The Basic Theory of the Space ProblemS7-3ThePrinciple StressThe Maximum and Minimum StressSet v as the direction of the principle stress, and , is the principlestress, so: T, =0Px =lo,Px=lox+mtyx+ntzxP, =mo,P, = ltxy +mo, + ntyPz =no,P,=ltxz+mtyz+nozI(ox -,)+mTyx +ntx =0LyT0x-0,=0Itxy +m(o, -0,)+ ntzy =0X0,-,?Tx0.ltxz +mt yz +n(o, -o,)= 0-[?+m?+n?=111
11 x x m yx n zx p = l + + y xy m y n zy p = l + + z xz m yz n z p = l + + Set v as the direction of the principle stress, and v is the principle stress, so: v =0 x v p = l py = m v pz = n v ( − ) + + = 0 x v m yx n zx l + ( − ) + = 0 xy m y v n zy l + + ( − ) = 0 xz m yz n z v l 1 2 2 2 l + m + n = 0 x v yx zx xy y v zy xz yz z v σ σ τ τ τ σ σ τ τ τ σ σ − − = − §7-3 The Principle Stress The Maximum and Minimum Stress
The Basic Theory of the Space ProblemTyax-a,T yx= 0T xyTzy9,-a,TxzTyzz-av013- Io, + I20,-I3 = 00203The invariants of the stress deviator tensor:Ii=0x+0, +αzI2 =ox,+o,,+0,x-Ttxz7[I,=0x0yo,+2tgtytx-0xt,-0yta-0,tgI=01+02+03I2=0102+0203+0301I3=01020312
12 = 0 − − − xz yz z v xy y v zy x v yx yz 2 3 0 2 1 3 v − I v + I v − I = 3 2 1 x y z I1 = + + 2 2 2 2 x y y z z x xy yz zx I = + + − − − 2 2 2 3 x y z 2 xy yz zx x yz y zx z xy I = + − − − The invariants of the stress deviator tensor: 1 = 1 + 2 + 3 I 2 = 1 2 + 2 3 + 3 1 I 3 = 1 2 3 I
The Basic Theory of the Space ProblemThe characteristic equation has three real roots, which representthe three principle stress of some pointFor the principle direction of the stress, substituting the i, 02, 03into the following three formulas:(ox-o)l+txm+txn=0T,l+(o, -o)m+tyμn=0txl+tμm+(o, -o)n=0and[2+m2+n2=1Then we can get the three principle stress directions13
13 The characteristic equation has three real roots, which represent the three principle stress of some point. and l 2+m2+n2=1 Then we can get the three principle stress directions. ( ) 0 ( ) 0 ( ) 0 + + − = + − + = − + + = l m n l m n l m n xz yz z xy y yz x xy xz For the principle direction of the stress, substituting the 1, 2, 3 into the following three formulas:
The Basic Theory of the Space ProblemMaximum shear stress020202011103030302-031-021-03TimaxT3maxT2max222Q1-3Maximum shear stress=Lmax214
14 Maximum shear stress 1 2 3 2 3 1max 2 − = ➢ Maximum shear stress 1 2 3 1 2 3 1 2 3max 2 − = 1 3 2max 2 − = 2 1 3 max − =
The Basic Theory of the Space ProblemThe property of stress invariant>Theprincipal stressandtheprincipaldirectionofstress> Property ofdependontheexternalforceandconstraintconditionsoftheinvariancestructure, and they are independent of the coordinate system> The three roots of the characteristic equation are determined.> The characteristic equation has three roots, that is, the> Property ofthree principal stresses of a point are all real Numbers.realnumbers> It can be proved by the properties of the cubic equation> Property ofAt any point, three different principal stress directionsorthogonalitare perpendicularto each other.y> The change of coordinate system leads to the change ofthe stress componentsbutthestress stateremainsthesame.> The stress invariant is the description of the stress state property15
15 ➢ The principal stress and the principal direction of stress depend on the external force and constraint conditions of the structure, and they are independent of the coordinate system. ➢ The three roots of the characteristic equation are determined. ➢ The characteristic equation has three roots, that is, the three principal stresses of a point are all real Numbers. ➢ It can be proved by the properties of the cubic equation. ➢ At any point, three different principal stress directions are perpendicular to each other. The property of stress invariant ➢ The change of coordinate system leads to the change of the stress components, but the stress state remains the same. ➢ The stress invariant is the description of the stress state property. ➢ Property of invariance ➢ Property of real numbers ➢ Property of orthogonalit y