Formula Sheet: Concrete Design

Table of Contents
1. Material Properties and Strength Parameters
2. Strength Reduction Factors (φ Factors)
3. Flexural Design of Beams
4. Shear Design
5. Development Length and Splices
6. Columns
7. Two-Way Slabs
8. Deflection Control
9. Prestressed Concrete
10. Load Factors and Load Combinations
11. Crack Control and Spacing of Reinforcement
12. Special Provisions for Seismic Design
13. Bearing Strength
14. Temperature and Shrinkage Reinforcement
15. Footings
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Material Properties and Strength Parameters

Concrete Compressive Strength

• Specified Compressive Strength: \( f'_c \)
  • 28-day compressive strength of concrete (psi or MPa)
  • Typical values: 3000-10,000 psi (20-70 MPa)
• Modulus of Elasticity: \[ E_c = 57,000\sqrt{f'_c} \text{ (psi)} \] \[ E_c = 4700\sqrt{f'_c} \text{ (MPa)} \]
  • \( E_c \) = modulus of elasticity of concrete (psi or MPa)
  • \( f'_c \) = specified compressive strength (psi or MPa)
  • Applies to normal weight concrete with density ≈ 145 pcf (2300 kg/m³)
• For Other Concrete Densities: \[ E_c = w_c^{1.5} \times 33\sqrt{f'_c} \text{ (psi)} \] \[ E_c = w_c^{1.5} \times 0.043\sqrt{f'_c} \text{ (MPa)} \]
  • \( w_c \) = unit weight of concrete (pcf or kg/m³)
  • Valid for \( w_c \) between 90-160 pcf (1440-2560 kg/m³)
• Modulus of Rupture (Flexural Tensile Strength): \[ f_r = 7.5\lambda\sqrt{f'_c} \text{ (psi)} \] \[ f_r = 0.62\lambda\sqrt{f'_c} \text{ (MPa)} \]
  • \( f_r \) = modulus of rupture (psi or MPa)
  • \( \lambda \) = modification factor for lightweight concrete
  • \( \lambda = 1.0 \) for normal weight concrete
  • \( \lambda = 0.85 \) for sand-lightweight concrete
  • \( \lambda = 0.75 \) for all-lightweight concrete

Steel Reinforcement Properties

• Yield Strength: \( f_y \)
  • Specified yield strength of reinforcement (psi or MPa)
  • Common values: Grade 40 (40 ksi), Grade 60 (60 ksi), Grade 75 (75 ksi)
• Modulus of Elasticity of Steel: \[ E_s = 29,000,000 \text{ psi} = 29,000 \text{ ksi} \] \[ E_s = 200,000 \text{ MPa} = 200 \text{ GPa} \]
  • \( E_s \) = modulus of elasticity of reinforcing steel

Strength Reduction Factors (φ Factors)

• Tension-controlled sections: φ = 0.90
  • Applies when net tensile strain \( \varepsilon_t \geq 0.005 \)
• Compression-controlled sections (tied): φ = 0.65
  • Applies when net tensile strain \( \varepsilon_t \leq 0.002 \)
  • For members with tied reinforcement
• Compression-controlled sections (spiral): φ = 0.75
  • Applies when net tensile strain \( \varepsilon_t \leq 0.002 \)
  • For members with spiral reinforcement
• Transition zone: Linear interpolation between φ values
  • Applies when \( 0.002 < \varepsilon_t="">< 0.005="">
  • For tied members: \[ \phi = 0.65 + 0.25\left(\frac{\varepsilon_t - 0.002}{0.003}\right) \]
  • For spiral members: \[ \phi = 0.75 + 0.15\left(\frac{\varepsilon_t - 0.002}{0.003}\right) \]
• Shear and torsion: φ = 0.75
• Bearing on concrete: φ = 0.65
• Post-tensioned anchorage zones: φ = 0.85
• Strut-and-tie models: φ = 0.75

Flexural Design of Beams

Fundamental Assumptions

• Strain distribution is linear across the depth
• Maximum usable concrete compression strain = 0.003
• Stress in steel below yield: \( f_s = E_s \varepsilon_s \)
• Stress in steel at/above yield: \( f_s = f_y \)
• Tensile strength of concrete is neglected in flexural calculations
• Concrete stress block uses equivalent rectangular distribution

Equivalent Rectangular Stress Block

• Stress block depth: \[ a = \beta_1 c \]
  • \( a \) = depth of equivalent rectangular stress block (in or mm)
  • \( c \) = distance from extreme compression fiber to neutral axis (in or mm)
  • \( \beta_1 \) = stress block factor
• Stress block factor β₁:
  • For \( f'_c \leq 4000 \) psi (28 MPa): \( \beta_1 = 0.85 \)
  • For \( f'_c > 4000 \) psi (28 MPa): \[ \beta_1 = 0.85 - 0.05\left(\frac{f'_c - 4000}{1000}\right) \geq 0.65 \text{ (psi)} \] \[ \beta_1 = 0.85 - 0.05\left(\frac{f'_c - 28}{7}\right) \geq 0.65 \text{ (MPa)} \]
• Stress in stress block: \( 0.85f'_c \)
• Compression force in concrete: \[ C = 0.85f'_c ab \]
  • \( C \) = resultant compression force (lb or N)
  • \( b \) = width of compression face (in or mm)
• Tension force in steel: \[ T = A_s f_y \]
  • \( T \) = resultant tension force (lb or N)
  • \( A_s \) = area of tension reinforcement (in² or mm²)

Singly Reinforced Rectangular Beams

• Equilibrium condition: \[ C = T \] \[ 0.85f'_c ab = A_s f_y \]
• Solving for stress block depth: \[ a = \frac{A_s f_y}{0.85f'_c b} \]
• Neutral axis depth: \[ c = \frac{a}{\beta_1} \]
• Nominal moment capacity: \[ M_n = A_s f_y \left(d - \frac{a}{2}\right) \] \[ M_n = 0.85f'_c ab\left(d - \frac{a}{2}\right) \]
  • \( M_n \) = nominal moment capacity (lb-in or N-mm)
  • \( d \) = effective depth from compression fiber to centroid of tension steel (in or mm)
• Design moment capacity: \[ \phi M_n \geq M_u \]
  • \( M_u \) = factored moment (lb-in or N-mm)
  • φ determined based on strain conditions

