ELLIPSE OF STRESS & STRAIN CLASS NOTES FOR MECHANICAL ENGINEERING

ELLIPSE OF STRESS & STRAIN

CLASS NOTES FOR MECHANICAL

ENGINEERING

 

Ellipse of stress finds resultant stress

and the angle of obliquity. In 2-D, it

is ellipse of stress . In 3-D it is ellipsoid

of stress. Actually all bodies are 3-D. All

stress systems are also 3-D. But to make

study easy to understand, first 2-D stress

system is studied. The stress is different

on each plane passing through a point.

Therefore, it is not justified to study any

one particular plane. There are actually

infinite planes. In order to fully understand

a  2D stress system,  study the stresses

across all possible plane orientations.

The axis of the ellipse are the two principal

stresses.

ELLIPSE OF STRESS

It is the locus of the resultant stress on the infinite inclined planes. It is a graphical method to find the resultant stress and the angle of obliquity on a given inclined plane.

 

 

ellipse-stress

CONSTRUCTION

Being a graphical method, there are a few steps of construction.

(i)  Taking O as center. Draw two concentric circles with radii equal to σ1 and σ2 respectively. Take σ1  greater of the  principal stresses.

(ii) Draw x-axis and y-axis as usual. Take X-axis as the plane of σ2 and y-axis as  the plane of σ1.

(iii)  Draw a line LM at an angle of θ with the plane of σ1 i.e. with the y-axis.

(iv) From O draw perpendicular to LM. It cuts the inner circle at P and the outer circle at N respectively.

(v)  From N draw a line NR perpendicular to X-axis to meet  at point R.

(vi) From point P draw a perpendicular to meet NR at point Q.

(vii) Join OQ.

(viii) OQ is resultant stress on the inclined plane.  Point Q traces its locus. Locus is an ellipse for different inclined planes.

(ix) Measure angle QON which is the angle of obliquity.

(x) Major principal stress becomes the semi major axis of the ellipse.

(xi) Minor principal stress becomes the semi minor axis of the ellipse.

Utility of the Ellipse of stress

(i) Ellipse of stress determines the state of stress on any inclined plane (any point) within the stressed body.

(ii) It determines the angle of obliquity on any inclined plane.

(iii) Higher angle of obliquity represents high shear stress on the inclined plane under consideration. It is in case of a ductile material.

(iv) Lower angle of obliquity represents lower shear stress on the inclined plane. It is a case of a brittle material.

 

ELLIPSE OF STRAIN

It is the locus of the resultant strain on the infinite inclined planes within a body under principle strains. It is a graphical method to find the resultant strain and the angle of obliquity on a given inclined plane (with the plane of major principal strain).

 

ellipse-strain

CONSTRUCTION

Being a graphical method, there are a few steps of construction.

(x)  Taking O as center, draw two concentric circles with radius equal to εand radius equal ε2 respectively. ( εis the greater  principal strain). Principal strains are the axis of the ellipse.

(xi)  Draw x-axis and y-axis as usual. X-axis as the plane of ε2 and y-axis as the plane of ε1.

(xii) Draw a line LM at an angle of θ with the plane of εi.e. with the y-axis.

(xiii) From O draw perpendicular to LM to cut the inner circle at P and the outer circle at N respectively.

(xiv) From N draw a line NR perpendicular to X-axis.

(xv)  From point P draw a perpendicular to meet NR at point Q.

(xvi) Join OQ.

(xvii) OQ Will be resultant strain on the inclined plane and point Q will trace its locus which will be an ellipse for different inclinations of the inclined plane.

(xviii) Measure angle QON which is the angle of obliquity.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

https://www.mesubjects.net/wp-admin/post.php?post=1136&action=edit          Mohr’s stress Circle

https://www.mesubjects.net/wp-admin/post.php?post=7556&action=edit           Principal stresses