Glossary

The slope of the Mohr-Coulomb failure envelope in effective stress space, describing how a soil gains strength as effective stress increases. It is a property of the envelope fitted to test data rather than a physical angle measurable on a grain.

What a Mohr circle actually tells you

The water contents at which a fine-grained soil changes consistency: the liquid limit between plastic and liquid behaviour, and the plastic limit between semi-solid and plastic. They are index tests on remoulded soil, so they classify a material rather than describe the state it is in on site.

Atterberg limits: the flow curve and the plasticity chart

A shape measure of the grading curve formed from the particle sizes at 10%, 30% and 60% passing. Together with the coefficient of uniformity it decides whether a coarse-grained soil is classified as well graded or poorly graded.

Soil classification from a grading curve

The ratio of horizontal to vertical effective stress in ground that has not deformed laterally. It sets the initial stress state of a numerical model, and a model started from the wrong value is already wrong before any load is applied.

The K0 procedure: initial stresses before anything moves

The ratio of the particle size at 60% passing to the size at 10% passing. It measures how wide the range of grain sizes is: a value near one describes a uniform soil, larger values a well graded one.

Soil classification from a grading curve

The parameter controlling how much a soil changes volume as it shears in a plastic model. It is separate from the friction angle: setting the two equal gives an associated flow rule, which is convenient mathematically and wrong physically, because it makes dense material expand without limit.

Dilatancy angle: what the choice of psi actually costs

The portion of total stress carried by contact between soil grains, equal to total stress minus pore water pressure. It governs strength and stiffness: two soils at the same total stress but different pore pressures behave differently, and only the effective stress predicts which.

Effective stress: a worked profile, uplift and settlement

The distribution of particle sizes in a soil, plotted as percentage passing against sieve size on a logarithmic axis. Because the size axis is logarithmic, reading a percentile off it is a logarithmic interpolation; treating it as linear is the most common error in classification.

Soil classification from a grading curve

The water content at the boundary between plastic and liquid behaviour, fixed by a standardised test rather than by a physical transition. In the Casagrande cup method it is read off a fitted flow curve at 25 blows, not taken from whichever single determination landed nearest 25.

Atterberg limits: the flow curve and the plasticity chart

A construction that plots the normal and shear stresses acting on every plane through a point as a circle in normal-shear stress space. Angles on the circle are twice the physical angles between planes, and a failure envelope tangent to the circle identifies the plane on which failure occurs.

What a Mohr circle actually tells you

The ratio of the greatest vertical effective stress a soil has ever carried to the stress acting on it now. A soil at an OCR above one has been unloaded, which raises both its horizontal stress and its stiffness relative to a normally consolidated soil at the same depth.

The K0 procedure: initial stresses before anything moves

A plot of plasticity index against liquid limit, divided by the empirical A-line and U-line, used to classify fine-grained soils. Position relative to the A-line separates clays from silts and organic soils; the U-line is an upper bound above which a result is treated as suspect.

Atterberg limits: the flow curve and the plasticity chart

The liquid limit minus the plastic limit: the width of the water-content range over which a soil behaves plastically. It is the vertical axis of the plasticity chart and the primary index separating clays from silts.

Atterberg limits: the flow curve and the plasticity chart

The pressure of water in the voids between soil grains. Unlike total stress, which follows from the weight of everything above, it is a boundary condition set by the groundwater regime and has to be established from observation rather than computed from the profile.

Effective stress: a worked profile, uplift and settlement

The shear strength of a fine-grained soil loaded fast enough that no drainage occurs, so its water content stays constant. It is not a fundamental soil property but a response to the current state, and it changes as the soil consolidates.

CPT interpretation: from qc and fs to Ic and su

A ground improvement technique in which soil is eroded by a high-pressure fluid jet and mixed in place with cement grout to form columns of treated ground. The treated body is built blind, so what governs design is less the strength achieved than whether the columns are where they were assumed to be.

Jet grout columns: overlap, spacing and verticality

The intersection between adjacent columns that makes a row of them continuous rather than a line of separate elements. The effective thickness of the resulting wall is the chord where the circles cross, and that chord shrinks faster than the spacing grows as the columns drift apart.

Jet grout columns: overlap, spacing and verticality

The permitted deviation of a drilled or jetted column from vertical, expressed as a proportion of depth. Because deviation accumulates with depth while the column diameter does not, a tolerance that is negligible at the surface can consume the whole overlap between adjacent columns further down.

