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Stress:Stress:Stress tensorStress tensor

Principal stresses and directionsPrincipal stresses and directions

Maximum shear stresses and directionsMaximum shear stresses and directions

Failure theoriesFailure theories

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StressStress

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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 tensorStress 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 бij 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 DirectionsPrincipal Stresses and Directions

Maximum Shear Stresses and Maximum Shear Stresses and DirectionsDirections

Failure TheoriesFailure 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 Stresses in dimensional bodies:bodies:

All real objects occupy three-dimensional All real objects occupy three-dimensional space. However, if two dimensions are space. However, if two dimensions are

very large or very small compared to the very large or very small compared to the others, the object may be modelled as others, the object may be modelled as

one-dimensional.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, 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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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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• 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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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 circleMohr's circleChristian Otto Mohr’s lifeChristian Otto Mohr’s 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 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.

Mohr's circle

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Vertical and horizontal Vertical and horizontal concentrated force on the concentrated force on the surface of the half-spacesurface 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).

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