Special cases of uncracked sections and calculation of displacements - Concrete Structures: Stresses and Deformations - page 70

3.1 Introduction
In the preceding chapter, we presented a method of analysis of the time-
dependent stresses and strains in composite sections composed of more than
one type of concrete or of concrete and structural steel sections with or
without prestressed or non-prestressed reinforcement. In the special case
when the section is composed of one type of concrete and the prestressed and
non-prestressed steel are situated (or approximately considered to be) in one
layer, the analysis leads to simpli

ed equations which are presented in this
chapter. Another special case which is also examined in this chapter, is a
cross-section which has reinforcement without prestressing and we will
consider the e

fi

ects
of cracking is excluded from the present chapter and deferred to Chapters 7
and 8.
The procedures of analysis presented in Chapter 2 and in the present chap-
ter give the values of the axial strain and the curvature at any section of a
framed structure at any time after loading. These can be used to calculate the
displacements (the translation and the rotation) at any section or at a joint.
This is a geometry problem generally treated in topics of structural analysis.
In Section 3.8 two methods, which will be employed in the chapters to follow,
are reviewed: the unit load theory based on the principle of virtual work and
the method of elastic weights. The two methods are applicable for cracked or
uncracked structures.

ff ects of creep and shrinkage. However, discussion of the e ff

3.2 Prestress loss in a section with one layer
of reinforcement
The method of analysis in the preceding chapter gives the loss of prestress
among other values of stress and strain in composite cross-sections with a
number of layers of reinforcements. When the total reinforcement, pre-
stressed and non-prestressed, are closely located, such that it is possible
to assume that the total reinforcement is concentrated at one

bre, it may
be expedient to calculate the loss of prestress by an equation - to be given
below - then

fi

fi nd the time-dependent strain and curvature by superposing the

e ect of the initial forces and the prestress loss.

Consider a prestressed concrete member with a cross-section shown in Fig.
3.1. The section has a total reinforcement area

ff

A st = A ns + A ps (3.1)

where A ns and A ps are the areas of the non-prestressed reinforcement and the
prestress steel, respectively. A reference point O is chosen at the centroid of the
concrete section . The total reinforcement A st is assumed to be concentrated in
one

fi bre at coordinate y st . The moduli of elasticity of the two types of