Stability of Steel Structures

OVERALL INSTABILITY EFFECTS IN BUILDING FRAMES SUBJECTED TO STATIC AND DYNAMIC LOADS

by

Le-Wu Lu

Paper prepared for presentation at the US-Japan Seminar on "Inelastic Instability of Steel Structures and Structural Elements," Tokyo, May 25 to June 2, 1981

Fritz Engineering Laboratory Lehigh University

Bethlehem, Pennsylyania

April 1981

Fritz Engineering Laboratory Report No. 471.1

OVERALL INSTABILITY EFFECTS IN BUILDING FRAMES SUBJECTED TO STATIC AND DYNAMIC LOADS

by

1 Le-Wu Lu

INTRODUCTION

Extensive research has been carried out since the early 1960's to study the effects of overall instability on the behavior and strength of structural frames. The initial work was concerned primarily with frames subjected to gravity load or gravity load in combination with lateral load which is assumed to increase monotonically from zero to its maximum (non-repeated or non-reversed). Design provisions applicable to building frames of regular geometrical configurations and subjected to usual combinations of gravity and lateral loads have been proposed. Optimum design methods that takes into account the effect of frame instability have also been developed. Subsequent work extended the study to include repeated and reversed lateral loads and dynamic responses due to seismic ground motions. Most recent work examined the problem of dynamic soil-structural interaction in the presence of the gravity load effect.

The studies on overall instability made before 1960 dealt mostly with elastic buckling problems associated with frames acted upon by static axial loads. Inelastic effects or the effects of primary bending moments were not considered in these studies, nor were the effects of dynamic load. A com­prehensive survey of the literature of the early work can be found in Ref. 1.

In this paper, a summary of the investigations of the various overall in­stability problems that have been carried out after 1960 is presented. The presentation deals first with frames under gravity load or combined loads, including both analysis and design problems. Frames subjected to repeated and reversed lateral load are next considered, with emphasis on experimental ~·:ark. The effect of overall instability on the dynamic response of frames .excited by earthquake motions is then examined. The final topic is on the dynamic response of soil-structure systems. The paper is intended to be an overview of the progress made on the subjects. No detailed discussion of a particular problem is given. The references cited should be consulted for complete information or detailed results.

1. Professor of Civil Engineering and Director, Building Systems Division, Fritz Engineering Laboratory, Lehigh University, Bethlehem, Pennsylvania.

FRAMES SUBJECTED TO GRAVITY LOADS

~e elastic and inelastic buckling strength of single and multi-story steel frames subjected for gravity loads acting on girders has been studied in References 2, 3, and 4. One of the principal effects of the girder loads is to introduce primary bending moments in all the members. These moments cause non-uniform yielding of the members and reduce the structure's overall lateral stiffness (in addition to the stiffness reduction due to axial lQ9_d). Buckling takes place when the overall stiffness becomes zero. The method of solution employed in Reference 3 was the typical eigenvalue approach, whereas the small fictitious lateral load approach was adopted in Reference 4. In the latter approach, the structure is assumed to have an initial geometrical imperfection (sway), and deflects continuously with. increasing load. The structure fails at the stability limit load. Figure 1 shows the load-deflection curve of an initially imperfect structure; the peak of the curve gives the "stability limit load" of the structure. The predictions made by the eigenvalue approach or the small-lateral load approach have been verified by experiments.

The study of the inelastic buckling strength of frames subjected to gravity loads acting on girders is important in the development of the plastic design meth.od for unbraced frames. If the girders are designed on the basis of the beam-mechanism condition with a load factor of 1.7 (U.S. practice), the stability requirement is that the entire structure (or a portion of it) should not buckle at a load less than 1.7 times the working gravity load. Because of the non-uniform yielding caused by the primary moment, the conventional frame buckling analysis cannot be rationally applied in determining the strength of the frame. The "effective column length factors" or the "K-factors" determined for perfectly straight frames whose members are subjected only to axial forces have no relevance to the problems. Only an inelastic frame buckling analysis which takes into consideration the effect of variable yielding in all the members can provide a solution to the problem of gravity load carrying capacity.

An approximate method of inelastic frame buckling analysis is given in Reference 5.

