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\DOI{10.5802/crmeca.276}
\datereceived{2024-10-19}
\daterevised{2024-11-26}
\dateaccepted{2024-11-20}
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\begin{document}

\begin{noXML}

\CDRsetmeta{articletype}{research-article}

\title{About the critical height of a vertical cut}

\alttitle{\`{A} propos de la hauteur critique d'une tranch\'{e}e verticale}

\author{\firstname{Jean} \lastname{Salen\c{c}on}\CDRorcid{0000-0001-8472-5946}}
\address{Hong Kong Institute for Advanced Study, City University, Hong Kong}
\email{Jean.salencon@academie-sciences.fr}

\keywords{\kwd{Strength criterion}
\kwd{Maximum resisting rate of work} 
\kwd{Tension cutoff}
\kwd{Vertical cut}
\kwd{Critical height}}

\altkeywords{\kwd{Crit\`{e}re de r\'{e}sistance} 
\kwd{Puissance r\'{e}sistante maximale}
\kwd{Troncature en traction}
\kwd{Tranch\'{e}e verticale}
\kwd{Hauteur critique}}


\begin{abstract}
The iconic problem of the stability analysis of a vertical cut is
revisited after Drucker's celebrated contribution, in order to assess
the sensitivity of the analysis to the soil tensile resistance. Within
the framework of the exterior approach of the theory of yield design,
the same virtual mechanisms as introduced by Drucker are reconsidered
and thoroughly implemented. It is observed that, whatever the value of
the friction angle, the drastic drop of the non-dimensional stability
factor observed by Drucker when the constituent soil does not sustain
tension remains confined within the immediate vicinity i.e., a few
percent, of zero tensile resistance. Despite the fact that this
conclusion cannot be considered as a final one since it is only based
upon a full implementation of Drucker's virtual collapse mechanisms,
we believe it may bring some relief as to the reliability of classical
analyses that do not take any tension cutoff into account, as a
response to Drucker's warning about ``the consequences of assuming
soil unable to take tension''.
\end{abstract}

\begin{altabstract}
On reprend le probl\`{e}me de l'analyse de stabilit\'{e} d'une
tranch\'{e}e verticale apr\`{e}s la contribution c\'{e}l\`{e}bre de
Drucker, afin d'\'{e}valuer la sensibilit\'{e} de l'analyse \`{a} la
r\'{e}sistance en traction du sol. Dans le cadre de l'approche
par l'ext\'erieur de la th\'{e}orie du calcul \`{a} la rupture, les
m\'{e}canismes virtuels introduits par Drucker sont r\'{e}examin\'{e}s
pour une mise en \oe{}uvre compl\`{e}te. On observe que, quelle que
soit la valeur de l'angle de frottement, la chute drastique du facteur
adimensionnel de stabilit\'{e} observ\'{e}e par Drucker lorsque le sol
n'offre aucune r\'{e}sistance \`{a} la traction reste confin\'{e}e dans
le voisinage imm\'{e}diat, c'est-\`{a}-dire quelques pour cent, de la
r\'{e}sistance nulle en traction. Bien que cette conclusion ne puisse
\^{e}tre consid\'{e}r\'{e}e comme d\'{e}finitive, puisqu'elle n'est
fond\'{e}e que sur une mise en \oe{}uvre compl\`{e}te des
m\'{e}canismes virtuels introduits par Drucker, nous pensons qu'elle
peut apporter un certain soulagement quant \`{a} la fiabilit\'{e} des
analyses classiques en r\'{e}ponse \`{a} l'avertissement de Drucker sur
\guillemotleft{} les cons\'{e}quences qui pourraient r\'{e}sulter d'une
hypoth\`{e}se de sol incapable de r\'{e}sister en
traction {}\guillemotright.
\end{altabstract}

\maketitle

\end{noXML}

\section{Yield design analysis for materials obeying a Coulomb or Tresca strength criterion with tension cutoff}

Tresca's and Coulomb's criteria with zero-tension cutoff were
considered by Drucker and Prager \cite{1,2} for the stability analysis
of a vertical cut, who, implementing a vanishing virtual velocity
field, identified a significant reduction of the critical height of the
cut. Salen\c{c}on and Pecker \cite{3}, Chatzigogos, Pecker and
Salen\c{c}on \cite{4} also referred to a Tresca criterion with
zero-tension cutoff for the determination of the ultimate bearing
capacity of shallow foundations under inclined eccentric loads.