Reinforcement Ratio

• Tension reinforcement ratio: \[ \rho = \frac{A_s}{bd} \]
  • \( \rho \) = reinforcement ratio (dimensionless)
• Balanced reinforcement ratio: \[ \rho_b = 0.85\beta_1\frac{f'_c}{f_y}\left(\frac{87,000}{87,000 + f_y}\right) \text{ (psi)} \] \[ \rho_b = 0.85\beta_1\frac{f'_c}{f_y}\left(\frac{600}{600 + f_y}\right) \text{ (MPa)} \]
  • \( \rho_b \) = balanced reinforcement ratio
  • At balanced condition: concrete crushes simultaneously as steel yields
  • Net tensile strain at balanced condition = 0.002
• Maximum reinforcement ratio: \[ \rho_{max} = 0.85\beta_1\frac{f'_c}{f_y}\left(\frac{0.003}{0.003 + 0.004}\right) \]
  • Ensures tension-controlled behavior with \( \varepsilon_t \geq 0.004 \)
• Minimum reinforcement for flexure: \[ A_{s,min} = \text{greater of:} \]
  • \[ A_{s,min} = \frac{3\sqrt{f'_c}}{f_y}bw d \text{ (psi)} \]
  • \[ A_{s,min} = \frac{0.25\sqrt{f'_c}}{f_y}bw d \text{ (MPa)} \]
  • OR
  • \[ A_{s,min} = \frac{200bw d}{f_y} \text{ (psi)} \]
  • \[ A_{s,min} = \frac{1.4bw d}{f_y} \text{ (MPa)} \]
  • \( b_w \) = web width (in or mm)

Doubly Reinforced Beams

• Total nominal moment: \[ M_n = M_{n1} + M_{n2} \]
  • \( M_{n1} \) = moment from tension steel and compression concrete
  • \( M_{n2} \) = moment from compression steel couple
• Moment from concrete compression: \[ M_{n1} = 0.85f'_c ab\left(d - \frac{a}{2}\right) \]
  • where \( a = \frac{(A_s - A'_s)f_y}{0.85f'_c b} \)
• Moment from compression steel: \[ M_{n2} = A'_s f'_s (d - d') \]
  • \( A'_s \) = area of compression reinforcement (in² or mm²)
  • \( f'_s \) = stress in compression steel (psi or MPa)
  • \( d' \) = distance from extreme compression fiber to centroid of compression steel (in or mm)
• Stress in compression steel: \[ f'_s = E_s \varepsilon'_s \leq f_y \] \[ \varepsilon'_s = 0.003\left(\frac{c - d'}{c}\right) \]
  • \( \varepsilon'_s \) = strain in compression steel

T-Beams and L-Beams

• Effective flange width: \( b_e \) is the smallest of:
  • \( b_w + 16t_s \)
  • \( b_w + \frac{L}{2} \) (for T-beams)
  • \( b_w + \frac{L}{12} \) (for L-beams)
  • Center-to-center spacing of beams
  • \( t_s \) = thickness of slab/flange (in or mm)
  • \( L \) = span length (in or mm)
• Case 1: Neutral axis in flange (a ≤ t_s):
  • Design as rectangular beam with width = \( b_e \)
  • \[ a = \frac{A_s f_y}{0.85f'_c b_e} \]
• Case 2: Neutral axis in web (a > t_s):
  • Separate compression force into flange and web components
  • \[ C_f = 0.85f'_c(b_e - b_w)t_s \]
  • \[ C_w = 0.85f'_c b_w a \]
  • \[ A_s f_y = C_f + C_w \]
  • Solve iteratively or use: \[ a = \frac{A_s f_y - 0.85f'_c(b_e - b_w)t_s}{0.85f'_c b_w} \]
  • \[ M_n = 0.85f'_c(b_e - b_w)t_s\left(d - \frac{t_s}{2}\right) + 0.85f'_c b_w a\left(d - \frac{a}{2}\right) \]

Shear Design

Shear Strength Requirements

• Design requirement: \[ \phi V_n \geq V_u \]
  • \( V_n \) = nominal shear strength (lb or N)
  • \( V_u \) = factored shear force (lb or N)
  • φ = 0.75 for shear
• Nominal shear strength: \[ V_n = V_c + V_s \]
  • \( V_c \) = shear strength provided by concrete (lb or N)
  • \( V_s \) = shear strength provided by shear reinforcement (lb or N)

Concrete Shear Strength (V_c)

• Simplified method for beams: \[ V_c = 2\lambda\sqrt{f'_c}b_w d \text{ (psi)} \] \[ V_c = 0.17\lambda\sqrt{f'_c}b_w d \text{ (MPa)} \]
  • \( b_w \) = web width (in or mm)
  • \( d \) = effective depth (in or mm)
  • λ = modification factor for lightweight concrete
• Detailed method for beams: \[ V_c = \left(1.9\lambda\sqrt{f'_c} + 2500\rho_w\frac{V_u d}{M_u}\right)b_w d \text{ (psi)} \] \[ V_c = \left(0.16\lambda\sqrt{f'_c} + 17\rho_w\frac{V_u d}{M_u}\right)b_w d \text{ (MPa)} \]
  • \( \rho_w = \frac{A_s}{b_w d} \)
  • \( \frac{V_u d}{M_u} \leq 1.0 \)
  • \( V_c \leq 3.5\lambda\sqrt{f'_c}b_w d \) (psi)
  • \( V_c \leq 0.29\lambda\sqrt{f'_c}b_w d \) (MPa)

Shear Reinforcement (Stirrups)

