Contents  1. Introduction to structural design  2. Loads  3. Wood  4. Steel 
Introduction to reinforced concrete  Material properties  Sectional properties  Design approaches  Construction systems  Tension elements  Columns  Beams  Connections 
Table A5.1: Dimensions of reinforced concrete beams, columns, and slabs
A. Cover requirements (from outside face of concrete to face of closest rebar)  

Interior  1½ in. (or ¾ in. for slabs) 
Exterior or exposed to ground  2 in. (or 1½ in. for No.5 bars or smaller) 
Formed directly to ground  3 in. 
B. Typical gross dimensions  

Beams and columns  Round to the nearest inch, or 2 in. increment, for all outside (gross) dimensions 
Slabs  Round to ½ in. increment (or 1 in. increment if over 6 in. thick) 
Table A5.2: Steel reinforcement — rebar — areas (in^{2}) for groups of bars
Designation and diameter  Number of bars  

Bar No.^{1}  SI Bar No.^{1}  Dia. (in.)  1  2  3  4  5  6  7  8  9  10  11  12  14  16 
3  10  0.375  0.11  No. 3 (10) bars are used primarily for ties and in slabs  
4  13  0.500  0.20  0.40  0.60  0.80  1.00  1.20  1.40  1.60  1.80  2.00  2.20  2.40  2.80  3.20 
5  16  0.625  0.31  0.62  0.93  1.24  1.55  1.86  2.17  2.48  2.79  3.10  3.41  3.72  4.34  4.96 
6  19  0.750  0.44  0.88  1.32  1.76  2.20  2.64  3.08  3.52  3.96  4.40  4.84  5.28  6.16  7.04 
7  22  0.875  0.60  1.20  1.80  2.40  3.00  3.60  4.20  4.80  5.40  6.00  6.60  7.20  8.40  9.60 
8  25  1.000  0.79  1.58  2.37  3.16  3.95  4.74  5.53  6.32  7.11  7.90  8.69  9.48  11.06  12.64 
9  29  1.128  1.00  2.00  3.00  4.00  5.00  6.00  7.00  8.00  9.00  10.00  11.00  12.00  14.00  16.00 
10  32  1.270  1.27  2.54  3.81  5.08  6.35  7.62  8.89  10.16  11.43  12.70  13.97  15.24  17.78  20.32 
11  36  1.410  1.56  3.12  4.68  6.24  7.80  9.36  10.92  12.48  14.04  15.60  17.16  18.72  21.84  24.96 
^{3}14  43  1.693  2.25  4.50  6.75  9.00  11.25  13.50  15.75  18.00  20.25  22.50  24.75  27.00  31.50  36.00 
^{3}18  57  2.257  4.00  8.00  12.00  16.00  20.00  24.00  28.00  32.00  36.00  40.00  44.00  48.00  56.00  64.00 
Notes:
1. Rebars in the US were traditionally designated by the nominal diameter (in.) multiplied by eight, so that a No. 3 bar, for example, has a nominal diameter of ⅜ in. Rebars are no longer marked using this designation (see Note 2).
2. Rebars are currently marked by the approximate number of millimeters in their diameter (SI units), although designation by nominal diameter (in.) multiplied by eight is still widely used in the US.
3. No. 14 and No. 18 bars are used primarily in columns.
Table A5.3: Reinforced concrete minimum width or diameter (in.) based on bar spacing
A. Minimum width (in.) for beams^{3}  

Designation  Number of bars in one line  
Bar No^{1}  SI Bar No^{2}  2  3  4  5  6  
4  13  6.33  8.17  10.00  11.83  13.67  
5  16  6.58  8.54  10.50  12.46  14.42  
6  19  6.83  8.92  11.00  13.08  15.17  
7  22  7.08  9.29  11.50  13.71  15.92  
8  25  7.33  9.67  12.00  14.33  16.67  
9  29  7.58  10.04  12.50  14.96  17.42  
10  32  7.83  10.42  13.00  15.58  18.17  
11  36  8.13  10.88  13.63  16.38  19.13 
B. Minimum width (in.) for tied columns^{3}  

Designation  Number of bars in one line  
Bar No^{1}  SI Bar No^{2}  2  3  4  5  6  
4  13  6.50  8.50  10.50  12.50  14.50  
5  16  6.75  8.88  11.00  13.13  15.25  
6  19  7.00  9.25  11.50  13.75  16.00  
7  22  7.25  9.63  12.00  14.38  16.75  
8  25  7.50  10.00  12.50  15.00  17.50  
9  29  7.94  10.75  13.56  16.38  19.19  
10  32  8.38  11.50  14.63  17.75  20.88  
11  36  8.81  12.25  15.69  19.13  22.56  
14  43  10.13  14.50  18.88  23.25  27.63  
18  57  11.88  17.50  23.13  28.75  34.38 
C. Minimum diameter (in.) for spiral columns^{3}  

