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V. ACI 318-14 Rectangular Beam without Torsion

Design of a doubly reinforced rectangular beam.

Reference

  1. ACI 318-14
  2. PCA Notes 2011 Example 6.2

Details

To determine the Required Area of Tension and Compression Reinforcement for given Concrete and Steel Properties with Given Factored Moment.

Design of Rectangular Beam with Compression Reinforcement-Doubly Reinforced

A beam cross-section is limited to size 14in X 23 in . Determine the required area of reinforcement for a factored moment Mu= 516 ft-kips. fc= 4000psi, fy=60,000 psi.,

Design Properties, available:

Validation

dcover = 2.5 in

Step 1: Check if compression reinforcement is required, using Ζ = 0.9.

See table 6-1 of PCA Notes (ref. 2).

b = 14 in

d = 20.5 in

Note: Assume d = dt

Mn = Mu/Ζ = 573.33 ft-kip

Rn = Mn / (b*d2) = 1.169(10)3 psi

Rnt = 911 psi

Note: Rnt = Maximum Rn for tension controlled section for that particular grade for concrete.

Rn > Rnt

Hence, both compression and tension reinforcement is required.

Step 2: Determine maximum moment with compression reinforcement,

Mnt

This condition corresponds to tension controlled limit (Ζ = 0.9).

Depth of neutral axis, c = 0.375dt = 0.375d = 7.687 in

β1 = 0.85 (from table 6.1 (ref. 2)

a = β1*c = 6.534 in

Compressive force, C = 0.85·fc·a·b = 311.036 kips

Tension force, T = C = 311.036 kips

Nominal moment strength resisted by the concrete section, without compression reinforcement, Mnt = T(d - a/2) = 446.669 ft-kips

Nominal moment strength resisted by the concrete section, with compression reinforcement, Mnc = Mn - Mnt = 126.664 ft-kips

Step 3: Determine area of tension steel

Area of tension steel required, As,nt , to develop Mnt

As,nt = T / fy = 5.184 in2

Step 4: Determine additional area of tension steel required

Area of tension steel required, As,additional , required to counteract additional moment due to T-C couple.

As,additional = (Mn - Mnt) / [ fy· (d - dcover)] = 1.407 in2

Total tension steel required:

As = As + As,nt = 6.591 in2

Comparison

Table 1. Comparison of results
Parameter STAAD.Pro Reference Difference Comments
Area Required (compression) (in2) 1.441 1.61 11.0 See note (*) below.
Area Required (tension) (in2) 6.591 6.355 3.6% See note (*) below.
Note: (*) The required steel calculation above assumes a single layer of bottom bars and the effective depth = 0.375*dt based on the single bar layer. The reference (ref. 2) calculations only considers a strength design and does not check for the minimum spacing requirements. STAAD.Pro however considers two layers of bars in the design to account for the minimum criteria and hence uses a neutral axis position based on the two bar layers. Hence the minor difference in the steel areas calculated.

STAAD Input

The file C:\Users\Public\Public Documents\STAAD.Pro CONNECT Edition\Samples\ Verification Models\10 Concrete Design\US\ACI\318-2014\ACI 318-14 Rectangular Beam Without Torsion.std is typically installed with the program.

STAAD SPACE
START JOB INFORMATION
***********************************************************************
* This example is a replica of example no 6.2 of PCA Notes 318-11.
***********************************************************************
ENGINEER DATE 29-Aug-18
JOB NAME PCA Example 6.2
JOB CLIENT Bentley Systems Inc.
ENGINEER NAME TK
END JOB INFORMATION
INPUT WIDTH 79
UNIT FEET KIP
JOINT COORDINATES
1 0 0 0; 2 8 0 0;
MEMBER INCIDENCES
1 1 2;
DEFINE MATERIAL START
ISOTROPIC CONCRETE
E 453600
POISSON 0.17
DENSITY 0.14999
ALPHA 5.5e-06
DAMP 0.05
TYPE CONCRETE
STRENGTH FCU 576
END DEFINE MATERIAL
MEMBER PROPERTY AMERICAN
1 PRIS YD 1.91667 ZD 1.16667
CONSTANTS
MATERIAL CONCRETE ALL
SUPPORTS
1 FIXED
LOAD 1 LOADTYPE Dead  TITLE LOAD CASE 1
JOINT LOAD
2 MZ 464.417
PERFORM ANALYSIS
PRINT SUPPORT REACTION ALL
START CONCRETE DESIGN
CODE ACI
UNIT INCHES KIP
MAXMAIN 8 ALL
MINMAIN 8 ALL
*MIMB 6 ALL
*MXMB 6 ALL
CLT 2 ALL
CLB 2 ALL
TRACK 2 ALL
DESIGN BEAM ALL
END CONCRETE DESIGN
FINISH

