Seismic analysis

An earthquake loads a slope with inertial forces that reverse many times a second. OGR Slip2D analyses it in three steps of increasing detail: the pseudo-static method, which replaces the shaking by constant body forces and returns a factor of safety; the yield seismic coefficient $K_y$, the horizontal coefficient that brings each surface to a target factor; and Newmark's rigid sliding block, which integrates an acceleration record above that coefficient to estimate the permanent displacement.

3.9.1The pseudo-static method#

The pseudo-static method, applied to landslides by Terzaghi (1950), represents the earthquake by two body forces proportional to the weight of the sliding mass, a horizontal one and a vertical one, whose coefficients are accelerations expressed as fractions of $g$ (Kramer, 1996). In the method of slices each slice carries its own share:

where

$W_s$

weight of the soil in the slice: its static weight, from the unit weights of the bands of its column (kN/m)

$k_h$

horizontal seismic coefficient, a magnitude, $0 \le k_h < 1$ (—)

$k_v$

vertical seismic coefficient, positive downwards, $|k_v| < 1$ (—)

Four conventions fix how these forces enter the equilibrium of every method in OGR Slip2D.

• The forces act on the soil only. The coefficients multiply the weight of the soil in the slice. A distributed or line load is a traction on the ground surface and carries no inertia here, and ponded water, which has no shear strength, carries none either. A fill whose inertia matters has to be drawn as a material.
• A positive $k_v$ acts downwards. The vertical load carried by the base of a slice is then
  $$W' = W_s\,(1 + k_v) + Q_v + W_w$$
  with $Q_v$ the vertical component of the external loads on the slice and $W_w$ the weight of the ponded water above it; a negative $k_v$ lightens the slice.
• The horizontal force always pushes the mass out of the slope. $k_h$ is a magnitude: every method applies $k_h W_s$ in the sense of sliding of the surface it is analysing, so a sign typed on the coefficient cannot turn the force round, and a negative value is refused.
• The horizontal force is proportional to the static weight, not to $W_s(1 + k_v)$: the horizontal and vertical accelerations of a mass are independent components. It acts at mid-height of the slice, $y_g = \tfrac12(\bar y_{top} + \bar y_{base})$, the centroid of a rectangular column; that elevation matters only to the methods that write a moment equation.

A closed form: the planar wedge#

On a planar slip surface the sliding mass is a single free body and the interslice forces cancel in the sum, so the pseudo-static factor of safety of a dry wedge has a closed form (Kramer, 1996):

where

$W$

weight of the wedge (kN/m)

$\beta,\ L$

inclination (°) and length (m) of the slip plane

$c',\ \phi'$

effective cohesion (kPa) and friction angle (°) on the plane

The methods that close global force equilibrium (Janbu's simplified method, the two Corps of Engineers methods and Lowe–Karafiath; Section 3.2) reproduce Eq. (3.9.3) to the tolerance of their solvers, for both signs of $k_v$, with and without $k_h$. The program's tests use it as the external anchor of the conventions listed above.

How the methods take the seismic forces#

Each method adds the two forces to its own equilibrium equations. In Bishop's simplified method on a circle, for instance, the vertical force joins the weight on both sides of the ratio and the horizontal force adds its moment about the centre to the driving side:

where

$W'$

vertical load on the base of the slice, Eq. (3.9.2) (kN/m)

$(x_c,\ y_c),\ R$

centre and radius of the circle (m)

$x,\ y_g$

abscissa of the slice and elevation of its mid-height (m); $(x - x_c)/R$ is $\sin\alpha$ at the slice's own abscissa, and the program takes this arm from the geometry

$b,\ u,\ \alpha$

width of the slice (m), pore pressure at its base (kPa) and inclination of the base (°)

The expression is written for the sense of sliding in which $\sum W'(x - x_c)/R$ is positive (the program reads the sense from the sign of the driving sum), and for a slope without ponded water, supports or tension crack, whose terms enter the same sums. The equations of every method are developed in Section 3.2.

OGR Slip2D takes $k_h$ and $k_v$ as given. Choosing them, and the factor of safety a pseudo-static analysis has to reach, is part of the design: codes such as EN 1998-5 derive the coefficients from the design ground acceleration of the site (CEN, 2004).

3.9.2The yield seismic coefficient#

A pseudo-static factor of safety answers for one chosen coefficient. The analysis can also be inverted, looking for the horizontal coefficient that brings a surface to the limit. That coefficient is the yield, or critical, seismic coefficient $K_y$, and $a_c = K_y\,g$ is the acceleration above which a rigid sliding mass starts to move (Newmark, 1965). OGR Slip2D obtains it in the first of the two ways Jibson (1993) lists, by repeating the limit-equilibrium analysis of each surface over horizontal coefficients until its factor of safety reaches the target:

where

$F(k_h)$

factor of safety of the surface with horizontal coefficient $k_h$, computed with the method of the analysis and the project's $k_v$

$F_t$

target factor of safety, 1.0 by default

The second way, $a_c = (F - 1)\,g\sin\alpha$ with $\alpha$ the thrust angle (Newmark, 1965), is not used: it replaces the equilibrium of the whole mass by a single angle. The search for $K_y$ runs as follows.