Designation  Number of bars in one line  
Bar No^{1}  SI Bar No^{2}  6  8  10  12  14  
4  13  8.50  9.73  10.97  12.23  13.49  
5  16  8.88  10.18  11.50  12.84  14.17  
6  19  9.25  10.63  12.03  13.44  14.86  
7  22  9.63  11.08  12.56  14.05  15.55  
8  25  10.00  11.53  13.09  14.66  16.23  
9  29  10.75  12.47  14.23  15.99  17.76  
10  32  11.50  13.42  15.36  17.32  19.29  
11  36  12.25  14.36  16.50  18.66  20.82  
14  43  14.50  17.18  19.91  22.65  25.41  
18  57  17.50  20.95  24.45  27.98  31.53 
Notes:
1. Rebars in the United States were traditionally designated by the nominal diameter (in.) divided by eight. Rebars are no longer marked using this designation.
2. Rebars are currently designated (and marked) by the approximate number of millimeters (SI units) in their diameter.
3. These minimum dimensions assume 1 in. maximum aggregate; 1½ in. cover (measured from outside face of rebar or spiral to face of concrete); and ½ in.diameter stirrups, ties, or spiral. Minimum widths or diameters are typically rounded up to nearest inch, or to the nearest even inch. The amount of column steel is also limited by the required reinforcement ratio, ρ_{g}, between 0.01 and 0.08.
Table A5.4: Specifications for steel ties and spirals in reinforced concrete columns
Ties  

Use minimum No. 3 bars to confine longitudinal steel up to No. 10; use
minimum No. 4 bars for No. 11, 14, and 18 longitudinal steel.
Centertocenter spacing of ties is the smaller of:


Spirals  
Use a continuous bar or wire of at least ⅜in. diameter, with the clear space measured between turns of the spiral no more than 3 in. and no less than 1 in or 4/3 of maximum aggregate size. A minimum ratio, ρ_{s}, of the volume of spiral steel to the volume of concrete inside the spiral (the
"core") is also specified: ρ_{s} = 0.45(A_{g} / A_{g} – 1)(f_{c}' / f_{y}) with f_{y} ≤ 60 ksi, A_{g} being the gross concrete area, and A_{c} being the area of the "core" within the spiral. 
Table A5.5: Reinforced concrete strength reduction factors, φ and α
Type of behavior  φ  α^{1} 

Bending  ^{2}0.9  n/a 
Axial tension  0.9  n/a 
Axial compression: spiral columns  ^{3}0.75  0.85 
Axial compression: tied columns  ^{3}0.65  0.80 
Shear  0.75  n/a 
Notes:
1. α accounts for unintended eccentricity or bending moment.
2. φ decreases linearly from value listed above at ε_{t} = 0.005 to 0.65 or 0.75 (for tied or spiral lateral reinforcement respectively) at ε_{t} = 0.002, where ε_{t} is the net tensile strain in the extreme tension steel (we assume "tensioncontrolled" sections in this text, with values of φ = 0.9 as shown).
3. φ increases linearly from value listed above at ε_{t} = 0.002 to 0.9 at ε_{t} = 0.005, where ε_{t} is the net tensile strain in the extreme tension steel for elements with combined compression and bending (this type of combined loading is beyond the scope of this book; in the problems considered herein, with only compressive stresses, the value of φ is as shown).
Table A5.6: "Shear" equations for reinforced concrete beams^{1}
Equation  

A. Capacity of steel stirrups^{2} (lb)  
B. Required stirrup spacing^{2} (in.)  
C. Capacity of concrete (lb)^{3}  V_{c} = 2bd 
D. Strength design equation^{2}  V_{u} ≤ φ(V_{c} + V_{s}) 
E. Required steel capacity (lb) from strength design equation^{2}  
F. Maximum stirrup spacing^{3} (in.)  For V_{s} ≤ 2V_{c}, the smaller of:

G. Design shear where no stirrups are needed^{2} (lb)  V_{u} = 0.5φV_{c} 
Notes:
1. Units are as follows:
b = cross section width, or "web" width for Tbeams (in.)
d = cross section effective depth (in.)
s = stirrup spacing (in.)
A_{v} = total stirrup bar area, equal to 2A_{s}, including both "prongs" (in^{2})
f_{y} = yield stress of steel stirrup (psi)
f_{c}' = cylinder strength of concrete (psi)
V_{u} = design (factored) shear force (lb)
V_{c} = capacity of concrete to resist shear (lb)
V_{s} = capacity of steel stirrups to resist shear (lb)
φ = 0.75 for shear (see Appendix Table A5.5)
2. Pound (lb) and pound per square inch (psi) units specified according to Note 1 may be changed to kips and ksi in these equations only.
3. The concrete cylinder strength f_{c}' must be in psi units in Appendix Table 5.6 part C (with the resulting value of V_{c} in lb units) and both the steel yield stress f_{y} and the concrete cylinder strength f_{c}' must be psi units in part F (with in. units resulting).
4. This limiting value for maximum spacing corresponds to the minimum amount of web steel required when V_{u} > 0.5φV_{c}, except that it does not apply when the total beam depth (thickness, h) is less than 10 in. or, for Tbeams, when the total beam depth is less than 24 in. and is also less than the larger of 2.5 × flange depth or 0.5 × the beam stem width.
Table A5.7: Approximate moment values for continuous reinforced concrete beams and slabs^{1}
End restraints for two spans  Positive moment  Negative moment  

