Special cases of uncracked sections and calculation of displacements - Concrete Structures: Stresses and Deformations - page 79

Table 3.1 Comparison of strains, curvatures and losses of prestress in two identical cross-
sections with and without non-prestressed reinforcement (Examples 2.2 and
3.1).

Without
non-prestressed
reinforcement

With
non-prestressed
reinforcement

Symbol
used

Axial strain immediately after prestress − 131×10 − 6 − 126×10 − 6 O

Curvature immediately after prestress − 192×10 − 6 m − 1 − 170×10 − 6 m − 1

Change in axial strain due to creep,
shrinkage and relaxation − 556×10 − 6 − 470×10 − 6 O

Change in curvature due to creep,
shrinkage and relaxation − 283×10 − 6 m − 1 − 128×10 − 6 m − 1

Axial strain at time t =∞ − 687×10 − 6 − 596×10 − 6 O + O

Curvature at time t =∞ − 475×10 − 6 m − 1 − 298×10 − 6 m − 1 +

Change in force in prestressed steel
(the loss) A ps ps − 243 kN − 208 kN

Axial force on concrete immediately
after prestress ∫ c ( t 0 )d A c − 1400 kN − 1329 kN

Axial force on concrete at t =∞ ∫ c ( t )d A c − 1157 kN − 878 kN

Change in force on concrete, P c ∫ [ c ( t 0 ) − 243 kN 451 kN

c ( t )]d A c

3.4 Reinforced concrete section without prestress:
effects of creep and shrinkage
The procedure of analysis of Section 2.5 when applied to a reinforced con-
crete section without prestress can be simpli ed as shown below.

Consider a reinforced concrete section with several layers of reinforcement
(Fig. 3.3), subjected to a normal force N and a bending moment M that
produce no cracking. The equations presented in this section give the changes
due to creep and shrinkage in axial strain, in curvature and in stress in con-
crete and steel during a period ( t

fi

t 0 ); where t > t 0 and t 0 is the age of concrete
at the time of application of N and M . The force N is assumed to act at
reference point O ch os en at the centroi d of the age-adjusted transformed sec-
tion of area A c plus [

−

α ( t , t 0 ) A s ]; where α ( t , t 0 ) is a ratio of elasticity moduli,

given by Equation (1.34).
Following the procedure of analysis in Section 2.5, two equations may be
derived for the changes in axial strain and in curvature during the period t 0 to
t (the derivation is given at the end of this section):

∆ ε O = η { φ ( t , t 0 )[ ε O ( t 0 ) + ψ ( t 0 ) y c ] + ε cs ( t , t 0 )} (3.15)

ψ ( t 0 ) + ε O ( t 0 ) y c + ε cs ( t , t 0 ) y c φ ( t , t 0 )

∆ ψ = κ (3.16)

r c r c

where ε O ( t 0 ) and ψ ( t 0 ) are instantaneous axial strain at O and curvature at age