HOW TO DESIGN CONCRETE STRUCTURES Columns
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How to design concrete structures using Eurocode 2
5. Columns Introduction
Designing to Eurocode 2
This should be redrafted as appropriate in each country
This guide is intended to assist engineers with the design of columns 1 and walls to Eurocode 2 . It sets out a design procedure to follow and gives useful commentary on the provisions within the Eurocode. Eurocode 2 does not contain the derived formulae; this is because it has been European practice to give principles and general application rules in the codes and for detailed application rules to be presented in other sources such as textbooks or guidance documents. The first guide in this series, How to design concrete structures using 2 Eurocode 2: Introduction , provides an overview of Eurocodes, including terminology.
Where NDPs occur in the text in this publication, recommended values in EN 1992 are used and highlighted in yellow. The UK values have been used for NDPs embedded in figures and charts and the relevant NDPs are scheduled separately to assist other users in adapting the www.eurocode2.info.) A figures and charts. (Derivations can be found at www.eurocode2.info.) full list of symbols related to column design is given at the end of this guide.
Design procedure A procedure for carrying out the detailed design of braced columns (i.e. columns that do not contribute to resistance of horizontal actions) is shown in Table 1. This assumes that the column dimensions have previously been determined during conceptual design or by using quick design methods. Column sizes should not be significantly different from those obtained using current practice. Steps 1 to 4 of Table 1 are covered by earlier guides in this series and the next step is therefore to consider fire resistance.
Fire resistance 3
Eurocode 2, Part 1–2: Structural fire design , gives a choice of advanced, simplified or tabular methods for determining determining fire resistance of columns. Using tables is the fastest method for determining the minimum dimensions and cover for columns. There are, however, some restrictions and if t hese apply further guidance can be obtained from specialist literature. The simplified method may give more economic columns, especially for small columns and/or high fire resistance periods. Rather than giving a minimum cover, the tabular method is based on nominal axis distance, a (see Figure 1). This is the distance from the centre of the main m ain reinforcing bar to the surface of the member.
It is a nominal (not minimum) dimension, and the designer should ensure that: a ≥ c nom nom + φ link link + φ bar bar /2.
Method A is slightly simpler and is presented in Table 2; limits of applicability are given in the notes. Similar data for load-bearing walls is given in Table 3.
For columns there are two tables given in Eurocode 2 Part 1–2 that present methods A and B. Both are equally applicable, although method A has smaller limits on eccentricity than t han method B.
For columns supporting the uppermost storey, the eccentricity will often exceed the limits for both methods A and B. In this situation Annex C of Eurocode 2, Part 1–2 may be used. Alternatively, consideration can be given to treating the column as a beam for determining the design fire resistance.
Column design A flow chart for the design of braced columns is shown in Figure 2. For slender columns, Figure 3 will also be required.
Structural analysis The type of analysis should be appropriate to the problem being considered. The following may be used: linear elastic analysis, linear elastic analysis with limited redistribution, plastic analysis and non-linear analysis. Linear elastic analysis may be carried out assu assuming ming cross sections are uncracked (i.e. concrete section properties), using linear stress-strain relationships and assuming mean values of long-term l ong-term elastic modulus. For the design of columns the elastic moments from the frame action should be used without any redistribution. For slender columns a non-linear analysis may be carried out to determine the second order moments; alternatively use the moment m oment magnification magnification method (Cl 5.8.7.3) or nominal curvature method (Cl 5.8.8) as illustrated in Figure 3.
Design moments The design bending moment is illustrated in Figure 4 and defined as:
M Ed Ed = max { M 02 02, M 0e 0e + M 2, M 01 01 + 0.5 M 2} where: = min {|M top M 01 01 top|, |M bottom bottom|} + e i NEd = max {|M top M 02 02 top|, |M bottom bottom|} + e i N Ed Ed = max {l o/400, h/30, 20} (units to be e i consistent with that used for moments). M top top, M bottom bottom = Moments at the top and bottom of the column = 0.6 M 02 M 0e 0e 02 + 0.4 M 01 01 ≥ 0.4 M 02 02 = N Ed M 2 Ed e 2 where N Ed Ed is the design axial load and e 2 is deflection due to second order order effects eff ects M 01 01 and M 02 02 should be positive if they give tension on the same side. A non-slender column can be designed ignoring second order effects and therefore the ultimate design moment, M Ed Ed = M 02 02. The calculation of the eccentricity, e 2, is not simple and is likely to t o require some iteration to determine the deflection at approxima approximately tely mid-height, e 2. Guidance is given in Figure 3.
