Structural Stability From Theory to Practice

Engineering Structures 22 (2000) 116–122www.elsevier.com/locate/engstruct

Structural stability: from theory to practice

W.F. Chen*

School of Civil Engineering, Purdue University, West Lafayette, IN 47907, USA

Received 27 July 1997; accepted 9 January 1998

Abstract

Over the past 40 years drastic improvements in our knowledge regarding the behavior, strength and design of steel buildingframes has been achieved. This paper provides several specific examples in which new knowledge has been implemented and betterdesign methods have been advanced in engineering practice. The directions of possible immediate implementation of some recentdevelopments in advanced analysis for practical frame design are outlined. 1999 Elsevier Science Ltd. All rights reserved.

Keywords:Advanced analysis; Buildings; Design; Effective length factor; Plasticity; Stability; Steel; Structural engineering

1. Introduction

What measuring stick should be used to assess theaccomplishment of structural stability research for steelbuilding frames in the past 40 years? Should it be thevolume of papers presented, or the number of journalarticles published, or the number of PhD theses producedor the number of new courses in the universities offered?I believe that the ‘bottom line’ for the structural engin-eering profession should be the amount of researchwhich finds its way into practice. The profession hasdeveloped in the past decades in important ways as adirect result of these intensive research activities world-wide. The SSRC-related ‘success stories’ that can beattributed to these developments fall into a number ofbroad categories. For the Symposium honoring ProfessorT.V. Galambos, it is the most appropriate occasion tosummarize these categories in this paper and provideseveral specific examples within each category wherenew knowledge has been implemented and, in somemeasure, a better understanding of the behavior of struc-tural members and systems has been developed and bet-ter design methods have been advanced. Directions ofpossible immediate implementations of some recentdevelopments for engineering practice are outlined here.

* Tel: 1 1-765-494-2254; Fax:1 1-765-496-1105

0141-0296/00/$ - see front matter 1999 Elsevier Science Ltd. All rights reserved.PII: S0141-0296 (98)00100-X

2. Behavior and design of structural members

Perhaps in no other area has there been such drasticimprovement in our knowledge regarding the behavior,strength and design of structural members including col-umns, beams and beam-columns using the mainframecomputing and finite element methods in the 1960s and1970s. There has been a steady flow of results fromSSRC research into the development of improved codesand standards governing the design of structural mem-bers in building codes. Major changes were made, forexample, in the design of biaxially loaded columns. Thestudies ranged from full-scale tests to complex finite-element analysis of beam-columns under various loadcombinations in plane and in space. The informationproduced has been implemented in AISC building codesand Euro-codes and has become standard practice in the1980’s. These and other related developments were sum-marized in a two-volume book by Chen and Atsuta in1976–77 [1] as well as the 1988 SSRC Guides editedby Galambos [2].

Another area where significant advances have beenmade is in the design of large fabricated cylindricalmembers as used in deep-water offshore structures. Sev-eral specific areas where the research results were instru-mental in bringing about major changes in API codesand other practices for engineering in offshore structuresinclude the effect of hydrostatic pressure on columnstrength, the beam-column strength and behavior con-sidering dent damage effects, and the assessment of the

117W.F. Chen/Engineering Structures 22 (2000) 116–122

strength of internally grout-repaired damaged members.This information and related developments were summa-rized in the books for example by Chen and Han [3] andChen and Toma [4] among others.

3. Structural design with K factors

In current engineering practice, the interactionbetween the structural system and its members is rep-resented by the effective length factor (Fig. 1). Thisclassical approach to structural design is describedclearly in the 1981 SSRC Technical Memorandum No.5 [5], which provides the basis for the development ofmodern steel design methods including the popular loadresistance factor design (LRFD) and allowable stressdesign (ASD) methods [6]. The effective length methodgenerally provides a good method for the design offramed structures. However, despite its popular use inthe past and present as a basis for design, the approachhas major limitations.

The first of these is that it does not give an accurateindication of the factor against failure, because it doesnot consider the interaction of strength and stabilitybetween the member and structural system in a directmanner. It is a well recognized fact that the actual failuremode of the structural system often does not have anyresemblance whatsoever to the elastic buckling mode ofthe structural system that is the basis for the determi-nation of the effective length factor K.