• Required when: \[ V_u > \phi V_c \]
• Shear strength of vertical stirrups: \[ V_s = \frac{A_v f_y d}{s} \]
  • \( A_v \) = area of shear reinforcement within spacing s (in² or mm²)
  • \( s \) = spacing of shear reinforcement along member (in or mm)
  • For two-legged stirrups: \( A_v = 2 \times A_{bar} \)
• Maximum shear reinforcement strength: \[ V_s \leq 8\sqrt{f'_c}b_w d \text{ (psi)} \] \[ V_s \leq 0.66\sqrt{f'_c}b_w d \text{ (MPa)} \]
• Minimum shear reinforcement area: \[ A_{v,min} = \frac{0.75\sqrt{f'_c}b_w s}{f_y} \text{ (psi)} \] \[ A_{v,min} = \frac{0.062\sqrt{f'_c}b_w s}{f_y} \text{ (MPa)} \]
  • OR
  • \[ A_{v,min} = \frac{50b_w s}{f_y} \text{ (psi)} \]
  • \[ A_{v,min} = \frac{0.35b_w s}{f_y} \text{ (MPa)} \]
  • Required when \( V_u > 0.5\phi V_c \)

Spacing Limits for Shear Reinforcement

• When \( V_s \leq 4\sqrt{f'_c}b_w d \) (psi) or \( V_s \leq 0.33\sqrt{f'_c}b_w d \) (MPa):
  • \( s_{max} = \min\left(\frac{d}{2}, 24 \text{ in}\right) \)
  • \( s_{max} = \min\left(\frac{d}{2}, 600 \text{ mm}\right) \)
• When \( V_s > 4\sqrt{f'_c}b_w d \) (psi) or \( V_s > 0.33\sqrt{f'_c}b_w d \) (MPa):
  • \( s_{max} = \min\left(\frac{d}{4}, 12 \text{ in}\right) \)
  • \( s_{max} = \min\left(\frac{d}{4}, 300 \text{ mm}\right) \)

Development Length and Splices

Development Length for Deformed Bars in Tension

• Basic development length: \[ l_d = \frac{f_y \psi_t \psi_e \psi_s}{25\lambda\sqrt{f'_c}}\frac{d_b}{(c_b + K_{tr})/d_b} \text{ (psi)} \] \[ l_d = \frac{f_y \psi_t \psi_e \psi_s}{2.1\lambda\sqrt{f'_c}}\frac{d_b}{(c_b + K_{tr})/d_b} \text{ (MPa)} \]
  • \( l_d \) = development length (in or mm)
  • \( d_b \) = nominal bar diameter (in or mm)
  • \( c_b \) = smaller of: distance from center of bar to nearest concrete surface, or half the center-to-center spacing of bars (in or mm)
  • \( K_{tr} \) = transverse reinforcement index
  • \( \frac{c_b + K_{tr}}{d_b} \leq 2.5 \)
• Simplified development length (conservative):
  • For #6 and smaller bars with clear spacing ≥ \( d_b \), cover ≥ \( d_b \), and stirrups:
    • \[ l_d = \frac{f_y \psi_t \psi_e}{25\lambda\sqrt{f'_c}}d_b \text{ (psi)} \]
    • \[ l_d = \frac{f_y \psi_t \psi_e}{2.1\lambda\sqrt{f'_c}}d_b \text{ (MPa)} \]
  • For all other cases:
    • \[ l_d = \frac{3f_y \psi_t \psi_e}{50\lambda\sqrt{f'_c}}d_b \text{ (psi)} \]
    • \[ l_d = \frac{f_y \psi_t \psi_e}{1.4\lambda\sqrt{f'_c}}d_b \text{ (MPa)} \]
• Modification factors:
  • \( \psi_t \) = reinforcement location factor
    • \( \psi_t = 1.3 \) for horizontal bars with > 12 in (300 mm) of fresh concrete below
    • \( \psi_t = 1.0 \) for all other cases
  • \( \psi_e \) = epoxy coating factor
    • \( \psi_e = 1.5 \) for epoxy-coated bars with cover < 3\(="" d_b="" \)="" or="" clear="" spacing="">< 6\(="" d_b="">
    • \( \psi_e = 1.2 \) for all other epoxy-coated bars
    • \( \psi_e = 1.0 \) for uncoated bars
    • \( \psi_t \psi_e \leq 1.7 \)
  • \( \psi_s \) = bar size factor
    • \( \psi_s = 0.8 \) for #6 and smaller bars
    • \( \psi_s = 1.0 \) for #7 and larger bars
• Transverse reinforcement index: \[ K_{tr} = \frac{40A_{tr}}{sn} \]
  • \( A_{tr} \) = total cross-sectional area of transverse reinforcement within spacing s (in²)
  • \( s \) = maximum spacing of transverse reinforcement (in)
  • \( n \) = number of bars being developed along the plane of splitting
  • Conservative to use \( K_{tr} = 0 \)
• Minimum development length: \( l_d \geq 12 \) in (300 mm)

Development Length for Deformed Bars in Compression

• Basic development length: \[ l_{dc} = \frac{f_y \psi_r}{50\lambda\sqrt{f'_c}}d_b \text{ (psi)} \] \[ l_{dc} = \frac{f_y \psi_r}{4.2\lambda\sqrt{f'_c}}d_b \text{ (MPa)} \]
  • \( l_{dc} \) = development length for bars in compression (in or mm)
  • \( l_{dc} \geq 8 \) in (200 mm)
• Modification factor ψ_r:
  • \( \psi_r = 1.0 \) for general case
  • \( \psi_r = 0.75 \) when reinforcement is enclosed by spiral ≥ 1/4 in diameter at pitch ≤ 4 in
• Length may be multiplied by:
  • \( \frac{A_s \text{ required}}{A_s \text{ provided}} \) when reinforcement exceeds requirement
  • Factor ≥ 0.67 (never reduce below 2/3 of calculated value)