Effective length Figure 5 gives guidance on the effective length of the column. However, for most real r eal structures Figures 5f and 5g only are applicable and Eurocode 2 provides two expressions to calculate the effective length for these situations. Expression (5.15) is for braced members and Expression (5.16) is f or unbraced members. In both expressions, the relative fl exibilities at either end, k 1 and k 2, should be calculated. The expression for k given in the Eurocode involves calculating the rotational stiffness of the restraining members making allowance for possible cracking. Once k 1 and k 2 have been calculated, the effective length factor, F , can be established from Table 4 for braced columns. The effective length is then l o = Fl . For a 400 mm square internal column supporting a 250 mm thick flat slab on a 7.5 m grid, the value of k could be 0.11, and therefore l o = 0.59l . In the edge condition condition k is effectively doubled and lo = 0.67 l . If the t he internal column had a notionally ‘pinned’ support at its base then l o = 0.77l . In the long term, Expressions (5.15) and (5.16) in the code will be beneficial as they are particularly suitable for incorporation into design software.
M 0e 0e
M 0e 0e
Slenderness Eurocode 2 states that second order effects may be ignored if they are less than 10% of the first order effects. As an alternative, if the slenderness ( λ ) is less than the slenderness limit ( λ lim lim), then second order effects may be ignored. Slenderness, λ = l o/i where i = radius of gyration and slenderness slendern ess limit.
where A = 1/(1+0.2 ϕ ef ef ) (if ϕ ef ef is not known, A = 0.7 may be used) (if ω , reinforcement ratio, is not B = known, B = 1.1 may be used) C = 1.7 – r m (if r m is not known, C = 0.7 may be used – see below) n = N Ed Ed / (Ac f cd cd) r m = M 01 01/M 02 02 , are the first order end moments, | M 02 M 01 M 01 02 02 02| ≥ | M 01 01| If the end moments M 01 01 and M 02 02 give tension on the same side, r m should be taken positive. Of the three factors A, B and C , C will have the largest impact on λlim and is the simplest to calculate. An initial assessment of λlim can therefore be made using the default values for A and B, but including a calculation for C (see Figure 6). C are are should be taken in determining C because the sign of the moments makes a significant difference. For unbraced members C should always be taken as 0.7.
Column design resistance For practical purposes the rectangular stress block used for the design of beams (see How to design concrete 4 structures using Eurocode 2: Beams ) may also be used for the design of columns (see Figure 7). However, the maximum compressive strain for concrete classes up to and including C50/60, when the whole section is in pure compression, is 0.00175 (see Figure 8a). When the neutral axis falls outside the section (Figure 8b), the maximum allowable strain is assumed to lie between 0.00175 and 0.0035, and may be obtained by drawing a line from the point of zero strain through the ‘hinge point’ of 0.00175 strain at mid-depth of the section. When the neutral axis lies within the section depth then the maximum compressive strain is 0.0035 (see Figure 8c).
The general relationship is shown in Figure 8d. For concrete classes above C50/60 the principles are the same but the maximum strain values vary. Two expressions can be derived for the area of steel required, (based on a rectangular stress block, see Figure 8) one for the axial loads and the other for the moments:
AsN/2 = (N Ed Ed – f cd cd b d c) / (σ sc sc – σ st st) where: AsN
= Area of reinforcement reinforcement required to resist axial load = Axial load N Ed Ed = Design value of c oncrete compressive compressive f cd cd strength ( ) = Stress in compression (and tension) σ sc σ sc st st reinforcement = Breadth of section b = Partial factor for concrete (1.5) γ c = Eff ective depth of concrete in compression d c = λ x ≤ h = 0.8 for ≤ C50/60 λ = Depth to neutral axis x = Height of section h
/2 – d c/2)] / [(h /2– /2–d c/2) (σ sc AsM/2 = [M – f cd b d c(h /2 cd b d sc -σ st st)] where: AsM = Total area of reinforcement required to resist moment Realistically, these can only be solved iteratively and therefore either computer software or column design charts (see Figure 9) may be used. A full range of design charts is available from the website www.eurocode2.info.
