advanced analysis of steel building frames

J. Construct. Steel Research 23 (1992) 1-29

Advanced Analysis of Steel Building Frames

M. J. Clarke, R. Q. Bridge, G. J. Hancock & N. S. Trahair

Centre for Advanced Structural Engineering, School of Civil and Mining Engineering, University of Sydney, New South Wales 2006, Australia

A B S T R A C T

The paper describes advanced analysis as defined in the Australian limit states design specification, AS4100-1990. Advanced analysis may be used for the second- order inelastic analysis and design of frames in which the members are compact and have full lateral restraint. Some aspects of the inclusion of residual stresses, geometrical imperfections and capacity factors in advanced analysis are discussed. An advanced analysis based on the finite element method and utilisin O a distributed plasticity formulation has been developed at the University of Sydney and is used to perform numerical studies of the behaviour of simple structural elements and frames, includin 9 the effects of residual stresses and geometrical imperfections. Based on the results of the analyses, some observations on the importance of including imperfections in the advanced analysis of steel building frames are made.

1. I N T R O D U C T I O N

The new Australian Standard for the limit states design of steel structures, AS 4100-1990,1 allows the use of advanced analysis for the analysis and design of frames in which the members are of compact cross-section (thereby preventing local buckling) with full lateral restraint (thereby preventing flexural-torsional buckling). The aim of advanced analysis is therefore to predict accurately the in-plane behaviour of two-dimensional frames comprising members of compact section.

At its present stage of development, advanced analysis is not practical nor widely available for use in routine design. Advanced analysis might, however, be used for a special or exceptional structure, 2 or for an existing structure whose capacity is in doubt, or in the investigation of a structural

1 J. Construct. Steel Research 0143-974X/92/$05-00 © 1992 Elsevier Science Publishers Ltd, England. Printed in Ma~ta

2 M . J . Clarke et al.

failure. Advanced analysis should be capable of predicting the maximum load carrying capacity of the structure, and the full-range load-deflection response. It should therefore model as many factors as possible that influence frame strength, including relevant material properties, residual stresses, geometrical imperfections, instability, connection behaviour, erection procedures and interaction with the foundations.

Advanced analysis provides accurate estimates of the moment and force distributions in the frame and therefore simplifies the design process. At least for the strength limit state, the design process simplifies to verifying that the structure can attain an equilibrium position under the factored design loads such that the section capacity requirements of the Standard for the members and the requirements of the Standard for the connections, both of which incorporate capacity (resistance) factors, are satisfied. Furthermore, the section capacity requirements of the Standard can be satisfied implicity in the advanced analysis if the analysis itself incorporates an accurate model of cross-sectional strength.

In AS 4100-1990, advanced analysis is distinguished from second-order plastic analysis by the more accurate modelling of material inelasticity and the inclusion of the effects of residual stresses and geometrical imperfections. Second-order plastic analysis is restricted to doubly symmetric I-sections which are compact, while advanced analysis is also applicable to other cross-sectional shapes such as rectangular and circular tubes, provided they are also compact.

The development of computer programs suitable for advanced analysis has followed two main directions. In the first approach, the effects of residual stresses, gradual yielding and geometrical imperfections are ac- commodated implicitly in the element formulation by the proper calibration of phenomenological models. The phenomenological models are typically incorporated in computer programs based on a concentrated plasticity formulation 3-6 and the resulting method of analysis is termed the modified plastic hinge or refined plastic hinge approach. Modified plastic hinge analyses have been developed which give accurate predictions of strength for practical frames using only one element per member. 5 The development of methods of advanced analysis based on plastic hinges is an active area of research of Task Group 29 'Second-Order Inelastic Analysis for Frame Design' of the Structural Stability Research Council (SSRC), USA.

The second approach to advanced analysis is variously referred to in the literature as distributed plasticity analysis, plastic-zone analysis, spread- of-plasticity analysis, distributed inelasticity analysis and elasto-plastic analysis. 5 These analyses are characterised by explicit modelling of the

Advanced analysis of steel building frames 3

spread of yielding within the members of a framework. Other factors that affect strength and stability, such as residual stresses and geometrical imperfections, are also explicitly modelled. Compared to the modified plastic hinge approach, distributed plasticity analyses are more generally applicable to cross-sectional shapes of arbitrary geometry, and the effects of specialised stress-strain behaviour and distributions of residual stresses, such as those occurring in cold-formed tubes, are more readily accom- modated. A distributed plasticity analysis is a useful research tool which can be used to calibrate and check the accuracy of simplified methods of inelastic analysis such as the modified plastic hinge methods, and for the establishment of design charts. Distributed plasticity advanced analysis suffers from the disadvantage of high computational expense and as such is not currently practical for routine analysis and design of steel structures.

Of those factors that should be modelled in advanced analysis, connection behaviour, erection procedures and foundation interaction are not readily quantified and are not included in the studies described in this paper. This paper is concerned with the advanced analysis of rigid-joined steel building frames, including the effects of material imperfections (residual stresses) and geometrical imperfections. A finite element nonlinear analysis developed at the University of Sydney 2 was used to perform the advanced analyses reported in this paper. The advanced analysis can include the effects of large displacements, residual stresses, material non- linearity, grad~tal yielding, elastic unloading, geometrical imperfections and non-propoktional loading on the strength of frames.

The effects of the distribution, shape and orientation of geometrical imperfections on the strength of simple braced and unbraced flames are examined using the advanced analysis. Suggestions are made as to what types of frames and loading conditions are likely to require careful consideration of the influence of imperfections on flame strength. The influence of frame out-of-plumbness can be included indirectly with the use of a notional horizontal force. The effectiveness of this approach is also discussed in the paper.

2 M E T H O D O F A D V A N C E D ANALYSIS

The advanced analysis employed in the studies reported in this paper is based on the finite element method using an isoparametric curved beam element, and is suited to the in-plane analysis of arches, beam-columns and plane frame structures such as stressed-arch (Strarch) frames. 2 The element formulation is essentially an implementation of the meridional

4 M . J . Clarke et al.

behaviour of the axisymmetric shell element employed by Teng and Rotter. 7

Element displacements are defined in terms of a global rectangular cartesian coordinate system. The global displacements at any point in the element are interpolated from the element nodal displacements using the Hermitian cubic shape functions. The advanced analysis includes total Lagrangian, updated Lagrangian and co-rotational formulations for the geometric non-linearity, s'9 although all the results reported in this paper have been computed using the total Lagrangian formulation.