End span  End span  
At interior support^{2}  At exterior support^{2}  
Discontinuous end unrestrained  w_{u} l_{n}^{2} / 11  n/a  n/a  
Discontinuous end restrained by spandrel girder  w_{u} l_{n}^{2} / 14  w_{u} l_{n}^{2} / 9  w_{u} l_{n}^{2} / 24  
Discontinuous end restrained by column  w_{u} l_{n}^{2} / 14  w_{u} l_{n}^{2} / 9  w_{u} l_{n}^{2} / 16  
End restraints for three or more spans  Positive moment  Negative moment  
Interior span  End span  Typical interior support^{2}  End span  
At interior support^{2}  At exterior support^{2}  
Discontinuous end unrestrained  w_{u} l_{n}^{2} / 16  w_{u} l_{n}^{2} / 11  n/a  n/a  n/a 
Discontinuous end restrained by spandrel girder  w_{u} l_{n}^{2} / 16  w_{u} l_{n}^{2} / 14  w_{u} l_{n}^{2} / 11  w_{u} l_{n}^{2} / 10  w_{u} l_{n}^{2} / 24 
Discontinuous end restrained by column  w_{u} l_{n}^{2} / 16  w_{u} l_{n}^{2} / 14  w_{u} l_{n}^{2} / 11  w_{u} l_{n}^{2} / 10  w_{u} l_{n}^{2} / 16 
Notes:
1. The units for uniformly distributed design load, w_{u}, are typically lb/ft or kips/ft; the units for clear span, l_{n}, are feet; and the resulting moment value, M_{u} is in ftlb or ftkips depending on the units chosen for the distributed load. These moment values are valid only for continuous reinforced concrete beams or slabs when the following conditions are met:
a. Lengths of adjacent spans do not differ by more than 20%.
b. The unfactored live load is less than or equal to 3 times the unfactored dead load.
2. The negative moment (at the face of support) can be taken as w_{u} l_{n}^{2} / 12 for slabs with clear spans no greater than 10 ft, and for beams framing into relatively stiff columns (specifically, the sum of column stiffness divided by the sum of beam stiffness at each end of the beam must be greater than 8). Stiffness is the product of modulus of elasticity and moment of inertia, neither of which are straightforward quantities for structural elements consisting of two materials bonded together. For normalweight concrete, the modulus of elasticity, E_{c} (psi), may be taken as 57,000, where the cylinder strength of concrete, f_{c}', is in psi units. The calculation of moment of inertia is left to the designer, with the American Concrete Institute (ACI) permitting any "reasonable and consistent assumptions." One suggestion is to use gross E_{c}I values for both beams and columns. Where E_{c} is the same for all members, a typical joint with columns and beams at all four orthogonal points, and constant width for column and beam sections, would qualify for the w_{u} l_{n}^{2} / 12 negative beam moment only when the column thickness at that joint becomes more than twice the beam thickness.
Table A5.8: Limits on steel ratio for "tensioncontrolled" reinforced concrete beams^{1,2,3}
f_{c}' (psi)  Limits on steel ratio, ρ_{min} – ρ_{max} 

3000  0.00333 – 0.01350 
4000  0.00333 – 0.01810 
5000  0.00354 – 0.02130 
Notes:
1. Values are for f_{y} = 60 ksi, φ = 0.9, and steel strain, ε_{t} = 0.005 for ρ_{max}.
2. Values for maximum steel ratio apply to all beams and oneway slabs; values for minimum steel ratio apply only to rectangular beams and negativemoment indeterminate Tbeams; the minimum steel ratio for positive moment Tbeams is ρ_{min} = 0.00333/ (b/b_{w}) for f_{c}' = 3000 psi and 4000 psi, and 0.00354/(b/b_{w}) for f_{c}' = 5000 psi; and the minimum steel ratio for oneway slabs is ρ_{min} = 0.00180/(h/d). For details, see Appendix Table A5.9 Part d.
3. It is permitted to reduce the amount of steel below the minimum values stipulated for beams, as long as the steel area provided is at least onethird greater than the steel area required by analysis.
Table A5.9: Values of R and ρ for reinforced concrete beams, Tbeams, and oneway slabs (using 60 ksi steel)^{1,2}
a) f_{c}' = 5 ksi and f_{y} = 60 ksi  

