CIVIL ENGINEERING FORMULAS

CONCRETE FORMULAS 131

Equating the compression and tension at the critical section yields

(5.43)

The criterion for compression failure is that the maximumstrain in the con-
crete equals 0.003 in/in (0.076 mm/mm). In that case,

(5.44)

where fssteel stress, ksi (MPa)
Esmodulus of elasticity of steel
29,000 ksi (199.9 GPa)

Balanced Reinforcing

Under balanced conditions, the concrete reaches its maximum strain of 0.003
when the steel reaches its yield strength fy. This determines the steel ratio for
balanced conditions:

(5.45)

Moment Capacity

For such underreinforced beams, the bending-moment capacity of ultimate
strength is

(5.46)

(5.47)

where and.

Shear Reinforcement

The ultimate shear capacity Vnof a section of a beam equals the sum of the
nominal shear strength of the concrete Vcand the nominal shear strength pro-
vided by the reinforcementVs; that is, VnVcVs. The factored shear force Vu
on a section should not exceed

(5.48)

wherecapacity reduction factor (0.85 for shear and torsion). Except for
brackets and other short cantilevers, the section for maximum shear may be
taken at a distance equal to dfrom the face of the support.

Vn(VcVs)

fy/fc aAsfy/0.85fc

0.90Asfyd

a

2 

Mu0.90[bd^2 fc(10.59)]

b

0.85 1 fc

fy



87,000

87,000fy

c

0.003

fs/Es0.003

d

c

pfy

0.85 1 fc

d