Special cases of uncracked sections and calculation of displacements - Concrete Structures: Stresses and Deformations - page 88

instantaneous values occurring at time t 0 and the other for the increments
developing during the period t 0 to t due to creep, shrinkage and relaxation.
With multi-stage prestress, the two steps are repeated for each prestress stage.
Assume that the prestress is applied at age t 0 and t 1 and we are interested in
the stress and strain at these two ages and at a later age t 2 . The analysis is to be
done in four steps to calculate the following:

1 ε O ( t 0 ) and ( t 0 ) are the instantaneous strain at reference point O and the

curvature immediately after application of the

ψ

fi rst prestress.

2 ∆ ε O ( t 1 , t 0 ) and ( t 1 , t 0 ) are the changes in strain at reference point O and

in curvature during the period t 0 to t 1 .

∆ ψ

3 ∆ ε O ( t 1 ) and ( t 1 ) are the additional instantaneous strain at reference

point O and curvature immediately after second prestress.

∆ ψ

4 ∆ ε O ( t 2 , t 1 ) and ( t 2 , t 1 ) are the additional change in strain at reference

point O and curvature during the period t 1 to t 2 .

∆ ψ

In each of the four steps, appropriate values must be used for the properties
of the cross-section, the modulus of elasticity of concrete, the shrinkage and
creep coe

cients and the relaxation; all these values vary according to the age
or the ages considered in each step.
It is to be noted that when the prestress is introduced in stage 2, instan-
taneous prestress loss occurs in the tendons prestressed in stage 1. This is
accounted for in the increments calculated in step 3.

3.8 Calculation of displacements
In various sections of Chapters 2 and 3 equations are given for calculation of
the axial strain and the curvature and the changes in these values caused by
temperature, creep and shrinkage of concrete and relaxation of prestressed
steel.
The present section is concerned with the methods of calculation of dis-
placements in a framed structure for which the axial strain and curvature are
known at various sections. The term 'displacement' is used throughout this
book to mean a translation or a rotation at a coordinate. A coordinate is
simply an arrow drawn at a section or a joint of a structure to indicate the
location and the positive direction of a displacement.
Once the axial strain and curvature are known, calculation of the dis-
placement is a problem of geometry and thus the methods of calculation are
the same whether the material of the structure is linear or non-linear and
whether cracking has occurred or not.