Shear carrying mechanism and behaviour of concrete

When a beam is loaded, shear force and bending moment together produce diagonal tension in the web. Concrete is weak in tension, so inclined cracks form at about 45$^\circ$ near the supports. After cracking, shear is carried by several mechanisms working together.

Shear transfer mechanisms (beam without stirrups)

1. Uncracked concrete compression zone: carries about 20-40% of the shear.
2. Aggregate interlock (friction) across the crack: rough crack faces transfer shear, about 35-50%.
3. Dowel action of longitudinal bars: bars crossing the crack resist transverse movement, about 15-25%.

          P
    ______|_______________
   |      |     inclined crack
   |  Vcz |  /   Vay (interlock)
   |______|_/____________
   o===o==/===o===o===o===   Vd (dowel)
  R

With shear reinforcement

Stirrups (and bent-up bars) cross the diagonal crack, carry tension, and hold the crack width small. This also improves aggregate interlock and dowel action. The behaviour is modelled as a truss (Ritter-Morsch analogy): concrete struts in compression and stirrups as vertical ties. Design shear strength:

$$V_u=V_c+V_{us},\qquad V_c=\tau_cbd,\qquad V_{us}=\frac{0.87f_yA_{sv}d}{s_v}$$

Failure conditions (depends on shear span ratio $a/d$)

Shear strength increases with concrete grade, percentage of tension steel $p_t$ (dowel action, smaller crack width) and axial compression, and decreases with depth (size effect). Since shear failures are brittle, beams are designed so flexural failure occurs first.