For the in-plane analysis of frames, the state of strain at a cross-section is defined by one component of membrane strain along the member reference line and one component of bending curvature. The strain- displacement relations are valid for large displacements and moderate rotations. Inelastic material behaviour is incorporated into the finite element nonlinear analysis through a distributed plasticity or plastic zone approach, which involves discretisation of the cross-section into a fine grid of monitoring points. The spread of yielding throughout the cross- section and along the length of members is therefore considered, and the effects of strain hardening and residual stresses are readily incorporated into the analysis. In the constitutive relationship for a point in the cross-section, an isotropic hardening theory of plasticity 1° has been adopted beyond the point of yield. The generalised stress resultants of axial force and bending moment are obtained by integrating the monitoring point stress distribution over the cross-section using Simpson's rule in two dimensions.

The nonlinear equilibrium equations are formulated using the principle of virtual displacements, s An incremental solution technique is used to solve the nonlinear equilibrium equations in conjunction with a number of iterative strategies that permit the load parameter as well as the nodal displacements to vary during the equilibrium iterations, and consequently enable load and displacement limit points to be overcome. A comprehen- sive review and assessment of these iterative strategies has been made by Clarke and Hancock. ll

3 SOME M O D E L L I N G CONSIDERATIONS IN ADVANCED ANALYSIS

3.1 Residual stresses

Hot-rolled structural members contain residual stresses as a consequence of different parts of the section experiencing different cooling rates during

Advanced analysis of steel buildino frames 5

the manufacturing process. For I-section members, the residual stresses are usually compressive at the flange tips. Compared to sections free of residual stresses, the magnitude and distribution of these compressive residual stresses influence the strength of compression members because of the earlier initiation of yielding and subsequent lowering of flexural strength that occurs. The effects of residual stresses should therefore be incorporated into advanced analysis. The effects of residual stresses are more severe for beam-columns bent about the minor axis than the major axis. 12

Residual stresses in hot-rolled I-sections have been measured by numer- ous researchers and analytical models proposed. The residual stress pattern employed by Galambos and Ketter la and Ketter 14 in their studies of the strength of steel beam-columns is shown in Fig. l(a). This distribution is consistent with measured residual stresses in American wide-flange column type sections resulting from cooling of the section during and after rolling.tS Researchers investigating the inelastic lateral buckling of beams

B

l

1

I

'I0 l

ogc= o.3fy

BT o1~q" = [ BT + t(D-2T) ] °RC

(a)

I

T4 I

~ t "l

I

I

h/!t ~--~~ ~0.35fy h/2 o.Sfy f,v,c

(b)

Fig. 1. Examples of residual stress distributions; (a) Residual stress distribution of Galam- bos & Ketter13; (b) Residual stress distribution of Bild & Trahair. 12

6 M.J . Clarke et al.

and beam-columns 16'17 represented the residual stresses induced in the section flanges after hot-rolling by parabolic distributions varying from 0.35fr compression at the flange tips to 0.5fr tension at the flange-web junction. The residual stress distribution in the web was represented by a quartic variation. However, based on achieving a better match to experi- mental distributions, the bilinear flange distribution and trilinear web distribution shown in Fig. l(b) was adopted by Bild and Trahair ~2 for studies of the in-plane strength of steel columns and beam-columns.

3.2 Geometrical imperfections

Material properties and residual stresses are relatively easily modelled and included in analytical methods that account for second-order effects. While these factors are important, geometrical imperfections may also have significant effects in some structures and they should therefore be included in advanced analysis. Design specifications include explicit or implicit fabrication tolerances for member out-of-straightness, and erection tolerances for local storey out-of-plumbness and global frame non- verticality. Of these, only member out-of-straightness has been closely examined and its effects included in column strength curves. The two types of imperfections, which may be termed member and frame imperfections, are illustrated in Fig. 2.

The modelling of geometrical imperfections for a frame is much more complex than for a single member. It is not just the magnitude, but also the shape, distribution and orientation of the imperfections that are significant. Researchers investigating the local buckling of thin-walled members have used the elastic buckling mode of the member as a basis for considering imperfections 18 on the assumption that this will have the worst effect on strength. This approach could be extended to frame structures. On the other hand, a judiciously selected pattern of imperfections can actually enhance the strength of a perfect frame.

An alternative approach to the modelling of imperfections could be based on physical considerations of erection tolerances for frames and fabrication tolerances for members (Fig. 2). As in other design specifica- tons, AS4100-1990 Stee l S t ruc tures I has specific requirements regarding the fabrication and erection tolerances for compression members. The out-of-straightness of a member bf length L about either principal axis shall not exceed the greater of 3 mm or L/IO00. The out-of-plumbness of a frame shall be such that the deviation of the top relative to the bottom of a compression member shall not exceed storey height/500. The align- ment of the frame shall be such that the deviation of any point above the

Advanced analysis of steel buildin# frames 7

50ram I I /

Envelope for ] plumbing of ! /

member

(a) Member (b) Local storey (e) Tolerances specified out-of-straightness out-of-plumbness in AS4100-1990 for imperfection A o/H plumbing of a bo/L compression member

Fig. 2. Member and flame geometric imperfections and flame erection tolerances.

base of a compression member from its correct position shall not exceed height/500 or the lesser of the following: 25 mm for a point up to 60 m above the base, or 25 mm plus 1 mm for every 3 m in excess of 60 m up to a maximum frame out-of-plumbness of 50 mm. From an analysis viewpoint, AS4100-1990 Clause 3.2.4 permits the frame out-of-plumbness imperfections to be represented by notional horizontal storey forces, each equal to 1//500 times the total design vertical loads applied at the floor level being considered.

It is interesting to note that design specifications usually neglect the member out-of-straightness in the elastic global analysis of the frame but include its effect on member design by the use of a column strength curve. Eurocode 319 suggests that member imperfections should be included when the axial force exceeds a defined limit. The basis of the column strength curve is an equivalent pin-ended column with an effective length which incorporates the strength-degrading effects of residual stresses and member imperfections. Eurocode 319 also recommends that frame imperfections should be included in the elastic analysis of the frame. Although the influence of the number of columns in a plane and the number of

8 M . d . Clarke et al.

storeys is considered, only limited guidance is given with respect to shape and distribution of imperfections. The Australian Standard AS4100- 19901 and the Canadian Standard CSA-S16.1-M.892° include the effect of frame imperfections through the use of an equivalent notional horizontal force.

