# V. IS456 2000-Doubly Reinforced Rectangular Beam

Verify the design of a rectangular beam per IS 456:2000.

## Details

A three story, concrete structure is subject to gravity loads only. Beam 59 is to be designed:
Section properties:
• Length of beam, L = 3,000 mm (clear span)
• Width of beam, B = 250 mm
• Depth of beam, D = 300 mm
• Clear cover = 30 mm
• Diameter of main reinforcement = 16 mm
• Diameter of stirrups = 8 mm
Material properties:
• Characteristic strength of concrete, fck = 20 N/mm2
• Yield strength of steel, fy = 415 N/mm2

Loads: total uniform force = 50 kN/m

## Validation

Effective Depth, d = 300 – 30 – 8 – (16/2) = 254 mm

Depth of Compression Reinforcement, d’ = 30 + (16/2) + 8 = 46 mm

## Design for Flexure

Limiting moment value, Mz,limit = 0.138×fck×B×d2 = 44.52 kN·m

From the STAAD.Pro analysis, Mz = 55.44 kN·m > Mz,limit at the start of the member (x = 0).

So, the actual moment is greater than limiting moment then the section will be design as doubly reinforced section.

From SP16 table 50 flexure, reinforcement percentage for doubly reinforced section:
• Percentage of tension reinforcement ρt = 1.212
• Percentage of compression reinforcement ρc = 0.288

Reinforcement at Start of Member (x = 0): top bars

Ast,top_a = 1.212×B×d/100 = 769.6 mm2

Provide 4 - 16 mm diameter bars. Actual Ast, top = 4×(π/4)×(16)2 = 804.2 mm2

Reinforcement at Start of Member (x = 0): bottom bars

Ast,bot_a = 0.288×B×d/100 = 182.9 mm2

Provide 2 - 16 mm diameter bars. Actual Ast, bot = 2×(π/4)×(16)2 = 402.1 mm2

From the STAAD.Pro analysis, Mz = 54.11 kN·m > Mz,limit at the end of the member (x = 3 m).

From SP16 table 50 flexure, reinforcement percentage for doubly reinforced section:
• Percentage of tension reinforcement ρt = 1.183
• Percentage of compression reinforcement ρc = 0.256

Reinforcement at End of Member (x = 3 m): top bars

Ast,top_b = 1.183×B×d/100 = 751.2 mm2

Provide 4 - 16 mm diameter bars. Actual Ast, top = 3×(π/4)×(16)2 = 804.2 mm2

Reinforcement at End of Member (x = 3 m): bottom bars

Ast,bot_b = 0.256×B×d/100 = 162.6 mm2

Provide 2 - 16 mm diameter bars. Actual Ast, bot = 2×(π/4)×(16)2 = 402.1 mm2

## Design for Shear

Provide percentage of tension steel, ρt = 1.266.

From Table 19 of IS 456:2000, for M20 grade concrete, shear stress: Tc = 0.673 N/mm2.

Factored shear force as per linear structural analysis in STAAD.Pro, Fy = 117.9 kN.

The nominal shear stress, Tv = 117.9 / (B×d) = 117.9×103 / (250 × 254) = 1.86 N/mm2.

The maximum shear stress from Table 20 of IS 456:2000 for M20 grade concrete, Tc,max = 2.8 N/mm2.

Shear strength of the reinforcement: Vus = Fy - Tc×B×d = 117.9 - 0.673×250×254×103 = 75.16 kN.

Provide 2 legged 8 mm diameter bar. So, the area of shear reinforcement, Asv = π/4 × 82×2 = 100.5 mm2.

Using formula of IS 456:2000 spacing for vertical stirrups: sv = 0.87×(fy ×Asv × d) / Vus = 122.8 mm.

So, provide 2 legged 8 mm diameter bar stirrups at 100 mm center to center.

## Results

Table 1. Comparison of results
Required area of steel at top of start section (mm2) 769.4 772.79 negligible
Required area of steel at bottom of start section (mm2) 182.6 186.11 negligible
Required area of steel at top of end section (mm2) 751.2 754.73 negligible
Required area of steel at bottom of end section (mm2) 162.3 166.44 negligible
Spacing of shear reinforcement at ends (mm) 100 100 none

