Transcript Shear Stresses in Concrete Beams

Shear Stresses in Concrete
Beams
Stresses Near Support
x
Near Top
y
Max. tensile stresses
a few degrees clockwise
x
Mid-height
Max tensile
stress 45°
y
Max. tensile stresses
a few degrees ccw
Near bottom
x
y
Shear Failure Path
Vertical reinforcing steel, shear stirrups, are added to resist
the portion of the shear force not resisted by the concrete.
Shear Strength of Concrete
The shear strength of a reinforced concrete beam is a result
of the dowel force, aggregate interlock, and shear
compression force.
Vc  2 f 'c bw d
f’c in psi
Shear Strength of Steel Stirrups
Vs 
Av f y d
where,
s = stirrup spacing
Av = total cross-sectional area
of the vertical legs of one stirrup
s
For equilibrium:
Vu   (Vc  Vs )
where, Ф = 0.75
For design:
s
Av f y d
Vs
Vs 
Vu

 Vc
ACI Limitations on Stirrup Spacing
1. s ≤ ½ d if Vs  4 f 'c bw d , f’c in psi. And for
Vu
f
'
b
d

 2 f 'c bw d
beams and girders if
c w

2. s ≤ ¼ d if 4 f 'c bwd  Vs  8 f 'c bw d
3. s 
4 Av f y
3bw f 'c
(upper limit on Vs)
,fy in ksi and Av in in2
4. s < 3 in. to allow placement of concrete
Shear Analysis
• Coefficients for shear given on p. 19 of notes or
Part 5 of Manual of Steel Construction
• If a beam or girder is supported in such a way
that vertical compression is induced in the
bottom, as will be the case if support is provided
by a column or a considerably deeper member,
the shear force a distance d from the support is
used for design.
Design of Shear Reinforcement
for Previous Example p. 21 notes
The partial office building floor plan shown had
beams spanning 30 ft and girders spanning 24 ft.
Design the slab, beams, and girders to support a
live load of 80 psf and a dead weight of 15 psf in
addition to the self weight of the structure. Use
grade 60 reinforcing steel and 4000 psi concrete.
30 ft
24 ft
24 ft
24 ft
30 ft
30 ft
30 ft