4 T H E O R E T I C A L A P P L I C A T I O N S O F A D V A N C E D ANALYSIS

4.1 Generation of column strength curve

The column strength curves included in design specifications are based on the analysis of an equivalent pin-ended column with an effective length. The advanced analysis described in Section 2 was used to generate solutions for a W8 × 31 wide-flange section 21 pin-ended column of length L buckling about the major axis. The columns analysed were assumed to have an elastic-perfectly-plastic stress-strain curve and the residual stress pattern described by Galambos and Ketter 13 with a maximum compressive residual stress of 0.3fy (Fig. l(a)). The columns were also assumed to have an initial geometrical imperfection that varied sinusoidally with a maximum value of 6oiL= 1/1000 at mid-length. The advanced analysis solutions are shown in Fig. 3, where they are compared with the ~b = - 1-0, 0.0 and + 1-0 column curves in AS4100-1990. It is observed in Fig. 3 that for this particular distribution of residual stresses and geometrical

I I [ I I I I I I l'Oi ~ 0.9 Euler

0.8

0.7

0.6

0.5 - -I.0

4 °

0.2 - n Finite element (6o/L = 1 / 1 ~ , oRc-- 0.3fy)

0.1

o I I t I t l P I I 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

1 L . L f y Column Slenderness Parameter X~ - ~-(7)VtE )

Fig. 3. Column strength curves for W8 x 31 section buckling about the strong axis.


imperfections, the advanced analysis solutions have a similar shape to but are slightly above the column curve for hot-rolled sections (~b = frO) over most of the column slenderness range.

4.2 Advanced analysis study of single cantilever column

To study the effects of geometrical imperfections and residual stresses on column strength, the cantilever column shown in Fig. 4 was analysed. The cantilever column permits the simple inclusion of both out-of-plumbness and out-of-straightness imperfections and hence provides insight into the more complex issue of the inclusion of geometrical imperfections in the analysis of framed structures. For most of the analyses, the column slenderness parameter 2c=(1/lr)(kL/r)x/(fr/E ) was chosen as unity since this is the value for which the squash load and elastic buckling loads coincide and produce the greatest imperfection sensitivity in columns (Fig. 3). The imperfection shapes considered are illustrated in Fig. 5 and consist of combinations of an out-of-plumbness imperfection of magnitude A0 and

-

1 Fig. 4. Cantilever column studied with advanced analysis.

k = 2

fy = 250 MPa

E = 200000 MPa

200 UC 46.2 section

i; 4 ,, III IV V

Fig. 5. Imperfection shapes for cantilever column.

10 M. J. Clarke et al.

a sinusoidal member out-of-straightness of amplitude 6o. The cross- sectional shape used in the analyses was a 200UC46.222 and the material behaviour was assumed to be elastic-perfectly-plastic with a yield stress of magnitude fr = 250 MPa. When included in the analysis, the distribution of residual stresses followed the Galambos and Ketter ~3 model with a compressive residual stress at the flange tips equal to 0.3fr (Fig. l(a)). Young's modulus was assumed to be 200 000 MPa.

The results of the finite element non-linear analyses and the column strengths obtained from AS 4100-1990 are summarised in Table 1. It is seen in Table 1 that both the geometrical imperfection and the residual stress distribution assumed in the analysis affects the computed strength of the column. The lowest strength estimate of the advanced analysis is Nc/N~=0"650 (column V with residual stresses) and the strength predicted by the column rules of AS 4100-1990 is 0.618. It is also interesting to note that columns IV and V differ only in the orientation of the

TABLE 1 Results for Strength of Cantilever Column

Slenderness Column Residual Imperfections Strenoth 2c 9eometr y stress ~ N c / N ~

f o I L Ao /L

0'5 , AS 4100 column strength , 0.886 1.0 , AS 4100 column strength , 0.618 1-0 I 0'0 0 0 1.000 b 1.0 II 0.0 0 1/500 0.750 1'0 III 0"0 I/IOOO 0 0"823 1.0 IV 0-0 1/10OO 1/500 0.801 1.0 V 0.0 1/1000 1/500 0.712 1.0 VI 0.0 1/10OO 0 0.732

1.0 II 0-3fy 0 1/500 0.681 1.0 III 0.3fy 1/10OO 0 0-727 1.0 IV 0.3fy 1/loo0 1/500 0.723 1-0 V 0.3fy 1/IOOO 1/500 0.650 1.0 VI 0.3fy 1/1000 0 0.666 1.0 VI 0.3fy 1/600 0 0.618 0.5 Vl 0.3fy 1/780 0 0.886

1.0 II 0-3fy 1/1200 1/500 c 0.654 1.0 II 0.3fy 1/IOOO 1/500 c 0.650 1.0 II f l U , 1/600 1/5oo c 0.632

a Compressive residual stress at flange tips. b Theoretical value. c Implemented by notational horizontal force of N*/5OO.


out-of-straightness imperfection in relation to the out-of-plumbness imperfection. The favourably oriented imperfection in column IV results in a strength of 0.723 compared to 0"650 for the unfavourable oriented imperfection in column V. The imperfection of column VI, which is in the buckling mode, results in an ultimate strength close to but slightly above the strength of column V. This close agreement is a result of the magnitudes of the chosen frame and member imperfections, for which Ao = k6o for the cantilever.

If the imperfection is assumed in the buckling mode (column VI) then it is seen in Table 1 that, for columns with slenderness parameters 2c=0"5 and 1.0, the out-of-straightnesses implied by the AS4100-1990 column curve are approximately 1/780 and 1/600, respectively. The implied out- of-straightness therefore varies with the length of the column.

For the cantilever column of slenderness 2c = 1.0, the results of using a notional horizontal force of magnitude N*/500, in which N* is the axial force, in lieu of the out-of-plumbness imperfection of magnitude Ao=L/500 are also given in Table 1 for different values of column out-of-straightness 60. For 6o/L = 1/1000, an identical strength of 0.650 is obtained by both methods.

4.3 Advanced analysis study of simple sway frame

The sway frame chosen to investigate the effects of geometrical imperfections on frame strength is shown in Fig. 6. The loading on this frame is non-proportional and comprises column loads P, followed by the horizontal load H. All members of the frame are W8 × 31 wide-flange sections 21 with L/rx = 40. This portal frame was originally analysed by E1-Zanaty et al., 2~ and is henceforth termed the E1-Zanaty frame, but has since been studied by other researchers using second-order elastic-plastic methods of analysis, incuding simple plastic hinge methods (using the yield surfaces described in Orbison et al. 24 and Duan and ChenZS), a modified plastic hinge method 3 and a refined plastic zone method. 26 The effect of geometrical imperfections on frame strength was not considered in any of the previous work. In this section of the paper, the advanced analysis described in Section 2 is used to study the influence of geometrical imperfections and residual stresses on frame strength. ~In order to ensure that accurate results were obtained with the advanced analysis, each member of the frame was subdivided into several elements. Short elements were used adjacent to the beam-column joints to model the formation of plastic hinges at these locations. As for the single cantilever column, the E1-Zanaty portal frame was anticipated to be sensitive to imperfections


A f

H

t

/ All members W8 x 31

/ L/rx = 40

/ fy = 248.2 MPa / / E = 200000 MPa

L

I I

/ I

I I

I

i I

I

i i

i I

I

i I

l I

Fig. 6. T h e E I - Z a n a t y p o r t a l f r ame .