The file C:\Users\Public\Public Documents\STAAD.Pro CONNECT Edition\Samples\ Verification Models\10 Concrete Design\India\IS456 2000-Doubly Reinforced Rectangular Beam.STD is typically installed with the program.

``````STAAD SPACE
START JOB INFORMATION
ENGINEER DATE 25-Jul-18
END JOB INFORMATION
INPUT WIDTH 79
*******************************************************************************
*This Problem is created to verify the design results produced by the program *
*with hand calculated design results for a rectangular(300 mm * 250 mm) Doubly*
*reinforced  beam section as per IS456:2000                                   *
*                                                                             *
*******************************************************************************
UNIT METER KN
JOINT COORDINATES
1 0 0 0; 2 0 3 0; 3 0 6 0; 4 0 9 0; 5 3 9 0; 6 6 9 0; 7 6 6 0; 8 6 3 0;
9 6 0 0; 10 3 6 0; 11 3 3 0; 12 3 0 0; 13 0 0 3; 14 0 3 3; 15 0 6 3; 16 0 9 3;
17 3 9 3; 18 6 9 3; 19 6 6 3; 20 6 3 3; 21 6 0 3; 22 3 6 3; 23 3 3 3; 24 3 0 3;
25 0 0 6; 26 0 3 6; 27 0 6 6; 28 0 9 6; 29 3 9 6; 30 6 9 6; 31 6 6 6; 32 6 3 6;
33 6 0 6; 34 3 6 6; 35 3 3 6; 36 3 0 6;
MEMBER INCIDENCES
1 1 2; 2 2 3; 3 3 4; 4 4 5; 5 5 6; 6 6 7; 7 7 8; 8 8 9; 9 3 10; 10 10 7;
11 2 11; 12 11 8; 13 5 10; 14 10 11; 17 11 12; 18 2 14; 19 3 15; 20 4 16;
21 5 17; 22 6 18; 23 7 19; 24 8 20; 25 10 22; 26 11 23; 27 13 14; 28 14 15;
29 15 16; 30 16 17; 31 17 18; 32 18 19; 33 19 20; 34 20 21; 35 15 22; 36 22 19;
37 14 23; 38 23 20; 39 17 22; 40 22 23; 41 23 24; 42 14 26; 43 15 27; 44 16 28;
45 17 29; 46 18 30; 47 19 31; 48 20 32; 49 22 34; 50 23 35; 51 25 26; 52 26 27;
53 27 28; 54 28 29; 55 29 30; 56 30 31; 57 31 32; 58 32 33; 59 27 34; 60 34 31;
61 26 35; 62 35 32; 63 29 34; 64 34 35; 65 35 36;
DEFINE MATERIAL START
ISOTROPIC CONCRETE
E 2.17185e+07
POISSON 0.17
DENSITY 23.5616
ALPHA 1e-05
DAMP 0.05
TYPE CONCRETE
STRENGTH FCU 27579
END DEFINE MATERIAL
MEMBER PROPERTY AMERICAN
4 5 9 TO 12 18 TO 26 30 31 35 TO 38 42 TO 50 54 55 59 TO 62 PRIS YD 0.3 ZD 0.25
1 TO 3 6 TO 8 13 14 17 27 TO 29 32 TO 34 39 TO 41 51 TO 53 56 TO 58 -
63 TO 65 PRIS YD 0.4 ZD 0.4
CONSTANTS
MATERIAL CONCRETE ALL
SUPPORTS
1 9 12 13 21 24 25 33 36 FIXED
4 5 9 TO 12 18 TO 26 30 31 35 TO 38 42 TO 50 54 55 59 TO 62 UNI GY -3.125
4 5 9 TO 12 18 TO 26 30 31 35 TO 38 42 TO 50 54 55 59 TO 62 UNI GY -1.875
4 5 20 22 44 46 54 55 UNI GY -7.875
21 30 31 45 UNI GY -15.75
9 TO 12 18 19 23 24 42 43 47 48 59 TO 62 UNI GY -45
25 26 35 TO 38 49 50 UNI GY -90
LOAD COMB 3 GENERATED IS:456/IS:800 GENRAL_STRUCTURES 1
1 1.5 2 1.5
LOAD COMB 4 GENERATED IS:456/IS:800 GENRAL_STRUCTURES 2
1 1.2 2 1.2
LOAD COMB 5 GENERATED IS:456/IS:800 GENRAL_STRUCTURES 3
1 1.5
LOAD COMB 6 GENERATED IS:456/IS:800 GENRAL_STRUCTURES 4
1 0.9
1 1.0 2 1.0
PERFORM ANALYSIS PRINT ALL
START CONCRETE DESIGN
CODE INDIAN
CLB 0.03 ALL
CLT 0.03 ALL
FC 20000 MEMB 59
FYMAIN 413688 MEMB 59
FYSEC 413688 MEMB 59
MAXMAIN 16 MEMB 59
MAXSEC 8 MEMB 59
MINMAIN 16 MEMB 59
MINSEC 8 MEMB 59
TRACK 2 MEMB 59
DESIGN BEAM 59
END CONCRETE DESIGN
FINISH
``````

```               ====================================================
IS 456 - 2000  B E A M   D E S I G N   R E S U L T S
====================================================
============================================================================
IS-456    L I M I T    S T A T E    D E S I G N
B E A M  N O.      59   D E S I G N  R E S U L T S
M20                    Fe415 (Main)               Fe415 (Sec.)
LENGTH:  3000.0 mm      SIZE:   250.0 mm X  300.0 mm   COVER: 40.0 mm
----------------------------------------------------------------------------------
SECTION |FLEXURE  (Maxm. Sagg./Hogg./Eqv. moments) |           SHEAR
(in mm) |    P       MZ       MX       ME   Load # |    VY       MX       VE  Load #
