Stepwise procedure for calculating Moment of resistance and compressive stresses in steel and concrete
While we proceed with the article series for “Doubly reinforced sections”, I would like to categorize the problems into different types. This will make your understanding of the concept better and concrete. I recommend that you practice enough to be able to understand and confidently solve the problems. This will also help you in real time when you would get into practice.
In our article series for doubly reinforced sections, we have covered the following:
Also check out: “Singly reinforced Sections” article series.
So let’s begin with different types of problems for “Doubly reinforced sections”.
Problem type 1
To find Moment of resistance (Mr)
In a problem where we have to find Mr, specific data is given so that you could calculate the Moment of resistance. The following data will be given in the problem. I suggest that you make notes of the points below.
Breadth of the beam = b
Effective depth of the beam = d
Area of tensile steel = Ast
Area of compressive steel = Asc
Permissible stress in concrete = σcbc
Permissible stress in steel = σst
Modular ratio = m
Four – step procedure to solving the problem:
Find xc by the following formula,
σcbc/ (σst/m) = xc/(d-xc)
Find x using the following formula,
bxx/2 + (1.5m – 1)Asc (x – d’) = m Ast (d – x)
Determine whether the section is under-reinforced or over-reinforced.
If x < xe; then the section is said to be under-reinforced
If x > xe; then the section is said to be over reinforced.
And finally the last step would be to calculate the Moment of resistance depending on the type of section (under reinforced or over reinforced)
For under reinforced sections, σst is already and σcbc is calculated by the following mentioned formula;
σcbc/ (σst/m) = x/(d-x)
σcbc = σcbc [(x – d’)/x]
To find actual stress in concrete σsc
σsc = 1.5m σcbc
If actual σsc < permissible compressive stress in steel, then take the actual σsc; if it is more, then take permissible compressive stress in steel.
Taking moment about tensile steel
Mr = bx(σcbc/2)[d – (x/3)] + (1.5m -1) Asc.σcbc (d – d’)
For over-reinforced sections, σcbc is known. Find σcbc by the following formula;
σcbc = σcbc [(x – d’)/x]
Problem type 2
To find actual compressive stress in concrete
We would be discussing a detailed step by step procedure for finding the actual compressive stresses in steel and concrete.
Problem type 3
Design of the section
I think this should be enough for all of you to be able to grasp at once. I suggest that you practice regularly until you get perfect with the calculations. The simplest thing you could do is follow the step by step procedure given in the post. Breaking down complex procedure into small steps is the easiest way to learn complex stuff. We would further proceed with a numerical example in which we would follow the above mentioned procedure and calculate the moment of resistance, compressive stresses in steel and concrete.