4

because the column slenderness parameter 2c=(1/n)(kL/r)x/(fr/E ) was approximately equal to unity.

Several different frame imperfection shapes have been investigated, as shown in Fig. 7. The imperfections were confined to the columns and included combinations of a sinusoidal member out-of-straightness of amplitude L/IO00, and a storey out-of-plumbness of L/500. These imperfection magnitudes were chosen in accordance with rules governing fabrication and erection tolerances specified in AS4100-1990. In Fig. 7, frame I corresponds to the perfect frame, frames II-V contain imperfection shapes in the same sense as the frame deformations under the horizontal load H, which therefore further degrade frame strength, and frame VI contains a favourable imperfection shape which enhances frame strength. For comparison with other analytical methods, the pattern of residual stresses (trR) proposed by Galambos and Ketter 13 and shown in Fig. l(a) with a maximum compressive residual stress at the flange tips of 0"3f r has been chosen. This magnitude is the same as that adopted by King et al. 3

for the modified plastic hinge solution, but is slightly lower than the distribution of magnitude fr/3 at the flange tips employed by White 26 in the plastic zone analysis.

Previous researchers studying the E1-Zanaty portal frame have reported results for the two cases of column loads of P/Py=0"4 and 0.6, in which Pr=Afy is the squash load of the section. The two additional cases of


Lt P P P P P P

" I ~ I

I II 1II

P P P P P P

iV V Vl

Fig. 7. Imperfection shapes for sway frame.

P/Py = 0"0 and 0.2 have been included in the present study. A complete set of results is given in Table 2, in which the maximum horizontal force has been non-dimensionalised with respect to the frame dimension L and the section plastic moment capacity Mp. The rigid plastic collapse load for the frame subjected to the horizontal load H only, and neglecting the reduction in plastic moment capacity due to axial force, is given by HL/Mp = 2. It is evident from Table 2 that for the cases P/Py = 0.2, 0.4 and 0.6, the unfavourable imperfections degrade frame strength compared to the perfect frame (frame I), and the degradation becomes greater as the column load increases. The unfavourable sway imperfection (frame III) appears to affect frame strength to a greater extent than the member out-of-straightness (frame II). In frame V, the out-of-plumbness imperfections of magnitude L/500 were modelled using a notional horizontal force of magnitude 2P/500. Frames IV and V are therefore nominally equivalent frames and the results in Table 2 indicate that virtually identical ultimate strengths were obtained with the advanced analysis. The use of a notional horizontal force to simulate frame out-of-plumbness geometrical imperfections therefore appears effective. The results for the favourable imperfection (frame VI) indicate that frame strength can be enhanced beyond the strength of the perfect frame.

Comparing the analyses of the single cantilever column (Fig. 5 and Table 1) with the analyses of the EI-Zanaty portal frame (Fig. 7 and Table 2), it is seen that the imperfection modes that have the worst effect on strength for each case (column V for the cantilever column and frame IV for the portal frame) are not the same. The difference in behaviour for the

14 M. d. Clarke et al.

TABLE 2 Results of Analyses for the El-Zanaty Portal Frame

Analysis Frame a Residual stress b

M a x i m u m horizontal force, H L / M p

P/Py = 0 0 P/Py = 0 2 P/Py = 0 4 P/Py = 0 6

SPH I c I 0.0 - - 0.802 0.351 SPH II d I 0.0 - - - - 0.784 0.340 MPH e I 0.0 - - - - 0.729 0.296 M P H I 0-3fy - - 0.657 0-221 PZ s I f r / 3 - - - - 0.627 0.193

AdvY I 0.0 1.992 1.355 0.709 0.282 Adv. I 0.3fy 1.991 1.318 0.655 0.219 Adv. II 0-3fy 1.991 1.315 0.642 0.186 Adv. III 0-3fy 1.991 1.286 0.591 0-117 Adv. IV 0.3f r 1.991 1.281 0-576 0.087 Adv. V 0.3fy 1.991 1.280 0.576 0.087 Adv. VI 0.3fy 1.991 1.357 0.738 0.350

a Refer to Fig. 7. b Compressive residual stress at flange tips. c Simple plastic hinge I (using yield surface of Orbison et al. z4)

a Simple plastic hinge II (using yield surface of Duan and Chen 25) e Modified plastic hinge (King et al.3). I Plastic zone (White26). g Advanced analysis (present).

two cases of column V (Fig. 5) and frame IV (Fig. 7) is due to the different boundary condition at the base of the column, and hence the different deformation mode that results.

A possible alternative method of generating suitable frame imperfections is to consider the buckled shape of the structure subjected to the vertical loads P only, and adopt frame imperfections in this mode, at least for the columns. This approach is illustrated in Fig. 8, where the imperfect

I I I

k L

V I I

k.k

I I

i

P P P P

V I I I I X X

Fig. 8. Imperfection shapes for column strength in sway frame.

Advanced analysis of steel building flames 15

columns in frame VIII follow the same sinusoidal shape as the equivalent pin-ended column of length kL (frame VII), in which k = 2.33 is the elastic effective length factor for the columns in the sway frame. This effective length factor was computed by solving the differential equation of buckling for an isolated sway column with a rotational restraint at one end of stiffness 6EI/L, in which EI and L are the flexural rigidity and length, respectively, of the beam in the E1-Zanaty frame. As indicated in Table 3, the strength of frame VIII subjected to column loads only (P/Py =0-652) is nearly identical to the equivalent pin-ended column of frame VII (P/Py = 0.640). Frame IX in Fig. 8 has the same imperfect shape of frame IV and therefore has both the out-of-straightness and out-of-plumbness imperfections. The ultimate load of this frame subjected to column loads only is P/Py=0"647. In frame X, the sway imperfection of frame IX has been replaced by the equivalent notional horizontal force of magnitude equal to 1//500 of the sum of the vertical loads acting on that storey. It is seen in Table 3 that the strength of frames X and IX are virtually identical.