Creep Depending on the assumptions used in the design, it may be necessary to determine the effective creep ratio φef (ref. Cl. 3.1.4 & 5.8.4). A nomogram is provided in the Eurocode (Figure 3.1) for which the cement strength class is required; however, at the design stage it often not certain which class applies. Generally, Class R should be assumed. Where the ground granulated blastfurnace slag (ggbs) exceeds 35% of the cement combination or where pulverized fuel ash (pfa) exceeds 20% of the cement combination, Class N may be assumed. Where ggbs exceeds 65% or where pfa exceeds 35%, Class S may be assumed.
Biaxial bending The effects of biaxial bending may be checked using Expression (5.39), which was first developed by Breslaer.
where:
M edz,y edz,y
=
M Rdz,y Rdz,y
=
a
=
N Rd Rd
=
Design moment in the respective respectiv e direction including second order effects in a slender column Moment of resistance in the respective direction 2 for circular circular and elliptical sections; sections; refer to Table 5 for rectangular sections Acf cd cd + Asf yd yd
Unbraced columns There is no comment made on the design of sway frames in Eurocode 2. However, it gives guidance on the effective length of an unbraced member in Expression (5.16). The value for C of 0.7 should always be used in Expression (5.13N). The design moments should be assessed including second order effects. The tabular method for fire resistance r esistance design design (Part 1–2) does not explicitly cover unbraced columns.
Walls When the section length of a vertical element is four times greater than its thickness it is defined as a wall. The design of walls does not differ significantly from that for columns except for the following: ■ The requirements for fire resistance (see Table 3). ■ Bending will be critical about the weak axis. ■ There are different rules for spacing and quantity of reinforcement (see below). There is no specific guidance given for bending about the strong axis for stability. Strut and tie method may be used (section 6.5 of the Eurocode).
Rules for spacing and quantity of reinforcement Maximum areas of reinforcement In Eurocode 2 the maximum nominal reinforcement area for columns and walls outside laps is 4%. However, this area can be increased provided that the concrete can be placed and compacted sufficiently. sufficiently. If required selfcompacting concrete may be used for particularly congested situations, where the reinforcing bars should be spaced to ensure that the concrete can flow around them.
Minimum reinforcement requirements The recommended minimum diameter of longitudinal reinforcement in columns is 12 mm. The minimum area of longitudinal reinforcement in columns is given by: As,min = 0.10 N Ed yd ≥ 0.002Ac Exp. (9.12N) The diameter Ed/f yd of the transverse reinforcement should not be less than 6 mm or one quarter of the maximum diameter of the longitudinal bars.
Note
Spacing requirements for columns
Particular requirements for walls
The maximum spacing of trans transverse verse reinforcement (i.e. links) in columns (Clause 9.5.3(1)) should not generally exceed: ■ 20 times the minimum diameter of the longitudinal bars. ■ the lesser dimension of the column. ■ 400 mm. At a distance within within the larger dimension of the column above or below a beam or slab these spacings should be multiplied by 0.6. The minimum clear distance between the bars should be the greater of the 1x bar diameter, aggregate size plus 5 mm or 20 mm.
The minimum area of longitudinal reinforcement reinforcement in walls is given by: As,min = 0.002Ac The distance between two adjacent vertical bars should not exceed the lesser of either three times the wall thickness or 400 mm. The minimum area of horizontal reinforcement in walls is the greater of either 25% of vertical reinforcement or 0.001 Ac. However, where crack control is important, early age thermal and shrinkage effects should be considered explicitly.
