# V.IS13920 2016-Doubly Reinforced Rectangular Beam

Verify the design of a rectangular beam per IS 13920:2016.

## 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

Check for Member Size

Width to Depth ratio of the member, B/D = 250/300 = 0.833 (Clause 6.1.1; IS 13920:2016)

Width of Beam, B = 250 mm > 200 mm (Clause 6.1.2; IS 13920:2016)

As per clause 6.1.3 of IS 13920:2016, depth of beam shall not have more than 1/4 of clear span. So, the allowable depth not more than, L/4 = 3000/4 = 750 mm > 300 mm. Hence ok. (Clause 6.1.3; IS 13920:2016)

Width of the supporting member, C2 = 400 mm and Breadth of the supporting member, C1 = 400 mm. So, the width of member, B, shall not exceed smaller of (C2) and (C1) = 400 mm or C2 + smaller of (C1) and (C2) = 800 mm. Hence ok. (Clause 6.1.4; IS 13920:2016)

Axial force from STAAD.Pro analysis, Fx = 6.805 kN

Factored axial compressive stress: Fx / (B×D) = 0.09 < 0.08fck = 1.6, Hence OK (Clause 6.1; IS 13920:2016)

## Design for Flexure

Minimum longitudinal reinforcement, рmin = 0.24×√(fck)/fy×B×d = 164.2 mm2

Maximum longitudinal reinforcement, рmax = 0.025×B×d = 1,588 mm2

Limiting moment value, Mz,limit = 0.138fck×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 рt = 1.19
• Percentage of compression reinforcement рc = 0.595

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

Ast,top_a = 1.19×B×d/100 = 755.7. mm2

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

Thus, clause 6.2.1(b) and 6.2.2 of IS 13920:2016 are satisfied.

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

Ast,bot_a = 0.595×B×d/100 = 377.8 mm2

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

Thus, clause 6.2.1(b) and 6.2.2 of IS 13920:2016 are satisfied.

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 рt = 1.161 and compression reinforcement рc = 0.581.

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

Ast,top_b = 1.161×B×d/100 = 737.4 mm2

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

Thus, clause 6.2.1(b) and 6.2.2 of IS 13920:2016 are satisfied.

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

Ast,bot_b = 0.225×B×d/100 = 368.7 mm2

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

Thus, clause 6.2.1(b) and 6.2.2 of IS 13920:2016 are satisfied.

As per clause 6.2.4 of IS 13920:2016, any intermediate section, on top or bottom face should have at least 804.2 mm2 /4 = 201.1 mm2 of steel as the main reinforcement. So the provided reinforcement satisfies this condition.

## Design for Shear

Calculation of resistance moment based on the area of steel provided:

Sagging resistance moment at start:

Mz,as = 0.87×fy×Ast,bot_a×d×[1 - Ast,bot_a×fy / (B×d×fck)] = 32.03 kN·m

Hogging resistance moment at start:

Mz,ah = 0.87×fy×Ast,top_a×d×[1 - Ast,top_a×fy / (B×d×fck)] = 54.37 kN·m

Sagging resistance moment at end:

Mz,bs = 0.87×fy×Ast,bot_b×d×[1 - Ast,bot_b×fy / (B×d×fck)] = 32.03 kN·m

Hogging resistance moment at end:

Mz,bh = 0.87×fy×Ast,top_b×d×[1 - Ast,top_b×fy / (B×d×fck)] = 54.37 kN·m

Simple shear from gravity load on the span:

Va = 1.2×50 kN/m × 3/2 = 90 kN

Factor shear force as per linear structural analysis in STAAD.Pro, Fy = 112.9 kN

Calculation of shear force due to formation of plastic hinge at both ends of the beam plus the factored gravity load on the span.

For sway to right:

Vu,ar = Va - 1.4×(Mz,as + Mz×b×h/3) = 49.68 kN

Vu,br = Va + 1.4×(Mz,as + Mz×b×h/3) = 130.3 kN

For sway to left:

Vu,al = Va + 1.4×(Mz,as + Mz×b×h/3) = 130.3 kN

Vu,bl = Va - 1.4×(Mz,as + Mz×b×h/3) = 49.68 kN

Design shear force at start: Vua = 130.3 kN

Design shear force at end: Vub = 130.3 kN

Design shear force, Vua = 130.3 kN (Considering clause 6.3.4(c) of IS13920:2016)

Provide 2 legged 8 mm diameter bar. Thus the area of shear reinforcement:

Av = 2×(π/4)×(8)2 = 100.5 mm2

Using the formula from IS 456:2000, the spacing for vertical stirrups is:

Sv = 0.87×fv×Asv×d / Vua) = 70.7 mm

Per clause 6.3.5 of IS13920:2016, the spacing of the links over a length of 2d at either end of the beam shall be the least of:
1. d/4 = 64 mm
2. bar diameter of the smallest bars = 6×16 = 96 mm
3. 100 mm
4. Sv = 71 mm

Thus, it should be not greater than 64 mm. Therefore, provide 2 legged - 8 mm links @ 60 mm center to center at left and right end over a length of 2d = 508 mm.

Per Clause 6.3.5.2 of IS13920:2016, the spacing of the stirrups in the mid-span shall not exceed d/2 = 127 mm. Thus, use a spacing of 125 mm elsewhere.

