Design of normal- and high-strength concrete columns

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Design of normal- and high-strengthconcrete columnsA. N. Beal, BSc, CEng, MICE, MIStructE, and N. Khalil, BSc, PhD, MACI, MASCE

& Research by Khalil to verify and develop

Beal's graphical method of buckling

analysis con®rms that Beal's method is

accurate and e�ective, giving results

which agree well with experimental data.

The method provides a powerful analytical

tool, allowing fast determination of the

column capacity for a wide range of

slenderness and loading conditions. It is

used here to investigate the behaviour of

normal- and high-strength concrete

columns under both axial and eccentric

loads and with a variety of slenderness

ratios and reinforcement proportions. The

results of this accurate analysis are used

as a benchmark for the assessment of code

of practice design rules. Existing BS 8110

design rules are found to give poor agree-

ment with both the theoretical analysis

and experimental results. New design

rules which give more consistent and

accurate results are proposed.

Keywords: buildings, structure & design;

columns; concrete structures

IntroductionThe analysis of slender reinforced concrete

columns is complicated because the buckling

analysis must take account of the non-linear

properties of the materials and also construction

imperfections. Exact analysis methods are gen-

erally complicated and unsuitable for everyday

design. Codes of practice recommend various

design methods for slender columns based on

capacity reduction factors, additional design

moments, or moment magni®cation, to account

for the e�ects of buckling. These methods are

approximate and largely empirical in nature.

There is now considerable interest in the use of

high-strength concrete in columns, which can

allow columns that are more slender to be used

to sustain high loads. Therefore one of the

objectives of the present study is to investigate

the theoretical behaviour of such columns, in

order to establish suitable rules for their design.

2. A simple graphical technique has been

proposed by Beal for rigorous analysis of pin-

ended slender reinforced concrete columns.1

Firstly, graphs of load eccentricity against

section curvature are prepared for various

values of applied axial load. Then a second set

of graphs is prepared which plot the de¯ected

shape of the column as it buckles, relating its

mid-height de¯ection to the maximum section

curvature. The de¯ection is calculated assum-

ing a sine curve for the de¯ected shape of the

column

Dh� 1

p2Lh

� �2 hR

�1�

where D is the de¯ection, L is the column

e�ective length, h is the column width in the

direction of buckling and R is the radius of

curvature. By overlaying the two graphs, the

section capacity can be established (see Fig. 1).

Once section moment±curvature relationships

are established, this technique allows rapid

determination of column capacity under any

combination of slenderness ratio and initial

eccentricity.

3. Beal's original paper set out the theoreti-

cal basis of the graphical analysis method.

Design recommendations were presented for

columns made from normal-strength concrete

(up to 50 N/mm2 cube strength) and it was

shown that these gave better agreement with

test results than existing UK codes of practice.

Compared with Cranston's long-term test

results,2 Beal's analysis gave ratios

(Ptest=Ptheory� of 0´97±1´97, with a mean of 1´28

and a coe�cient of variation of 0´16, a con-

siderable improvement over existing design

methods (see later). In the discussion of Beal's

paper,3 Drs Maisel and Beeby showed that a

computer program devised by Cranston for non-

linear buckling analysis gave very similar

results to the graphical analysis, for both peak

load and column de¯ection (see Figs 12 and 13

in Reference 3). In the same discussion, Dr

Wong described an alternative graphical analy-

sis method which he and Professor Kong have

developed; he con®rmed that in his opinion this

should also give similar results.

4. At Leeds University, Dinku4 investigated

the proposed analytical method in the light of

existing experimental data reviewed by Cran-

ston. Dinku developed computer programs to

generate load eccentricity±curvature graphs in

terms of capacity ratio (P=Po), based on stress±

strain relationships for concrete and steel. His

research veri®ed that the graphical analysis

method predicts the capacity of concentrically

or eccentrically loaded pin-ended slender

columns quickly and accurately.

5. The work of Beal and Dinku encouraged

further research by Khalil, which is summar-

ized here. Following her veri®cation of the

accuracy of the graphical analysis method,

A. N. Beal,

Associate, Thomason

Partnership,

Leeds

N. Khalil,

Assistant Professor,

Department of Civil

Engineering,

University of

Balamand, Tripoli,

Lebanon

345

Proc. Instn Civ.

Engrs Structs &

Bldgs, 1999, 134,

Nov., 345 ± 357

Paper 11847

Written discussion

closes 31 May

2000


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it was used to carry out a comprehensive

theoretical investigation of the behaviour of

reinforced concrete columns made from both

normal- and high-strength concrete. The e�ects

of creep, cracking, initial imperfections and

non-linear stress±strain curves were all taken

into account in the analysis, which covered a

wide range of load eccentricities and also the

full practical range of column slenderness. The

results from existing (BS 8110) design rules

were compared with the answers from the

accurate analysis. The results from the accurate

analysis were then used as a benchmark for

developing new design recommendations for

both normal- and high-strength concrete

columns. These new design recommendations

give more consistent and reliable results.

Research by Khalil6. The computer programs developed by

Dinku to generate graphs for eccentricity

against curvature for di�erent capacity ratios

(P=P0) were limited in their application and

were based on CP 1105 stress±strain relation-

ships for concrete and steel. Therefore it was

necessary to generalize these programs to deal

with di�erent locations of reinforcement and to

take account of long-term e�ects and also to

update them using the BS 8110 stress±strain

diagrams for concrete and steel.

7. In the test programme, eleven reinforced

concrete columns with slenderness ratios

between 18 and 63 were tested under short-term

load and eight similar columns tested under

sustained load. Details of the experimental

investigation including descriptions of test

columns, concrete properties, test procedures

and discussion of the observations made are

fully covered elsewhere.6,7 In these tests, the

experimental buckling load and mid-height

eccentricity at the point of instability were

measured and compared with theoretical pre-

dictions (Table 1).

8. In the long-term tests, a load of 60% of

short-term failure load was applied to 28-day-

old columns and maintained for a period of 90

days. If at the end of this period the column had

not failed, it was then subjected to an increas-

ing short-term load until failure. Applying this

short-term load at the end of the loading period

will tend to produce a rather higher apparent

strength than if the entire load were long term,

so the tests indicate a good agreement between

the theoretical predictions and experimental

results. Close agreement was also obtained

between theoretical and actual column de¯ec-

tions at failure.

