Geomechanics project

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Geomechanics project
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• Stress
• Stress tensor
• Principal stresses and directions
• Maximum shear stresses and directions
• Failure theories

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• Stress

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• Stress, defined as force per unit area, is a
  measure of the intensity of internal forces
  acting within a body across imaginary internal
  surfaces, as a reaction to external applied
  forces and body forces.
• Stress is often broken down into its shear and
  normal components as these have unique physical
  significance.
• Stress is to force as strain is to deformation.

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• Stress tensor

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• Stress is a second-order tensor with nine
  components, but can be fully described with six
  components due to symmetry in the absence of body
  moments.
• In N dimensions, the stress tensor ?ij is
  defined by

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• The transformation relations for a second-order
  tensor like stress are different from those of a
  first-order tensor, which is why it is misleading
  to speak of the stress 'vector'. Mohr's circle
  method is a graphical method for performing
  stress (or strain) transformations.

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• Principal Stresses and Directions
• Maximum Shear Stresses and Directions
• Failure Theories

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• Failure Theories
• This section uses the functionality in Structural
  Mechanics to consider three fundamental failure
  criteria
• maximum normal stress theory
• maximum shear stress theory
• distortion energy theory

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• Stresses in dimensional bodies
• All real objects occupy three-dimensional space.
  However, if two dimensions are very large or very
  small compared to the others, the object may be
  modelled as one-dimensional.

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• For one-dimensional objects, the stress tensor
  has only one component and is indistinguishable
  from a scalar. The simplest definition of stress,
  s F/A, where A is the initial cross-sectional
  area prior to the application of the load, is
  called engineering stress or nominal stress.
• In one dimension, conversion between true stress
  and nominal (engineering) stress is given by
• strue (1 ee)(se)
• The relationship between true strain and
  engineering strain is given by
• etrue ln(1 ee).

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Stress in two-dimensional bodies

• Augustin Louis Cauchy was the first to
  demonstrate that at a given point, it is always
  possible to locate two orthogonal planes in which
  the shear stress vanishes. These planes in which
  the normal forces are acting are called the
  principal planes, while the normal stresses on
  these planes are the principal stresses.

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• The two dimensional Cauchy stress tensor is
  defined as
• Then principal stresses s1,s2 are equal to

• Mohr's circle is a graphical method of extracting
  the principal stresses in a 2-dimensional stress
  state. The maximum and minimum principal stresses
  are the maximum and minimum possible values of
  the normal stresses.

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Stress in three dimensional bodies

• the stress has two directional components one
  for force and one for plane orientation in three
  dimensions these can be two forces within the
  plane of the area A, the shear components, and
  one force perpendicular to A, the normal
  component.
• This gives rise to three total stress components
  acting on this plane.

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• Mohr's circle
• Christian Otto Mohrs life

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Mohr's life

• Christian Otto Mohr (October 8, 1835 - October 2,
  1918) was a German civil engineer, one of the
  most celebrated of the nineteenth century.
• Starting in 1855, his early working life was
  spent in railroad engineering for the Hanover and
  Oldenburg state railways, designing some famous
  bridges and making some of the earliest uses of
  steel trusses.

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• Mohr was an enthusiast for graphical tools and
  developed the method, for visually representing
  stress in three dimensions, previously proposed
  by Carl Culmann.
• In 1882, he famously developed the graphical
  method for analysing stress known as Mohr's
  circle and used it to propose an early theory of
  strength based on shear stress.

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Mohr's circle

• Mohr's circles provide a planar representation
  of a three-dimensional stress state. Mohr's
  circle may also be applied to three-dimensional
  stress. In this case, the diagram has three
  circles, two within a third.

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• Vertical and horizontal concentrated force on the
  surface of the half-space

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Vertical concentrated force on the surface of
half space
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• . On account of symmetry, the displacement of the
  point N in picture is defined by two components
• w (r, z) the vertical displacement of the
  point N
• u r (r, z) the horizontal radial displacement
  of the point N

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• The line load p?, acting on the surface of a
  half space, can be divided in an infinite
  number of elementary forces p? dy. Since it is
  assumed that the soil mass is ideally elastic,
  the resultant produced by force p? dy

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Horizontal line load of infinite length on the
surface of half space

• The problem of determination of stresses and
  displacements produced by a concentrated
  horizontal load acting on the surface of a half
  space was studied by Cerruti (1882).