Further guidance and advice ■
■ ■
Guides in this series cover: cov er: Introduction to Eurocodes, Eurocodes, Getting started, Slabs, Beams, Columns, Foundations, Foundations, Flat slabs and Deflection. For free downloads, details of other publications and more information on Eurocode 2 visit www.eurocode2.info This guide is taken from The Concrete Centre’s publication, How to design concrete structures using Eurocode 2 (Ref. CCIP-006) For information on all the new Eurocodes visit www.eurocodes.co.uk
References 1 2 3 4
EN 1992–1–1, Eurocode 2: Design of concrete structures. General rules and rules for buildings . . NARAYANAN, R S & BROOKER, O. How to design concrete structures using Eurocode 2: Introduction . The Concrete Centre, 2005. 2005. EN 1992–1–2. Eurocode 2: Design of concrete structures. General rules – structural fire design . MOSS, R M & BROOKER, O. How to design concrete structures using Eurocode 2: Beams . The Concrete Centre, 2006.. 2006
Additional references for precast construction 1. 2. 3.
EN 13225 - Linear structural elements EN 12794 - Foundation piles EN 13369-Common rules for precast concrete products
Acknowledgements This guide was originally published by BCA and The Concrete Centre in the UK. The authors of the original publication were R M Moss BSc, PhD, CEng, MICE, MIStructE AND O Brooker BEng, CEng, MICE, MIStructE
Europeanised versions of Concise EC2 and How To Leaflets Convention used in the text 1. Nationally determined parameters that occur in the the text have been highlighted highlighted yellow yellow 2. Text is highlighted highlighted in pink pink indicates indicates that some action is required required on the part of the country adapting the documents for its use
Tables & Charts: Word versions Table 1 Column design procedure Step
Task
1 2
Determine design life Assess actions on the column
3
Determine which combinations combinations of actions apply Assess durability requirements and determine concrete strength Check cover requirements for appropriate fire resistance period Calculate min. cover for durability, durability, fire and bond requirements Analyse structure structure to obtain critical moments and axial forces
4 5 6 7
8 9
Check slenderness Determine area of reinforcement required
10
Check spacing of bars
Further guidance guidance Chapter in this publication st art ed 2: Gett ing st st art ed 2: Gett ing st
Int roducti roducti on t o 1: Int Eurocodes st art ed 2: Gett ing st 2: Getting started and Table 2 st art ed 2: Gett ing st 2: Getting started and ‘Structural analysis’ section See Figures 2 and 3 See Figures 2 and 3 ‘Rules for spacing’ section
Standard NA to EN 1990 Table NA.2.1 EN 1991 (10 parts) and National Annexes NA to EN 1990
EN 1992–1–2 EN 1992–1–1 Cl 4.4.1 EN 1992–1–1 section 5
EN 1992–1–1 section 5.8 EN 1992–1–1 section 6.1 EN 1992–1–1 sections 8 and 9
Note NA = National Annex
Tabl e 2 Mi Mi nimum column column dimensi dimensi ons and axi s di stances f or f i r e resi resi stance Standard fire resistance R 60 R 90 R 120
Minimum dimensions (mm) Column width bmin/axis distance, a, of the main bars Column exposed on more than one side Column exposed on one side (µ f f i = 0.7) µ f f i = 0.5 µ f f i = 0.7 200/36 250/46 155/25 300/31 350/40 300/45 350/53 155/25 a a 400/38 450/40 350/45a 350/57a 175/35 a a 450/40 450/51 a b 450/75 295/70
