Sensitivity of semi-rigid frames to initial imperfections

J. Construct. Steel Research 18 (1991) 309--316

Sensitivity of Semi-rigid Frames to Initial Imperfections*

Franti~ekWald CzechTechnicalUniversity, 16629 Prague 6, Thakurova 7, Czechoslovakia

ABSTRACT The effects of imperfections are presented for the limit state concept design philosophy of structural steel buildings with flexible beam-column connections. The sensitivity of calibrating frames with four types of connections was tested on a second-order elastic in-plane model with semirigid connections expressed as a second-order experimental equation. It is demonstrated that for design application of real frames the sensitivity of semirigid frames to imperfections is quite adequate for rigid ones.

INTRODUCTION In a limit states approach to design, it is essential to consider all structural components which may affect the limit state behaviour of the structure. The force-deformation behaviour of the beam-to-column connections in a steel building frame can have a substantial effect on the structural behaviour of the frame. This fact has been recognized at least since 1917, when Wilson and Moore conducted tests on riveled connections. Yet hundreds of tests have been conducted on beam-to-column connections. However, it is rather difficult to assess the behaviour of the various types of connections accurately. The most commonly used connections exhibit nonlinear behaviour. As a result, nonlinear structural analysis techniques are often entailed for a semirigid procedure. We have a variety of design methods for rigid frame design, each with its *Presented at the International Colloquium on Stability of Steel Structures held in Budapest, Hungary, 1990. 3O9 Y. Construct. Steel Research 0143-974X/91/$03-50 (~) 1991 Elsevier Science Publishers Ltd,

England. Printed in Great Britain

310

Frantidek Wald

own advantages of use and accuracy. Each of them gives reasonable results, but many are very complicated. It seems that second-order elastic direct methods are most accurate for many cases and best for computer supported design. The effects of imperfections will be allowed for in frame analysis by means of an equivalent geometric imperfection in the form of an initial sway. Taking into consideration connection flexibility in the analysis and design process represents an important step towards the manifestation of the limit states concept. The problem of second-order direct design is in calibrating the imperfections, which we know for local elements and need for frames. Against the background of this information, probabilistic methods may be used to derive and define the maximum load-carrying capacity of a given type of structural member. The shape of dangerous imperfections is after first buckling occurs and is for building frames similar to sway. 1 On the other hand, for computer supported frame design there is no reason for separation of frames into sway and nonsway, or rigid and partially restrained.

IMPERFECTIONS Among the various factors that affect the strength of a column in a framework, the following are considered to be important: initial crookedness end restraints, residual stresses, load eccentricities, variation in mechanical properties of material over cross-section, stress-strain characteristics of material, loading, and unloading and reloading of yielded fibres. All of them can be treated as a geometric imperfection--the initial sway of frame. This calibration from known element imperfection to unknown imperfection of frame establishes t h e accuracy of the method and its limits of use. 1The value of imperfection should be smaller when we are including the residual stresses in the frame model and the same when we are taking into account the end restraints.

C A L I B R A T I N G FRAMES For sway frames with rigid joints a set of models was established, which should be used as a calibrating system in order to check the reliability of different computer programs or new simplified approaches or approximations for the ultimate limit state calculation. 2 For frames with semirigid joints it will be necessary to have the same set. In this study we used the Vogel set 2 with rigid and semirigid connections for its simplicity and common use (Fig. 1).

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M O D E L L I N G OF SEMIRIGID C O N N E C T I O N S We know that linear models are simple, but inaccurate, polynomials may encounter a negative stiffness, and cubic B-spline needs a large amount of test data. Power models have too many parameters and exponentials cannot fit well test curves that do not flatten out near the final loadings. Because all these methods can easily be implemented in a computerized method of analysis, the exponential combined piece-wise model for calculation, 3'4 as refined, was used in thisstudy. The need for a simple description of connection rigidity influence was solved by introducing the modified initial stiffness. 5 Five types of connections are used in the analysis. They are labelled connections S, H, T, E (Fig. 2). Connection S is a single web angle connection, and connection H is a header plate connection, both tested by Wald and Janda, 6 connection T is a top and seated angle connection with double web cleats tested by Azizinamin ettal.,71and connection E is an extended end plate connection tested by Johnson and Walpole. 8 The size of the connection T - E is not exact for this frame, but is very close to the optimum, and rotational deformation of any connection never exceeds 0.05 rad.

Frantigek Wald

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M O D E L L I N G OF B E A M - C O L U M N E L E M E N T The behaviour of flexible connections is represented by a discrete nonlinear rotational spring M = Rc(0r, M)4~r, where R¢(~br, M) is a function corresponding to the secant stiffness of the connection. In the unloading case the connection behaviour is assumed to unload linearly, following the initial stiffness Ri of connection. The relative rotation On at node i is expressed as the rotational degree of freedom of the member minus rotational degree of freedom of the connection. Substituting this into the element stiffness matrix formulation: 9

IK~ + K~l{d} + {Ff} = {F} we have a reasonable base for iterative solution. Because we have applied a polynomial of the third degree for the displacement function and only a linear for axial deformation along the element we have to cast the compressed element into four elements.

