UDC 624.072.28.016

A simple design method for composite columns R. P. Johnson, M A , CEng, FIStructE, FICE Professor of Civil Engineering, University of Warwick

D. G. E. Smith, MSc, DIC, CEng, MICE Scott Wilson Kirkpatrick & Partners

Synopsis The next British Code of Practice for composite structures in buildings, now in preparation, will not include composite columns. A simplified design method for such columns, originally developed for the draft for public comment of this Code, has been revised to take account of subsequent work and the publication of BS 5400: Part 5. This method and other recommendations on composite columns for buildings are now presented in Code form, with explanations, comparisons with other methods, and a worked example.

Notation

is the cross-sectional area of concrete is the cross-sectional area of reinforcement is the cross-sectional area of steel section is the overall dimension of steel section measured perpendicular to y axis (b, in the Bridge Code) is the lesser of h, and hy as here defined is the overall depth of steel section measured perpendicular to x axis is the modulus of elasticity of steel * are the eccentricity of load about x and y axes, respectively is the term in expression for design ultimate load, with subscripts S, y , 1,2,3, etc., as appropriate is the characteristic concrete cube strength is the characteristic strength of reinforcement is the nominal yield strength of structural steel is the design yield strength of structural steel is the greater of h, and 5. as here defined is the overall depth of column perpendicular to major axis of bending is the overall depth of column perpendicular to minor axis of bending is the reduction factor for slenderness of axially loaded column, as in BS 5400: Part 5 is the length of column between centres of end restraints is the effective length of a column, with subscript .Y or y as appropriate is the design failure load of column calculated by the method of BS 5400: Part 5 is the design failure load of column calculated by the present method, including substitution of K, for F , , as noted are the axial failure loads, taking account of slenderness is the design failure load of column subjected to a constant design moment M, is the ,design failure load of a column subjected to a constant design moment M, is the strength of a column in biaxial bending is the squash load of a column is the greatest radius of gyration of the steel section is the least radius of gyration of the steel section is the concrete contribution factor is the ratio of lesser to greater bending moment at the ends of a column length, positive for single curvature bending are the slenderness functions, as defined in BS 5400: Part 5

The Structural EngineerlVolume 58NNo. 3lMarch 1980

Introduction In almost all current methods, there are three stages in designing a column in a framed building structure: an analysis of the frame to determine the end moments and axial load for the column length considered; choice of an effective length for this column; and a check that a column with an assumed cross-section and this effective length can resist the calculated actions, applied at its ends.

It is generally agreed that the stage 3 check should be based on an ultimate-strength analysis. For composite columns, an appropriate method of high accuracy is available: the modified Basu and Sommerville method., given in BS 5400: Part 53. It is hereafter referred to as the method of the Bridge Code.

The first two stages of the design are straightforward for rigid-jointed frames that remain elastic. The assumption of linear-elastic behaviour is inconsistent with the method likely to be used in stage 3, but is usually conservative in this context4. In buildings, most steel frames and many composite frames, however, have so-called simple joints. Moment-rotation relationships for these are at present unpredictable, and vary widely. Various empirical methods are used for stages 1 and 2 in the design of columns in such frames, giving results with uncertain and often excessive margins of safety. Where it is accepted that these stages of the design method will inevitably be approximate, there is little advantage in using a very accurate method for stage 3 alone; designers will prefer a shorter method, even if it is less accurate. The Basu and Sommerville method was therefore not included in the draft for public comment of Part 3 of the new BS 4495.

Possible alternatives were the cased strut method6, and a development of the method given for reinforced concrete columns in CP 110 1972. Neither was used, for reasons discussed below. Instead, a new and simple method was devised, and thoroughly checked against more accurate methods.

The comment most frequently made on the draft Part 3 was that it was too long. Its critics considered that about 90% of the composite construction actually used in buildings was in the form of simply-supported beams. A working part of the drafting committee, set up to revise Part 3, therefore decided to omit from it all reference to columns and frames. This material may appear in a Part 5 of the new BS 449, but probably not for several years. The new design method for columns is therefore given here (Appendix l), and its derivation and validation are explained. There is a worked example in Appendix 2.

It is hoped that designers of composite columns will try it out, and will send their comments to the British Standards Institution, for the attention of Sub-Committee CSB/27/3. The method as presented here is applicable to universal column sections (rolled H-sections) encased in normal-density concrete, but not to encased universal beam sections, filled tubes, or columns made with lightweight-aggregate concrete.

For short, braced, reinforced concrete columns with loads at low eccentricity, CP 110 gives reduced squash load expressions that provide the simplest possible design method. Similar expressions have been derived for composite columns, including filled tubes. These are also given in Appendix 1, and discussed in this paper.

The cased strut method

The method given in BS 449j of allowing for concrete encasement, generally known as the cased strut method, is popular with designers because it is simple. It is essentially an elastic design method for steel columns, modified to take some account of the contributions of the surrounding concrete to the minor-axis slenderness and the strength in axial compression, and is always very conservative.

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The modified cased strut method given in the draft Part 1 of the new BS 449 makes provision for concrete of grades 20 to 30, rather than the single grade 21 used previously; and it takes full account of concrete covers up to 75 mm, compared with 50 mm for slenderness and 75 mm for strength in the current BS 449. This extends the usefulness of the encasement in reducing minor-axis slenderness: though for steel flanges more than 400 mm wide it may still be beneficial to take ry as the radius of gyration of the steel section alone, rather than to use the value given for the encased section, 0.2hY. In flexure, concrete is assumed to prevent local or lateral buckling of the steel member but is otherwise ignored, so that the bending strength of the column is taken as the plastic moment of resistance of the steel section, irrespective of its slenderness.

This method, although less conservative than before, still gives design strengths that are low or very low when compared with results of tests, except for axially loaded columns. A rational design method for composite columns should give strengths that tend to those for a reinforced concrete column of the same size and reinforcement, as the proportion of structural steel in the member tends to zero. The method of the bridge Code is almost compatible in this respect with that of CP 110; but the cased strut method is not. For these reasons it was decided to provide an alternative method in the draft Part 3.