FRAMES SUBJECTED TO COMBINED GRAVITY AND LATERAL LOADS

Figure 2 shows the lateral load vs. lateral deflection relationships (solid line) of a two-bay, three-story steel frame tested under constant gravity loads and monotonically increasing lateral loads (Reference 6). For each story, the total gravity load acting through the story deflection produces a secondary overturning moment, also called the "P-6 moment". The effect of these moments is to reduce the overall strength and stiffness in resisting the lateral loads and eventually lead to instability failure. The failure load is again referred to as the "stability limit load". The significance of the .P-6 moment was first observed in a study on restrained columns permitted to sway (Reference 7). To assess properly the load-carrying capacity of the structure, a second-order, elastic-plastic analysis including the effects of the P-6 moment is required.

Two types of computerized second-order analysis have been developed, the main difference being the ways that the stiffness of the individual members are evaluated. The first type uses plastic hinge idealization and assumes that the member stiffness changes abruptly at the formation of each hinge (References 8 and 9). In the second type, yielding is assumed to develop gradually and continuously and spreads in the highly stressed regions·of each member. This spread of yielding is taken into account in evaluating the member stiffness (References 6 and 10). The first approach is simpler and capable of handling rather large frames. Predictions based on this approach have found to agree reasonably well with experimental results, as illustrated in Figure 2. Additional comparisons of theoretical and experimental results can be found in References 6 and 11. The second approach is more accurate but requires much larger amount of computing effort. The applicability of this approach to tall or complex frames has not yet been demonstrated.

DESIGN METHODS TO ACCOUNT FOR OVERALL INSTABILITY EFFECTS

Several design methods which take into account in a direct manner the effects of overall instability have been developed and applied to building design. When the plastic method is adopted in the design, the basis for strength calculation is the §tability limit load as described above •. For the case of combined gravity and lateral loads, the specified load factor against instability failure is 1.30 (U.S. practice). A three-step design process is often required:

(1) Performing a preliminary design, including an estimated P-~ moment for each story at the ultimate load. This can be accomplished by using the simple computerized procedure described in Ref. 12.

(2) Analyzing the preliminary design structure by the plastic subassemblage methods developed in References .13 and 14. The analysis is carried out story-by-story and for each story the P-~ moment can be included directly. The results may show weaknesses in certain stories (load factor less than 1.30) which can strengthened by using larger members. The complete load-deformation relationship of each story is also obtained from the analysis. At the end of this step, all the individual stories should show a load factor of 1.30 or higher.

(3) Second-order, elastic-plastic analysis of the whole structure using one of the methods described above. A minimum load factor of 1.30 should be achieved. Further adjustment of member sizes may be necessary in order to meet all the design requirements.

Before the development of the computerized second-order, elastic-plastic analysis which allows the direct determination of the stability limit load, an empirical formula, known as the Merchant-Rankine formula, had been proposed for estimating this limit load (Reference 15). The basic form

of the formula is

= +.· 1 l.p

in which ASL is the stability limit load factor .. ACR the overall elastic buckling load factor, and Ap the plastic limit load factor. Although it is widely used in Europe, the formula has certain drawbacks in its application:

(1) It is basically an empirical formula and was first proposed as an extension of the Rankine formula to simple frames. Its development is based mostly on intuition and on small scale model experiments. The applicability of this formula to tall building frames is not yet verified (especially frames with considerable amount of column shortening).

(2) In applying the formula, two separate analyses must be performed to determine the elastic buckling load and the plastic limit load. For relatively complex frames, considerable computing effort ·is usually required for these analyses. On the other hand, if a second-order, elastic-plastic analysis program is available, the stability limit load can be determined .. directly by performing only one analysis.

(3) If the \sL given by the formula is less than the required, it is not easy for the designer to detect which portions of the frame are particularly weak and require strengthening. Further­more, it is impossible to guarantee that the structure would not fail locally (member failure or story failure) at a load factor below the \sL calculated for the structure as a whole.

Because of these drawbacks, the usefulness of the Merchant-Rankine formula in practical design appears to be very limited.

