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Reinforcement Ratio

The amount of steel reinforcement in concrete members should be limited. Over-reinforcing (the placement of too much reinforcement) will not allow the steel to yield before the concrete crushes and there is a sudden failure.

The reinforcement ratio in concrete beam design is th following fraction:

The reinforcement ratio , ρ, must be less than a value determined with a concrete strain of 0.003 and tensile strain of 0.004 (minimum). When the strain in the reinforcement is 0.005 or greater, the section is tension controlled. (For smaller strains the resistance factor reduces to 0.65 because the stress is less than the yield stress in the steel.)

 

Maximum Reinforcement

Based on the limiting strain of 0.005 in the steel, x(or c) = 0.375d so

α = β₁ (0.375d) to find As-max

The values of β₁ are presented in the following Table:

 

Minimum Reinforcement

Minimum reinforcement is provided even if the concrete can resist the tension, in order to control cracking.

Minimum required reinforcement:

but not less than

where:

f_y is the yield strength in psi

b_w is the width of the web of a concrete T-Beam cross section

d is the effective depth from the top of a reinforced concrete beam to the centroid of the tensile steel

 

Cover for Reinforcement

Cover of concrete over/under the reinforcement must be provided to protect the steel from corrosion. For indoor exposure, 1.5 inch is typical for beams and columns, 0.75 inch is typical for slabs, and for concrete cast against soil, 3 inch minimum is required.

 

Bar Spacing

Minimum bar spacings are specified to allow proper consolidation of concrete around the reinforcement. The minimum spacing is the maximum of 1 in, a bar diameter, or 1.33 times the maximum aggregate size.

 

Effective width b_eff

In case of T-Beams or Gamma-Beams, the effective slab can be calculated as follows:

For interior T-sections, b_eff is the smallest of:

L/4, bw + 16t, or center to center of beams

 

For exterior T-sections, bE is the smallest of

bw + L/12, bw + 6t, or bw + ½(clear distance to next beam)

 

When the web is in tension the minimum reinforcement required is the same as for rectangular

sections with the web width (bw) in place of b.

When the flange is in tension (negative bending), the

minimum reinforcement required is the greater value of

where:

f_y is the yield strength in psi

b_w is the width of the web of a concrete T-Beam cross section

b_eff is the effective flange width

Compression Reinforcement

If a section is doubly reinforced, it means there is steel in the beam seeing compression. The force in the compression steel that may not be yielding is

Cs = As´(f´s - 0.85f´c)

The total compression that balances the tension is now: T = Cc + Cs.

The moment taken about the centroid of the compression stress is Mn = T(d-a/2)+Cs(a-d’)

where As‘ is the area of compression reinforcement

d’ is the effective depth to the centroid of the compression reinforcement

Because the compression steel may not be yielding, the neutral axis x must be found from the force equilibrium relationships, and the stress can be found based on strain to see if it has yielded.