Page 203 - Civil Engineering Formulas

CONCRETE FORMULAS 139

an I- or T-beam may be designed by the following formulas, which ignore the
compression in the stem, as is customary:
I
k (5.82)
1 f s /nf c
where kd distance from extreme compression surface to neutral axis, in (mm)
d distance from extreme compression surface to centroid of tensile
steel, in (mm)
f stress in tensile steel, ksi (MPa)
s
f stress in concrete at extreme compression surface, ksi (MPa)
c
n modular ratio E /E
s c
Because the total compressive force C equals the total tension T,
1 bt
C f c (2kd t) T A s f s (5.83)
2 kd
2ndA s bt 2
kd (5.84)
2nA s 2bt

2
2
where A area of tensile steel, in (mm ); and t flange thickness, in (mm).
s
The distance between the centroid of the area in compression and the cen-
troid of the tensile steel is
t (3kd 2t)
jd d z z (5.85)
3(2kd t)
The moment resistance of the steel is
M s Tjd A s f s jd (5.86)
The moment resistance of the concrete is
f c btjd
M c Cjd (2kd t) (5.87)
2kd
In design, M and M can be approximated by
c
s
M s A s f s d t (5.88)
2

1 t
M c f c bt d (5.89)
2 2
derived by substituting d t/2 for jd and f /2 for f (1 t/2kd), the average
c
c
compressive stress on the section.