## Concrete Beam Design A Concrete Beam at the Balanced Condition

### Flexural Design: Find Moment Capacity Given Concrete Beam Cross Section

1. Find the depth of the equivalent rectangular stress block (a):

a = {A_sf_y\over{0.85f'_c b}}

where:

As = Area of the tension steel.
fy = The yield strength of the steel (e.g. 40 ksi, 60 ksi)
f'c = The compression strength of the concrete (e.g. 4 ksi, 6 ksi)
b = the width of the concrete beam

2. Check to see whether or not the section is tension controlled:

If \left({a\over d}\right) \le \left({a_b\over d}\right)      then the section will be assumed to be tension controlled.

where:

a = from section 1
d = depth of to the rebar
{a_b\over d} = {β_1}{\left(87,000\over 87,000 + f_y\right)}

Note: For f'c ≤ 4 ksi, β1 = 0.85

3. Compute the nominal moment capacity (Mn):

 Most common concrete beam equation: M_n = A_sf_y\left(d-{a\over2}\right)

where:

Mn = The nominal moment
d = the depth of the concrete beam (note: that this is not the same as the height of the beam)

4. Compute the factored moment capacity (řMn):

Above in Section #2 it was determined whether or not the section is tension controlled. The results from Section #2 will determine what ř (reduction factor) you should use. This will then be compared to the factored moment you find acting on the beam.

For a beam to work:

řM_n \ge Factored Moment

where:

ř = the Reduction Factors for concrete

### References

1. American Concrete Institute, "ACI 318", 2005

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