Hooked Bar Development Length

• Standard hook:
  • 90° bend + 12\( d_b \) extension, OR
  • 180° bend + 4\( d_b \) extension (minimum)
• Development length for standard hooks in tension: \[ l_{dh} = \frac{0.02f_y \psi_e}{\lambda\sqrt{f'_c}}d_b \text{ (psi)} \] \[ l_{dh} = \frac{f_y \psi_e}{5.5\lambda\sqrt{f'_c}}d_b \text{ (MPa)} \]
  • \( l_{dh} \geq 8d_b \) and ≥ 6 in (150 mm)
• Modification factors for hooked bars:
  • Side cover ≥ 2.5 in AND for 90° hook, cover on bar extension ≥ 2 in: multiply by 0.7
  • Hook enclosed vertically/horizontally within ties/stirrups spaced ≤ 3\( d_b \): multiply by 0.8
  • Excess reinforcement: multiply by \( \frac{A_s \text{ required}}{A_s \text{ provided}} \)
  • For lightweight concrete: use appropriate λ value
  • Combined factor ≥ 0.7 (never reduce below 70% of calculated value)

Lap Splices

• Tension lap splices - Class A: \( l_{lap} = 1.0l_d \)
  • Required when \( A_s \text{ provided} \geq 2 \times A_s \text{ required} \) AND ≤ 50% of bars spliced at any location
• Tension lap splices - Class B: \( l_{lap} = 1.3l_d \)
  • Required for all other cases
• Compression lap splices: \( l_{lap} = l_{dc} \)
  • When \( f_y \leq 60,000 \) psi: minimum \( l_{lap} = 12 \) in
  • When \( f_y > 60,000 \) psi: minimum \( l_{lap} = (0.0005f_y)d_b \geq 12 \) in
• Minimum lap splice length: 12 in (300 mm)

Columns

Reinforcement Limits

• Longitudinal reinforcement ratio: \[ \rho_g = \frac{A_{st}}{A_g} \]
  • \( A_{st} \) = total area of longitudinal reinforcement (in² or mm²)
  • \( A_g \) = gross cross-sectional area of column (in² or mm²)
  • \( 0.01 \leq \rho_g \leq 0.08 \)
• Minimum number of longitudinal bars:
  • Rectangular or circular tied columns: 4 bars minimum
  • Spiral columns: 6 bars minimum
• Clear spacing between longitudinal bars:
  • ≥ 1.5\( d_b \)
  • ≥ 1.5 in (40 mm)
  • ≥ (4/3) × maximum aggregate size

Tied Columns

• Minimum tie size:
  • #3 ties for longitudinal bars #10 and smaller
  • #4 ties for longitudinal bars #11 and larger, and bundled bars
• Maximum tie spacing:
  • ≤ 16 × (diameter of longitudinal bar)
  • ≤ 48 × (diameter of tie bar)
  • ≤ least dimension of column
• Tie arrangement:
  • Every corner bar and alternate longitudinal bar shall have lateral support from corner of tie with included angle ≤ 135°
  • No bar shall be farther than 6 in clear on each side from laterally supported bar

Spiral Columns

• Minimum spiral reinforcement ratio: \[ \rho_s = 0.45\left(\frac{A_g}{A_c} - 1\right)\frac{f'_c}{f_{yt}} \]
  • \( \rho_s \) = ratio of volume of spiral reinforcement to core volume
  • \( A_c \) = area of core measured to outside of spiral (in² or mm²)
  • \( f_{yt} \) = yield strength of spiral reinforcement ≤ 100 ksi (700 MPa)
• Spiral details:
  • Minimum spiral size: #3 (10M)
  • Clear spacing: 1 in to 3 in (25 mm to 75 mm)
  • Anchorage: 1.5 extra turns at each end
• Volumetric spiral ratio: \[ \rho_s = \frac{4A_{sp}}{D_s s} \]
  • \( A_{sp} \) = area of spiral bar (in² or mm²)
  • \( D_s \) = diameter of spiral measured center-to-center of spiral bar (in or mm)
  • \( s \) = pitch of spiral (in or mm)

Axial Load Capacity

• Maximum design axial load (tied column): \[ \phi P_{n,max} = 0.80\phi\left[0.85f'_c(A_g - A_{st}) + f_y A_{st}\right] \]
  • φ = 0.65 for tied columns
  • 0.80 factor accounts for accidental eccentricity
• Maximum design axial load (spiral column): \[ \phi P_{n,max} = 0.85\phi\left[0.85f'_c(A_g - A_{st}) + f_y A_{st}\right] \]
  • φ = 0.75 for spiral columns
  • 0.85 factor accounts for accidental eccentricity

Slenderness Effects

• Effective length factor k:
  • Depends on end conditions and bracing
  • For braced (non-sway) frames: 0.5 ≤ k ≤ 1.0
  • For unbraced (sway) frames: k ≥ 1.0
• Slenderness ratio: \[ \frac{kl_u}{r} \]
  • \( l_u \) = unsupported length of column (in or mm)
  • \( r \) = radius of gyration (in or mm)
• Radius of gyration approximation:
  • Rectangular columns: \( r = 0.30h \)
  • Circular columns: \( r = 0.25h \)
  • \( h \) = overall dimension in direction of bending (in or mm)
• Slenderness may be neglected if:
  • Braced frames: \( \frac{kl_u}{r} \leq 34 + 12\left(\frac{M_1}{M_2}\right) \)
  • Unbraced frames: \( \frac{kl_u}{r} \leq 22 \)
  • \( \frac{M_1}{M_2} \) = ratio of smaller to larger end moment (negative for single curvature)

Two-Way Slabs

Minimum Slab Thickness (without deflection calculations)

• For slabs without interior beams spanning between supports:
  • Without drop panels: \[ h_{min} = \frac{l_n(\alpha_{fm} + 0.2)}{1500} \geq \frac{l_n}{30} \]
  • With drop panels: \[ h_{min} = \frac{l_n(\alpha_{fm} + 0.2)}{1800} \geq \frac{l_n}{36} \]
  • Absolute minimum: 5 in (125 mm) for slabs without drop panels
  • Absolute minimum: 4 in (100 mm) for slabs with drop panels
• Variables:
  • \( l_n \) = clear span in long direction (in or mm)
  • \( \alpha_{fm} \) = average value of \( \alpha_f \) for all beams on edges of panel
  • \[ \alpha_f = \frac{E_{cb}I_b}{E_{cs}I_s} \]
  • \( E_{cb} \) = modulus of elasticity of beam concrete
  • \( E_{cs} \) = modulus of elasticity of slab concrete
  • \( I_b \) = moment of inertia of beam about centroidal axis
  • \( I_s \) = moment of inertia of slab (width = beam spacing)

