)12.74 At a point in a stressed body, the principal stresses are

iented as shown inFrg.Pl2.74. Use Mohr's circle to determine:

.l the stresses onplane a-a.

-- ) the stresses on the horizontal and vertical planes at the point.

, r the absolute maximum shear stress at the point.

a

/ 2oo MPa

T

er: I tu"-'(i)

FIGURE P12,74

>:

12.10 c¡ru¡nAL srATE oF srREss ATA PoINT

1e general three-dimensional state of stress at a point was previously introduced in

::ction I2.2. This state of stress has three normal stress components and six shear

.:ress components, as illustrated in Fig. 12.11 .The shear stress components shown on.s. 12.11 are not all independent, however, since moment equilibrium requires that

Iyr:fxy Ty7:Tzy T*z:Tu

-he stresses shown in Fig. 12.11 are all positive according to the normal and shear stress

.-sn conventions outlined in Section 12.2.

NORMAL AND SHEAR STRESSES

:rpressions for the stresses on any oblique plane through the point, in terms of stresses on

.re reference x, y, and z planes, can be developed with the aid of the free-body diagram

::rowfl i1 Fig. l2.I8a. The n axis is normal to the oblique (shaded) face. The orientation

n axis

(a) Tetrahedron free-body diagram (á) Orientation of n axis defined

FIGURE 12.17

J:I

I

I

:TGURE 12.18 Tetrahedron for deriving principal stresses on an oblique plane

488sTRESs TRANSFORMATIONS

The terms:

/: cosam= cosB

,?: cosyare called direction cosines.

Fr: & dA: ordA.l + ty*dA.m* rudA.nFr : Sy dA : on dA. m -t t, dA. n * r, dA. IF, : S, dA : o, dA . n * r* dA . t + .cu, dA . m

of the n axis can be def,ned by. three angres a, §, and y, as shown inFig. r2.rgb. The are¿of the oblique face of the tetrahedral erement is defined to be dA. Area"s of the ¿ y, and :faces are thus dA cosa, dAcos p, and dAcosy, respectivery: rrr"'r*riánt force F on ü;oblique face is s d4, where s is the resultant stress on the area. The resultant stress s::related to the normal and shear stress components on the oblique face by the expression;

(r2.2:

tt¡. and

Expanspal stre

rvhere

The conlvere displane srare the p

of nume;

Fo¡STTCSSCS

the relar

sir,es thr;edin_e C.

for the :¡Eqr-

The forces on the ¿ y, and z faces are shown as three components, the magnitude of eac:being the product of the area by the appropriate stress. If we use r, m, andÍ to represen:cosa, cosB, and cos7, respectively, the firce equilibrium equations in the ¿ y, and :directions are

./ + ,S) .m I Sr.a; thereforeany oblique plane through thc

tiIUI

i:(

I rhese ¡elationships can be established by considering the volume of the tetrahedron in Fig. 12.1ga. The , r.lume Vof thetetrahedroncan.be,expressed^a1 v: iarae: +i;;,:-!";;dAr"= !aza,t,.Ho*¿:the distance dn from the origin to.the center ofthe obli{ue face can also be expressed as dn : dx cos u = d., :,0 = dzcos 1. Thus, the areas ofthe tetrahedron fu."s "un

be expressed as dA,: d.l cos a', alf,J; ;; ;.dAr: dAcosy.

from which the three orthogonal components of the resurtant stress are

&:ar.l*.cyx.rfl*tzr.nSy=?ra.l*on.m*,cr.nSr:rrr.l*qr.m*or.n

The normal component on of the resultant stress ,S equals S,from Eq. (a), the following equation of the normal stress onpoint is obtained:

(12:5

(o, - or)l ! Íy*m * run: O

(or-or)m*ron*rrl:0(o, - or)n * rrrl * rrrm: 0

This set of equations has a nontrivial solution only if the determinant of the coefflcients ¡

The shear stress ?,r on the oblique plane can be obtained from the relation §: o] + t:For a given problem, the values of ,s and onwilbe obtained r."* nqr. trr and (12.25).