R and ρ for positivemoment Tbeams and slabs  R and ρ  
ρ_{min} defined as A_{s} / (bd)
R ρ 
R ρ 
R ρ 
ρ_{min} for pos.moment Tbeam with b/b_{w} = 11.50
0.0184 0.000308  0.2071 0.00354  0.7449 0.01375 
ρ_{min} for pos.moment Tbeam with b/b_{w} = 11.00
0.0193 0.000322  0.2190 0.00375  0.7570 0.01400 
ρ_{min} for pos.moment Tbeam with b/b_{w} = 10.50
0.0202 0.000337  0.2332 0.00400  0.7690 0.01425 
ρ_{min} for pos.moment Tbeam with b/b_{w} = 10.00
0.0212 0.000354  0.2474 0.00425  0.7810 0.01450 
ρ_{min} for pos.moment Tbeam with b/b_{w} = 9.50
0.0223 0.000373  0.2614 0.00450  0.7929 0.01475 
ρ_{min} for pos.moment Tbeam with b/b_{w} = 9.00
0.0235 0.000393  0.2754 0.00475  0.8047 0.01500 
ρ_{min} for pos.moment Tbeam with b/b_{w} = 8.50
0.0249 0.000416  0.2894 0.00500  0.8165 0.01525 
ρ_{min} for pos.moment Tbeam with b/b_{w} = 8.00
0.0265 0.000443  0.3033 0.00525  0.8282 0.01550 
ρ_{min} for pos.moment Tbeam with b/b_{w} = 7.50
0.0282 0.000472  0.3172 0.00550  0.8399 0.01575 
ρ_{min} for pos.moment Tbeam with b/b_{w} = 7.00
0.0303 0.000506  0.3310 0.00575  0.8516 0.01600 
ρ_{min} for pos.moment Tbeam with b/b_{w} = 6.50
0.0326 0.000545  0.3448 0.00600  0.8632 0.01625 
ρ_{min} for pos.moment Tbeam with b/b_{w} = 6.00
0.0353 0.000590  0.3585 0.00625  0.8747 0.01650 
ρ_{min} for pos.moment Tbeam with b/b_{w} = 5.75
0.0368 0.000616  0.3721 0.00650  0.8862 0.01675 
ρ_{min} for pos.moment Tbeam with b/b_{w} = 5.50
0.0385 0.000644  0.3857 0.00675  0.8976 0.01700 
ρ_{min} for pos.moment Tbeam with b/b_{w} = 5.25
0.0402 0.000674  0.3992 0.00700  0.9090 0.01725 
ρ_{min} for pos.moment Tbeam with b/b_{w} = 5.00
0.0423 0.000708  0.4127 0.00725  0.9203 0.01750 
ρ_{min} for pos.moment Tbeam with b/b_{w} = 4.75
0.0445 0.000745  0.4262 0.00750  0.9316 0.01775 
ρ_{min} for pos.moment Tbeam with b/b_{w} = 4.50
0.0470 0.000787  0.4396 0.00775  0.9428 0.01800 
ρ_{min} for pos.moment Tbeam with b/b_{w} = 4.25
0.0497 0.000833  0.4529 0.00800  0.9472 0.01810 
ρ_{min} for pos.moment Tbeam with b/b_{w} = 4.00
0.0528 0.000885  0.4662 0.00825  0.9539 0.01825 
ρ_{min} for pos.moment Tbeam with b/b_{w} = 3.75
0.0563 0.000944  0.4794 0.00850  0.9650 0.01850 
ρ_{min} for pos.moment Tbeam with b/b_{w} = 3.50
0.0602 0.001011  0.4926 0.00875  0.9761 0.01875 
ρ_{min} for pos.moment Tbeam with b/b_{w} = 3.25
0.0648 0.001089  0.5057 0.00900  0.9871 0.01900 
ρ_{min} for pos.moment Tbeam with b/b_{w} = 3.00
0.0702 0.001180  0.5188 0.00925  0.9981 0.01925 
ρ_{min} for pos.moment Tbeam with b/b_{w} = 2.75
0.0765 0.001287  0.5318 0.00950  1.0090 0.01950 
ρ_{min} for pos.moment Tbeam with b/b_{w} = 2.50
0.0841 0.001416  0.5447 0.00975  1.0198 0.01975 
ρ_{min} for pos.moment Tbeam with b/b_{w} = 2.25
0.0933 0.001573  0.5576 0.01000  1.0306 0.02000 
ρ_{min} for pos.moment Tbeam with b/b_{w} = 2.00
0.1049 0.001770  0.5705 0.01025  1.0413 0.02025 
ρ_{min} for slabs with thickness, h = 12
0.1162 0.001964  0.5833 0.01050  1.0520 0.02050 
ρ_{min} for slabs with thickness, h = 11
0.1171 0.00198  0.5961 0.01075  1.0626 0.02075 
ρ_{min} for slabs with thickness, h = 10
0.1183 0.002  0.6088 0.01100  1.0732 0.02100 
ρ_{min} for slabs with thickness, h = 9
0.1198 0.002025  0.6214 0.01125  1.0838 0.02125 
ρ_{min} for slabs with thickness, h = 8
0.1216 0.002057  0.6340 0.01150  1.0858 0.02130 
ρ_{min} for slabs with thickness, h = 7
0.1241 0.0021  0.6465 0.01175  — — 
ρ_{min} for slabs with thickness, h = 6
0.1276 0.00216  0.6590 0.01200  — — 
ρ_{min} for slabs with thickness, h = 5.5
0.1300 0.0022  0.6714 0.01225  — — 
ρ_{min} for slabs with thickness, h = 5
0.1329 0.00225  0.6838 0.01250  — — 
ρ_{min} for slabs with thickness, h = 4.5
0.1366 0.002314  0.6962 0.01275  — — 
ρ_{min} for slabs with thickness, h = 4
0.1416 0.0024  0.7084 0.01300  — — 
ρ_{min} for slabs with thickness, h = 3.5
0.1485 0.00252  0.7206 0.01325  — — 
ρ_{min} for slabs with thickness, h = 3
0.1589 0.0027  0.7328 0.01350  — — 
b) f_{c}' = 4 ksi and f_{y} = 60 ksi  

