Page 201 - Bridge and Highway Structure Rehabilitation and Repair

176 SECTION 2 STRENGTHENING AND REPAIR WORK

1. Ultimate moment M 3 )M n
u
3 A f (d 6 a /2) (4.41)
s
y
where a 3 A f /0.85 f 1 b and A 3 (M /))/f jd) (4.42)
c
y
s
y
u
s
Parabola is replaced by an equivalent rectangle (0.85 fc1 8 a) in area
2. Plastic hinge zone for substructure: Assume a pattern of plastic hinges at locations of peak
bending moments. At ultimate loads, failure is likely to occur at these locations.
Equilibrium equations can be expressed in terms of factored moments and moments of
resistance and solved on the computer as a matrix.
4.16.2 Shear Behavior of Reinforced Concrete Beams
1. Ritter-Morsch model:
The original truss model was proposed by Ritter in 1899 and was developed by Morsch
in 1906 for bent-up bars in place of vertical stirrups. Basic assumptions are:
• Shear forces are resisted by shear reinforcement only
• Angle of inclination of diagonal struts is constant at 45°.
2. ACI 318-95 model:
ACI 318 Model is developed for beams with normal concrete strength
i.e. f 1 : 41.5 MPa (6 ksi).
c
f 1 value 9 41.5 MPa (6 ksi) (but : 69.2 MPa, i.e.,10 Ksi) can be used, provided the mini-
c
mum web reinforcement is increased.
3. Plasticity model:
Lampert and Thurlimann in 1968 proposed a modification to the Ritter-Morsch truss by

using a variable value of angle of cracking (Figure 4.19).

4. Modified compression fi eld theory:
One of the many significant changes in design approach is the use of modifi ed compres-

sion field theory (MCFT) for shear design of reinforced concrete and prestressed concrete

girders. The Ritter-Morsch truss method was used for most of the twentieth century. Research
at Northwestern University by the author, Edwin Rossow, and S.P. Shah has verifi ed the
validity of the method proposed by M. P. Collins. Diagonals represent concrete struts and

verticals represent steel stirrups. Modifications to the Ritter-Morsch constant angle truss
need to be introduced.
The compatibility model (MCFT) was recently proposed by M. P. Collins and further devel-
oped by T. C. Hsu. It proposes a variable angle of cracking (0). It considers the important aspect
of compatibility between concrete and steel stirrups. It explains the biaxial state of stress at a
diagonal crack better than the plasticity model. The compatibility model is based on the modifi ed

compression field theory for concrete beams. Canadian and U.S. contributions to the ultimate
load design method use Mohr’s circle approach for principal stress and principal planes.
In prestressed concrete sections, both axial compressive stress and fl exural compressive
stresses are introduced. The law of superposition is applicable, and tensile stress from applied
load is reduced or cancelled by axial compressive stress from prestressed tendons.

Figure 4.19 Reinforced or prestressed concrete beam idealized as a varying angle hybrid truss.