The second and perhaps the most serious limitation isprobably the rationale of the current two-stage processin design: elastic analysis is used to determine the forcesacting on each member of a structural system, whereasinelastic analysis is used to determine the strength ofeach member treated as an isolated component. There isno verification of the compatibility between the isolatedmember and the member as part of a frame. The individ-

Fig. 1. Interaction between a structural system and its componentmembers.

ual member strength equations as specified in specifi-cations are not concerned with system compatibility. Asa result, there is no explicit guarantee that all memberswill sustain their design loads under the geometric con-figuration imposed by the framework.

The other limitations of the effective length methodinclude the difficulty of computing a K factor, which isnot user-friendly for a computer-based design, and theinability of the method to predict the actual strength ofa framed member, among many others. To this end, thereis an increasing awareness of the need for practicalanalysis/design methods that can account for the com-patibility between the member and system. With therapid increase in the power of desktop computers anduser-friendly software in recent years, the developmentof an alternative method to a direct design of structuralsystem without the use of K factors becomes moreattractive and realistic. The real challenge is making thistype of new approach to design work and competitivein engineering practice. An extensive research on thistopic, now known as advanced analysis to design, hasbeen made at several universities around the world formany years, and significant advancements have beenmade, although much more remains to be done.

4. Advanced analysis to design

Extensive research has been devoted to the develop-ment and validation of several advanced analysismethods. A promising technique of developing high-order beam elements (making only one or two necessaryto describe the behavior along a members’ length) is inprogress at Cornell University [7]. An intensely rigorousmethod using workstations and super-computers to solvethousands of degrees of freedom has been in develop-ment around the world for many years [8,9]. Simple cali-bration techniques and practical approaches have beenresearched here at Purdue [10]. Intermediate solutionsinclude plastic-zone, quasi-plastic hinge, elastic–plastichinge methods and various modifications thereof. All insome way account for residual stresses, geometricimperfections, non-linearities and moment redistributionthroughout a structure. Briefly they are outlined below:

1. Plastic zone [8,9]:a. Discretized finite elements along the length and

through the cross-section.b. Captures the incremental load-versus-deflection

response considering the second-order geo-metric distortion.

c. A constant residual stress pattern is assumed.d. The spread of plasticity is traced.

2. Quasi-plastic zone [7]:a. A compromise between plastic zone and elastic

plastic hinge methods.

118 W.F. Chen/Engineering Structures 22 (2000) 116–122

b. The spread of plasticity is considered by flexi-bility coefficients.

c. A simplified residual stress pattern is used.d. The fully plastic cross-section is calibrated to

the plastic zone solution.e. There is no potential to upgrade this from its

current two-dimensional restriction.

3. Elastic–plastic hinge [11]a. Zero length plastic hinges.

b. No spread of yielding through the cross-section,or along the length.

c. No consideration of residual stresses.d. Second-order geometric effects can be con-

sidered.

4. Refined plastic hinge method [12]:a. A step up from the elastic plastic model for

two dimensions.b. Distributed plasticity-smooth stiffness degra-

dation of a hinge.c. Inelasticity is considered indirectly by forces

rather than strains. Tangent (Et) modulus is usedto describe the effect of residual stresses.

d. Stiffness degradation function is used for grad-ual yielding.

e. Connection flexibility can be modelled usingrotational springs.

5. Practical refined plastic hinge method [13]:a. The refined model (4. above) is made practical

by calibration to the LRFD empirical codeequations.

b. A separate modification of tangent modulus (Et)is imposed to consider geometric imperfections.

c. The CRC tangent modulus model is usedallowing residual stresses to be considered sep-arately.

For the last method to work effectively on popularcommercial programs in use in design offices todayfurther changes are needed. The Purdue method [14] wasdeveloped to perform designs, using simple modifi-cations to elastic parameters familiar to LRFD users,comparable to those achieved by traditional code pro-cedures—but a unique program was required. Theadvanced capabilities included two modifications formaterial non-linearity and one for geometric imperfec-tions. The degradation of stiffness due to gradual yield-ing of a cross-section subject to flexural moments wasdefined as shown in Fig. 3. Residual stresses wereaccounted for by reducing the tangent modulusEt asplotted in Fig. 2. The last non-linear effect, geometricimperfections, was considered by applying a furtherreduction toEt. Details of this development will be sum-marized in the following section.