R and ρ for positivemoment Tbeams and slabs  R and ρ  
ρ_{min} defined as A_{s} / (bd)
R ρ 
R ρ 
R ρ 
ρ_{min} for pos.moment Tbeam with b/b_{w} = 11.50
0.0174 0.000290  0.1939 0.00333  0.7135 0.01350 
ρ_{min} for pos.moment Tbeam with b/b_{w} = 11.00
0.0181 0.000303  0.2058 0.00354  0.7249 0.01375 
ρ_{min} for pos.moment Tbeam with b/b_{w} = 10.50
0.0190 0.000317  0.2176 0.00375  0.7362 0.01400 
ρ_{min} for pos.moment Tbeam with b/b_{w} = 10.00
0.0199 0.000333  0.2315 0.00400  0.7475 0.01425 
ρ_{min} for pos.moment Tbeam with b/b_{w} = 9.50
0.0210 0.000351  0.2454 0.00425  0.7587 0.01450 
ρ_{min} for pos.moment Tbeam with b/b_{w} = 9.00
0.0221 0.000370  0.2593 0.00450  0.7698 0.01475 
ρ_{min} for pos.moment Tbeam with b/b_{w} = 8.50
0.0234 0.000392  0.2731 0.00475  0.7809 0.01500 
ρ_{min} for pos.moment Tbeam with b/b_{w} = 8.00
0.0249 0.000416  0.2868 0.00500  0.7919 0.01525 
ρ_{min} for pos.moment Tbeam with b/b_{w} = 7.50
0.0265 0.000444  0.3004 0.00525  0.8028 0.01550 
ρ_{min} for pos.moment Tbeam with b/b_{w} = 7.00
0.0284 0.000476  0.3140 0.00550  0.8137 0.01575 
ρ_{min} for pos.moment Tbeam with b/b_{w} = 6.50
0.0306 0.000512  0.3275 0.00575  0.8245 0.01600 
ρ_{min} for pos.moment Tbeam with b/b_{w} = 6.00
0.0331 0.000555  0.3409 0.00600  0.8352 0.01625 
ρ_{min} for pos.moment Tbeam with b/b_{w} = 5.75
0.0346 0.000579  0.3543 0.00625  0.8459 0.01650 
ρ_{min} for pos.moment Tbeam with b/b_{w} = 5.50
0.0361 0.000605  0.3676 0.00650  0.8565 0.01675 
ρ_{min} for pos.moment Tbeam with b/b_{w} = 5.25
0.0378 0.000634  0.3809 0.00675  0.8670 0.01700 
ρ_{min} for pos.moment Tbeam with b/b_{w} = 5.00
0.0397 0.000666  0.3941 0.00700  0.8775 0.01725 
ρ_{min} for pos.moment Tbeam with b/b_{w} = 4.75
0.0418 0.000701  0.4072 0.00725  0.8879 0.01750 
ρ_{min} for pos.moment Tbeam with b/b_{w} = 4.50
0.0441 0.000740  0.4202 0.00750  0.8982 0.01775 
ρ_{min} for pos.moment Tbeam with b/b_{w} = 4.25
0.0467 0.000784  0.4332 0.00775  0.9085 0.01800 
ρ_{min} for pos.moment Tbeam with b/b_{w} = 4.00
0.0496 0.000833  0.4461 0.00800  0.9126 0.01810 
ρ_{min} for pos.moment Tbeam with b/b_{w} = 3.75
0.0529 0.000888  0.4590 0.00825  — — 
ρ_{min} for pos.moment Tbeam with b/b_{w} = 3.50
0.0566 0.000951  0.4718 0.00850  — — 
ρ_{min} for pos.moment Tbeam with b/b_{w} = 3.25
0.0609 0.001025  0.4845 0.00875  — — 
ρ_{min} for pos.moment Tbeam with b/b_{w} = 3.00
0.0659 0.001110  0.4971 0.00900  — — 
ρ_{min} for pos.moment Tbeam with b/b_{w} = 2.75
0.0719 0.001211  0.5097 0.00925  — — 
ρ_{min} for pos.moment Tbeam with b/b_{w} = 2.50
0.0790 0.001332  0.5222 0.00950  — — 
ρ_{min} for pos.moment Tbeam with b/b_{w} = 2.25
0.0876 0.001480  0.5347 0.00975  — — 
ρ_{min} for pos.moment Tbeam with b/b_{w} = 2.00
0.0984 0.001665  0.5471 0.01000  — — 
ρ_{min} for slabs with thickness, h = 12
0.1158 0.001964  0.5594 0.01025  — — 
ρ_{min} for slabs with thickness, h = 11
0.1167 0.001980  0.5716 0.01050  — — 
ρ_{min} for slabs with thickness, h = 10
0.1179 0.002000  0.5838 0.01075  — — 
ρ_{min} for slabs with thickness, h = 9
0.1193 0.002025  0.5959 0.01100  — — 
ρ_{min} for slabs with thickness, h = 8
0.1212 0.002057  0.6080 0.01125  — — 
ρ_{min} for slabs with thickness, h = 7
0.1237 0.002100  0.6200 0.01150  — — 
ρ_{min} for slabs with thickness, h = 6
0.1271 0.002160  0.6319 0.01175  — — 
ρ_{min} for slabs with thickness, h = 5.5
0.1294 0.002200  0.6438 0.01200  — — 
ρ_{min} for slabs with thickness, h = 5
0.1323 0.002250  0.6556 0.01225  — — 
ρ_{min} for slabs with thickness, h = 4.5
0.1360 0.002314  0.6673 0.01250  — — 
ρ_{min} for slabs with thickness, h = 4
0.1410 0.002400  0.6789 0.01275  — — 
ρ_{min} for slabs with thickness, h = 3.5
0.1478 0.002520  0.6905 0.01300  — — 
ρ_{min} for slabs with thickness, h = 3
0.1581 0.002700  0.7021 0.01325  — — 
c) f_{c}' = 3 ksi and f_{y} = 60 ksi  

