Hibbeler

Cheatsheet Content

1. Introduction to Retaining Walls Retaining walls are civil engineering structures designed to resist the lateral pressure of soil or other granular materials, thereby preventing the collapse or erosion of an earth mass that is at a higher elevation than the ground in front of the wall. They are commonly used in various applications such as basements, bridge abutments, roads, and landscaping. Common types of retaining walls include: Gravity Walls: Rely solely on their own mass for stability against overturning and sliding. Often made of mass concrete, masonry, or stone. Cantilever Walls: The most common type of reinforced concrete retaining wall. Consist of a vertical stem and a horizontal base slab (footing). The stem acts as a cantilever fixed to the base. Counterfort/Buttress Walls: Used for high walls where cantilever walls become uneconomical. They feature triangular cross-walls (counterforts on the retained side, buttresses on the exposed side) that tie the stem and base together, acting as tension/compression elements. Sheet Pile Walls: Made of steel, vinyl, or timber piles driven into the ground. Diaphragm Walls: Cast in-situ concrete walls, often used for deep excavations. This cheatsheet primarily focuses on the design of Cantilever Reinforced Concrete Retaining Walls as per BS 8110. 2. Design Procedure Overview The design of a cantilever retaining wall involves several stages, combining geotechnical principles for stability and structural engineering principles for reinforcement design: Data Collection and Site Investigation: Gather information on soil properties, site conditions, groundwater levels, and applied loads (surcharge). Preliminary Sizing: Estimate initial dimensions for the stem, base slab (toe and heel) based on empirical rules or experience. Stability Checks (Serviceability Limit State - SLS): Check against overturning. Check against sliding. Check bearing pressure on the underlying soil. (These checks use unfactored loads and soil properties). Structural Design (Ultimate Limit State - ULS): Design the stem for bending and shear. Design the toe slab for bending and shear. Design the heel slab for bending and shear. (These checks use factored loads and material strengths). Detailing: Specify reinforcement arrangements, concrete cover, construction joints, and drainage provisions. 3. Material Properties (BS 8110-1:1997, Sections 2 & 3) Concrete: Characteristic compressive strength: $f_{cu}$ (e.g., C25/30, C30/37, C40/50). The first number is cylinder strength, second is cube strength. BS 8110 uses cube strength. Design strength for ULS: $f_{cd} = 0.67 f_{cu} / \gamma_{mc}$. Partial safety factor for concrete: $\gamma_{mc} = 1.5$ (for flexure and axial load). Unit weight of reinforced concrete: $\gamma_c \approx 24 \text{ kN/m}^3$. Steel Reinforcement: Characteristic yield strength: $f_y$ (e.g., 460 N/mm$^2$ for high yield deformed bars, 250 N/mm$^2$ for mild steel plain bars). Design strength for ULS: $f_{yd} = f_y / \gamma_{ms} = 0.87 f_y$. Partial safety factor for steel: $\gamma_{ms} = 1.05$. 4. Soil Properties & Earth Pressures Geotechnical Parameters (from site investigation): Unit weight of retained soil: $\gamma_s$ (or $\gamma_{fill}$ if different). Angle of internal friction of retained soil: $\phi$ (or $\phi_{fill}$). Cohesion of retained soil: $c$. Often assumed zero for granular backfill. Unit weight of foundation soil: $\gamma_{f}$ (below footing). Angle of internal friction of foundation soil: $\phi_{f}$. Allowable bearing pressure of foundation soil: $P_{allow}$ (or $q_{all}$). Coefficient of friction between concrete and foundation soil: $\mu = \tan(k \phi_f)$, where $k$ is typically $2/3$ to $1$. Active Earth Pressure (Rankine's Theory - for vertical wall and horizontal backfill): Coefficient of Active Earth Pressure: $K_a = \tan^2(45^\circ - \phi/2)$. Lateral pressure at depth $h$: $P_a = K_a \gamma_s h$. Total active force (for height $H$): $F_a = 0.5 K_a \gamma_s H^2$, acting at $H/3$ from the base. Surcharge Load ($q_s$): Uniform surcharge on the retained soil surface (e.g., traffic load, stored material). Equivalent height of soil: $h_e = q_s / \gamma_s$. Equivalent lateral pressure due to surcharge: $P_{surcharge} = K_a q_s$, acting uniformly over the full height $H$. Total force due to surcharge: $F_{qs} = K_a q_s H$, acting at $H/2$ from the base. Passive Earth Pressure: Coefficient of Passive Earth Pressure: $K_p = \tan^2(45^\circ + \phi_f/2)$. Used to enhance sliding resistance, typically provided by a key or embedded toe. Often ignored for conservative design unless a positive key is provided. 