Design of Singly reinforced Beam with Numerical Examples

We know concrete is strong in compression and weak in tension. When the load is applied to the beam, it gets compressed on the upper part and tensioned at the bottom part. Hence to overcome this tension force, we need a steel rod at the bottom of the beam.

So, when we provide steel rods in the bottom of the beam then, this type of beam is known as a singly reinforced beam but when we provide steel rods on the upper and bottom part of the beam then it will be called a doubly reinforced beam.

Here, we will discuss singly reinforced beams, Please read the article below for steps of designing singly reinforced beams.

Singly reinforced beam design- Step by Step Numerical

Singly reinforced Numerical Example -1

1. A rectangular reinforced concrete beam is to be simply supported on two walls of 125 mm width with a clear span of 6.0 m. The characteristics live load is 12 kN/m and the grade of concrete is M20. Also, use Fe 415 steel. What is the effective span of the Beam? Design a suitable section for bending and determine the necessary tensile steel. 

Solution:

From given,

• Beam is simply supported with width of support = 125 mm
• Clear span = 6.0 m
•  Live load = 12 kN/m
•  fck = 20 N/mm²
•  fy = 415 N/mm²
• Then, effective span (leff) = ?

Design suitable section, also check in deflection:

Assume depth of beam (d) = 

                                                  d = 500 mm

• Calculation of limiting value of bending moment (Mu_ilim)

From ANNEX G of code IS 456:2000;

M_uilim  =      

Or,    [M_uilim = 0.36 fck bx_uimax  X (d-0.42 x_uimax)] -96-P

also, from clause 38.1 of code IS 456:2000;

For Fe 415 steel; —–> (P-70) 

X_uimax = 0.48d

  = 0.48 X 500

  = 240 mm

Or, M_uilim   = 0.36 X 20 X 250 X 240 X (500-0.42 X 240)

• M_uilim = 172.454 kN-m

• Comparison of Mu with Mu_ilim

Since, Mu < Mu_ilim

Hence, the section can be designed as singly reinforced section.

Calculation of area of reinforcement

From, Annex ‘G’ of Code IS 456:2000;

or, 601.5233 = Ast-1.66 × 10¹ (Ast)²

By solving: Ast=677.782 mm 

Provide, diameter of bar = 20 mm

Then, number of bars required =

   = 2.1585

    = 3

Provide 3-20 mm ∅ bars with 

Ast, provided       = 

 = 942 mm² > Ast, calculated ok

Check:

Minimum reinforcement required is

Ast,min      

.’. Ast,min = 256.024 mm² < 942 mm² 

O.K.

Maximum reinforcement

(Ast) max = 4% of bD

= 4/100 x 250 x 550

= 5500mm²  > 942mm²

   O.K.

Design summary: 

b = 250mm

d = 500mm

D = 550mm

3-20 mm ∅ bars

Singly reinforced Numerical Example -2

2. A rectangular Beam is to be simply supported on supports of 230 mm width. The clear span of the beam is 6m. The Beam is to have a width of 300 mm. The characteristics superimposed load is 12 kN/m. Using M20 concrete and Fe 415 steel, design the beam.

Solution:

From given:

• Beam is simply supported on supports of width = 230 mm
• clear span of beam   = 6 m
• width of beam (b)     = 300 mm 
• superimposed load   = 12kN/m
• fck                                = 20 N/mm²
• fy                 = 415 N/mm²

Assumption of depth of beam (d) 

Assume, depth of beam (d) =  1/12^th  to  1/15^th of span

= 1/12 X 6000 to 1/15 X 6000

= 500 to 400

Take, d = 400 

Provide effective cover of the beam   = 500

Then, overall depth of beam       = d + eff. Cover 

       = 400+50

      .’. D = 450 mm   

Effective span calculation: 

From clause 22.2 (a) of code IS 456:2000

For supply support Beam:

The effective span of the beam should be taken as smaller of following two:

1. Clear span + effective depth of beam 

= 6000+400

or,   

     b) c/c of support  

    = 6000+ 230/2 + 230/2

  = 6230mm

  = 6.23m

.’. Effective span  (𝜗 

_eff.)  = 6.23m

• Calculation of total loas, total factored load and factored bending moment

From given, 

Imposed load = 12 kM/m

Self wt. Of beam = 𝛾 

_RCC bD x 1

 = 20×0.3×0.45×1

 = 3.375

.’. Total load = 12+3.375

 = 15.375 kN/m

.’. factored total load = 1.5 x 15.375

.’. W_u    =   23.06 kN/m 

Now ,

  Factored bending moment (mu) is given by 

Mu    = {Wu (𝜗 

_eff.)}/8

= {23.06 x (6.23)²}/8

.’. Mu = 111.878 kM/m 

Also, the maximum shear force occurs at the support and is given by 

V_u =    (wu 𝜗 

_eff. )/2    =   (23.06 x 6.23)/2

.’. V_u  = 71.832kN

• Calculation of limiting value of Bending moment (mu_ilim)