----------------------------------------------------------------------------------
0.0 |    0.00     0.00     0.00     0.00     1 |  112.94     0.77   117.85     3
|    0.00   -55.44     0.77   -56.44     3 |
250.0 |    0.00     0.00     0.00     0.00     1 |   94.19     0.77    99.10     3
|    0.00   -29.55     0.77   -30.54     3 |
500.0 |    0.00     0.00     0.00     0.00     1 |   75.44     0.77    80.35     3
|    0.00    -8.35     0.77    -9.34     3 |
750.0 |    0.00     8.17     0.77     9.16     3 |   56.69     0.77    61.60     3
|    0.00     0.00     0.00     0.00     1 |
1000.0 |    0.00    20.00     0.77    20.99     3 |   37.94     0.77    42.85     3
|    0.00     0.00     0.00     0.00     1 |
1250.0 |    0.00    27.14     0.77    28.13     3 |   19.19     0.77    24.10     3
|    0.00     0.00     0.00     0.00     1 |
1500.0 |    0.00    29.60     0.77    30.59     3 |    0.44     0.77     5.35     3
|    0.00     0.00     0.00     0.00     1 |
1750.0 |    0.00    27.36     0.77    28.36     3 |  -18.31     0.77    23.21     3
|    0.00     0.00     0.00     0.00     1 |
2000.0 |    0.00    20.44     0.77    21.44     3 |  -37.06     0.77    41.96     3
|    0.00     0.00     0.00     0.00     1 |
2250.0 |    0.00     8.84     0.77     9.83     3 |  -55.81     0.77    60.71     3
|    0.00     0.00     0.00     0.00     1 |
2500.0 |    0.00     0.00     0.00     0.00     1 |  -74.56     0.77    79.46     3
|    0.00    -7.46     0.77    -8.45     3 |
2750.0 |    0.00     0.00     0.00     0.00     1 |  -93.31     0.77    98.21     3
|    0.00   -28.44     0.77   -29.43     3 |
3000.0 |    0.00     0.00     0.00     0.00     1 | -112.06     0.77   116.96     3
|    0.00   -54.11     0.77   -55.11     3 |
----------------------------------------------------------------------------
STAAD SPACE                                              -- PAGE NO.   11
SUMMARY OF REINF. AREA (Sq.mm)
----------------------------------------------------------------------------
SECTION |          TOP            |         BOTTOM          |    STIRRUPS
(in mm) | Reqd./Provided reinf.   | Reqd./Provided reinf.   |   (2 legged)
----------------------------------------------------------------------------
0.0 |  772.79/ 804.25( 4-16d )|  186.11/ 402.12( 2-16d )|  8d  @ 100 mm
250.0 |  384.93/ 402.12( 2-16d )|    0.00/ 402.12( 2-16d )|  8d  @ 100 mm
500.0 |  153.61/ 402.12( 2-16d )|    0.00/ 402.12( 2-16d )|  8d  @ 100 mm
750.0 |    0.00/ 402.12( 2-16d )|  129.04/ 402.12( 2-16d )|  8d  @ 100 mm
1000.0 |    0.00/ 402.12( 2-16d )|  251.83/ 402.12( 2-16d )|  8d  @ 100 mm
1250.0 |    0.00/ 402.12( 2-16d )|  349.91/ 402.12( 2-16d )|  8d  @ 100 mm
1500.0 |    0.00/ 402.12( 2-16d )|  385.60/ 402.12( 2-16d )|  8d  @ 100 mm
1750.0 |    0.00/ 402.12( 2-16d )|  353.09/ 402.12( 2-16d )|  8d  @ 100 mm
2000.0 |    0.00/ 402.12( 2-16d )|  257.70/ 402.12( 2-16d )|  8d  @ 100 mm
2250.0 |    0.00/ 402.12( 2-16d )|  129.04/ 402.12( 2-16d )|  8d  @ 100 mm
2500.0 |  153.61/ 402.12( 2-16d )|    0.00/ 402.12( 2-16d )|  8d  @ 100 mm
2750.0 |  368.69/ 402.12( 2-16d )|    0.00/ 402.12( 2-16d )|  8d  @ 100 mm
3000.0 |  754.73/ 804.25( 4-16d )|  166.44/ 402.12( 2-16d )|  8d  @ 100 mm
----------------------------------------------------------------------------
SHEAR DESIGN RESULTS AT DISTANCE d (EFFECTIVE DEPTH) FROM FACE OF THE SUPPORT
-----------------------------------------------------------------------------
SHEAR DESIGN RESULTS AT   452.0 mm AWAY FROM START SUPPORT
VY =    79.04 MX =     0.77 LD=    3
Provide 2 Legged  8d  @ 100 mm c/c
SHEAR DESIGN RESULTS AT   452.0 mm AWAY FROM END SUPPORT
VY =   -78.16 MX =     0.77 LD=    3
Provide 2 Legged  8d  @ 100 mm c/c
============================================================================
```