Equilibrium (Plasticity) Truss Model - Unified Theory of Concrete Structures - page 25

Figure 2.2 Equilibrium in element shear

defined in the axis (horizontal axis) is in the direction of the

longitudinal steel bars with a uniform spacing of s . The transverse steel bars are arranged in
the t axis (vertical axis) with a uniform spacing of s t . After cracking, the concrete is separated
by diagonal cracks into a series of concrete struts, as shown in Figure 2.2(b). The cracks are
defined in the r

− t coordinate, where the

d coordinate, where the r axis is normal to the cracks and the d axis is in
the direction of the diagonal concrete struts. The r

−

− d coordinate is oriented at an angle α r

with respect to the t coordinate. The diagonal concrete struts, the longitudinal steel bars

and the transverse steel bars form a truss which is capable of resisting the shear flow q .
Equilibrium in the longitudinal direction is shown by the force triangle on the right face
of the shear element, Figure 2.2(b). The shear flow q pointing upward is resisted jointly by
a longitudinal steel force n

−

and a diagonal concrete force, ( σ d h )sin α r . The steel force n is

defined as the longitudinal steel force per unit length, A f / s , where A is the cross-sectional

area of one longitudinal steel bar and f is the stress in the longitudinal steel bars. The concrete
force (

σ d h )sin α r represents the diagonal concrete stress σ d acting on a thickness of h and a

width of sin α r relationship is shown by the geometry in Figure 2.2(a). From this

force triangle the shear flow q can be related to the longitudinal steel force n by the geometry:

α r .Thesin

q = n cot α r (2.4)

Similarly, equilibrium in the transverse direction is shown by the force triangle on the top
face of the shear element, Figure 2.2(b). The shear flow q pointing to the right is resisted jointly
by a transverse steel force n t and a diagonal concrete force ( σ d h ) cos α r . The steel force n t is

defined as the transverse steel force per unit length, A t f t / s t , where A t is the cross-sectional

area of one transverse steel bar and f t is the stress in the transverse steel bars. The concrete
force ( σ d h ) cos α r represents the diagonal concrete stress σ d , acting on a thickness of h and