For the case of P//Py= 0"6, the load-deflection curves for the frame are shown in Fig. 9 for different methods of analysis, imperfection shapes and the presence or absence of residual stresses. The horizontal deflection A (see Fig. 6) plotted in Fig. 9 is measured relative to the initially imperfect frame geometry. The computed frame ultimate load decreases as the plasticity formulation in the analysis methods becomes more refined (curves 1-5). For the geometrically perfect frame with residual stresses (frame I + aR), similar results are obtained for the plastic zone approaches of White 26 and the present advanced analysis (curves 5 and 7 in Fig. 9). The plastic zone ultimate loads are significantly below the ultimate loads predicted by the simple plastic hinge analyses.

The results shown in Table 2 highlight the importance of including geometrical imperfections and residual stresses in the advanced analysis of

TABLE 3 Results of Analyses of Column Strength for the

EI-Zanaty Frame

Frame Residual Maximum column stesss" load, P/Py

VII 0"3fy 0"640 VIII 0.3fy 0-652 IX 0"3fy 0.647 X 0"3fy 0-646

° Compressive residual stress at flange tips.


0.40 r 1 I I

0.35

0.30

~ 0 . 2 5

i 0.20

0.15

0.10

0.05

1 - j

- ~PY =° ' ° 14 2

3

0 5 10 15 20 25

Lateral Deflection A/L (xl03 )

Key to curves

1 Simple plastic hinge I 6 Advanced (Frame i) 2 Simple plastic hinge II 7 Advanced (Frame I ÷ o R) 3 Modified plastic hinge 8 Advanced (Frame II + OR) 4 Modified plastic hinge (oR) 9 Advanced (Frame 111 + OR) 5 Plastic zone (Frame I +o R ) 10 Advanced(Frame IV + OR)

Fig. 9. Load-deflection curves for EI-Zanaty portal frame.

sway frames which have columns with a slenderness parameter 2e around unity and are subjected to high axial loads (P/Py = 0.4, 0.6). At the lower level of axial force (P/Py = 0.2), the influence of imperfections is reduced.

4.4 Advanced analysis study of simple braced frame

The frame shown in Fig. 10 was employed to investigate the influence of imperfections and residual stresses on the strength of a simple braced frame. The non-proportional loading on the structure comprises the column loads P followed by the uniformly distributed beam load w. All members of this braced frame were a W4 x 13 section 21 with L/rx= 100. As for the sway frame, the chosen braced frame was expected to be imperfection sensitive as the column slenderness parameter 2c = (1/nXkL/r) × x/(fy/E) was approximately equal to unity.

In the analyses of the braced frame shown in Fig. 10, symmetry was utilised and to ensure accurate results with the advanced analysis, the column and beam were subdivided into several elements. Short elements


w

'I

All members W4 x 13

l-/rx = 100

i fy = 248.2 MPa

E = 200000 MPa

L

Fig. 10. Braced portal frame.

were used adjacent to potential plastic hinge locations. The uniformly distributed load w was modelled by consistent nodal forces.

The imperfection shapes which have been analysed using the advanced analysis are shown in Fig. 11, and consist of both a sinusoidal member out-of-straightness of amplitude L/IO00 and a storey out-of-plumbness of L/500. Frame I is the geometrically perfect frame, frames II-IV contain unfavourable imperfections and frame V contains a favourable imperfection which opposes the frame deformations. As for the sway frame, the Galambos and Ketter la distribution of residual stress shown in Fig. l(a) with a maximum compressive residual stress of 0.3fy was employed in the analyses.

Advanced analyses have been performed for each of the frames shown in Fig. 11 for column loads P/Py of 0.0, 0.2, 0.4 and 0.6, and uniformly distributed beam loading w. The results are given in Table 4, from which it

P P P P P

1 I1 HI IV V

F i g . 11. I m p e r f e c t i o n s h a p e s fo r b r a c e d f r a m e :


TABLE 4 Results of Analyses for the Simple Braced Portal Frame

Analysis Frame a Residual stress b

Maximum beam load, wL2/Mp

P/Py = 0.0 P/Py = 0.2 P/Py = 0.4 P/Py = 0.6

Adv. c I 0-0 15.71 13.61 10.30 4.85 Adv. I 0.3fr 15.69 13.31 9.52 3.83 Adv. II 0.3fy 15.68 13-10 8.63 2.28 Adv. III 0.3fy 15.68 13-30 9.52 3.79 Adv. IV 0-3fr 15.68 13.09 8.64 2.28 Adv. V 0-3fr 15.69 13.51 10.33 5-37

a Refer to Fig. 11. b Compressive residual stress at flange tips. c Advanced analysis.

is evident that the unfavourable member out-of-straightness has a strength-degrading effect which increases in severity as P/Py increases, but the storey out-of-plumbness has virtually no effect. For P/Py = 0.0 the rigid plastic collapse load of the frame, neglecting the reduction in Mp due to axial force, is wL2/Mv=16. The computed solutions for P/Py=O'O are slightly below wL2/Mv= 16 on account of the reduced plastic moment capacity of the columns because of axial force, and are essentially unaffected by the frame imperfections. A favourable member out-of- straightness imperfection increases the strength of the frame beyond that of the geometrically perfect frame.

The load-deflection curves for the braced frame are shown in Fig. 12 for the case of P/Py=0.6. Virtually identical load-deflection responses are obtained for frames I and III, and for frames II and IV, indicating that the storey out-of-plumbness has negligible effect on frame behaviour.

The results shown in Table 4 highlight the importance of including member imperfections and residual stresses in the advanced analysis of braced frames which have columns with a slenderness parameter 2c around unity and are subjected to high axial loads ( P / P y = 0.4, (~6). At the lower level of axial force (P/Py --- 0"2), the influence of member imperfections is reduced.

5 P R A C T I C A L A P P L I C A T I O N S O F A D V A N C E D ANALYSIS

5.1 Introduction

The preceding sections of this paper have illustrated the effects of residual stresses and geometrical imperfections on the strength of columns and

Advanced analysis of steel building fiames 19

I 1 I I I I

5 -- ~ Advanced analysis I ~Y I (Frame I)

Advanced analysis / / ] (FrameI+ oR) ~ ~ ]

Advanced analysis y / / (Frame II + oR) ~ ] /

o 0 1 2 3 4 5 6

Central Deflection of Beam A/L ( xl0 3 )

Fig. 12. Load-deflection curves for braced portal frame.

