. The frame’s inelastic buckling mode is shown in the figure along with the column a and the load at incipient buckling cPn = P. The column inelastic K factor, back-calculated by setting SRF x the column elastic flexural buckling load Pe to cPn = P from the inelastic buckling solution, is shown as well. This result matches with a traditional iterative calculation based on AISC a values. The inelastic K of 0.861 is smaller than a K factor of 0.894 back-calculated from an elastic buckling analysis of the frame. Because of the column inelastic stiffness reduction, the rotational restraint provided by the elastic beam at the top of the column is increased, re-sulting in a 2.6 % increase in the column axial resistance for this problem (a small but measureable increase). A buckling analysis with the inelastic SRF incorporated integrally within the calculations captures these attributes of the response directly and explicitly.

W16x40, Lb = 9.144 m 

W14x120, Lc = 14.17 m 

Fy = 345 MPa

Columna= 0.633cPn = 5130 kN

K = 0.861 

P

Figure 2. Column inelastic buckling analysis example

5. Inelastic Lateral Torsional Buckling (LTB) Analysis using the Stiffness Reduction Obtained from the AISC LTB Strength Curves ltb Generally, one can write the factored AISC LRFD beam LTB design resistance as

0.9b n b b e b ltb eM R M R M (13)

where Me represents the theoretical beam elastic LTB resistance, Me represents the beam inelastic LTB re-sistance, and Rb is the web bend buckling strength reduc-tion factor, equal to 1.0 for a compact or noncompact web I-section. The term ltb is the SRF corresponding to the nominal AISC LTB strength curves. The derivation of this factor parallels the derivation of the basic column SRF, a, presented in Section 2. To keep the presentation succinct, the derivation of ltb is not provided here. Rather, just the resulting equations for ltb are summarized. These equa-tions are as follows. For all types of I-section members,

when ,b L

yc

R Fm

F where ,u

b yc

Mm

M

1ltb (14a)

However, for compact and noncompact web I-section

members with . ,b max LTBL

yc b yc

MFm

F M

4 2

2

2 2 26.76 2

ltb

yc

Y X

FX m Y

E

(14b)

and the corresponding stiffness reduction factor is

SRF = 0.9 ltb (14c)

where:

. 0.9b max LTB pc ycM R M (15)

is the so-called LTB “plateau strength,” equal to 0.9Mp for a compact-web cross-section,

11

1.951

pc p p ycr

t t tL

pc yc

m

R L L FLY m

r r r EF

R F

(16)

and

2 xc oS hX

J (17)

(The additional common variables in these equations are defined in the glossary at the end of the paper.) Conversely,

for slender-web I-sections with ,b L b max

yc b yc

R F Mm

F M

the following simpler form is obtained in comparison to Eq. (14b):

2

1.1 1.1h

ycbltb

b LLh

yc

mR

FRm

R FFR

F

(18a)

and the corresponding stiffness reduction factor is

SRF = 0.9 Rb ltb (18b)

where Rh is the hybrid cross-section factor, which is not addressed in the ANSI/AISC 360 Specification, but is addressed by similar strength equations in the AASHTO LRFD Specifications (AASHTO 2015). Furthermore, for compact- and noncompact-web sections, one can write

max .

max

max. max max.

/u u

b yc b LTB pc

u upc pc

b LTB b LTB

M Mm

M M R

M M MR R

M M M

(19)

where maxbM is the general “plateau strength” taken as

the minimum of the independent flexural strengths calcu-lated for LTB, flange local buckling (FLB), and tension flange yielding (TFY) as applicable:

max max. . .min , ,b b LTB b n FLB b n TFYM M M M (20)

The advantage of writing m as shown in Eq. (19) is that

the ratio max

u

b

M

M

varies from zero to a maximum value of

1.0, as does u

c ye

P

P

. These ratios facilitate the description

of an interpolated beam-column net SRF discussed sub-sequently. For slender-web I-sections, one can write

max . /

u u

b yc b LTB b h

M Mm

M M R R

max

max. max max.

u ub h b h

b LTB b LTB

M M MR R R R

M M M

(21)

where

Mmax.LTB = Rh Myc (22)

Figures 3 and 4 illustrate how ltb varies relative to the well-known column stiffness reduction factor a for rep-resentative beam- and column-type wide flange sections respectively. The behavior of ltb for slender-web I-sections is similar to that shown for the beam-type W21x44 section.