When structures are designed by the allowable-stress method, only working load response is calculated and the determination of ultimate strength is usually not part of the design process. It is therefore not possible to account for the overall instability effects at the ultimate load level. A major design problem is: How to include the instability effects at the working load so that the structure would have sufficient strength at the ultimate? Another related but more basic question is: What is the stability limit load of a structure designed according to the present allowable-stress provisions? To provide answers to these questions, a major analytical study of the strength and behavior of multi-story steel frames was under­taken (Refs. 16, 17, & 18). The frames selected in the study were carefully designed according to the usual design practice. _The study covered many design conditions and structural analysis assumptions: stress-controlled design vs. drift-controlled design, proportional loading vs. non-proportional loading, inclusion vs. exclusion of overall instability effects, gravity load vs. combined gravity and lateral loads, etc. The following are a few

of the conclusions reached:

(1) Under proportionally increasing gravity and lateral loads, the presence of P-~ moment reduces the load-carrying capacity by about 12 to 22%. That is, AsL is about 12 to 22% less than Ap·

(2) Under non-proportional loads, the stress-controlled designs can achieve a lateral.load factor between 2.06 and 3.06 when the gravity load is maintained at the working level.

(3) Frames designed to meet a drift limitation of 0.002 have large lateral load-carrying capacity. The lateral load factor varies from a low of 1.45 for a proportional loading case to more than 4.0 for a non-proportional loading case, depending on the amount of gravity load.

(4) The average story P-~ moment at the factored load level (load factor = 1.30) is about 14 to 20% of the lateral load moment in the stress-controlled designs and 7 to 14% in the drift-controlled designs (drift limitation of 0.002).

(5) When gr.avity load alone acts on the frames, the results show that overall buckling, as characterized by bifurcation of the structure as a whole, is very unlikely. Instead, failure under gravity load is more of a localized phenomenon involving individual stories or members.

The results of this study strongly suggest that it might be possible to separate steel frames into two categories: (1) one in which overall instability is not of primary concern, and (2) one in which overall instability should be fully accounted for. The frames in the first category are referred to as "stiff frames" and those in the second category as "flexible frames". The conditions which characterize the inherently stiff frames are (Ref. 19):

(1) The story stiffness is such that LV/~F > 7 LP/h .

(2) The ratio Mel~ > 0.25

(3) The column axial stress is such that fa/1.33 Fa< 0.75 or fa/0.8Fy < 0.75

(4) The column slenderness ratio less than 35

In the above equations, LV is the story shear, ~F the first-order story deflection due to rv, LP the total gravity load, Me the gravity load moment, M the lateral load moment, and h the story height. Other notations follow t~ose used in the AISC Specification. The first condition is most important and has the effect of limiting P-~ effects to an acceptably small value.

For the design of the flexible frames, a rational approach is to apply the so-called "P-~ method" (Refs. 20, 21, 22). For each story, an estimated P~~ moment is introduced which has the effect of amplifying the bending moments and axial forces in both columns and girders. These amplified

values can be calculated using an amplification factor

1• Ar = --~....,....-

LP~F 1 - LVh

The above expression may be modified to include inelastic and other non-linear effects.

1

1 - a

in which a is an empirical factor. Discussions of the appropriate a values for use in design can be found in References 20~ 21~ and 22.

OPTIMUM DESIGN CONSIDERING OVERALL INSTABILITY EFFECTS

A great deal of research has been done on the optimum (minimum weight) design of multi-story frames by applying the techniques of mathematical programming. Most available methods tackle the problem either as a linear problem or as a non-linear problem. The linear formulation can be applied if the design is based on the plastic limit load. When ov:erall instability effects are included in the design, geometrical compatibility conditions must also be satisfied. The resulting problem becomes non-linear.

In Refe;rence 23~ a general procedure for the minimum weight design of multi-story steel frames subjected to combined gravity and lateral loads was developed. An equilibrium approach based on the deformed geometry of the structure was adopted in formulating the optimizati·on problem and the design constraints were derived d:trectly from the AISC Specification formulas. The effects of member and overall instability were taken into account. The required compatibility conditions were included in the constraints by the use of a set of -compatibility coefficients. By. incorporating these coefficients in the formulation~ it was possible to maintain the linearity of the design space so that the optimization problem remains as a linear problem, and hence, large systems can be handled efficiently.