4

i3 E

I

I 7 8

simple portal frame structures. Most of these examples were chosen deliberately to be sensitive to imperfections and would represent very atypical designs. For the simple sway portal frame for example, the elastic buckling load factor of the frame corresponding to column loads of 0.6Py is approximately 1.5.

With the aim of examining the effects of residual stresses and geometrical imperfections in steel frameworks of a more practical nature, the four frames shown in Fig. 13 were analysed with the advanced analysis. In a previous paper 27 the frames of Fig. 13 were analysed using elastic and plastic methods of analysis, and in these analyses the actual yield strengths corresponding to Grade 250 steel were used. Details of the frame members are given in Table 5. The first three frames are unbraced frames, the first of which is a typical two-storey frame while the second is an atypically slender two-storey frame. The third structure is a single-storey pitched roof portal frame which is typical of many industrial structures. The fourth frame is a rigid-jointed roof truss, which acts as a braced flexural frame under primary bending actions. Each frame is subjected to a simplified set of loads, which represent the design loads caused by dead, live and wind forces.

Guidelines for inclusion of geometrical imperfections in the analyses of the frames of Fig. 13 are given in Section 5.2 following. As for the previous examples, the residual stresses in the members were assumed to have the Galambos and Ketter ~ 3 distribution (Fig. l(a)) with a compressive residual stress at the flange tips of 0"3ft. The elastic modulus for all members of the frames was assumed to be 200000MPa. The steel was assumed to be Grade 250 and elastic-perfectly-plastic. The precise values of yield stresses used in the analyses are given in Table 5.

20 M. .I. Clarke et al.

2i II I o Ioo 50004

16 F • :~ ~5000 10000

5 0 0 0 5 0 0 0 5 0 0 0 5 0 0 0

(a) (b) 16,0 8.7 18.3

87 8.7 18.3 • 18.3 9.35 1.2 14.15

1.2 1.2 1.2 1.2 1.2 1.2 ~ 3 0 0 0 54 2.2

"" 3,2 12.o ~ 4000

8 @ 300O = 24000 (c) 2O

20 20

511110

51)1111

t~ =!_ -!~ -!= -!~ _!_ -!_ -L -! 8@500O=4O000

(d) (Dimensions in ram, loads in kN)

Fig. 13. Frames analysed with advanced analysis; (a) Two-storey frame; (b) Slender frame; (c) Portal frame; (d) Rigid-joined truss.

5.2 Inclusion of geometrical impedections

The examples of columns and simple portal frames in previous sections of this paper have demonstrated that residual stresses and geometrical imperfections can affect the strength of structures which are slender and subjected to high axial loads. It was shown that for the advanced analysis of a cantilever column of length L of a particular geometrical slenderness (2c = 1-0) and cross-section (200UC46.2) that inclusion of an out-of-straightness imperfection of magnitude go~L= 1/1000 and an out-of-plumbness imperfection of magnitude Ao/L = 1/500 (or an equivalent notional horizontal force) together with a defined distribution of residual stresses, results in an estimate of column strength which is similar to that obtained according to the column strength rules in AS 4100-1990. The values of 6o/L = 1/1000 and

Advanced analysis of steel building flames

TABLE 5 Sections and Yield Stresses Employed in Analysis of the Frames of Fig. 13

21

Member Location Section A I f y (ram 2) (ram 4) (MPa)

(a) Two-storey frame A,E Lower column 200UC52.2 6640 52"6 × 106 250 B,D Upper column 150UC30.0 3820 17.4 x 106 260 F Lower beam 310UB40.4 5150 85-2 × 106 260 C Upper beam 200UB25.4 3230 23.6 × 106 260

(b) Slender frame A,E Lower column 150UC37.2 4740 22.2 × 106 260 B,D Upper column 100UC14.8 1890 3-19 × 106 260 F Lower beam 200UB25.4 3230 23.6 × 106 260 C Upper beam 150UBI4.0 1790 6.66 × 106 260

(c) Portal frame A,D Column 360UB44.7 5700 121 × 106 260 B,C Rafter 360UB44.7 5700 121 × 106 260

(d) Rigid-jointed truss A,D Top chord 250UC72.9 9290 38'7 x 106 250 B,C Top chord 200UC52.2 6640 17"7 × 10 6 250 E,F Bottom chord 100UC14.8 1890 1.14 × 106 260 G,I Web 150UC23.4 2980 4'03 × 106 260 H Web 100 UC14.8 1890 1"14 × 106 260

Ao/L = 1/500 are consistent with the tolerances specified in AS 4100-1990 for fabrication of members and erection of the frame, respectively. It is also evident from Table 1, however, that for a given residual stress distribution, the out-of-straightnesses implied in the column curve of AS 4100-1990 varies with the geometrical slenderness (for slendernesses of 2 ,=0.5 and 1-0, the out-of-straightness implied are 3o/L = 1/780 and 1/600 respectively). Also, if the manufacturing process is assumed to produce members with a small constant initial curvature, then the equivalent out-of-straightness imperfection ~o/L varies linearly with the length, as shown in Fig. 14.

Ideally, one would like to incorporate the aforementioned theoretical aspects of imperfections into practical advanced analysis of frames. For the purpose of advanced analysis of the frames of Fig. 13, the following guidelines were adopted for the inclusion of geometrical imperfections:

(1) Member out-of-straightness imperfections rio were assumed to be of magnitude L/1000.

(2) Frame out-of-plumbness imperfections were assumed to be of magnitude L/500.

2 2 M. J. Clarke et al.

Radius R

Fig.

~ _ L L - 8R

14. Equivalent out-of-straightness imperfection for a curvature.

member of uniform initial

(3) The imperfections were always oriented so that they were thought to be predominantly detrimental to the frame strength.

(4) Sway frames were analysed with both out-of-plumbness and out- of-straightness imperfections. Braced frames were assumed to contain out-of-straightness imperfections only.

5.3 Incorporation of capacity factors in advanced analysis

In the limit states strength method of design of AS 4100-1990, the design actions S* acting on the members of the frame are compared with the design section and member capacities ~bRu, and design is satisfactory if S* ~< ~bRu, in which the capacity factor ~b and the nominal capacities Ru are specified in AS 4100-1990.

When elastic analysis and design is used, the section and member axial and moment capacities are simply reduced by the capacity factor ~b. When first- or second-order plastic analysis is used, the section plastic moment capacity allowing for the presence of axial force is also factored down by ~b. This is essentially equivalent to adopting a factored down material yield stress of ~bfy. When advanced analyses is used, however, it is not clear how the capacity factors should be incorported into the analysis and design procedure. Adoption of a reduced yield stress ~bfy will certainly reduce the section axial and moment capacities correctly, but will not in general have the correct effect on the member capacities where failure of the member is also influenced by instability.

