8.2.1 Second group limit states design includes: cracking design;
crack opening design; deformation analysis.
8.2.2 Cracking design is performed when it is necessary to provide crack absence (see 4.3) and as additional for crack opening and deformation analysis.
91 8.2.3 In the cracking design, to avoid cracking, the load safety factor γf is assumed as follows
γf > 1,0 (as in strength design). In crack opening and deformation analysis (including crack
design as additional) one assumes the load safety factor γf = 1,0.
Cracking design of reinforced concrete members
8.2.4 Cracking design of reinforced concrete members is performed considering:
М > Мcrc; (8.116)
where М – bending moment due to external load about axis normal to the plane of moment and passing through the centroid of the reduced cross-section;
Мcrc – bending moment sustained by normal section while cracking, determined by formula
(8.121).
Cracking for centrally tensioned members is determined considering:
N > Ncrc, (8.117)
where N – axial tension force due to external load;
Ncrc – axial tension force sustained by a member while cracking and determined in
compliance with 8.2.13.
8.2.5 Crack opening design is performed in case conditions (8.116) or (8.117) are complied. Design of reinforced concrete members is carried out for short-term and long-term crack opening.
Short-term crack opening is determined due to combined permanent and temporary (long- term and short-term) loads, long-term – only due to permanent and temporary long-term loads (4.6).
8.2.6 Crack opening design is performed considering:
acrc acrc,ult, (8.118)
where acrc – crack width due to external load, determined according to 8.2.7, 8.2.15 - 8.2.17.
acrc,ult – ultimate crack width.
Values of acrc,ult are assumed equal to:
а) from providing safety condition for reinforcement of А240 ... А600, В500 classes: 0,3 mm – at long-term cracking;
0,4 mm – at short-term cracking;
of А800, А1000, Вр1200 – Вр1400, К1400, К1500 (К-19) and К1500 (К-7), К1600 classes with diameter 12 mm:
92 0,2 mm – at long-term cracking;
0,3 mm – at short-term cracking;
of Вр1500, К1500 (К-7), К1600 classes with diameter 6 and 9 mm: 0,1 mm – at long-term cracking;
0,2 mm – at short-term cracking;
b) from the condition of limiting the permeability of structures 0,2 mm – at long-term cracking;
0,3 mm – at short-term cracking.
8.2.7 Design of reinforced concrete members should be performed taking account of long- term and short-term crack opening of normal and inclined cracks.
The width of long-term crack opening is determined as follows
acrc = acrc1, (8.119)
while the width of short-term crack opening is determined as follows
acrc = acrc1 + acrc2 - acrc3, (8.120)
where acrc1 – crack width due to long-term action of permanent and temporary long-term
loads;
acrc2 – crack width due to short-term action of permanent and temporary (long-term and
short-term) loads;
acrc3 – crack width due to short-term action of permanent and temporary long-term loads.
Calculation of cracking moment for cracks normal to the longitudinal axis
8.2.8 Generally bending moment Мсrс resulting to cracking is determined by deformation
model according to 8.2.14.
Cracking moment, considering inelastic strains of tensile concrete for members of rectangular, T-section and I-section with reinforcement located at top and bottom sides, is permitted to be determined in compliance with guidelines listed in 8.2.10 - 8.2.12.
8.2.9 Cracking moment is permitted to be determined neglecting inelastic strains of tensile concrete according to 8.2.11, assuming Wpl = Wred in formula (8.121). In case conditions (8.118)
or (8.139) are not complied, then cracking moment should be determined considering inelastic
strains of tensile concrete.
8.2.10 Cracking moment considering inelastic strains of tensile concrete is determined in accordance with the following assumptions:
93 sections remain plane after deformations;
stress diagram in the compression zone of concrete is assumed of triangular form as for elastic body (figure 8.17);
stress diagram in the tension zone of concrete is assumed of trapezoidal form with stresses not exceeding design values of tensile resistance of concrete Rbt.ser;
strain of edge tensioned concrete fibre is assumed equal to the ultimate value εbt,ult due to
short-term loading (8.1.30); for two-sign strain diagram in section εbt,ult = 0,00015;
stresses in reinforcement are assumed according to strains as for elastic body.
