Simplified prediction of deflections - Concrete Structures: Stresses and Deformations - page 306

χφ ( t , t 0 ); where E s is the modulus of elasticity of steel; E c ( t 0 ) is the modulus of

elasticity of concrete at time t 0 ; φ and χ are creep and aging coe cients,

functions of the ages t 0 and t (see Section 1.7).
Equations (9.1)-(9.4) are applicable to uncracked sections in state 1 or fully
cracked sections in state 2, employing coe

cients κ s1 , κ φ 1 and κ cs1 for state 1

and κ cs2 for state 2.

For state 2, cracking is assumed to occur at t 0 due to the bending moment
M . The concrete in tension is ignored; thus, the cross-section in state 2 is
composed of the area of concrete in compression plus the area of the
reinforcement. For T or rectangular cross-sections, the depth of the compres-
sion zone may be determined by Equation (7.16). The geometrical properties
of the cracked section are assumed to undergo no further changes during the
period of creep and shrinkage.
The graphs in Figs F.1 to F.10 of Appendix F give the values of the

κ s2 , κ φ 2 and

κ -

coe cients in the two states for rectangular cross-sections. For easy reference,

the variables in these graphs are listed in Table 9.1. Expressions for the coef-
fi cients for a general cross-section are also given in Appendix F. These are

derived from Equations (2.16) and (3.16).

9.3 Deflection prediction by interpolation
between uncracked and cracked states
In a simpli fi ed procedure suggested in Section 9.4, the probable maximum

de ects of creep and

shrinkage, is predicted by empirical interpolation between lower and upper
bounds, D 1 and D 2 . The values of D 1 and D 2 are determined, assuming the
member to have a constant cross-section in states 1 and 2, respectively. An
empirical coe

fl ection in reinforced concrete members, including the e ff

cient ζ is employed to determine the probable de fl ection

between the two limits D 1 and D 2 . The di ed

procedure and the method discussed in Chapter 8 is that the interpolation is
performed on the de

ff erence between this simpli fi

fl ection at one section to be de fi ned below, rather than on

the curvature at various sections of the member.
The interpolation coe cient used in Chapter 8 depends on the value of the

bending moment and the cracking moment at the section considered (see
Equation (8.41) ). Here the interpolation coe

ection is based on
the bending moment at one section which is referred to as the ' determinant '
section. Similarly, the properties of the cross-section in states 1 and 2 will be
based on the reinforcement at the determinant section.
If we apply any of Equations (C.4), (C.8), (C.12) or (C.16) to calculate the

cient for de fl

de ection in a simple beam in terms of curvature at various sections, it

becomes evident that the maximum de

fl

ection is largely dependent upon the
curvature at mid-span. This is so because the largest curvature is at this
section and this value is multiplied by the largest coe

fl

cient in each equation.
Thus, for a simple beam, the determinant section is considered at mid-span.