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.0 0.2 0.4 0.6 0.8 1.0 1.2

Beam LTB TauFactor

Column TauFactor

,

u u

c ye c max

P M

P M

0.223

Figure 3. Column and beam factors for a W21x44 representative beam-type wide flange section

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.0 0.2 0.4 0.6 0.8 1.0 1.2

Beam LTB TauFactor

Column TauFactor

,

u u

c ye c max

P M

P M

0.180

Figure 4. Column and beam factors for a W14x257 representative column-type wide flange section

The LTB inelastic stiffness reduction factor, ltb, is

generally somewhat larger (i.e., reduces the capacity less) than the corresponding column inelastic stiffness reduc-

tion factor, a, for a given normalized load ratio u

c ye

P

P

or

max

.u

b

M

M

It should be noted that based on the AISC LTB

strength curves, I-section beams still have significant effective inelastic stiffness when uM reaches the plateau

resistance bMmax. For the above W21x44 and W14x257 examples, ltb = 0.223 and 0.180 respectively when this level of loading is reached.

In addition to the above, if the internal moment at in-cipient buckling, Mu, is larger than the corresponding bMmax at the most critically loaded cross-section, based on an analysis in which the ltb values are calculated using the corresponding internal moments throughout the length of the members, the “plateau strength” has been reached at the critical cross-section; hence, the design strength is the applied load level at which the internal moment at the critical cross-section is equal to bMmax.

Since both of the above example cross-sections are doubly symmetric and have compact flanges, bMmax = bMmax.LTB. For doubly-symmetric sections having a non-compact or slender compression flange, or singly- symmetric sections with a noncompact or slender com-pression flange and hc > h (usually associated with the compression flange being smaller than the tension flange), flange local buckling (FLB) governs the plateau re-sistance and bMmax = bMmax.FLB. For singly-symmetric sections with hc < h, either FLB or tension flange yielding (TFY) can govern for the plateau resistance. For doubly- or singly-symmetric sections with compact webs and compact flanges, bMmax = bMmax.LTB = bMmax.FLB = bMmax.TFY = bMp, or as stated in the AISC Specification, the limit state is “yielding” and the other limit states do not apply.

For proper calculation of the LTB resistance from a buckling analysis, several requirements must be satisfied: 1. In the context of doubly-symmetric I-section members,

the buckling analysis software must rigorously include the contributions from warping rigidity ECw as well as the St. Venant torsional rigidity GJ and the lateral bending rigidity EIy.

2. In addition, for singly-symmetric I-section members, the buckling analysis must account rigorously for the behavior associated with the shear center differing from the cross-section centroidal axis, which relates to the monosymmetry factor, βx, in analytical equations for the LTB resistance of these types of beams.

3. The SRF = 0.9 Rb ltb should be applied equally to each of the elastic stiffness contributions GJ, ECw and EIy, at a given cross-section, for the execution of the buckling analysis. Physically, it can be argued that the effective reduction in the St. Venant torsional rigidity of an inelastic beam is not as large as the reduction in the effective EIy and ECw values. However, the use of an equal reduction on all three rigidities (at a given cross-section) is simple and sufficient. Furthermore, equal reduction on all three cross-section rigidities reproduces the beam LTB resistance from the AISC Specification equations exactly for cases involving uniform bending and simply-supported end condi-tions.

4. A separate SRF of 0.9 x 0.877 x a should be applied to the elastic stiffness contributions EA, EIx and EQx (where Qx is the first moment of the area about the reference axis of the cross-section, equal to zero when the reference axis is the cross-section centroidal axis, but non-zero in cases such as singly-symmetric section members, where it is common for the shear center to be taken as the reference axis in the structural analysis). For beam members subjected to zero axial load, a = 1. Beam-column members are addressed subsequently. Furthermore, it should be noted that for singly- symmetric sections, the moment about the centroidal axis must be used generally in the calculation of ltb.

5. The internal force state upon which the buckling analysis is based is to be determined using the elastic properties of the structure, using the rules specified in Chapter C or Appendix 7 of the AISC Specification. If the Effective Length Method of design is employed, the nominal elastic properties of the structure are used in the above load-deflection analysis to determine the internal forces. If the Direct Analysis Method of de-sign is employed, all the cross-section elastic stiff-nesses are reduced generally by 0.8b. In this case, AISC Chapter C also gives rules permitting the use of b = 1 if additional system initial geometric imperfec-tion effects are included in the load-deflection analysis. Also, AISC Chapter C and Appendix 7 specify re-quired nominal initial imperfections of the points of intersection of the members in the structure (i.e., “system imperfections”, or equivalent Notional loads, corresponding to these imperfections.