Additional moment methods

Slender reinforced concrete columns are designed in accordance with C P 1 lo7 by adding to the design bending moment about the relevant axis an additional moment that is a function of the slenderness and the axial load, and then checking that the column cross-section can resist the axial load plus the enhanced bending moment.

Work on a similar method for composite columns is illustrated in Fig 1. The full lines show ultimate strengths for a typical composite column in single-curvature minor axis bending in terms of the axial load, N , applied end moment M , and slenderness ratio l,,/hy, computed by the method of the Bridge Code. This can be taken as a standard against which approximate methods are checked.

When M / M , = 0, the additional moments at ultimate load for slendernesses l ey /hy of 12 and 20 (for example) are given by the intercepts AB and CD, respectively. From such intercepts this approximate expression for the additional moment M , for this column can be deduced:

Other studies considered the influences of the axis of bending and the factor K defined in clause 3.5.7.4 of ref. 7; but-no formulae for M , could be found that gave predictions of acceptable aecuracy throughout the ranges of the relevant variables.

THE NEW DESIGN METHOD

The preceding studies led to the conclusion that it was probably not possible to find an acceptable method for composite columns in which the effects of slenderness and of different cross-sectional geometries were treated separately.

The authors then began to develop an integrated method in which both types of variable were considered together. It was hoped that this would agree closely with the method of the Bridge Code over a wide range of composite cross-sections, and be less conservative, under all conditions, than the cased strut method.

Parameters

In preference to the slenderness functions and flexural strengths adopted as design parameters in the Bridge Code, simpler variables were selected. Slenderness was represented, as in C P 110, by the ratio of the effective column length to the overall depth of the section but, unlike C P 110, this had to be combined with properties of the cross-section. It was obviously essential to use as few such properties as possible.

The overall dimensions of the concrete section entered into the slendernesses and the concrete contribution factor x,. Longitudinal reinforcement in the casing was neglected because large-diameter bars are seldom, if ever, used in composite columns.

The precision needed to describe the geometry of the steel section was determined by a parametric survey of all universal beam and column sections. Four non-dimensional parameters were selected which adequately describe the geometry of symmetrical steel sections. Extreme values of these are given in Tables 1 and 2. Three of them are seen to be interrelated; sections with high D / B ratios (and consequently relatively small flanges) have low r , /D and r, /B ratios. These varied by no more than 7% of the mean for universal columns, and were expected to produce a much smaller variation in the properties of a composite section. The variation for universal beams was much larger.

TABLE; l-Extreme section properties for universul columns

It follows that the additional moment for a column with ley/hy = 12 at some lesser load, such as N / N , = 0.5, is given by the intercept EF. A point on an approximate design curve for this column can be found by deducting an equal intercept (GH) from the curve for ley/hy = 0. The dashed curves in Fig 1, constructed in this way, are seen to give poor correlation with the computed curves.

Section 1 r , /D 1 r y / B I D / B I A J B D

0 0.2 0.4 0.6 MIM,

10

Fig. I . Additionul nronrmt mrthodjor minor-usis bmding (Jconrpositr columns

356 x 368 x 129

0.427 1 5 2 x 1 5 2 ~ 2 3 0.420 2 5 4 x 2 5 4 ~ 1 3 2 0.390 3 5 6 x 4 0 6 ~ 6 3 4 0.439 0.255

0.259 0.255 0241

0.966 1.111 1.059 l m0

0.126 0.40 1 0.232 0.128

mean variation k

0.4 14 I 6% 0.250 4/0 1.038 7% 0.263 52% TABLE 2.- -Extreme section properties for universal beams

3 0 5 x 1 0 2 ~ 2 5 3 0 5 x 1 6 5 ~ 4 0 203 x 133 x 30 305x 127x48 4 0 6 x 1 4 0 ~ 3 9

mean variation f.

0.387 0.425 0.420 0.403 0.400

0.406

0.189 0.222 0.228 0.2 14 0.194

0.209 10%

3.00 1.84 1.54 2.48 2.80

2.27 32x

0.101 0.102 0.138 0.1 56 0.075

0.122 28%

To simplify the design method, it was decided to exclude universal beams, because they are rarely used in composite columns for buildings. This made it possible to neglect the variation in these three ratios, provided that account was taken of whether bending was about the major or the minor axis.

The strength of the composite section in axial compression was represented by two variables only-the concrete contribution factor and the squash load Nu of a short column, both of which are defined in Appendix 1 . The factor a, is as in the Bridge Code, but the steel term in the definition of N u is0.93Asf;, whereas in the Bridge Code it is0.91 A&. Thedifference arises because, in such definitions, nominal or characteristic strengths of materials are used in Part 5 of the Bridge Code. following CP 110, and y m for structural steel is taken as 1.10 (i.e., 1/0.91). In the draft Part 1 of the new BS 449,

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however, the design yield strength of steel fyd is used, where Ad = fy/y,,,, and it is stated that, when f, is known, ym can be taken as 1/093. In the computer studies described below, design strengths NB based on the present method were calculated, and compared with strengths N , that were calculated by the method of the Bridge Code, except that the steel term in Nu was again taken as 093A,f,. The values off, used were 245 N/mm2 for grade 43 and 345 N/mmz for grade 50 steel, as given in BS 4360.

The steel section was assumed to be symmetrically placed within the rectangular concrete section. If, in practice, there is additional concrete cover on one face only, it would be on the safe side to neglect it when using this method. Only two variables in addition to a, and Nu were then needed to represent the flexural properties of the cross-section: the ratios D/h, and B/h,.

For loading, only single-curvature bending ( p = + 1) was considered initially, as this can be defined by the axial load N and the eccentricity ratios e,/h, and e,/h,. Allowance for the effects of other types of bending is given by enhancement ratios for N (clause 7.4.1), which are appropriate for a triangular distribution of moments ( p = 0) and conservative for double- curvature bending. They arc based on a reassessment of the results of computer studies done for the Bridge Code.