5. Preliminary Sizing (Typical Empirical Ratios) These are initial estimates; final dimensions are determined by stability and structural checks. $H$ = total height from base of footing to top of wall. Base width ($B$): $0.5H$ to $0.7H$. Toe projection ($T$): $0.2B$ to $0.3B$. Heel projection ($L_H$): $B - T - t_s$ (where $t_s$ is stem thickness at base). Base thickness ($t_f$): $0.08H$ to $0.1H$, or $H/12$ to $H/15$. Minimum $300 \text{ mm}$. Stem thickness ($t_s$): Tapers from $t_f$ at the base to $200-300 \text{ mm}$ at the top. Minimum $200 \text{ mm}$. 6. Stability Checks (Serviceability Limit State - SLS) These checks ensure the overall stability of the wall against gross movement. Use unfactored (characteristic) loads and average soil properties. Overturning Stability: Calculate all moments about the toe (point 'O') of the footing. Resisting Moment ($M_R$): Sum of moments due to vertical forces (self-weight of wall, soil above heel, surcharge above heel). Overturning Moment ($M_O$): Sum of moments due to horizontal forces (active earth pressure, surcharge pressure). Factor of Safety (FoS) against Overturning: $FoS_{OT} = M_R / M_O$. Requirement: $FoS_{OT} \ge 2.0$ (common practice). Sliding Stability: Calculate all horizontal forces. Resisting Force ($F_R$): Sum of friction force under the base + passive resistance (if considered). Friction force = $\mu \times \sum V$, where $\sum V$ is the sum of all vertical forces on the base. Sliding Force ($F_S$): Sum of active earth pressure and surcharge pressure. Factor of Safety (FoS) against Sliding: $FoS_{SL} = F_R / F_S$. Requirement: $FoS_{SL} \ge 1.5$ (if passive resistance included), or $FoS_{SL} \ge 2.0$ (if passive resistance ignored, more conservative). If $FoS_{SL}$ is insufficient, consider: increasing base width, adding a shear key, or sloping the base. Bearing Pressure Check: Calculate the resultant vertical load ($\sum V$) and its position from the toe. Locate the eccentricity of the resultant vertical force from the center of the base: $e = (B/2) - (M_R - M_O) / \sum V$. Check that the eccentricity falls within the middle third of the base: $e \le B/6$. This ensures no tension at the heel. Calculate maximum and minimum bearing pressures: $q_{max/min} = \frac{\sum V}{B} (1 \pm \frac{6e}{B})$. Requirement: $q_{max} \le P_{allow}$ (allowable bearing pressure) and $q_{min} \ge 0$ (no uplift). 7. Structural Design (Ultimate Limit State - ULS) Design individual components (stem, toe, heel) for bending and shear using factored loads and material partial safety factors. Load Factors (BS 8110 Table 2.1): Dead loads (self-weight of wall, soil): $1.4$. Imposed loads (surcharge): $1.6$. Earth pressure: $1.4$. Ultimate Bearing Pressure: Calculate the upward soil pressure under the base using ultimate vertical and horizontal loads. This is needed for designing the toe and heel. 7.1. Design of Stem The stem acts as a vertical cantilever fixed at the top of the base slab. Critical section for bending and shear: At the junction with the base slab (face of the base). Design Moment ($M_{stem}$): Due to factored active earth pressure and factored surcharge pressure. $M_{stem} = (1.4 \times 0.5 K_a \gamma_s H_{stem}^2) \times (H_{stem}/3) + (1.4 \times K_a q_s H_{stem}) \times (H_{stem}/2)$. $H_{stem}$ = height of stem from top of base slab. Design Shear Force ($V_{stem}$): $V_{stem} = 1.4 \times 0.5 K_a \gamma_s H_{stem}^2 + 1.4 \times K_a q_s H_{stem}$. Flexural Design (for a 1m strip): Calculate effective depth $d = t_s - \text{cover} - \text{bar diameter}/2$. $K = M_{stem} / (f_{cu} b d^2)$ (where $b = 1000 \text{ mm}$). If $K \le K' = 0.156$, no compression reinforcement is required. Lever arm $z = d [0.5 + \sqrt{0.25 - K/0.9}] \le 0.95d$. Area of tension steel required $A_s = M_{stem} / (0.95 f_y z)$. Select appropriate bar size and spacing for the main vertical reinforcement on the earth side. Shear Check: Design shear stress $v = V_{stem} / (bd)$. Compare $v$ with