9. To provide a general indication of the

validity and accuracy of the proposed theory,

comparisons with other published tests were

carried out. These are shown in Table 2.

The BS 8110 method10. It should be noted that the `additional

moment' theory used for slender column design

in BS 811012 is open to serious theoretical

objections (see }79±82 of Reference 3).

11. BS 8110 estimates the buckling de¯ec-

tion of a column at failure on the basis that

(a) the maximum concrete compressive strain

is 0´0035

(b) the maximum steel tensile strain is 0´002

(c) initial imperfections are allowed for by an

initial eccentricity of 0´05h unless the

applied moment exceeds this, in which case

the allowance for imperfections is zero

(d ) at higher compressive stresses, the

assumed buckling de¯ection is reduced

using an interaction formula by a factor K ,

on the assumption that the steel tensile

strain at failure will not exceed 0´002.

12. Based on this estimate of column buck-

ling de¯ection, an additional moment is calcu-

lated which is added to the applied design

moment and the column section is then

Table 1. Comparison between theoretical predictions and Khalil's test results

Short-term tests Long-term tests

Description Ptest/Ptheory etest/etheory Ptest/Ptheory etest/etheory

Mean 0´99 1´17 1´26 1´15

Range 0´86±1´22 1´01±1´38 1´05±1´53 0´97±1´33

Coe�cient of variation 0´13 0´10 0´14 0´09

Number of tests 11 8

Table 2. Comparison with published test results (Ptest/Ptheory)

Short-term tests Long-term tests

Statistical parameter Pancholi8 Dracos9 Ramu10 Goyal11

Minimum 0´72 0´83 0´91 1´00

Maximum 1´20 1´14 1´46 1´32

Mean 0´89 0´98 1´17 1´15

Standard deviation 0´13 0´09 0´14 0´07

Coe�cient of variation 0´15 0´09 0´12 0´06

Number of tests 29 36 29 20

346

Fig. 1. Overlay of

graphs to establish

slender column and

capacity

BEAL AND KHALIL


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designed to resist the total combined moment.

The section strength is calculated on the basis

of full plasticity, with a maximum concrete

strain of 0´0035 and allowable steel strains in

excess of yield.

13. However, the following points should be

noted.

(a) Buckling failure in a slender column

typically occurs at stresses and strains

well below the values associated with

fully plastic material failure.

(b) As noted earlier, the assumed concrete

strain limit of 0´0035 is for short-term

loads; under long-term loading (which is

more relevant to real structures), creep

increases the maximum strain to two to

three times this value.

(c) For 460 N/mm2 steel reinforcement, the

tensile strain at yield is 0´0023Ðnot 0´002

as assumedÐand in any case the reinforce-

ment can be stressed past its yield point,

developing strains far in excess of 0´002.

(d ) The `¯at rate' allowance of 0´05h to cover

initial imperfections in an axially loaded

column is too crude and it is a mistake to

ignore the e�ects of initial imperfections on

the strength of an eccentrically-loaded

column.

14. It is strange that the BS 8110 additional

moments are calculated on the assumption that

the steel tensile strain is always less than its

elastic limit (0´002) whereas the concrete com-

pressive strain is taken up to its plastic limit

(0´0035). To compound the problem, these

moments are then applied to a section whose

strength is calculated on the assumption that

both the steel and the concrete can be stressed

right up to their plastic limits. Then the

calculation is done on the basis of short-term

material properties, despite the fact that real

columns carry long-term loads. Thus the BS

8110 analysis is not based on a consistent,

logical assessment of column behaviour in

either elastic or plastic conditions. The failure

to allow properly for the e�ects of creep is a

serious omission.

15. Some of the assumptions underlying the

BS 8110 method are very conservative but

others are over optimistic. Therefore when com-

pared with an accurate rigorous analysis, the

results are likely to vary unpredictably, being

overconservative in some cases and possibly

unsafe in others. Most published comparisons

between the BS 8110 method and test results

consider only short-term tests and these often

show reasonable correlations, albeit with high

coe�cients of variation. However, comparisons

with long-term test results (which are more

relevant to real design) show that, in tests, a

disturbing number of failures have occurred at

loads well below those predicted by the code.

Cranston's original report on the development

of the code design method2 included a table of

experimental results from long-term load tests

and in Reference 1 these were compared with

the results from CP 110 (which is very similar

to BS 8110) and Beal's analysis. The results are

summarized in Table 3. As can be seen, the CP

110 method gave poor prediction of experi-

mental results, with a high coe�cient of varia-

tion and test columns failing at as little as 56%

of the predicted load. Beal's analysis gave much

better results, with a reduced coe�cient of

variation and test `failures' almost eliminated.

16. It should be noted that the draft Euro-

code EC2 for concrete design adopts a di�erent

formulation of the `additional moment' method

from BS 8110. The EC2 treatment is more

logical in some respects, particularly its treat-

ment of initial imperfections (where the initial

imperfection is taken as L/400), and when

compared with an accurate theoretical analysis

it gives more consistent and reasonable results

than BS 8110.13 However, it still lacks the ®rm

logical basis that is an essential requirement

for any important structural calculation.

17. In principle, the problems identi®ed in

the BS 8110 method could be dealt with by

recalculating the additional moments to take

proper account of steel post-yield behaviour

and concrete creep to give appropriate values

for use with plastic ultimate section strength

but the results would be extremely conservative

and unlikely to be acceptable. The problem is

that by the time that a column develops its full

plastic moment resistance at mid-height, it has

usually buckled so badly that it can carry very

little load.

18. As an alternative, both the buckling

de¯ection and the section strength could be

calculated on the basis of elastic theory. This

would be more logical and appropriate, as

buckling usually occurs at stresses which are in

the elastic range, so an elastic analysis would

probably give better results than plastic theory.

However, the results would still be conservative

and concrete designers would need to relearn

elastic theory, which is distinctly unfashionable

among present-day engineers.

