In 3d state of stress , the independent stress component required to define state of stress at a point are
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Advanced Mechanics of Materials and Applied Elasticity: Analysis of Stress
By Ansel C. Ugural and Saul K. Fenster
Jul 20, 2011
Contents
␡
1.1 Introduction
1.2 Scope of Treatment
1.3 Analysis and Design
1.4 Conditions of Equilibrium
1.5 Definition and Components of Stress
1.6 Internal Force-Resultant and Stress Relations
1.7 Stresses on Inclined Sections
1.8 Variation of Stress Within a Body
1.9 Plane-Stress Transformation
1.10 Principal Stresses and Maximum in-plane Shear Stress
1.11 Mohr's Circle for Two-Dimensional Stress
1.12 Three-Dimensional Stress Transformation
1.13 Principal Stresses in Three Dimensions
1.14 Normal and Shear Stresses on an Oblique Plane
1.15 Mohr's Circles in Three Dimensions
1.16 Boundary Conditions in Terms of Surface Forces
1.17 Indicial Notation
References
Problems
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This chapter is from the book
This chapter is from the book
Advanced Mechanics of Materials and Applied Elasticity, 5th EditionAdvanced Mechanics of Materials and Applied Elasticity, 5th Edition
Learn More Buy
This chapter is from the book
Advanced Mechanics of Materials and Applied Elasticity, 5th EditionAdvanced Mechanics of Materials and Applied Elasticity, 5th Edition
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1.13 Principal Stresses in Three Dimensions
For the three-dimensional case, it is now demonstrated that three planes of zero shear stress exist, that these planes are mutually perpendicular, and that on these planes the normal stresses have maximum or minimum values. As has been discussed, these normal stresses are referred to as principal stresses, usually denoted s 1, s 2, and s 3. The algebraically largest stress is represented by s 1, and the smallest by s 3: s 1 > s 2 > s 3.
We begin by again considering an oblique x' plane. The normal stress acting on this plane is given by Eq. (1.28a):
Equation a
037equ01.jpg
The problem at hand is the determination of extreme or stationary values of s x' . To accomplish this, we examine the variation of s x' relative to the direction cosines. Inasmuch as l, m, and n are not independent, but connected by l 2 + m 2 + n 2 = 1, only l and m may be regarded as independent variables. Thus,
Equation b
037equ02.jpg
Differentiating Eq. (a) as indicated by Eqs. (b) in terms of the quantities in Eq. (1.26), we obtain
Equation c
037equ03.jpg
From n 2 = 1 – l 2 – m 2, we have u2202.gifn/u2202.gifl = –l/n and u2202.gifn/u2202.gifm = –m/n. Introducing these into Eq. (c), the following relationships between the components of p and n are determined:
Equation d
037equ04.jpg
These proportionalities indicate that the stress resultant must be parallel to the unit normal and therefore contains no shear component. It is concluded that, on a plane for which s x' has an extreme or principal value, a principal plane, the shearing stress vanishes.
It is now shown that three principal stresses and three principal planes exist. Denoting the principal stresses by s p , Eq. (d) may be written as
Equation e
037equ05.jpg
These expressions, together with Eq. (1.26), lead to
Equation 1.31
01equ31.jpg
A nontrivial solution for the direction cosines requires that the characteristic determinant vanish:
Equation 1.32
01equ32.jpg
Expanding Eq. (1.32) leads to
Equation 1.33
01equ33.jpg
where
Equation 1.34a
01equ34a.jpg
Equation 1.34b
01equ34b.jpg
Equation 1.34c
01equ34c.jpg
The three roots of the stress cubic equation (1.33) are the principal stresses, corresponding to which are three sets of direction cosines, which establish the relationship of the principal planes to the origin of the nonprincipal axes. The principal stresses are the characteristic values or eigenvalues of the stress tensor t ij . Since the stress tensor is a symmetric tensor whose elements are all real, it has real eigenvalues. That is, the three principal stresses are real [Refs. 1.8 and 1.9]. The direction cosines l, m, and n are the eigenvectors of t ij .
It is clear that the principal stresses are independent of the orientation of the original coordinate system. It follows from Eq. (1.33) that the coefficients I 1, I 2, and I 3 must likewise be independent of x, y, and z, since otherwise the principal stresses would change. For example, we can demonstrate that adding the expressions for s x' , s y' , and s z' given by Eq. (1.28) and making use of Eq. (1.30a) leads to I 1 = s x' + s y' + s z' = s x + s y + s z . Thus, the coefficients I 1, I 2, and I 3 represent three invariants of the stress tensor in three dimensions or, briefly, the stress invariants. For plane stress, it is a simple matter to show that the following quantities are invariant (Prob. 1.27):
Equation 1.35