Strength reduction: what the factor of safety reduces

There are two ways to compute a factor of safety for a slope, and they define the factor differently.

Limit equilibrium assumes a failure surface, divides the mass above it into slices, and takes the ratio of available shear resistance to mobilised shear stress on that surface. The factor is a property of the surface, and the analysis is repeated over many surfaces to find the critical one.

Strength reduction divides the strength parameters by a trial factor and asks a finite element analysis whether the slope still stands:

$$c_F = \frac{c'}{F}, \qquad \tan\varphi_F = \frac{\tan\varphi'}{F}$$

$F$ is increased until the analysis fails to converge, and the last converged value is the factor of safety. No failure surface is assumed; whatever mechanism forms, forms.

The definitions coincide, and the reason is worth working out on a problem simple enough to check by hand.

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The infinite slopePermalink to “The infinite slope”#

A slope of constant angle $\beta$, failing on a plane parallel to the surface at depth $z$, has a closed-form factor of safety:

$$F = \frac{c'}{\gamma z \cos^2\beta \tan\beta} + \frac{\tan\varphi'}{\tan\beta}$$

Two terms: a cohesion contribution that falls with depth, and a friction contribution that does not depend on depth at all. With $\beta = 20°$, $\varphi' = 32°$, $c' = 5$ kPa, $z = 3$ m and $\gamma = 19.0$ kN/m³:

$$\gamma z \cos^2\beta \tan\beta = 19.0 \times 3.0 \times 0.8830 \times 0.3640 = 18.319$$ $$F = 0.2729 + 1.7168 = 1.990$$

What the factor is reducingPermalink to “What the factor is reducing”#

Now apply the strength reduction definition. Divide by $F = 1.990$:

$$c_F = \frac{5}{1.990} = 2.5129\ \text{kPa}, \qquad \varphi_F = \arctan\frac{\tan 32°}{1.990} = 17.43°$$

and evaluate the same closed form with that pair. It returns exactly 1.

That is the content of the method, and it is worth stating plainly: the factor of safety is the number by which the strength can be divided before the slope is exactly in limiting equilibrium. It is not a load factor, it is not a margin on the driving forces, and it does not scale the geometry or the unit weight. The one thing it is is the reserve in the strength — which, on a slope, is where the uncertainty actually lives.

Note also what the reduction does to the balance between the two terms. Cohesion is divided by 1.990; the friction term is divided by the same factor applied to $\tan\varphi'$, so the mobilised friction angle falls from 32° to 17.43°, a reduction of 14.57° rather than a proportional one. The two mechanisms are reduced at the same rate in the quantities that enter the strength envelope, not in the angles.

Why the two methods agree, and when they do notPermalink to “Why the two methods agree, and when they do not”#

Strength reduction and a rigorous limit equilibrium method — Bishop’s, Spencer’s, anything that satisfies both force and moment equilibrium — normally agree to within a few percent, and where they disagree the reason is identifiable.

Non-associated flow. Limit equilibrium implicitly assumes the material is on the point of sliding along the surface, which corresponds to associated flow. A finite element analysis with $\psi < \varphi$ finds a lower factor, for the reason set out in the article on dilatancy — the same Davis reduction that lowers a bearing capacity by a third lowers a slope factor too, by less because the confinement is lower.

Progressive failure. Limit equilibrium mobilises the peak strength everywhere on the surface at once. A finite element analysis mobilises it where the strain is, so on a strain-softening material the peak is passed at one end of the mechanism while the other end is still elastic, and the computed factor is lower — correctly.

Mechanism freedom. Where the critical mechanism is not close to any of the shapes searched — a compound surface through a thin weak layer, or a mechanism controlled by a structural element — limit equilibrium can miss it entirely, and strength reduction finds it because it was never told what to look for.

Convergence as a failure criterion. This is the weakness of the method. “The analysis stopped converging” is a numerical event, and it depends on the tolerance, the iteration limit and the mesh as well as on the physics. A defensible strength reduction analysis reports the criterion used, and checks that the factor is stable under mesh refinement — for which the Grid Convergence Index is the appropriate tool, since the quantity of interest is a scalar output of a series of runs.

What actually moves the numberPermalink to “What actually moves the number”#

Return to the infinite slope and turn on steady seepage parallel to the surface, with the water table at the ground surface. The effective normal stress on the failure plane falls, and the friction term is scaled by the buoyant fraction of the unit weight:

$$F = \frac{c'}{\gamma_{sat} z \cos^2\beta \tan\beta} + \frac{\gamma'}{\gamma_{sat}}\cdot\frac{\tan\varphi'}{\tan\beta}$$

With $\gamma_{sat} = 20.5$ kN/m³ and therefore $\gamma' = 10.69$ kN/m³, the ratio $\gamma'/\gamma_{sat} = 0.5215$ and

$$F = 0.2530 + 0.8953 = 1.148$$

A fall of 42.3%, from a change to the pore pressure alone. The geometry, the strength and the depth are untouched.

Which puts the whole exercise in proportion. The choice between strength reduction and limit equilibrium moves a factor of safety by a few percent, and it is worth getting right. Where the water stands moves it by forty, and it is the term that has to be measured rather than assumed.