8.2.11 Cracking moment considering inelastic strains of tensile concrete is determined by the following formula
Mcrc = Rbt.ser Wpl ± N ex, (8.121)
where Wpl – elastoplastic section modulus for edge tensioned concrete fibre determined in
accordance with 8.2.10;
ех – distance from application point of axial force N (located in the centroid of the reduced
section) to core point, the farthest one from the tension zone, the cracking of which is verified. In formula (8.121) sign “plus” is assumed for compression axial force N, sign “minus” – for tension force.
1 – Centroid position of a reduced cross-section
Figure 8.17 – Scheme of stress-strain diagram of a section while cracking verification due to
bending moment (a), bending moment and axial force (b)
For rectangular sections and T-sections with a flange located in the compression zone, value Wpl while the moment acts in the plane of symmetry is permitted to be assumed equal to
94 Wpl = 1,3Wred, (8.122)
where Wred – elastic reduced section modulus of the tension zone of a section determined in
accordance with 8.2.12.
8.2.12 Section modulus Wred and distance ех are determined by the following formulas:
(8.123)
(8.124) where Ired – second moment of area of a reduced section about its centroid
Ired = I + Is α + I's α; (8.125)
I, Is, I's – second moment of areas of concrete, tensile reinforcement and compressive
reinforcement respectively;
Ared – reduced cross-sectional area determined as follows
Ared = A + As α + A's α; (8.126)
α – reduction coefficient for reinforcement to concrete
A, As, A's – cross-sectional areas of concrete, tensile and compressive reinforcement
respectively;
уt – distance from the most tensioned concrete fibre to the centroid of a reduced cross-
section
here St,red – first moment of area of a reduced cross-section about the most tensioned
concrete fibre
Section modulus Wred is permitted to be determined neglecting reinforcement.
8.2.13 Force Ncrc while cracking in centrally tensioned members is determined by the
formula
Ncrc = Ared Rbt,ser. (8.127)
8.2.14 Cracking moment based on non-linear deformation model is performed according to basic provisions listed in 6.1.24 and 8.1.20 - 8.1.30, but considering concrete in the tension zone
95 of a normal section, determined from stress-strain diagram of tensile concrete according to
6.1.22. Design characteristics of materials are assumed for second group limit states.
Мсrс is determined from system of equations given in 8.1.20 - 8.1.30, assuming concrete strain
εbt,max at member tensioned side due to external load that is equal to ultimate tensile concrete
strain εbt,ult determined according to 8.1.30.
Calculation of crack widths for cracks normal to the longitudinal axis
8.2.15 Width of normal cracks acrc,i (i = 1, 2, 3 - см. 8.2.7) is determined by the following
formula
(8.128) where σs – stress in longitudinal tensile reinforcement in a normal section with a crack
resulting from the respective external load determined in compliance with 8.2.16;
ls – basic (not considering surface reinforcement type) distance between adjacent normal
cracks determined in compliance with 8.2.17;
ψs – coefficient considering non-uniform distribution of tensile reinforcement strains
between cracks; coefficient ψs may be assumed equal to 1; in case the condition (8.118) is not
complied, then ψs is determined by formula (8.138);
φ1 – coefficient considering durability of loading, assumed equal to: 1,0 – at short-term loading;
1,4 – at long-term loading;
φ2 – coefficient considering profile of longitudinal reinforcement, assumed equal to: 0,5 – for ribbed-profile and reinforcing wires;
0,8 – for plain reinforcement;
φ3 – coefficient considering type of loading, assumed equal to: 1,0 – for bending and eccentrically compressed members; 1,2 – for tensioned members.
8.2.16 Stress σs in tensile reinforcement of bending members is determined as follows
(8.129) where Ired, yc – second moment of area and height of the compression zone of the reduced
96 of concrete, sectional areas of tensile and compressive reinforcement in compliance with
8.2.27. In respective formulas one should assume αs2 = αs1.
For bending members ус = х (figure 8.18), where х – height of the compression zone of
concrete, determined in compliance with 8.2.28 at αs2 = αs1.
