All materials have some component of frictional strength and therefore are stronger in compression than tension.

**Mohr-Coulomb Criterion (Revisited)**

All materials have some component of frictional
strength and therefore are stronger in compression than tension. For EPS materials
(e.g., steel), this dif-ference is insignificant, but for cast iron, rock,
concrete, or soil, frictional strength predominates and cannot be neglected.
The Mohr-Coulomb crite-rion introduced in previews pages has the fundamental
form:

which plots as a straight line in stress space. Thus
the shear strength, *s*, is a linear function of the normal stress σ* _{n}*
on the potential slip plane where tan ϕ is the coefficient of
friction.

Since very “brittle” materials such as concrete have
little tensile strength and granular materials like sand have none, we will, in
this session pages, reverse the arbitrary sign conventions of elasticity. Thus,
compression will be positive and shear stresses will be positive if the signs
of the subscripts are mixed. The con-vention for plotting Mohr’s Circle is
shown in Figure 12.1. Also, throughout, it will be assumed that we have a
plane-strain situation where the out-of-plane normal stress, σ* _{y}*
, is intermediate. That is:

Therefore the slip
planes on which τ' *s* are in the *x*'*z*
plane and only the largest Mohr’s Circle need be plotted in future figures.

One fundamental
difference introduced by the M-C criterion is the possibility of two potential
failure states and two sets of slip surfaces, “active” and “passive” at a
point, even when one principal stress component is the same. This is easily
seen in the triaxial “test” situation shown in Figure 12.2. Assuming this M-C
material is initially at constant hydrostatic pressure σ_{x}* ^{i}* + σ

It is useful to go through a formal plastic analysis
for either the active or passive case (although the result is obtained by
simply drawing Mohr’s Circle touching the strength envelope as in Figure
12.2c). Consider, for example, the passive case illustrated again in Figure
12.3 where the stress field is constant (b any size). Note that the active case
would give the same stressed element rotated 90 o where σ* _{x}* would now be the
minimum principal stress and σ

which apply in the
principal stress orientation for either the active or passive case. Thus only
when σ* _{x}*
or σ

Finally, it is important to reemphasize that the
block in Figure 12.3 can be thought of as a differential element (*b* = *db*)
and therefore the analysis and equations are general for a point. However, they
only apply on a finite scale in a stress field where the orientation of the
principal stresses (the isoclinic field) is constant. This is not the normal
situation. Thus a straight slip surface as an overall failure mechanism for
limit analysis, while it may be a nice guess, can seldom be exact.

Demonstrating why, because of frictional strength,
encased concrete columns are so much stronger.

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Civil : Principles of Solid Mechanics : Slip Line Analysis : Mohr-Coulomb Criterion (Revisited) |

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