Na An

in indicial notation this can be rewritten as nr Qrs — 'A.r^b s

In

litial (X1)

SH S11=tn1

Plane

Principal Plane

Figure 2.5: Principal Stresses or

in matrix notation this corresponds to n ([ct] — A[I]) = 0 (2.28)

where I corresponds to the identity matrix. We really have here a set of three homogeneous algebraic equations for the direction cosines n.

27 Since the direction cosines must also satisfy n1 + n2 + n| = 1 (2.29)

they can not all be zero. hence Eq.2.28 has solutions which are not zero if and only if the determinant of the coefficients is equal to zero, i.e

an — A

^12

<713

021

0"22 — A

^23

=0

(2.30)

^31

^32

^33

— A

arS —

ASrs

|=0

(2.31)

— AI

=0

(2.32)

28 For a given set of the nine stress components, the preceding equation constitutes a cubic equation for the three unknown magnitudes of A.

29 Cauchy was first to show that since the matrix is symmetric and has real elements, the roots are all real numbers.

30 The three lambdas correspond to the three principal stresses a(i) > a(2) > a(3). When any one of them is substituted for A in the three equations in Eq. 2.28 those equations reduce to only two independent linear equations, which must be solved together with the quadratic Eq. 2.29 to determine the direction cosines n®r of the normal n® to the plane on which a® acts.

31 The three directions form a right-handed system and n3 = nixn2 (2.33)

32 In 2D, it can be shown that the principal stresses are given by:

0 0

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