Posts Tagged Numerical examples for doubly reinforced sections
Numerical example for determining stresses in steel and concrete
In our article series for “Design of Doubly reinforced sections”, we covered the following:
In our previous article, we discussed a detailed 6 step procedure for determining stresses in steel and concrete. Now we shall move on with a numerical example in which we will use the 6 step procedure to solve the problem.
Problem Type two: Determining stresses in steel and concrete using the 6 step procedure
A rectangular beam is 200mm wide and 480mm deep. It has to resist a bending moment of 100 kN-m. The reinforcedment consists of four 25mm ⏀ bars on tension side and three 22mm⏀bars on compression side. The centres of bars being 30mm from the top and bottom edges of the beam. Find the stresses set up in steel and concrete. m=18.66
Given data is as follows:
Breadth of the beam = b = 200mm
Effective depth of the beam = d = 480 – 30 = 450mm
Distance of compressive steel from the top edge of the beam to the centre of the steel = d’ = 30mm
Bending moment = M = 100kN-m
Modular ratio = m = 18.66
Area of tensile steel = Ast = 4 π/4 x 25 x 25 = 1964 mm2
Area of compressive steel = Asc = 4 π/4 x 22 x 22 = 1140 mm2
bx.x/2 + (1.5m – 1)Asc (x – d’) = mAst(d-x)
200x2/2 + (1.5×18.66 – 1) 1140 (x – 30)
= 18.66 x 1964 x (450 – x)
Therefore, x2 + 674.17x – 174147 = 0
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)
Moment of Resistance calculations | Doubly reinforced sections
In our article series for “Doubly reinforced sections”, we have covered the following:
Now we shall move on with a solved example. This will help you understand the methods in a better way. I suggest that you do them yourselves too. Practice will help you make your concepts more concrete and clear.
An reinforced concrete 300mm x 600mm effective dimensions is provided with tensile and compressive reinforcement of 1256mm2 each. The compressive steel is placed 30mm from the top edge of the beam. If σcbc = 7N/mm2, σst = 190N/mm2 and m = 13.33, find the moment of resistance of beam by following two methods:
1) Elastic theory method
2) Steel beam theory method