From a general viewpoint, Coulomb or Tresca strength criteria with a
zero-tension cutoff, which include cohesionless Coulomb's criterion,
illustrate the concept of strength criteria for which the zero-stress
state lies on the boundary of the domain of resistance, as considered
by Fr\'{e}mond and Friaa \cite{5}. For such criteria, it turns out that,
when implementing the exterior approach of the Yield design theory, the
minimization process often results in ``vanishing'' virtual velocity
fields, such as those implemented in \cite{2} or \cite{6}.

These criteria stand as particular cases of intrinsic curve type
strength criteria with a tension cutoff set at a value $T$, which may
either be zero, with the meaning that no tension can be sustained by
the material, or positive allowing some tensile stresses. As a
``tribute'' to D.~C.~Drucker (1918--2001), this contribution is devoted
to a more detailed analysis of the minimization process implemented in
\cite{2} for the stability analysis of a vertical cut, in the cases of a
Tresca or Coulomb strength criterion when a positive tension cutoff $T$
tends to zero, in order to assess the sensitivity of the analysis to
the soil tensile resistance.


\section{Critical height of a vertical cut} 

In terms of non-ordered principal stresses, a Coulomb criterion defined
by a friction angle $\phi $ and a cohesion $C$, with a tension cutoff
denoted by $T$ such that $0\leq T< H=C\cot \phi $, can be written as 
\def\theequation{\thesection.\arabic{equation}}
{\begin{eqnarray}\label{eqn2.1}
f(\underline{\underline{\sigma }})=\mathrm{Max}
\{\sigma _{i}(1+\sin \phi )-\sigma _{j}(1-\sin \phi )-2C\cos \phi ,
\sigma _{i}-T  \mid i,j=1,2,3\}\leq 0,
\end{eqnarray}}\unskip
as shown in Figure~\ref{fig1}.


\begin{figure}
\includegraphics{fig01}
\caption{\label{fig1}Coulomb's criterion with a tension cutoff set at a value $T$
in the $(\sigma ,\tau )$ plane of the Mohr stress representation.}
\end{figure}