19. The problem is that the buckling be-

haviour of a reinforced concrete column is too

complex to be accurately modelled by a simple

theoretical analysis. To achieve better results

than present-day codes, a semi-empirical

approach is the most realistic way forward. It

Table 3. Ratio (test/theory) for CP110/BS 8110 and

Beal's analysis compared with Cranston's published

long-term test results1

Method Mean Range

Coe�cient of

variation

CP 110/BS 8110 0´94 0´56±1´42 0´22

Beal's analysis 1´28 0´97±1´97 0´16

347

DESIGN OF NORMAL-

AND HIGH-STRENGTH

CONCRETE COLUMNS


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would be di�cult and costly to test all possible

types of column and loading in the laboratory.

However, if a practical, accurate method of

theoretical analysis is established and proved

to give satisfactory results, it is then possible

to carry out a comprehensive investigation of

column behaviour and the e�ects all the various

factors can have on column strength can be

analysed and quanti®ed. The results of the

accurate theoretical analysis will then provide a

benchmark against which the results from

various design rules can be assessed and the

design methods can be optimized by adjusting

the values of key parameters in order to give

the best results.

Safety factors20. Before proceeding further, it should also

be noted that there are problems in applying

the partial factor system favoured by current

limit state codes to the analysis and design of

slender concrete columns. Codes such as BS

8110 and EC2 divide the structure's safety

factor into two parts: a load factor gL applied to

loads; and a materials factor gm applied to the

material strength (e.g. concrete cube strength or

steel yield stress). However, Young's modulus

for concrete is proportional only to the square

root of the cube strength and for steel it does

not vary with yield stress at all. Thus in these

codes there is e�ectively only a reduced partial

safety factor applied to concrete sti�ness and

no factor at all on the sti�ness of reinforcing

steel. As the strength of a slender column

depends primarily on its sti�ness, BS 8110's

partial factor theory would lead to the conclu-

sion that a slender column should have a lower

safety factor than a stocky column, which is

surely wrong.

21. Buckling failure can occur suddenly,

with little warning and minimal scope for

bene®cial redistribution of load and there is a

strong case for saying that the safety factor

against this type of failure should be at least as

high as that for other more ductile failure

modes such as ¯exural bending. To achieve a

consistent safety factor, it is necessary to adopt

either a global overall safety factor on the

column strength (as used in permissible stress

codes), or, if partial factors are to be used, the

`materials' factor should be applied to the

member strength (as in the American `resistance

factor') rather than to the material strength

(as in the BS 8110 and EC2 `materials

factors').

22. (It should be noted that only part of BS

8110's `material factor' for 1´5 on concrete

should be considered as a safety factor. Because

of inferior placing and curing conditions, con-

crete in real structures has typically only about

75% of the strength of laboratory test cubes.14

The e�ective safety factor on concrete in a real

BS 8110 design is thus about 1´56 0´75= 1´12,

which compares with the steel partial factor of

1´15 (reduced to 1´05 in the most recent revision

of BS 8110).)

Analysis of normal- and high-strengthconcrete columns23. Following the research work by Khalil

outlined earlier, the graphical analysis method

was used to investigate slender column beha-

viour over the full realistic range of concrete

strengths, including high-strength concrete.

The theoretical column load capacity was

calculated for a wide range of slenderness

ratios and load eccentricities, taking into

account the e�ects of initial imperfections and

concrete creep.

24. The analysis covered concrete cube

strengths from 20 N/mm2 to 100 N/mm2. The

BS 811012 stress±strain curve (Fig. 2) was used

to de®ne concrete short-term behaviour, but for

concrete with cube strengths over 60 N/mm2 the

peak compressive strain was limited to

e=0´00357 (( fcu7 60)/50 000). As columns in

real structures generally support predominantly

long-term loads, this was the condition con-

sidered in the analysis. The concrete short-term

stress±strain curve was modi®ed for long-term

loading by the creep factor f, with a value of

2´0 for cube strengths up to 30 N/mm2. For

higher concrete strengths this was reduced to

fhsc = fp(40/( fcu + 10)), giving creep factors

ranging from 2´00 for 20 N/mm2 concrete to 1´21

for 100 N/mm2 concrete. Combining the various

factors, the concrete ultimate long-term com-

pressive failure strain ranged from 0´0105 for

20 N/mm2 concrete to 0´00595 for 100 N/mm2

concrete.

Parabolic curve

5·5√fcu/γm kN/mm2

2·4 × 10–4√fcu/γm

0·67 fcu/γm

Str

ess

Strain 0·0035

fy/γm N/mm2

fy/γm N/mm2

200 kN/mm2

Strain

Tension

Compression

Str

ess

348

Fig. 2. Short-term

design stress±strain

curve for normal-

weight concrete

according to BS 8110:

Part 1 (fCu is in

N/mm2)

Fig. 3. Short-term

design stress±strain

curve for

reinforcement

according to BS 8110:

Part 1 ( fy is in

N/mm2)

BEAL AND KHALIL


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25. Steel behaviour was assumed to follow

the BS 8110 stress±strain curve (see Fig. 3) and

allowance was made in the analysis for dis-

placement of concrete by the steel. The section

calculations were based on a 4 bar reinforce-

ment arrangement with d � 0�8h, which should

reasonably cover most real columns without

being unduly conservative.

26. Calculations were run using a spread-

sheet which was developed to run on Lotus 123

or Microsoft Works to calculate the moment±

curvature relationship for each section at

various values of axial load. This spreadsheet

is designed to allow any value of concrete

strength, creep factor or concrete peak strain

to be inserted, reinforcement e�ective depth to

be varied and either 4 bar or 6 bar reinforce-

ment arrangements considered. To use the

spreadsheet, material properties and a value

of peak concrete strain are ®rst selected and

then the neutral axis position is varied (by trial

and error) until the load eccentricity and

curvature are shown which correspond to the

required value of axial load (P � 0�1Po, 0�2Po,

etc., where Po is the section ultimate

compressive strength). By repeating the

calculations for a range of neutral axis

positions and peak strain values, families of

curves can be generated which de®ne the

section ¯exural behaviour under di�erent axial

loads.

27. The allowance for column initial imper-

fections was taken as an initial bow of 0´002L,

which in the absence of better information

was felt to be a reasonable value for use in

practical design. This is equivalent to an initial

bow of 6 mm on a braced 3 m high column, or

12 mm out of plumb on a 3 m high unbraced

column.

