Civil Engineering Reference
In-Depth Information
7.2.2 Various Definitions of Lever Arm Area, A
o
When Rausch (1929) derived the basic torsion equation (7.68) assuming
45
◦
, a reinforced
concrete member after cracking was idealized as a space truss with linear, one-dimensional
members. Each diagonal concrete strut is idealized as a straight line lying in the center surface
of the hoop bars. Hence, the lever arm area
A
o
is defined by the area within the center surface
of the hoop bars. This definition of
A
o
, commonly denoted as
A
1
, has been adopted by the
German Code (German Standard DIN 4334, 1958) and others. Using the bending analogy, this
definition is equivalent to assuming that the resultant lever arm
jd
is defined as the distance
between the centroid of the tension bars and the centerline of the stirrups in the compression
zone. In terms of torsional strength this assumption is acceptable near the lower limit of
the total steel percentage of about 1%, but becomes increasingly nonconservative with an
increasing amount of steel, as shown in Figure 7.10. For a large steel percentage of 2.5-3%
near the upper limit of under-reinforcement (both the longitudinal steel and stirrups reach
α
r
=
700
ACI CODE (Hsu)
RAUSCH
600
B6
500
B5
TEST CURVE
D4
B4
400
D3
SERIES B
(SOLID)
SERIES D
(HOLLOW)
B3
300
10''
5''
10''
B2
T
cr
D2
D1
B1
200
Minimum
reinforcement
100
UNDER-
REINFORCED
PARTIALLY
OVERREINFORCED
COMPLETELY
OVERREINFORCED
0
100
200
300
400
500
600
700
A
t
f
ty
s
(A
1
), (in. k)
Figure 7.10
Comparison of Rausch's formula and ACI code formula with tests (1 in.
=
25.4 mm;
1in.k
=
113 N m)
