Geotechnical Systems & Piles

Cheatsheet Content

### Foundations - **Definition:** A structure that transmits loads to underlying soils. - **Importance:** Transfers superstructure loads to soil, often concrete, steel, or wood. Geotechnical engineers design based on soil properties. - **Shallow Foundations:** Often the least expensive type. - **Key Design Considerations:** - **Ultimate Limit State:** Foundation must not collapse or become unstable. - **Serviceability Limit State:** Settlement must be within tolerable limits. - _Settlement often governs shallow foundation design._ - **Practical Application:** Sizing a safe foundation requires estimating soil bearing capacity and settlement. ### Bearing Capacity Terminology - **Embedment Depth ($D_f$):** Depth below ground surface where the foundation rests. - **Ultimate Bearing Capacity ($q_{ult}$):** Maximum vertical pressure soil can support. - **Allowable/Safe Bearing Capacity ($q_a$):** Working pressure ensuring safety against shear collapse. Usually a fraction of $q_{ult}$. - **Factor of Safety (FS):** Ratio of ultimate bearing capacity to allowable bearing capacity ($FS = q_{ult} / q_a$). In geotechnical engineering, FS is typically between 2 and 5. ### Basic Concepts of Soil Behavior - **Footing Analysis:** A contact problem between soil and footing (often concrete). - **Soil Treatment:** Idealized as elastic, linear elastic-perfectly plastic, elastoplastic, or rigid-perfectly plastic material. - **Failure Mechanism:** - **Uncontained Plastic Flow:** Plastic zones appear in soil, leading to surface heaving. - **Slip Surface:** The boundary between plastic and non-plastic/non-deforming zones in rigid-perfectly plastic material. - **Heaving Influences:** Overburden pressure, strain-hardening ability. - **Contained Plastic Flow:** Failure may not be well-defined, and a distinct failure mechanism might not develop. ### Conventional Bearing Capacity Failure - **Assumption:** Rigid punch creates two plastic zones. - **Symmetry:** Each zone is symmetrical about a vertical plane through the footing center (for vertical loading). - **Wedge Formation:** One zone is a logarithmic spiral wedge merging with a triangular wedge. ### Bearing Capacity Analysis (Limit Equilibrium Method) - **Goal:** Derive ultimate vertical stress (soil's bearing capacity). - **Essential Steps:** 1. Select a plausible failure mechanism/surface. 2. Determine forces acting on the failure surface. 3. Use equilibrium equations (from statics) to find collapse/failure load. - **Example: Strip Footing on Stiff Clay (Undrained Shear Strength, $s_u$)** - **Step 1:** Assume a semi-circular failure mechanism. - **Step 2:** Draw free body diagram. Shear stress on failure surface is $s_u$. Normal and shear stresses assumed uniform. - **Step 3:** Use equilibrium equations (e.g., moment about A) to solve for $P_u$. Neglect soil mass in failure wedge. - **Exact Solution:** $P_u = 5.14 B s_u$ (for semi-circular mechanism). - **Mechanism Selection:** The chosen failure mechanism should yield the least value of $P_u$. Multiple plausible mechanisms may need to be analyzed. ### Issues with Limit Equilibrium Method - **Soil Model:** Is soil truly a rigid-perfectly plastic material? - **Deformation:** Is soil deformation negligible? - **Failure Mechanism:** Predicting the exact failure mechanism for soils is difficult. ### Modes of Bearing Capacity Failure - **General Shear Failure:** - Well-defined failure pattern with continuous slip surface from footing edge to ground surface. - Characterized by a clear failure plane and sudden collapse. - **Local Shear Failure:** - Failure pattern defined *only immediately below* the foundation. - Consists of a wedge and slip surfaces starting at footing edges. - Significant vertical compression under footing, but no catastrophic collapse or tilting. - **Punching Shear Failure:** - Vertical movement of footing with compression of soil directly underneath. - Continued penetration by vertical shear around footing perimeter. - Soil outside loaded area remains largely uninvolved. ### Terzaghi Bearing Capacity Equations (1943) - Derived from Prandtl's (1920) failure mechanism and limit equilibrium. - For a footing at depth $D_f$ below ground level of a homogeneous soil. - **Assumptions:** - Soil is semi-infinite, homogeneous, isotropic, weightless, rigid-plastic. - Embedment depth $D_f \le B$ (width of footing). - General shear failure occurs. - Soil above footing base replaced by surcharge stress ($\gamma D_f$). - Footing base is rough. - Angle in wedge is $\phi$ (later found to be $\frac{\pi}{4} + \frac{\phi}{2}$ by Vesic, 1973). - Shear strength of soil above footing base is negligible (Meyerhof, 1951, later considered it). - **Ultimate Bearing Capacity ($q_{ult}$):** - **Prandtl (1921) & Reissner (1924):** - $q_{ult} = cN_c + qN_q$ (weightless soil) - $q_{ult} = 0.5 B \gamma N_\gamma$ (for $c=0, q=0$) - **Terzaghi (1943):** - $q_{ult} = cN_c + qN_q + 0.5 B \gamma N_\gamma$ - Where $N_c, N_q, N_\gamma$ are dimensionless bearing capacity factors. - $q = \gamma D_f$ (overburden pressure). - **Bearing Capacity Factors:** - **$N_q$:** $e^{\pi \tan \phi'} \tan^2 (45^\circ + \frac{\phi'}{2})$ - **$N_c$:** $(N_q - 1) \cot \phi'$ - **$N_\gamma$:** - $2(N_q + 1) \tan \phi'$ (Vesic, 1970) - $1.8(N_q - 1) \tan \phi'$ (Hansen) - $(N_q - 1) \tan(1.4 \phi')$ (Meyerhof) - $2(N_q - 1) \tan \phi'$ (Euro Code-7) ### Bearing Capacity Equations (Centric Vertical Loads) - **Fine-grained soils (short-term / Total Stress Analysis - TSA):** - $q_{ult} = 5.14 s_u s_c d_c$ (where $s_u$ is undrained shear strength) - **All soils (long-term / Effective Stress Analysis - ESA):** - $q_{ult} = \gamma D_f N_q s_q d_q + 0.5 \gamma B N_\gamma s_\gamma d_\gamma$ - **Net Ultimate Bearing Capacity ($q_{u,net}$):** - $q_{u,net} = \gamma D_f (N_q - 1) s_q d_q i_q b_q g_q + 0.5 \gamma B' N_\gamma s_\gamma d_\gamma i_\gamma b_\gamma g_\gamma$ - **$N_q, N_\gamma$**: Bearing capacity factors (functions of $\phi'$) - **$s_c, s_q, s_\gamma$**: Shape factors - **$d_c, d_q, d_\gamma$**: Embedment depth factors - **$i_c, i_q, i_\gamma$**: Load inclination factors - **$b_c, b_q, b_\gamma$**: Base inclination (tilt) factors - **$g_c, g_q, g_\gamma$**: Ground inclination factors - **$B'$**: Width of equivalent footing ### For Meyerhof Equation (Shape, Depth, Inclination Factors) | Factor | Value | Condition | |--------|-------|-----------| | $s_c$ | $1 + 0.2 K_p (B'/L')$ | Any $\phi$ | | $s_q = s_\gamma$ | $1 + 0.1 K_p (B'/L')$ | $\phi' > 10^\circ$ | | $s_q = s_\gamma$ | $1$ | $\phi' = 0^\circ$ | | $d_c$ | $1 + 0.2 \sqrt{K_p} (D_f/B')$ | Any $\phi'$ | | $d_q = d_\gamma$ | $1 + 0.1 \sqrt{K_p} (D_f/B')$ | $\phi' > 10^\circ$ | | $d_q = d_\gamma$ | $1$ | $\phi' = 0^\circ$ | | $i_c = i_q$ | $(1 - \alpha/90^\circ)^2$ | Any $\phi'$ | | $i_\gamma$ | $(1 - \alpha/90^\circ)^2$ | $\phi' > 10^\circ$ | | $i_\gamma$ | $0$ (for $\alpha > 0$) | $\phi' = 0^\circ$ | - $K_p = \tan^2 (45^\circ + \phi'/2)$ - $\alpha$: Angle of resultant R measured from vertical. ### Groundwater Effects on Long-Term Bearing Capacity - **Situation 1:** Groundwater level at a depth $\ge B$ below footing bottom. No effect. - **Situation 2:** Groundwater level within depth $B$ below footing bottom. - $q_{u,net} = \gamma D_f (N_q - 1) s_q d_q + 0.5 [\gamma z + (\gamma_{sat} - \gamma_w) (B-z)] N_\gamma s_\gamma d_\gamma$ - **Situation 3:** Groundwater level within embedment depth $D_f$. - $q_{u,net} = [\gamma z + (\gamma_{sat} - \gamma_w) (D_f - z)] (N_q - 1) s_q d_q + 0.5 (\gamma_{sat} - \gamma_w) B N_\gamma s_\gamma d_\gamma$ - A rise in groundwater level to/above ground surface can reduce long-term bearing capacity by about half. ### Eccentric Loads - **Definition:** Resultant load location is not coincident with the footing centroid. - **Eccentricities:** $e_B = M_x / V_n$ and $e_L = M_y / V_n$, where $V_n$ is vertical load, $M_x, M_y$ are moments about x and y axes. - **Footing Width Correction:** Dimensions are adjusted to align load center with centroid. - Effective width $B' = B - 2e_B$, Effective length $L' = L - 2e_L$. - Effective area $A' = B'L'$. - **Stresses from Eccentric Loads:** For a vertical load $V_n$ at eccentricity $e$ (moment $Ve$), stresses are: - $\sigma = V_n/A \pm (V_n e y)/I$ where $I$ is moment of inertia, $y$ is distance to outer edge, $A$ is cross-sectional area. - For rectangular section ($B \times L$): $I = B^3 L / 12$, $Z = I/(B/2) = B^2 L / 6$. - **Along Width ($e_B$):** - $\sigma_{max} = \frac{V_n}{A} (1 + \frac{6e_B}{B})$ - $\sigma_{min} = \frac{V_n}{A} (1 - \frac{6e_B}{B})$ - **Along Length ($e_L$):** - $\sigma_{max} = \frac{V_n}{A} (1 + \frac{6e_L}{L})$ - $\sigma_{min} = \frac{V_n}{A} (1 - \frac{6e_L}{L})$ - **Minimum Width for Eccentric Loads:** Since soil has no tensile strength, $\sigma_{min}$ cannot be less than zero. - Eccentricity limits: $e_B \le B/6$ and $e_L \le L/6$. - Minimum width/length: $B_{min} = 6e_B$, $L_{min} = 6e_L$. ### Soil Compressibility and Scale Effects - **Terzaghi (1943):** Proposed using reduced strength characteristics ($c^*, \phi^*$) in bearing capacity equations. - **Vesic (1970) for $\phi^*$:** $\phi^* = \tan^{-1}(\beta \tan \phi')$ where $\beta = 0.67 + D_r - 0.75 D_r^2$ for $0 \le D_r \le 0.67$. - **Compressibility Factor ($\zeta_{qc}$, Vesic 1970):** - $\zeta_{qc} = \exp [\{(-4.4 + 0.6 B'/L') \tan \phi'\} + \{(3.07 \sin \phi') (\log_{10} 2I_r) / (1+\sin \phi')\}]$ - $\zeta_{sc} = 0.32 + 0.12 B'/L' + 0.60 \log_{10} I_r$ - $\zeta_{qc} = \zeta_{\gamma c}$ - **Rigidity Index ($I_r$):** $I_r = G / (c + q \tan \phi')$ where $G$ is shear modulus. - **Critical Rigidity Index ($I_r)_{crit}$:** - $(I_r)_{crit} = \frac{1}{2} \exp [\frac{1}{2} (3.30 - 0.45 B'/L') \cot (45^\circ - \phi'/2)]$ - If $I_r > (I_r)_{crit}$, soil is incompressible. ### Pile Foundations - **Definition:** Slender structural members (steel, concrete, timber, plastic, composites) transmitting loads deep into soil. - **Key Terms:** - **Skin Friction Stress ($f_s$):** Frictional/adhesive stress on pile shaft. Also called shaft friction or adhesive stress. - **End Bearing Stress ($f_b$):** Stress at the base or tip of the pile. Also called point resistance or tip resistance. - **Ultimate Load Bearing Capacity ($Q_{ult}$):** Maximum vertical load a pile can transfer to soil. - **Allowable Load Bearing Capacity ($Q_a$):** Working load ensuring safety against shear collapse. - **Factor of Safety (FS):** Ratio of ultimate to allowable load capacity (2 to 3 typically). - **When Used:** - Insufficient near-surface soil bearing capacity. - Estimated settlement exceeds tolerable limits. - Excessive differential settlement due to soil variability. - Structural loads include lateral loads, moments, uplift forces. - Difficult/expensive to excavate for shallow foundations. ### Types of Piles - **Material & Construction:** Cast-in-place concrete, precast/prestress concrete, steel H-plate, steel