FRAMES SUBJECTED TO REPEATED AND REVERSED LATERAL LOAD

Figure 3 shows the load-deflection relationship of a simple portal frame tested under constant gravity loads and reversed lateral load (Ref. 24). The maximum lateral loads at.tained in the two directions are significantly different. When the direction of the lateral load is reversed, the residual P-~ moment acts to oppose the applied load, causing an apparent increase of the maximum load. This increase depends on the magnitude of the lateral deformation accepted by the frame prior to reversing the load.

When the load cycles are repeated continuously in both directions, the usual symmetrical hysteresis loops are again obtained. Figure 4 shows the hysteresis loops of another single story frame tested under controlled lateral displacements (Ref. 25). The maximum test load is about 40% higher than that calculated for the monotonic loading condition. The loops appear to be very stable even after the attainment of the maximum load. The test also showed significant influence of strain hardening occurring at the plastic hinges.

Similar behavior has also been observed in the subsequent tests of multi-story frames and composite frames (Refs. 25 and 26). As long as individual members do not fail locally, the stable nature of the hysteresis loops is always maintained and unaffected by ov:erall instability.

FRAMES SUBJECTED TO SEISMIC GROUND MOTIONS

The first detailed study of the overall instability effects in frames subjected to earthquake motions was reported in References 27 and 28. A simple portal frame with either elasto-plastic or bilinear hysteretic characteristics was analyzed for its dynamic response. The results show that gravity load increases the amount of plastic drift significantly and may lead to collapse.

Another study has examined the. seis.mic response of a 10-story and a 25-story single bay steel frame (Ref. 29). All columns in the frames were assumed to remain elastic and girders could behave either elastically or plastically. In the plastic range, the moment-curvature relationship of the Ramberg - Osgood type was used. The lateral deflections found from the analysis were relatively small, indicating the stiff character of the frames selected in the study. The following is one of the major conclusions: The P-~ effect influenced the elastic response by as much as 10%; whereas its effect on the inelastic response was insignificant, the change being in the order of approximately 1%. Because of the relatively high stiffness of the frames, it is unlikely that the region of negative slope on the load-deflection diagram (after attainment of the stability limit load) was ever ~eached in the analysis.

A simplified second-order analysis and design procedure has been proposed in Reference 30. According to this method, the second-order quantities c~n be determined by using the corresponding quantities calculated by the first-order analysis and the coeffj_cient C given by

1 c

in which N is the column axial load, Nb the buckling axial load, and Sa(TI) and Sa(TII) the spectral accelerations corresponding to periods

,.

TI (calculated by the first-order theory) and TII (calculated by the second-order theory), The first part, alway greater than unity, is similar to the amplification factor used in beam-column desig-n and is related to the factor of safety against overall buckling. The second part may be greater or less than unity, depending on the ground

· excitation and the dynamic characteristics of the structure such as the fundamental period of vibration and damping ratio.

SEISMIC RESPONSE ON SOIL-STRUCTURE SYSTEMS

A study was carried out recently to investigate the effect of gravity load on the dynamic response of a simple soil-structure system (Refs. 31 and 32). The basic problem examined is shown in Figure 5. A rigorous formulation of the problem in .both time and frequency domain was developed and numerical calculations were per­formed covering several major parameters. The calculations were made in frequency domain and the input motion was assumed to be the uniform, horizontal free-fiel4 motion with harmonic time dependence. The following is some of the findings of the pa-rametric study:

11) Gravity load leads to a simultaneous softening of the structure and the soil. Significant softening of the soil, which results in a more pronounced interaction effect, occurs in the structure is relatively tall.

12) Gravity effect causes an additional increase in the peak spectral amplitudes and an accompanying decrease in-the resonant frequencies.

13) Gravity effect tends to broaden the effective frequency band which is relatively narrow in soil-structure systems.

This study assumes that the basic characteristics of the structure and those of the soil do not change with time nor with the level of ground input. Inelastic action in the structure is not considered.

SUMMARY

The recent investigations of the effects of overall instability on the static and dynamic response of building frames have been discussed. Only two-dimensional frames subjected to in~plane loads or ground accelerations are considered, but some of the general observations may also be valid for three-dimensional frames and other structural systems. The various design approaches and procedures that have been

·proposed to account for the instability·effects are reviewed and commented upon. The preferred approaches would be those which can take into account in a direct manner the effect of the secondary moment.