R 240 Note The table is taken from EN 1992–1–2 Table 5.2a (method (method A) and is valid under the following conditions: 1 The effective length of a braced column under fire conditions ( l0,f of l0,f 0,f i) ≤ 3 m. The value of l 0,f i may be taken as 50% of the actual length for intermediate floors and between 50% and 70% of the actual length for the upper floor column. 2 The first order eccentricity under fire conditions should be ≤ 0.15b 0.15b (or h). Alternatively use method B (see Eurocode 2 Part 1–2, Table 5.2b). The eccentricity under fire conditions may be taken as that used in normal temperature design. 3 The reinforcement area outside lap locations does not exceed 4% of the concrete cross section. 4 µ f f i is the ratio of the design axial load under fire conditions to the design resistance of the column at normal temperature conditions. µ f f i may conservatively be taken as 0.7. Key a Minimum 8 bars b Method B may used which indicates 600/70 for R 240 and µ f fi = 0.7. See EN 1992–1–2 Table 5.2b
Tabl e 3 Mi Mi nimum nimum reinf orced orced concret concret e wall dimens dimensions and and axis dist ances ances f or l oad-beari oad-beari ng for fi r e resist resist ance Standard fire resistance
Minimum dimensions (mm) Wall thickness/axis distance, a, of the main bars Wall exposed on one Wall exposed on two side (µ f f i = 0.7) sides (µ f f i = 0.7) a 130/10 140/10a 140/25 170/25 160/35 220/35 270/60 350/60
REI 60 REI 90 REI 120 REI 240 Notes 1 The table is taken from EN 1992-1-2 Table 5.4. 2 See note 4 of Table 2. Key a Normally the requirements of EN 1992–1–1 will determine the cover
Table 4 Ef Ef f ecti ve leng lengtt h f act act or, F , f or br aced aced columns columns k2
k1 0.10 0.59 0.10 0.62 0.20 0.64 0.30 0.66 0.40 0.67 0.50 0.69 0.70 0.71 1.00 0.73 2.00 0.75 5.00 0.76 9.00 Pinned 0.77
0.20 0.62 0.65 0.68 0.69 0.71 0.73 0.74 0.77 0.79 0.80 0.81
0.30 0.64 0.68 0.70 0.72 0.73 0.75 0.77 0.80 0.82 0.83 0.84
0.40 0.66 0.69 0.72 0.74 0.75 0.77 0.79 0.82 0.84 0.85 0.86
0 .50 0.67 0.71 0.73 0.75 0.76 0.78 0.80 0.83 0.86 0.86 0.87
0.70 0.69 0.69 0.73 0.73 0.75 0.75 0.77 0.77 0.78 0.78 0.80 0.80 0.82 0.82 0.85 0.85 0.88 0.88 0.89 0.89 0.90 0.90
Table 5 Value of a for rectangular sections NEd/NRd 0.1 0.7 a 1.0 1.5 Note Linear interpolation may be used
1.0 2.0
1.00 0.71 0.74 0.77 0.79 0.80 0.82 0.84 0.88 0.90 0.91 0.92
2.00 0.73 0.77 0.80 0.82 0.83 0.85 0.88 0.91 0.93 0.94 0.95
5.00 0.75 0.79 0.82 0.84 0.86 0.88 0.90 0.93 0.96 0.97 0.98
9 .00 Pinned 0.76 0.76 0.77 0.80 0.80 0.81 0.83 0.83 0.84 0.85 0.85 0.86 0.86 0.86 0.87 0.89 0.89 0.90 0.91 0.91 0.92 0.94 0.94 0.95 0.97 0.97 0.98 0.98 0.98 0.99 0.99 0.99 1.00
Figure 2 Flow Flow chart f or br aced aced column column desi desi gn
Start
Initial column size may be determined using quick design methods or through iteration.
Determine the actions on the column using an appropriate analysis method. The ultimate axial load, NEd and the ultimate moments are Mtop and M bottom (Moments from analysis)
Greek beta Determine the effective length, l0, using either: 1. Figure 5 or 2. Table 4 or 3. Expression (5.15) from BS EN 1992–1–1
Determine first order moments (see Figure 4) M 01 = Min {|M {|M top|, |M |M bottom|} + ei NEd M 02 = Max {|M {|M top|, |M |Mbottom|} + ei NEd Where ei = Max {l {l0/400, h/30, 20} (units to be consistent with that for bending moment). M 01 and M 02 should have the same sign if they give tension on the same side.
Determine slenderness, , from either: = l0/i where i = radius of gyration = 3.46 l0/h for rectangular sections (h (h = overall depth) = 4.0 l0/d for circular sections (d (d = column diameter)
Determine slenderness limit, 15.4 C
lim,
from:
lim
n (See ‘Slenderness’ section on page 5 for explanation)
Yes Is
≤
lim?