pipe, timber. - **Geometry:** Tapered, corrugated, straight drilled, belled drilled. - **Resistance Mechanism:** Piles are classified based on how they derive their capacity. ### Pile Installation - **Impact Hammers:** Pile driven by a ram striking the pile head. - **Vibratory Drivers/Extractors:** Drop hammers, air/steam hammers, diesel hammers, hydraulic impact hammers. - **Drilled Shafts:** Installed in a pre-bored hole. - **Key Points:** - Installation method affects structural integrity, soil properties, and load capacity. - Can cause structural damage, remold soil, and reduce load capacity. - Simple drop hammers, steam, or pneumatic hammers are common. ### Bearing Capacity Assumption (Piles) - **Ultimate Load Capacity ($Q_{ult}$):** Conventionally has two parts: - **Skin Friction ($Q_f$):** Due to friction/adhesion along the shaft. - **End Bearing ($Q_b$):** At the base or tip of the pile. - $Q_{ult} = Q_f + Q_b - W_p$ (where $W_p$ is pile weight, often neglected in calculations for $Q_{ult}$ as it's balanced by buoyant forces in saturated soil or is small compared to $Q_f+Q_b$). - **Pile Classification:** - **Friction Pile:** $Q_f > 80\%$ of $Q_b$. - **End-Bearing Pile:** $Q_b > 80\%$ of $Q_f$. - **Floating Pile:** End bearing is neglected. ### Load Transfer (Piles) - **Mobilization:** Full skin friction and end bearing are not mobilized at the same displacement. - **Full Skin Friction:** Requires 5-10 mm vertical displacement (depends on soil strength/stiffness, independent of pile length/diameter). - **Full End Bearing (Driven Piles):** Requires 8-10% of pile tip diameter displacement. - **Full End Bearing (Drilled Shafts/Bored Piles):** Requires $\approx 30\%$ of pile tip diameter displacement (when slip/failure zones like shallow foundations form). - **Importance:** Different safety factors can be applied to skin friction and end bearing. - **Coarse-grained soils:** Load transfer is approximately linear with depth (higher loads at top, lower at bottom). - **Fine-grained soils:** Load transfer is non-linear and decreases with depth (more compression at top, less at bottom due to elastic compression). ### Methods for Determining Pile Load Capacity (Driven Piles) - **Statics:** $\alpha$ and $\beta$ methods (semi-empirical). - **Pile Load Test:** (ASTM D 1143) - **Pile Driving Formulae.** - **Wave Analysis:** (Pile driving analysis) ### Pile Load Test - **Purposes:** - Determine load capacity of single pile. - Verify static load capacity estimates. - Determine settlement at working loads. - Obtain load transfer information (skin friction, end bearing). - Verify pile length. - Check structural integrity (effects of installation method). - **Setup:** Reaction beam, settlement gauges, Dywidag bars, hydraulic ram, reaction piles, test pile. - **Interpretation of Failure Load:** - **Well-defined:** Clear ultimate load. - **Ill-defined:** Intersection of tangents at beginning and end of load-displacement curve. - **Key Points:** - Provides load capacity and settlement at working load for a specific site. - Various criteria/techniques for allowable load capacity. - Requires accurate measurements and careful installation. ### Skin Friction (Statics Method) - $Q_f = \sum_{i=1}^{j} (f_s)_i \times (\text{perimeter})_i \times (\text{length})_i$ - $j$: Number of soil layers within embedded length. - **TSA ($\alpha$ method):** - $f_s$: Lower of $0.5 \sqrt{\sigma'_{zo}}$ or $0.5 s_u^{0.75} (\sigma'_{zo})^{0.25}$ (Randolph and Murray, 1985). - **ESA ($\beta$ method):** - $f_s = \beta \sigma'_{zo}$ - **Fine-grained soils (Burland, 1973):** $\beta = (1 - \sin \phi'_{cs}) (OCR)^{0.5} \tan \delta'$ - **Coarse-grained soils (Poulos, 