In general, the Chapter C and Appendix 7 re-quirements entail that a general inelastic nonlinear buckling analysis must be used to assess the member resistances. This is a buckling analysis in which the pre-buckling displacement effects are considered, via the use of a second-order load-deflection analysis to determine the internal forces. Given a selected level of applied load and the corresponding internal forces, the SRFs are calculated for the buckling analysis, and then the buckling analysis is performed to evaluate the member resistances. If the buckling eigenvalue is greater than 1.0, the buckling resistance is greater than the current load level.

In some cases, the member failure mode deter-mined from the buckling analysis may involve an overall buckling of the entire structural system; how-ever, in many situations, the member failure will in-volve a localized member buckling involving several unbraced lengths in the vicinity of a critical region, or failure by reaching a FLB or TFY limit state.

For problems involving only beam members or involving concentrically loaded columns with negli-gible pre-buckling displacements, an inelastic linear buckling analysis provides an acceptable solution. This type of bucking analysis entails the use of a first-order elastic analysis to determine the system in-ternal forces, followed by the calculation of the cor-

responding SRFs and the execution of the buckling analysis. The software SABRE2 (White et al. 2015) automates

and satisfies all the above requirements for general doubly- or singly-symmetric I-section members with prismatic or non-prismatic stepped and/or tapered geom-etries, as well as frames composed of these types of members. Figure 5 shows a snapshot of the main window of SABRE2. The SABRE2 software provides streamlined capabilities for defining all attributes of the above types of problems, as well as the definition of out-of-plane point and panel bracing along the lengths of the members.

Figure 5. Screen shot of a clear-span frame analysis being conducted using the SABRE2 software (White et al. 2015).

A sufficient number of elements per member must be employed in the above LTB solutions. For frame elements based on thin-walled open-section beam theory and cubic Hermitian interpolation of the transverse displacements and twists along the element length (the type of frame element employed by SABRE2), four elements within each unbraced length tend to be sufficient. In addition, for inelastic buckling cases involving a moment gradient, the variation of the inelastic stiffness along the member length must be captured. Eight elements with each span between the major-axis bending support locations tends to be sufficient to capture the variation in the SRFs for problems that do not have any reversal of the sign of the moment within the span. Sixteen elements within each span between major-axis bending support locations tends to be sufficient for problems involving fully-reversed curvature bending. The frame elements in SABRE2 use a five-point Gauss-Lobatto numerical integration along their length to capture the variations in the SRFs along the length of each element.

Obviously, the above inelastic LTB solutions are not manual engineering solutions. However, for that matter, neither is the general second-order elastic analysis of an indeterminate frame. Although engineers can conduct approximate analysis to perform initial sizing of the members in an indeterminate frame structure, commonly they do not rely on these analyses, manual moment dis-tribution calculations, etc. for final design at this day and time. With the appropriate software implementation of the

above a and ltb calculations using a frame element based on thin-walled open-section beam theory, the above procedure is quite easy to apply. The software performs the appropriate elastic matrix analysis of the structure to determine the required member internal forces. Then it performs an inelastic eigenvalue buckling analysis based on these forces to evaluate the design. If the software automatically handles the internal inelastic stiffness reductions based on the magnitude of the internal forces, as is performed in SABRE2, the inelastic buckling anal-ysis is relatively straightforward to apply.

This approach can be quite powerful to provide highly accurate consideration of end restraints, continuity across braced points, general moment gradient and finite bracing stiffness effects on the LTB resistance of beam and frame members. One key attribute of the power of this approach is that, similar to the a approach for column buckling, once one has determined the load level corresponding to incipient inelastic buckling using the ltb factor, the in-ternal forces in the model at the buckling load correspond precisely to the design moment resistances bMn. This approach allows the consideration of any and all restraints from bracing and member end conditions to be directly and automatically considered in the design assessment, by including them in the structural analysis model. Regard-ing the assessment of the required stiffnesses for stability bracing, this assessment is accomplished as a direct and integral part of the calculation of the member LTB re-sistances. If the buckling eigenvalue is greater than 1.0 from the buckling analysis, with the internal element stiffnesses calculated based on the ltb equations given the internal forces at a load level , then the beam has suffi-cient design strength for LTB at that load level. In addition, SABRE2 calculates the bMmax values associated with flange local buckling and tension flange yielding, as applicable, and checks these. If the critical beam cross- section reaches bMmax based on Eq. (20), with > 1 and with the reference load taken as the required loading from a given LRFD load combination, the system maximum load is governed by reaching this “plateau resistance” prior to the occurrence of LTB. (Note that other limit states such as web crippling, connection limit states, etc. must be checked separately, just as they would be in ordinary design.)

