55Centre line

Cover

Outer edge of effective cross section, circumference, u

t_ef z_i

T_Ed

t_ef/2

Figure 9.1

Notations used in Section 9

6.3.1

6.3.2

Exp. (6.30)

Fig. 6.11

The torsional capacity of a solid rectangular section with shear reinforcement on the outer periphery T_Rd,maxmay be deduced from the general expression:

T_Rd,max= 2vf_cdk₂b³sin q cos q where

k₂ = coefficient obtained from Table 9.1 b = breadth of the section (< h, depth of section)

Torsional resistance governed by the area of closed links is given by:

T_Rd = A_sw/s) = T_Ed/(2A_kcot q) f_ywd where

A_sw = area of link reinforcement

f_yw,d = design strength of the link reinforcement s = spacing of links

Additional longitudinal reinforcement distributed around the periphery of the section should be provided and the area of this reinforcement should be obtained from the following expression:

2A_sl = T_Edu_kcot q/(f_yd2A_k) = (A_sw/s) u_kcot²q where

T_Ed = applied design torsion u_k = perimeter of the area A_k Assuming f_yd= f_ywd

9.3 Combined torsion and shear

In solid sections the following relationship should be satisfied:

(T_Ed/T_Rd,max) + (V_Ed/V_{Rd, max}) ≤ 1.0 where

T_Rd,max = 2mf_cdA_kt_ef,isin q cos q as in Section 9.2.

V_Rd,max = v_wzmf_cd(cot q + cot a)/(1 + cot²q) as in Section 7.3.2.

Table 9.1 Values of k₂

h/b 1

k₂ 0.141

2 0.367

3 0.624

4 0.864

Exp. (6.28) ENV 1992-1-1 Exp. (4.43)^[21]

Exp. (6.29)

Serviceability

57

10 Serviceability

10.1 Introduction

The common serviceability limit states considered are:

■Stress limitation.

■Crack control.

■Deflection control.

For the UK, explicit checks on concrete stresses at serviceability are not normally required, unless lower values for partial factor g_cthan those shown in Section 2, Table 2.3 are used.

Similarly, the steel stress need not be checked unless the values for g_sare smaller than those indicated.

In compression members, the provision of links in accordance with detailing rules ensures that there is no significant longitudinal cracking.

Cracking and deflection may be verified by either following calculation procedures or by observing the rules for bar diameters and bar spacing and span-to-effective-depth ratios. This publication does not consider calculation methods.

10.2 Control of cracking

Cracks may be limited to acceptable widths by the following measures:

■Provide a minimum amount of reinforcement, so that the reinforcement does not yield immediately upon formation of the first crack (see Section 10.3).

■Where restraint is the main cause of cracking, limit the bar diameter to that shown in Table 10.1. In this case any level of steel stress may be chosen but the chosen value must then be used in the calculation of A_s,minand the size of the bar should be limited as shown.

■Where loading is the main cause of cracking, limit the bar diameter to that shown in Table 10.1 or limit the bar spacing to that shown in Table 10.2.

When using either table the steel stress should be calculated on the basis of a cracked section under the relevant combination of actions.

In the absence of specific requirements (e.g. water-tightness), the limiting calculated crack width w_maxmay be restricted to 0.3 mm in all exposure classes under quasi-permanent load combinations. In the absence of specific requirements for appearance, this limit may be relaxed to, say, 0.4 mm for exposure classes X0 and XC1.

In building structures subjected to bending without significant axial tension, specific measures to control cracking are not necessary where the overall depth of the member does not exceed 200 mm.

7.2, NA &

PD 6687^[7]

7.3.3(2)

7.3.1(5)

& NA

10.3 Minimum reinforcement areas of main bars

If crack control is required, the minimum area of reinforcement in tensile zones should be calculated for each part (flanges, web, etc.) as follows:

A_s,min= k_ck f_ct,effA_ct/s_s where

k_c = a coefficient to allow for the nature of the stress distribution within the section immediately prior to cracking and for the change of the lever arm as a result of cracking

= 1.0 for pure tension and 0.4 for pure bending

k = a coefficient to allow for the effect of non-uniform self-equilibrating stresses, which lead to a reduction of restraint forces

= 1.0 for web heights or flange widths ≤ 300 mm and k = 0.65 when these dimensions exceed 800 mm. For intermediate conditions interpolation may be used

f_ct,eff = mean value of the tensile strength of concrete effective at the time cracks may be first expected to occur at the appropriate age. f_ct,eff= f_ct,m(see Table 3.1) A_ct = area of concrete in that part of the section which is calculated to be in the

tension zone i.e. in tension just before the formation of the first crack s = absolute value of the maximum stress permitted in the reinforcement Table 10.1

Maximum bar diameters for crack control Steel stress (MPa)

0.4 mm 0.3 mm 0.2 mm

Maximum bar size (mm) for crack widths of

160

Maximum bar spacing for crack control Steel stress (MPa)

0.4 mm 0.3 mm 0.2 mm

Maximum bar spacing (mm) for maximum crack widths of

160

Serviceability

59

10.4 Minimum area of shear reinforcement

10.4.1 Beams

The minimum area of shear reinforcement in beams A_sw,minshould be calculated from the following expression:

A_sw,min/(sb_w sin a) ≥ 0.08 f_ck^0.5/f_yk where

s = longitudinal spacing of the shear reinforcement b_w = breadth of the web member

a = angle of the shear reinforcement to the longitudinal axis of the member.