Jet grout columns: overlap, spacing and verticality

A mooring configuration in which the line hangs as a catenary and part of its length rests on the seabed. Because the line leaves the touchdown point horizontal, the anchor sees pure horizontal load, which is what allows a drag anchor to be used at all.

Catenary mooring: chain length, tension and anchor load

A single large-diameter tubular pile driven into the seabed to support an offshore wind turbine. It behaves as a cantilever governed by lateral stiffness rather than axial capacity, and it is stiff enough relative to its length that methods calibrated on slender piles do not describe it.

Monopile p-y curves: what PISA was needed to fix

A nonlinear relation between lateral soil resistance per unit length of pile and lateral displacement at a given depth, used to represent the soil as a series of independent springs. That independence is the assumption which fails for a stiff, large-diameter pile, because such a pile rotates rather than bends.

Monopile p-y curves: what PISA was needed to fix

A design method for laterally loaded monopiles that replaces the single load-transfer curve with four soil reaction components, including distributed moment and base reactions, each calibrated against field tests and three-dimensional finite element analysis. It supplies the reactions a spring model has no term for.

Monopile p-y curves: what PISA was needed to fix

An inverted steel bucket installed in the seabed by pumping water out of it, so that the resulting pressure difference drives it down. Installation is bounded at both ends: too little suction and it stops short of depth, too much and the soil plug inside fails upward.

Suction caissons: the suction needed, and the limit

The location at which a mooring line first meets the seabed. Its position fixes how much chain hangs suspended and therefore the tension at the top, and it migrates as the floating structure offsets, which is the source of a catenary system restoring stiffness.

Catenary mooring: chain length, tension and anchor load

An idealisation for a body and loading with rotational symmetry about a vertical axis, such as a circular tank or a single pile, in which the model solves one radial slice. Stress decays with depth faster than under plane strain, because the load is finite in every horizontal direction rather than in only one.

Plane strain vs axisymmetric: the depth the stress reaches

A method that solves a continuum problem by dividing the domain into elements, approximating the unknown field inside each by shape functions, and assembling the element equations into one global system. In geotechnics it is what allows staged construction, arbitrary geometry and nonlinear soil behaviour to be treated together.

Gauss quadrature in finite elements: how many points

Numerical integration that samples an integrand at optimally placed points with matching weights, so an n-point rule integrates a polynomial of degree 2n-1 exactly. Element stiffness integrals are evaluated this way, and the point count chosen decides whether an element is exactly integrated, over-integrated or under-integrated.

Gauss quadrature in finite elements: how many points

An idealisation in which strain normal to the modelled plane is zero, appropriate to a structure long enough that its ends do not govern — an embankment, a retaining wall, a strip footing. It is not a slice through a three-dimensional problem but a different problem, and it loads the ground more deeply than a finite structure of the same width does.

Plane strain vs axisymmetric: the depth the stress reaches

Using fewer quadrature points than exact integration of the element stiffness would require. It is cheaper and can relieve locking, but it risks admitting deformation modes that produce no strain at any sampling point and therefore cost no energy, which the element then cannot resist at all.

Gauss quadrature in finite elements: how many points

The interpolation function expressing a field inside an element in terms of its nodal values. Its polynomial order sets how the element can deform, which quadrature rule integrates it exactly, and how it behaves when the material is nearly incompressible.

Gauss quadrature in finite elements: how many points

A way of obtaining a factor of safety from a finite element model by dividing the strength parameters by a trial factor and raising it until the analysis will no longer converge. What it reduces is strength, not load, so the factor it returns answers a different question from a load factor and the two are not interchangeable.

Strength reduction: what the factor of safety reduces

The spuriously stiff response of low-order elements when the material is nearly incompressible, as soil is under undrained conditions or at fully developed plastic flow. The element cannot represent a deformation that changes shape without changing volume, so it resists one that ought to be free.

Dilatancy angle: what the choice of psi actually costs

The exact derivative of the stress-update algorithm with respect to strain, as distinct from the derivative of the continuum constitutive law. Newton iteration converges quadratically only when the Jacobian is consistent with the algorithm actually used, and substituting the elastic or continuum tangent can cost orders of magnitude in iteration count.

The consistent tangent: 7 iterations against 887

The assumed relation between stress and strain for a material, which is what turns a set of equilibrium equations into a solvable problem. Writing one down and implementing it correctly are separate problems, and the yield surface is the easy half.