The exterior approach of the yield design theory states that the rate
of work by external forces exerted on the structure whose stability is
under concern shall not be superior to the maximum resisting rate of
work in any virtual velocity field $\underline{U}$. Such virtual
velocity fields $\underline{U}$ can be piecewise continuous and
continuously differentiable over $\Omega $, with strain rate
$\underline{\underline{d}}$ and velocity discontinuities
$\underline{V}$ across jump surfaces $\Sigma $ with normal
$\underline{n}$. The maximum resisting rate of work is then defined as 
{\begin{eqnarray}\label{eqn2.2}
\mathrm{P}_{\mathrm{mr}}(\underline{U})=\int _{\Omega }\uppi (\underline{\underline{d}})
\,\mathrm{d}\Omega +\int _{\Sigma }\uppi (\underline{V},\underline{n})\,\mathrm{d}\Sigma 
\end{eqnarray}}\unskip
with
{\begin{eqnarray}\label{eqn2.3}
\uppi (\underline{\underline{d}})=\mathrm{Sup}\{\underline{\underline{\sigma }}\colon 
\underline{\underline{d}} \mid  f (\underline{\underline{\sigma }})\leq 0\},
\uppi (\underline{V},\underline{n})=\mathrm{Sup}\{\underline{V}\cdot 
\underline{\underline{\sigma }}\cdot
\underline{n} \mid  f (\underline{\underline{\sigma }})\leq 0\}, 
\end{eqnarray}}\unskip
For a Coulomb criterion with tension cutoff as written in (\ref{eqn2.1}), the
expressions for (\ref{eqn2.3}) can be found in \cite{7} in the forms
{\begin{eqnarray}\label{eqn2.4}
\left\{
\begin{array}{rcl}
\underset{\phi ,C,T}{\uppi }(\underline{\underline{d}}) &=& +\infty 
\quad\mbox{if }\mathrm{tr}\,\underline{\underline{d}}< (| d_{1}| +| d_{2}| +| d_{3}| )\sin \phi \\
\underset{\phi ,C,T}{\uppi }(\underline{\underline{d}})
&=& C (| d_{1}| +| d_{2}| +| d_{3}| -\mathrm{tr}\,\underline{\underline{d}})\tan 
(\uppi /4+\phi /2)\\
&&+\,\displaystyle \frac{T}{1-\sin \phi }(\mathrm{tr}\,\underline{\underline{d}}-(| d_{1}| +| d_{2}| +| d_{3}| )
\sin \phi )  \quad\mbox{if }\mathrm{tr}\,\underline{\underline{d}}\geq 
(| d_{1}| +| d_{2}| +| d_{3}| )\sin \phi 
\end{array} 
\right.
\end{eqnarray}}\unskip
and
{\begin{eqnarray}\label{eqn2.5}
\left\{
\begin{array}{rcl}
\underset{\phi ,C,T}{\uppi }(\underline{V},\underline{n}) &=& +\infty 
\quad\mbox{if } \underline{V} \cdot \underline{n}< | \underline{V}| \sin \phi \\
\underset{\phi ,C,T}{\uppi }(\underline{V}, \underline{n}) &=& C
(| \underline{V}| -\underline{V} \cdot \underline{n}) \tan (\uppi /4+\phi /2)\\
&&+\, \displaystyle \frac{T}{1-\sin \phi }(\underline{V} \cdot\underline{n}-
| \underline{V}| \sin \phi )  \quad\mbox{if }\underline{V}\cdot
\underline{n}\geq | \underline{V}| \sin \phi .
\end{array}  
\right.
\end{eqnarray}}\unskip
Corresponding expressions in the case of a Tresca criterion with
tension cutoff are obtained  making $\phi =0$ in these equations.


The same virtual collapse mechanism as first devised by Drucker and
Prager is considered in this analysis as a 2-dimensional problem, with
notations displayed in Figure~\ref{fig2}. Non dimensional factors
$\varepsilon $ and $\tau $ are introduced with the constraints
{\begin{eqnarray}
&\displaystyle
0< \varepsilon =e/h\leq \tan (\uppi /4-\phi /2)/2,
\label{eqn2.6}\Seqnsplit
&\displaystyle
0\leq \tau =T/C< \cot \phi .
\label{eqn2.7}
\end{eqnarray}}\unskip
It is worth recalling that the material resistance to simple tension
($\sigma _{1}=\rho ,\sigma _{2}=0$) is $\rho =2C\tan (\uppi /4-\phi
/2)$, (Figure~\ref{fig1}), so that the reduction factor on the
resistance to tension introduced by the tension cutoff is just
{\begin{eqnarray}\label{eqn2.8}
T/\rho =(\tau /2)\tan (\uppi /4+\phi /2).
\end{eqnarray}}\unskip

\begin{figure}
\includegraphics{fig02}
\caption{\label{fig2}Virtual collapse mechanism considered in this analysis.}
\end{figure}

The description of the virtual collapse mechanism in Figure~\ref{fig2}
goes as follows.