A more rational way to incorporate the capacity factors ~b in advanced analysis might be to consider the design actions S* as being characteristic of the complete frame rather than an individual member, and similarly the nominal resistance Ru to be the capacity of the complete frame, rather than the capacities applicable to individual members. Design would then be satisfactory for member strength if the inequality S* ~< ~bR,, is satisfied in an

Advanced analysis of steel buildino flames 23

integral sense for the frame. In AS 4100-1990, the capacity factors ~b are equal to 0.9 for all section strengths and all member strengths, which include stability effects. Therefore, ~b = 0-9, could also be adopted as the capacity factor for the whole frame in advanced analysis. The preceding discussion has ignored the effects of connection capacity on the strength of the frame. Although not considered in this paper, the effects of connection behaviour and strength should in general also be considered in advanced analysis and approximate allowances made for the capacity factors of the connections.

5.4 Results of advanced analyses

For the two-storey frame shown in Fig. 13(a), the imperfection modes included in the advanced analyses are shown in Fig. 15, and the results of the analyses (design ultimate strengths) are given in Table 6. It is evident that the inclusion of residual stresses and geometrical imperfections had virtually no effect for this frame.

For the slender frame shown in Fig. 13(b), the imperfection modes included in the advanced analyses are shown in Fig. 15. The results of the analyses of the geometrically perfect frame with no residual stresses and the geometrically imperfect frames with residual stresses are given in Table 6. The combined effect of residual stresses and geometrical imperfections is seen to reduce the design ultimate strength (~b=0.9) of the frame from 1.04 to 1.02. Even though this frame is atypically slender (the elastic buckling load factor is approximately 3.4), imperfections and residual stresses evidently have only a small effect on the ultimate frame strength.

The ultimate strength of the slender frame computed using a reduced yield stress of 0.9fy is 1-06 (Table 6). This figure is higher than the design strength value of 1.02 which was computed using the full yield strength fy and a capacity factor of 0-9 on the frame strength. The use of a reduced

I 11 III

60/L = 1/1000 AoAI = 1/500

F i g . 15. Frame geometries analysed for two-storey frame and slender frame.


TABLE 6 Results of Advanced Analyses of the Frames of Fig. 13

Frame Geometry Residual Design

stress ~ ultimate strength b

Two-storey frame I 0.0 1-21 (Figs 13(a)and 15) I 0.3fy 1-20

1I 03fy 120 III 03fy 120

Slender frame I 0-0 1.04 (Figs 13(b) and 15) I 0"3fy 1.03

II 0.3fy 1.02 II 0.3fy 1.06 c III 0"3f, 1.02

Portal frame I 0.0 1.28 (Figs 13(c)and 16) II 0"3fy 1.28

III 0-3fy 1.28

Rigid-jointed truss I 0.0 1.85 (Figs 13(d)and 17) I 0.3fy 1.79

II 0.3fy 1.75

a Compressive residual stress at flange tips. b Computed by factoring the nominal ultimate strength by ~b=0.9. c Nominal ultimate strength evaluated using a reduced yield stress of 0'9fy.

yield stress evidently does not account for the reduction in strength of members due to instability in the same manner as a global capacity factor.

The effect of the sequence of load application on ultimate strength was also studied for the slender frame. Frame II of Fig. 15 was analysed by first applying the vertical loads at a load level of 1.02/~b = 1.13 times the design loads. The design horizontal loads were then incremented until the ultimate load of the frame was attained. The maximum horizontal load resisted by the frame was computed to be 1.13 times the design horizontal loads, indicating that for this frame, the sequence of load application had no noticeable effect on strength.

The imperfect frame geometries employed in the advanced analyses of the portal frame are shown in Fig. 16. However, the results shown in Table 6 indicate that despite the presence of material and geometrical imperfections, the design ultimate strength of the frame was unaltered from the strength of the perfect frame. The high elastic buckling load factor of approximately 13.3 for this frame explains the insensitivity of this frame to imperfections.


f I

H

Ill

(80/L)mcmber = 111000 Ao/H = 1/500

Fig. 16. Frame gcomctrics analyscd for portal frame.

I

II

(60/L)membc , = 1/1000

Fig. 17. Frame gcomctrics analysed for rigid-jointed truss.

The rigid-jointed truss differs from the previous three frames analysed using advanced analysis in that it is a braced framc and the members arc assumed to contain out-of-straightness irnpcrfcctions only. For each member of length L, the out-of-straightness impcrfcction was assumed to bc of magnitude ~o/L= 1/1000, and oriented predominantly in the same sense as thc frame deformations (Fig. 17). Thc orientation of the


imperfection in the vertical web member was completely arbitrary. The elastic buckling load factor for the truss was computed to be approximately 3.6 and first-order plastic collapse of the frame occurred through beam mechanisms in the upper top chord members. The results of the advanced analyses are given in Table 6, in which it is seen that the combined effect of residual stresses and geometrical imperfections reduces the design ultimate frame strength by approximately 5% compared to the perfect frame.

6 C O N C L U S I O N S

The method of advanced analysis described in the Australian Standard AS4100-1990 Steel Structures is the most refined of all the methods of structural analysis permitted. From a theoretical viewpoint it is also the most rational since it provides accurate estimates of the moment and force distributions in the frame at the strength limit state and simplifies the design phase of the analysis/design cycle. With advanced analysis, it is no longer necessary to perform member or section capacity checks, because the effects of the material and geometrical imperfections and of the material and geometrical nonlinearities have already been included in the analysis.

The relevance of including residual stresses and geometrical imperfections in advanced analysis has been studied in this paper. The simple unbraced and braced portal frames studied were anticipated to be, and subsequently shown to be, sensitive to imperfections. The conclusions can be summarised as follows:

(1) Both out-of-straightness geometrical imperfections and residual stresses affect the strength of columns and therefore should be considered in advanced analysis.

(2) For the simple sway portal frame studied, the out-of-plumbness imperfections reduced frame strength to a greater extent than the member out-of-straightness imperfections.

(3) The strength of the simple braced portal frame was affected most by the member out-of-straightness and was virtually unaffected by storey out-of-plumbness.

(4) Both the simple sway and braced portal frames became more imperfection sensitive as the column axial loads increased.

(5) The notional horizontal forces specified in Clause 3.2.4 of AS 4100- 1990 can be used successfully in place of out-of-plumbness imperfections in advanced analysis.


