combined stresses and mohr_s circle

8/8/2019 Combined Stresses and Mohr_s Circle

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Combined Stresses and MohrsCircle

Material in this lecture was taken from chapter 4 of Mott, Machine Elements in Mechanical Design, 2003

General Case of CombinedStresses

Two-dimensional stress condition

Mott, Machine Elements in Mechanical Design, 2003

General Case of CombinedStresses cont

The normal stresses, x and y, could be dueto a direct tensile force or to bending. If thenormal stresses were compressive (negative),the vectors would be pointing in the oppositesense, into the stress element.

The shear stress could be due to direct shear,torsional shear, or vertical shear stress. Thedouble-subscript notation helps to orient thedirection of shear stresses. For example, xyindicates the shear stress acting on theelement face that is perpendicular to the x-axis and parallel to the y-axis.


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General Case of CombinedStresses cont

A positive shear stress is one that tends torotate the stress element clockwise

In the first figure, xy is positive and yx isnegative. Their magnitudes must be equalto maintain the element in equilibrium.

With the stress element defined, theobjectives of the remaining analysis are todetermine the maximum normal stress,and the planes on which these stressesoccur.

Maximum Normal Stresses

The combination of the applied normaland shear stresses that produces themaximum normal stress is called themaximum principle stress, 1.


Maximum Normal Stresses

The minimum principle stress, 2equals:



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Maximum Normal Stressescont

The angle of inclination of the planes onwhich the principle stresses act, calledprinciple planes, can be found from:


)]/(2arctan[2

1 yxxy =

Measured from the positive X axis

Maximum Normal Stressescont

The angle ismeasured from thepositive x-axis of theoriginal stresselement to themaximum principlestress, 1. Then theminimum principle

stress, 2, is on theplane 90o from 1. Mott, Machine Elements in Mechanical Design, 2003

Maximum Shear Stress

On a different orientation of the stresselement , the maximum shear stress

will occur.



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The angle of inclination of the elementon which the maximum occurs iscomputed as follows:



The angle between the principle stresselement and the maximum shear stress

element is always 45o.

On the maximum shear stress element,there will be normal stresses of equalmagnitude acting perpendicular to theplanes on which the maximum shearstresses are acting.


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Average Normal Stress

The average of two applied normalstresses:


General Procedure for Analyzingany Combined Stress


General Procedure



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Example 4.1


Example 4.1


Example 4.1



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Example 4.1


Example 4.1


Mohrs Circle

Because of the many terms and signsinvolved, and the many calculations

required in the computation of the principlestresses and the maximum shear stress,there is a rather high probability of error.Using the graphic aid Mohrs circle helps tominimize errors and gives a better feelfor the stress condition at the point ofinterest.


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Mohrs Circle

After Mohrs circle has been constructed, itcan be used for the following:


Mohrs Circle

The data needed to construct Mohrscircle are the same as those needed tocompute the preceding values, becausethe graphical approach is an exactanalogy to the computations.

Mohrs Circle

Mohrs circle is actually a plot of the combinationof normal and shearing stresses that exist on a

stress element for all possible angles oforientation of the element. This method isparticularly valuable in experimental stressanalysis work because the results obtained frommany types of standard strain gageinstrumentation techniques give the necessaryinputs for the creation of Mohrs circle.


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Methodology


Methodology


Methodology


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Methodology

Display of Results from MohrsCircle


Example 4-2



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Example 4-2


Example 4-2


Example 4-2


combined stresses and mohr_s circle

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