Mohr-Coulomb return mapping: the corners and the apex

A plasticity model whose yield surface is a smooth cone in principal stress space, often used as a numerically convenient substitute for Mohr-Coulomb. Being a cone fitted to a hexagon, it can match at some loading directions but is necessarily wrong at others, and which match is chosen decides where the error falls.

Drucker-Prager: four ways to match Mohr-Coulomb

The rule fixing the direction of the plastic strain increment once yielding occurs. An associated rule takes that direction normal to the yield surface, which for soil overpredicts dilation; a non-associated rule uses a separate plastic potential, at the cost of a non-symmetric stiffness matrix.

Dilatancy angle: what the choice of psi actually costs

A nonlinear empirical failure criterion for rock masses, written in terms of intact rock strength and parameters describing how jointed and how disturbed the mass is. Its curvature in stress space is the essential difference from a linear criterion, and it is why rock strength does not reduce to a single friction angle.

Hoek-Brown tunnel: the closed-form ground reaction curve

A perfectly plastic model whose yield criterion is a linear failure envelope in normal-shear stress space, forming an irregular hexagonal pyramid in principal stress space. Its edges and apex are singular points at which the surface has no unique normal, which is what makes implementing it harder than the equation suggests.

Mohr-Coulomb return mapping: the corners and the apex

The algorithm that takes a trial stress computed elastically, finds it outside the yield surface, and returns it to the surface while updating plastic strain. On a surface with corners the single-surface return can produce a state that violates the ordering of principal stresses, so the corner and apex cases must be handled explicitly.

Mohr-Coulomb return mapping: the corners and the apex

The boundary in stress space separating states a material sustains elastically from states at which plastic flow occurs. A stress state outside it is inadmissible, and the task of a plasticity algorithm is to return any such state to the surface.

Mohr-Coulomb return mapping: the corners and the apex

A problem whose answer is known independently, used to test a solver. Its value rests entirely on that independence: a benchmark whose reference value was produced by the same code proves nothing about that code.

Reproducing a PLAXIS 2D benchmark to within 0.03%

An uncertainty estimate for a discretised solution, computed from results on three systematically refined meshes and reported as a band on the finest result. It turns a single finite element number into a number with an error bar, which is what makes it comparable with a measurement.

Grid Convergence Index: an error bar on a FEM result

The convergence rate a solution sequence actually achieves, computed from results on three meshes, as opposed to the order the discretisation is theoretically meant to deliver. A wide gap between the two is evidence that something — a singularity, too coarse a mesh, an implementation error — is not behaving as the theory assumes.

Grid Convergence Index: an error bar on a FEM result

Estimating the exact solution of a discretised problem by extrapolating results from successively refined meshes to zero element size. It underlies the grid convergence index and requires the solutions to lie in the asymptotic range, which has to be checked rather than assumed.

Grid Convergence Index: an error bar on a FEM result

Checking that the governing equations are the right ones for the physical problem — that the mathematics describes the world. It is answered against measurement, and no amount of verification substitutes for it.

Geotechnical FEM verification: the KATAI 2D benchmark suite

Checking that a model solves its governing equations correctly — that the code does the mathematics it claims to do. It is answered against closed-form solutions and published benchmarks, and it is a separate question from whether those equations describe reality.

Geotechnical FEM verification: the KATAI 2D benchmark suite

The diffusivity in the consolidation equation, combining permeability and compressibility into the single parameter that sets how fast excess pore pressure dissipates. Its units are area over time, which is why halving the drainage path quarters the time.

Terzaghi consolidation: the series and its approximations

The time-dependent settlement of a saturated soil as load transfers from the pore water to the soil skeleton and water drains away. The rate is governed by how fast water can leave, which makes it a diffusion problem rather than an elastic one.

Terzaghi consolidation: the series and its approximations

The proportion of the ultimate consolidation settlement reached at a given time. Its exact form is an infinite Fourier series; the two expressions textbooks give are approximations to that series, each accurate over part of the range and neither over all of it.

Terzaghi consolidation: the series and its approximations

The annulus of soil around an installed vertical drain whose permeability has been reduced by the installation itself. Ignoring it underestimates the time needed to reach a target degree of consolidation, because water must cross the damaged soil before it reaches the drain.

Vertical drains: the smear zone doubles the time

The dimensionless time of consolidation theory, formed from the coefficient of consolidation, the elapsed time and the square of the drainage path length. It is what lets one solution serve every layer thickness and every soil.