The final list of variables for use in the design method was thus as follows:

for slenderness I,,/h,, ley/hy

for the cross-section ac, Nu, Dlh,, Blh,

for the loading N , exlhx7 eylhy

Computer tests

The new method was developed, essentially, by trial and error. Ultimate loads for individual column lengths obtained by the first trial method were computed, and compared with those given by the method of BS 5400: Part 53, slightly modified, which was assumed to be correct. The main discrepancies were noted, the method was modified to minimise them, and the computations were repeated, until all results agreed with the correct results to within the tolerances discussed below. The restrictions that had to be placed on the scope of the method, to keep it simple, also emerged during this process.

The two most dissimilar structural steel sections were selected from Tables 1 and 2 for each type of section considered. These were:

universal columns-356 x 406 x 534 uc and 356 x 368 x 129 uc universal beams-305 x 102 x 25 U B and 203 x 133 x 30 UB.

Variations in the ratio of the depth of the structural steel section to the depth of the encased section, D/h, and B/h,, were considered over the range 055 to 085. Steel grades 43 and 50 and concrete grades 20 and 40 were considered. Steel strut curves for materials less than 40 mm thick were used, irrespective of the actual thickness. The same thickness of concrete cover was assumed about each axis, a convenient restriction adopted to limit the proliferation of variables. Greatest attention was paid to columns with a minor axis slenderness ratio ley/hy of 12, slightly above the norm, but values of 4,8, 16, 20, 25, and 30 were also considered. The columns were analysed as pin- ended, so that the ratio of lex/hx to ley/hy ranged from 0.91 to 1.03.

The following eccentricities of load were examined: for minor axis-e,/h, values of 0,0-04,010,030,060, and 1.00 for major axis-e,/h, values of 0,004,O 10,030,0*60, 1-00, and 1.50 for biaxial bending--e,/h, and eJh, values of 010 and 010, 060 and

The highest of these ratios corresponded to bending moments of between 70% and 100% of the flexural strength of the cross-sections studied. In total, the investigation considered 56 columns with universal column cores and 12 with universal beams, with between 6 and 15 loading cases for each. Properties of 28 of these columns are giveninTables 3 and 4.

Tolerances

The conservative assumptions given below were made in the computer tests. Together, they may be employed to justify an occasional unsafe error in the new method, when compared with that of the Bridge Code, which for slender columns has itself been shown to be conservative in comparison with test resultslO*.

-As in the design of steel columns, it is proposed that the effective lengths of composite columns should be based on the distances between beam centre lines. This neglects the additional stiffness present at joints, which is of particular benefit in slender columns. (In reinforced concrete design, effective lengths are based on the clear

0.60, 1-00 and 1.00, 1.00 and 1.50

lex/hx or Ieylhy

Fig 2. Values of FI, and FI,

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distance between beams'.) -The slenderness reduction factors incorporated in the new method

are conservative on average by 2%, when compared with the computer tests.

-Reinforcing bars are normally located close to a surface. of the column section and so, for a given area of steel, are more efficient in resisting bending moment than is the structural steel section. In the analyses, the reinforcement was treated as additional steel with a contribution to the flexural strength proportionate to that of the structural steel section. This assumption causes the greatest reduction in the predicted column strength when the eccentricity of loading is high. It was acceptable for the new method because this still showed a great improvement over the cased strut method, which is particularly conservative for columns with this type of loading.

Each of these assumptions may not be relevant to the design of every column, a fact that was taken into account in the assessment of tolerances. The conservative limit was determined by the accuracy of the cased strut method, which is discussed later. The limits selected were:

TABLE 3-Ratios NB/N, for encased universal columns with &.,,/h, = 12.0

exercise of curve fitting, for which reciprocal functions appeared best suited.

Axially loaded columns The design.expressions ((A.6) and (A.?) in Appendix 1) simplify to

N = N , = F , N , . . . . (2) where F , is the lower of F , , and F, , . It takes account of the influence of slenderness on the ultimate load and is equivalent to the factor K , in the Bridge Code. I t was found that curves defined by two variables only (given in Fig 2) could be fitted to the computed data such that the ratios F , / K , differed from unity by much less than the tolerances given above for all members to which the method is applioable". F , , was expressed solely in terms of the slenderness ley/hy and the lumped parameter a,B/h,. F , , was expressed in terms of lJh, and D/h;; the influence of a, was found to be negligible.

Curves for F , , and F , , were given in the draft for public comment only for. slendernesses below 20 because this is the limit of the range that has been thoroughly checked. Values for the range 20 to 30 are shown as dashed lines

Minor- axis bending, e,/h, Major- axis bending, e,/h, Biaxial bending denoted by (ey/hy)/(e,/hx)

0-10/0.10 060/0-60 1.0/1.0

090 0 9 1 090 1.02 0.97 0.9 5 1.04 1.10 1.04 1-09 1.13 1.14 1.10 1.18 1.19 1-01 0.97 095 1.05 1.06 1.07 1.02 0.98 0.96 1-03 1.02 1.02 094 0-99 0-97 1.03 1.01 0-99 1.02 1 .00 1 * 0 0 0.94 0.99 0.98

004 0.60 1 a * 004 0.60 1 .00

055

065

075

085

016 0.28 0 50 0 59 067 021 0.50 016 043 006 012 0.36 044

1-05 099 1 -00 1 * 0 0 1 * 0 0 1.00 0.99 0-98 1 -00 0.94 0.95 099 1 .00

1.02 1.03 1.02 1 a03 1-03 1.03 1-02 1.03 1 *02 1-04 1.03 1.02 l * 0 0

0.94 0.97 1.03 l .07 1.12 099 1-04 1.01 1.03 1.05 1.04 1.03 1 * 0 0

092 095 1.01 l .08 1.13 097 1 e 0 4 099 1.03 1.03 1.02 1-03 1.01

0.98 0.99 1.02 1.04 1.05 099 1.02 0.99 1.00 099 099 0 99 099

0-89 0.97 1.07 1.19 1.22 0.95 1.07 095 1 .00 0.94 0.96 0.96 096

0.87 0.96 1.08 1.21 1 *24 0.94 1.08 0.94 1 .00 093 0.95 0.95 0.96

* e,/h, eccentricities of 1.50 were ulso lewdfor some sections hut results were similur to those giretl for 1.00.

