Reinforced Concrete Design Basics

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1. Material Properties (BS 8110 / Eurocode 2) Characteristic Compressive Strength of Concrete ($F_{cu}$): Given: $F_{cu} = 25 \, \text{N/mm}^2$ Design Compressive Strength ($f_{cd}$): BS 8110: $f_{cd} = 0.67 F_{cu} / \gamma_c = 0.67 \times 25 / 1.5 = 11.17 \, \text{N/mm}^2$ Eurocode 2: $f_{cd} = \alpha_{cc} F_{cu} / \gamma_c = 1.0 \times 25 / 1.5 = 16.67 \, \text{N/mm}^2$ (for $f_{ck}$) Characteristic Yield Strength of Reinforcement ($F_y$): Given: $F_y = 410 \, \text{N/mm}^2$ Design Yield Strength ($f_{yd}$): BS 8110: $f_{yd} = F_y / \gamma_s = 410 / 1.05 = 390.48 \, \text{N/mm}^2$ Eurocode 2: $f_{yd} = F_y / \gamma_s = 410 / 1.15 = 356.52 \, \text{N/mm}^2$ Partial Safety Factors: Concrete ($\gamma_c$): 1.5 (for ultimate limit state) Steel ($\gamma_s$): 1.05 (BS 8110) / 1.15 (Eurocode 2) 2. Design of Beams (Rectangular Section) 2.1. Flexural Design (Ultimate Limit State) Assumptions (BS 8110): Strain in concrete at failure = 0.0035 Strain in steel at failure $\ge 0.002$ Stress block: Rectangular, $0.45 F_{cu}$ over $0.9x$ depth Moment of Resistance ($M_u$): $M_u = 0.156 F_{cu} b d^2$ (for singly reinforced, $x/d \le 0.5$) $M_u = 0.87 F_y A_s z$ Lever arm ($z$): $z = d - 0.45x$ or $z = d \left[0.5 + \sqrt{0.25 - \frac{K}{0.9}}\right]$ where $K = \frac{M}{F_{cu} b d^2}$ Area of Steel ($A_s$): $A_s = \frac{M}{0.87 F_y z}$ Neutral Axis Depth ($x$): $x = \frac{0.87 F_y A_s}{0.45 F_{cu} b}$ For singly reinforced beam, $x/d \le 0.5$ (BS 8110) or $x/d \le 0.45$ (Eurocode 2 for $F_{cu} \le 50$) Minimum and Maximum Reinforcement: Minimum $A_s$: $0.24\%$ of $b d$ (BS 8110 for high yield steel) or $0.13\%$ (Eurocode 2 for $F_y = 500$) Maximum $A_s$: $4\%$ of gross cross-sectional area 2.2. Shear Design Design Shear Stress ($v$): $v = \frac{V}{b d}$ where $V$ is design shear force Allowable Shear Stress ($v_c$): Depends on $100 A_s/(b d)$, $F_{cu}$, and effective depth $d$. BS 8110 formula: $v_c = 0.79 (100 A_s/(b d))^{1/3} (F_{cu}/25)^{1/3} (400/d)^{1/4} / \gamma_m$ (with limits) Shear Reinforcement (Links/Stirrups) ($A_{sv}$): Required if $v > v_c$ $A_{sv} / s_v = \frac{(v - v_c) b}{0.87 F_{yv}}$ (BS 8110) $s_v$: spacing of links, $F_{yv}$: yield strength of links (e.g., $250 \, \text{N/mm}^2$) Maximum spacing: $0.75 d$ Maximum Shear Stress: $v_{max} = 0.8 \sqrt{F_{cu}}$ or $5 \, \text{N/mm}^2$ (whichever is smaller) 3. Design of Slabs 3.1. Flexural Design Similar to beams, but usually designed as $1 \, \text{m}$ wide strips. Effective depth ($d$) is crucial for deflections. Minimum Reinforcement: Main bars: $0.13\%$ of $b h$ (high yield) Distribution bars: $0.13\%$ of $b h$ (high yield) Maximum Spacing: Main bars: $3 d$ or $750 \, \text{mm}$ (whichever is smaller) Distribution bars: $3.5 d$ or $750 \, \text{mm}$ (whichever is smaller) 3.2. Deflection Control Achieved by limiting the span/effective depth ratio ($L/d$). Basic $L/d$ ratio is modified by factors for tension reinforcement, compression reinforcement, and flanged sections. For simply supported beams/slabs: Basic $L/d \approx 20$ (for $F_y = 410 \, \text{N/mm}^2$) For continuous beams/slabs: Basic $L/d \approx 26$ (for $F_y = 410 \, \text{N/mm}^2$) 4. Design of Columns 4.1. Axially Loaded Short Columns Ultimate Axial Load Capacity ($N_u$): $N_u = 0.4 F_{cu} A_c + 0.75 F_y A_{sc}$ (BS 8110) $A_c$: net area of concrete, $A_{sc}$: area of steel reinforcement Minimum Reinforcement: $0.4\%$ of gross cross-sectional area Maximum Reinforcement: $6\%$ (in general), $10\%$ (at laps) Links (Ties): Diameter $\ge$ quarter of largest main bar diameter, or $6 \, \text{mm}$ (whichever is greater) Spacing $\le$ 20 times the diameter of smallest main bar, or minimum column dimension, or $400 \, \text{mm}$ (whichever is smallest) 4.2. Columns with Bending Requires interaction charts (M-N interaction diagrams) or more complex iterative calculations. 5. Foundations (Pad Footings) Bearing Pressure: Check design bearing pressure against allowable soil bearing pressure. Flexural Design: Designed as an inverted cantilever slab. Moment at column face: $M = \frac{q (L-a)^2}{8}$ (for uniformly loaded cantilever) $q$: net upward soil pressure, $L$: footing dimension, $a$: column dimension Punching Shear: Critical around the column perimeter. Check shear stress at $1.5 d$ from column face (BS 8110). $v_{punch} = \frac{V_{punch}}{u d}$ where $u$ is the critical perimeter. One-Way Shear: Check at distance $d$ from column face. 6. General Detailing Rules Cover to Reinforcement: Depends on exposure conditions, fire resistance, and aggregate size. e.g., $20-50 \, \text{mm}$ for internal elements. Bar Spacing: Minimum horizontal distance between bars: $h_{agg} + 5 \, \text{mm}$ or bar diameter (whichever is greater). Maximum spacing as per flexural design. Laps and Anchorage: Lap length and anchorage length depend on bar diameter, concrete strength, and steel stress. Typical lap lengths are $30-40$ times bar diameter.