V. ACI 318-08 Rectangular Singly Reinforced Beam

Flexural and shear reinforcement design of a singly reinforced concrete beam per the ACI 318-2008 code.

Details

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
f_yt= 60,000 psi
cover = 2.5 in
D = 16 in
b_w = 10 in

Validation

Ag = D×b_w = 16 × 10 = 160 in²

d = D - cover = 16 - 2.5 = 13.5 in

Bending Reinforcement

ϕ = 0.9

Minimum reinforcement of flexural members [Cl 10.5]

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

β_{1} = 0.85 - 0.05 (\frac{{f^{'}}_{c} - 4, 000}{1, 000}) = 0.85

[Cl 10.2.7]

C_{\max} = \frac{0.003}{0.003 + 0.005} d = 5.063 in

a_{\max} = β_{1} C_{\max} = 4.303 in

The ultimate moment:

M_{u} = \frac{w_{u} l^{2}}{8} = \frac{9.736 {(12)}^{2}}{8} = 121.7 ft⋅k

The depth of the compression block is:

a = d - \sqrt{d^{2} - 2 \frac{M_{u}}{0.85 {f^{'}}_{c} φ b_{w}}} = 4.183 in

Hence, a < a_max.

The area of the tensile steel reinforcement required is:

A_{s, r e q' d} = \frac{M_{u}}{φ f_{y} (d - a / 2)} = 2.371 {in}^{2}

Use 2-#10 bars: A_s = 2.53 in²

Shear Reinforcement

ϕ = 0.75

Limit of concrete strength: $\sqrt{{f^{'}}_{c}} = 63.2 < 100$ , hence OK. [Cl 11.1.2]

ρ_{w} = \frac{A_{s}}{b_{w} d} = 0.018

λ = 1

Shear and moment at a distance, d, from the support:

V_{u 1} = w (\frac{l}{2} - d) = 9.736 (\frac{10}{2} - 1.125) = 37.73 kip

M_{u 1} = \frac{w d}{2} (l - d) = \frac{9.736 \times 1.125}{2} (10 - 1.125) = 48.6 ft⋅k = 583.2 in⋅k

Per Cl 11.2.2.1 equation 11.5, the unfactored shear capacity of concrete is:

V_{c} = (1.9 λ \sqrt{{f^{'}}_{c}} + 2, 500 ρ_{w} V_{u 1} \frac{d}{M_{u 1}}) b_{w} d = 21.48 kip

ϕV_c = 0.75×21.48 = 16.11 kip

The maximum shear capacity of the reinforcement is:

{φ V}_{s} = 8 \sqrt{{f^{'}}_{c}} b_{w} d = 0.75 \times 68.31 = 51.23 kip

[Cl 11.4.7.9]

The minimum shear reinforcement , Av/s, for any load combination is:

A_{v s, \min} = {}_{\max}{| \begin{matrix} 50 \frac{b_{w}}{f_{y}} = 0.008 {in}^{2} /in \\ \frac{b_{w}}{f_{y}} 0.75 \sqrt{{f^{'}}_{c}} = 0.0003 {in}^{2} /in \end{matrix}}

[Cl 11.4.6.3 and R11.4.7]

A_{v s, r e q' d} = \frac{V_{u 1} - φ V_{c}}{φ f_{y t} d} = 0.036 {in}^{2} /in

Use 2-legged #4 stirrups (A_v = 0.39 in² per stirrup).

s_max = d/2 = 13.5 /2 = 6.75 in

Use maximum spacing of 6.75 in: A_vs/s = 0.39 / 6 = 0.058 in²/in

V_{s} = \frac{V_{u 1} - φ V_{c}}{φ} = 28.83 kip

Results

Table 1. Comparison of results
Parameter	Hand Calculation	STAAD.Pro	Difference	Comments
Design moment, M_u (ft·k)	121.7	121.7	none
Tension reinforcement required, A_s,re'd (in²)	2.371	2.41	1.6%	OK
Design shear, V_u1 (kip)	37.73	37.73	none
Shear reinforcement, A_v/s (in²/in)	0.058	0.058*	none
V_c (kip)	21.48	21.34	negligible
V_s (kip)	28.83	28.97	negligible

*Shear reinforcement provided by STAAD: A_vs/s = 0.39 / 6.7 = 0.058 in²/in

STAAD Input

The file C:\Users\Public\Public Documents\STAAD.Pro CONNECT Edition\Samples\ Verification Models\10 Concrete Design\US\ACI 318-08 Rectangular Singly Reinforced Beam.STD is typically installed with the program.