-at normal eccentricities of the total column load (i.e. eJh, and e,/h, not exceeding about 0.6 for uniaxial bending): 5% unsafe to 10% safe

-at higher eccentricities in uniaxial bending and in biaxial bending with D/h, and Blh, ratios exceeding 070 (the usual case): unsafe to 15% safe; and with lower Dlh, and Blh, ratios: 10% unsafe to 25% safe.

Derivation of design expressions

The derivation of the design expressions involved a complex

TABLE 4-Ratios NB/N, for encased unicersal columns of various slendernesses

in Fig 2, as they may be useful for preliminary designs, which should finally be checked by the method of the Bridge Code.

Definition of axes In Part 5 of the Bridge Code, the agreed European definition of slenderness is used, and the x-axis of bending, also called the major axis, is chosen such that A, 4 E,,, where 1, and i, are the slenderness functions for the two axes. In the simpler method presented here, the rather long calculation of slenderness functions is omitted, so that the only practicable definition of the x (major) axis is that it is the major axis of the steel cross-section (clause 7.1.8).

Eccentric loading The influence of load eccentricities was studied first for pin-ended columns with slenderness ley/hy = 12 and eccentricity ratios e,/h, and e,/h, of 0.04 and 0.60. This led to the terms F , and F , in equations (A.6) and (A.7) which are functions solely of the eccentricity ratios and of the cross-section properties a, and either Dlh, or B/hy. The terms F , and F , were found to apply to larger eccentricities of load but not to columns of greater slenderness.

Typical results are given in Table 3 in terms of the ratio NB/NA, where N B is the design ultimate load given by the new method, but using K , values from the Bridge Code, and N A is the value given by the Bridge Code. With N B so calculated the comparison of the effect of the variables in F , and F , for different columns is clearer than if F , is substituted for K , . The substitution would not significantly affect the results as F , I K , values are close to unity, as can be seen in Tables 3 and 4.

Unsafe errors exceeding loo/;, (entries exceeding 1.10) are shown in Table 3 when 2, = 0.59 and 0-67. This led to the limit rc 0.50 in clause 7.4 (Appendix 1).

The effect of reducing the slenderness ratio from 12 to 4 is shown in Table 4.

The Structural EngineedVolume 58NNo. 3IMarch 1980

leylhy " C e,lh, = 0.04

090 0.90 0-9 1 0 9 1 0-92 1.05 1.01 l .05 1.02 1.04 1.05 1 e02 1.03 1-02 l .04

e,lh, = 060

1.00 1 .00 0.99 1.00 0-95 l .04 0-92 1-07 0.9 5 1.06 1.09 097 1.10 1 .00 1.1 1

e,lh, = 0.60

1.02 0.99 0.97 0.99 0.89 1.02 0.85 1.04 0.87 0.97 0.98 082 0.98 0.83 0.94

eJh; =

0.60

1.01 0.99 0.98 1 .00 0.92 1.03 0 8 8 1.07 0-90 1.03 l .05 0.88 1.06 0.9 1 1.04

e,/h, =

4

20

30

055 0.65 0.65 0.75 0-75 055 0.65 065 075 075 0.55 0-65 0-65 075 075

0- 50 0.2 1 0 50 0.16 0.43 0 50 0.2 1 0.50 0.16 0.43 0 50 0.2 1 0.50 0 16 0.43

1 .00 1 * 0 0 1 .00 1 -00 1 .00 1 .00 1.03 1-00 099 1.01 0.99 1.02 1 .00 096 1-03

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0.5

0.4

0 -3

F pin ends on y-axis . pin hnded column

l. I range of Axhy

0 2 - in tables 3and 4 N - N" 0.610

0.15 - 010.6

I ,/' ------

c- 0.1 - - proposed method --- bridge code method

4- L,

1

=,y pin ends on x -axis

------ l

I 0.0410 0/0.04 e

load eccentricities 2 and 2 hx hy

t 0 .'6IO

010.6

x and yhave to be interchanged when using bridge code method

0.5 0.6 0.7 0.8 Q9 1.0 1.1 1.2 1.3

A x l h Slender coluntns For eccentrically loaded columns with slenderness ratios exceeding 12 it was found that F, and F , over-corrected for the effect of moment, as is evident from the curves for the 'uncorrected method in Fig 5(a). The method is therefore safe in this region, and can be used. It is analogous to design in CP 110 when the K correction is neglected. However, unlike CP 110, the correction for slenderness used here does not involve iteration and is so simple to calculate and apply that its neglect can seldom be justified. The corrected values NB agree much better with the values N, found by the method of the Bridge Code (Table4 and Fig. 5(a)). The last three lines of the table show that unsafe errors slightly exceeding 10% occur in pin-ended columns when ley/hy = 30.

Biaxially loaded columns Equation (A.8) in clause 7.4.4 is provided for use when neither e, nor e, can be assumed to be zero. It is of similar form to the interaction formula given in clause 11.3.6 of BS 5400: Part 5, which has been found to be conservative12 for biaxial bending of slender columns. Typical results of the calculations by which it has been checked are given in Tables 3 and 4.

Biaxial failure under major axis loading

When the load applied to a column length is eccentric in the stronger plane . of bending (that associated with the higher value of K , ) and the slenderness for buckling in that plane is much less than that for minor-axis buckling, failure in a biaxial mode is possible. In developing relevant design methods, one meets the problem that, in the original Basu and Sommerville method, the strengths for uniaxial bending about the two axes do not converge on the same value as er and ey tend to zero.

In the Bridge Code these two problems have been resolved and allowance has been made for construction tolerances by requiring (clause 11.3.5 of Part 5 ) that, for columns subjected to major axis bending but free to buckle about the minor axis, a nominal minor axis eccentricity of 0 0 3 6 must be considered. Thus, all such columns have to be designed for biaxial bending.