R and ρ for positivemoment Tbeams and slabs  R and ρ 
ρ_{min} defined as A_{s} / (bd)
R ρ 
R ρ 
ρ_{min} for pos.moment Tbeam with b/b_{w} = 11.50
0.0173 0.000290  0.1920 0.00333 
ρ_{min} for pos.moment Tbeam with b/b_{w} = 11.00
0.0181 0.000303  0.2151 0.00375 
ρ_{min} for pos.moment Tbeam with b/b_{w} = 10.50
0.0189 0.000317  0.2287 0.00400 
ρ_{min} for pos.moment Tbeam with b/b_{w} = 10.00
0.0199 0.000333  0.2423 0.00425 
ρ_{min} for pos.moment Tbeam with b/b_{w} = 9.50
0.0210 0.000351  0.2557 0.00450 
ρ_{min} for pos.moment Tbeam with b/b_{w} = 9.00
0.0221 0.000370  0.2691 0.00475 
ρ_{min} for pos.moment Tbeam with b/b_{w} = 8.50
0.0234 0.000392  0.2824 0.00500 
ρ_{min} for pos.moment Tbeam with b/b_{w} = 8.00
0.0248 0.000416  0.2955 0.00525 
ρ_{min} for pos.moment Tbeam with b/b_{w} = 7.50
0.0265 0.000444  0.3086 0.00550 
ρ_{min} for pos.moment Tbeam with b/b_{w} = 7.00
0.0284 0.000476  0.3217 0.00575 
ρ_{min} for pos.moment Tbeam with b/b_{w} = 6.50
0.0305 0.000512  0.3346 0.00600 
ρ_{min} for pos.moment Tbeam with b/b_{w} = 6.00
0.0331 0.000555  0.3474 0.00625 
ρ_{min} for pos.moment Tbeam with b/b_{w} = 5.75
0.0345 0.000579  0.3602 0.00650 
ρ_{min} for pos.moment Tbeam with b/b_{w} = 5.50
0.0360 0.000605  0.3728 0.00675 
ρ_{min} for pos.moment Tbeam with b/b_{w} = 5.25
0.0378 0.000634  0.3854 0.00700 
ρ_{min} for pos.moment Tbeam with b/b_{w} = 5.00
0.0396 0.000666  0.3979 0.00725 
ρ_{min} for pos.moment Tbeam with b/b_{w} = 4.75
0.0417 0.000701  0.4103 0.00750 
ρ_{min} for pos.moment Tbeam with b/b_{w} = 4.50
0.0440 0.000740  0.4226 0.00775 
ρ_{min} for pos.moment Tbeam with b/b_{w} = 4.25
0.0466 0.000784  0.4348 0.00800 
ρ_{min} for pos.moment Tbeam with b/b_{w} = 4.00
0.0495 0.000833  0.4470 0.00825 
ρ_{min} for pos.moment Tbeam with b/b_{w} = 3.75
0.0527 0.000888  0.4590 0.00850 
ρ_{min} for pos.moment Tbeam with b/b_{w} = 3.50
0.0564 0.000951  0.4710 0.00875 
ρ_{min} for pos.moment Tbeam with b/b_{w} = 3.25
0.0608 0.001025  0.4828 0.00900 
ρ_{min} for pos.moment Tbeam with b/b_{w} = 3.00
0.0657 0.001110  0.4946 0.00925 
ρ_{min} for pos.moment Tbeam with b/b_{w} = 2.75
0.0716 0.001211  0.5063 0.00950 
ρ_{min} for pos.moment Tbeam with b/b_{w} = 2.50
0.0787 0.001332  0.5179 0.00975 
ρ_{min} for pos.moment Tbeam with b/b_{w} = 2.25
0.0873 0.001480  0.5294 0.01000 
ρ_{min} for pos.moment Tbeam with b/b_{w} = 2.00
0.0979 0.001665  0.5408 0.01025 
ρ_{min} for slabs with thickness, h = 12
0.1151 0.001964  0.5522 0.01050 
ρ_{min} for slabs with thickness, h = 11
0.1160 0.001980  0.5634 0.01075 
ρ_{min} for slabs with thickness, h = 10
0.1172 0.002000  0.5746 0.01100 
ρ_{min} for slabs with thickness, h = 9
0.1186 0.002025  0.5857 0.01125 
ρ_{min} for slabs with thickness, h = 8
0.1204 0.002057  0.5966 0.01150 
ρ_{min} for slabs with thickness, h = 7
0.1229 0.002100  0.6075 0.01175 
ρ_{min} for slabs with thickness, h = 6
0.1263 0.002160  0.6184 0.01200 
ρ_{min} for slabs with thickness, h = 5.5
0.1286 0.002200  0.6291 0.01225 
ρ_{min} for slabs with thickness, h = 5
0.1314 0.002250  0.6397 0.01250 
ρ_{min} for slabs with thickness, h = 4.5
0.1351 0.002314  0.6503 0.01275 
ρ_{min} for slabs with thickness, h = 4
0.1399 0.002400  0.6607 0.01300 
ρ_{min} for slabs with thickness, h = 3.5
0.1467 0.002520  0.6711 0.01325 
ρ_{min} for slabs with thickness, h = 3
0.1569 0.002700  0.6814 0.01350 
d) Notes for minimum and maximum steel ratio, ρ_{min} and ρ_{max} 