Reduction coefficient for reinforcement to concrete αs1 is determined by the following
formula
(8.130) where Eb,red – reduced modulus for compressive concrete considering inelastic deformations
of compressive concrete and determined by the following formula
(8.131) Concrete strain εb1,red is assumed equal to 0,0015.
Stress σs is permitted to be determined by the formula
(8.132) where zs – distance from the centroid of tensile reinforcement to the application point of the
resultant of forces in the compression zone of concrete.
1 – centroid position of a reduced cross-section
Figure 8.18 – Scheme of stress-strain diagram of a member with cracks due to bending
97 For members with rectangular cross-sections with no compressive reinforcement available (or neglecting it), zs is determined as follows
(8.133) For members with rectangular, T- (with a compressed flange) and I-cross-section, zs is
assumed equal to 0,8h0.
Stress σs in tensile reinforcement due to bending moment M and axial force N is determined
by the following formula
(8.134) where Ared, ус – reduced cross-sectional area of a member and the distance from the most
tensioned concrete fibre to the centroid of the reduced section, determined according to general design rules of geometric characteristics of elastic member sections considering sectional area of only compression zone of concrete, sectional areas of tensile and compressive reinforcement in compliance with 8.2.28, assuming reduction coefficient for reinforcement to concrete αs1.
Stress σs is permitted to be determined by the formula
(8.135) where es − distance from the centroid of tensile reinforcement to the application point of
axial force N taking into account eccentricity equal to M/N.
For members with rectangular sections with no compressive reinforcement available (or neglecting it), zs is permitted to be determined by formula (8.133), where хт – height of the
compression zone of concrete considering axial force, determined according to 8.2.28, at αs2 =
αs1.
For members with rectangular, T- (with a compressed flange) and I-cross-section, zs is
permitted to be taken equal to 0,7h0.
In formulas (8.134) and (8.135) sign “plus” is assumed for tension axial force, sign “minus” – for compression axial force.
Stresses σs should not exceed Rs,ser.
8.2.17 Basic distance between cracks ls is determined as follows
98 and is assumed not less than 10ds and 10 cm and not more than 40ds and 40 cm.
Here Аbt – sectional area of tensile concrete;
As – sectional area of tensile reinforcement;
ds – nominal reinforcement diameter.
Аbt is determined by the height of the concrete tension zone xt using design rules of cracking
moment in compliance with 8.2.8 - 8.2.14.
In any case Аbt is assumed equal to the sectional area while its height is not less than 2а and
not more than 0,5h.
8.2.18 Coefficient ψs is determined by the formula
(8.137) where σs,crc – stress in longitudinal tensile reinforcement in a section with a crack
immediately after normal cracks appeared, is determined according to 8.2.16, assuming in the respective formulas М = Мcrc;
σs – the same as mentioned above due to the considered loading.
Coefficient ψs for bending members is permitted to be assumed as follows
(8.138) where Мcrc is determined by formula (8.121).
Deformation analysis of reinforced concrete members
8.2.19 Deformation analysis of reinforced concrete members is performed taking into account service requirements for structures.
Deformation analysis should be performed for:
permanent, temporary long-term and short-term loads (see 4.6) when strains are limited by technological and structural requirements;
permanent, temporary long-term loads when deformations are limited by esthetic requirements.
8.2.20 Ultimate deformations of members are assumed in compliance with SP 20.13330 and regulatory documents for different structures.
Deflection analysis of reinforced concrete members
8.2.21 Deflection analysis of reinforced concrete members is performed considering the following condition:
99 f fult, (8.139)
where f – deflection of a reinforced concrete member due to external load; fult – ultimate deflection of a reinforced concrete member;
Deflections of reinforced concrete structures are determined according to general rules of structural mechanics depending on bending, shear and axial deformation characteristics of reinforced concrete member in sections lengthwise (curvature, shear angle etc.).
In cases when deflections of reinforced concrete members generally depend on bending deformations, deflections are determined according to rigidity characteristics given in 8.2.22
and 8.2.31.