Direct Design Method - Limitations

• Minimum of 3 continuous spans in each direction
• Panels must be rectangular with long/short span ratio ≤ 2
• Successive span lengths (center-to-center) in each direction cannot differ by more than 1/3 of longer span
• Columns may be offset maximum 10% of span from either axis
• All loads must be due to gravity only (uniformly distributed)
• Service live load ≤ 2 × service dead load
• For panels with beams, relative stiffness ratio in two perpendicular directions must be between 0.2 and 5.0

Direct Design Method - Total Static Moment

• Total factored static moment: \[ M_0 = \frac{w_u l_2 l_{n}^2}{8} \]
  • \( M_0 \) = total factored static moment (lb-ft or N-m)
  • \( w_u \) = factored load per unit area (psf or kPa)
  • \( l_2 \) = length of span transverse to \( l_n \), measured center-to-center of supports (ft or m)
  • \( l_n \) = clear span length in direction moment is determined (ft or m)

Direct Design Method - Moment Distribution

• Negative factored moment at exterior support:
  • Exterior edge unrestrained: \( M_{neg} = 0 \)
  • Slab with beams or edge restrained: \( M_{neg} = 0.26M_0 \)
  • Slab without beams between interior supports: \( M_{neg} = 0.26M_0 \)
  • With edge beam: \( M_{neg} = 0.30M_0 \) (when \( \alpha_f \geq 0.8 \))
• Positive factored moment at midspan:
  • \( M_{pos} = 0.52M_0 \) for end span
  • \( M_{pos} = 0.35M_0 \) for interior span
• Negative factored moment at interior support:
  • \( M_{neg} = 0.70M_0 \) for interior support of end span
  • \( M_{neg} = 0.65M_0 \) for all interior supports
• Adjustment: Sum of positive and average negative moments must equal \( M_0 \)

Direct Design Method - Lateral Distribution

• Column strip: Design strip with width = lesser of:
  • 0.25 × smaller span (\( l_1 \) or \( l_2 \))
  • Distance to adjacent column strip
• Middle strip: Portion of slab bounded by two column strips
• Distribution factors for interior negative moment:
  • Column strip: 75% (slabs without beams) to 90% (with beams, \( \alpha_{f1}l_2/l_1 \geq 1.0 \))
  • Middle strip: remainder
• Distribution factors for exterior negative moment:
  • Column strip: 100% (edge fully restrained)
  • Middle strip: 0% (edge fully restrained)
  • Varies based on edge support conditions and \( \alpha_f \)
• Distribution factors for positive moment:
  • Column strip: 60% to 90%
  • Middle strip: remainder
  • Depends on beam configuration and stiffness ratios

Punching Shear in Two-Way Slabs

• Critical section: Located at distance \( d/2 \) from column face
  • \( d \) = effective depth of slab
  • Perimeter \( b_0 \) minimizes perimeter-to-area ratio
• Two-way shear stress: \[ v_u = \frac{V_u}{b_0 d} \]
  • \( v_u \) = factored shear stress (psi or MPa)
  • \( V_u \) = factored shear force transferred to column (lb or N)
  • \( b_0 \) = perimeter of critical section (in or mm)
• Concrete two-way shear strength (lowest of three equations):
  • \[ v_c = \left(2 + \frac{4}{\beta}\right)\lambda\sqrt{f'_c} \text{ (psi)} \]
  • \[ v_c = \left(\frac{\alpha_s d}{b_0} + 2\right)\lambda\sqrt{f'_c} \text{ (psi)} \]
  • \[ v_c = 4\lambda\sqrt{f'_c} \text{ (psi)} \]
  • For MPa: divide coefficients by 12
  • β = ratio of long side to short side of column
  • \( \alpha_s \) = 40 for interior columns, 30 for edge columns, 20 for corner columns
• Shear strength requirement: \[ \phi v_c \geq v_u \]
  • φ = 0.75 for shear

Moment Transfer at Slab-Column Connections

• Fraction of unbalanced moment transferred by flexure: \[ \gamma_f = \frac{1}{1 + \frac{2}{3}\sqrt{\frac{b_1}{b_2}}} \]
  • \( b_1 \) = width of critical section measured parallel to axis of bending (in or mm)
  • \( b_2 \) = width of critical section measured perpendicular to axis of bending (in or mm)
• Fraction of unbalanced moment transferred by eccentricity of shear: \[ \gamma_v = 1 - \gamma_f \]
• Combined shear stress: \[ v_u = \frac{V_u}{b_0 d} + \frac{\gamma_v M_u c}{J_c} \]
  • \( M_u \) = factored unbalanced moment (lb-in or N-mm)
  • \( c \) = distance from centroid of critical section to point of maximum stress (in or mm)
  • \( J_c \) = property of critical section analogous to polar moment of inertia (in⁴ or mm⁴)

Deflection Control

Minimum Thickness (one-way slabs and beams without deflection calculations)

• Simply supported: \( h_{min} = \frac{l}{20} \)
• One end continuous: \( h_{min} = \frac{l}{24} \)
• Both ends continuous: \( h_{min} = \frac{l}{28} \)
• Cantilever: \( h_{min} = \frac{l}{10} \)
• Notes:
  • \( l \) = span length (in or mm)
  • \( h \) = overall thickness (in or mm)
  • For \( f_y \neq 60,000 \) psi (420 MPa), multiply by \( (0.4 + \frac{f_y}{100,000}) \) (psi) or \( (0.4 + \frac{f_y}{700}) \) (MPa)