28. As previously described, graphs were

prepared, plotting the results of the section

behaviour calculations as load eccentricity (e/h)

against section curvature (h/R) for various

values of axial load P (expressed as a propor-

Table 4(a). Ratio of slender column capacity to short column `squash' capacity (P=Po) for 20 N/mm2 and

r= 0´8% (see Fig. 4(a))

20 N/mm2 r=0´8% d=0´8h

L/h e=0 e=0´1 e=0´2 e=0´3 e=0´4 e=0´5

0 1´00 1´00 1´00 1´00 1´00 1´00

5 0´93 0´89 0´91 0´95 0´95 0´91

10 0´84 0´80 0´75 0´78 0´77 0´75

15 0´73 0´64 0´59 0´61 0´61 0´58

20 0´58 0´47 0´44 0´44 0´46 0´45

25 0´41 0´33 0´31 0´35 0´34 0´37

30 0´30 0´23 0´23 0´25 0´28 0´29

40 0´16 0´13 0´13 0´16 0´18 0´21

50 0´1 0´085 0´09 0´11 0´12 0´15

60 0´055 0´055 0´065 0´065 0´08 0´095

Table 4(b). Ratio of slender column capacity to short column `squash' capacity (P=Po) for 20 N/mm2 and r= 4%

(see Fig. 4(b))

20 N/mm2 r=4% d=0´8h

L/h e=0 e=0´1 e=0´2 e=0´3 e=0´4 e=0´5

0 1´00 1´00 1´00 1´00 1´00 1´00

5 0´93 0´93 0´93 0´93 0´98 1´00

10 0´85 0´83 0´81 0´83 0´86 0´90

15 0´75 0´71 0´70 0´71 0´74 0´79

20 0´61 0´56 0´57 0´60 0´62 0´65

25 0´44 0´43 0´46 0´48 0´53 0´56

30 0´32 0´34 0´36 0´40 0´43 0´48

40 0´18 0´20 0´24 0´26 0´30 0´34

50 0´11 0´14 0´17 0´20 0´20 0´23

60 0´08 0´09 0´11 0´14 0´14 0´17

1·0

0·8

0·6

0·4

0·2

0

P/P

0

0 10 20 30 40 50 60L/h(a)

0 10 20 30 40 50 60L/h(b)

20 N/mm2, 0·8% 20 N/mm2, 4%

00·10·30·5

e/h 00·10·30·5

e/h

349

Fig. 4. Relationship

between slender

column capacity

L/h and short

column `squash'

capacity P=Po for:

(a) 20 N/mm2,

r = 0´8%; and

(b) 20 N/mm2 and

r= 4%

DESIGN OF NORMAL-

AND HIGH-STRENGTH

CONCRETE COLUMNS


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tion of the section strength in pure compres-

sion, Po). Another graph was then prepared (on

tracing paper) showing the relationship

between buckling de¯ection and section curva-

ture for various slenderness ratios, assuming

that the buckled shape of the column follows a

sine curve: D=h � �L=h�2�h=R�=p2 (see equation

(1)). By overlaying this buckling de¯ection

graph on the section behaviour graphs, the

column capacities for various slenderness ratios

could be read o� directly.

29. As noted earlier, where applied

moments are signi®cant, for an exact solution

an iterative approach is required which takes

into account the di�erence between the circular

curvature induced by the applied end moments

and the sinusoidal curvature induced by buck-

ling. The de¯ection induced by circular curva-

ture is

Dh� 1

8Lh

� �2 hR

�2�

The results presented in this paper have all

been calculated by the exact iterative method.

30. The results of the theoretical analysis

are summarized in Tables 4±8 (Figs 4±8 refer).

These are expressed as the ratio of slender

column capacity to short column `squash'

capacity (P=Po). These results show the e�ect

that buckling and initial imperfections have on

the load capacity (with no safety factors

applied) for each column analysed. They give a

precise analytical benchmark against which

proposed design methods for slender columns

can be assessed.

31. In carrying out the analysis, one inter-

esting point was apparent at the outset: when

high strength concrete is used, an eccentrically

loaded section may be able to carry a greater

long-term load than its short-term capacity,

because the concrete's limited short-term strain

capacity restricts the steel stress which can be

developed. As it is usual to calculate the


r= 0´8% (see Fig. 5(a))

40 N/mm2 r=0´8% d=0´8h

L/h e=0 e=0´1 e=0´2 e=0´3 e=0´4 e=0´5

0 1´00 1´00 1´00 1´00 1´00 1´00

5 0´92 0´91 0´88 0´91 0´84 0´86

10 0´80 0´74 0´71 0´69 0´64 0´66

15 0´66 0´58 0´50 0´49 0´46 0´49

20 0´50 0´38 0´35 0´33 0´35 0´37

25 0´35 0´24 0´23 0´25 0´26 0´31

30 0´25 0´18 0´18 0´18 0´20 0´27

40 0´12 0´10 0´10 0´11 0´13 0´16

50 0´06 0´06 0´06 0´065 0´085 0´08

60 0´04 0´04 0´04 0´045 0´06 0´08


(see Fig. 5(b))

40 N/mm2 r=4% d=0´8h

L/h e=0 e=0´1 e=0´2 e=0´3 e=0´4 e=0´5

0 1´00 1´00 1´00 1´00 1´00 1´00

5 0´92 0´91 0´92 0´96 0´98 1´00

10 0´83 0´78 0´77 0´80 0´86 0´87

15 0´70 0´65 0´62 0´67 0´71 0´75

20 0´56 0´51 0´51 0´55 0´58 0´63

25 0´39 0´36 0´39 0´43 0´45 0´51

30 0´29 0´27 0´30 0´34 0´37 0´39

40 0´15 0´17 0´20 0´23 0´25 0´27

50 0´09 0´10 0´12 0´14 0´18 0´18

60 0´06 0´08 0´09 0´11 0´13 0´13

1·0

0·8

0·6

0·4

0·2

0

P/P

0

0 10 20 30 40 50 60L/h(a)

0 10 20 30 40 50 60L/h(b)

40 N/mm2, 4%

00·10·30·5

e/h 00·10·30·5

e/h

40 N/mm2, 0·8%

350


between slender

column capacity

L/h and short

column `squash'

capacity P=Po for:

(a) 40 N/mm2,

r = 0´8%; and

(b) 40 N/mm2,

r = 4%

BEAL AND KHALIL


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strength of a reinforced concrete section on the

basis of short-term loading, short-term section

strengths have been used for the comparisons

in Tables 4±8.

