Partial factors (Eurocode 7) — OGR Slip2D documentation

Eurocode 7 verifies a slope by applying partial factors to the actions and to the strength of the ground instead of asking for a global factor of safety. OGR Slip2D can apply the factors of the design approaches of EN 1997-1 to a copy of the model before the analysis. The number it then reports is an over-design factor, and what this version does not apply is listed under Limitations of this version.

3.12.1Limit-state design#

In limit-state design the stability of a slope is a geotechnical (GEO) ultimate limit state: the design effect of the actions must not exceed the design resistance (CEN, 2004; Frank et al., 2004),

E_d \le R_d

and the design values come from characteristic or representative values through partial factors, material properties being divided by theirs and actions multiplied by theirs:

X_d = \dfrac{X_k}{\gamma_M}, \qquad F_d = \gamma_F\,F_{rep}

where

$X_k,\ \gamma_M$: characteristic value of a material property ( $c'$ , $\tan\phi'$ , $c_u$ …) and its partial factor
$F_{rep},\ \gamma_F$: representative value of an action and its partial factor, $\gamma_G$ for permanent and $\gamma_Q$ for variable actions

EN 1997-1 groups the factors in sets (A1 and A2 on actions, M1 and M2 on materials, R1, R2 and R3 on resistances) and combines them in three design approaches. A limit-equilibrium analysis carried out with design values measures how far the design resistance exceeds the design effect of the actions; Frank et al. (2004) call that ratio the over-design factor, and the limit state is satisfied when it is at least one:

\text{ODF} = \dfrac{R_d}{E_d} \ge 1

3.12.2Design approaches and presets#

Table 3.12.1 gives the factors of the four Eurocode 7 presets as the program stores them.

Table 3.12.1 Partial factors of the presets of OGR Slip2D (DesignStandardSettings.PRESETS), in the program's order:

\gamma_G,\ \gamma_Q,\ \gamma_{c'},\ \gamma_{\tan\phi'},\ \gamma_\gamma,\ \gamma_{R;e},\ \gamma_{cu},\ \gamma_\tau

\gamma_\tau

divides the whole shear strength of the models that are not written in

c'

and

\tan\phi'

, and is not an EN 1997-1 factor. The permanent-action factor

\gamma_G

is stored but not applied in this version (D226).

Preset	Permanent	Variable	Cohesion	tan friction	Unit weight	Resistance	Undrained	Other models
DA1-C1	1.35	1.5	1.0	1.0	1.0	1.0	1.0	1.0
DA1-C2	1.0	1.3	1.25	1.25	1.0	1.0	1.4	1.25
DA2	1.35	1.5	1.0	1.0	1.0	1.1	1.0	1.0
DA3	1.35	1.5	1.25	1.25	1.0	1.0	1.4	1.25

The sets behind the presets are those of EN 1997-1, Annex A: DA1-C1 combines A1, M1 and R1; DA1-C2 combines A2, M2 and R1; DA2 combines A1, M1 and R2, whose $\gamma_{R;e} = 1.1$ is the factor on the earth resistance of a slope; DA3 combines M2 and R3 with A2 on geotechnical actions (the actions on a slope are geotechnical) and A1 on structural ones. Two columns are decisions of the program: $\gamma_{cu} = 1.4$ is the M2 factor on undrained strength, and $\gamma_\tau$ is set equal to the $\gamma_{\tan\phi'}$ of the same set, so that an envelope that cannot be split into $c'$ and $\tan\phi'$ receives the same strength reduction. Besides the four presets, a standard can be Custom, with the eight factors entered by hand, or None.

3.12.3How the factors are applied#

The factors are applied to a copy of the model before any analysis runs, so the solver remains a plain limit-equilibrium engine and the user's model is never changed. Material factors divide and action factors multiply. The resistance factor $\gamma_{R;e}$ divides every strength category as well, which is dividing the whole resisting side:

c'_d = \dfrac{c'}{\gamma_{c'}\,\gamma_{R;e}}, \qquad \tan\phi'_d = \dfrac{\tan\phi'}{\gamma_{\tan\phi'}\,\gamma_{R;e}}

c_{u,d} = \dfrac{c_u}{\gamma_{cu}\,\gamma_{R;e}}, \qquad \tau_d(\sigma'_n) = \dfrac{\tau(\sigma'_n)}{\gamma_\tau\,\gamma_{R;e}}

\gamma_d = \gamma_\gamma\,\gamma, \qquad Q_d = \gamma_Q\,Q

where

$\gamma$: unit weight and saturated unit weight of every material (kN/m³)
$Q$: magnitude of every distributed and line load (kPa, kN/m)
$\tau(\sigma'_n)$: shear strength of a model in the ‘other models’ category at the normal stress $\sigma'_n$

The friction angle is factored through its tangent, as EN 1997-1 does: dividing $\tan 30^\circ$ by 1.25 gives $\phi'_d = 24.79^\circ$ , not the $24.0^\circ$ obtained by dividing the angle. The angle $\phi^b$ of the unsaturated strength goes with $\tan\phi'$ , and the two strength envelopes of a rapid-drawdown analysis have their intercept divided like $c'$ and the tangent of their slope like $\tan\phi'$ . Each strength model declares which category its parameters belong to (Table 3.12.2); a model that declares none is left as it is, and the analysis says so.

