Simplified prediction of deflections - Concrete Structures: Stresses and Deformations - page 305

Figure 9.1 Curvature at a reinforced concrete cross-section subjected to bending moment.

ε cs ( ∆ ψ ) cs = −

d

κ cs (9.3)

where ψ c is the instantaneous curvature at a hypothetical uncracked concrete

section without reinforcement:

M
E c ( t 0 ) I g

ψ c = (9.4)

I g = moment of inertia of gross concrete section about an axis

through its centroid

E c ( t 0 ) = modulus of elasticity of concrete at time t 0

ψ ( t 0 ) = instantaneous curvature

( ∆ ψ ) φ and ( ∆ ψ ) cs = curvature changes caused by creep and by shrinkage

ε cs = ε cs ( t , t 0 ) = value of free shrinkage during the period considered.

Following the sign convention adopted through this topic, positive
strain represents elongation; hence, shrinkage of concrete is a negative
quantity.
A positive bending moment produces positive curvature (Fig. 9.1). In a
cross-section with top and bottom reinforcements, shrinkage is restrained by
the reinforcement and the result is smaller shrinkage at the face of the section
with heavier reinforcement. In a simple beam subjected to gravity load, the
heavier reinforcement is generally at the bottom. Thus, the curvature due to
shrinkage is of the same sign as the curvature due to the positive bending
moment due to load. For the same reason, in a cantilever with heavier
reinforcement at the top, curvatures due to shrinkage and due to gravity load
are cumulative.
κ s , cients depending on the geometri-

cal properties of the cross-section, the ratio

κ and κ cs are dimensionless coe

α ( t 0 ) = E s / E c ( t 0 ) and the product