If desired, rather than solving for the beam LTB load given bracing stiffnesses, one can consider a given LRFD applied factored loading Mu (with = 1) and then solve for the ideal bracing stiffnesses necessary to develop this factored load level at buckling. The ideal bracing stiff-nesses are then multiplied by 2/ to obtain the required bracing stiffnesses for design. 6. Validation and Demonstration of the LTB Stiffness Reduction Equations Consider the LTB resistance of a suite of W21x44 beams having torsionally and flexurally simply-supported end conditions and unbraced lengths ranging from zero to 6.096 m. Figure 6 shows the results for the uniform

bending case as well as a basic moment gradient case involving an applied moment at one end and zero moment at the other end of the beams. All of the buckling analysis calculations in this example, and in the subsequent ex-amples are performed using the SABRE2 software (White et al., 2015).

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

Mu / bM

p

Lb /Lp

Uniform Moment AISC & SABRE2

Moment Gradient AISC

Moment Gradient SABRE2

Figure 6. Lateral torsional buckling design resistances for W21x44 beams (Fy = 345 MPa), calculated using the AISC Specification equations and using buckling analysis with the corresponding stiffness reduction factor 0.9ltb (Rb = 1.0)

The following observations can be made from this

LTB study: 1. The buckling analysis results for the LTB resistance

under uniform bending match exactly with the calcu-lations from the AISC Specification Section F2 equa-tions. Therefore, only one curve is shown for the uni-form bending case in Fig. 6.

2. The buckling analysis results for the LTB resistance under the moment gradient fit closely with the calcu-lations from AISC Specification Section F2 using a moment gradient factor Cb = 1.75. However, this rig-orous LTB curve is slightly different from the one obtained using the Section F2 LTB equations. The differences between these curves are important, and may be explained as follows: a. For longer unbraced lengths, where the beam is

elastic and ltb = 1, the buckling load determined from SABRE2 is approximately 6 % larger than the capacity determined from the AISC Section F2 equations with Cb taken as 1.75. The SABRE2 solution is a more accurate assessment in this case. The 1.75 value for Cb is a lower-bound approx-imation developed by Salvadori (1955). The SABRE2 solution is approximately 11 % larger than the solution with Cb = 1.67 obtained from AISC Eq. (F1-1) for this problem. AISC Eq. (F1-1), originally developed by Kirby and Nethercot (1979), gives a “lower” lower-bound solution than Professor Salvadori’s equation for this problem.

b. For intermediate unbraced lengths at which the maximum moment at incipient buckling is larger than bFLSxc = 0.9(0.7FySxc) (equal to 290 kN-m for

the W21x44), the inelastic buckling analysis solu-tion is again fully consistent with the AISC Section F2 equations, but is a more accurate assessment of the LTB resistance than the direct use of the AISC Section F2 equations. In this case, as the buckling resistance increases above bFLSxc, some reduction in the LTB resistance occurs due to the onset of yielding at the locations where the internal moment is largest. The approach taken in AISC Chapter F is to simply scale the uniform bending LTB resistance by Cb, but with a cap of bMmax on the flexural re-sistance. As discussed by Yura et al. (1978), this approach tends to over-predict the true response to some extent in the vicinity of where the elastic or inelastic LTB design strength curve intersects

bMmax, although the approximation is considered to be acceptable. The LTB resistances obtained from the buckling analysis are slightly smaller than those obtained directly from the AISC Chapter F2 equations in the vicinity of the location where the LTB resistance reaches the plateau resistance bMmax, reflecting the more rigorous accounting for inelastic stiffness reduction effects on the LTB re-sistance in the buckling analysis based solution.

7. Roof Girder Design Example Figure 7 shows a roof girder design example adapted from a suite of example problems developed by the AISC Ad hoc Committee on Stability Bracing (AISC 2002). The girder has a 21.3 m span and is subjected to gravity load-ing applied from outset roof purlins connected to its top flange and spaced at 1.52 m. The girder ends are assumed to be flexurally and torsionally simply supported.

308 kN‐m 308 kN‐m

3.05 m4.57 m 3.05 mSYM

21.3 m

A

A1.52 m (TYP)

Fy = 345 MPa

Frame spacing = 7.62 m

Panel bracing at top flange:Lbr = 1.52 m, br = 0.876 kN/mm

Torsional bracing:T = 723 kN‐m/rad

320 kN‐m230 mm purlinIx = 524 cm

4

51 mm

762 mm

610 x 3.78 mm

152 x 9.52 mm

roof girder

A = 1.92 cm2

rz = 6.25 mmIyc = 281 cm

4

Sx = 1110 mm3

rt = 39.1 mm

Figure 7. Roof girder example, adapted from (AISC 2002).