The method now proposed for columns in buildings is much simpler. The problem of lack of convergence of uniaxial strengths was overcome by using

The Structural Engineer/Volume 58A/No. 3/March 1980

Fig 3. Vuriution of design strength with rutio of slendernesses. tuking uccount of nominal eccentricities of louding

1.1, 1.6 1.0

the same value of F , (the lower of F,, and F,,)for both major and minor axis bending. This use of F , , in design for major-axis bending causes a conservative error that increases with the ratio of minor axis to major axis slenderness, and thus is greatest when minor-axis failure under major-axis loading is most likely. The computer tests showed that, for columns with slenderness ratios less than 20, there is then no need to consider a nominal minor-axis eccentricity in design for major-axis bending, and so no need to design such columns for biaxial bending: the use of F , in place of K,, and K, , thus solves both problems.

Columns with intermediate minor-axis restraint

As explained in the paragraph 'Definition of axes', the minor or y axis of the column length is taken in this method as the weaker axis of bending of the

0.85

0 a80

NIN ,, 0.68

0 e60

0-50

0.40 0 4 8 12

I, Ih

Fig 4 . Reduced squush loud' design of short columns

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steel cross-section, whereas in the Bridge Code it is the weaker axis of buckling of the composite member. In what follows, only the first of these definitions is used.

It is fairly common in practice for a long column to be braced at midheight about the y-axis only, with the result that ie,ihx > /ey/hy. In the Bridge Code, slenderness functions 1 are defined as the ratio of the effective length to the length at which the Euler load and the squash load are equal. For the column considered here, 1, would probably exceed L, (so that I , would be called A,, and vice versa).

Tables 3 and 4 give typical results from computer tests on pin-ended columns for which 1,/1, ranged from 0.67 to 0.95. When 1, exceeds I,, the method as originally proposed5 is unsafe, and so has been modified by relating le/h in clause 7.5 to lex/h, (rather than ley/hy) when F , , > F, , . This is accurate enough when neither slenderness (I/h) exceeds 20.

The effect o f variation in 1,/1, inslender columns was considered by making a separate sudy of a series of columns with a steel section as in line 1 of Table l , and with h, = 479 mm, h, = 491 mm, feu = 40 N/mm2, f , = 345 N/mm2, a, = 0.426, and Nu = 9-22 MN. The slendernesses le,/hx and ley/hy were varied from 12 to 30.8 and 12 to 30, respectively, such that the range 0.3 l,/iy < 2.0 was explored. The eccentricity ratios (e/h) considered were 004 and 060 about each axis in turn.

. 0 3 l , 1 1 ~ 1 1 , 1 ~ 1 1 1 1 , I l I l l l l I l I I I I l l 5 10 15 20 2 5 30

( a ) PROPOSED METHOD AND BRIDGE CODE METHOD

le y / h y

Fig 5 . Compurison of design methods for composite columt1.s

In addition, account was taken of the rules for minimum eccentricity of loading, but using 0.04 rather than 0.03 for the Bridge Code method (so that the combination 0.6/0 was taken as 0.6/0 for the present method and as 0.610.04 for the Bridge Code).

Typical results are shown in Fig 3. The new method gives values N , / N , somewhat above 1.10 for eccentricities 0.04/0 with E.,/Iy < 1 and for O / O N with E.,/;C, > 1. The r e m n in both cases is that, for the slender columns considered here, the design load given by the Bridge Code is significantly reduced by the conservative biaxial bending formula, which enters the calculation through the use of a nominal eccentricity about the other axis.

Short braced columns with loads at low eccentricity

For very short columns with loads at the low eccentricity of0-04 h, or 0.04 h,, the new method is about 10% conservative, as shown by lines 1 to 5 of Table 4. A simpler and less conservative method, using reduced squash load expressions similar to those in C P 110, is therefore given in clauses 7.2 and 7.3. It enables many, if not most, of the column lengths that occur in practice to be designed for axial load only, and is also applicable to concrete-filled

90

tubes. Its derivation for encased H-sections is now explained. For convenience, shortcolumns are defined (clause 7.1.9) as in CP 110. In

the squash load method in C P 110, no account is taken of the effects of instability. Two reduced squash loads are given, 0.89 Nu and 078 Nu approximately, which correspond to load eccentricities of about 0046 h, and 0 10 h,, respectively.

In the new method, the design formulae for axially loaded columns are based on computations with loading at an eccentricity of 0.04 h, in the direction that gives the lowest design load, and on 010 h, for columns supporting an approximately symmetrical arrangement of beams. The method of the Bridge Code was used in the computations, with single- curvature bending. Both this assumption (i.e. p = + l ) and the use of column lengths between beam centre-lines make the method over-conservative for short columns, so the calculated ultimate loads were increased by 5% for a slenderness of 8 and by 10% for a slenderness of 12. The resulting design loads, given in clauses 7.2 and 7.3, are shown in Fig 4 as multiples of Nu, with the ranges of slenderness within which they are applicable. The upper limits given for N / N , correspond to the computed values when ley/hy = 4.

Even with the correction noted above, the values turn out to be lower (in terms of Nu) than those in C P 110, although the expressions used for Nu in the two methods are identical when A, is taken as zero. The reason is that the

N

(b) CASED STRUT AND BARE STEEL METHODS

4 -Yt6

l52 x 152 U C 23

Fig 6 . Cro.ss-sc.ction ofcdumtl

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present method takes account of slenderness effects in short columns, whereas the method of C P 110 does not, presumably because this unsafe approximation is considered to be offset by other approximations that are conservative. The minimum eccentricity was taken as 0.04 h,, rather than 0046 h,,, in order to reduce this discrepancy between the two methods.

