mohr circle

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يتم رسم دوائر مور إما بإعطائك مصفوفة للإجهاد و حساب الإجهادات الرئيسية و بعد ذلك ترتيبها و تعيين النقاط على المحاور الإحداثية و رسم الدوائر و سوف أقوم بالقريب العاجل إن شاء الله بإرفاق مثال محلول عن رسم الدوائر
 

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د.محمد باشراحيل

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المهندس معتز
هذا موضوع عن الإجهاد والإنفعال
وهذا رابطه


وهذه مقتبسة من المشاركة 6 .

Plane stress

Figure 7.1 Plane stress state in a continuum.


A state of plane stress exist when one of the three principal
6581887e1c4b23b9ac384772c33f908b.png
, stresses is zero. This usually occurs in structural elements where one dimension is very small compared to the other two, i.e. the element is flat or thin. In this case, the stresses are negligible with respect to the smaller dimension as they are not able to develop within the material and are small compared to the in-plane stresses. Therefore, the face of the element is not acted by loads and the structural element can be analyzed as two-dimensional, e.g. thin-walled structures such as plates subject to in-plane loading or thin cylinders subject to pressure loading. The other three non-zero components remain constant over the thickness of the plate. The stress tensor can then be approximated by:
f7fa9e7a53a7d6594608fda54606e91d.png
. The corresponding strain tensor is:
240f22c1c109fb3938e3dae4faa6e448.png
in which the non-zero
e744302d4661fac8ea13a481f86ceda7.png
term arises from the Poisson's effect. This strain term can be temporarily removed from the stress analysis to leave only the in-plane terms, effectively reducing the analysis to two dimensions.

[edit] Plane strain

If one dimension is very large compared to the others, the principal strain in the direction of the longest dimension is constrained and can be assumed as zero, yielding a plane strain condition. In this case, though all principal stresses are non-zero, the principal stress in the direction of the longest dimension can be disregarded for calculations. Thus, allowing a two dimensional analysis of stresses, e.g. a dam analyzed at a cross section loaded by the reservoir.
[edit] Stress transformation in plane stress and plane strain

Consider a point
fe6b6f2dcae650d9e649d40ca981681d.png
in a continuum under a state of plane stress, or plane strain, with stress components
2ae261ebbae2209c05dba0e52a94c8d4.png
and all other stress components equal to zero (Figure 7.1, Figure 8.1). From static equilibrium of an infinitesimal material element at
fe6b6f2dcae650d9e649d40ca981681d.png
(Figure 8.2), the normal stress
4bc84aa01952ddfb37a7c055c7cc84e4.png
and the shear stress
7c1c920a206cbf2744fa3e1b5ee13889.png
on any plane perpendicular to the
6373accf16c083723e8abae2f5401af2.png
-
bfb6488d6c250ac5aeed1bbf139baaa5.png
plane passing through
fe6b6f2dcae650d9e649d40ca981681d.png
with a unit vector
2966606b4719807ecbf366c17d2bb9e4.png
making an angle of
0a5000fe8b6b5570dd5a1ce00b828ef6.png
with the horizontal, i.e.
170e976a229cb1aa373bcfe79e9c591d.png
is the direction cosine in the
6373accf16c083723e8abae2f5401af2.png
direction, is given by:
464ab497467706191b9ee2c4b6238d8f.png
7642947c65fc6c5d5d0ba790366dcfda.png
These equations indicate that in a plane stress or plane strain condition, one can determine the stress components at a point on all directions, i.e. as a function of
0a5000fe8b6b5570dd5a1ce00b828ef6.png
, if one knows the stress components
2ae261ebbae2209c05dba0e52a94c8d4.png
on any two perpendicular directions at that point. It is important to remember that we are considering a unit area of the infinitesimal element in the direction parallel to the
bfb6488d6c250ac5aeed1bbf139baaa5.png
-
8576a3c704dbd2ab0ffed196359d8de5.png
plane.

Figure 8.1 - Stress transformation at a point in a continuum under plane stress conditions.



Figure 8.2 - Stress components at a plane passing through a point in a continuum under plane stress conditions.