The top flange in this problem is braced at the purlin locations by light-weight roof deck panels, having a shear panel stiffness of 0.876 kN/mm. Flange diagonal braces are provided from the purlins to the bottom flange at the mid-span of the girder plus at two additional locations on

each side of the mid-span with a spacing of 3.05 m be-tween each of these positions. These diagonal braces restrain the movement of the bottom flange relative to the top flange, and therefore they are classified as torsional braces. The provided elastic torsional bracing stiffness is modeled as T = 723 kN-m/rad. These torsional braces combine with the panel lateral bracing from the roof deck to provide out-of-plane stability to the roof girder. The above bracing stiffnesses are divided by 2/ and the corresponding reduced stiffnesses are employed for ine-lastic buckling analysis, per AISC Appendix 6 (AISC 2010 and 2015).

Figure 8 shows the governing overall lateral-torsional buckling mode for this roof girder, determined using SABRE2. The lines shown with a diamond symbol in the horizontal plane at the top flange level represent the shear panel bracing from the roof deck, and the circular lines at the mid-span and at two locations on each side of the mid-span represent the torsional bracing from the roof purlins and the framing of a flange diagonal to the bottom flange of the girder. The arrow symbols indicate zero displacement constraints. Figure 9 shows the variation in the net SRF along the length of the girder obtained from the SABRE2 solution.

Figure 8. Governing overall lateral-torsional buckling mode for the roof girder

The applied load scale factor on the required vertical

gravity load at incipient inelastic buckling of the girder is = 1.010. Therefore, the girder and its bracing system are sufficient to support the required LRFD loading. One can observe a noticeable lateral deformation within the adja-cent shear panels on each side of the mid-span torsional brace in Fig. 8. This indicates that the light roof panel bracing is providing slightly less than full bracing at the first braced point on each side of the mid-span. Never-theless, the overall design strength is slightly larger than the required strength from the LRFD loading. Figure 9 shows that significant yielding is developed both at the mid-span and at the girder ends when the girder reaches its maximum design resistance.

It is important to note that the accurate assessment of the combined bracing stiffnesses is somewhat challenging for this problem using any method other than the SABRE2 buckling analysis. The basic requirements specified in AISC Appendix 6 do not address combined

lateral and torsional partial bracing. The AISC (2015) Appendix 1 provisions provide guidance for the use of advanced load-deflection analysis methods for the general stability design. However, the application of these meth-ods necessitates the modeling of an appropriate initial out-of-alignment of the girder braced points as well as out-of-straightness of the girder flanges between the braced points. The geometric imperfections needed to evaluate the different bracing components are in general different for each of the bracing components, and the geometric imperfections necessary to evaluate the max-imum strength of the girder are in general different from those necessary to evaluate the bracing components. One can rule out the need to perform many of these analyses by identifying the girder critical unbraced lengths as well as the critical bracing components. However, short of the type of buckling analysis provided by SABRE2, it can be difficult to assess which unbraced lengths and which bracing components are indeed the critical ones. SABRE2 provides not only an assessment of the adequacy of the bracing system stiffnesses, but it also provides a “direct” check of the member design resistance given the mem-ber’s bracing restraints and end boundary conditions.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.2 0.4 0.6 0.8 1

Net SRF

Normalized position along girder length

Figure 9. Variation of the net stiffness reduction factor (SRF) along the length of the roof girder at its maximum design resistance corresponding to = 1.01

The only shortcoming of the above buckling analysis

approach in the context of the above type of design problem is that this approach does not provide any direct estimate of the bracing strength requirements. However, based on numerous results from experimental testing and from refined FEA simulation of experimental tests, it is recommended that the simple member force percentage rules of Appendix 6 can be used to specify the minimum required strengths for the different bracing components (AISC 2015). 8. Stiffness Reduction Factors for Beam-Columns Traditional beam-column strength interaction equations utilize a simple interpolation between the member axial strength in the absence of bending, cPn, and the member flexural strength in the absence of axial loading, bMn. Given the stiffness reduction factors SRF = 0.9 x 0.877 x

a e

g

A

A for axial load only and SRF = 0.9 Rb ltb for bend-

ing only, one might expect that a simple interpolation between these stiffness reduction factors would provide an accurate representation of the net SRF for beam- column members. The authors have found that the fol-lowing interpolation between the cross-section column and beam net SRF values provides an accurate charac-terization of I-section beam-column strengths: 1. The unity check value with respect to the cross-section

maximum strength is obtained using the equations

max

8

9u u

c ye c

P MUC

P M

for 0.2u

c ye

P

P

(23a)

and

max2

u u

c ye c

P MUC

P M

for 0.2u

c ye

P

P

(23b)