Universal beams and other steel sections

The accuracy of the new method for concrete-encased universal beam sections was also studied. It gave strengths between 20% and 50% above the correct values for columns loaded at large minor-axis eccentricities, though the error was much reduced for the wider universal beam sections. For ey/hy = 0.04 and most cases of major-axis bending. the method gave ratios N , / N , in the range 0.8 to 1.1.

The new method is considered to be applicable to all rolled steel H or I sections with r , /D 2 0.39 and r y / B 2 0.24. Provision is made in clause 7.4(2) only for universal column sections mainly because other sections are seldom used in concrete-encased composite columns.

Comparisons with other design methods

The strengths of a particular composite column section and of its steel core as determined by various methods are compared in Fig 5, for wide ranges of eccentricities of loading and minor-axis slenderness. The design strengths N are divided by the squash load N u , as calculated by the proposed method. The curves for zero eccentricity represent limiting values, and are not applicable in practice. Only single-curvature bending of pin-ended columns is considered, and the column section is that described in Columns with intermediate minor-axis restraint.

The curves labelled bare steel in Fig 5 were calculated in accordance with the current BS 4496. It was assumed that the ends of the column were torsionally restrained but free to warp, so that the same l,/r, ratios were used for the derivation of both axial strengths and bending strengths. For this situation, the draft Part 1 of the new BS 449* gives lower strengths, as the new methods are more sensitive to the poor lateral-torsional restraint and adverse bending-moment distribution assumed here. In more typical situations, the new draft would give strengths similar to, or higher than, those from ref. 6.

The other curves in Fig 5(b) show that the modified cased strut method gives strengths generally loo/, to 157; higher than those found by the cased strut method given in BS 449, except that neither method gives much advantage over the bare steel method for short columns loaded at high eccentricities.

Fig 5(a) shows that, for the particular column cross-section and strengths of materials studied, the proposed method everywhere gives strengths that agree closely with those from the longer method of the Bridge Code3. Similar results are obtained for columns of grade 43 steel and grade 30 concrete. The dashed lines show the extent to which the new method becomes conservative at slendernesses exceeding 12 if the optional adjustments to F , and F , (Appendix 1, clause 7.5) are not made.

The new method gives strengths significantly higher than the modified cased strut method, as shown by corresponding pairs of curves in Fig 5(a) and (b). The margin is a minimum of 87; for axially loaded short columns, but increases to 30/, at nominal eccentricities and to over 100% at the maximum eccentricities for which the method is applicable. The advantages over the bare steel method are of course greater still, being about SS?, for axially loaded columns, 40(1/0 for e,/h, = 0.6. and 140:,;, for ey/hy = 0.6.

Conclusions

Studies made for the draft for public comment of the new Code of Practice for the use of composite structures in buildings showed that all of the design methods then available for composite columns were either too complex or too conservative. The new method presented in this paper provides a combination of simplicity and economy appropriate for the design of column lengths in buildings. It consists of reduced squash load expressions of wide applicability to short braced columns, and a general method for columns resisting significant bending moments that is applicable to encased universal column (H) sections.

The authors would welcome comments from designers on whether there is any demand for a simplified method of the type here presented, or for some developments of it that have been completed, which take account of lightweight concrete casings, universal beam sections, and back-to-back channels.

The Structural EngineerlVolume 58NNO. WMarch 1980

Acknowledgements

The authors are grateful to Dr. I. M. May for his comments on this work. They acknowledge with thanks the financial support for consultancy drafting provided by the Department of the Environment, subsequent assistance from Constrado, and the facilities provided by Scott Wilson Kirkpatrick & Partners and the University of Warwick.

References 1. Johnson, R. P.: Composite structures of steel and concrete, Vol. I :

Beams, columns, frames, and applications in building, Crosby Lockwood Staples, London, 1975

2. Johnson. R. P,, and Buckby, R. J.: Composite structures ofsteel und concrete, Vol. 2: Bridges, Crosby Lockwood Staples, London, January 1979

3. BS 5400: Steel, concrete, and composite bridges: Part 5 : Code of practicefor design of composite bridges, British Standards Institution, London, May 1979

4. May, I. M.: Restrained composite columns, PhD thesis, University of Warwick, October 1976

5. British Stundurd ,for the LI.W of structural steel in building: Part 3: Composite construction, draft for public comment, British Standards Institution, August 1976

6. BS449: Part 2: The use ofstructuralsteel in building, British Standards Institution, 1969

7. CP 110: The structural use ( fconcrete: Part I : Design, materials and Mwrkmanship, British Standards Institution, 1972

8. British Standardfor the use of structural steel in building: Part I : Simple construction and continuous construction, draft for public comment, British Standards Institution, November 1977

9. Anderson, D. : Design methods for composite columns, Technical Paper 130, BSI Sub-Committee B/l16/5, August 1976

10. Virdi, K . S. , and Dowling, P. J , : A unified design method for composite columns, Publications, IABSE, Vol. 36-11, 1976, pp. 165 184

1 1 . Smith, D. G. E., and Johnson, R. P.: A new design method for composite columns, Technical Puper 223, BSI Sub-Committee B/20/5, April 1976

12. Dowling, P. J., Chu, H. F., and Virdi, K. S. : The design of composite columns for biaxial bending, Prelim. Report, 2nd lnt. Colloquium, Stability o f s t e e l Structures, Liege, April 1977, pp. 165- 174

Appendix I The design of lengths of composite columns in buildings for known axial loads and end moments

The following clauses are based on the draft for public comment (August 1976) of Part 3 (Composite construction) of the British Standard for the use of structural steel in buildings, which is intended to replace BS 449. Some modifications have been made as a result ofcomments and subsequent work on these clauses and on the column clauses in BS 5400: Purr 5 , , and clauses not within the scope of the paper have been omitted.

