## 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 (A _{s}):**

*where:*

**M**= The factored moment

_{u}**ø**= the Reduction Factors for concrete

**f**= The yield strength of the steel (e.g. 40 ksi, 60 ksi)

_{y}**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 (A _{c}):**

*where:*

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

_{c}3. **Solve for the depth of the compressive block (a):**

*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 A _{c} is achieved.*)

4. **Resolve for the nominal moment (M _{n}):**

*where:*

**M**= The nominal moment

_{n}5. **Find the required Area of steel for the tension side (A _{s-req'd}):**

*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 ε*_{t}:ε_t = {{\left(d-c\over c\right)} (0.003)}

*where:*

**ε**= the tensile strain in the concrete (normally between 0.001 and 0.005)

_{t}*6c) Your ø factor can be found depending on what ε you have:*

An example of this can be found here.

### References

- American Concrete Institute, "ACI 318", 2005