2. The above UC value is employed in the a and ltb

equations instead of the ratios u

c ye

P

P

and max

.u

c

M

M

3. The angle

max

/atan

/u c y

u b

P P

M M

(24)

is calculated. This angle is the position of the current force point within a normalized x-y interaction plot of the axial and moment strength ratios for a given cross-section.

4. The net SRF representing the beam-column response is determined using the interpolation equation

0.9 0.877 0.9190 90

ea b ltbo o

g

ASRF R

A

(25)

This SRF value is applied to ECw, EIy and GJ. The sep-

arate column SRF = 0.9 x 0.877 x a e

g

A

A, with a obtained

directly from the column strength ratio ,u

c ye

P

P

is applied

to EIx, EQx and EA, as discussed previously in Section 5.

In addition, it should be noted that the area ratio e

g

A

Ain the

above expressions is determined directly from the axial force uP as discussed previously in Section 3. The fol-

lowing section demonstrates the quality of the beam- column strength estimates obtained from the above ap-proach for several basic validation cases.

9. Beam-Column Examples Figures 10 and 11 show the beam-column strength curves obtained from SABRE2 for several suites of flexurally and torsionally simply-supported beam-columns sub-jected to uniform primary bending moment and a linear primary moment gradient loading respectively. The fol-lowing observations may be made from these plots:

1. For the shorter members, the strength envelopes are essentially equal to the fully-effective cross-section plastic strength curves (i.e., Eqs. 23 with Pye = Py = Fy

Ag) at smaller axial load values. This result corre-sponds to the mechanics of the beam-column strength problem reported by Cuk et al. (1986) and approxi-mated by a combination of Eqs. (H1-1) and (H1-2) in the AISC (2010 and 2015) Specification.

2. For larger axial load values, the strength envelopes “peel away” from the above in-plane strengths at a particular axial load level and approach the out- of-plane column strength of these simply-supported members in the limit of zero bending and pure axial compression. The strength envelopes in these regions are slightly convex due to the loss of effectiveness of the W21x44 webs with increasing axial force. That is, the W21x44 webs are slender under uniform axial compression.

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.2 0.4 0.6 0.8 1.0

cPn /

c Py

bMn /bMp

Fully‐Effective Cross‐Section Plastic Strength

L = 0.914 m

L = 1.52 m

L = 3.05 m

L = 4.57 m

Figure 10. Beam-column strength curves obtained from SABRE2 for several suites of flexurally and torsionally simply-supported W21x44 members subjected to uniform primary bending moment and uniform axial compression

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.2 0.4 0.6 0.8 1.0

cPn/

cPy

cMn /cMp

Fully‐Effective Cross‐Section Plastic Strength

L = 2.29 m

L = 4.57 m

L = 3.05 m

Figure 11. Beam-column strength curves obtained from SABRE2 for several suites of flexurally and torsionally simply-supported W21x44 members subjected to uniform axial compression and moment gradient loading from an applied bending moment at one end, zero bending mo-ment at the opposite end

3. The members with longer lengths are not able to develop bMn = bMp for the case of zero axial load; rather, their bending resistance is governed by out-of-plane lateral torsional buckling. The moment gradient loadings allow the development of bMn = bMp for larger member unbraced lengths in the case of zero axial load.

4. For the longest W21x44 members considered (i.e., L = 4.57 m), subjected to uniform primary bending mo-ment, the failure is entirely due to elastic beam and beam-column lateral torsional buckling. In this case, the resulting strength envelope ranges between 0.9 x 0.877 Pe for pure axial compression and zero bending and 0.9 Rb Me for pure bending and zero axial com-pression, where Pe is the theoretical out-of-plane flexural buckling load and Me is the theoretical lateral torsional buckling moment for pure bending. The shape of the strength curve in this case matches ex-actly with the theoretical beam-column elastic LTB resistance, scaled by an interpolated reduction factor ranging from 0.9 x 0.877 to 0.9 Rb.