COMPUTATION TECHNIQUE For a prescribed load increment {R} the structure displacement increment {D,} was solved from the equation {Rs} = [K] ' ' {D~}. This displacement increment was then added to cumulative structure displacement evaluated at the end of the previous calculation cycle to update the displacement configuration of the structure. The difference between the internal and external force vectors gives an unbalanced force vector which is used as Rs to correct for D,. The load control Newton-Rapson iterative technique was used to trace the load-deflection curve of the frame.

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314

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NUMERICAL STUDIES The sensitivity of imperfections for three typical frames I, II and III 2 was examined on the elastic limit load. The behaviour of frames with connection types S, H, T and E was compared to the rigid one R. T h e m a x i m u m drift column m o m e n t s are normalized by the rigid frame solutions. Herein the value is the ratio of the difference between the rigid result and the real semirigid behaviour to the rigid results: The imperfec-

Sensitivity of semirigid frames to initial imperfections

315

tion was parametrized as a real value ratio to the height of the frame. Figures 3 to 5 illustrate the variations of analysis results for each frame. The circle-solid line represents the variation of maximum drifts and the star-solid line expresses the variation of maximum column moments. Examining the values enclosed in Figs 3-5, we caffsay that the sensitivity of the semirigid frames to the geometric imperfections is very similar to the rigid ones to some boundaries of rigidity. These are the frames reasonably designed to the serviceability limit states (H/300 resp. 1/500 Eurocode 3, 1989). To show these boundaries is for this particular mode expressed the same variation, Fig. 6. Modified initial stiffness Cko was used for each frame with imperfections after Eurocode No. 3--1988. The stiffness was expressed as relative C = Cko L/E1, where L is length, E is modulus of elasticity and I is moment of inertia of the connected beam.

CONCLUSIONS The most complex, accurate and reasonably difficult design model for rigid and semirigid frames is a direct second-order elastic design. There is no reason for using some approximate method with difficulties on boundaries. All thesemodels are necessarily tested on calibrating frames. The sensitivity of semirigid frames to imperfections is very similar to the rigid frames for reasonably stiff frames. This means that frames which have recommended limits for horizontal deflection have the same sensitivity to imperfections as rigid ones.

REFERENCES 1. Eurocode No. 3, Design of Steel Structures, Final Draft Dec. 1988, Liaison Engineers, Feb. 1989. 2. Vogel, U., Calibrating Frames, Vergleichsberechnungen an verschieblichen Rahmen, Stahlbau 54/1985, S. 295-301. 3. Kishi, N. and Chen, W. F., Steel connection data bank program. CE-STR-8618, Purdue Univ., W. Lafayette, IN, 1986. 4. Kishi, N. and Chen, W. F., Data base of steel beam-to-column connections, CE-STR-86-26, Purdue Univ., W. Lafayette, IN, 1986. 5. Kishi, N., Chen, W. F., Matsuoka, K. G. & Nomachi, S. G., Moment-Rotation Relation of Top- and Seat-Angle with Double Web-Angle Connections and the Behaviour, Strength and Design of Steel Structures, (R. Bjorhovde, et al. Elsevier, London, 1988, pp. 121-34. 6. Wald, F. & Janda, P., Rigidity of Beam to Column Connection. Experimental Research and Testing of Steel Structures International Conference, Prague, 1989.

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7. Azizinamini, A., Bradburn, J. H. & Radziminski, J. B., Static and Cyclic Behaviour of Semi-rigid steel beam-column connections, Univ. of South Carolina, Columbia, 1985. 8. Johnston, N. D. & Walpole, W. R., Bolted End-Plate Beam to Column Connections under Earthquake Type Loading. Research Report 81-7, Department of Civil Engineering, University of Canterbury, Christchurch, N6'kv Zealand, 1981. 9. Chen, W. F. & Lui, E. M., Structural Stability--Theory and Implementation, Elsevier, New York, 1987, 490 pp.

BIBLIOGRAPHY Bjorhovde, R., Brozetti, J. & Colson, A., Classification of connections, Connections in Steel Structures, Elsevier, London, 1987, pp. 388-96. Jones, S. W., Kirby, P. A. and Nethercot, D. A., Modeling of semi-rigid connection behaviour and its influence on steel column behaviour. Joints in Structural Steelwork, eds J. H. Hewlett, W. M. Jenkins and R. Stainsby, Pentech Press, Plymouth, UK, 1981, pp. 5.73-5.78. Jones, S. W., Nethercot, D. A. & Kirby, P. A., Influence of connection stiffness on column strength. Struct. Eng., 65A(11) (1987) 399-405. Morris, G. A., Packer, J. A., Beam-to-column connections in steel frames. Can. J. Cir. Eng., 14 (1987) 68-76. Dalen, K. V., Dalen, M. V., Discussion; Nethercot, D. A., Zandonini, R., Discussion; Can. J. Or. Eng., 15 (1988) 280-4. Nethercot, D. A., Kirby, P. A. & Davison, J. B., Structural Performance of Steel Frames with Semi-Rigid Connections, IABSE, Congress Report, Helsinki, 1988, pp. 681-6. Shen, Z. Y. & Lu, L. W., Analysis of initially crooked, end restrained steel column. J. Construct. Steel Res., 3 (1983) 10-18.