Mat Foundations and Pile Foundations – Key Concepts
Mat Foundations and Pile Foundations – Key Concepts
Combined Footings and Mat Foundations (6.1–6.7)
Combined footings support a line of two or more columns; mat foundations (raft) support several columns/walls on a single concrete slab.
Common types of combined footings:
Rectangular combined footing
Trapezoidal combined footing
Strap footing
Mat foundations are advantageous on soils with low bearing capacity or when spread footings would encroach on property lines; mats may be supported by piles to control settlement.
Rectangular combined footing (design steps):
Area:
Location of resultant:
For uniform soil pressure, resultant passes through foundation centroid; length is chosen accordingly (conceptually, width determined after setting length). In the text: L=2igl(L_2+Xigr) and the width follows from area, with .
Trapezoidal combined footing (6.2) design idea: determine area from net allowable pressure, location of resultant from column loads, and solve for base dimensions using geometric relations (Eq. 6.6 and 6.7). For a trapezoid, with shape constraints.
Strap footing (cantilever) (6.3): connects an eccentrically loaded base to an interior column’s foundation to balance moments; used when space or loading demands restraining differential settlement.
Common mat foundation types (6.4): flat plate, flat plate thickened under columns, beams-and-slab, flat plates with pedestals, slab with basement walls as part of the mat.
Mats may be pile-supported or used over high water tables to control buoyancy.
Bearing capacity of mats (6.4): use the same ultimate bearing capacity equation as shallow foundations (Section 3.6):
Gross ultimate capacity: qu = c'NcF{cs}F{cd}F{ci} + qNqF{qs}F{qd}F{qi} + frac{1}{4}etaar{y}BNF{ys}F{yd}F{yi}
Net ultimate capacity:
For mats on clay, FS guidance: not less than about 3 under dead load or max live load; for extreme conditions, 1.75–2; for mats on sand, FS around 3 is typical.
For saturated clays, a special form of the net ultimate capacity is given, incorporating undrained cohesion $c_u$ (Eq. (6.8)–(6.11)); for sands, net allowable pressure can be estimated from standard penetration numbers (Eq. (6.12)–(6.13)); simplified raft guidance yields:
Net allowable pressure for lightly simplified raft:
Net allowable pressure in English units:
Design condition: the applied foundation pressure $q$ must satisfy In practice, serviceable raft settlements are constrained (typical raft settlement 50 mm, differential ~19 mm) to ensure performance.
Example 6.1 and 6.3 illustrate calculating net ultimate bearing capacity and safety factors for mats on clay with given soil properties and loads; compensated foundations (6.7) reduce net soil pressure by deeper basements (compensation principle).
Compensated foundation design (6.7): deeper base beneath a larger portion of the structure can yield a more uniform net soil pressure; partially compensated foundations use a safety formulation (Eq. 6.21–6.22) to assess FS.
Pile Foundations (11.2–11.25)
Piles are deep foundations used when shallow foundations are unsuitable due to highly compressible soils, horizontal loading, expansive soils, uplift conditions, or when piles reach a stronger layer.
Primary pile types (11.3):
Point bearing piles: reach bedrock/strong stratum; capacity mainly from tip resistance .
Friction piles (skin friction): rely on shear along the shaft; length is driven into soft/weak soils; can be very long if no bedrock is reached.
Compaction piles: compact soil near ground surface to improve density.
Pile installation methods (11.4): various hammers (drop, single-acting, double-acting, diesel) plus possible jetting/augering; cushions may be used to reduce impact.
Load transfer mechanism (11.5): total pile load $Q$ is sum of shaft friction $Qs$ plus tip load $Qp$; at ultimate load, $Q = Q_u$ and shaft friction mobilizes at smaller displacements than tip resistance.
Pile capacity framework (11.6): ultimate pile capacity
Total:
Point bearing (tip) capacity at the pile tip can be expressed as for driven piles in a given soil, with effective vertical stress $q'$ at the tip and bearing-capacity factors $N^$, $N^{**}$ adjusting for shape/depth.
Friction (skin) resistance: Q_s = iggl( ext{sum over depth } ziggr) p rac{dL}{dz} f(z) where $p$ is pile perimeter, and $f(z)$ is unit skin friction along the length.
Meyerhof’s method for $Qp$ (11.7): $qp$ increases with embedment until a critical ratio $(L/D){cr}$; beyond this, $qp$ tends to a maximum value; practical estimation uses standard penetration (N60) data and a critical embedment ratio.
Vesic’s method (11.8): point bearing $Q_p$ via cavity-expansion theory:
Qp = Ap ty N^* + ext{(friction term)} with $N^$ and $N^{*}$ depending on soil and effective stress; for clays, $qp$ relates to undrained cohesion $cu$ below the pile tip.
Janbu’s method (11.9): another approach to compute $Q_p$ using $c'N^* + q'N^$ with $N^, N^{*}$ from assumed failure surfaces; provides alternative $N^$ values via geometry of the failure surface.
Frictional resistance in sand (11.12): $Qs = rac{pL f(z)}{pz}$; unit friction $f(z)$ varies with depth and installation method; typical $f$ increases with depth and then levels off; empirical correlations link $f$ to $N{60}$ (Meyerhof) and cone tests (Nottingham–Schmertmann) (11.42–11.46).
Frictional resistance in clay (11.13): several methods:
λ-method: $f_{av} = \lambda ( \sigma' + 2 c )$ (mean over embedment);
a-method: $f_{ad} = a \\bar{\sigma} f$, with adhesion factor $a$ depending on soil stiffness;
β-method (drained/remolded state): $f = K \, an \phi' \, \\sigma^+$; with $K$ earth-pressure coefficient depending on OC/NC state.
For normally consolidated clays, conservative forms use $K \,=\, 1 - \sin \phi'$, etc.
Piles resting on rock (11.14): point-bearing on rock uses $q_p = q (N+1)$ with $N = \tan^2(45 + \frac{\phi}{2})$; typical table values given for rock strength.
Negative skin friction (11.21): downward drag caused by consolidation or water-table changes; formulas to compute down-drag $Q_n$ using soil-pile interaction and pore pressure conditions; cases include clay fill over granular, granular fill over clay, and downdrag zone treatments.
Group piles (11.22): loads in groups reduce capacity due to overlapping stress fields; group efficiency $n$ defined as
with $Qg$ the group capacity and $Q{all}$ the single-pile capacity; spacing $d$ and geometry affect whether piles act as a block or individually.
Several empirical equations exist (Converse–Labarre, Los Angeles Group Action, Seiler–Keeney) to estimate $n$ depending on pile spacing and configuration (Figure 11.42).
Ultimate capacity of group piles in saturated clay (11.23–11.25): compute group capacity by summing contributions from individual piles or by treating the group as a block; choose the lesser value between methods and apply factor of safety (FS).
Step 1: compute sum of per-pile capacities using layered-clay data (Eq. 11.123).
Step 2: treat group as a block with dimensions, compute block skin resistance and bearing capacity factor $N^*$ (from Fig. 11.44).
Step 3: compare results; take the smaller as the allowable group capacity.
Example-style practice topics include estimating $Q_u$ via Meyerhof/Vesic/Janbu methods, group efficiency calculations, and negative skin friction assessments.
Quick-reference Formulas (selected)
Rectangular combined footing area and location:
Width relation from area:
Trapezoidal combined footing: area and resultant location given by Eqs. (6.6)–(6.7) (details depend on $B1$, $B2$, $L$, $X$).
Net/allowable raft pressure on soils (sands):
Design check: $$q \