# mohr circle

#### Eng_moatazabbas

##### عضو جديد
سلام عليكم ورحمه الله وبركاته

برجاء اريد المساعدة فى اى معلومات عن دائرة مور فى تحليل الاجهادات

### مواضيع مماثلة

#### أحمد السماوي

##### عضو جديد
ثنائية الأجهاد أم ثلاثية ...؟؟؟

#### General michanics

##### عضو جديد
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

#### Eng_moatazabbas

##### عضو جديد
اسف جدا على التاخير فى الرد

ولكن اريد ثنائية الاحداثيات

#### د.محمد باشراحيل

##### إستشاري الملتقى
سلام عليكم ورحمه الله وبركاته

برجاء اريد المساعدة فى اى معلومات عن دائرة مور فى تحليل الاجهادات

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 , 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: . The corresponding strain tensor is: in which the non-zero 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.

 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.
 Stress transformation in plane stress and plane strain

Consider a point in a continuum under a state of plane stress, or plane strain, with stress components and all other stress components equal to zero (Figure 7.1, Figure 8.1). From static equilibrium of an infinitesimal material element at (Figure 8.2), the normal stress and the shear stress on any plane perpendicular to the - plane passing through with a unit vector making an angle of with the horizontal, i.e. is the direction cosine in the direction, is given by:  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 , if one knows the stress components 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 - 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 equal to zero. Thus we have: and we obtain This equation defines two values which are apart (Figure 8.3). The same result can be obtained by finding the angle which makes the normal stress a maximum, i.e. The principal stresses and , or minimum and maximum normal stresses and , respectively, can then be obtained by replacing both values of into the previous equation for . This can be achieved by rearranging the equations for and , first transposing the first term in the first equation and squaring both sides of each of the equations then adding them. Thus we have where which is the equation of a circle of radius centered at a point with coordinates , called Mohr's circle. But knowing that for the principal stresses the shear stress , then we obtain from this equation:    Figure 8.3 - Transformation of stresses in two dimensions, showing the planes of action of principal stresses, and maximum and minimum shear stresses.

When the infinitesimal element is oriented in the direction of the principal planes, thus the stresses acting on the rectangular element are principal stresses: and . Then the normal stress and shear stress acting on a plane making an angle of with the principal directions can be obtained by making . Thus we have  Then the maximum shear stress occurs when , i.e. (Figure 8.3): Then the minimum shear stress occurs when , i.e. (Figure 8.3): 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, , and ordinate, , 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 with components . 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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