1. If $F(0) \le F_t$, the surface reaches the target without any earthquake and $K_y = 0$.
2. Otherwise $k_h$ grows from zero in steps of 0.05. After each solve a secant through the last two points predicts the crossing, and the next coefficient may jump towards it by at most two steps, so that the first crossing cannot be stepped over. A coefficient at which the method returns no factor of safety is passed over, not taken as a crossing.
3. The first coefficient with $F \le F_t$ closes a bracket, which the Illinois variant of regula falsi (Dowell & Jarratt, 1971) narrows until $|F - F_t| \le 10^{-7}$ or the bracket is narrower than $10^{-5}$. The value reported is the upper end of the final bracket, where $F \le F_t$.
4. If no coefficient up to 1.0 reaches the target, the surface has no $K_y$. A surface costs at most 40 solves.

3.9.3Newmark's sliding block#

Newmark (1965) modelled the sliding mass as a rigid-plastic block: it moves with the ground while the ground acceleration stays below $a_c$, and slides relative to the ground when the acceleration exceeds it. While the block slides, its acceleration relative to the ground is

where

$a(t)$

horizontal ground acceleration of the record, positive in the direction that drives the block downslope

$u,\ \dot u,\ \ddot u$

displacement, velocity and acceleration of the block relative to the ground

and the permanent displacement is the double integral of $\ddot u$ over the time the block slides. OGR Slip2D integrates it with the scheme of Wilson & Keefer (1983) as Jibson (1993) publishes it, step by step. Both integrations are trapezoidal, because at a given time step a different quadrature gives a different answer:

with $\Delta t$ the sampling interval of the record. Three rules complete the scheme.

• Starting. A block at rest (relative velocity below $10^{-4}$ cm/s) starts to slide only when $a(t)$ exceeds $a_c$; otherwise $\ddot u = 0$.
• Sliding. The resistance acts in the direction of sliding, $\ddot u = a - a_c$, so the block decelerates once $a(t)$ drops below $a_c$ and goes on moving on its own inertia.
• Stopping. When $\dot u_i \le 0$ the block re-attaches: $\dot u_i$ and $\ddot u_i$ are set to zero.

Upslope movement is excluded by default, which is Newmark's assumption that the resistance to sliding upslope is very large. With upslope displacement allowed, the program integrates a symmetric two-sided block, with the same $a_c$ in both directions, and reports the net displacement, which can be negative. Wilson and Keefer's own two-sided variant, which gives the upslope direction a larger critical acceleration through the thrust angle, needs a datum the program does not have and is not implemented.

A record is a signed acceleration, and which sign drives the slope depends on how the instrument was oriented. Four options resolve it: the record as recorded (direct polarity), the record negated (inverse), the mean of the two displacements, or the larger of the two, which is the default. A scale factor multiplies the accelerations of the record and leaves $a_c$ unchanged.

3.9.4Acceleration records#

Records are entered in Loading → Seismic Records from a plain-text file, in either of the two layouts Jibson (1993) names for the input of a Newmark integration. Which one a file uses is read from the number of values on its first data line.

• Pairs of time and acceleration, one pair per line. The interval is taken from the first two times; a record whose step departs from it by more than $10^{-6}$ of the interval is imported with a note, because the integration assumes a constant step and the program does not resample it. A time column that does not increase is refused.
• A single column of accelerations at a constant interval, which then has to be given.

Blank lines, lines starting with #, ; or %, and lines that do not read as numbers are skipped, so a file can keep its header; commas are read as separators. The values may be in g, cm/s² or m/s². They are converted to g on import and stored in g inside the project file, so a project carries its own records. A usable record has at least two samples, a positive interval and finite values; its duration is $(n - 1)\,\Delta t$ for $n$ samples, and its peak ground acceleration is the largest $|a|$.

3.9.5Limitations#

• A constant force for a transient load. The pseudo-static method stands a few seconds of oscillating acceleration in for one constant coefficient. Its factor of safety says whether the mass can stand that coefficient, not how far it moves when it cannot (Kramer, 1996).
• A rigid block with a constant $K_y$. $K_y$ is computed once, from the strengths and pore pressures of the model, and held constant through the record. Loss of strength or build-up of pore pressure during the shaking, and the dynamic response of the slope itself, are outside the model.
• One horizontal record. The vertical component of the ground motion is not integrated; the project's $k_v$, if any, only enters the solve for $K_y$.

3.9.6References#

• CEN (2004). EN 1998-5:2004, Eurocode 8: Design of structures for earthquake resistance — Part 5: Foundations, retaining structures and geotechnical aspects. European Committee for Standardization, Brussels.
• Dowell, M. & Jarratt, P. (1971). A modified regula falsi method for computing the root of an equation. BIT Numerical Mathematics, 11(2), 168–174.
• Jibson, R.W. (1993). Predicting earthquake-induced landslide displacements using Newmark's sliding block analysis. Transportation Research Record, 1411, 9–17.
• Kramer, S.L. (1996). Geotechnical Earthquake Engineering. Prentice Hall, Upper Saddle River, NJ.
• Newmark, N.M. (1965). Effects of earthquakes on dams and embankments. Géotechnique, 15(2), 139–160.
• Skempton, A.W. (1954). The pore-pressure coefficients A and B. Géotechnique, 4(4), 143–147.
• Terzaghi, K. (1950). Mechanism of landslides. In Paige, S. (ed.), Application of Geology to Engineering Practice (Berkey Volume). Geological Society of America, New York, 83–123.
• Wilson, R.C. & Keefer, D.K. (1983). Dynamic analysis of a slope failure from the 6 August 1979 Coyote Lake, California, earthquake. Bulletin of the Seismological Society of America, 73(3), 863–877.