Revising BS 811032. A comparison between the results of the

accurate analysis and designs in accordance

with the BS 8110 rules is presented in Table

9(a) and Fig. 9(a) and Table 9(b) and Fig. 9(b).

(The ratios give the strength of the slender

column relative to that of a column with

L/h=0.) This is done for an axially loaded

column made from 80 N/mm2 concrete, with

r=0´8% and 4%, taking into account the

additional moment for buckling in the BS 8110

calculation and also the factor K which reduces

this at high axial loads. The optimum value of

K has been calculated iteratively in accordance

with BS 8110. A K=1 line is also shown added

in Tables 9(a) and (b) to show the e�ect of

setting the factor K as a constant 1´0, instead of

being reduced by iteration.

33. As can be seen, the BS 8110 `exact'

analysis overestimates the strength of a column

with 0´8% steel by up to 95% and for 4% steel

it overestimates the strength by up to 80%. The

worst problems are with slenderness ratios

around L/h=20, where the BS 8110 factor K

acts to eliminate buckling e�ects from the

calculation, leading to a calculated column

strength which is far in excess of the true

value. If setting K=1 in all cases, the results

from the BS 8110 become more realistic but still

the column strength is overestimated by 10±

30% for slenderness ratios of 10 and above.

34. Table 10 shows the adjusted values of

additional load eccentricity (including allow-

ance for initial imperfections) which would be

required to give results which agree with the

rigorous theoretical analysis. As can be seen,

the additional eccentricities which are required

to match the results of the accurate analysis are

generally higher than the present values


r= 0´8% (see Fig. 6(a))

60 N/mm2 r=0´8% d=0´8h

L/h e=0 e=0´1 e=0´2 e=0´3 e=0´4 e=0´5

0 1´00 1´00 1´00 1´00 1´00 1´00

5 0´91 0´90 0´89 0´87 0´80 0´77

10 0´79 0´72 0´66 0´62 0´54 0´57

15 0´63 0´53 0´45 0´40 0´40 0´44

20 0´46 0´35 0´29 0´28 0´29 0´36

25 0´31 0´23 0´18 0´18 0´22 0´28

30 0´22 0´15 0´15 0´15 0´18 0´19

40 0´11 0´075 0´065 0´08 0´095 0´12

50 0´05 0´05 0´04 0´055 0´065 0´07

60 0´04 0´03 0´025 0´035 0´05 0´05


(see Fig. 6(b))

60 N/mm2 r=4% d=0´8h

L/h e=0 e=0´1 e=0´2 e=0´3 e=0´4 e=0´5

0 1´00 1´00 1´00 1´00 1´00 1´00

5 0´93 0´91 0´92 0´97 1´00 1´00

10 0´81 0´75 0´76 0´78 0´84 0´85

15 0´66 0´60 0´59 0´63 0´68 0´69

20 0´51 0´45 0´45 0´49 0´55 0´55

25 0´37 0´32 0´35 0´38 0´42 0´41

30 0´26 0´23 0´26 0´27 0´33 0´35

40 0´13 0´14 0´16 0´18 0´20 0´22

50 0´075 0´09 0´11 0´13 0´14 0´17

60 0´055 0´065 0´075 0´085 0´10 0´13

1·0

0·8

0·6

0·4

0·2

0

P/P

0

0 10 20 30 40 50 60L/h(a)

0 10 20 30 40 50 60L/h(b)

60 N/mm2, 0·8% 60 N/mm2, 4%

00·10·30·5

e/h 00·10·30·5

e/h

351


between slender

column capacity

L/h and short

column `squash'

capacity P=Po for:

(a) 60 N/mm2,

r = 0´8%; and

(b) 60 N/mm2,

r = 4%

DESIGN OF NORMAL-

AND HIGH-STRENGTH

CONCRETE COLUMNS


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recommended in BS 8110. However, it is inter-

esting that for axially loaded columns with

slenderness up to L/h=25, the required value

of additional load eccentricity is fairly constant

for all values of concrete strength and

reinforcement percentage. Above L/h=25, the

results diverge, with more heavily reinforced

columns requiring higher additional moments

than lightly reinforced columns for satisfactory

results. However, if a degree of conservatism

can be accepted in the design of lightly

reinforced slender columns, a single set of

additional moments could be speci®ed which

would give satisfactory and safe results for

columns of all concrete strengths and reinforce-

ment percentages. The proposed values of

eccentricity (Table 11) can be calculated from

the formula

eaddh� 0�05� 0�00023 L

h

� �2�3�3�

Reduction factor method35. An alternative approach for slender

column design is to use simple capacity reduc-

tion factors to cover the e�ects of buckling. As

can be seen from Tables 12 and 13, a single set

of load reduction factors can be speci®ed for

each concrete strength and this gives a simple

design method which gives rather more eco-

nomical and consistent results for axially

loaded columns than the additional moment

approach. (Note: the values quoted would need

to be adjusted to re¯ect the di�erence between

site concrete strength and laboratory test

cubesÐthey equate to `speci®ed cube strength'

values of 130 N/mm2, 80 N/mm2, 50 N/mm2 and

25 N/mm2 respectively.)

Eccentric loading36. There remains the problem of design for

eccentric loading. As can be seen from the

tables and graphs which show the results of

the accurate analysis, heavily reinforced


r= 0´8% (see Fig. 7(a))

80 N/mm2 r=0´8% d=0´8h

L/h e=0 e=0´1 e=0´2 e=0´3 e=0´4 e=0´5

0 1´00 1´00 1´00 1´00 1´00 1´00

5 0´92 0´90 0´89 0´87 0´79 0´83

10 0´79 0´73 0´64 0´58 0´50 0´56

15 0´62 0´50 0´42 0´38 0´36 0´45

20 0´45 0´34 0´26 0´25 0´28 0´36

25 0´31 0´22 0´16 0´18 0´21 0´27

30 0´21 0´15 0´12 0´13 0´14 0´18

40 0´09 0´08 0´07 0´065 0´085 0´12

50 0´06 0´06 0´035 0´05 0´05 0´09

60 0´04 0´025 0´025 0´035 0´035 0´06


(see Fig. 7(b))