It is not possible to perform beam-column strength cal-culations with this level of rigor by a second-order load-deflection analysis combined with the traditional application of separate manual strength interaction equa-tions. SABRE2 incorporates the AISC member strength equations ubiquitously within a buckling analysis, via calculated net stiffness reduction factors (SRFs), to pro-vide a more rigorous characterization of the member resistances. 10. Conclusions Traditional design of structural steel members and frames involves the use of a second-order elastic load-deflection analysis to estimate member internal forces, followed by the application of separate “manual” member resistance equations. These “manual” equations commonly require the calculation of various design strength factors, such as different member effective length factors (K) corre-sponding to ideal flexural and torsional column buckling as well as beam lateral torsional buckling (often estimated based on ideal elastic buckling, but sometimes determined using more tedious inelastic buckling estimates), and moment gradient and load height modifiers, which are represented by the term Cb in the AISC (2010 and 2015) provisions. These factors become increasingly tedious to calculate and increasingly tenuous in terms of the result-ing accuracy for problems involving general loadings and general member intermediate (flexible bracing) and end (flexible rotational and/or translational) restraints. The assessment of the adequacy of flexible intermediate lateral and/or torsional bracing generally requires the consideration of the member inelastic stiffness properties at the strength limit. Emerging advanced design proce-dures focus on an explicit analysis of the second-order load-deflection response of geometrically imperfect members and systems; however, these procedures are relatively expensive to conduct and can require the con-sideration of a tremendous number of load combinations

times geometric imperfection patterns to perform a com-plete assessment of an overall structural system.

This paper presents a comprehensive method for the design of structural steel members and systems via the use of buckling analysis combined with appropriate column, beam and beam-column strength reduction factors (SRFs). The net SRFs are derived from the current ANSI/AISC 360 Specification (AISC 2015) column, beam and beam-column strength provisions. The resulting proce-dure provides a rigorous check of member design re-sistances, accounting for continuity effects across braced points, as well as lateral and/or rotational restraint from other framing including a wide range of types and con-figurations of stability bracing. Although the emphasis of the examples presented in this paper is on more basic demonstration and validation problems involving pris-matic members, the recommended procedure is particu-larly powerful for the design assessment of frame utilizing general stepped and/or tapered I-section members. 11. Glossary of Terms A Gross cross-sectional area Ae Effective cross-section area corresponding to a given

axial compression force, accounting for plate local buckling effects

Ag Gross cross-sectional area Cb Beam moment gradient and load height factor Cw Warping constant E Modulus of elasticity of steel, taken equal to 200 GPa G Shear modulus of elasticity of steel, taken equal to

77.2 GPa FL Magnitude of flexural stress in the compression

flange at which flange local buckling or lateral tor-sional buckling is taken to be influenced by yielding in the AISC Specification

Fcr Column critical buckling stress Fe Member internal axial compression stress at incipient

elastic buckling of the member or structural system Fy Specified minimum yield stress Fyc Specified minimum yield stress of the compression

flange I Moment of inertia in the plane of bending Ix Moment of inertia about the major principal axis Iy Moment of inertia about the minor principal axis Iyc Moment of inertia of the compression flange about

the axis of the web J St. Venant torsional constant K Column or beam effective length factor L Member length Lbr Member unbraced length between the braced points Lp Prismatic beam unbraced length limit within which

the AISC nominal lateral torsional buckling (LTB) resistance under uniform bending is equal to the plateau resistance Mmax.LTB

Lr Prismatic beam unbraced length limit beyond which the nominal AISC lateral torsional buckling (LTB) resistance is taken as the theoretical elastic LTB re-sistance

Me Elastic lateral torsional buckling moment of the member for the case of zero axial compressive force

Me Member maximum buckling moment equal to ltb Me Mmax Largest potential moment that can be developed

in the beam cross-section considering the three po-tential governing limit states of compression flange yielding, compression flange local buckling, and tension flange yielding

Mmax.FLB Largest potential moment that can be developed in the beam cross-section considering only the com-pression flange local buckling limit state

Mmax.LTB Largest potential moment that can be developed prior to lateral torsional buckling for sufficiently short member lengths, considering only the lateral torsional buckling limit state; commonly referred to as the LTB “plateau resistance,” and referred to in the AISC Specification as the limit state of yielding or compression flange yielding

Mmax.TFY Largest potential moment that can be developed in the beam cross-section considering only the ten-sion flange yielding limit state

Mn Member nominal flexural strength Mp Cross-section plastic bending moment Mu Cross-section internal moment corresponding to a

given ASCE LRFD load combination Myc Moment at nominal yielding of the extreme fiber of

the compression flange Pe Member internal axial compression force at incipient

elastic buckling of the member or structural system Pe Member axial force equal to a Pe Pn Member nominal axial resistance Pu Cross-section internal axial load corresponding to a

given ASCE LRFD load combination Py Cross-section yield axial load, equal to AgFy Pye Yield axial load based on the effective cross-section

area, taken equal AeFy

Qx First moment of the gross cross-sectional area about the reference axis of the cross-section