8.2.22 Deflections for bending members with the same section along the member length without cracks are determined according to general rules of structural mechanics using cross- section rigidity, determined by formula (8.143).
Curvature calculation of reinforced concrete members
8.2.23 The curvature of bending, eccentrically compressed and eccentrically tensioned members to calculate their deflections is determined:
а) for members or their fragments where the tension zone has no cracks normal to the longitudinal axis – according to 8.2.24, 8.2.26;
b) for members or their fragments where the tension zone has cracks – according to 8.2.24,
8.2.25 and 8.2.27.
Members or their fragments are considered without cracks if the cracking does not occur [i.e. the condition (8.116) is not complied] due to total load, including permanent, temporary long-term and short-term loads.
The curvature of reinforced concrete members with and without cracks may be also determined based on deformation model according 8.2.32.
8.2.24 Total curvature of bending, eccentrically compressed and eccentrically tensioned members are determined by the following formulas:
for fragments without cracks in the tension zone
(8.140) for fragments with cracks in the tension zone
100 In formula (8.140):
− curvature due to short-term loads, due to permanent and temporary long-term loads respectively.
In formula (8.141):
− curvature due to short-term action of total load which causes deformations under consideration;
− curvature due to short-term action of permanent and temporary long-term loads;
− curvature due to long-term action of permanent and temporary loads.
Curvatures and are determined according to 8.2.25.
8.2.25 Curvature 1/r of reinforced concrete members due to respective loads (8.2.24) is determined as follows
(8.142) where М – bending moment due to external load (taking account of moment due to axial force N) relative to axis normal to the plane of bending moment and passing through the centroid of a reduced cross-section;
D – bending rigidity of a reduced cross-section of a member, determined by the formula D = Eb1 Ired, (8.143)
where Еb1 – deformation modulus of compressed concrete according to loading durability
and taking into account if cracks are available or not;
Ired – second moment of area of the reduced cross-section about its centroid, determined
taking into account if cracks are available or not.
Deformation modulus of concrete Еb1 and second moment of area of a reduced section Ired
for members without cracks in the tension zone and with cracks are determined in compliance with 8.2.26 and 8.2.27 respectively.
Reinforced concrete member rigidity in fragment without cracks in the tension zone
8.2.26 The rigidity of reinforced concrete member D in the fragment without cracks is determined by formula (8.143).
101 Second moment of area Ired of the reduced cross-section of a member about its centroid is
determined the same as for the solid body, according to the general rules of mechanics of materials, taking into account total sectional area of concrete and sectional areas of reinforcement with reduction coefficient for reinforcement to concrete α.
Ired = I + Is α + Is α, (8.144)
where I − second moment of area of concrete section about the reduced cross-section centroid;
Is, I's − second moments of sectional areas of tensile and compressive reinforcement
respectively about the reduced cross-section centroid; α – reduction coefficient for reinforcement to concrete,
(8.145) Value I is determined according to general design rules of geometric characteristics of elastic member sections.
Second moment of area Ired is permitted to be determined neglecting reinforcement.
Values of concrete deformation modulus in formulas (8.143), (8.145) are assumed equal to: at short-term loading
Eb1 = 0,85 Eb; (8.146)
at long-term loading
(8.147) where φb,cr – is assumed according to Table 6.12.
Reinforced concrete member rigidity in fragment with cracks in the tension zone
8.2.27 The rigidity of reinforced concrete member in fragments with cracks in the tension zone is determined according to the following assumptions:
sections remain plane after deformation;
stresses in the compression zone of concrete are determined the same as for elastic body; tensile concrete work in section with normal crack is neglected;
tensile concrete work in fragment between adjacent normal cracks is considered with coefficient ψs.
The rigidity of reinforced concrete member D in fragments with cracks is determined by formula (8.143) and assumed not exceeding the rigidity without cracks.
102 Compressive concrete deformation modulus Еb1 is assumed equal to the reduced
deformation modulus Еb,ser, determined by formula (6.9) considering design concrete resistance
Rb,ser for respective loads (long-term and short-term action).
