CELE Reinforced & Prestressed Concrete — Reinforced Concrete Footings, Bond and DevelopmentStudy Notes

Footings must be sized from the **service (working) load** and the allowable soil bearing capacity $q_a$ (in kPa, typically provided by the geotechnical engineer or local code). The required plan area is: $$A_{\text{req}} = \frac{P_{\text{service}}}{q_a}$$ For a square footing, the side length is $B = \sqrt{A_{\text{req}}}$. Rectangular footings are sized similarly: $L \times B = A_{\text{req}}$. **Critical Distinction:** Sizing uses **service** (unfactored, working) loads. Once the plan dimensions are fixed, all subsequent structural design (shear, flexure, development) uses the **factored** design load $P_u = 1.4P_\text{DL} + 1.7P_\text{LL}$ (or per NSCP 2015 load factors). **Net Upward Design Pressure:** With factored load and plan area $A_\text{footing}$: $$q_u = \frac{P_u}{A_\text{footing}}$$ This $q_u$ represents the upward pressure exerted by the soil on the footing base during design (ultimate) conditions and is used in shear and flexure calculations. **Practical Note (PRC Common Error):** Candidates often confuse the sizing phase (service-load-based) with the design phase (factored-load-based). Always size first with service load and $q_a$; then apply load factors and use $q_u$ for structural checks.

Punching shear is a **critical failure mode** for footings. The column (or wall) tends to "punch" through the footing along a perimeter at a distance $d/2$ (effective depth) from the column faces. This is a two-way (or three-way) shear around the column. **Critical Perimeter for Square Column:** For a square interior column of side $c$ and footing effective depth $d$, the critical perimeter is: $$b_o = 4(c + d)$$ This perimeter is measured at $d/2$ from all four column faces, forming a square outline. **Nominal Punching Shear Strength:** The **critical equation** for square/compact columns (per NSCP 2015 / ACI 318) is: $$V_c = 0.33 \, \lambda \sqrt{f'_c} \, b_o d$$ where: - $f'_c$ = concrete compressive strength (MPa) - $\lambda$ = 1.0 (normal-weight concrete); 0.75 (lightweight); typically 1.0 in design - $b_o$ = critical perimeter (mm) - $d$ = effective depth (mm) - $V_c$ is in Newtons; divide by 1000 for kN The **design shear strength** is $\phi V_c$ with $\phi = 0.75$ for shear (NSCP 2015). **Punching Demand (Factored Shear Force):** The upward soil pressure acts over the entire footing area. The punching demand is the total upward pressure **outside** the critical perimeter: $$V_u = q_u \left[ A_{\text{footing}} - (c + d)^2 \right]$$ Note: $(c + d)^2$ is the area inside the critical perimeter where shear stress is not computed. **Design Check:** $$\phi V_c \ge V_u \quad \Rightarrow \quad 0.75 V_c \ge V_u$$ If this check fails, either (a) increase footing thickness $d$, (b) increase concrete strength $f'_c$, or (c) add punching shear reinforcement (stirrups or shear studs). **NSCP 2015 Alternative Equations (for reference):** ACI 318 / NSCP also permits two alternative expressions involving the aspect ratio $\beta_c$ (column perimeter / column width) and the position $\alpha_s$. These typically yield higher $V_c$ values for rectangular columns and are not governing for square columns.

In addition to punching shear, the footing must be checked for **one-way (beam) shear** along any vertical plane at a distance $d$ from the column face across the full width of the footing. **Critical Section for Beam Shear:** The critical section is at distance $d$ from the **face** of the column (not from the center). For a square column of side $c$ on a footing of width $B$, the cantilevered length is: $$\ell_{\text{cant}} = \frac{B - c}{2}$$ The one-way shear check is performed at distance $d$ from the column face: $$d_\text{section} = \ell_{\text{cant}} - d = \frac{B - c}{2} - d$$ **One-Way Shear Force:** $$V_u = q_u \times B \times d_\text{section}$$ (Here $B$ is the footing width perpendicular to the shear plane; for a square footing, both directions have the same $V_u$ by symmetry.) **Nominal One-Way Shear Strength:** $$V_c = 0.17 \, \lambda \sqrt{f'_c} \, B d$$ Design strength: $\phi V_c = 0.75 V_c$ (shear, $\phi = 0.75$). **Design Check:** $$\phi V_c \ge V_u$$ **Relationship Between the Two Shear Checks:** For typical footings with moderate overhang, **punching shear** is usually more critical (less margin). One-way shear typically governs only for very thin, heavily overhanging footings. It is still checked to satisfy code requirements.