MAGNITUDE AND ORIENTATION OF PRINCIPAL STRESSESA principal plane was previously defined as a plane on which the shear stress r,, is zero. T:¡:normal stress o, on such a plane was defined as a principal stress d.p. Irthe oblique plane :;Fig. 12.18 is a principal pla.ne, then S : o, and S. : o)1, Sn : ooá, S, : oon. When theocomponents are substituted into Eq. (a), the equations

"u, u"i"*rl'tt"n to produce the follo; -ing homogeneous linear equations in terms of the direction cosines t, m, ind n:

\ andn is equal to zero. Thus, 489GENERAL STATE OF STRESS

ATA POINT(r2.26)

=.rpansion of the determinant yields the following cubic equation for determining the princi-ral stresses:

4-rr4*I2oo-\:o (12.21)

It:o**orlo,12: oro, * oro, * oro, - ti - ,j, - ,',13: 0*o16r*2rrryxrzx- b*r3r* ort2u+ ozTz-y)

The constants 11, 12, and 13 are stress invariants. Recall that stress invariants for plane stress The roots ofBq. (12.27) canbe;i ere discussed in Section 12.6 and that the invariants 11 and 12 were given in Eq. (12.9) for readily estimated by plotting a

:lanestresswhereo.: tyr: ru:0. Equation (12.27) alwayshasthreerealroots,which graphoftheleft-handsideof the

rre the principal stresses at a given point. The roots of Eq. (12.21) can be found by a number equation as a function of o.

-rf numerical methods.For given values of ox,6y,...,rz,Eq. (12.27) gives three values of the principal

itresses op1, op\> and ooz. Substituting these values for or, in turn, into Eq. (b) and usingüe relation

12+m2*n2:1 (")

-¡ives three sets of direction cosines for the normals to the three principal planes. The pre-.'eding discussion verifies the existence of three mutually perpendicular principal planes:or the most general state of stress.

Equation (b) can also be rewritten in matrix form as

Observe that the trivial solution (l : m : n: 0) is not possible for this equationsince the direction cosines must satisfy Eq. (c). This equation can be solved as astandard eigenvalue problem. The three eigenvalues correspond to the three principalstresses art opz, and opz. The eigenvector that corresponds to each eigenvalue con-sists of the direction cosines {1, m, n} of the normal to the principal plane. In devel-oping equations for maximum and minimum normal stresses, the special case will beconsidered in which x,r: 'tyz: Tzr:0. No loss in generality is introduced byconsidering this special case, since it involves only a reorientation of the reference ¿_1, z axes to coincide with the principal directions. Given that the x, y, and z planesare now principal planes, the stresses ox, oy, o. become opr 6p2, and ooz.SolvingEq. (a) for the direction cosines yields

l(o- - or¡ ry* rzx

| ,, loy - op) rzy

| ,*, ry, (o, - op)-0

(t2.28)

l'".i:' oi!,,s ,,:i,) ]{r}: {[]

,s,opl

s)m:¿O^)

s.6"1

492STR ESS TRANSFORMATIONS

P12.77 At a::- = 60MPa -

-120MPa.: -_.tr l the norma

normal is o:.y, and ; ar;.

: r the principaat the poin:

P12.78 Ar a ;,::0,o.:[.- S ksi. Deterru.-. Ihe norm¿l

lormal n:¡¡,:- :e princi:.'

:i rhe poiil:

,i2.79 -\r . ¡-" = -l\fPa T.

- ._ lr

. *13 nLr=l: :--* i _ _. ,._tá l-,-r-::- -:= ^;--.--_-_ fL.r.!._a I

": ihe p...-::

3"2.80 .{: " :,-- = Ái,¡ \l|. I-* l.fP:.. - = -

Assume ort ) oo2> 06 > 0

(a) Principal stress element

FIGURE 12-20

(á) Mohr's circle

The largest magnitude from these three possible results is ?u6. -u^; hence, the ab::.r.ii*maximum shear stress can be expressed as

---"*-o*¡, Ij'o'": :_

_ 2_.- ;- --

which confirms Eq' (12.18) regarding the absolute maximum shear stress magnitude Thrumaximum shear stress acts on the plane bisecting the angle between the maximur ¡1uu

minimum principal stresses.

