Stress, by definition, is a vector as zuch in tbe real world one would expect 3 dimensions o be required o fully model a sfess field. Hower/er, sDe can often reduce complexsystems o a 2 dimensional tress ield where it is assumed hat the stress n one direction is zero. In this handout he classic anatysis of a 2D stress ield will be examined, n particular he application of Mohr's stress circle to the uo6"1513ading f 2D stress. Further more firndamental xamples f the application of 2D stress nnalysis wiii be presented. 2D Stress heory Consider a rectangular element, the applied forces to the structure (of which the element s a part) give rise o two orthogonal direct stresses a nominally horizontal direct stress, a nominally vertical direa stress and a shear stress as llustrated n Figure l). Note that the orientation of the triangle ABC may be rotated zuch that it fits the criteria above, he criteria does not mean hat the triangle ABC can only lie in one direction. Further more, consider a plane at any angle 0 relative o horizontal plane (note hat of signifies he shess acting on the y plane andoy signifies he stress acting he x plane). t should be noted that o;, o, and rry are sfresses hat have been directly applied o the strucfure, og and re a.re tresses hicb arise rom that loading on ctDl plane. If it is assumed hat the triangle ABC has unit width and tlur the hypoteneus s unit lengtlr, then the area cjf AC is l, the area of AB is cosO and the area of BC is sixCI. Utilising Newton's second aw of motion" and in partiorlar d'Alembert's expressions or og and r0 may be obtained. Figure 1 - 2D stress model -rl
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( o , - o , ) .- l ' ' l s i n 2 9 - t co s? A| . , . 2 0 - r c o s 2 A . . . . . . ( 2 )\ z )
(The readershouldattempt o proveboth equations and2)
Upon close exarninatiorgquation1 revealrthat it'is"not"the complex'equationt
seems. ndeed, t is but an eguation or a circle (as llustrated n Figure 2) whose ures
are t and o, whose centre
lies on ((o*+or)/2,0).
Note that within the circle,
the anglss are twice what
they are in the real
structure (2e), further
more note that the angles
are measured from a
different datum to that
normd$ orperienced,hatis they aremeasuredrom
the o* plane.
It is interesting to note
tlat the combinationof the two orthogonal stressanda shearstresshas ncreasedhe
direct stresses ctingon the system.This fact is highlighted y the horizontaloris, this
is the direct stressacting on the syster4 and anyplanealigningwith this aris bas zero
shear stressaaing on it but ma,ximumdirect stresses, ny two orthogonalplanesaligning with this axis are called the PRINCIPAL PIIWES, and the dLect stress o1
Considera drive shaft,a tran$nissionshaft in a car or a PTO shaft n a tractor for
example. f the shaft has an externaldiarneterof 50 *, is constructed rom thickwalled tubing whereF7 mm and is transmitting 6 kW at 3600 rev/min; determine he
principalstressesn the system.
A circular shaft in torsion may be modelledusing the torsion eguation, his may be
rearrangedo yield :-
t : TrlI
T = P / w =6000 = 15.92Nm
3 6 0 0 x 2 x n / 6 0
. zr(o4-6+)z(.054-.0364)r :
32 r/-
therefore, heappliedshearstresso the zurfaceof the shaft s15 .92.025
t:ff i :0.887 MN/m2
It is noted hat for a systemactingunder orsion only, the systems in pureshear,hatis o" andoy areboth zero. An examination f equation4 with this informationshows
that the principlestresses r€ o1 - '1-!,e'2: 0 ando3 : -1. An examination f equation3 shows hat he principalplanesie at 45o to the x-y planes. hat s as llustratednFieure l . l
Figure l.l
This correspondso a stress ircleas llustrated n FigureBl.2
3. A thin walled steel tube of 150 mm internaldiameterand wall thicknes2mm
containsa gaspressurisedo I I bar. Determine he tensilestressand shearstress
actingon a helical seem nclinedat25o o the cross ection.(24.4&7.92MN/m2)
4. A shortcolumn s loadedaxiallywith a compressiveorceof l5kN. The column s
also subjectedo a torsional oad of 150 Nm. ffthe column is constnrcted rom ahollow steel ube of 250 mm internal diameterand wall thickness25 mm determine
a) Thecompressivend shearstressesctingon a rectangular lement aligned
with the long axisof the column)on he surfaceof the column.
b) Theprincipal stresses.c) The maximrrm shearstress.
1. Astateof two-dimensionalstra in ise*=70010{,er= -600 10-6 ndy"y 300x 10-6.Calculateheprincipalstrainsn magnitude nddirection.Check he results singMohr'scircle construction.
[Ans: 717x 10-6,617x 10-6]
2. The followingstrainsare ecorded y wo straingauges,heiraxesbeingat right angles:
E*= 390x 10-6; v= -!20 x 10{. Find he values f the stresses* ando, actingalongthese xes f the elevant lastic onstantsreE = 208GN/m2andv = 0.3.
[Ans: 80.9MNVm2;-0.69MN/m2]
3. The followingstrainswere ecorded n a rectangulartrain osette:-
E " = 4 5 0 x10 -6 , u=230x1 0 { ;e = 0
Determine:-
(a) theprincipa-ltrains nd hedirections f theprincipal trainaxes;
(b) theprincipalstressesf E = 200GN/m2 ndv = 0.3
[Ans: 450.05x 10-6 t i" clockwiserom A, -0.055 105at 91" clockwiserom A; 98
MN/m2l
4. Thevaluesof straingiven n problem were ecorded n a 60" delta osettegauge.Whata-re ow thevaluesof theprincipal tninsand heprincipal [esses.Check heanalyticalsolutionby means f Mohr's circles.