For vertical links sin a = 1.0.

10.4.2 Flat slabs

In slabs where punching shear reinforcement is required, the minimum area of a link leg, A_sw,min should be calculated from the following expression:

A_sw,min(1.5 sin a + cos a)/(s_rs_t) ≥ 0.08 f_ck^0.5/f_yk where

s_rand s_t= spacing of shear reinforcement in radial and tangential directions respectively (see Figure 12.5)

10.5 Control of deflection

10.5.1 General

The deflection of reinforced concrete building structures will normally be satisfactory if the beams and slabs are sized using the span-to-effective-depth ratios. More sophisticated methods (as discussed in TR58^[23]) are perfectly acceptable but are beyond the scope of this publication.

10.5.2 Basic span-to-effective-depth ratios

Basic span-to-effective-depth ratios are given in Table 10.3.

This table has been drawn up on the assumption that the structure will be subject to its design loads only when the concrete has attained the strength assumed in design, f_ck. If the structure is to be loaded before the concrete attains f_ck, then a more detailed appraisal should be undertaken to take into account the loading and the strength of the concrete at the time of loading.

Table 10.3 is subject to the following conditions of use:

■Values for span-to-effective-depth ratios have been calculated using the criterion that the deflection after construction is span/500 for quasi-permanent loads.

■Table values apply to reinforcement with f_yk= 500 MPa, s_s= 310 MPa, and concrete class C30/37 (f_ck= 30).

■Values of basic span-to-effective-depth ratios may be calculated from the formulae:

l/d = K[11 + 1.5f_ck^0.5p₀/p + 3.2 f_ck^0.5(p₀/p – 1)^1.5] if p₀≤ p or

l/d = K[11 + 1.5 f_ck^0.5p₀/(p – p') + f_ck^0.5(p'/p₀)^0.5/12] if p₀> p

9.2.2(5)

& NA

9.4.3(2)

7.4.1, 7.4.2

& NA

7.4.3

PD 6687^[7]

7.4.2(2)

Exp. (7.16a)

Exp. (7.16b)

where

l/d = limit span-to-effective-depth

K = factor to take into account different structural systems p₀ = reference reinforcement ratio = f_ck^0.5/1000

p = required tension reinforcement ratio = A_s,req/b_d

For flanged sections, p should be based on an area of concrete above centroid of tension steel

p’ = required compression reinforcement ratio = A_s2/b_d

■When the area of steel provided A_{s, prov}is in excess of the area required by calculation A_s,req, the table values should be multiplied by 310/s_s= (500/f_yk) (A_s,prov/A_s,req) ≤ 1.5 where s_s= tensile stress in reinforcement in midspan (or at support of cantilever) under SLS design loads in MPa.

■In flanged beams where b_eff/b_wis greater than 3, the table values should be multiplied by 0.80. For values of b_eff/b_wbetween 1.0 and 3.0 linear interpolation should be used.

■The span-to-effective depth ratio should be based on the shorter span in two-way spanning slabs and the longer span in flat slabs.

■When brittle partitions liable to be damaged by excessive deflections are supported on a slab, the table values should be modified as follows:

a) in flat slabs in which the longer span is greater than 8.5 m, the table values should be multiplied by 8.5/l_eff; and

b) in beams and other slabs with spans in excess of 7 m, the table values should be multiplied by 7/l_eff.

Table 10.3

Basic ratios of span-to-effective-depth, l/d, for members without axial compression

Beams

Structural system

Slabs

Simply supported

End span of continuous beams

Interior spans of continuous beams N/A

Cantilever

One- or two-way spanning slabs simply supported End span of one-way spanning continuous slabs

or

two-way spanning slabs continuous over one long edge Interior spans of continuous slabs

Flat slab (based on longer span) Cantilever

K

1.0

1.3

1.5

1.2 0.4

Highly stressed concrete ρ ρ = 1.5%

14

18

20

17 6

Lightly stressed concrete ρ ρ = 0.5%

20

26

30

24 8 Table 7.4N

& NA

In document Concise Eurocode 2 (Page 65-71)