1988):** $\beta = (1 - \sin \phi'_{cs}) \tan \delta'$ - $\delta'$: Soil-pile interface friction angle. ### Driven Piles - Skin Friction Factors (TSA, Tomlinson, 1987) - Adhesion factor ($\alpha_u$) varies with undrained shear strength ($s_u$) and pile length/diameter ratio. - Generally decreases with increasing $s_u$. - For sands, $\alpha_u$ typically ranges from 0.25 to 1.0. - For clays, $\alpha_u$ varies with stiffness. - **Typical Range of Interfacial Friction Angle ($\delta'$):** - **Material | Steel | Concrete | Timber** - $\delta'$ | $\frac{2}{3}\phi'_{cs}$ to $0.8 \phi'_{cs}$ | $0.9 \phi'_{cs}$ to $1.0 \phi'_{cs}$ | $0.8 \phi'_{cs}$ to $1.0 \phi'_{cs}$ ### End Bearing ($\alpha$ method) - **TSA:** $Q_b = f_b A_b = N_c (s_u)_b A_b$ - $f_b$: Base resistance stress. - $N_c$: Bearing capacity coefficient. - $(s_u)_b$: Undrained shear strength at base. - $A_b$: Cross-sectional area of base. - **For $\frac{L}{D} \ge 3$ and $(s_u)_b \ge 25 \text{ kPa}$:** $N_c = 9$. - **For $(s_u)_b \le 25 \text{ kPa}$:** $N_c = 6$. ### End Bearing ($\beta$ method) - **ESA:** $Q_b = f_b A_b = N_q (\sigma'_{zo})_b A_b$ - $f_b = N_q (\sigma'_{zo})_b$: Base resistance stress. - $N_q$: Bearing capacity coefficient (function of $\phi'$). - $(\sigma'_{zo})_b$: Vertical effective stress at base. - $A_b$: Cross-sectional area of base. - **Janbu (1976) Equation for $N_q$:** - $N_q = (\tan \phi' + \sqrt{1 + \tan^2 \phi'})^2 \exp(2 \psi_p \tan \phi')$ - $\psi_p$: Angle of pastification. - **Budhu Equation for $N_q$:** - $N_q = 0.6 \exp(0.126 \phi'_{cs})$ (where $\phi'_{cs}$ is in degrees). ### Pile Group Failure Modes - **Block Failure:** Occurs when pile spacing is small, causing the group to fail as a single unit. - **ESA:** $(Q_{ult})_{gb} = \sum_{i=1}^{j} \{\beta_i (\sigma'_z)_i \times (\text{perimeter})_{ig} \times L_i\} + N_q (\sigma'_z)_b (A_b)_g$ - **TSA:** $(Q_{ult})_{gb} = \sum_{i=1}^{j} \{(a_u)_i (s_u)_i \times (\text{perimeter})_{ig} \times L_i\} + N_c (s_u)_b (A_b)_g$ - Where $(\text{perimeter})_{ig}$ is the perimeter of the group in layer $i$, and $(A_b)_g$ is the base area of the group. - **Single Pile Failure:** Each pile mobilizes its full load capacity individually. - $(Q_{ult})_{gs} = n Q_{ult}$ (where $n$ is total number of piles, $Q_{ult}$ is single pile ultimate capacity). - **Efficiency Factor ($\eta_e$):** Ratio of group load capacity to total individual pile capacity. - $\eta_e = (Q_{ult})_g / (n Q_{ult})$ - **Janbu (1976) for $s/D$ ratio and friction angle:** - $\frac{s}{D} = 1 + 2 \sin \psi_p (\tan \phi' + \sqrt{1 + \tan^2 \phi'}) \exp(2 \psi_p \tan \phi')$ ### Site Characterization (Objective) - **Importance:** Adequate knowledge of ground conditions is crucial for analysis, design, and construction of geotechnical systems. - **Consequences of Inadequate Investigation:** Project delays, soil failures, cost over-runs. - **Cost-Effectiveness:** Site investigation should be part of the design process; its cost rarely exceeds 0.5% of project costs. ### Soil Exploration Methods - **Test Pit:** - **Method:** Dug by hand or machine (backhoe). - **Advantages:** Excellent shallow-depth stratigraphy, large disturbed soil samples, large undisturbed block samples, field tests at bottom. - **Disadvantages:** Depth limited (~6m), uneconomical for deep pits, difficult below groundwater/in rock, scarring. - **Drilling/Boring:** - **Method:** Drilling vertical/inclined circular holes. - **Common Procedures:** Wash boring, auger boring, percussion drilling, rotary drilling. - **Summary of Methods:** - **Geophysical Methods (GPR, Seismic, EM):** Non-destructive, quick, provides stratigraphy. No soil samples, limited