Determine the Capacity of a Reinforced Concrete Beam with Tension Reinforcement

In the last post, I talk about how to determine the required reinforcing for a rectangular beam. To elaborate more on the same topic, I am going to show you how to actually calculate out the capacity (using my handy dandy flowchart).

This most likely is just a refresher for many of you but it doesn't hurt to get more familiar with the calculation (especially if you haven't designed concrete for awhile).

The Goal

To determine the moment capacity, , without having to memorize anything – you just need to follow the flowchart.

Flowchart

Given

• (or ) Provided reinforcing steel (or steel ratio).
•  Specified compressive strength. This is typically 3000, 4000, or 5000 psi.
• Specified yield strength of reinforcement. Usually 60,000 psi for new buildings and 40,000 psi for older buildings.
• Width of the beam.
• Usually the total beam depth – cover – 1/2 of the rebar diameter.

Determine

• Moment capacity of the section.

Quick Check

I've demonstrated the following “quick check” in an earlier post:

   

If we set and rearrange the equation, we get:

   

where the units for is [in²] and is [in].

Step-by-Step

#	Equation	Action	Notes/Explanation
1		Calculate	This calculates the size of the compression stress block.
2		Calculate	This is the ratio between the T-C moment arm and d. It'll be used in step [8] to obtain the moment capacity.
3		Calculate	Location of the neutral axis from top fiber. See previous post/flowchart step [5] for the calculation of .
4		Calculate	This is the strain in the tension reinforcement.
5		Check	This checks whether the section is tension or compression controlled per er ACI 318, 9.3.2.2.
6		Calculate if answer in [5] is no (i.e., compression controls).	If compression controls, you have to reduce the factor based on this formula.
7		Calculate if answer in [5] is yes (i.e., tension controls).	This is the factor for tension controlled section.
8		Calculate	The moment capacity.
9		Calculate (Optional)	This is the reinforcement ratio that will cause “balanced strain condition” which is when these two events occur at the same time: Tension reinforcement yields. Strain in the concrete reaches 0.003. In terms of design, you just want to make sure that your reinforcement ratio is less than this calculated value to prevent brittle failures.

Example

Given

• (4-No.8)
• 
• 
• 
• 

Quick Check

   

Remember that this quick check is just an estimate. The point is just to make sure that we didn't mess up the actual calculation somewhere along the way.

We'll verify the real capacity in the table below.

Use the Flowchart

#	Equation	Results	Notes/Explanation
1		6.1961 in
2		0.8451
3		7.2895 in	is calculated to be 0.85 for .
4		0.0052
5		Yes, tension controls.
6		Not applicable
7		0.90
8		240 kip-ft	This is pretty close to the quick check (253 kip-ft) which means we probably didn't screw up arithmetically.
9		0.0214	Compares with , balanced reinforcement ratio is greater; therefore we will not get brittle failure which is good.

Done!

There you have it. Is this helpful? Let me know in the comments below.