Ellipse of stress

Ellipse of stress

Introduction

Ellipse of stress is used to find resultant stress and the angle of obliquity on any plane within a stressed body. In 2-D, it is called ellipse of stress . In 3 D it is called 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 state of 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 represent and understand a  2D or 3-D stress system,  the stresses across all possible plane orientations passing through that point must be considered. 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 within a body under principal stresses. It is a graphical method to find the resultant stress and the angle of obliquity on a given inclined plane (with the plane of major principal stress).

 

 

ellipse-stress

CONSTRUCTION

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

(i)  Taking O as center, draw two concentric circles with radius equal to σ1 and radius equal σ2 respectively.( σ1 is the greater of the  principal stresses given, assumed)

(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 to cut 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 Will be resultant stress on the inclined plane and point Q will trace its locus which will be an ellipse for different inclinations of the inclined plane.

(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 value of angle of obliquity represents high shear stress on the inclined plane under consideration, may be a case of ductile material.

(iv) Lower value of angle of obliquity represents lower shear stress on the inclined plane under consideration, may be a case of brittle material.