Fig. 2. Moment-curvature relationship for a perfect plastic and workhardening hinge.

Fig. 3. Member tangent stiffness degradation derived from the CRCcolumn curve.

Two difficulties were noted with regard this method,which, being changed, would result in a practical methodof using current software to achieve the same types ofanalysis.

1. The choice of using stability functions to account forthe P-D effect does not translate well into the finiteelement world. Stability functions use small defor-mations theory implicitly to capture an effect thatmany finite element analyses account for explicitly.The advantage of this technique is only one elementis needed per member. This is becoming less and lessof a driving issue with the speed of computer analysistoday, and the price of having more elements alongthe length (while capturing behavior taking place out-of-alignment with the centre-line of the members) issmall. This allows the finite element method (as itexists in most all structural analysis packages) to beused as they stand.


2. The degradation in stiffness modification factors wasapplied by the Purdue team to coefficients in the stiff-ness matrix of each element. It was required that thefundamental solution algorithms be re-coded in orderto consider this non-linear behavior ‘simply’. Thisdifficulty will be overcome with this method by tak-ing advantage of the iterative capabilities inherent (orsoon to be implemented) in common analysis pack-ages [15]. Non-linearities due to material behaviorfast becoming available in commercial codes, andusing this to define a failure criterion would presenta convenient and practicable ‘advanced analysis’method.

5. Structural design without K factor

In the following, I shall briefly summarize a practicalsolution to the problem by simply modifying an elasticprogram with a modified tangent modulus based on thefamiliar CRC column strength equation together with arefined plastic hinge concept. These modifications con-sider the following key behavioral effects of a steelmember: second-order, gradual yielding associated withresidual stresses and flexure and geometric imperfec-tions. To meet the current LRFD requirements, thesemodifications have been calibrated against the LRFDspecification.

5.1. Second-order effects

To capture second-order effects, the simplified stab-ility functions reported by Chen and Lui [16] is adopted.The incremental force-displacement relationship of amember may be written as, in the usual notations

3MA

MB

P4 5

EII 3S1 S2 0

S2 S1 0

0 0 A/I43uA

uB

e4 (1)

where S1 and S2 are stability functions, for in-planebending of a prismatic beam-column.

5.2. Cross-section plastic strength

The LRFD cross-section plastic strength curves areadopted for both strong and weak-axis bending

PwcPy

189

MwbMp

5 1.0 forP

wcPy

$ 0.2 (2a)

P2wcPy

1M

wbMp

5 1.0 forP

wcPy

, 0.2 (2b)

The reduction factorsw are selected as 0.85 for axialstrength and 0.9 for flexural strength just as the LRFDspecification does.

5.3. Residual stresses

The CRC tangent modulus is employed to account forthe gradual yielding effect due to residual stresses alongthe length of members under axial loads. In thisapproach, the elastic modulusE instead of the momentof inertia I is reduced to account for the reduction of theelastic portion of the cross-section, because the reductionof elastic modulus is easier to implement than that ofthe moment of inertia for different sections. Thereduction rate in stiffness for both strong and weak axisis taken to be the same and this reduction is reflected bythe CRCEt as (Fig. 2).

Et 5 1.0E for P # 0.5Py (3a)

Et 5 4PPy

ES1 2PPyD for P > 0.5Py (3b)

5.4. Distributed plasticity

When idealized plastic hinges are formed at the mem-ber ends, the elastic stiffness at the ends will be reducedabruptedly to zero (Fig. 3). To represent a gradual tran-sition from the elastic stiffness at the onset of yieldingto the stiffness associated with a full plastic hinge at theends, the parameterh representing a gradual stiffnessreduction associated with flexure is introduced with 0#h # 1.0 according to the parabolic expression (Fig. 4)

h 5 4a(1 2 a) for a > 0.5 (4)

Fig. 4. Parabolic plastic hinge stiffness degradation function withao 5 0.5 based on LRFD sectional strength equation.