Second moment of area Ired of the reduced cross-section of a member about its centroid is
determined according to the general rules of mechanics of materials, taking into account only sectional area of concrete, sectional areas of compressive reinforcement with reduction coefficient αs1 and tensile reinforcement with reduction coefficient αs2
Ired = Ib + Is αs2 + I's αs1, (8.148)
where Ib, Is, I's – second moments of sectional areas of compression zone of concrete, tensile
and compressive reinforcement about the centroid of the reduced cross-section neglecting tension zone of concrete.
Is and I's are determined according to general rules of mechanics of materials, assuming the
distance from the most compressive concrete fibre to the centroid of the reduced (with reduction coefficients αs1 and αs2) cross-section neglecting tension zone of concrete (figure
8.19); for bending members
yст = xт,
where хт – average height of the compression zone of concrete considering the effect of
tensile concrete between cracks and determined in compliance with 8.2.28 (figure 8.19).
Ib and уст are determined according to general design rules of geometric characteristics of
elastic member sections.
Coefficients αs1 and αs2 are determined in accordance with 8.2.30.
8.2.28 The neutral axis position (average height of concrete compression zone) for bending moments is determined from expression:
Sb0 = αs2 Ss0 - as1 S's0, (8.149)
where Sb0, Ss0 and S's0 – first moments of area of the concrete compression zone, tensile and
compressive reinforcement about neutral axis.
For rectangular sections only with tensile reinforcement, the height of the compression zone is determined as follows
(8.150) where
For rectangular sections with tensile and compressive reinforcement, the height of the compression zone is determined by the following formula
103 (8.151) where
The height of the compression zone for T-sections (with a compressed flange) and I-sections is determined as follows
(8.152)
where
A'f – sectional area of a compressed flange overhangs.
1 – centroid position of a reduced cross-section neglecting tension zone of concrete
Figure 8.19 – Reduced cross-section (а) and scheme of stress-strain diagram of a member
with cracks (b) for deformation analysis for bending moment
The neutral axis position (height of the compression zone) for eccentrically compressed and tensioned members is determined as follows
(8.153) where yN – distance from the neutral axis to the application point of the axial force N, with
104 Ib0, Is0, I's0, Sb0, Ss0, S's0 – second moments of area and first moments of area of the
compression zone of concrete, tensile and compressive reinforcement about the neutral axis. The height of the compression zone for rectangular sections for bending moments М and axial force N is permitted to be determined by the following formula
(8.154) where хМ – the height of the compression zone of a bending member, determined by
formulas (8.149) - (8.152);
Ired, Ared – second moment of area and reduced cross-sectional area, determined for the full
section (neglecting cracks).
Geometric characteristics of a section are determined according to general design rules of elastic member sections.
In formula (8.154) sign “plus” is assumed for compression axial force, sign “minus” – for tension axial force.
8.2.29 The rigidity of bending reinforced concrete members is permitted to be determined by the following formula
D = Es,redAsz(h0 - xm), (8.155)
where z – distance from the centroid of tensile reinforcement to the application point of the resultant of forces in the compression zone.
Value z for rectangular section if no compressive reinforcement is available (or neglecting it) is determined by the following formula
(8.156) Value z for members with rectangular, T- (with a compressed flange) and I-cross-section is assumed equal to 0,8h0.
8.2.30 Reduction coefficients for reinforcement to concrete are assumed equal to: for compressive reinforcement
(8.157) for tensile reinforcement
105 where Eb,red – reduced deformation modulus of compressive concrete, determined by
formula (6.9) at short-term and long-term loading, substituting Rb to Rb,ser;
Es,red – reduced deformation modulus of tensile reinforcement, determined considering
effect of tensile reinforcement work between cracks, as follows
Es,red = Es/ψs. (8.159)
Coefficient ψs is determined by formula (8.138).
It is permitted to assume ψs = 1 and therefore αs2 = αs1. If the condition (8.139) is not
complied, the calculation is carried out considering coefficient ψs, determined by formula
(8.138).
8.2.31 Deflections of reinforced concrete members may be determined according to general rules of structural mechanics using bending rigidity characteristics D instead of curvature (1/r)