Circular Representation of State of Stress Assignment Help

Assignment Help: >> Principal Stresses and Strains - Circular Representation of State of Stress

Circular Representation of State of Stress:

From a state of stress defined by the components sx, sy and txy, we express the stress components on an arbitrary plane as

 2091_Circular Representation of State of Stress.png

                       t= - (sx-sy )/2)sin 2q + txy cos 2q

which we might rewrite in general as

 

s =  a + b cos 2q + c sin 2q       ---------(i)

                t = - b sin 2q + c cos 2q        -------------(ii)

Now Let us try to establish direct relationship between s and t by eliminating q between

Eqs. (i) & (ii).

To make simplify the effort, let us take the origin of coordinate at (a, 0), so that the new variable   = (s - a) is let for building the relationship.

 357_Circular Representation of State of Stress1.png--------------(iii)

t = - b sin 2q + c cos 2q   -------------- (iv)

By Squaring & adding, we obtain

561_Circular Representation of State of Stress2.png

+ b2  sin 2  2q + c2  cos2  2q - 2bc cos 2q sin 2q

= b2  (cos2  2q + sin 2  2q) + c2  (sin 2  2q+ cos2  2q)

i.e.

87_Circular Representation of State of Stress3.png

Since b and c are constants let b2  + c2  = r 2

∴        2209_Circular Representation of State of Stress4.png

Eq. shows that if s and t, the normal and shear stress components on any arbitrary plane are plotted as coordinates, the locus of the point will be a circle whose centre will

(sx  + s y)/2     , 0  and radius will be

1727_Circular Representation of State of Stress5.png

In other terms, the state of stress at particular point might be represented by a circle & a point on the circle represents the normal & shear stress components, on some plane as horizontal & vertical coordinates.

Analysis of State of Stress Interpretation of the state of stress
Free Assignment Quote

Assured A++ Grade

Get guaranteed satisfaction & time on delivery in every assignment order you paid with us! We ensure premium quality solution document along with free turntin report!

All rights reserved! Copyrights ©2019-2020 ExpertsMind IT Educational Pvt Ltd