Immediate Deflection

• Effective moment of inertia: \[ I_e = \left(\frac{M_{cr}}{M_a}\right)^3 I_g + \left[1 - \left(\frac{M_{cr}}{M_a}\right)^3\right]I_{cr} \leq I_g \]
  • \( I_e \) = effective moment of inertia (in⁴ or mm⁴)
  • \( M_{cr} \) = cracking moment (lb-in or N-mm)
  • \( M_a \) = maximum moment in member at stage deflection is computed (lb-in or N-mm)
  • \( I_g \) = moment of inertia of gross concrete section about centroidal axis (in⁴ or mm⁴)
  • \( I_{cr} \) = moment of inertia of cracked section transformed to concrete (in⁴ or mm⁴)
• Cracking moment: \[ M_{cr} = \frac{f_r I_g}{y_t} \]
  • \( f_r \) = modulus of rupture = \( 7.5\lambda\sqrt{f'_c} \) (psi) or \( 0.62\lambda\sqrt{f'_c} \) (MPa)
  • \( y_t \) = distance from centroidal axis to extreme tension fiber (in or mm)
• For continuous beams: Use average \( I_e \) for positive and negative moment regions:
  • Simple/cantilever spans: Use \( I_e \) at midspan
  • Continuous spans: \( I_e = 0.70I_{e,midspan} + 0.15(I_{e,end1} + I_{e,end2}) \)

Long-Term Deflection

• Additional long-term deflection: \[ \Delta_{LT} = \lambda_{\Delta} \Delta_i \]
  • \( \Delta_{LT} \) = additional long-term deflection
  • \( \Delta_i \) = immediate deflection due to sustained load
  • \( \lambda_{\Delta} \) = time-dependent factor
• Time-dependent factor: \[ \lambda_{\Delta} = \frac{\xi}{1 + 50\rho'} \]
  • ξ = time-dependent factor for sustained loads
    • ξ = 1.0 for 3 months
    • ξ = 1.2 for 6 months
    • ξ = 1.4 for 12 months
    • ξ = 2.0 for 5 years or more
  • \( \rho' = \frac{A'_s}{bd} \) = compression reinforcement ratio

Deflection Limits

• Flat roofs not supporting/attached to nonstructural elements: \( \Delta_{max} = \frac{l}{180} \)
• Floors not supporting/attached to nonstructural elements: \( \Delta_{max} = \frac{l}{360} \)
• Roof/floor construction supporting/attached to nonstructural elements likely to be damaged: \( \Delta_{max} = \frac{l}{480} \)
• Roof/floor construction supporting/attached to nonstructural elements not likely to be damaged: \( \Delta_{max} = \frac{l}{240} \)

Prestressed Concrete

Prestressing Steel Properties

• Modulus of elasticity of prestressing steel:
  • \( E_{ps} = 28,500 \) ksi (197,000 MPa) for stress-relieved strands and wires
  • \( E_{ps} = 30,000 \) ksi (207,000 MPa) for low-relaxation strands and wires
• Specified tensile strength: \( f_{pu} \)
  • Typical: 250 ksi or 270 ksi (1725 MPa or 1860 MPa)
• Yield strength: \( f_{py} \approx 0.85f_{pu} \) to \( 0.90f_{pu} \)

Prestress Losses

• Types of losses:
  • Elastic shortening (ES)
  • Creep of concrete (CR)
  • Shrinkage of concrete (SH)
  • Relaxation of prestressing steel (RE)
  • Friction (pretensioned: none; post-tensioned: FR)
  • Anchorage seating/slip (post-tensioned: AS)
• Total prestress loss: \[ \Delta f_{pT} = \Delta f_{pES} + \Delta f_{pCR} + \Delta f_{pSH} + \Delta f_{pRE} + \Delta f_{pFR} + \Delta f_{pAS} \]
• Effective prestress after losses: \[ f_{pe} = f_{pi} - \Delta f_{pT} \]
  • \( f_{pi} \) = initial prestress immediately after transfer
  • \( f_{pe} \) = effective prestress after all losses

Friction Loss (Post-Tensioned)

• Prestress at distance x from jacking end: \[ f_{px} = f_{pj} e^{-(\mu\alpha + Kx)} \]
  • \( f_{pj} \) = prestress at jacking end (psi or MPa)
  • \( f_{px} \) = prestress at distance x from jacking end (psi or MPa)
  • μ = coefficient of friction (wobble coefficient)
  • α = sum of absolute values of angular changes (radians)
  • K = wobble friction coefficient per unit length (1/ft or 1/m)
  • x = distance from jacking end (ft or m)
• Simplified (for small values of μα + Kx <> \[ f_{px} \approx f_{pj}(1 - \mu\alpha - Kx) \]

Anchorage Seating Loss (Post-Tensioned)

• Prestress loss due to anchorage slip: \[ \Delta f_{pAS} = \frac{E_{ps} \Delta_{set}}{L} \]
  • \( \Delta_{set} \) = anchorage set/slip (in or mm)
  • \( L \) = length of tendon affected (in or mm)

Elastic Shortening Loss

• For pretensioned members: \[ \Delta f_{pES} = \frac{E_{ps}}{E_c}f_{cgp} \]
  • \( f_{cgp} \) = concrete stress at center of gravity of prestressing steel due to prestress force and self-weight
• For post-tensioned members (simultaneous stressing): \[ \Delta f_{pES} = \frac{n}{2}\frac{E_{ps}}{E_c}f_{cgp} \]
  • \( n \) = number of identical tendons
• For post-tensioned members (sequential stressing): Average loss varies from 0 (last tendon) to full value (first tendon)