After confirming that shear capacity is adequate, the footing must be designed for **flexure** (bending moment). The critical section for moment in a footing is at the **face of the column**. **Critical Section for Moment:** The footing acts as a cantilever beam extending from the column face to the edge. At the column face, the cantilevered slab width is $\frac{B - c}{2}$ (for a square footing and square column). **Flexural Demand (Factored Moment):** The upward soil pressure over the cantilevered length $\ell = \frac{B - c}{2}$ creates a moment: $$M_u = q_u \times B \times \frac{\ell^2}{2} = q_u \times B \times \frac{1}{2} \left( \frac{B - c}{2} \right)^2$$ Simplifying: $$M_u = \frac{q_u \times B \times (B - c)^2}{8}$$ For footings extending in both directions symmetrically, apply this calculation in each direction. **Required Flexural Reinforcement:** Using the standard beam design formula, the required tension steel area is: $$A_s = \frac{M_u}{\phi f_y \left( d - \frac{a}{2} \right)}$$ where $a = \frac{A_s f_y}{0.85 f'_c B}$ (depth of compression block). Alternatively, use the design coefficient $R_u = \frac{M_u}{B d^2}$ and solve iteratively or use ACI tables. **For simplicity in hand calculations** (as in PRC exams), assume $a \approx d/5$ or use the approximate formula: $$A_s \approx \frac{M_u}{0.9 f_y d}$$ **Minimum Reinforcement:** Per ACI 318 / NSCP 2015, the minimum reinforcement ratio is: $$\rho_{\min} = \frac{1.4}{f_y} = \frac{1.4}{415} = 0.00337$$ So $A_{s,\min} = \rho_{\min} \times B \times d = 0.00337 B d$. **Bar Layout (Reinforcement Distribution):** For a square footing, bars run in both directions. Typically, place the required $A_s$ near the tension face (bottom). The bars should be: - Distributed uniformly across the width $B$ - Spaced not more than $d$ or $300$ mm apart (NSCP/ACI) - Adequately covered and developed (covered in Bond and Development section) **Common Practice:** In footings, reinforce both the bottom (tension) in both orthogonal directions. Some designs also use top bars (negative reinforcement) near the column to control crack control if loads are eccentric.

Reinforcing bars embedded in concrete develop their yield strength through **bond stress** (adhesion and friction) along the bar surface. The bar must be long enough to transfer the full yield force to the concrete. This embedment length is called the **development length** $\ell_d$. **Why Development Matters in Footings:** In a footing, bars placed near the bottom are anchored by embedment into the footing depth. If $\ell_d$ is too short, the bars cannot reach their yield force, and the footing will fail prematurely (bar slip). The development length depends on: - Bar diameter $d_b$ - Concrete strength $f'_c$ - Yield strength $f_y$ - Bar position (top/bottom, confined/unconfined) - Cover and spacing **Simplified Development Length Formula (NSCP 2015 / ACI 318):** For **tension bars** (normal-weight concrete, $\lambda = 1.0$): $$\ell_d = \frac{f_y \psi_t \psi_e}{K \lambda \sqrt{f'_c}} d_b$$ where: - $f_y$ = yield strength of bar (MPa), typically 415 or 500 for Philippine steel - $d_b$ = bar diameter (mm) - $f'_c$ = concrete strength (MPa) - $K$ = coefficient depending on bar size: - $K = 2.1$ for bars $\le 20$ mm diameter ("small bars") - $K = 1.7$ for bars $> 20$ mm diameter ("large bars") - $\psi_t$ = top-bar factor: - $\psi_t = 1.3$ for top bars (cast with fresh concrete above) - $\psi_t = 1.0$ for bottom bars - $\psi_e$ = epoxy-coating factor: - $\psi_e = 1.5$ for epoxy-coated bars (rare in footings) - $\psi_e = 1.0$ for uncoated bars (normal) - $\lambda$ = 1.0 for normal-weight concrete **Minimum Development Length:** Regardless of calculation, $\ell_d \ge 300$ mm per NSCP 2015 / ACI 318. **Modified (Favorable) Conditions:** The above formula applies when bars have **adequate cover and spacing**: - Clear cover $\ge d_b$ and - Clear spacing between bars $\ge d_b$ and - Minimum stirrups present If these conditions are **not met**, the coefficients $K$ are reduced (less-favorable conditions): - $K = 1.4$ for small bars ($\le 20$ mm) - $K = 1.1$ for large bars ($> 20$ mm) **Hooks and Splices:** - **Hooks** (90° or 180°) reduce the required development length. A **standard 180° hook** can develop a bar in approximately $0.7 \ell_d$ (straight embedment). Hooks are used when straight length is constrained. - **Mechanical anchorages** (plates, caps) can also reduce $\ell_d$. - **Splices** (lap, welded, mechanical) must develop both bars; lap-splice length typically $= 1.3 \ell_d$ or per code tables. **Application to Footings:** In a footing, bars are placed near the bottom (tension face). The development length is measured from the **critical section** (at the column face, where moment is maximum and bars reach peak stress) to the **end of the bar**. Typically: $$\text{Available length} = \text{half footing width} - \text{cover at edge}$$ For example, if $B = 2.70$ m and cover = 75 mm: $$\text{Available length} = \frac{2.70}{2} - 0.075 = 1.35 - 0.075 = 1.275 \text{ m} = 1275 \text{ mm}$$ If the calculated $\ell_d > 1275$ mm, the straight bar cannot fit; use hooks or reduce bar diameter. **Common Board-Exam Pitfall:** - Forgetting to add $\psi_t$ for top bars (very common; $\psi_t = 1.3$ increases $\ell_d$ by 30%) - Using $K = 2.1$ for all bars (only for $\le 20$ mm) - Omitting the 300 mm minimum - Calculating available length incorrectly (not accounting for cover or bar bend radius)