Zone \xboxed{3} remaining motionless, the virtual velocity field
$\underline{U}$ in zone \xboxed{1} defined as $\Omega A'B'B$ consists
of an anticlockwise rigid body rotational motion, with angular velocity
$\omega $ about point $\Omega $. This implies that zone \xboxed{1}
separates from zone \xboxed{3} with a velocity discontinuity
$\underline{V}(y)$ along the $\xi $ axis when crossing
$\Omega B$, whose magnitude is
{\begin{eqnarray}\label{eqn2.9}
V(y)=\omega [h(1-\varepsilon \tan (\uppi /4+\phi /2))+y]. 
\end{eqnarray}}\unskip
The velocity field is continuous across $\Omega A'$ and across $\Omega
A$. Complying with these boundary conditions, the velocity field in
zone \xboxed{2}, delimited by $\Omega AA'$, is defined as follows:
referring to the $\alpha $ and $\beta $ lines in Figure~\ref{fig2},
$\underline{U}$ is constant along any $\beta $ line and normal to
$\Omega A'$ with magnitude $\omega \,x^{\alpha }$, where $x^{\alpha }$
denotes the abscissa of the considered $\beta $ line along $\Omega
\alpha$\footnote{Cf. \cite{2,7}.}. The virtual
strain rate in zone \xboxed{2} is constant, with principal axes along
the $\xi $ and $y$ axes and principal values 
{\begin{eqnarray}\label{eqn2.10}
\begin{cases} 
d_{\xi \xi}=(\omega /2)\tan (\uppi /4+\phi /2)\\
d_{yy}=-(\omega /2)\tan (\uppi /4-\phi /2),
\end{cases} 
\end{eqnarray}}\unskip
so that 
{\begin{eqnarray}\label{eqn2.11}
\mathrm{tr}\,\underline{\underline{d}}=d_{\xi \xi }+d_{yy}=\omega 
\tan \phi =(| d_{\xi \xi }| +| d_{yy}| )\sin \phi .
\end{eqnarray}}\unskip
Implementing the yield design exterior approach in such a virtual
velocity field calls for the computation of the rate of work by
external forces $\mathrm{P}_{\mathrm{e}}(\omega ,\gamma ,h,\varepsilon
)$, where $\gamma $ stands for the 2-D specific weight of the soil, and
the maximum resisting rate of work $\mathrm{P}_{\mathrm{mr}}(\omega
,C,\phi ,\tau ,h,\varepsilon )$, in the whole bulk of the soil. Since
zone \xboxed{3} remains motionless, it does not contribute to any of
these two quantities.

The rate of work by external forces $\mathrm{P}_{\mathrm{e}}(\omega
,\gamma ,h,\varepsilon )$ results from the contribution of zones
\xboxed{1} and \xboxed{2}. With
{\begin{eqnarray}
&\displaystyle
\mathrm{P}_{\mathrm{e}}^{1}(\omega ,\gamma ,h,\varepsilon )=
(\gamma \omega \, h^{3}\varepsilon ^{2}/2)(1-(5/3)\varepsilon 
\tan (\uppi /4+\phi /2))
\label{eqn2.12}\Seqnsplit
&\displaystyle
\mathrm{P}_{\mathrm{e}}^{2}(\omega ,\gamma ,h,\varepsilon )=
(\gamma \omega \,h^{3}\varepsilon ^{3}/3)\tan (\uppi /4+\phi /2),
\label{eqn2.13}
\end{eqnarray}}\unskip
$\mathrm{P}_{\mathrm{e}}(\omega ,\gamma ,h,\varepsilon )$ results 
in\fnTxt[2]{Previous works did not require a complete computation
of this quantity.} 
{\begin{eqnarray}\label{eqn2.14}
\mathrm{P}_{\mathrm{e}}(\omega ,\gamma ,h,\varepsilon )=
(\gamma \,\omega \,h^{3}\varepsilon ^{2}/2)(1-\varepsilon 
\tan (\uppi /4+\phi /2)).\fnRef{2}
\end{eqnarray}}\unskip