The principal directions (Figure 8.3), i.e. orientation of the planes where the shear stress components are zero, can be obtained by making the previous equation for the shear stress
7c1c920a206cbf2744fa3e1b5ee13889.png
equal to zero. Thus we have:
2211f37b7bb0a8ce0cb529ddd19cd6bd.png
and we obtain
0b0aa65a26e32960a788a2faf99daf8a.png
This equation defines two values
785610fed60fb565e73666c2acf1ab68.png
which are
f20f319dbb8dd31624b6377bf6cecbd5.png
apart (Figure 8.3). The same result can be obtained by finding the angle
0a5000fe8b6b5570dd5a1ce00b828ef6.png
which makes the normal stress
4bc84aa01952ddfb37a7c055c7cc84e4.png
a maximum, i.e.
3696febe1a15e1723a0afe2443f28afb.png

The principal stresses
78f3fd038d105a34b56d914a2fd9d499.png
and
f6d67717ab44d06e79ceb02a72ce5dfe.png
, or minimum and maximum normal stresses
12e870c316e206a1e6aa061c194cb6e3.png
and
72dfe70a55152ff7109b02eb5a2effd9.png
, respectively, can then be obtained by replacing both values of
785610fed60fb565e73666c2acf1ab68.png
into the previous equation for
4bc84aa01952ddfb37a7c055c7cc84e4.png
. This can be achieved by rearranging the equations for
4bc84aa01952ddfb37a7c055c7cc84e4.png
and
7c1c920a206cbf2744fa3e1b5ee13889.png
, first transposing the first term in the first equation and squaring both sides of each of the equations then adding them. Thus we have
129d8cd805c51a932abec44d9278cf67.png
where
1ec1e07d2cdac0309f69195afd137b6d.png
which is the equation of a circle of radius
3b8a9cb05773d251ae3891482c5599da.png
centered at a point with coordinates
b6468372d603f706ec51b7825dc6e679.png
, called Mohr's circle. But knowing that for the principal stresses the shear stress
e2e950057a7e19a878327b4d09357611.png
, then we obtain from this equation:
d250eba2c2754c819f4b94feb4d82c92.png
582fb5788adbde36223356dbff238e18.png

Figure 8.3 - Transformation of stresses in two dimensions, showing the planes of action of principal stresses, and maximum and minimum shear stresses.



When
502d20d07fa06330929b4c06a0af9166.png
the infinitesimal element is oriented in the direction of the principal planes, thus the stresses acting on the rectangular element are principal stresses:
b75224ae1b3df556cbb274311c4e70de.png
and
3729849afd4b977bfc3fef0632dd2aff.png
. Then the normal stress
4bc84aa01952ddfb37a7c055c7cc84e4.png
and shear stress
7c1c920a206cbf2744fa3e1b5ee13889.png
acting on a plane making an angle of
0a5000fe8b6b5570dd5a1ce00b828ef6.png
with the principal directions can be obtained by making
502d20d07fa06330929b4c06a0af9166.png
. Thus we have
b18cbdc796000e6b347019359ab0b5fe.png
122afead2c62a7aa632defe2203a5973.png
Then the maximum shear stress
2652df78374e4f9d6c98fa6fa719c627.png
occurs when
7a44a70a3012569b40a71f483d51dd39.png
, i.e.
c8338fb707c139d7981b4ff2ba069d17.png
(Figure 8.3):
970806b58b2614f0ae7844f69ca5d381.png
Then the minimum shear stress
5764155c7b08ebec76dc339025e465e2.png
occurs when
adddfbc5d1935b6daa5b4f0a764a5b24.png
, i.e.
724b6a54ac3a241f7225dda1e64f0a17.png
(Figure 8.3):
dfb2264a20be62eeaede6830629d18ff.png
[edit] Mohr's circle for stress

The Mohr's circle, named after Christian Otto Mohr, is a two-dimensional graphical representation of the state of stress at a point. The abscissa,
4bc84aa01952ddfb37a7c055c7cc84e4.png
, and ordinate,
7c1c920a206cbf2744fa3e1b5ee13889.png
, of each point on the circle are the normal stress and shear stress components, respectively, acting on a particular cut plane with a unit vector
2966606b4719807ecbf366c17d2bb9e4.png
with components
14753f7a98c95d1a3193c9eb45053cb2.png
. In other words, the circumference of the circle is the locus of points that represent state of stress on individual planes at all their orientations.
Karl Culmann was the first to conceive a graphical representation for stresses while considering longitudinal and vertical stresses in horizontal beams during bending. Mohr's contribution extended the use of this representation for both two- and three-dimensional stresses and developed a failure criterion​
 
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