Transcript Reinforced Concrete Flexural Members
Reinforced Concrete Flexural Members
Reinforced Concrete Flexural Members
Concrete is by nature a continuous material Once concrete reaches its tensile strength ~400 psi, concrete will crack.
Stress in steel will be ~ 4000 psi.
Design Criteria
• Serviceability – Crack width limits – Deflection limits • Strength – must provide adequate strength for all possible loads
A s area of steel in tension zone A s ’area of steel in compression zone d distance from center of tension reinforcement to outermost point in compression d’ distance from center of compression reinforcement to outermost point in compression
Strain and Stress in Concrete Beams
Strain Stress jd T C T C f ε c ε s f s d cracked concrete ε c ε s cracked concrete ε s > ε y ε c =0.003
cracked concrete f s cracked concrete f s =f y c c f c f c =f’ c M = Tjd = Cjd T = A s f s where j is some fraction of the ‘effective depth’, d at failure, T = A s F y C = T = force in As’ and concrete
Stress in Concrete at Ultimate
ACI 318 approximates the stress distribution in concrete as a rectangle 0.85
f’ c where a wide and = β 1 c.
‘a’ high, C concrete = 0.85
f’ c a b w C steel = A’ s f’ s A s f y = 0.85
f’ c a b w + A’ s f’ s
Definitions
• β 1 shall be taken as 0.85 for concrete strengths f’c up to and including 4000 psi. For strengths above 4000 psi, β 1 shall be reduced continuously at a rate of 0.05
for each 1000 psi of strength above 4000 psi, but β 1 shall not be taken less than 0.65.
• b w = width of web • f’ s = stress in compression reinforcement (possibly f y )
With
No
Compression Steel…
A s f y = 0.85f’ c a b w
a
A s f y
0 .
85
f
'
c b w jd
d
a
2
j
1
a
2
d
For most beams, 5/6 ≤ j ≤ 19/20
Moment Equation
recall, M = Tjd = Cjd and T = A s F y φ = 0.9 for flexure M u ≤ ΦM n =0.9Tjd = 0.9A
s f y jd substituting 5/6 ≤ j ≤ 19/20 0.75A
s f y d ≤ M u ≤ 0.85A
s f y d
Reinforcement Ratio
A s b w d
Reinforcement ratio for beams '
A
'
s b w d
Compression reinforcement ratio
Design Equations
A s
M u
0 .
85
f y d A s
M u
0 .
75
f y d A s
M u
0 .
80
f y d
For positive moment sections of T-shaped beams, and for negative moment sections of beams or slabs where ρ ≤ ⅓ ρ b.
For negative moment sections where ρ ≥ ⅔ ρ b and for positive moment sections without a T flange and with ρ ≥ ⅔ ρ b.
For intermediate cases where ⅓ ρ b < ρ < ⅔ ρ b regardless of the direction of bending.
Balanced Reinforcement Ratio, ρ
b To insure that steel tension reinforcement reaches a strain ε s ≥ f y /E s before concrete reaches ε = 0.003 (steel yields before concrete crushes) the reinforcement ratio must be less than ρ b . Where ρ b is the balanced reinforcement ratio or the reinforcement ratio at which the steel will yield and the concrete will crush simultaneously.
b
0 .
319 1
f c
'
f y
For rectangular compression zones (negative bending) For positive bending (T-shaped compression zone) reinforcement ratio is usually very low (b very large) b = effective flange width, least of: b w + half distance to the adjoining parallel beam on each side of the web ¼ the span length of the beam b w + 16 h f
Balanced Reinforcement Ratio
ρ b for rectangular compression zone F y, ksi f’ c = 3000 psi 4000 5000 6000 40 0.0203
0.0271
0.0319
0.0359
50 60 0.0163
0.0136
0.0217
0.0255
0.0287
0.0181
0.0213
0.0239
Note: if ρ > ρ b can add compression reinforcement to prevent failure due to crushing of concrete.
Depth of Beam for Preliminary Design
The ACI code prescribes minimum values of h, height of beam, for which deflection calculations are not required.
Minimum values of h to avoid deflection calculations Type of beam construction beams or joists one way slabs simply supported one end continuous both ends continuous l l /16 /20 l l /18.5
/24 l l /21 /28 cantilever l l /8 /10
Preliminary Design Values
ρ ≤ 5/3 ρ b practical maximum reinforcement ratio For typical d/b w ratios:
d
3 2 .
5
M f y
b u
Beam Analysis
ACI 318 Approximate Moments and Shears
Compression Reinforcement
If ρ > ρ b must add compression reinforcement to prevent failure due to crushing of concrete
A
'
s
(
A s
b w d
b
)
f y f
'
sb f
'
sb
87 1 8
d
' 3
d
Crack Control
For serviceability, crack widths, in tension zones, must be limited.
ACI 318 requires the tension reinforcement in the flanges of T-beams be distributed over an effective flange width, b, or a width equal to 1/10 span, whichever is smaller. If the effective flange width exceeds 1/10 the span, additional reinforcement shall be provided in the portions of the flange.
outer
Flexure Design 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 30 ft 30 ft 30 ft 24 ft 24 ft 24 ft