Allowable Stresses

• Prestressing steel (immediately after transfer):
  • Due to tendon jacking force: ≤ 0.94\( f_{py} \) but not > 0.80\( f_{pu} \)
  • Immediately after transfer: ≤ 0.82\( f_{py} \) but not > 0.74\( f_{pu} \)
• Prestressing steel (service conditions after losses): ≤ 0.80\( f_{py} \)
• Concrete (at transfer before losses):
  • Compression in precompressed tension zone: ≤ 0.60\( f'_{ci} \)
  • Compression elsewhere: ≤ 0.45\( f'_{ci} \)
  • Tension (without bonded reinforcement): ≤ 3\( \lambda\sqrt{f'_{ci}} \) (psi) or ≤ 0.25\( \lambda\sqrt{f'_{ci}} \) (MPa)
  • Tension (with sufficient bonded reinforcement): ≤ 6\( \lambda\sqrt{f'_{ci}} \) (psi) or ≤ 0.50\( \lambda\sqrt{f'_{ci}} \) (MPa)
  • \( f'_{ci} \) = concrete strength at time of initial prestress
• Concrete (service conditions after losses):
  • Compression due to prestress + sustained loads: ≤ 0.45\( f'_c \)
  • Compression due to prestress + total load: ≤ 0.60\( f'_c \)
  • Tension in precompressed tension zone: ≤ 6\( \lambda\sqrt{f'_c} \) (psi) or ≤ 0.50\( \lambda\sqrt{f'_c} \) (MPa) if bonded reinforcement provided
  • Tension in precompressed tension zone: ≤ 7.5\( \lambda\sqrt{f'_c} \) (psi) or ≤ 0.62\( \lambda\sqrt{f'_c} \) (MPa) if calculations show deflection acceptable

Flexural Strength - Bonded Tendons

• Stress in prestressing steel at nominal strength: \[ f_{ps} = f_{pu}\left(1 - \frac{\gamma_p}{\beta_1}\frac{\rho_p f_{pu}}{f'_c}\right) \]
  • \( \rho_p = \frac{A_{ps}}{bd_p} \)
  • \( A_{ps} \) = area of prestressing steel (in² or mm²)
  • \( d_p \) = distance from extreme compression fiber to centroid of prestressing steel (in or mm)
  • \( \gamma_p \) = factor for type of prestressing steel
    • \( \gamma_p = 0.55 \) for \( f_{py}/f_{pu} \geq 0.90 \)
    • \( \gamma_p = 0.40 \) for \( f_{py}/f_{pu} \geq 0.85 \)
    • \( \gamma_p = 0.28 \) for \( f_{py}/f_{pu} \geq 0.80 \)
• For members with bonded reinforcement in addition to prestressing: \[ f_{ps} = f_{pu}\left(1 - \frac{\gamma_p}{\beta_1}\left[\rho_p\frac{f_{pu}}{f'_c} + \frac{d}{d_p}\left(\omega - \omega'\right)\right]\right) \]
  • \( \omega = \frac{\rho f_y}{f'_c} \) (tension reinforcement index)
  • \( \omega' = \frac{\rho' f_y}{f'_c} \) (compression reinforcement index)
• Depth of stress block: \[ a = \frac{A_{ps}f_{ps} + A_s f_y - A'_s f'_s}{0.85f'_c b} \]
• Nominal moment strength: \[ M_n = A_{ps}f_{ps}\left(d_p - \frac{a}{2}\right) + A_s f_y\left(d - \frac{a}{2}\right) - A'_s f'_s\left(d' - \frac{a}{2}\right) \]

Minimum Bonded Reinforcement in Prestressed Members

• Required when: \( f_{pe} > 0.50f_{pu} \)
• Minimum area: \[ A_s = 0.004A \]
  • \( A \) = area of that part of cross-section between flexural tension face and center of gravity of gross section (in² or mm²)
  • Bonded reinforcement must be distributed uniformly over precompressed tension zone

Shear in Prestressed Members

• Concrete shear strength: \[ V_c = \left(0.6\lambda\sqrt{f'_c} + 700\frac{V_u d_p}{M_u}\right)b_w d_p \text{ (psi)} \]
  • \( V_c \leq 5\lambda\sqrt{f'_c}b_w d_p \) (psi)
  • \( \frac{V_u d_p}{M_u} \leq 1.0 \)
• Minimum concrete shear strength: \[ V_{c,min} = 2\lambda\sqrt{f'_c}b_w d_p \text{ (psi)} \]
• For members with effective prestress ≥ 40% of tensile strength of flexural reinforcement:
  • Lesser of:
    • \[ V_{ci} = 0.6\lambda\sqrt{f'_c}b_w d_p + V_d + \frac{V_i M_{cr}}{M_{max}} \]
    • \[ V_{cw} = \left(3.5\lambda\sqrt{f'_c} + 0.3f_{pc}\right)b_w d_p + V_p \]
  • \( V_d \) = shear force at section due to unfactored dead load
  • \( V_i \) = factored shear force at section due to externally applied loads
  • \( M_{cr} \) = cracking moment
  • \( M_{max} \) = maximum factored moment at section
  • \( f_{pc} \) = compressive stress in concrete after allowable losses at centroid of section (or junction of web and flange when centroid in flange)
  • \( V_p \) = vertical component of effective prestress force at section

Load Factors and Load Combinations

Strength Design Load Combinations (LRFD)

• 1.4D
• 1.2D + 1.6L + 0.5(L_r or S or R)
• 1.2D + 1.6(L_r or S or R) + (L or 0.5W)
• 1.2D + 1.0W + L + 0.5(L_r or S or R)
• 1.2D + 1.0E + L + 0.2S
• 0.9D + 1.0W
• 0.9D + 1.0E

• D = dead load
• L = live load
• L_r = roof live load
• S = snow load
• R = rain load
• W = wind load
• E = earthquake load

Crack Control and Spacing of Reinforcement

Minimum and Maximum Spacing

• Minimum clear spacing between parallel bars:
  • ≥ \( d_b \) (bar diameter)
  • ≥ 1 in (25 mm)
  • ≥ (4/3) × maximum aggregate size
• Maximum spacing of flexural reinforcement (closest to tension face):
  • For beams and one-way slabs: \[ s_{max} = 15\left(\frac{40,000}{f_s}\right) - 2.5c_c \]
  • But not greater than: \[ s_{max} = 12\left(\frac{40,000}{f_s}\right) \]
  • \( c_c \) = clear cover from nearest surface in tension to surface of flexural tension reinforcement (in)
  • \( f_s \) = calculated stress in reinforcement at service loads (psi)
  • May take \( f_s = \frac{2}{3}f_y \) for simplicity

