Equilibrium (Plasticity) Truss Model - Unified Theory of Concrete Structures - page 51

2.2.4 Axial Tension-Shear-Bending Interaction
The effect of axial tension on the yield strength of a member can easily be included in the
interaction relationship. Only a brief discussion is required in this section. The effect of axial
tension will be illustrated by its interaction with shear and bending. In the truss model, it is
assumed that the axial tension is resisted only by the longitudinal steel bars. Consequently, the
axial tension does not destroy the internal equilibrium of the beam truss action under shear and
bending. The only addition to the equilibrium condition is the tensile forces in the bottom and
top stringers. In other words, Equations (2.52) and (2.59) for shear and bending interaction
should include a simple new term due to the axial tension force P :

P
2 +

M
d v

V
2

N b =

+ α r

tan (2.95)

P
2 −

M
d v

V
2

N t = + tan α r (2.96)

The other equilibrium equations remain valid for the forces in the transverse steel bars and
in the concrete struts.
Using the same rational approach as in Section 2.2.1., but including the new term for P ,the

P V interaction relationships for the two modes of failure can be derived. For the first

mode of failure we have

− M −

V
V o

2

P
P o +

M
M o +

R = 1 (2.97)

The second mode of failure will give

1
R

1
R

V
V o

2

P
P o

M
M o

− + = 1 (2.98)

=
0.5. It can be seen that P and M have a straight line relationship, but the relationship between
P and V is nonlinear.

Equations (2.97) and (2.98) are shown in Figure 2.15 as two interaction surfaces for R

2.3 ACI Shear and Torsion Provisions
The shear and torsion provisions of the ACI Code were significantly revised in 1995 (ACI
318-95). With minor changes, it continued to be used up to the present day (ACI 318-08). The
code provisions after 1995 were based on the equilibrium (plasticity) truss model derived in
Sections 2.1 and 2.2. The background of the 1995 ACI Code provisions was explained in a
paper by Hsu (1997). In this section, we will elaborate on the ACI design procedures and will
provide a design example.