## 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) 384.7 386.39 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) 375.6 377.37 negligible
Spacing of shear reinforcement at ends (mm) 60 60 none

The file C:\Users\Public\Public Documents\STAAD.Pro CONNECT Edition\Samples\ Verification Models\10 Concrete Design\India\IS 13920\2016\IS13920 2016-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 IS13920:2016                                 *
*                                                                             *
*******************************************************************************
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
PRINT SUPPORT REACTION
PRINT MEMBER FORCES
START CONCRETE DESIGN
CODE IS13920 2016
GLD 7 MEMB 59
CLB 0.03 ALL
CLT 0.03 ALL
FC 20000 MEMB 59
FYMAIN 415000 MEMB 59
FYSEC 415000 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-13920:2016  B E A M   D E S I G N   R E S U L T S
====================================================
EUDL CONSIDERED ON MEMBER #     59 IS  50.00 N/MM.
============================================================================
IS-13920   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. Sagging/Hogging moments)|           SHEAR
(in mm) |     P        MZ        MX   Load Case  |    VY       MX  Load Case
----------------------------------------------------------------------------
0.0 |     0.00      0.00      0.00     1     |  112.94     0.77      3
|     0.00    -55.44      0.77     3     |
250.0 |     0.00      0.00      0.00     1     |   94.19     0.77      3
|     0.00    -29.55      0.77     3     |
500.0 |     0.00      0.00      0.00     1     |   75.44     0.77      3
|     0.00     -8.35      0.77     3     |
750.0 |     0.00      8.17      0.77     3     |   56.69     0.77      3
|     0.00      0.00      0.00     1     |
1000.0 |     0.00     20.00      0.77     3     |   37.94     0.77      3
|     0.00      0.00      0.00     1     |
1250.0 |     0.00     27.14      0.77     3     |   19.19     0.77      3
|     0.00      0.00      0.00     1     |
1500.0 |     0.00     29.60      0.77     3     |    0.44     0.77      3
|     0.00      0.00      0.00     1     |
1750.0 |     0.00     27.36      0.77     3     |  -18.31     0.77      3
|     0.00      0.00      0.00     1     |
2000.0 |     0.00     20.44      0.77     3     |  -37.06     0.77      3
|     0.00      0.00      0.00     1     |
2250.0 |     0.00      8.84      0.77     3     |  -55.81     0.77      3
|     0.00      0.00      0.00     1     |
2500.0 |     0.00      0.00      0.00     1     |  -74.56     0.77      3
|     0.00     -7.46      0.77     3     |
2750.0 |     0.00      0.00      0.00     1     |  -93.31     0.77      3
|     0.00    -28.44      0.77     3     |
3000.0 |     0.00      0.00      0.00     1     | -112.06     0.77      3
|     0.00    -54.11      0.77     3     |
*** DESIGN SHEAR FORCE AT SECTION     0.0 IS   133.30 KN.
- CLAUSE 6.3.3 OF IS-13920
*** DESIGN SHEAR FORCE AT SECTION  3000.0 IS   133.30 KN.
- CLAUSE 6.3.3 OF IS-13920
Note :
MOMENT OF RESISTANCE IS CALCULATED BASED ON THE AREA OF STEEL PROVIDED.
IF AREA OF STEEL PROVIDED IS MUCH HIGHER COMPARED TO AREA OF STEEL
REQUIRED MOMENT OF RESISTANCE WILL INCREASE WHICH MAY INCREASE DESIGN
SHEAR FORCE.
----------------------------------------------------------------------------
STAAD SPACE                                              -- PAGE NO.   35
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 )|  386.39/ 402.12( 2-16d )|  8d  @  60 mm
250.0 |  384.93/ 402.12( 2-16d )|  193.20/ 402.12( 2-16d )|  8d  @  60 mm
500.0 |  193.20/ 402.12( 2-16d )|  193.20/ 402.12( 2-16d )|  8d  @  60 mm
750.0 |  193.20/ 402.12( 2-16d )|  193.20/ 402.12( 2-16d )|  8d  @ 100 mm
1000.0 |  193.20/ 402.12( 2-16d )|  251.83/ 402.12( 2-16d )|  8d  @ 100 mm
1250.0 |  193.20/ 402.12( 2-16d )|  349.91/ 402.12( 2-16d )|  8d  @ 100 mm
1500.0 |  193.20/ 402.12( 2-16d )|  385.60/ 402.12( 2-16d )|  8d  @ 100 mm
1750.0 |  193.20/ 402.12( 2-16d )|  353.09/ 402.12( 2-16d )|  8d  @ 100 mm
2000.0 |  193.20/ 402.12( 2-16d )|  257.70/ 402.12( 2-16d )|  8d  @ 100 mm
2250.0 |  193.20/ 402.12( 2-16d )|  193.20/ 402.12( 2-16d )|  8d  @ 100 mm
2500.0 |  193.20/ 402.12( 2-16d )|  193.20/ 402.12( 2-16d )|  8d  @  60 mm
2750.0 |  368.69/ 402.12( 2-16d )|  193.20/ 402.12( 2-16d )|  8d  @  60 mm
3000.0 |  754.73/ 804.25( 4-16d )|  377.37/ 402.12( 2-16d )|  8d  @  60 mm
----------------------------------------------------------------------------
SHEAR DESIGN RESULTS AT DISTANCE 2d (TWICE EFFECTIVE DEPTH) FROM FACE OF THE SUPPORT
SHEAR DESIGN RESULTS AT   704.0 mm AWAY FROM START SUPPORT
VY =    60.14 MX =     0.77 LD=    3
Provide 2 Legged  8d  @  60 mm c/c
SHEAR DESIGN RESULTS AT   704.0 mm AWAY FROM END SUPPORT
VY =   -59.26 MX =     0.77 LD=    3
Provide 2 Legged  8d  @  60 mm c/c
CONFINING REINFORCEMENT  : Provide reinforcement as per CL 9.2.1 b)
of IS-13920:2016 at member start.
CONFINING REINFORCEMENT  : Provide reinforcement as per CL 9.2.1 b)
of IS-13920:2016 at member end.
********************END OF BEAM DESIGN RESULTS********************
```