Matrices Review Matri Multiplication : When the number of columns of the first matri is the same as the number of rows in the second matri then matri multiplication can be performed. Here is an eample of matri multiplication for two matrices Here is an eample of matrices multiplication for a matri When A has dimensions mn, B has dimensions np. Then the product of A and B is the matri C, which has dimensions mp.
Transpose of Matrices : The transpose of a matri is found b echanging rows for columns i.e. Matri A (aij) and the transpose of A is: A T (aij) Where i is the row number and j is the column number. For eample, The transpose of a matri would be: n the case of a square matri (mn), the transpose can be used to chec if a matri is smmetric. For a smmetric matri A A T
The Determinant of a Matri : Determinants pla an important role in finding the inverse of a matri and also in solving sstems of linear equations. Determinant of a matri Assuming A is an arbitrar matri A, where the elements are given b: Determinant of a matri The determinant of a matri is more difficult
nverse Matri For a matri the matri inverse is Eample: Cos A Sin Sin Cos A A Cos Sin Cos Sin Sin Cos Sin Cos A T A Cos Sin For a matri
XY Y X X ( ) [T ] Coordinate Transformations T ] [ ] [ T XY Y X Y
Theor of Matri Method for Stress Calculations in -D From equations of rotational transformation of ais, we obtain the following: T inversel Hence T Y T Y T AC Area A AB A BC A A X X B C
Ug and force equilibrium equation, we obtain epressions for stress transformations as follows: {} Σ{ F} F F F {} F F F AB BC AC {} A A A A {} A A Canceling area A out and pre-multipling b transformation T (where T T T the identit matri. The order of the matri multiplication does matter in the final outcome., we have
{} A A
For the forces in the X ais we will use the same procedure. Y Y X X B C D BD Area A BC A CD A { } { } {} {} {} Σ BD BC CD A A A A A A F F F F F F F
Combining the above epressions T T T or
State of Stresses in Three Dimensions K The general three dimensional state of stress consists of three unequal principal stresses acting at a point (triaial state of stresses). L The plane JKL is assumed to be a principal plane and is the principal stress acting normal to the plane. - J - Letα, β and γ are the angles between the vector and the, and ais respectivel and α l β m γ
Under equilibrium conditions ( ) ( ) l l l m m ( ) m l m l m l l l m m m As, l and m are different than ero (non-trivial solution) l m The determinant must be equal to ero Solution of the determinant results in a cubic equation in
The eigenvalues of the stress matri are the principal stresses. The eigenvectors of the stress matri are the principal directions. m l > > m l The three roots are the three principal stresses,,.,, and are nown as stress invariants as the do not change in value when the aes are rotated to new positions.
has been seen before for the two dimensional state of stress. t states the useful relationship that the sum of the normal stresses for an orientation in the coordinate sstem is equal to the sum of the normal stresses for an other orientation
( ) 6, A A ± α α ( ) ( ) A Cos A α Stress nvariants for Principal Stresses,, ma ma The solution are the eigenvalues of the stress tensor
Eample: determine the principal stresses for the state of stress (in MPa). Solution: The solution are the eigenvalues of the stress tensor; Substituting: ( 4) ( 4) ( 8) 4 4 ( ( 8) ) ( ( ( ) ) (( 4) ( 4)) ) 8 One solution -8MPa is a principal stress because and are ero, then the other two principal stresses are eas to find b solving the quadratic equation inside the square bracets for (4) ± 6 ( ) ± ( ) 4(4) 6MPa -6MPa
4 4 8 8,6 6,8, A 96.4 96.4Cos 96.4Cos 96.4Cos Cos(α ) (.5).86 8 (.5 6) 59.99 8 8 α.5 6. (.5 6) 79. 9 6 6 8
Eample : Determine the maimum principal stresses and the maimum shear stress for the following triaial stress state. Solution Stress _ Tensor [ ] 4 4 4 MPa 4 5 4 5 5-5 MPa 5 MPa -895 MPa
Solution to Eample 6-5.8 MPa 4 6.5 MPa Sigm a (MPa) - -8-6 -4-4 6 8-65. MPa -4-6 -8 Stress (MPa) ma 65.MPa / (65. 6.5MPa 5.8) 58.5MPa 5.8MPa
Mohr s Circles for -D Analsis Mohr s circles can mae visualiation of the stress condition clearer to the designer. Note that the principal stress values are alwas ordered b convention so the is the largest value in the tensile direction and is the largest value in the compressive direction. Note also that there is one dominant pea shear stress in this diagram. Be forewarned the principal stresses and this pea shear stress are going to pla a strong role in determining the factor of safet in mechanical design.
A Mohr s circle can be generated for triaial stress states, but it is often unnecessar, as it is sufficient to now the values of the principal stresses. The principal stresses must be ordered from larger to smaller.
Compare -D and -D Mohr s Circle. f is ero, does it have an effect in D?
Q P α α Consider then the plane will be an angle α from, in the direction of (clocwise). Point P Consider then the plane will be an angle α from, in the direction of (clocwise). Point Q The required sstem of stresses, fall within P and Q. Loci determined b the center in
β β Consider then the plane will be an angle β from, in the direction of (anticlocwise). Point R Consider then the plane will be an angle β from, in the direction of (clocwise). Point S The required sstem of stresses, fall within R and S. Loci determined b the center in
Eample: Use Mohr s Circle to obtain the principal stresses and maimum shear of a component subjected to the following stresses: 9tension tension 5compression 4ccw ccw counterclocwise
Stress on ANY nclined Plane (-D) S The stress on a plane (S) can be decomposed into its normal component (S n ) and its shear component (S s ). S S n s S s s n s f α, β and γ are the angles between the vector S n and the, and ais respectivel and α l β m γ then
[ ] [ ] T T T t can be proved that
Mohr s Circles for -D Analsis There is no eas Mohr s circle graphical solution for problems of triaial stress state. Solution for maimum principal stresses and maimum shear stress is analtical. Consider the, and ais to coincide with the ais of the principal stresses, and. f α, β and γ are the angles between the normal to the plane and the, and ais respectivel and α l β m l m γ We would lie to find graphicall the normal stress and shear stress on the plane.
Octahedral Plane and Stresses An octahedral plane is a plane that maes three identical angles with the principal planes. ( ) ( ) ( ) ( ) 6 9 n n n n n n n n op op op op m l m m l l m l
Mean and Deviatoric Stresses When describing the materials behavior of metals, one concludes that in certain cases some stress components pla a more important role than other components. Plastic behavior of metals, is reported to be independent of the average (mean) normal stress. M M M op M M Mean stress matri Deviatoric stress M M M The shear components do not change
D Deviatoric Stress Matri Deviatoric stresses pla an important role in the theor of plasticit. The influence the ielding of ductile materials. The principal stresses obtained onl from the deviatoric matri is M M M D P,
Eample For a given stress matri representing the state of stress at a certain point [ ] MPa Find the stress invariant, the principal stresses, the principal directions, the octahedral stress and the shear stress associated with the octahedral stress. Solution: 5 5 5 5.4. 5.4
( ) ( ) ( ) 5.4 5.4 5.4 m l m l m l.4. 5.4 [ ] MPa ( ) ( ) ( ) m l m l m l.67.48.657 m l m l.555.8 m l.546.64.74 m l 5 Mean 9 8 8 5) ( 6 5 9 6 9 op op op