wherea is the force-state parameter obtained from thelimit state surface corresponding to the member ends(Fig. 5)

a 5PPy

189

MMp

forPPy

$29

MMp

(5a)

a 5P

2Py

1MMp

forPPy

#29

MMp

(5b)

similar to the LRFD plastic sectional strengthexpressions. This is known as the refined plastic hingeconcept. It reflects the distributed plasticity effects asso-ciated with bending actions at the member ends. Whenthese refined plastic hinges are present at both ends ofa member, the incremental elastic force–displacementrelationship as given in Eq. (1) can now be modified toinclude both the inelasticity within the member by usingEt instead ofE and the distributed plasticity at the endsby using the refined plastic hinge concept using thehparameter as

3MA

MB

P4 5 (6)

EtIL 3

hAFS1 2S2

2

S1

(1 2 hB)G hAhBS2 0

hAhBS2 hBFS1 2S2

2

S1

(1 2 hA)G 0

0 0AI

4 3uA

uB

e4

wherehA and hB are stiffness reduction factor as givenin Eq. (4) at end A and end B, respectively. Details ofthis development are given elsewhere [14].

Fig. 5. Smooth stiffness degradation for a work-hardening plastichinge based on LRFD sectional strength curve.

Fig. 6. CRC and reduced tangent modulus for members with geo-metrical imperfection.

5.5. Geometric imperfections

The degradation of member stiffness due to geometricimperfections may be simulated by a further reductionof member stiffness. This may be achieved simply by afurther reduction of the tangent modulusEt as (Fig. 6)

E9t 5 0.85Et (7)

Herein, the reduction factor 0.85 is used to reducefurther the CRCEt as given in Eq. (3a) and (3b) toinclude the effect of geometric imperfections (Fig. 6).Thus, if the modulusE in the Euler buckling formula isreplaced byE9

t, the column strength curve as specifiedby the LRFD specification will be obtained within amaximum error of no more than 5% (Fig. 7). The furtherreduced modulusE9

t is applicable for both braced andunbraced members and frames.

Fig. 7. Comparison of column strength curves with further reducedtangent modulus method.


6. Behavior and design of structural system

When I first began to work in structural stability over35 years ago, evaluation of the first-order response of astructural system was a significant problem. Thisincludes linear elastic analysis and simple plastic analy-sis, and the progress made to the present state-of-the-art,which deals routinely with second-order inelastic analy-sis of complicated structural systems having hundredsof thousands of degrees of freedom, is miraculous. Themodelling of all types of structural systems of high-risebuildings can now be handled quickly and efficiently onrelatively inexpensive computers. The primary limitationis a sufficient understanding of the response of some sec-ondary structural elements such as concrete floor slabs,composite joints and walls that make up that system todevelop simple but realistic models that can be incorpor-ated into the analysis programs.

The advent of personal computers, particularly in thecomputing and graphics performance of engineeringworkstations, has made more sophisticated methods ofanalysis feasible in design practice. While the use offirst-order analysis for elastic or plastic design is still thenorm of engineering practice, a new generation of codeshas emerged that recommends the second-order theoryas the preferred method of analysis. The basic theory forsecond-order inelastic analysis is well established anddocumented in open literature. The real challenge ismaking this type of analysis work in engineering prac-tice. The advanced analysis approach to design as illus-trated in Fig. 8 can predict more accurately the possiblefailure modes of a structure, exhibit a more uniform levelof safety, and provide a better long-term serviceabilityand maintainability.

7. Seismic design with structural fuse

The analytical capability of tracing the performanceof a structure into the non-linear range required by seis-mic loads is also available [17]. Advanced analysis com-bines the theory of stability with the theory of plasticityand traces the gradual plastification of members with

Fig. 8. Analysis and design methods.

rigid or flexible joints in a steel frame (Chen and Sohal,[18]). The power of this tool is as follows: with theability to predict the actual moment distribution at loadlevels that require members to sustain their plasticmoment capacity, ‘breaks’ can be strategically locatedthroughout a structure. These structural ‘fuses’ can bedesigned to fail themselves without the risk of the build-ing as a whole falling down, while leaving the majorityof the connections in satisfactory condition (AISC, [19]).This would not only limit the amount of post-quakerepair necessary, but would also indicate where thefailed connections were and thus greatly reduce theexpense of ‘exploratory procedures’.