Rb Cross-section bend buckling strength reduction factor, AASHTO (2015) notation, represented by the symbol Rpg in the AISC Specification

Rh Hybrid girder factor from AASHTO (2015) Rpc AISC LRFD web plastification factor, or effective

plastic section modulus considering web slenderness effects

Sx Elastic section modulus Sxc Elastic section modulus to the extreme fiber of the

compression flange for major-axis bending SRF Stiffness reduction factor employed for the buckling

analysis UC Unity check value with respect to the cross-section

maximum strength, given by Eqs. (23) X Cross-section major-axis flexural to St. Venant tor-

sional property ratio, taken as /xc oS h J

Y Intermediate factor used in calculating ltb for com-pact and noncompact web I-sections, given by Eq. (16)

bf Width of flange

d Full nominal depth of the cross-section f Member axial stress equal to Pu /Ae h Clear distance between the flanges less the fillet

radius for rolled I-sections; clear distance between the flanges for welded I-sections

hc Twice the distance from the centroid of the cross-section to the inside face of the compression flange less the fillet radius for rolled I-sections, and to the inside face of the compression flange for welded I-sections

ho Distance between the flange centroids m normalized cross-section moment Mu / bMyc rt radius of gyration of the compression flange plus

one-third of the web area in compression due to the application of major-axis bending moment alone

Applied design load scale factor Mu Cross-section internal moment corresponding to a

given value of the load scale factor Pu Cross-section axial force corresponding to a given

value of the load scale factor T Provided torsional bracing stiffness br Provided lateral bracing stiffness x Cross-section monosymmetry factor i Ideal bracing stiffness, defined as the bracing stiff-

ness at which the member or structure and its bracing system buckle at the required design load

AISC LRFD resistance factor on bracing stiffness, equal to 0.75

b AISC LRFD resistance factor for flexure c AISC LRFD resistance factor for axial compression Eigenvalue obtained from the buckling analysis,

equal to the multiple of the current loading corre-sponding to buckling, given the stiffness properties associated with the current loading state

a Column inelastic stiffness reduction factor not in-cluding the additional factors 0.9 x 0.877 x Ae /Ag

b Stiffness reduction factor applied to the member flexural rigidity for a second-order load-deflection analysis per the AISC Direct Analysis Method

ltb Beam lateral torsional buckling stiffness reduction factor not including the additional factors 0.9Rb

Angle of the force point within the normalized x-y interaction plot of the cross-section axial and moment strength ratios, given by Eq. (24)

12. References AASHTO (2015). AASHTO LRFD Bridge Design Speci-

fications, 7th Edition with 2015 Interim Revisions, American Association of State Highway and Trans-portation Officials, Washington, DC.

AISC (2015). Specification for Structural Steel Buildings, Draft AISC 360 Specification 2016, Ballot 4, August.

AISC (2010). Specification for Structural Steel Buildings, ANSI/AISC 360-10, American Institute of Steel Con-struction, Chicago, IL.

AISC (2002). “Example Problems Illustrating the Use of the New Bracing Provisions – Section C3, Spec and

Commentary”, Ad hoc Committee on Stability Bracing, November.

ASCE (1997). Effective Length and Notional Load Ap-proaches for Assessing Frame Stability: Implications for American Steel Design, American Society of Civil Engineers, Reston, VA.

Cuk, P.E., Rogers, D.F. and Trahair, N.S. (1986). “Inelas-tic Buckling of Continuous Steel Beam-Columns,” Journal of Constructional Steel Research, 6(1), 21-52.

Kirby, P.A. and Nethercot, D.A. (1979). Design for Structural Stability, Wiley, New York.

Salvadori, M.G. (1955). “Lateral Buckling of Beams,” Transactions ASCE, Vol. 120, 1165.

White, D.W., Jeong, W.Y. and Toğay, O. (2015). “Sabre 2,” <white.ce.gatech.edu/sabre> (Aug. 3, 2015).

Yura, J.A., Galambos, T.V. and Ravindra, M.K. (1978). “The Bending Resistance of Steel Beams,” Journal of the Structural Division, ASCE, 104(ST9), 1355-1370.

Ziemian, R. (2014). “MASTAN2 v3.5,” <www.mastan2. com > (Aug 3, 2015).