80 N/mm2 r=4% d=0´8h

L/h e=0 e=0´1 e=0´2 e=0´3 e=0´4 e=0´5

0 1´00 1´00 1´00 1´00 1´00 1´00

5 0´92 0´91 0´93 0´97 1´00 1´00

10 0´79 0´75 0´76 0´81 0´85 0´86

15 0´64 0´58 0´59 0´62 0´65 0´66

20 0´49 0´42 0´45 0´50 0´51 0´52

25 0´34 0´30 0´33 0´36 0´37 0´42

30 0´23 0´22 0´24 0´28 0´31 0´31

40 0´12 0´13 0´15 0´17 0´20 0´24

50 0´07 0´08 0´095 0´12 0´14 0´16

60 0´05 0´055 0´07 0´09 0´085 0´10

1·0

0·8

0·6

0·4

0·2

0

P/P

0

0 10 20 30 40 50 60L/h(a)

0 10 20 30 40 50 60L/h(b)

80 N/mm2, 0·8% 80 N/mm2, 4%

00·10·30·5

e/h 00·10·30·5

e/h

352


between slender

column capacity

L/h and short

column `squash'

capacity P=Po for:

(a) 80 N/mm2,

r = 0´8%; and

(b) 80 N/mm2,

r = 4%

BEAL AND KHALIL


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slender columns (r=4%) behave reasonably

consistently as the applied load eccentricity

varies (although the reduction factor does not

fall with increasing load eccentricity, as normal

interaction formulae would predict). Therefore

for these columns the same reduction factors or

additional moments already determined for the

design of axially loaded columns would also be

suitable for eccentric loading. However, for

lightly reinforced columns the load capacities

under eccentric loading are more seriously

reduced by the e�ects of slenderness. This issue

is not addressed by existing codes (except the

IStructE Recommendations for the Permissible

Stress Design of Reinforced Concrete Building

Structures15).

37. Table 14 compares the P=Po values

for the proposed `additional moment' and

`reduction factor' methods with the results

of the accurate analysis for an eccentrically

loaded 100 N/mm2 column with r=0´8%

under both short-term and long-term loading.

38. As can be seen, both the reduction factor

method and the additional moment method

considerably overestimate the strength of the

column. In principle, this could be a serious

problem: lightly reinforced columns are often

subjected to some bending. However, it is worth

thinking about the circumstances which lead to

the development of moments in concrete

columns.

39. In a braced frame, column moments

generated by beams and slabs occur primarily

at the column ends, away from the maximum

buckling de¯ection. Furthermore, if any buck-

ling deformation does occur in the column, this

will act to reduce the end moments. Therefore it

is reasonable to assume that for columns in

braced frames the problem is unlikely to have

serious consequences. In sway frames, the main

source of column moments is commonly wind

load, which is a short-term loading, so the

e�ects of creep will be less relevant. In normal

strength concrete, the higher short-term


r= 0´8% (see Fig. 8(a))

100 N/mm2 r=0´8% d=0´8h

L/h e=0 e=0´1 e=0´2 e=0´3 e=0´4 e=0´5

0 1´00 1´00 1´00 1´00 1´00 1´00

5 0´92 0´93 0´85 0´78 0´75 0´72

10 0´79 0´74 0´65 0´48 0´46 0´48

15 0´60 0´51 0´38 0´34 0´32 0´39

20 0´43 0´34 0´24 0´26 0´25 0´30

25 0´30 0´20 0´15 0´17 0´16 0´18

30 0´20 0´14 0´11 0´11 0´11 0´12

40 0´09 0´07 0´055 0´055 0´055 0´06

50 0´055 0´035 0´035 0´035 0´035 0´035

60 0´04 0´02 0´02 0´02 0´02 0´025

Table 8(b). Ratio of slender column capacity to short column `squash' capacity (P=Po) for 100 N/mm2 and

r= 4% (see Fig. 8(b))

100 N/mm2 r=4% d=0´8h

L/h e=0 e=0´1 e=0´2 e=0´3 e=0´4 e=0´5

0 1´00 1´00 1´00 1´00 1´00 1´00

5 0´92 0´95 0´96 1´00 1´00 1´00

10 0´79 0´78 0´79 0´83 0´85 0´87

15 0´63 0´60 0´61 0´64 0´69 0´66

20 0´47 0´42 0´44 0´46 0´50 0´50

25 0´33 0´29 0´31 0´34 0´38 0´39

30 0´23 0´22 0´24 0´27 0´28 0´31

40 0´125 0´13 0´15 0´17 0´19 0´23

50 0´07 0´075 0´09 0´11 0´13 0´15

60 0´05 0´055 0´065 0´075 0´095 0´095

1·0

0·8

0·6

0·4

0·2

0

P/P

0

0 10 20 30 40 50 60L/h(a)

0 10 20 30 40 50 60L/h(b)

100 N/mm2, 0·8%100 N/mm2, 4%

00·10·30·5

e/h 00·10·30·5

e/h

353


between slender

column capacity

L/h and short

column `squash'

capacity P=Po for:

(a) 100 N/mm2,

r = 0´8%; and

(b) 100 N/mm2,

r = 4%

DESIGN OF NORMAL-

AND HIGH-STRENGTH

CONCRETE COLUMNS


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sti�ness generally increases the column

strength su�ciently to compensate for the

theoretical strength shortfall under eccentric

loading, so again there is unlikely to be a

serious problem.

40. The problem is only serious in normal-

strength concrete if a lightly reinforced column

resists a permanent applied moment from (say)

a cantilever beam, or a permanent lateral load.