Some of the new aspects that can be further con-sidered in design practice when performance based (viaadvanced analysis) becomes standard practice are:

Semi-rigid connections and thus partly restrainedmembers have been researched extensively but thisknowledge has not been able to be easily implementedinto code-based design practice. Unless computermethods are adopted, small practical use can be garneredfrom the new knowledge of connection rotation charac-teristics and their effect on the global behavior of frames.

Furthermore, three-dimensional behavior is a naturalextension. Even only partly considering out-of-planebehavior (e.g. lateral torsional buckling) becomespossible.

8. Summary

The practical design method presented here introducesthe potential (and emphasizes the necessity) of advancedanalysis procedures. Long accustomed to isolated mem-ber-by-member capacity checks, one analysis now con-siders all components and their interdependence. Theglobal analysis provides information on the failure modeand thus allows an assessment of damage sustained atcollapse loads. If the damage can be predicted, it can becontrolled by design procedures calibrated to maintainadequate performance criteria (as opposed to the Codes’traditional safety levels). Only when design engineersare assured of the validity and convinced on the practi-cality of performance-based analysis and design will thisand other advanced capabilities be implemented in day-to-day offices and the results of many years of researchbe granted their true fulfillment: a place in Practice.

References

[1] Chen WF, Atsuta T. Theory of beam-columns, Vol. 1, In-planebehavior and design, 1976, Vol. 2, Space behavior and design,1977. New York: McGraw–Hill.

[2] Galambos TV, editor. Guide to stability design criteria for metalstructures, 4th edn. Wiley–Interscience, 1988.


[3] Chen WF, Han DJ. Tubular members in offshore structures. Lon-don: Pitman, 1985.

[4] Chen WF, Toma S. Analysis and software of cylindrical mem-bers. Boca Raton, FL: CRC Press, 1996.

[5] SSRC. General principles for the stability design of metal struc-tures. Technical Memorandum No. 5, Civil Engineering, ASCE,February 1981, pp. 53–54.

[6] Chen WF, Lui EM. Structural stability: theory and implemen-tation. New York: Elsevier, 1987.

[7] Deierlein GG. Steel-framed structures. Progress in structuralengineering and materials, Vol. 1, No. 1. London: CRC Ltd.,September 1997.

[8] Clarke MJ, Bridge RQ, Hancock GJ, Trahair NJ. Advancedanalysis of steel building frames. Journal of Constructional SteelResearch 1992;23(1–3):1–30.

[9] McGuire W. Computer-aided analysis. In: Dowling PJ, HardingJE, Bjorhovde R, editors. Constructional steel design—and inter-national guide. Elsevier Applied Science, 1992:915–932.

[10] Chen WF, Toma S. Advanced analysis of steel frames. BocaRaton, FL: CRC Press, 1994.

[11] White DW, Chen WF. Plastic-hinge based methods for advancedanalysis and design of steel frames. Bethlehem, PA: StructuralStability Research Council, Lehigh University, 1993.

[12] Liew JYR, White DW, Chen WF. Second-order refined plastichinge analysis for frame design: Parts 1 and 2. Journal of Struc-tural Engineering, ASCE 1993;119(11):3196–237.

[13] Kim SE, Chen WF. Practical advanced analysis for braced steelframe design and practical advanced analysis of unbraced steelframe design. Journal of Structural Engineering, ASCE1996;122(11):1259–74.

[14] Chen WF, Kim SE. LRFD steel design using advanced analysis.Boca Raton, FL: CRC Press, 1997.

[15] Chen WF, Han DJ. Plasticity for structural engineers. New York:Springer–Verlag, 1988.

[16] Chen WF, Lui EM. Stability design of steel frames. Boca Raton,FL: CRC Press, 1992.

[17] White DW, Chen WF, editors. Proceedings of the US–JapanSeminar on Innovations in Stability Concepts and Methods forSeismic Design in Structural Steel [special issue]. EngineeringStructures 1997;20(4–6).

[18] Chen WF, Sohal I. Plastic design and second-order analysis ofsteel frames. New York: Springer-Verlag, 1995.

[19] AISC. Load of resistance factor design specification for structuralsteel buildings. Chicago, IL: American Institute of Steel Con-struction, 1997.

Documents

Structural Stability From Theory to Practice