STAAD SPACE
START JOB INFORMATION
ENGINEER DATE 06-May-19
JOB CLIENT Bentley Systems Inc.
END JOB INFORMATION
INPUT WIDTH 79
UNIT FEET KIP
JOINT COORDINATES
1 0 0 0; 2 10 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 1.333 ZD 0.8333
CONSTANTS
MATERIAL CONCRETE ALL
SUPPORTS
1 PINNED
2 FIXED BUT FX MZ
LOAD 1 LOADTYPE None  TITLE LOAD CASE 1
MEMBER LOAD
1 UNI GY -9.736
PERFORM ANALYSIS
PRINT MEMBER PROPERTIES
PRINT SUPPORT REACTION ALL
START CONCRETE DESIGN
CODE ACI 2008
UNIT INCHES KIP
TRACK 2 ALL
DESIGN BEAM ALL
END CONCRETE DESIGN
FINISH

STAAD Output

   =====================================================================
          BEAM  NO.     1 DESIGN RESULTS - FLEXURE PER CODE ACI 318-08
   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=    121.70 KIP-FT  AT  5.00 FT, LOAD    1|
   |   REQD STEEL=  2.41 IN2, RHO=0.0180, RHOMX=0.0214 RHOMN=0.0033 |
   |   MAX/MIN/ACTUAL BAR SPACING= 10.00/ 2.54/ 4.73 INCH           |
   |   REQD. DEVELOPMENT LENGTH = 48.53 INCH                        |
   |----------------------------------------------------------------|
     Cracked Moment of Inertia Iz at above location =    1816.05 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        1/    0
     0.83        0.642/    0.000        37.19/      0.00        1/    0
     1.67        1.215/    0.000        67.61/      0.00        1/    0
     2.50        1.700/    0.000        91.28/      0.00        1/    0
     3.33        2.071/    0.000       108.18/      0.00        1/    0
     4.17        2.307/    0.000       118.32/      0.00        1/    0
     5.00        2.388/    0.000       121.70/      0.00        1/    0
     5.83        2.307/    0.000       118.32/      0.00        1/    0
     6.67        2.071/    0.000       108.18/      0.00        1/    0
     7.50        1.700/    0.000        91.28/      0.00        1/    0
     8.33        1.215/    0.000        67.61/      0.00        1/    0
     9.17        0.642/    0.000        37.19/      0.00        1/    0
    10.00        0.000/    0.000         0.00/      0.00        1/    0
          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=   37.73 KIP  Vc=   21.34 KIP  Vs=   28.97 KIP
    Tu=    0.00 KIP-FT  Tc=   1.95 KIP-FT  Ts=   0.00 KIP-FT  LOAD     1
    NO STIRRUPS ARE REQUIRED FOR TORSION.
    REINFORCEMENT IS REQUIRED FOR SHEAR.
    PROVIDE NUM. 4 2-LEGGED STIRRUPS AT  6.7 IN. C/C FOR   47. IN.
      STAAD SPACE                                              -- PAGE NO.    6
    AT END   SUPPORT - Vu=   37.73 KIP  Vc=   21.34 KIP  Vs=   28.97 KIP
    Tu=    0.00 KIP-FT  Tc=   1.95 KIP-FT  Ts=   0.00 KIP-FT  LOAD     1
    NO STIRRUPS ARE REQUIRED FOR TORSION.
    REINFORCEMENT IS REQUIRED FOR SHEAR.
    PROVIDE NUM. 4 2-LEGGED STIRRUPS AT  6.7 IN. C/C FOR   47. 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    |
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 |_________|  |_________|  |_________|  |_________|  |_________|   |_________|
   ********************END OF BEAM DESIGN**************************