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California State University, Northridge College of Computer Science and
Engineering Department of Civil and Manufacturing Engineering STRAIN
TRANSFORMATION Submitted to: Nazaret Dermendjian Submitted for: Applied
Mechanics 340 Date requested: April 6, 1999 Date due: April 27, 1999 Date
submitted: April 27, 1999 The following report will be on Strain Transformation.
Strain transformation is similar to stress transformation, so that many of the
techniques and derivations used for stress can be used for strain. We will also
discuss methods of measuring strain and material-property relationships. The
general state of strain at a point can be represented by the three components of
normal strain, Îx, Îy, Îz, and three components of shear strain, gxy, gxz, gyz.
For the purpose of this report, we confine our study to plane strain. That is,
we will only concentrate on strain in the x-y plane so that the normal strain is
represented by Îx and Îy and the shear strain by gxy . The deformation on an
element caused by each of the elements is shown graphically below. Before
equations for strain-transformation can be developed, a sign convention must be
established.
As seen below, Îx and Îy are positive if they cause elongation in
the the x and y axes and the shear strain is positive if the interior angle
becomes smaller than 90°. For relative axes, the angle between the x and x'
axes, q, will be counterclockwise positive. If the normal strains Îx and Îy and
the shear strain gxy are known, we can find the normal strain and shear strain
at any rotated axes x' and y' where the angle between the x axis and x' axis is
q. Using geometry and trigonometric identities the following equations can be
derived for finding the strain at a rotated axes: Îx' = (Îx + Îy)/2 + (Îx -
Îy)cos 2q + gxy sin 2q (1) gx'y' = [(Îx - Îy)/2] sin 2q + (gxy /2) cos 2q (2)
The normal strain in the y' direction by substituting (q + 90°) for q in Eq.1.
The orientation of an element can be determined such that the element's
deformation at a point can be represented by normal strain with no shear strain.
These normal strain are referred to as the principal strains, Î1 and Î2 . The
angle between the x and y axes and the principal axes at which these strains
occur is represented as qp. The equations for these values can be derived from
Eq.1 and are as followed: tan 2qp = gxy /(Îx - Îy) (3) Î1,2 = (Îx -Îy)/2 ± {[(Îx
-Îy)/2]2+ (gxy/2)2 }1/2 (4) The axes along which maximum in-plane shear strain
occurs are 45° away from those that define the principal strains and is
represented as qs and can be found using the following equation: tan 2qs = -(Îx
- Îy) / 2 (5) When the shear strain is maximum, the normal strains are equal to
the average normal strain. These values are determined from the following
equations: gmax / 2 = {[(Îx - Îy) / 2]2 + (gxy / 2)2}1/2 (6) Îavg = (Îx + Îy) /
2 (7) We can also solve strain transformation problem using Mohr's circle. The
coordinate system used has the abscissa represent the normal strain Î, with
positive to the right and the ordinate represents half of the shear strain g/2
with positive downward. Determine the center of the circle C, which is on the Î
axis at a distance of Îavg from the origin.
Please note that it is important to
follow the sign convention established previously. Plot a reference point A
having coordinates (Îx , gxy / 2). The line AC is the reference for q = 0. Draw
a circle with C as the center and the line AC as the radius. The principal
strains Î1 and Î2 are the values where the circle intersects the Î axis and are
shown as points B and D on the figure below. The principal angles can be
determined from the graph by measuring 2qp1 and 2qp2 from the reference line AC
to the Î axis. The element will be elongated in the x' and y' directions as
shown below. The average normal strain and the maximum shear strain are shown as
points E and F on the figure below. The element will be elongated as shown. To
measure the normal strain in a tension-test specimen, an electrical-resistance
strain gauge can be used. An electrical-resistance strain gauge works by
measuring the change in resistance in a wire or piece of foil and relates that
to change in length of the gauge. Since these gauges only work in one direction,
normal strains at a point are often determined using a cluster of gauges
arranged in a specific pattern, referred to as a strain rosette.
Using the
readings on the three gauges, the data can be used to determine the state of strain,Îx, Îy, gxy, at that point using geometry and trigonometric identities.
It is important to note that the strain rosettes do not measure strain that is
normal to the free surface of the specimen. Mohr's circle can then be used to
solve for any in plane normal and shear strain of interest. It is important to
mention briefly material-property relation ships. Note that it is assumed that
the material is homogeneous, isotropic, and behaves in a linear elastic manner.
If the material is subject to a state of triaxial stress, sx, sy, sz,(not
covered in this report) associated normal strains Îx, Îy, Îz, are developed in
the material. Using principals of superposition, Poisson's ratio, Îlat = -nÎlong
, and Hooke's law, as it applies in the uniaxial direction Î = s/E , the normal
stress can be related to the normal strain. Similar relationships can be
developed between shear stress and shear strain. This report was a brief summary
of strain transformation and the related topics of strain gauges and
material-property relationships. It is important to realize that this report was
confined to in plane strain transformation and that a more complete study would
involve shear strain in three dimensions, then material-property relationships
could be developed further. Also, theories of failure were not covered in this
report.
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