7. Composite columns 7. I General 7.1.1 Scope. Design methods are given for concrete-encased steel sections and concrete-filled circular and rectangular hollow steel sections, all of which take account of the composite action between the various elements forming the cross-section. The extent to which these methods are conservative is related to their complexity. Bending about the two principal axes of a column is considered separately for each axis, and separate provision is made for bending about both axes simultaneously. The methods given in 7.2 to 7.5 assume fully composite action for the whole of the load. 7.1.2 Materials ( 1 ) Steel. In columns formed from concrete-encased steel sections, the structural steel section should be either:

(a) a rolled steel joist or universal section of grade 43 or 50 steel that complies with the requirements of BS 4: Part 1 : Hot rolled sections; or

(b) a symmetrical I-section fabricated from grade 43 or 50 steel; or (c) two channels in accordance with BS 4: Part 1 in contact or spaced

apart not less than 40 mm, nor more than half their depth, and

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complying with the appropriate requirements of Part 1 (of the new BS 449); or

(d) structural hollow sections. Concrete-filled hollow steel sections may be either rectangular or circular and should :

(a) comply with the requirements of BS 4848 : Part 2, and (b) have a wall thickness of not less than:

bf Jf,13Es

for each wall in a rectangular hollow section (RHS); or

for circular hollow sections (CHS) where

bf is the external dimension of the wall of the RHS D is the outside diameter of the CHS E, is Youngs modulus of elasticity of steel f, is the nominal yield strength of the steel

(2) Concrete. The concrete should be of normal density in accordance with C P 110. The characteristic strength should be not less than 20 N/mm2, except that in concrete filled tubes designed in accordance with BS 5400: Part 5 the associated minimum strength should apply. (3 ) Reinforcement. Steel reinforcement should be in accordance with the relevant clauses on materials and detailing of C P 110.

7.1.3 Concrete cover in cased sections. The concrete cover to the structural section should be not less than 50 mm. The concrete cover to the reinforcement should be not less than the minimum permitted in C P 110, nor for longitudinal bars less than 25 mm.

7.1.4 Minimum reinforcement in cased sections. The longitudinal and transverse reinfokement should each have a cross-sectional area not less than that of 5 mm diameter bars at a pitch of 200 mm. At least four longitudinal bars are required to support the transverse reinforcement.

7.1.5 Composite action. No consideration need be given to longitudinal shear stresses in a composite column that is not subjected to lateral loading along its length, provided that the connections between the steel members of the adjacent beams and columns are sufficiently strong to transmit the full shear force without assistance from the concrete.

In other situations the design of beam-column joints should be such that the longtudinal shear stress at the steel-concrete interface is not excessive.

7.1.6 Design methods for columns at the ultimate limit state ( 1 ) Concrete-encased steel columns. These columns may be designed by the following methods, provided that the concrete encasement and longitudinal reinforcement considered in design are symmetrical about the principal axes of symmetry of the steel section.

For short braced columns (as defined in 7.1.9), in situations which restrict the magnitude of the applied moments, by the methods of 7.2 and 7.3. For short and slender columns, in situations in which it is considered acceptable to neglect both the reinforcement and any concrete cover in excess of 75 mm, by the method of Part 1 (i.e. the cased strut method). The method becomes progressively more conservative as the ratio of bending moment to axial load increases. For short and slender columns, with universal column steel sections, by the method of 7.4 and 7.5. This method takes some account of the reinforcement. For short and slender columns, by the method of BS 5400: Part 5, but in accordance with 7.1.7 to 7.1.12. This method takes full account of the reinforcement. In applying this method, account should be taken of the different factors of safety for materials (7,) incorporated in the expressions.

(2) Concrete filled hollow tubes. These columns may be designed by the following methods, provided that any reinforcement considered in design is symmetrical about the principal axes of symmetry of the steel section.

(a) For short braced columns (as defined in 7.1.9) in situations that restrict the magnitude of the applied moments, by the methods of 7.2 and 7.3.

92

(b) For short and slender columns, by the method of BS 5400: Part 5 but in accordance with 7.1.8 to 7.1.12. This method will generally give greater strengths than (a) for short columns of circular section, as account is taken of the confinement of the concrete. In applying this method, account should be taken of the different factors of safety for materials (7,) incorporated in the expressions.

7.1.7 Limit state of serviceability

in : Tensile cracking of concrete. No check for crack control need be made

(a) concrete-filled hollow steel sections, or (b) concrete-encased steel sections, provided that the design axial load

at the ultimate limit state is greater than 0.20fc,A,, where the symbols are as defined in 7.2.

Where the design axial load in concrete-encased steel sections is less than the value given in (b) above, the column should be considered as a beam for the purpose of crack control.

7.1.8 Axes of encased column sections. For the purposes of 7.2,7.3,7.4 and 7.5, the majorand minor axes should be taken as the major and minor axes of the structural steel section.

7.1.9 Short and slender columns. A column may be considered as short when neither of the ratios lex/hx and l,,,/h, exceeds 12. I t should otherwise be considered as slender. In these expressions

l,, is the effective length in respect of the major axis l,, is the effective length in respect of the minor axis h, is the overall depth perpendicular to the major axis h, is the overall depth perpendicular to the minor axis

7.1.10 Efective length of a column (The determination of effective lengths is not within the scope of the paper, so this clause is omitted. Clause 11.224 of BS 5400: Part 5 (Effective length) could be used with the methods given here.)

7.1.11 Slenderness limits j o r columns. The effective length l,, should not exceed the least of: 40h,, 250r,, and 100b2/h. The effective length l,, should not exceed the least of: 40h,, 250r,, and 100b2/h. In these expressions

b is the lesser of h, and h, h is the greater of h, and h, r , and ry are the radii of gyration of the steel member alone

and the other symbols are as defined in 7.1.9.

7.1.12 Moments and forces in columns (The analysis of frames is not within the scope of this paper so this clause is omitted.)

1.2 Short braced axially loaded columns

To allow for eccentricity due to construction tolerances the ultimate axial load for a short column, which by the nature of the structure cannot be subjected to significant bending moments, should not exceed N given by the lesser of

N = [1.03 -0.03(l,/h)]Nu

and N = 0-85Nu

where for encased sections

Nu = A,fyd + 0*87A,f,, + 0.45A,fcu . . . . (A.2) or, for concrete filled hollow sections,

Nu = A,& + 0-87Arf,,, + 0*53Acfcu . . . . (A.3) and l,/h is the greater of lcx/hx and l,,/hy, which must not exceed 12. In the previous expressions,

A, is the cross-sectional area of the structural steel section A, is the cross-sectional area of reinforcement A, is the area of concrete in the cross-section

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fyd is the design yield strength of the structural steel f, is the characteristic yield strength of the reinforcement f,, is the characteristic 28-day cube strength of the concrete

and the other symbols are as defined in 7.1.9.

