[BS en 13084-2-2007] -- Free-standing Chimneys. Concrete Chimneys

BRITISH STANDARD

Free-standing chimneys — Part 2: Concrete chimneys

The European Standard EN 13084-2:2007 has the status of a British Standard

ICS 91.060.40

12&23 Mcr

NOTE 3 At the beginning of the range the regions of disturbance from adjacent cracks touch each other. If now the moment is further increased, the steel stress will also increase, but the reduction of steel stress due to concrete in tension between cracks is constant. Therefore the line for range c is parallel to the "pure state II" line.

20

EN 13084-2:2007 (E)

Key a)

Range a:

M < Mcr

b)

Range b:

M = Mcr

c)

Range c:

M > Mcr

where:

Mcr

is the bending moment at cracking, see Equation (A.2)

M∆T

is the bending moment from temperature difference ∆T

ML

is the bending moment from other actions (loading)

k∆T

is the curvature due to temperature difference ∆T, see Equation (A.7)

I k cr

is the curvature state I at cracking, see Equation (A.3)

II k cr

is the curvature state II at cracking, see Equation (A.4)

1 k cr

is the curvature, when the formation of cracks is completed, see Equation (A.6)

keff

is the effective curvature

k

II

is the curvature state II

∆k

is the stiffening effect, see Equation (A.5)

Ecm

is the modulus of elasticity of concrete

I

I

is the 2 moment of area at state I

I

II

is the 2 moment of area at state II, see Equation (A.1)

Ieff

nd nd

is the effective 2

nd

moment of area

state I: uncracked cross-section state II: cracked cross-section Figure A.1 — Moment-curvature-relation

21

EN 13084-2:2007 (E)

nd

II

The 2 moment of area at state II, I , of a cross-section under the action of the characteristic values of a bending moment, M, and an axial force, N, may be calculated as follows:

∆x  II  II  I = I 0 1 + e  

(A.1)

where:

I 0II

nd

is the 2 moment of area at state II under the action of pure bending;

∆x

e=

is the difference between the position of the neutral axis of the cross-section at state I and state II respectively under the action of pure bending;

M N

is the eccentricity of the characteristic value of the axial force.

The bending moment at cracking may be calculated as follows:

N  M cr = W I  f ctm - I   A

(A.2)

where: I

is the section modulus at state I;

W I

A

is the area of cross-section at state I;

fctm

is the mean value of the tensile strength;

N

is the characteristic value of axial force.

I

k cr = II

k cr =

M cr I E cm × I

(A.3)

M cr II E cm × I

(A.4)

(

∆ k = 0,4 k IIcr - k Icr k 1cr = k IIcr - ∆ k

22

)

(A.5) (A.6)

EN 13084-2:2007 (E)

A.2 Calculation procedure The characteristic value of the restraint bending moment, M∆T, due to the temperature difference between the inner and outer surfaces of the concrete wall can be calculated according to the following steps: 1)

Evaluation of effects of actions

The following effects of actions have to be determined

NL

is the characteristic value of axial force due to actions (loading);

ML is the characteristic value of bending moment due to actions (loading); k∆T is the curvature due to temperature difference ∆T, see Equation (A.7).

k ∆T =

αΤ × ∆ T

(A.7)

t

where:

∆T is the characteristic value of the temperature difference, ∆T;

2)

-5

-1

αT

is the coefficient of thermal expansion, for concrete αT = 10 K ;

t

is the wall thickness;

Check of range a

Range a applies if the following requirement is satisfied:

M L + k ∆T × E cm × I

I

≤ M cr

In this case the characteristic value of the bending moment, M∆T, can be calculated from the Equation (A.8):

M ∆T = α Τ × ∆ T × E cm × I t 3)

I

(A.8)

Check of range b

Range b applies if the following requirements are satisfied: I M L + k ∆T × E cm × I > M cr and

M L × k1 + k ≤ k1 cr ∆T cr M cr In this case the characteristic value of the bending moment, M∆T, is as given in Equation (A.9): M ∆T = M cr - M L

(A.9)

23

EN 13084-2:2007 (E)

4)

Check of range c

Range c applies if the requirements of range a and range b are not satisfied. nd

The characteristic value of the bending moment, M∆T, and the effective 2 determined by Equations (A.10) and (A.11):

M ∆T =

α Τ × ∆T t

× E cm × I eff

M ∆T + M L = M ∆T + M L - ∆ k II E cm × I E cm × I eff II

where I may be calculated in accordance with Equation (A.1).

24

moment of area, Ieff, may be

(A.10)

(A.11)

EN 13084-2:2007 (E)

Annex B (normative) Limitations of crack widths

It has to be verified that, under the action of the bending moment, M, and the axial force, N, the maximum design crack width in accordance with Table 1 will not be exceeded at the serviceability limit state. For assessing the crack width the steel stresses σs and σsr at state II (cracked cross-section) have to be determined. In this calculation the mean compressive strength of concrete may be used and a linear material law for concrete, described by the mean modulus of elasticity, Ecm, may be applied. The design crack width, wk, shall be calculated by Equation (B.1): 0,89

 σ 0,88 × d s   sr  = 3,5 wk 2  3 f cm  

×

σ s - 0,4 × σ sr Es

(B.1)

where:

σs

is the stress in Newtons per square millimetres in the tension reinforcement calculated on the basis M ; of a cracked cross-section (state II) under actual loading conditions with the eccentricity e = N

σsr is the stress in Newtons per square millimetres in the tension reinforcement calculated on the basis

of a cracked cross-section (state II) under actions maintaining the actual eccentricity e, at which the tensile strength of the concrete, fctm in accordance with Equation (1), in the uncracked cross section (state I) is reached;

Es

fcm

= fck + 8 in Newtons per square millimetres, mean compressive strength of concrete;

fck

is the characteristic compressive strength of concrete in Newtons per square millimetres;

ds

is the diameter of the reinforcing bar in millimetres;

is the modulus of elasticity of reinforcing steel in Newtons per square millimetres.

When using light-weight aggregate concrete the symbols fcm, fctm and fck shall be replaced by flcm, flctm and flck respectively in Equation (B.1) and in the legend.

25

BS EN 13084-2:2007

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