V. ACI 318-11 Rectangular Singly Reinforced Beam

Design of a singly reinforced concrete beam per the ACI 318-2011 code.

References

ACI 318-11
PCA Notes 2011: Example 7.1

Details

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

Select a rectangular beam size and required reinforcement to carry service load moments M_DL = 56 ft-kips and M_LL= 35 ft-kips. Select reinforcement to control flexural cracking.

A 10' simply supported beam is subjected to an ultimate uniform load of 9.736 k/ft.

f'_c = 4,000 psi
f_y = 60,000 psi
cover = 2 in

Validation

This is a complete design procedure for rectangular sections with tension reinforcement only, a minimum beam depth will be computed using the maximum reinforcement permitted for tension-controlled flexural members, ρ_t..

d_cover = dist. from bottom of beam up to center of bottom bar = C_c + 1/2(bar diameter) = 2.5 in = 63.5 mm

Determine the maximum tension-controlled reinforcement ratio fro material strengths from Table 6-1 of PCA Notes on ACI 318-11.

ρ_t. = 0.01806

Compute bd² required:

M_u = 1.2·M_DL + 1.6·M_LL = 123 ft·kips

R_{n} = ρ_{t} f_{y} (1 - 0.5 ρ_{t} \frac{f_{y}}{0.85 {f'}_{c}}) = 910.9 psi

{b d^{2}}_{r e q' d} = \frac{M_{u}}{ϕ R_{n}} = 1, 803 {in}^{3}

Size members so that bd²_(prov) ≥ bd²_(req'd).

b = 10 in (column width)

d = \sqrt{\frac{1, 803}{10}} = 13.43 in

Total beam depth, D > d + d_cover = 15.93 in; use a D = 16 in; d = 13.5 in.

ρ_{p r o v} = \frac{0.85 {f'}_{c}}{f_{y}} (1 - \sqrt{1 - \frac{2 R_{n, p r o v}}{085 {f'}_{c}}}) = 0.018

A_{s} = ρ_{prov} \cdot b \cdot d_{prov} = 2.406 {in}^{2}

Verify the design.

T = A_sf_y = 2.406×60 = 144.36 kips

Design moment strength, ϕM_n = D_m

Strength reduction factor, ϕ = 0.9

D_{m} = ϕ [A_{s} f_{y} (d_{prov} - \frac{a}{2})] = 123.2 ft·kip

D_m > M_u, hence design OK.

Compute minimum reinforcement

A_{s, \min} = {}_{\max}{| \begin{matrix} \frac{200}{f_{y}} b d = 0.45 {in}^{2} \\ \frac{3 \sqrt{{f^{'}}_{c}}}{f_{y}} b d = 0.427 {in}^{2} \end{matrix}}

Since A_s > A_s,min, the design satisfies the minimum reinforcement criteria.

Compute the reinforcement provided

Use 2-#10 bars: A_s = 2.53 in²

Check Bar Spacing

End bar diameter, ⌀ = 1.27 in, two bars in a layer

f_{s} = \frac{2}{3} f_{y} = 40 {(10)}^{3} psi

Maximum spacing allowed:

s_{\max} = {}_{\min}{| \begin{array}{l} 15 (\frac{40,000}{f_{s}}) - 2.5 C_{c} \\ 12 (\frac{40,000}{f_{s}}) \end{array}} = 10 in

Spacing provided:

s_{p r o v} = \frac{b - (2 C_{c} + 2 \times 0.5 \emptyset)}{N - 1} = 4.73 in

Results

Table 1. Comparison of results
Parameter	Hand Calculation	STAAD.Pro	Difference
Area of steel required (in²)	2.406	2.44	negligible
Maximum bar spacing (in)	10	10	none
Bar spacing (in)	4.73	4.73	none

STAAD Input

The file C:\Users\Public\Public Documents\STAAD.Pro CONNECT Edition\Samples\ Verification Models\10 Concrete Design\US\ACI-318-11 Singly Reinforced Rectangular Beam.std is typically installed with the program.

STAAD PLANE RECTANGULAR CONCRETE BEAM DESIGN PER ACI 318 2011
START JOB INFORMATION
ENGINEER DATE 18-December-19
JOB NAME PCA Example 7.1
JOB CLIENT Bentley Systems Inc.
ENGINEER NAME TK
END JOB INFORMATION
INPUT WIDTH 79
UNIT FEET KIP
JOINT COORDINATES
1 0 0 0; 2 10 0 0;
MEMBER INCIDENCES
1 1 2;
UNIT INCHES KIP
MEMBER PROPERTY AMERICAN
1 PRIS YD 16 ZD 10
DEFINE MATERIAL START
ISOTROPIC MATERIAL1
E 3150
POISSON 0.17
END DEFINE MATERIAL
CONSTANTS
MATERIAL MATERIAL1 ALL
SUPPORTS
1 2 PINNED
UNIT FEET KIP
LOAD 1 DEAD LOAD
MEMBER LOAD
1 UNI GY -4.48
LOAD 2 LIVE LOAD
MEMBER LOAD
1 UNI GY -2.8
LOAD COMB 3 ACI 318R-02
1 1.2 2 1.6 
PERFORM ANALYSIS
UNIT INCHES KIP
START CONCRETE DESIGN
CODE ACI 2011
*MINMAIN 8 ALL
*MAXMAIN 9 ALL
TRACK 2 ALL
DESIGN BEAM 1
END CONCRETE DESIGN
FINISH