A systematic approach ensures accurate footing design and helps avoid exam errors: **Step-by-Step Workflow:** **Phase 1: Sizing (Service Loads)** 1. Obtain column load (D + L) and allowable soil bearing $q_a$ from geotechnical report. 2. Compute required area: $A_{\text{req}} = P_{\text{service}} / q_a$. 3. Select plan dimensions $B \times L$ (typically square, $B = \sqrt{A}$). 4. Choose footing thickness $h$ (typically 0.3–0.6 m for light to medium loads; 0.8–1.2 m for heavy industrial loads). Rule of thumb: $h \approx (B - c) / 4$ to $(B - c) / 2$ for moderate overhang. **Phase 2: Structural Design (Factored Loads)** 1. Apply load factors: $P_u = 1.4 P_{\text{DL}} + 1.7 P_{\text{LL}}$ (NSCP 2015). 2. Compute design pressure: $q_u = P_u / A$. 3. **Two-way shear check:** Compute $b_o = 4(c + d)$, $V_c = 0.33\lambda\sqrt{f'_c}b_o d$, demand $V_u = q_u[A - (c+d)^2]$. Verify $\phi V_c \ge V_u$ ($\phi = 0.75$). 4. **One-way shear check:** Compute $V_u$ at distance $d$ from column face, $V_c = 0.17\lambda\sqrt{f'_c}Bd$. Verify $\phi V_c \ge V_u$. 5. **Flexure design:** Compute $M_u = q_u B (B-c)^2 / 8$. Find $A_s$ using $A_s = M_u / (0.9 f_y d)$. Compare to $A_{s,\min} = 0.00337 B d$; use the larger. 6. Select bar size and spacing. Verify spacing $\le 300$ mm. 7. **Development length:** For each bar, compute $\ell_d$ using the formula. Check that available embedment $\ge \ell_d$. If not, use hooks or reduce bar diameter. **Phase 3: Detailing** 1. Show bar layout (plan and section views). 2. Specify cover (typically 40–50 mm for interior footings, 75 mm for exposed surfaces). 3. Provide clear dimensions, bar size, spacing, and embedment lengths. 4. Note any hooks or bends. **Common Pitfalls (PRC Board Exams):** | Pitfall | Impact | Prevention | |---------|--------|------------| | **Confusing service and factored loads** | Wrong footing size or design pressure | Always label: sizing uses service $P$; design uses $P_u$ | | **Using gross area for punching demand** | Overestimating $V_u$, incorrect conclusion | Subtract $(c+d)^2$ from footing area; this is the area outside the critical perimeter | | **Omitting one-way shear check** | Missing a potential failure mode | Always check both punching and beam shear; list both in solutions | | **Wrong critical section for moment** | Using wrong $\ell$ in $M_u$ calculation | Moment is at the **column face**, not at centerline or edge | | **Applying $\ell_d$ from edge instead of from critical section** | Underestimating required embedment | In footings, development is measured from the column face (critical section) toward the edge | | **Forgetting $\psi_t = 1.3$ for top bars** | Underestimating $\ell_d$ for top bars (can cause exam errors) | Top bar factor **always** 1.3; bottom bar always 1.0; **check bar position** | | **Using $K = 2.1$ for all bars** | Wrong $\ell_d$ for bars $> 20$ mm | Split: $K = 2.1$ for $\le 20$ mm; $K = 1.7$ for $> 20$ mm | | **Minimum $\ell_d$ of 300 mm** | Can be critical for small bars in thick footings | Always take max($\ell_d$ calc, 300 mm) | | **Inadequate available length** | Bars cannot develop; foundation fails | Always compute available embedment and compare to required $\ell_d$ | | **Ignoring unfavorable conditions** | Underestimating $\ell_d$ when cover/spacing are tight | Check code requirements for cover and spacing; if not met, use unfavorable $K$ | **Exam Strategy:** 1. **Label all quantities clearly.** Examiners grade step-by-step; unclear notation loses points. 2. **Show load factor application.** Write "$P_u = 1.4(800) + 1.7(500) = ...$" explicitly. 3. **Two-way AND one-way shear.** Never omit either check, even if one is clearly adequate. 4. **Unit consistency.** Keep dimensions in mm (or all in m), convert stresses to MPa, forces to kN. Show conversions: "$b_o = 3600$ mm $= 3.6$ m." 5. **Development length table or formula.** Write the formula and plug in values step-by-step, showing the coefficient choice ($K = 2.1$ or $1.7$, $\psi_t$, etc.). 6. **Available vs. required.** Explicitly compare available embedment to $\ell_d$; state whether hooks are needed.