Column is slender (refer to Figure 3).
No Column is not slender. M Ed = M 02 Use column chart (see Figure 9) to find As required for NEd and M Ed. Alternatively, solve by iteration or any other recognized method used in the country
Check rules for spacing and quantity of reinforcement (see page 7)
Figure 3 Flow char char t f or slender slender columns (nominal (nominal curv at ure met met hod)
Start Determine Kr from Figure 9 or from Kr = (n (nu - n) / (n (nu - nbal) ≤ 1 where: n = NEd / (Ac f cd cd), relative axial force NEd = the design value of axial force nu = 1 + nbal = 0.4 = As,est f yd (Ac f cd yd / (A cd) As,est = the estimated total area of steel Ac = the area of concrete Calculate Kϕ from Kϕ = 1 + ϕ ef ef ≥ 1 where: ϕ ef ef = the effective creep ratio = 0.35 + f ck ck/200 - /150 = the slenderness ratio. See section on creep (page 6) Revise value of As,est
Calculate e2 from
e 2
0 .1
K r K f yd 0.45 d E s
l 0
2
where Es = Elastic modulus of reinforcing steel (200 GPa)
M 0e = 0.6 M 02 + 0.4 M 01 ≥ 0.4 M 02 M 2 = NEd e2 M Ed = Max {M { M 02, M 0e + M 2, M 01 + 0.5 M 2}
Use column chart to find As,req’d for NEd and M Ed Alternatively, solve by iteration or by any other recognized method used in the country
No Is As,req’d ≈ As,est? Yes Check detailing requirements
Select ed symbol symbol s Symbol 1/r 0 1/r a A Ac As B c C d e2 ei Es f cd cd f ck ck l l0 Kr Kϕ M 01,M02 M2 M 0e M Ed M Eqp n nbal nu NEd rm x z α cc cc
β ε yd yd γ m λ λ lim lim µ fifi ϕ ef ef ϕ ( ∞,t0) ω |x| Max. {x,y+z}
Definition Reference curvature Curvature Axis distance for fore resistance Factor for determining slenderness limit Cross sectional area of concrete Area of total column reinforcement Factor for determining slenderness limit Factor depending on curvature distribution Factor for determining slenderness limit Effective depth Second order eccentricity Eccentricity due to geometric imperfections Elastic modulus of reinforcing steel Design value of concrete compressive strength Characteristic cylinder strength of concrete Clear height of compression member between end restraints Effective length Correction factor depending on axial load Factor taking into account creep First order moments including the effect of geometric imperfections |M |M 02| ≥ |M 01| Nominal second order moment Equivalent first order moment Ultimate design moment First order bending moment under quasi-permanent loading Relative axial force Value of n of n at maximum moment of resistance Factor to allow for reinforcement in the column Ultimate axial load Moment ratio Depth to neutral axis Lever arm Coefficient taking account of long term effects on compressive strength and of unfavourable effects resulting from the way load is applied Factor Design value of strain in reinforcement Partial factor for material properties Slenderness Slenderness limit Degree of utilisation in a fire Effective creep ratio Final creep coefficient to Cl 3.1.4 Mechanical reinforcement ratio Absolute value of x of x The maximum of values x or y + z
Value ε yd yd/(0.45 d) Kr Kϕ 1/r 0 1 / (1+0.2 ϕ ef ef ) bh
10 (for constant cross-section) cross-sectio n) 1,7 – r m (1/r (1/r)l0/c 200 GPa α cc cc f ck ck/γ c
NEd e2 0.6 M 02 + 0.4 M 01≥ 0.4 M 02
NEd/(A /(A cf cd cd) 0.4 1 + ω M 01/M 02 (d – z)/0.4 1.0
0.35 + f ck /150 ck/200 - λ /150 f yd yd/Es 1.15 for reinforcement (γ s) 1.5 for concrete (γ c) l0/i NEd,fi/NRd ϕ ( ∞,t0) M Eqp/M Ed A s f yd /(A c f cd yd/(A cd)