STAAD Output

                       STAAD.PRO CONCRETE DESIGN - (ACI-318-14)   v2.0                
                       ***********************************************
Units: KIP   , INCHES (Unless Noted Otherwise)
 Member :      1 
DESIGN SUMMARY
-----------------------------------------------------------------------------
| Status         :   Pass         Type    : Beam            Length:   96.000 |
| Critical Ratio :   0.910        Criteria: Flexure                          |
| Critical Clause:   9.5.2                                                   |
-----------------------------------------------------------------------------
CROSS SECTION
-------------------------------------------------------
| Shape: Rectangular | Width:   14.00 | Depth:   23.00 |
-------------------------------------------------------
DESIGN INPUTS
-------------------------------------------------------------------------
| Concrete | Fc           4.000 |                     | Ec     0.360E+04 |
| Steel    | Fy(main)    60.000 | Fy(trans)    60.000 | Es     0.290E+05 |
| Cover    | Top          2.000 | Bottom        2.000 | Sides      1.500 |
-------------------------------------------------------------------------
CRITICAL STRENGTH RESULTS
-----------------------------------------------------------------------
| Category |    Demand     |  Min Capacity  |  Max Capacity  |  Ratio  |
-----------------------------------------------------------------------
| Axial    |          0.000|       -871.565 |        554.580 |   0.000 |
| Flexure  |       5573.004|      -2586.136 |       6122.188 |   0.910 |
| Shear Y  |          0.000|        -21.019 |         21.019 |   0.000 |
| Shear Z  |          0.000|        -22.720 |         22.720 |   0.000 |
| Torsion  |          0.000|          0.000 |          0.000 |   0.000 |
-----------------------------------------------------------------------
LONGITUDINAL BAR DETAILS AT CROSS SECTIONS
---------------------------------------------------------------------------------
| Distance | Position |  Ast-reqd  |  Ast-prov  | No(s)bars| Size | No of Layers |
---------------------------------------------------------------------------------
|    0.000 |     Top  |      1.610 |      2.370 |     3    |  # 8 |       1      |
|          |  Bottom  |      6.355 |      7.900 |    10    |  # 8 |       2      |
|   24.000 |     Top  |      1.610 |      2.370 |     3    |  # 8 |       1      |
|          |  Bottom  |      6.355 |      7.900 |    10    |  # 8 |       2      |
|   48.000 |     Top  |      1.610 |      2.370 |     3    |  # 8 |       1      |
|          |  Bottom  |      6.355 |      7.900 |    10    |  # 8 |       2      |
|   72.000 |     Top  |      1.610 |      2.370 |     3    |  # 8 |       1      |
|          |  Bottom  |      6.355 |      7.900 |    10    |  # 8 |       2      |
|   96.000 |     Top  |      1.610 |      2.370 |     3    |  # 8 |       1      |
|          |  Bottom  |      6.355 |      7.900 |    10    |  # 8 |       2      |
---------------------------------------------------------------------------------
      STAAD SPACE                                              -- PAGE NO.    6
LONGITUDINAL BAR LAYOUT
---------------------------------------------------------------------------
|                |    Bars     |      Location     | Distance  |   Anchor  |
|       Position | Nums | Size | Start  |   End    | From Face | Start End |
---------------------------------------------------------------------------
|            Top |    3 | # 8  |    0.00|    96.00 |     2.50  |  Yes  Yes |
|         Bottom |    5 | # 8  |    0.00|    96.00 |     2.50  |  Yes  Yes |
|         Bottom |    5 | # 8  |    0.00|    96.00 |     4.50  |  Yes  Yes |
---------------------------------------------------------------------------
TRANSVERSE BAR LAYOUT
------------------------------------------------------------------------------------
|                                |        Asv        |     Rebar Specification      |
| Zone | Dir.|  From   |    To   |  Reqd.  |  Prov.  | Nums | Size | Spacing | Legs |
------------------------------------------------------------------------------------
|    1 |  Y  |    0.00 |   96.00 |    0.03 |    0.03 |    8 |  # 4 |  13.71  |   2  |
|    1 |  Z  |    0.00 |   96.00 |    0.03 |    0.03 |    8 |  # 4 |  13.71  |   2  |
------------------------------------------------------------------------------------
------------------------ Member:     1 Design Ends ------------------------