allowable concrete shear stress $v_c$ from BS 8110 Table 3.9 (based on $100 A_s/(bd)$, $d$, and $f_{cu}$). If $v \le v_c$, no shear reinforcement is needed. Otherwise, increase stem thickness. Also check $v \le v_{max} = 0.8 \sqrt{f_{cu}}$ or $5 \text{ N/mm}^2$. Minimum and Distribution Reinforcement: Provide horizontal distribution bars on both faces and vertical bars on the exposed face. 7.2. Design of Toe Slab The toe slab acts as a cantilever fixed at the face of the stem. Critical section for bending and shear: At the face of the stem. Forces acting on the toe: Upward ultimate soil pressure (from ULS bearing pressure analysis) and downward self-weight of the toe. The soil pressure profile under the base needs to be re-evaluated for ultimate loads. Design Moment ($M_{toe}$): Integrate the net ultimate pressure diagram over the toe projection from the stem face. Design Shear Force ($V_{toe}$): Integrate the net ultimate pressure diagram over the toe projection from the stem face. Flexural Design: Calculate $A_s$ (bottom reinforcement) for $M_{toe}$ per meter width. Shear Check: Check $v = V_{toe} / (bd)$ against $v_c$. 7.3. Design of Heel Slab The heel slab acts as a cantilever fixed at the face of the stem. Critical section for bending and shear: At the face of the stem. Forces acting on the heel: Upward ultimate soil pressure, downward self-weight of the heel, downward weight of the soil above the heel, and downward surcharge above the heel. Design Moment ($M_{heel}$): Sum of moments about the stem face due to all ultimate loads. Design Shear Force ($V_{heel}$): Sum of forces about the stem face. Flexural Design: Calculate $A_s$ (top reinforcement) for $M_{heel}$ per meter width. Shear Check: Check $v = V_{heel} / (bd)$ against $v_c$. 8. Shear Key Design (If required for sliding) If sliding stability is insufficient, a shear key can be added below the heel or stem. The key engages passive earth pressure in front of it. Design the shear key as a small cantilever for bending and shear due to the passive pressure. 9. Reinforcement Detailing (BS 8110-1:1997, Section 3.12) Main Reinforcement: Stem: Main vertical bars on the earth-retained side (tension face). Horizontal distribution bars on both faces. Toe: Main horizontal bars at the bottom (tension face). Heel: Main horizontal bars at the top (tension face). Minimum Reinforcement (BS 8110-1:1997, Table 3.25): For high yield ($f_y = 460 \text{ N/mm}^2$): $0.13\% bh$. For mild steel ($f_y = 250 \text{ N/mm}^2$): $0.24\% bh$. Maximum Spacing (BS 8110-1:1997, Section 3.12.11.2.7): Main reinforcement: $3d$ or $750 \text{ mm}$, whichever is smaller. Distribution reinforcement: $3.5d$ or $450 \text{ mm}$, whichever is smaller. Concrete Cover (BS 8110-1:1997, Table 3.4): For faces in contact with ground: $75 \text{ mm}$. For faces exposed to weather: $50 \text{ mm}$. For faces not exposed to weather: $25 \text{ mm}$ (for $f_{cu} \ge \text{C35}$ in non-aggressive environment) or $35 \text{ mm}$ (for $f_{cu} \ge \text{C25}$). Development Length (Anchorage) and Lap Lengths: Ensure adequate lengths for bars to develop their full strength and for splicing. Construction Joints: Typically provided at the base-stem junction. For long walls, vertical movement joints (expansion/contraction) are required at intervals (e.g., $10-15 \text{ m}$), and vertical construction joints (day joints) are needed for practical construction. Use shear keys and/or dowel bars at joints to transfer shear and maintain alignment. 10. Drainage Provisions Effective drainage is CRITICAL to prevent the build-up of hydrostatic pressure behind the wall, which significantly increases lateral forces and can lead to failure. Weep Holes: Small openings through the stem, typically $100 \text{ mm}$ diameter at $1-2 \text{ m}$ spacing, just above ground level in front of the wall. Granular Backfill: A free-draining granular layer (e.g., gravel, crushed stone) immediately behind the stem, extending to the full height of the retained soil. This prevents fine soil from clogging drains. Perforated Drain Pipe (French Drain): A pipe laid at the base of the granular backfill, connected to an outlet, to collect and discharge water. A filter fabric (geotextile) should be placed between the granular backfill and the retained soil to prevent migration of fines.