Skin Reinforcement

• Required when: Effective depth \( d > 36 \) in (900 mm) for beams
• Minimum area: \[ A_{sk} \geq 0.012(d - 30)s \text{ (in-lb)} \] \[ A_{sk} \geq 0.001(d - 750)s \text{ (mm-N)} \]
  • \( A_{sk} \) = total area of skin reinforcement per unit height on each side face (in²/ft or mm²/m)
  • \( s \) = spacing of skin reinforcement ≤ \( d/6 \) and ≤ 12 in (300 mm)
• Distribution: Uniformly distributed along both side faces of member for distance \( d/2 \) nearest flexural tension reinforcement

Special Provisions for Seismic Design

Special Moment Frames

• Transverse reinforcement spacing in columns:
  • Over length \( l_0 \) from joint face: \( s_0 \leq \) smallest of:
    • 8 × diameter of smallest longitudinal bar
    • 24 × diameter of hoop bar
    • One-half minimum member dimension
    • 12 in (300 mm)
  • \( l_0 \) = largest of: one-sixth clear span, maximum cross-sectional dimension, 18 in (450 mm)
• Beam transverse reinforcement in potential plastic hinge zones:
  • First hoop at ≤ 2 in (50 mm) from face of support
  • Maximum spacing ≤ minimum of:
    • \( d/4 \)
    • 8 × diameter of smallest longitudinal bar
    • 24 × diameter of hoop bar
    • 12 in (300 mm)
  • Extends from face of support over distance = 2h (twice member depth)

Confinement Reinforcement

• Volumetric ratio of confining reinforcement (rectangular hoops): \[ \rho_s = \frac{A_{sh}}{s h_c} \]
  • \( A_{sh} \) = total area of hoop reinforcement within spacing s (in²)
  • \( h_c \) = core dimension measured center-to-center of confining reinforcement (in)
  • \( s \) = spacing of transverse reinforcement (in)
• Minimum confinement: \[ \rho_s \geq 0.3\left(\frac{A_g}{A_{ch}} - 1\right)\frac{f'_c}{f_{yh}} \]
  • \( A_g \) = gross area of column section (in²)
  • \( A_{ch} \) = cross-sectional area of column core measured to outside of transverse reinforcement (in²)
  • \( f_{yh} \) = yield strength of hoop reinforcement ≤ 100 ksi

Bearing Strength

Concrete Bearing on Concrete

• Nominal bearing strength (full area loaded): \[ P_{nb} = 0.85f'_c A_1 \]
  • \( P_{nb} \) = nominal bearing strength (lb or N)
  • \( A_1 \) = loaded area (in² or mm²)
• Nominal bearing strength (partial area loaded): \[ P_{nb} = 0.85f'_c A_1\sqrt{\frac{A_2}{A_1}} \leq 1.7f'_c A_1 \]
  • \( A_2 \) = maximum area of supporting surface geometrically similar to and concentric with loaded area (in² or mm²)
  • \( \sqrt{\frac{A_2}{A_1}} \leq 2 \)
• Design bearing strength: \[ \phi P_{nb} \geq P_u \]
  • φ = 0.65 for bearing
  • \( P_u \) = factored bearing load (lb or N)

Temperature and Shrinkage Reinforcement

Slabs Where Flexural Reinforcement Extends in One Direction

• Minimum ratio of reinforcement area to gross concrete area:
  • Slabs with Grade 40 or 50 deformed bars: \( \rho_{t\&s} = 0.0020 \)
  • Slabs with Grade 60 deformed bars or welded wire reinforcement: \( \rho_{t\&s} = 0.0018 \)
  • Slabs with reinforcement \( f_y > 60,000 \) psi: \[ \rho_{t\&s} = \frac{0.0018 \times 60,000}{f_y} \geq 0.0014 \]
• Maximum spacing: Lesser of 5h or 18 in (450 mm)
  • \( h \) = slab thickness

Footings

Bearing Pressure and Soil Interaction

• Uniform bearing pressure (concentric load): \[ q = \frac{P}{A} \]
  • \( q \) = bearing pressure (psf or kPa)
  • \( P \) = column load (lb or N)
  • \( A \) = footing area (ft² or m²)
• Linear bearing pressure (eccentric load): \[ q = \frac{P}{A} \pm \frac{M_x y}{I_x} \pm \frac{M_y x}{I_y} \]
  • \( M_x, M_y \) = moments about respective axes (lb-ft or N-m)
  • \( I_x, I_y \) = moments of inertia of footing area (ft⁴ or m⁴)
  • For rectangular footing: \( I = \frac{bL^3}{12} \)

One-Way Shear (Beam Shear)

• Critical section: Located at distance \( d \) from face of column or wall
  • \( d \) = effective depth of footing
• Concrete one-way shear strength: \[ V_c = 2\lambda\sqrt{f'_c}b_w d \text{ (psi)} \] \[ V_c = 0.17\lambda\sqrt{f'_c}b_w d \text{ (MPa)} \]
  • \( b_w \) = width of critical section (in or mm)
• Design requirement: \[ \phi V_c \geq V_u \]

Two-Way Shear (Punching Shear)

• Critical section: Located at \( d/2 \) from column face
• Same provisions as two-way slabs (see Punching Shear section above)

Flexural Design of Footings

• Critical section for moment:
  • At face of column for footings supporting concrete column
  • Halfway between face and centerline of wall for footings supporting masonry wall
• Minimum depth at edge: ≥ 6 in (150 mm) for footings on soil
• Minimum reinforcement: Same as temperature and shrinkage reinforcement
  • \( A_{s,min} = 0.0018bh \) for Grade 60 reinforcement

Transfer of Force at Base of Column

• Bearing on concrete: Use bearing strength provisions (see Bearing Strength section)
• When bearing strength is exceeded: Provide dowels or mechanical connectors
• Minimum dowel area: \[ A_{s,dowel} = 0.005A_g \]
  • \( A_g \) = gross area of supported column (in² or mm²)
  • Minimum 4 dowels
• Dowel development:
  • Into column: compression development length \( l_{dc} \)
  • Into footing: full development length (tension or compression as applicable)