design parameters. - **Power Augers:** Quick, used in uncased holes, good for undisturbed samples, no drilling mud. Depth limited (~15m), difficult in deep drilling/stiff soils, site accessibility needed. - **Wash Boring:** Quick, low equipment cost, good for uncased holes, groundwater location easy. Depth limited (~30m), slow in stiff clays/gravels, difficult to get accurate GWL, no undisturbed samples. - **Rotary Drills:** Quick, drills any soil/rock, deep drilling possible (7500m), good for undisturbed samples. Expensive, site accessibility needed, difficult to get GWL, time for setup/cleanup. ### Soil Sampling - **Method:** Inserting sampling device or excavating soil. - **Types:** - **Disturbed Samples:** Particle distribution and void space disturbed. Most sampling causes disturbance. - **Purpose:** Visual identification, reconstitute for lab tests. - **Example:** Auger sample (thick-walled sampler). - **Undisturbed Samples:** Particle distribution and void space maintained. - **Purpose:** Specialized lab tests. - **Example:** Block samples (cohesive), frozen samples (cohesionless). - **Characteristics:** Expensive, requires careful handling. - **High-Quality Samples:** Small amount of disturbance. - **Purpose:** Lab tests. - **Method:** Large diameter and/or thin-walled samplers. - **Characteristics:** Economical, requires careful handling. - **Soil Samplers:** Pushed or hammered into soil. - **Minimize Disturbance:** Reduce wall thickness, sharpen shoe, use inner/outer clearances, use piston. - **Thick-walled:** Area ratio $\ge 10\%$. - **Thin-walled:** Area ratio $ ### In-Situ Testing Techniques - **Standard Penetration Test (SPT):** - **Applicability:** Coarse-grained, non-gravelly soils. - **Data Type:** Phenomenological (strength). - **Method:** 63.5-kg hammer falling 0.76m, driving split-barrel sampler (50mm OD, 35mm ID) 0.456m in 3 increments. $N$-value is blows for last 0.304m. - **Corrections:** Need to correct $N$ for energy efficiency (typically 60%). - **Corrected $N_{cor}$:** $N_{cor} = C_R C_S C_B C_E N$ (where $C_R, C_S, C_B, C_E$ are rod length, sampler type, borehole diameter, and energy correction factors). - **Overburden Pressure Correction ($C_N$, Peck et al. 1974):** $C_N = 0.77 \log_{10} (1916 / \sigma'_{zo})$ ($C_N \le 2$, $\sigma'_{zo} > 24 \text{ kPa}$). - **For $N > 15$ (negative porewater pressure):** $N' = 15 + \frac{1}{2}(N - 15)$. - **Empirical Relationships:** - $\phi' = \tan^{-1} ((N_1)_{60} / (12.2 + 20.3 \times \sigma'_v/p_a))^{0.34}$ - $s_u = 4.5 (N_1)_{60}$ in kPa - **Advantages:** Useful for stratigraphy/bedrock, inspect soil, quick/simple, widely available, penetrates dense materials. - **Disadvantages:** Performance errors (hammer, cleaning, GWL), unreliable for coarse gravels, boulders, soft clays, silts, mixed soils. - **Compactness of Coarse-Grained Soils (based on $N_{cor}$):** - 0-4: Very loose - 4-10: Loose - 10-30: Medium - 30-50: Dense - >50: Very dense - **Cone Penetration Test (CPTu):** - **Applicability:** Soils finer than gravel. - **Data Type:** Phenomenological (deformation modulus, strength, electrical, chemical, hydraulic). - **Method:** Continuous in-situ test, measures cone resistance ($q_c$), sleeve friction ($f_s$), pore water pressure ($u_2$). - **Characteristics:** Reliable for soil stratigraphy/engineering parameters, quasi-static insertion (60° apex angle), measurements every 25-50mm penetration. - **Mainly empirical data interpretation.** - **Empirical Relationships:** - $s_u = (q_c - \sigma_z) / N_k$ - $N_k = 19 - (PI - 10)/5$ (for $PI > 10$) - $\phi'_p = 35^\circ + 11.5 \log (I_c / (30 \sigma'_{zo}))$ (for $25^\circ