41. Therefore the additional moment

method produces reasonable results for lightly

reinforced columns in braced frames, or resist-

ing wind sway moments but can overestimate

their strength when resisting permanent

Table 9(a). Slender column strength compared with section strength (80 N/mm2, 0´8%)

L/h

5 10 15 20 25 30 40 50 60

Theory 0´92 0´79 0´62 0´45 0´31 0´21 0´09 0´06 0´04

BS 8110 (exact K) 0´88 0´88 0´88 0´88 0´40 0´24 0´07 0´04 0´02

BS 8110 (K=1) 0´88 0´88 0´74 0´58 0´40 0´24 0´07 0´04 0´02

Table 9(b). Slender column strength compared with section strength (80 N/mm2, 4%)

L/h

5 10 15 20 25 30 40 50 60

Theory 0´92 0´79 0´64 0´49 0´34 0´23 0´12 0´07 0´05

BS 8110 (exact K) 0´88 0´88 0´88 0´88 0´56 0´32 0´18 0´11 0´07

BS 8110 (K=1) 0´88 0´88 0´74 0´58 0´43 0´32 0´18 0´11 0´07

Table 10. Adjusted values of additional load eccentricity (including allowance for initial imperfections) required

to give results which agree with theoretical analysis

BS 8110

Add. ecc. required to match accurate analysis

total add.100 N/mm2 60 N/mm2 40 N/mm2 20 N/mm2

L/h ecc. (K=1) 0´8% 4´0% 0´8% 4´0% 0´8% 4´0% 0´8% 4´0%

5 0´05 0´03 0´03 0´04 0´03 0´04 0´03 0´03 0´03

10 0´05 0´08 0´07 0´10 0´08 0´09 0´07 0´07 0´06

15 0´11 0´17 0´15 0´18 0´17 0´17 0´14 0´13 0´11

20 0´20 0´27 0´25 0´28 0´27 0´27 0´24 0´22 0´20

25 0´31 0´37 0´39 0´40 0´42 0´39 0´41 0´37 0´37

30 0´45 0´47 0´57 0´49 0´64 0´49 0´60 0´51 0´57

40 0´80 0´68 0´95 0´72 1´25 0´77 1´30 0´79 1´22

50 1´25 0´85 1´67 1´25 1´88 1´36 1´89 1´21 1´94

60 1´80 1´04 1´97 1´53 2´57 1´80 2´90 1´89 2´87

Table 11. Proposed revised values of additional

eccentricity

L/h

Additional

eccentricity

5 0´06

10 0´10

15 0´17

20 0´28

25 0´43

30 0´62

40 1´16

50 1´91

60 2´88

354

Fig. 9. BS 8110

comparison for:

(a) 80 N/mm2,

r = 0´8%; and

(b) 80 N/mm2,

r = 4%

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applied moments. However, as Table 14 shows,

for lightly reinforced high-strength concrete

columns designed by the reduction factor

method there is a problem, even under wind

loads.

42. It should be noted that the proposed

design rules are generally more conservative

than the present BS 8110 recommendations,

so an eccentrically loaded high-strength

concrete column designed to BS 8110 could be

quite seriously underdesigned. Further work is

necessary to work out a full solution to this

problem but the proposed design rules, outlined

later, include recommendations to ensure

acceptable results. At present, it would be

prudent to draw attention to the situation and

to recommend lower factors for the reduction

factor method in situations where a column is

required to resist permanent moments other

than the incidental end moments associated

with normal frame action. These reduced

values of reduction factor could be based on

the factors for e=0´3h and r=0´8%, with

values for other load eccentricities and re-

inforcement percentages being obtained by

interpolation. For the additional moment

method, `additional additional moments' are

necessary.

Conclusions43. Analysis of the theoretical behaviour of

columns under long-term loading shows that

the existing recommendations of BS 8110 are

seriously inadequate. It is proposed that recom-

mendations for slender column design are

revised and based on either modi®ed additional

moments, or else on an alternative approach

based on column capacity reduction factors. In

either case, caution is needed where a lightly

reinforced high-strength column is subjected to

applied moments. Recommendations for both

design methods are summarized in the follow-

ing subsections.

Recommendations for design methods

44. It should be noted that where design

concrete strength exceeds 60 N/mm2, greater

minimum links and minimum reinforcement are

required than for normal strength concrete.16

Additional moment method

45. Design of slender columns can be

carried out satisfactorily using the additional

moment method of BS 8110 if the additional

moments in Table 15 are used in place of those

recommended in the Code. Table 15 includes an

allowance for initial imperfections, so the Code

allowance of 0´05h need not be applied and the

Code factor K should be set as 1´0 in all cases.

The values of additional load eccentricity

proposed in Table 15 replace those in BS 8110

and are suitable for all concrete strengths from

20 N/mm2 to 100 N/mm2. They are suitable for

the design of axially loaded columns and also

Table 12. Capacity reduction factors

100 N/mm2 60 N/mm2 40 N/mm2 20 N/mm2

L/h 0´8% 4´0% 0´8% 4´0% 0´8% 4´0% 0´8% 4´0%

5 0´92 0´92 0´91 0´93 0´92 0´92 0´93 0´93

10 0´79 0´79 0´79 0´81 0´80 0´83 0´84 0´85

15 0´60 0´63 0´63 0´66 0´66 0´70 0´73 0´75

20 0´43 0´47 0´46 0´51 0´50 0´56 0´58 0´61

25 0´30 0´33 0´31 0´37 0´35 0´39 0´41 0´44

30 0´20 0´23 0´22 0´26 0´25 0´29 0´30 0´32

40 0´09 0´125 0´11 0´13 0´12 0´15 0´16 0´18

50 0´055 0´07 0´05 0´075 0´06 0´09 0´10 0´11

60 0´04 0´05 0´04 0´055 0´04 0´06 0´055 0´08

Table 13. Proposed reduction factors (axial loads)

L/h 100 N/mm2 60 N/mm2 40 N/mm2 20 N/mm2

5 0´92 0´92 0´92 0´93

10 0´79 0´79 0´80 0´84

15 0´60 0´63 0´66 0´73

20 0´43 0´46 0´50 0´58

25 0´30 0´32 0´35 0´41

30 0´20 0´22 0´25 0´30

40 0´09 0´11 0´12 0´16

50 0´06 0´06 0´06 0´10

60 0´04 0´04 0´04 0´06

Table 14. P=Po for eccentrically loaded column (100 N/mm2, 0´8%)

True column capacity

L/h

(e=0)

Red. factor

e=0´3h

long term

e=0´3h

short term

Proposed

add. mom. method

5 0´92 0´78 0´85 0´88

10 0´79 0´48 0´70 0´70

15 0´60 0´34 0´46 0´52

20 0´43 0´26 0´31 0´32

25 0´30 0´17 0´22 0´21

30 0´20 0´11 0´15 0´13

40 0´09 0´055 0´075 0´065

50 0´055 0´035 0´05 0´035

60 0´04 0´02 0´04 0´025

355

DESIGN OF NORMAL-

AND HIGH-STRENGTH

CONCRETE COLUMNS


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for columns subjected to normal frame moments

in a braced frame, or columns in a sway frame

which is subjected to wind moments. Where a

column is required to resist a signi®cant long-

term moment in the region of its maximum

buckling de¯ection (i.e. the mid-height of a

braced column or the ends of an unbraced

column), 4% reinforcement should always be

provided. A formula for the proposed additional

eccentricities is

eaddh� 0�05� 0�00023 L

h

� �2�3�4�

Reduction factor method

46. If a capacity reduction factor method is

preferred, the following factors may be used.