Although very extensive research has already been carried out, as evid.enced by the large number of articles published during the la'st decade, many problems· still remain to be investigated. The

stability of frames interconnected by flexible floor diaphragms is a problem of considerable practical importance. Consensus is yet to be reached on hcn.r to include the overall instability effects in the conventional allowable-stress design. Research is needed to extend,: the P-~ method to three-dimensional structures in which the effect of tensional motion may be important. The behavior of tube systems and core-supported systems has received only limited attention, especially in the inelastic range. More detailed_studies of the significance (or insignificance) of the gravity effect in frames subjected to earth­quake excitations are needed. Rational stability provisions for aseismic building design are not available at the present time. The interrelation betl-Jeen the gravity effect and soil-structure interaction effect is another area for future research.

LOAD, A p

IB I I I D

I I

c

-- G --G'

p

p

p

7lr

- ~-~h I

ill 7.7

o' 0~~---------------------------------

~ l--eo

H

(kips)

SWAY DEFLECTION, ~h

:FIG. 1 Behavior of Frame Under Gravity Load

8

6

4

0 2 4 6 B

SWAY 6. (in.)

Story Heioht•IOfl.

Boy Spocino •15 ft.

10

FIG •. 2 Behavior of Frame Under Combined Gravity and Lateral Loads

--jYf­

·~ HI p n -1

' West East

---Initial application --First cycle -·-·Second cycle -··-Final application

East

40

10

. I I . i/

// II

// 20 /

i 30 i

I

West

FIG. 3 Unsymmetrical Load-Deformation Relationships

6P 6P I I 20

AH·~H 10

3 2

10

20 H· WEST (kips)

FIG. 4 Large Amplitude Load-Deformation Hysteresis Loops

h

r:.--L--:J ~--r--~ I I I I I I I I I I I I I

FIG. 5 Gravity Effect in Soil-Structure System

REFERENCES

1. Lu, L. W. A SURVEY OF LITERATURE ON THE STABILITY OF FRAMES, Welding Research Council Bulletin No. 81, September, 1962

2. Lu, L. W. STABILITY OF FRAMES UNDER PRIMARY BENDING MOMENTS, Journal of the Structural Division, ASCE, Vol. 89, No. ST3, June, 1963

3. Lu, L. W. INELASTIC BUCKLING OF STEEL FRAMES, Journal of the Structural Division, ASCE, Vol. 91, No. ST6, December, 1965

4. McNamee, B. M. and Lu, L. W. INELASTIC MULTI-STORY FRAME BUCKLING, Journal of the Structural Division, ASCE, Vol. 98, No. ST7, July, 1972

5. Cheong-Siat-Moy, F. INELASTIC SWAY BUCKLING OF MULTISTORY FRAMES, Journal of the Structural Division, ASCE, Vol. 102, No. STl, January, 1976

6. Yarimci, E. INCREMENTAL INELASTIC ANALYSIS OF FRAMED STRUCTURES AND SOME EXPERIMENTAL VERIFICATIONS, Ph.D. Dissertation, Lehigh University, 1966

7. Levi, V., Driscoll, G. C. Jr., and Lu, L. W. ANALYSIS OF RESTRAINED COLUMNS PERMITTED TO SWAY, Journal of the Structural Division, ASCE, Vol. 93, No. STl, February, 1967

8. Parikh, B. P. THE ELASTIC-PLASTIC ANALYSIS AND DESIGN OF UNBRACED MULTI-STORY STEEL FRAMES, Ph.D. Dissertation, Lehigh University, 1966

9. Kim, S. W. ELASTIC-PLASTIC ANALYSIS OF UNBRACED FRAMES, Ph.D. Dissertation, Lehigh University, 1971

10. Alvarez, R. J. and Birnstiel, C. INELASTIC ANALYSIS OF MULTISTORY MULTIBAY FRAMES, Journal of the Structural Division, ASCE, Vol. 95, No. STll, November, 1969

11. Arnold, P., Adams, P. F., and Lu, L. W. STRENGTH AND BEHAVIOR OF AN INELASTIC HYBRID FRAME, Journal of the Structural Division, ASCE, Vol. 94, No. STl, January, 1968