STAAD Output

          ACI 318-11   BEAM  NO.     1   DESIGN RESULTS 
          =============================================
   LEN -  10.00FT.  FY - 60000.  FC - 4000.  SIZE - 10.00 X 16.00 INCHES
   LEVEL    HEIGHT     BAR INFO       FROM            TO         ANCHOR 
          FT.    IN.              FT.     IN.     FT.     IN.   STA  END
   _____________________________________________________________________
     1    0 + 2-3/4    2-NUM.10     0 + 0-0/0     10 + 0-0/0    YES  YES
   |----------------------------------------------------------------|
   |   CRITICAL POS MOMENT=    123.20 KIP-FT  AT  5.00 FT, LOAD    3|
   |   REQD STEEL=  2.44 IN2, RHO=0.0183, RHOMX=0.0214 RHOMN=0.0033 |
   |   MAX/MIN/ACTUAL BAR SPACING= 10.00/ 2.54/ 4.73 INCH           |
   |   REQD. DEVELOPMENT LENGTH = 48.52 INCH                        |
   |----------------------------------------------------------------|
     Cracked Moment of Inertia Iz at above location =    1817.34 inch^4
   REQUIRED REINF. STEEL SUMMARY :
   -------------------------------
   SECTION     REINF STEEL(+VE/-VE)     MOMENTS(+VE/-VE)    LOAD(+VE/-VE)
   (FEET)         (SQ. INCH)                (KIP-FEET)
     0.00        0.000/    0.000         0.00/      0.00        3/    1
     0.83        0.650/    0.000        37.64/      0.00        3/    0
     1.67        1.231/    0.000        68.44/      0.00        3/    0
     2.50        1.723/    0.000        92.40/      0.00        3/    0
     3.33        2.101/    0.000       109.51/      0.00        3/    0
     4.17        2.341/    0.000       119.78/      0.00        3/    0
     5.00        2.423/    0.000       123.20/      0.00        3/    0
     5.83        2.340/    0.000       119.78/      0.00        3/    0
     6.67        2.101/    0.000       109.51/      0.00        3/    0
     7.50        1.723/    0.000        92.40/      0.00        3/    0
     8.33        1.231/    0.000        68.44/      0.00        3/    0
     9.17        0.650/    0.000        37.64/      0.00        3/    0
    10.00        0.000/    0.000         0.00/      0.00        3/    1
          B E A M  N O.     1 D E S I G N  R E S U L T S - SHEAR
    AT START SUPPORT - Vu=   38.19 KIP  Vc=   21.42 KIP  Vs=   29.50 KIP
    Tu=    0.00 KIP-FT  Tc=   1.95 KIP-FT  Ts=   0.00 KIP-FT  LOAD     3
    NO STIRRUPS ARE REQUIRED FOR TORSION.
    REINFORCEMENT IS REQUIRED FOR SHEAR.
    PROVIDE NUM. 4 2-LEGGED STIRRUPS AT  6.7 IN. C/C FOR   46. IN.
      RECTANGULAR CONCRETE BEAM DESIGN PER ACI 318 2011        -- PAGE NO.    4
    AT END   SUPPORT - Vu=   38.19 KIP  Vc=   21.42 KIP  Vs=   29.50 KIP
    Tu=    0.00 KIP-FT  Tc=   1.95 KIP-FT  Ts=   0.00 KIP-FT  LOAD     3
    NO STIRRUPS ARE REQUIRED FOR TORSION.
    REINFORCEMENT IS REQUIRED FOR SHEAR.
    PROVIDE NUM. 4 2-LEGGED STIRRUPS AT  6.7 IN. C/C FOR   46. IN.
  ___   1J____________________  120.X  10.X  16_____________________   2J____ 
 |                                                                           |
 |                                                                           |
 |                                                                           |
 |  8#4 C/C  7                                                 8#4 C/C  7    |
 | 2#10H 2.63   0.TO  120.                                                   |
 ||=========================================================================||
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 |         |  |         |  |         |  |         |  |         |   |         |
 |         |  |         |  |         |  |         |  |         |   |         |
 |         |  |         |  |         |  |         |  |         |   |         |
 |         |  |         |  |         |  |         |  |         |   |         |
 | 2#10    |  | 2#10    |  | 2#10    |  | 2#10    |  | 2#10    |   | 2#10    |
 |   OO    |  |   OO    |  |   OO    |  |   OO    |  |   OO    |   |   OO    |
 |         |  |         |  |         |  |         |  |         |   |         |
 |_________|  |_________|  |_________|  |_________|  |_________|   |_________|
   ********************END OF BEAM DESIGN**************************