Minimum steel ratio for rectangular beams and negativemoment, indeterminate Tbeams:
b / b_{w} = 1.0 For f_{c}' = 3 or 4 ksi, ρ_{min} = 0.00333 For f_{c}' = 5 ksi, ρ_{min} = 0.00354 
Minimum steel ratio for positivemoment Tbeams:
The effective width, b, of a positivemoment Tbeam is smaller of the following: b = web width + ¼ clear beam span b = centerline distance between beams b = web width + 16 times slab thickness 
Minimum steel ratio for negativemoment determinate Tbeams (e.g., precast sections and cantilevers):
For b / b_{w} ≥ 2, ρ_{min} = 0.00667(b_{w} / b) for f_{c}' = 3 or 4 ksi; and ρ_{min} = 0.00708(b_{w} / b) for f_{c}' = 5 ksi. For b / b_{w} < 2, ρ_{min} = 0.00333 for f_{c}' = 3 or 4 ksi; and ρ_{min} = 0.00354(b / b_{w}) for f_{c}' = 5 ksi. 
Minimum steel ratio for 1way slabs:
ρ_{min} = 0.00180(h/d) for slabs; the same steel ratio applies to shrinkage and temperature control steel perpendicular to slab longitudinal bar. 
Maximum steel ratio: ρ_{max} = 0.01350 for f_{c}' = 3 ksi ρ_{max} = 0.01810 for f_{c}' = 4 ksi ρ_{max} = 0.02130 for f_{c}' = 5 ksi 
Notes:
1. M_{u} ≤ φbd^{2}R, where φ = 0.9, R = ρf_{y} (1 – 0.5882ρf_{y} / f_{c}'), and ρ = A_{s} / bd. When using this table, R, f_{y}, and f_{c}' are in ksi units; b is the effective flange width (or effective width); b_{w} is the beam stem or web width; and d is the effective depth, all in inch units. For positivemoment Tbeams, results are valid only when the compressive stress block depth, a = ρf_{y} d/(0.85f_{c}') ≤ slab thickness, h. Steel strain at failure, ε_{t} = 0.005 for ρ_{max} (that is, only tensioncontrolled sections are considered).
2. The values for ρ_{min} tabulated for slabs assume that the slab effective depth, d, is equal to the slab thickness, h – 1 (inch units).
Table A5.10: Development length in inches, l_{d}, for 60 ksi deformed bars in tension, uncoated, normalweight concrete, with adequate spacing and/or stirrups and at least 12 in. of fresh concrete below the tension bars^{1,2,3,4,5}
f_{c}' (psi)  Bar number ["inlb" designation, with nominal diameter (in.) = bar number/8]  