The results will tend to be rather more

economical than those from the additional

moment method for normal strength concrete

but care is needed for columns of high-strength

concrete. The quoted factors apply to

(a) columns with 4% reinforcement for

all conditions of slenderness and load

eccentricity

(b) all columns with 0´8% reinforcement sub-

jected to axial loads and to moments

induced by beam bending in a braced frame

(c) wind sway moments in columns with a

design concrete cube strength not exceed-

ing 50 N/mm2.

For other conditions (i.e. for lightly reinforced

columns subjected to long-term moments which

cannot be relieved by redistribution and also

for lightly reinforced columns of high-strength

concrete in sway frames) the reduced factors

quoted in brackets should be applied. These

reduced factors apply for r=0´8% and for load

eccentricity 50´3h. Values for e4 0´3h and

r=between 0´8% and 4%, may be obtained by

interpolation. (Table 16 refers.)

References1. BEALEAL A. N. The design of slender columns.

Proceedings of the Institution of Civil Engineers,

Part 2, 1986, 81, September, 397±414.

2. CRANSTONRANSTON W. B. Analysis and design of re-

inforced concrete columns. Research Report 20.

Cement and Concrete Association, Slough, 1972.

3. Discussion: Proceedings of the Institution of Civil

Engineers, Part 2, 1987, 83, June, 483±496.

4. DINKUINKU A. Design of Pin-ended Slender Columns.

MSc dissertation, University of Leeds, October

1987.

5. BRITISHRITISH STANDARDSTANDARDS INSTITUTIONNSTITUTION. CP 110: Part 1.

The Structural Use of Concrete: Design, Materials

and Workmanship. BSI, London, 1972.

6. KHALILHALIL N. J. Tests on Slender Reinforced Concrete

Columns (in press).

7. KHALILHALIL N. J. Slender Reinforced Concrete

Columns. PhD thesis, University of Leeds,

September 1991.

8. PANCHOLIANCHOLI V. R. The Instability of Slender

Reinforced Concrete Columns. PhD thesis,

University of Bradford, September 1997.

9. DRACOSRACOS A. Long Slender Reinforced Concrete

Columns. PhD thesis, University of Bradford,

August 1982.

10. RAMUAMU P., GRENACHERRENACHER M., BAUMANNAUMANN M. and

THURLIMANNHURLIMANN B. Versuche an Gelenkig Gelagerlen

Stahlbetonstutzen Unterdauerlast (Long-term

Tests on Pin-ended Reinforced Concrete Columns).

Institute fur Baustatik Eidgenossische Tech-

nische Hochschule, Zurich, May 1969, Bericht No.

6418-1.

11. GOYALOYAL B. B. Ultimate Strength of Reinforced

Concrete Columns under Sustained Load . PhD

thesis, University of Dundee, 1970.

Table 15. Proposed revised additional eccentricity values for BS 8110 additional moment method

L/h

5 10 15 20 25 30 40 50 60

Add. eccentricity 0´06 0´10 0´17 0´28 0´43 0´62 1´16 1´91 2´88

Table 16. Proposed design values for capacity reduction factors

Proposed reduction factors

L/h 120 N/mm2 80 N/mm2 50 N/mm2 25 N/mm2

5 0´92 (0´78) 0´92 (0´87) 0´92 (0´91) 0´93 (0´95)

10 0´79 (0´48) 0´79 (0´62) 0´80 (0´69) 0´84 (0´78)

15 0´60 (0´34) 0´63 (0´40) 0´66 (0´49) 0´73 (0´61)

20 0´43 (0´26) 0´46 (0´28) 0´50 (0´33) 0´58 (0´44)

25 0´30 (0´17) 0´32 (0´18) 0´35 (0´25) 0´41 (0´35)

30 0´20 (0´11) 0´22 (0´15) 0´25 (0´18) 0´30 (0´25)

40 0´09 (0´06) 0´11 (0´08) 0´12 (0´11) 0´16 (0´16)

50 0´06 (0´04) 0´06 (0´06) 0´06 (0´07) 0´10 (0´11)

60 0´04 (0´02) 0´04 (0´04) 0´04 (0´05) 0´06 (0´07)

356

BEAL AND KHALIL


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On: Thu, 09 Jun 2011 09:26:49

12. BRITISHRITISH STANDARDSTANDARDS INSTITUTIONNSTITUTION. BS 8110: Part 1.

Structural Use of Concrete: Code of Practice

for Design and Construction. BSI, London,

1985.

13. BEALEAL A. N. Draft Eurocode 2: Is this the future of

concrete design?. Proceedings of the Institution of

Civil Engineers, Structures and Buildings, 1993,

99, Nov., 337±385.

14. PLOWMANLOWMAN et al. Cores, cubes and the speci®ed

strength of concrete. The Structural Engineer,

1974, 52, No. 11.

15. INSTITUTIONNSTITUTION OFOF STRUCTURALTRUCTURAL ENGINEERSNGINEERS. Recom-

mendations for the Permissible Stress Design of

Reinforced Concrete Building Structures. IStructE,

London, 1985.

16. CONCRETEONCRETE SOCIETYOCIETY. Design Guidance for High

Strength Concrete. Concrete Society, London,

1998, Technical Report 49.

Please email, fax or post your discussion contributions to The Publisher: email:

[email protected]; fax: 0171 799 1325; or post to Lesley Wilson, Journals

Department, Institution of Civil Engineers, 1±7 Great George Street, London

SW1P 3AA.

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