7.3 Short braced columns supporting an approximately symmetrical arrangement of beams

The ultimate axial load for a short column of this type, where (a) the beams are designed for uniformly distributed imposed loads,

(b) the spans of beams on opposite sides of the column and at the same and

level do not differ by more than 15% of the longer, should not exceed N given by the lesser of

N = C0.80 - 0.025(Ie/h)]N, . . . . (A.4)

and N = 0.68N,

where the symbols are as defined in 7.2.

7.4 Short columns resisting moments and axial forces

Concrete-encased columns may be designed in accordaxe with 7.4.1 to 7.4.4 provided that the following conditions are satisfied.

The column is short as defined in 7.1.9. The steel core is a universal column section. The concrete contribution factor a, does not exceed 0.50, where

a, = 0-45A,fc,/N, . . . .(AS)

and the squash load N u is as given in 7.2. The eccentricities of load about the x-axis and y-axis do not exceed 1.5hX and 1.0hY, respectively. Neither D/h, nor B/h, are less than 0.50, where D is the overall depth of the steel section perpendicular to the x-axis, and B is the flange breadth of the steel section.

The other symbols in this clause are as defined in 7.1.9 and 7.2.

7.4.1 Design eccentricities ojusiuljorres. Design strengths given in 7.4.2 to 7.4.4 are for columns subjected to single-curvature bending. The eccentricity of loading about each axis should be taken as the greater of the values for the two ends of the column, subject to conditions (1 ) to ( 3 ) below. (1) Where the applied load is eccentric about one axis only, the eccentricity about that axis should be taken as not less than 0.04b, where b is the lesser of h, and h, as defined in 7.1.9. No nominal eccentricity about the other axis need be considered. ( 2 ) Where the applied load is eccentric about both axes, neither eccentricity should be taken as less than 0.04b. (3) Where 7.4 is used for an axially loaded column, the eccentricity of loading should be taken as 0.04b about the axis that gives the lower strength. For a column subject to double-curvature bending, or with at least one end prevented from rotation, in the plane or planes considered, the design strengths given in 7.4.2 to 7.4.4 may be increased as follows:

(1) for lex/hx and ley/hy 8, by 57; (2) for 8 < lex/h, and ley/hy 12, by 10%.

7.4.2 Design for bending about the major usis. The design ultimate load on a column should not exceed the design strength N , given by

. . (A.6)

where

2.3ex/h,

( 1 - a,)D/h;+ ~ , / 3 F , = -

N , is as defined in 7.2 F , is the lower value of F , , and F , , F , , and F , , are the strut load slenderness reduction factors given in Fig 2

for the appropriate values of l,,/h,, l,,/h,, a,, Dlh,, and Blh, e, is the eccentricity of load about the x-axis

and the other symbols are as defined in 7.1.9 and 7.4.

7.4.3 Design j b r bending about the minor axis. The design ultimate load on a column should not exceed the design strength N , given by

NUF, l + F 3

N , = - ... I

where

e, is the eccentricity of load about the y-axis and the remaining terms are as defined in 7.4.2.

. (A.7)

7.4.4 Design for biaxial bending. The design ultimate load on a column should not exceed the design strength N , , given by

where N u , F , , and F , are as defined in 7.4.2, and F , is as defined in 7.43.

7.5 Slender columns

A column that is slender as defined in 7.1.9 may be designed conservatively by the methods given for short columns in 7.4. Alternatively, providing that

(1 ) both lex/hx 20 and ley/hy < 20, and (2) the conditions of 7.4 (2) to ( 5 ) are satisfied,

the design strength of the column may be calculated from 7.4.1 to 7.4.4, with F , replaced by F , and F , replaced by F,, given by:

where

/ , /h = ley/hy (but not less than 12) when F , , F , , l,/h = 0.7ie,/h, (but not less than 12) when F , , > F , ,

and the symbols are as defined in 7.4.1 and 7.42. The substitutions for F , and F , are both applicable when a column is slender about one axis only.

Appendix 2 Worked example : slender encased H-section

For the column shown in Fig 6: f:,, = 30 N/mm2, fyd = 329 N/mm2, fry = 425 N/mm2, l,, = 3.2 m, l,, = 5.2 m. Design ultimate loads: N = 580 kN, M , = 43 kNm, M , = 0. The strength of the column is checked as follows. The symbol (j refers to clauses in Appendix l . From (37.44 l), ey/hy = 0. From data, e,/h, = 431480 x 0.30 = 0.30. For cross-section, A, = 29.57 cm, A , = 8.04 cmi, A, = 742.4 cm2. From (A.2) in 67.2. N u = 2273 kN. As B = 152 mm, h, = 260 mm, then B/h, = 0.585. From (AS) in (37.4, a, = 0.441, so a,B/h, = 0.258. From Fig 2 with ley/hy = 5.210.26 = 20, F , , = 0.60. As D = 152 mm, h, = 300 mm, then Dlh, = 0.5 1. From Fig 2 with lex/hx = 3.210.3 = 10.7, F , , > F, , , so F , = 0.60.

2.3 X 0.3

0.559 X 0.51 +0.147 From 67.4.2, F - ____ .- ~ = 1.592 2 -

From (37.5, F , = 1.592[1-(8/37)] = 1.248. From 47.4.2, N , = 2273 X 0.6i2.248 = 607 kN. Thus N , exceeds N , so the column length is strong enough. The method of BS 5400: Part 5 is much longer, and gives N , , = 6 0 1 kN when ey/hy (nominal minor-axis eccentricity) is taken as 0.04.

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