The importance of imperfections on structures of a more practical nature subjected to a realistic set of design loads was also studied in the paper. Simplified distributions of imperfections were proposed based on the guidelines provided in AS 4100-1990 relating to fabrication tolerances for members and erection tolerances for column and frame out-of- plumbness. The combined effects of residual stresses and geometrical imperfections were found to have negligible effect on the ultimate strength of the two-storey frame, the slender frame and the portal frame. For the rigid-jointed truss, the effects of imperfections resulted in a 5% decrease in ultimate strength. The fact that each member of the truss was bent about the weak axis rather than the strong axis may have accounted for the more substantial loss in strength that occurred for the truss compared to the other frames studied.

At the current state of research, it is difficult to define more detailed guidelines for the distribution and orientation of geometrical imperfections which should be employed in the advanced analysis of steel frameworks. To some extent it is left to the skill and experience of the analyst/designer to judge the importance of imperfections on the frame ultimate strength and account satisfactorily for the effects of imperfections in advanced analysis.

One possible approach is to orient all the imperfections unfavourably, if indeed that is possible. This approach, however, may be regarded as over-conservative from a reliability viewpoint. A reliability-based approach for imperfection distribution and orientation would therefore constitute a more rational approach and would be consistent with the theory used in the determination of load factors and capacity reduction factors employed in limit states design.

R E F E R E N C E S

1. Standards Australia, AS 4100-1990 Steel Structures, Sydney, 1990. 2. Clarke, M. J. & Hancock, G. J., Finite-element nonlinear analysis of stressed-

arch frames, J. Structural Engg, ASCE, 117 (1991) 2819-37. 3. King, W. S., White, D. W. &Chen, W. F., A modified plastic hinge method for

second-order inelastic analysis of steel rigid frames. Structural Engineering Report No. CE-STR-90-13, School of Civil Engineering, Purdue University, West Lafayette, IN, 1990.

4. Liew, J. Y. R., White, D. W. &Chen, W. F., Modified plastic hinge solutions for calibration frames. Structural Engineering Report No. CE-STR-91-13, School of Civil Engineering, Purdue University, West Lafayette, IN, 1991.

5. White, D. W., Liew, J. Y. R. &Chen, W. F., Second-order inelastic analysis for frame design: A report to SSRC Task Group 29 on recent research and the


perceived state-of-the-art. Structural Engineering Repot No. CE-STR-91-12, School of Civil Engineering, Purdue University, West Lafayette, IN, 1991.

6. Ziemian, R. D., White, D. W., Deierlein, G. G. & McGuire, W., One approach to inelastic analysis and design. In Proc. 1990 National Steel Construction Conf., American Institute of Steel Construction, Chicago, IL, 1990, pp. 19-1-19-19.

7. Teng, J. G. & Rotter, J. M., Elastic-plastic large deflection analysis of axisymmetric shells. Computers and Structures, 31 (1989) 211-33.

8. Bathe, K.-J., Finite Element Procedures in Engineering Analysis. Prentice-Hall, Englewood Cliffs, NJ, 1982.

9. Mattiasson, K., Bengtsson, A. & Samuelsson, A., On the accuracy and efficiency of numerical algorithms for geometrically nonlinear structural analysis. In Finite Element Methods for Nonlinear Problems, ed. P. G. Bergan, K. J. Bathe & W. Wunderlich. Springer Verlag, Berlin, 1986.

10. Mendelson, A., Plasticity: Theory and Application. Macmillan, New York, 1968. 11. Clarke, M. J. & Hancock, G. J., A study of incremental-iterative strategies for

non-linear analyses. Int. J. Numerical Methods in Engg, 29 (1990) 1365-91. 12. Bild, S. & Trahair, N. S., In-plane strengths of steel columns and beam-

columns. J. Construct. Steel Research, 13 (1989) 1-22. 13. Galambos, T. V. & Ketter, R. L., Columns under combined bending and

thrust. J. Engg Mechanics Div., ASCE, 85(EM2) (1959) 1-30. 14. Ketter, R. L., Further studies on the strength of beam-columns. J. Structural

Div., ASCE, $7(ST6) (1961) 135-52. 15. Ketter, R. L., Kaminsky, E. L. & Beedle, L. S., Plastic deformation of

wide-flange beam-columns. Trans. ASCE, 120 (1955) 1028. 16. Bradford, M. A. & Trahair, N. S., Inelastic buckling of beam-columns with

unequal end moments. J. Construct. Steel Res., 5 (1985) 195-212. 17. Trahair, N. S., Inelastic lateral buckling of beams. In Developments in the

Stability and Strength of Structures, Vol. 2, Beams and Beam-Columns. Applied Science Publishers, London, 1983, Chap. 2.

18. Davids, A. J. & Hancock, G. J., Nonlinear elastic response of locally buckled thin-walled beam-columns. Thin-Walled Structures, 5 (1987) 211-26.

19. Commission of the European Communities, Eurocode No. 3, Design of Steel Structures, Part l--General Rules and Rules for Buildings, edited draft, issue 3, Brussels, April 1990.

20. Canadian Standards Association, Steel Structures (Limit States Design), CAN/CSA-S16.1-M.89, Toronto, Ontario, 1989.

21. Load and Resistance Factor Design (lst edn). American Institute of Steel Construction, Inc., Chicago, IL, 1986.

22. Hot Rolled and Structural Products, (1991 edn). BHP Steel, Melbourne, Australia, 1991.

23. E1-Zanaty, M. H., Murray, D. W. & Bjorhovde, R., Inelastic behaviour of multistory steel frames. Structural Engineering Report No. 83, The University of Alberta, Edmonton, Alberta, Canada, April 1980.

24. Orbison, J. G., McGuire, W. & Abel, J. F., Yield surface applications in non-linear steel frame analysis. Computer Methods in Applied Mechanics and Engineering, 33 (1982) 557-73.

25. Duan, L. & Chen, W. F., Design interaction equation for steel beam- columns. J. Structural Eno0, ASCE, 115 (1989) 1225-43.

Advanced analysis of steel building flames 29

26. White, D. W., Material and geometric nonlinear analysis of local planar behaviour in steel frames using interactive computer graphics. Department of Structural Engineering Report No. 86-4, Cornell University, Ithaca, NY, June 1986.

27. Bridge, R. Q., Clarke, M. J., Hancock, G. J. & Trahair, N. S., Trends in the analysis and design of steel building frames. In Proc. 2nd National Structural Engineering Conf., The Institution of Engineers, Australia, Adelaide, 1990, pp. 12-18.

advanced analysis of steel building frames

Documents