Since the virtual velocity field is continuous across $\Omega A$ and
$\Omega A'$, the maximum resisting rate of work
$\mathrm{P}_{\mathrm{mr}}(\omega ,C,\phi ,\tau ,h,\varepsilon )$
results from the addition of the maximum resisting rate of work along
$\Omega B$, which can be calculated from (\ref{eqn2.5}), and the maximum
resisting rate of work developed in zone \xboxed{2}, to be computed
from (\ref{eqn2.4}) with (\ref{eqn2.11}):
{\begin{eqnarray}\label{eqn2.15}
\mathrm{P}_{\mathrm{mr}}(\omega ,C,\phi ,\tau ,h,\varepsilon )=
\mathrm{P}_{\mathrm{mr}}^{\Omega B}(\omega ,C,\phi ,\tau ,h,\varepsilon )+
\mathrm{P}_{\mathrm{mr}}^{2}(\omega ,C,\phi ,\tau ,h,\varepsilon ). 
\end{eqnarray}}\unskip
We get\fnTxt[3]{As in \cite{2,8}.}
{\begin{eqnarray}
&\displaystyle
\mathrm{P}_{\mathrm{mr}}^{\Omega B}(\omega ,C,\phi ,\tau ,h,\varepsilon )=
(\omega h^{2}/2)C\,\tau (1-\varepsilon \tan (\uppi /4+\phi /2))^{2}
\label{eqn2.16}\Seqnsplit
&\displaystyle
\mathrm{P}_{\mathrm{mr}}^{2}(\omega ,C,\phi ,\tau ,h,\varepsilon )=
\omega \,h^{2}\varepsilon ^{2}C\,\tan (\uppi /4+\phi /2).\fnRef{3}
\label{eqn2.17}
\end{eqnarray}}\unskip

Finally, implementing the yield design exterior approach results in the
fundamental equation that can be written as
{\begin{eqnarray}\label{eqn2.18}
\frac{\gamma h}{C}\leq \frac{2\tan (\uppi /4+\phi /2)}
{1-\varepsilon \tan (\uppi /4+\phi /2)}+\tau 
\frac{1-\varepsilon \tan (\uppi /4+\phi /2)}{\varepsilon ^{2}}, 
\end{eqnarray}}\unskip
to be minimized with respect to $\varepsilon $, with $\tau $ as a
parameter, within constraints (\ref{eqn2.6}) and (\ref{eqn2.7}).
The first term on the right hand side of this inequation, which results
from $\mathrm{P}_{\mathrm{mr}}^{2}(\omega ,C,\phi ,\tau
,h,\varepsilon )$ and does not depend on $\tau $, decreases with
$\varepsilon $, while the second term, which only depends on $\tau $,
increases when $\varepsilon $ decreases. Thus, for any value of $\tau $
complying with (\ref{eqn2.7}), the minimization process will adjust the
geometrical variable $\varepsilon $ to the value $\varepsilon
_{\mathrm{m}}(\tau )$ that yields the best balance between these two
contributions, resulting in the best upper bound for $\gamma \,h/C$.
Referring to $T/\rho =\tau \tan (\uppi /4+\phi /2)/2$ instead of $\tau $,
Equation~(\ref{eqn2.18}) can be written as 
{\begin{eqnarray}\label{eqn2.19}
\frac{\gamma h}{C}\tan (\uppi /4-\phi /2)\leq 
\frac{2}{1-\varepsilon \tan (\uppi /4+\phi /2)}+2\frac{T}{\rho }
\frac{1-\varepsilon \tan (\uppi /4+\phi /2)}{\varepsilon ^{2}
\tan ^{2}(\uppi /4+\phi /2)}, 
\end{eqnarray}}\unskip
or, introducing
{\begin{eqnarray}
&\displaystyle
E(\phi )=\varepsilon \tan (\uppi /4+\phi /2),
\label{eqn2.20}\Seqnsplit
&\displaystyle
\frac{\gamma h}{C}\tan (\uppi /4-\phi /2)\leq \frac{2}{1-E(\phi )}+2\frac{T}{\rho }
\frac{1-E(\phi )}{[E(\phi )]^{2}}
\label{eqn2.21}
\end{eqnarray}}\unskip