From ANNEX “G” of Code IS 456:2000;

Also, from clause 38.1 of code 456:2000;

For, Fe 415 steel 

X_{ui max}  = 0.48 x d 

   = 0.48 x 400

X_{ui max}  = 192mm

.’. M_{ui lim} = 0.36 x 20 x 300 x 192 x (400 – 0.42 x 192)

.’. M_{ui lim} = 132.445kN/m 

• Comparison of Mu with  M_{ui lim}  

Since, Mu (111.878) < M_{ui lim} (132.445)

Hence, the section can be designed as singly reinforced section 

• Calculation of area of Reinforcement:

From ANNEX “G” of Code IS 456:2000; 

Or, 774.671 = Ast – 1.729 x 10^-4 (Ast)²

By solving, we get     Ast = 921.487mm²

Provider diameter of bars = 20mm 

Then number of bars required is 

 = 2.9346

= 3

.’. Provide 3 bars of 20 mm diameter with Ast, 

Provided = 3 x (π/4) x (20)²

         = 942mm² > 921.487mm² 

• Design of  shear reinforcement 

From clause 40.1 of code IS 456:2000; 

Nominal shear stress in beam is 

                     Where, b = total width

 d = effective depth 

                     = 0.598 N/mm² 

Also, percentage of reinforcement in beam is 

Pt = 0.785 

Then, from table 19 of code IS 456:2000;  

Design shear strength of concrete

For M20 concrete and Pt = 0.785

Pt	N/mm2
At 0.75	0.56
At 1.0	0.62
At 0.785	?

By using linear interpolation 

.’. = 0.5684 N/mm² 

Again, from table 20 of code IS 456:2000; maximum shear stress              (max) for M20 concrete is 

= 2.8 N/mm²

Thus, 

< <max

Hence, shear force resisted by concrete 

= x bd

= 0.5684 x 300 x 400

= 68208 N

.’. Shear force to be resisted by reinforcement is 

V_us      = V_u – x bd

 = 71832 – 68208

V_us = 3624 N

Now, provide 2 legged 6mm dia vertical stirrups and Fe 250 steel 

Then, from clause 40.4 of code 456:2000, 

For vertical stirrups 

Where, fy = 250 N/mm²

V_us    = 3624 N

A_sv    = 2 x π/4 x (6)²

            = 56.52 mm²   

d       = 40 mm (depth of beam)

sv      = spacing of stirrups to be calculated then 

3624 = (0.87 x 250 x 56.25 x 400)/2

.’. sv  = 1356.854 mm

But, maximum spacing is permitted at a inimum of two: 

1. 0.75 d = 0.75 x 400 = 300 mm
2. 300 mm 

.’. provide S_v = 300 mm 

Hence, provide 2 legged 6mm vertical stirrups @300 mm c/c spacing 

• Check for deflection: 

From clause 23.2.1 of code IS 456:2000; 

For simply supported beam 

Basic value of span of effective depth ratio = 20

Also, factors for no compressive steel (k₁) = 1 

   for not flanged section (k₂) = 1 

And for modification factor (k₃) for tensil steel 

From figure 4 of same clause,

                    fs   = 265.4585 N/mm²

    And also , Pt = 0.785

Then, from figure 4 of same clause, 

For fs      =      235.4585 and 

       Pt     =      0.785

Modification factor (k₃)  = 1.18

Then, maximum permitted span to effective depth ratio I.e; 

                              = 1 x 1 x 1.18 x 20

= 23.6

Also,

        =  15.575

Thus 

Hence, deflection control is satisfactory.

• Check for minimum reinforcement 

The minimum reinforcement is given by 

Ast_min = (0.85 bd)/fy

  = (0.85 x 400 x 300) / 415

  = 245.783 mm²   < 942 mm² 

• Check for maximum reinforcement 

The maximum reinforcement is given by

Ast_max  = 4% of bD 

    = 4/100 x 300 x 450

    = 5400mm² > 942mm²

• Details of reinforcement are given below: 

I hope this article on the design of singly reinforced beams remains helpful for you.

Happy learning

Civil Concept

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