Table 3.12.2 Which factor reaches each strength model. Every factor in the second column is multiplied by

\gamma_{R;e}

Category	Factor	Strength models and the parameters divided
$c'$ and $\tan\phi'$	$\gamma_{c'},\ \gamma_{\tan\phi'}$	Mohr–Coulomb; drained–undrained; anisotropic linear (both directions); C/φ function; anisotropic strength function (every row). Snowden anisotropic linear and generalized anisotropic: each function or rule by its own model
$c_u$	$\gamma_{cu}$	Undrained; SHANSEP ( $A$ , $S$ and the minimum); vertical stress ratio ( $K$ and the minimum); the three undrained models that vary with depth (reference value, rate and cut-off)
Shear strength, other models	$\gamma_\tau$	Generalized and classic Hoek–Brown and Barton–Bandis (a divisor of $\tau$ ); power curve and hyperbolic (the parameters that scale $\tau$ ); shear–normal and discrete functions ( $\tau$ of every point)
None	—	Infinite strength and no strength, which are modelling devices rather than strengths

With every material factor equal to the same $\gamma$ and $\gamma_{R;e} = 1$ , every envelope is divided by $\gamma$ at every normal stress, so the factored model returns exactly $F/\gamma$ in every method: the strength reduction by which the limit-equilibrium factor of safety is defined (Section 3.1.1). Figure 3.12.1 shows the design envelopes of a Mohr–Coulomb soil under the presets.

Figure 3.12.1 Characteristic Mohr–Coulomb envelope (

c' = 10

kPa,

\phi' = 30^\circ

) and the design envelopes OGR Slip2D analyses under DA1-C2 and DA3 (

c'

and

\tan\phi'

divided by 1.25) and under DA2 (both divided by

\gamma_{R;e} = 1.1

). Under DA1-C1 the envelope is the characteristic one. Computed with OGR Slip2D 0.1.235.

In OGR Slip2D

Partial factors are set on the Design Standard page of Analysis → Project Settings: Apply partial factors (off by default, because switching it on changes every result previously compared against), the Design standard, and the eight factors, which are editable when the standard is Custom. The analysis reports what was factored, material by material, with the notes of the factor report.

Source: ogr_core/project/design_factors.py · ogr_core/project/settings.py · ogr_core/materials/strength_model.py · ogr_core/materials/builtin_models.py

3.12.4What the reported number means#

With a design standard on, the partial factors have already been applied to the inputs, so the number the analysis reports is not a factor of safety: it is the over-design factor of Eq. (3.12.3), and the design check requires it to be at least 1. The interface, the command line and the API label it as an over-design factor and report which factors were applied. For a slope whose materials are all of the $c'$ – $\phi'$ kind and which carries no external loads, the over-design factor is $F/1.25$ under DA1-C2 and DA3, $F/1.1$ under DA2, and $F$ itself under DA1-C1, whose material factors are all 1.0.

A standard switched on with every factor left at 1.0 changes nothing, and the analysis says so, because the label would otherwise suggest a factored result.

3.12.5Limitations of this version#

Not applied in this version (D226)

The program's source and its factor report state these limitations, which this version does not correct.

The permanent-action factor $\gamma_G$ is not applied. The weight of the soil and any permanent load carry no action factor, and every distributed and line load takes the variable-action factor $\gamma_Q$ . Under DA1-C1, whose material factors are all 1.0, a slope with no external loads therefore returns exactly its unfactored factor of safety, and under DA2 it returns $F/\gamma_{R;e} = F/1.1$ . Every run with $\gamma_G \ne 1$ adds a note saying so.
DA3 carries the action factors of set A1 (1.35 and 1.5) where a slope, whose actions are geotechnical, takes those of A2 (1.0 and 1.3). As $\gamma_G$ is not applied in any case, the practical effect is that external loads are multiplied by 1.5 instead of 1.3.
Every load is treated as unfavourable: $\gamma_Q$ multiplies all distributed and line loads, including any whose effect is stabilising.
Not factored: support capacities, pore pressures, ponded water, the water in a tension crack and the seismic coefficients.

Read the over-design factor of a run with these points in mind, in particular under DA1-C1, DA2 and DA3, where the $\gamma_G = 1.35$ of set A1 is not part of the number.

3.12.6Partial factors in the other analyses#

Every analysis reads the factored copy: the search for the critical surface, the transient and rapid-drawdown stages and the drawdown sweep, and the back analysis of support force, which then returns the force needed to reach the target over-design factor (Section 3.10.6). A statistical run samples the model and factors each sample after drawing it, so that a sampled cohesion is factored like the deterministic one; its probability of failure is then the probability that the over-design factor falls below one (Section 3.11). With Compute Ky on, the reported number is a seismic coefficient of the factored model, and the interface and the API label it as one (Section 3.9.2).

3.12.7References#

CEN (2004). EN 1997-1:2004, Eurocode 7: Geotechnical design — Part 1: General rules. European Committee for Standardization, Brussels.
Frank, R., Bauduin, C., Driscoll, R., Kavvadas, M., Krebs Ovesen, N., Orr, T. & Schuppener, B. (2004). Designers' Guide to EN 1997-1, Eurocode 7: Geotechnical design – General rules. Thomas Telford, London.