12. Driscoll, G. C., Jr., Armdcost, J. 0. III, and Hansell, W. C. PLASTIC DESIGN OF MULTISTORY FRAMES BY CO}~UTER, Journal of the Structural Division, ASCE, Vol. 96, No. STl, January, 1970

13. Daniels, J. H. and Lu, L. W. PLASTIC SUBASSEMBLAGE ANALYSIS FOR UNBRACED FRAMES, Journal of the Structural Division, ASCE, Vol. 98, No. ST8, August, 1972

14. Cheong-Siat-Moy, F., and Lu, L. W. STIFFNESS AND STRENGTH DESIGN OF MULTISTORY FRAMES, IABSE Memoires, Vol. 36-II, 1976

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16; Lu, L. W., Ozer, E., Daniels, J. H., Okten, 0. s.· and Morino, S. STRENGTH AND DRIFT CHARACTERISTICS OF STEEL FRAJffiS, Journal of the· Structural Division, ASCE, Vol. 103, No. STll, November, 1977

17. Lu, L. W. and Ozer, E. LATERAL DEFLECTION AND STABILITY OF TALL BUILDINGS, Proceedings, Malaysian Conference on Tall Buildings, Kuala Lumpur, December, 1974

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21. Wood, B. R., Beaulieu, D. and Adams, P. F. COLUMN DESIGN BY P-DELTA METHOD, Journal of the Structural Division, ASCE, Vol. 102, No. ST2, February, 1976

22. Wood, B. R., Beaulieu, D. and Adams, P. F. FURTHER ASPECTS OF DESIGN BY P-DELTA METHOD, Journal of the Structural Division, ASCE, Vol. 102, No~ S~3, March, 1976

23. Ying, M. K. and Lu, L. W. OVERALL INSTABILITY AND OPTIMUM DESIGN OF FRAMES, Preprint No. 2267, ASCE National Meeting on Structural Engineering, Cincinnati, Ohio, 1974

24. Arnold, P., Adams, P. F. and Lu, L. W. THE EFFECT OF INSTABILITY ON THE CYCLIC BEHAVIOR OF A FRAME, Proceedings, RILEM Symposium on "Effects of Repeated Loading on Materials and Structures", Mexico City, September, 1966

25. Carpenter, L. D. and Lu, L. W. REPEATED AND REVERSED LOAD TESTS ON FULL-SCALE STEEL FRA}ffiS, Proceedings, Fourth World Conference on Earthquake Engineering, Santiago, Chile, January, 1969

26. Carpenter~ L. D. and Lu, L. W. HYSTERETIC BEHAVIOR OF FULL-SCALE STEEL AND COMPOSITE FRA}mS, Proceedings, Sixth European Conference on Earthquake Engineering, Dubrovnik, Yugoslavia, September, 1978

27. Jennings, P. C. and Husid, R. COLLAPSE OF YIELDING STRUCTURES DURING EARTHQUAKES, Journal of the Engineering Mechanics Division, ASCE, Vol. 94, No. EM5, October, 1968

28. Husid, R. THE EFFECT OF GRAVITY ON THE COLLAPSE OF YIELDING STRUCTURES WITH · EARTHQUAKE EXCITATION, Proceedings, Fourth World Conference on Earthquake Engineering, ~antiago, Chile, January, 1969

29. Gael, S. C. P-~ AND AXIAL COLUMN DEFORMATION IN ASEISMIC FRA}mS, Journal of the Structural Division, ASCE, Vol. 95, No. ST8, August, 1969

30. Karadogan, H. F. EFFECTS OF P-~ MOMENTS AND ROTATORY INERTIAS ON THE BEHAVIOR OF BUILDING STRUCTURES, Proceedings, International Colloquium on "Stability of Structures under Static and Dynamic Loads", Washington, D. C., May, 1977

31. Aydinoglu, M. N. and Lu, L. W. THE EFFECT OF STRUCTURAL INSTABILITY ON DYNAMIC RESPONSE OF SOIL- . STRUCTURAL SYST~1S, Proceedings, Seventh World Conference on Earthquake Engineering, Istanbul, Turkey, September, 1969

32. Aydinoglu, M. N. and Lu. L. W. GRAVITY EFFECT ON DYNAMIC SOIL-STRUCTURE INTERACTION, Fritz Engineering Laboratory Report No. 431.1, June, 1981