9.3.1 Second group limit states design includes: cracking design;
crack opening design; deformation analysis.
9.3.2 Cracking design is performed when it is necessary to provide crack absence and as additional for crack opening and deformation analysis.
Crack absence requirements should be met for prestressed structures with ensured impermeability in case of fully tensioned section (being under pressure of liquid or gases, affected by radiation etc.), for unique structures with high durability requirements and for structures operated in aggressive medium.
9.3.3 In cracking design to avoid cracking, the load safety factor γf is assumed as follows γf >
1,0 (as for the strength design). In crack opening and deformation analysis (additional cracking design included) the load safety factor should be assumed as γf = 1,0.
9.3.4 The design of bending prestressed members by second group limit states is performed both at eccentric compression for combined action of forces due to external load М and axial force Np equal to the tendon force Р.
Cracking design of prestressed reinforced concrete members
9.3.5 Cracking design of prestressed bending members is performed according to basic provisions stated in chapter 8.2 and considering guidelines 9.3.6 - 9.3.10.
Calculation of cracking moment for cracks normal to the longitudinal axis
9.3.6 Generally bending moment Мсrс of cracking is determined based on deformation model
according to 9.3.10. Cracking moment, considering common sections (rectangular and T-section with reinforcement located at top and bottom sides with a compressed flange) is permitted to be determined in compliance with 9.3.7.
9.3.7 Cracking moment is calculated considering inelastic strains of tensile concrete in accordance with 9.3.8.
Cracking moment is permitted to be determined neglecting inelastic strains of tensile concrete, assuming in formula (9.36) Wpl = Wred. In case conditions (8.118) or (8.139) are not
complied, then cracking moment should be determined considering inelastic strains of tensile concrete.
9.3.8 Cracking moment of prestressed bending members considering inelastic strains of tensile concrete is determined as follows
119 Mcrc = Rbt,ser Wрl ± Р еяр, (9.36)
where Wpl – section modulus of the reduced section for the end tensioned fibre taking into
account provisions in 8.2.10;
eяр = еор + r – ех – distance from application point of prestressing force Р to the core point,
the farthest one from the tension zone, the cracking of which is verified; еор – the same for the centroid of the reduced section;
r – distance from the centroid of the reduced section to the core point,
(9.37) In formula (9.36) sign “plus” is assumed when directions of moments Р еяр and external
bending moment М are opposite; sign “minus” – when directions are the same. Wred and Ared are determined in compliance with 8.2.
For rectangular sections and T-sections with a compressed flange, Wpl for moment in the
plane of symmetry axis is permitted to be determined according to formula (8.122).
9.3.9 Force Ncrc while cracking in centrally tensioned members is determined by formula
(8.131) 8.2.
9.3.10 Cracking moment based on non-linear deformation model is performed according to basic provisions listed in 6.1.24, 9.2.13 - 9.2.15, but considering concrete in the tension zone 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 9.2.13 - 9.2.15, assuming concrete strain
εbt,max at tensile side of a member due to external load is equal to the ultimate tensile concrete
strain εbt,ult, determined according to 8.1.30.
Calculation of crack widths for cracks normal to the longitudinal axis
9.3.11 Width of normal cracks is determined by formula (8.128), where stresses σs in tensile
reinforcement with bending prestressed members due to external load are determined by the following formula
(9.38) where Ired, Ared, ус – second moment of area, reduced cross-sectional area of a member and
the distance from the most compressed fibre to the centroid of the reduced section, to be determined taking account of sectional area in the compression zone of concrete, sectional
120 areas of tensile and compressive reinforcement in compliance with 8.2.27. In respective formulas one should assume αs2 = αs1.
Np – prestressing force (9.3.4);
Мр – bending moment due to external load and prestressing force determined according to
the formula
Mp = M ± Np eop, (9.39)
where еор – distance from the application point of prestressing force Np to the centroid of
the reduced section.
