## Concrete Beam Design

### Flexural Design: Find Tension Steel Given Dimensions Only

Fig 1: A Concrete Beam at the Balanced Condition

1. Estimate your Area of Steel (As):

A_s = {M_u\over {øf_y(d-λ})}

where:

Mu = The factored moment
ø = the Reduction Factors for concrete
fy = The yield strength of the steel (e.g. 40 ksi, 60 ksi)
d = the depth of the concrete beam (note: that this is not the same as the height of the beam)
λ = a\over 2 --> for this step of the problem assume λ = 0.1*d

2. Solve for the area of concrete in compression (Ac):

A_c = {f_yA_s\over{0.85f'_c}}

where:

f'c = the compressive strength of concrete (e.g. 4 ksi, 6 ksi)

3. Solve for the depth of the compressive block (a):

a = {A_c\over b}

where:

b = base of the beam

(Note: that this is only applicable to rectangular beams. If a T-section or a beam with a notch is used then the depth of the compressive block must be found by solving for the depth (from the top) at which the area Ac is achieved.)

4. Resolve for the nominal moment (Mn):

M_n = A_sf_y(d- λ)

where:

Mn = The nominal moment

5. Find the required Area of steel for the tension side (As-req'd):

A_{s-req'd} = {{\left(M\over øM_n\right)} A_s}

where:

M = The factored moment acting on the beam
 From here you must ensure you have used the correct ø factor. This will need to be verified for work related calculations; but for P.E.? related studying, problems will most likely never go past step 5.

6. Verify that you used the correct ø factor for Step 5:

• 6a) find c:
c = {a\over β_1}

where:

a = was found in Step 3 above
• 6b) find εt:
ε_t = {{\left(d-c\over c\right)} (0.003)}

where:

εt = the tensile strain in the concrete (normally between 0.001 and 0.005)
• 6c) Your ø factor can be found depending on what ε you have:
An example of this can be found here.

### References

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

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