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1. FFrreeee VViiddeeoo LLeeccttuurreess
ffoorr MMBBAA
BByy::
vviiddeeoo..eeddhhoollee..ccoomm

2. video.edhole.com

3. SSttrreessss::
SSttrreessss tteennssoorr
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MMaaxxiimmuumm sshheeaarr ssttrreesssseess aanndd ddiirreeccttiioonnss
FFaaiilluurree tthheeoorriieess
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4. SSttrreessss
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5. • 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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6. video.edhole.com

7. SSttrreessss tteennssoorr
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8. • 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
бij is defined by:
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9. • 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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10. Principal SSttrreesssseess aanndd DDiirreeccttiioonnss
MMaaxxiimmuumm SShheeaarr SSttrreesssseess aanndd DDiirreeccttiioonnss
FFaaiilluurree TThheeoorriieess
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11. • 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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12. SSttrreesssseess iinn ddiimmeennssiioonnaall
bbooddiieess::
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tthhee oobbjjeecctt mmaayy bbee mmooddeelllleedd aass oonnee--
ddiimmeennssiioonnaall..
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13. • For one-dimensional objects, the stress tensor
has only one component and is indistinguishable
from a scalar. The simplest definition of stress, σ
= 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
• σtrue = (1 + εe)(σe)
• The relationship between true strain and
engineering strain is given by
• εtrue = ln(1 + εe).
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14. 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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15. • 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.
• The two dimensional
Cauchy stress tensor is
defined as:
• Then principal stresses
σ1,σ2 are equal to:
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16. 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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17. MMoohhrr''ss cciirrccllee
CChhrriissttiiaann OOttttoo MMoohhrr’’ss lliiffee
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18. 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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19. • 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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20. 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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21. VVeerrttiiccaall aanndd hhoorriizzoonnttaall
ccoonncceennttrraatteedd ffoorrccee oonn tthhee
ssuurrffaaccee ooff tthhee hhaallff--ssppaaccee
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22. Vertical concentrated force on
the surface of half – space
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23. • . 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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24. • 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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25. 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).
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