APPLICATION OF MOHR'S C¡RCLE TO THREE.DIMENSIONALSTRESS ANALYSIS

In Fig. 12.20a, the principal stresses opt, opz, and oú at a point are shown on a stres: rtrment. we will assume that the principal stresses have been ordered so that oot ) -t-opz and that all three are greater than zero. Furthermore, observe that the prinóipal ;.ii:m*represented by the stress element are rotated with respect to the -r-)-¿ axes. Usl:. mru

three principal stresses, Mohr's circle can be plotted to visually represent the r": r,uuri,

stress combinations possible at the point (Fig. l2.2ob). Stress combinations for alj ¡-",,,i¡,,ble planes plot either on one of the circles or in the shaded area. From Mohr's circ-: mrabsolute maximum shear stress magnitude given by Eq. (12.30) is evident.

fabs max

Stress combination on arbitrary plane

PROBLEMS

P12.75 At a point in a stressed body, the known stresses are;, : 4OMPa(T), or, : 20MPa(C), o,:2QMPa(T), r,, :- +0 MPa, r): : 0, and r u : + 30 MPa. Determine:¡ r the normal and shear stresses on a plane whose outward

normal is oriented at angles of 40",75', and 54. with the ¿ y,and ¿ axes, respectively.

r r the principal stresses and the absolute maximum shear stressat the point.

P'12,76 At a point in a stressed body, the known stres;:. ¡Nrr

o,: l4ksi(T), or: l2ksi(T), o.: 10ksi(T), r,,: *4k:--4 ksi. and t* : 0. Determine:(a) the normal and shear sffesses on a plane whose c-:.n,uu@:

normal is oriented at angles of 40.,60", and,66.2 uir_ -ru ,,,,

y, and z axes, respectively.(b) the principal stresses and the absolute maximum she¡: .ryrsr.

at the point.

492STRESS TRANSFORMATIONS

Assume op1 >opz>orl>0

(a) Principal stress element

FIGURE 12.20

1á) Mohr's circle

The largest magnitude from these three possible results is ro6. -u*; hence, the ¡:'" ruruü'

maximum shear stress can be expressed as

which confirms Eq. (12.18) regarding the absolute maximum shear stress magnii-:: -iiltff

maximum shear stress acts on the plane bisecting the angle between the maxi;-----r ,lrmttttt

minimum principal stresses.

APPLICATION OF MOHR"S CIRCLE TO THREE-DIMENSIONALSTRESS ANALYSIS

In Fig. 12.20a, the principal stresses 6ot, 6pz, and opz at a point are shown on a s-::. ::rlrLrr,

ment. We will assume that the principal stresses have been ordered so that 6pt ) '06 and that all three are greatü than zero. Furthermore, observe that the princip- : -rue

represented by the stress element are rotated with respect to the x-y-¿ &XeS. L:,: lllllfir

three principal stresses, Mohr's circle can be plotted to visually represent the ir iluri

stress combinations possible at the point (Fig. 12.20b). Stress combinations for -- : 'r,rn,

ble planes plot either on one of the circles or in the shaded area. From Mohr's c.:: : fllttr.

absolute maximum shear stress magnitude given by Eq. (12.30) is evident.

\--l

J]j't"

Tabs max

Stress combination on arbitrary plane

PROBLEMS

P12,75 At a point in a stressed body, the known stresses are

o*:40MPa(T), oy : 2OMPa(C), oz:21MPa(T), r,r, :+40 MPa, ryz : 0, and r - : +30 MPa. Determine:(a) the normal and shear stresses on a plane whose outward

normal is oriented at angles of 40', 75', and 54o with the ¿ ),and z axes, respectively.

(b) the principal stresses and the absolute maximum shear stress

at the point.

P12.76 At a point in a stressed body, the known sr.¡., iumi

o, : 14 ksi (T), or : 12 ksi (T), o. : 10 ksi (T), t,, : -1- j r ,

-4 ksi. and r- : 0. Determine:

(a) the normal and shear stresses on a plane whose - -. , lltffll

normal is oriented at angles of 40", 60", and 66.2' \,,. -: r'ly, and z axes, respectively.

(b) the principal stresses and the absolute maximum sh;.: -"rw

at the point.