3  4  5  6  7  8  9  10  11  14  18  
3000  27  36  45  53  62  71  80  90  100  121  161 
4000  23  31  39  46  54  62  70  78  87  104  139 
5000  21  28  34  41  48  55  62  70  78  93  124 
Notes:
1. Bars must have a clear space between them at least equal to twice the bar diameter, that is, at least equal to 2d_{b}, and a clear cover at least equal to the bar diameter, d_{b}. Alternatively, if adequate stirrups or ties are used throughout the development length region to confine the bars and prevent splitting of the concrete, the minimum clear spacing requirement may be reduced to db. For bars not meeting these conditions, multiply values by 1.5.
2. Values assume "top" bars in tension with at least 12 in. of freshlyplaced concrete below them; for "bottom" bars in tension (that is, bars placed for positive moment in beams), divide values by 1.3.
3. The development length may be reduced if the steel bar area provided is greater than the bar area required by multiplying the tabular value by the ratio of bar area required to bar area provided.
4. All of the modifications mentioned in Notes 1 and 2 are cumulative; that is, a value may be multiplied by one or more of the applicable modification factors.
5. In any case, the development length, l_{d}, cannot be less than 12 in.
Table A5.11: Development length for 60 ksi standard hooks in inches, l_{dh}, for uncoated bars, normalweight concrete^{1,2,3,4,5,6,7,8}
f_{c}' (psi)  Bar number ["inlb" designation, with nominal diameter (in.) = bar number/8]  

3  4  5  6  7  8  9  10  11  14  18  
3000  7  11  16  21  26  32  38  46  53  70  108 
4000  7  11  15  19  24  30  36  43  50  66  101 
5000  7  10  14  19  24  29  35  41  48  63  98 
Notes:
1. Development lengths must be multiplied by 1.333 where lightweight concrete is used.
2. Development lengths must be multiplied by 1.20 where epoxycoated or zinc and epoxy dualcoated reinforcement is used.
3. Unlike the case for deformed bars in tension (Appendix Table A5.10), the development length may not be reduced if the steel bar area provided is greater than the bar area required.
4. For hooked bars no greater in size than No. 11, the development length may be divided by 1.6 if the centerline spacing between hooked bars is at least 6d_{b} or if the total area of ties or stirrups confining the hooked bars is not less than 0.4 times the total area of the hooked bars.
5. For hooked bars no greater in size than No. 11, the development length can be divided by 1.25 if the cover on the side of the hooked bars (i.e., perpendicular to their longitudinal direction) is at least equal to 6d_{b} or if the hooked bars are inside a column core with side cover at least equal to 2.5 in.
6. All of the modifications mentioned in Notes 1 and 2 are cumulative; that is, a value may be multiplied by one or more of the applicable modification factors.
7. A 90° hook must be extended a distance of 12d_{b} below the bent portion of the bar, which in turn is defined by an inner radius that cannot be less than 3d_{b} for bars smaller than No. 9; 4db for No. 9, No. 10, and No. 11 bars; and 5d_{b} for No. 14 and No. 18 bars.
8. In any case, the development length for hooks, l_{dh}, cannot be less than 8d_{b} or 6 in.
Table A5.12: Development length in inches, l_{dc}, for 60 ksi deformed bars in compression^{1,2,3,4}
f_{c}' (psi)  Bar number ["inlb" designation, with nominal diameter (in.) = bar number/8]  

3  4  5  6  7  8  9  10  11  14  18  
3000  9  11  14  17  20  22  25  28  31  38  50 
4000  8  10  12  15  17  19  22  25  27  33  43 
5000  8  9  12  14  16  18  21  23  26  31  41 
Notes:
1. Values may be multiplied by 0.75 where adequately confined by a spiral or ties (specifically, with a minimum ¼ in. spiral at no more than a 4 in. pitch; or with No. 4 ties spaced at no more than 4 in. on center).
2. Values may be multiplied by the ratio of required steel bar area to provided steel bar area, except in cases where the anchorage is required to reach the yield stress, f_{y}, or in certain highrisk seismic zones.
3. All of the modifications mentioned in Notes 1and 2 are cumulative; that is, a value may be multiplied by one or both of the applicable modification factors.
4. In any case, the development length for compression, l_{dc}, cannot be less than 8 in.
Table A5.13: Recommended minimum thickness (in.) of reinforced concrete beams and slabs for deflection control^{1,2}
Beams  L / 16  L / 18.5  L / 21  L / 8 
Slabs  L / 20  L / 24  L / 28  L / 10 
Notes:
1. Beam diagram symbols in top row of tables represent the following conditions (from left to right): simplysupported; one end pinned and one end continuous; both ends continuous; and cantilever.
2. L = span (in.)
© 2020 Jonathan Ochshorn; all rights reserved. This section first posted November 15, 2020; last updated November 15, 2020.