Bending and Axial Loads - Unified Theory of Concrete Structures - page 73

the stress equilibrium condition can be established using the stress diagram in Figure 3.1(c).
Bernoiulli's strain compatibility condition is represented by the strain diagram in Figure 3.1(b).
According to Bernoulli linear strain distribution, as shown in Figure 3.1(b), a bending
moment M will induce a maximum compressive strain

ε c at the top surface and the tensile

strain ε s in the rebar. The neutral axis (NA), which indicates the level of zero strain, is located

at a distance kd from the top surface. Assuming a perfect bond exists between the rebars and
the concrete, a compatibility equation can be established in terms of the three variables, ε c , ε s

and k . Using the geometric relationship of similar triangles we have:
ε s
ε c =

1 − k

(3.1)

k

In the stress diagram of Figure 3.1(c), the linear stress distribution in the compression zone
is obtained from the linear strain distribution (Figure 3.1b), according to Hooke's law. The
stresses vary linearly from the maximum of f c at the top surface to zero at the neutral axis.
Furthermore, the resultant force C 3) kd from the

top surface. By neglecting the tensile stresses of concrete below the neutral axis, the tensile
resistance is concentrated in the rebar tensile force T

= (1 / 2) f c kbd is located at a distance (1 /

= A s f s . Equilibrium of T = C gives:

1
2

A s f s = f c kbd (3.2)

In Figure 3.1(c), the rebar tension force T and the concrete compression force C constitute
an internal couple. The lever arm of the couple jd is (1 − k / 3) d . Equilibrium of internal and

external moments gives:

A s f s d 1

k
3

M = − (3.3)

Hooke's laws for concrete and steel, as shown in Figure 3.1(e) and (f), furnish two additional
equations:

f c = E c ε c (3.4)

f s = E s ε s (3.5)

Using these five equations, (3.1)-(3.5), we can solve five unknowns, f s , f c , ε s , ε c and k .

φ

The solution will allow us to calculate the curvature using the compatibility condition:

ε c
kd

ε s

φ = or (3.6)

(1 − k ) d

curve).
The detailed solution procedures of the five equations depend on the type of bending
problems described in the next section.

Equations (3.3) and (3.6) establish the moment curvature relationship ( M - φ

3.1.1.3 Types of Bending Problem

In a singly reinforced concrete member, the bending action involves nine variables, b , d , A s ,
M , f s , f c , ε c and k , as indicated in Figure 3.1(a)-(c). The five equations (3.1)-(3.5), derived

from Navier's three principles, can only be used to solve five unknown variables. Therefore,
four of the nine variables must be given before the remaining five unknown variables can be

ε s ,