Terzaghi consolidation: the series and its approximations

A set of tests with known expected results, run on every change, whose purpose is to detect that something previously correct has stopped being so. For engineering software it is the mechanism that makes a verification claim survive the next commit instead of describing one release.

Geotechnical FEM verification: the KATAI 2D benchmark suite

The margin within which a computed result is accepted as matching its reference. A tolerance chosen after seeing the result records nothing; one derived from the discretisation and rounding expected, and stated in advance, is a claim the test can genuinely fail.

Geotechnical FEM verification: the KATAI 2D benchmark suite

A measure of how far a relative error in the data of a linear system can be amplified in its solution. It bounds the accuracy attainable in floating-point arithmetic whatever the algorithm, so an ill-conditioned system limits what any solver can deliver.

Floating point: why two runs of one model disagree

Entries that become non-zero during factorisation of a sparse matrix at positions which were zero in the original. Fill-in is what determines the memory and time a direct solve actually costs, and it depends on the order in which unknowns are eliminated rather than on the matrix alone.

Sparse direct solvers: ordering, fill-in and the cost

A permutation of the unknowns chosen to minimise fill-in during factorisation, computed by heuristics such as minimum degree or nested dissection. It is not a marginal tuning parameter: the same matrix eliminated in two orders can differ by orders of magnitude in cost.

Sparse direct solvers: ordering, fill-in and the cost

An iterative scheme that solves a nonlinear system by repeatedly linearising it about the current estimate and solving for a correction. Its quadratic convergence depends on the Jacobian being the true derivative of the residual that is actually being driven to zero.

The consistent tangent: 7 iterations against 887

A matrix in which most entries are zero, stored so that only the non-zeros are held and operated on. Finite element stiffness matrices are sparse because a node couples only to the nodes it shares an element with.

Sparse direct solvers: ordering, fill-in and the cost

The property that exact addition has and floating-point addition does not: summing the same values in a different order can give a different result, because every intermediate is rounded. It is why a parallel reduction, a reordered assembly loop or a different compiler can move the last digits of an answer.

Floating point: why two runs of one model disagree

The property that a calculation returns the same answer when repeated by someone else, on other hardware, from the stated inputs. It is stronger than repeatability, which asks only that one machine agree with itself, and in floating-point work it has to be designed for rather than assumed.

Floating point: why two runs of one model disagree

The spacing between adjacent representable floating-point numbers at a given magnitude, used as the natural unit for measuring rounding error. Expressing a discrepancy in ULPs states how many representable values apart two results are, which is scale-independent in a way an absolute difference is not.

Floating point: why two runs of one model disagree

The empirical divisor relating net cone resistance to undrained shear strength. It is not measured on a routine project, so it is taken from correlation, and the value chosen scales the derived strength in direct proportion.

CPT interpretation: from qc and fs to Ic and su

A site investigation test in which an instrumented cone is pushed into the ground at constant rate while cone resistance, sleeve friction and usually pore pressure are logged continuously. It yields a dense profile, but it measures resistance rather than soil type or strength, both of which are inferred from it.

CPT interpretation: from qc and fs to Ic and su

Measured cone resistance adjusted for the pore pressure acting on the unequal end areas of the cone. The correction is minor in sand and substantial in soft clay, where the raw reading understates resistance.

CPT interpretation: from qc and fs to Ic and su

An SPT blow count corrected to a 60% hammer energy ratio, the reference efficiency to which published correlations are referred. The same soil tested with two rigs of differing efficiency gives two raw counts and one N60.

SPT: N60, (N1)60, and five routes to a friction angle

N60 corrected further to a reference vertical effective stress, so that counts taken at different depths can be compared. Without it, one relative density reads as a rising blow count for no reason other than depth.

SPT: N60, (N1)60, and five routes to a friction angle

A single number combining normalised cone resistance with normalised friction ratio, used to place a CPT reading in a soil behaviour zone. It classifies how the soil responds to the cone, which is related to, but not the same as, a classification by grading and plasticity.

CPT interpretation: from qc and fs to Ic and su

A borehole test counting the blows needed to drive a split-spoon sampler a set distance with a falling hammer. The raw count depends heavily on the equipment delivering the blows, so it is corrected before use rather than read as a property of the soil.

SPT: N60, (N1)60, and five routes to a friction angle