Sign “minus” in formula (9.39) is assumed when directions of moments М and Np eop are
opposite, and sign “plus”– when the directions are the same.
Stress σs is permitted to be determined according to the formula
(9.40) where z – distance from the centroid of the same reinforcement in the tension zone to the point where the resultant of forces is applied in the compression zone of a member;
esp – distance from the centroid of the same reinforcement to the application point of force
Np.
For members with rectangular cross-sections with no compressive reinforcement available (or neglecting it), z is determined by the formula
z = h0 - xN/3. (9.41)
where xN – height of the compression zone determined in accordance with 8.2.28 taking
account of prestressing force Np.
For members with rectangular, T- (with a compressed flange) and I-cross-section, zs is
permitted to be assumed equal to 0,7h0.
Stresses σs, determined by formulas (9.38) and (9.40), should not exceed (Rs,ser - σsp).
Deformation analysis of prestressed reinforced concrete members
9.3.12 Deformation analysis of prestressed members is performed in compliance with 8.2.19
- 8.2.32 considering additional guidelines 9.3.13 - 9.3.15.
9.3.13 Total curvature of bending prestressed members for calculation of their deflections is determined according to 8.2.24, while values of curvatures, and in formulas
121 While calculating the curvature, it is permitted to consider effects from shrinkage deformations and concrete creep during prestressing.
9.3.14 Curvature 1/r of bending prestressed members due to respective loads is determined by the formula
(9.42) where М − bending moment due to external load;
Np and еор – prestressing force and its eccentricity about the centroid of the reduced cross-
section of a member;
D – bending rigidity of the reduced cross-section of a member, determined according to 8.2, as for eccentrically compressed by prestressing force member taking account of bending moment due to external load (figure 9.3).
1 – centroid position of a reduced cross-section neglecting tension zone of concrete
Figure 9.3 – Reduced cross-section (а) and scheme of stress-strain diagram of a bending
prestressed member with cracks (b) in its deformation analysis
9.3.15 Curvature of bending prestressed members is permitted to be determined as follows
(9.43) where zp – distance from the application point of prestressing force to the application point
122 z – distance from the centroid of tensile reinforcement to the application point of resultant of forces in the compression zone;
xN – height of the compression zone considering prestressing.
The height of the compression zone for bending members without prestress is determined according to 8.2.28 by multiplying μs by
zp and z are permitted to be obtained, assuming the distance from the application point of
resultant of forces in the compression zone to the most compressive fibre of a section equal to 0,3h0.
Curvature calculation of prestressed members based on non-linear deformation model
9.3.16 The full curvature of bending prestressed members in fragments without cracks in the tension zone of a section is determined by formula (8.140), in fragments with cracks in the tension zone − by formula (8.141).
Curvatures from formulas (8.140) and (8.141) are obtained from system of equations (9.26) - (9.34) considering guidelines 9.2.13. The stress in prestressing reinforcement crossing cracks, for members with normal cracks in the tension zone, is determined by the following formula
(9.44) for ordinary reinforcement
Here εsi(j),crc – strain in tensile reinforcement in the section with a crack due to external load
immediately after normal cracks appeared;
εsi(j) – average strains of tensile reinforcement crossing cracks in the referred stage.
εspi – strain of prestressing reinforcement.
Short-term stress-strain diagrams of compressive or tensile concrete are used in the design while calculating the curvatures due to short-term loading. Long-term stress-strain diagrams of concrete with design characteristics for second group limit states are used while calculating the curvatures due to long-term loading.
123
10 Structural requirements 10.1 Principal rules
10.1.1 To provide safety and serviceability of concrete and reinforced concrete structures both requirements established from analysis and structural requirements for geometric dimensions and reinforcement have to be implemented.
Structural requirements are set for the following cases: design cannot guarantee adequate resistance of a structure to external loads and actions;
structural requirements establish limiting conditions within which design provisions may be used;
structural requirements implement manufacturing process of concrete and reinforced concrete structures.