It follows that, for any given value of $\phi$, with $T/\rho $ as a
parameter, the minimum of the right hand of
(\ref{eqn2.21}) with respect to $\varepsilon $ is
obtained for the same value of $E(\phi)=\varepsilon \tan (\uppi /4+\phi
/2)$ denoted by $E_{\mathrm{m}}(T/\rho )$, where
$E_{\mathrm{m}}=(e/h)_{\mathrm{m}}\tan (\uppi /4+\phi /2)$,
(Figure~\ref{fig3}). 

\begin{figure}
\includegraphics{fig03}
\caption{\label{fig3}$(e/h)_{\mathrm{m}}\tan (\uppi /4+\phi /2)$ as a
function of $T/\rho $.}
\end{figure}

The corresponding value of the right hand member of (\ref{eqn2.21}) is
the best upper bound for the left hand member of this equation. It is
capped by the best available upper bounds in the absence of a tension
cutoff, i.e., $3.83$ whatever $\phi$, (3.77649 for $\phi =0$) as
usually agreed upon \cite{9,10,11,12}.


The results of the minimization process are displayed in
Figure~\ref{fig4}, where $(\gamma \,h/C)\tan (\uppi /4-\phi /2)$ is
plotted as a function of $\tau =T/\rho =\tan (\uppi /4+\phi /2)T/2C$. 


\begin{figure}
\includegraphics{fig04}
\caption{\label{fig4}Normalized results of the minimization process on
Equation~(\ref{eqn2.21}).}
\end{figure}


It comes out, through this normalization process, that all the results
fit on a parabolic like curve with a horizontal axis and a summit on
the vertical axis at $\gamma \,h\tan (\uppi /4-\phi /2)/C=2$, which is
the critical value obtained by Drucker and has also been established as
a lower bound estimate through a simple three-zone relevant stress
field.


It also appears that the mechanisms in Figure~\ref{fig2} only prevail
over the classical log-spiral one for small values of $T/\rho $, namely
when $T/\rho \leq 0.07$, while the drastic drop to the critical and
exact value $\gamma \,h/C=2\tan (\uppi /4+\phi /2)$ for $\tau =0$ remains
confined within the immediate vicinity of $T/\rho =0$. From a practical
viewpoint, if a 20\% drop is considered as acceptable, $T/\rho $ must
be superior to 3\%, which means that the tensile resistance of the soil
shall not be less than $0.06C\tan (\uppi /4-\phi /2)$. It is worth
insisting on the fact that these results are valid whatever the value
of $\phi $. 

Obviously, strictly speaking, the conclusions enounced here above
cannot be considered as final since they only result from a full
implementation of Drucker's virtual collapse mechanisms.
Nevertheless, it can be observed that, from a physical viewpoint, these
mechanisms, when $T/\rho \searrow 0$, with $E_{\mathrm{m}}(T/\rho
)\searrow 0$, are perfectly suited to matching the zero normal stress
constraint, which is imposed as a boundary condition along the
stress-free vertical and horizontal edges of the cut, with the
vanishing tensile resistance of the material. Hence, as a response to
Drucker's warning about ``\textit{the consequences of assuming soil
unable to take tension}''~\cite{2}, we believe that the analysis that
has been developed here may bring some relief as to the reliability of
classical analyses that do not take any tension cutoff into account.

In the particular case of a purely cohesive frictionless material
($\phi =0$), numerical static lower bound approaches have been
developed, assuming full tensile resistance of the material ($\rho
=2C$), which culminated with $\gamma h/C\geq 3.77522$ as a lower
bound estimate in \cite{11}. Checking the sensitivity of such approaches
to a reduction of the tensile resistance would, obviously, help to
comfort the preceding conclusion.

\section*{Declaration of interests}
The authors do not work for, advise, own shares in, or receive funds
from any organization that could benefit from this article, and have
declared no affiliations other than their research organizations.

\section*{Acknowledgements} 
The author is indebted to Professor Alain Pecker and the reviewer for
their valuable comments and suggestions.

\back{}

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