Effects of temperature - Concrete Structures: Stresses and Deformations - page 361

∆ N = σ restraint d x d y (10.18)

∆ M x = σ restraint y d x d y (10.19)

∆ M y = σ restraint x d x d y (10.20)

where N is a normal force at the centroid.

To remove the arti

∆

fi cial restraint, we apply on the cross-section the forces

−∆ N , −∆ M x and −∆ M y , producing at any point the stress

∆ N

A

∆

M x
I x

∆

M y
I y

∆ σ = − + y + x (10.21)

where A is the area of the cross-section; I x and I y are moments of inertia
about centroidal axes x and y .
The self-equilibrating stresses are given by superposition:

σ = σ restraint + ∆ σ (10.22)

The normal strain at the centroid O and the curvatures in the yz and xz
planes respectively are:

− ∆

N
EA

∆ ε O = (10.23)

∆

M x
EI x

∆ ψ x = − (10.24)

∆

M y
EI y

∆ ψ y = − (10.25)

Substitution of Equations (10.17-20) in the last three equations would
show that the values ∆ ε O , ∆ ψ x and ∆ ψ y are independent of the value of E .

10.8 Continuity stresses
Equations (10.23-25) give the axial strain and the curvatures at any cross-
section of a statically determinate beam. These can be used to calculate the
displacements at member ends. If these displacements are not free to occur, as
for example in a continuous structure, statically indeterminate forces develop,
producing continuity stresses which must be added to the self-equilibrating
stresses to produce the total stresses at any section. Analysis of the statically
indeterminate forces can be performed by the general force or displacement
methods (see Sections 4.2 and 5.2).