Macromechanical Analysis of a Lamina: Definitions Review and Stress Analysis
Explore the Macromechanical Analysis of a Lamina with Dr. Autar Kaw from the University of South Florida, covering definitions, stress analysis on infinitesimal areas, forces on various planes, and displacement equations. Learn about stresses on infinitesimal cuboids, normal and shearing strains, and displacement in different directions for a comprehensive understanding of laminas in mechanical engineering.
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Chapter 2 Macromechanical Analysis of a Lamina Part 2 Review of Definitions Dr. AutarKaw Department of Mechanical Engineering University of South Florida, Tampa, FL 33620 Courtesy of the Textbook Mechanics of Composite Materials by Kaw
lim P n , = n 0 A A lim P s = s 0 A A FIGURE 2.5 Stresses on infinitesimal area on an arbitrary plane
P = lim A , x x 0 A lim P y , = xy 0 A A lim P z = xz 0 A A FIGURE 2.6 Forces on an infinitesimal area on the y-z plane
, = xy yx , = yz zy = zx xz FIGURE 2.7 Stresses on an infinitesimal cuboid
FIGURE 2.8 Normal and shearing strains on an infinitesimal area in the x-y plane
u = u(x,y,z) = displacement in x-direction at point (x,y,z), - AB lim A B = v = v(x,y,z) = displacement in y-direction at point (x,y,z), x 0 AB AB w = w(x,y,z) = displacement in z-direction at point (x,y,z) Where: = P 2 + 2 ( ( A B A P B ) ) 2 2 u + x u [ ( ) ( ] ) [ ( - ) ( ] ) = + x x, y x, y + v + x x, y v x, y , AB = x
Substituting 1 2 / 2 2 lim ( ) ( ) ( ) ( ) u x + + - u x,y v x + + - v x,y 1 1 = + + - x 0 x x x 1 2 / 2 2 u v 1 1 = + + - x x x u u 1 < < = x x x v 1 < < x
u = u(x,y,z) = displacement in x-direction at point (x,y,z), - AD lim A D = v = v(x,y,z) = displacement in y-direction at point (x,y,z), y 0 AD AD w = w(x,y,z) = displacement in z-direction at point (x,y,z) Where: = D 2 2 ( ( A D A Q + Q ) ) 2 2 [ ( ) ( ] ) [ ( ) ( ] ) = v x,y + , v x,y + u x,y + , - u x,y , y + AD = y
Substituting 1/2 2 2 lim ( v - ) ( )} ( u - ) ( ) v x, y + y x, y u x, y + y x, y = y + 1 + - 1 y 0 y y 1 2 / 2 2 v u 1 1 = + + - y y y u 1 v < < y = y y v 1 < < y
= + xy 1 2 Where: lim P B = , 1 0 AB A P B v = ( - ) ( ) P + x x, y v x, y , u = u - x ( ) + ( ) A P + x x, y x, y
= + xy 1 2 Where: D lim Q = , 2 0 AD A Q u = D ( u - ) ( ) Q x, y + y x, y , = + y) v(x, - y A Q v(x, y + y)
( u - ) ( ) u x, y + y x, y ( v - ) ( ) v + x x, y x, y Substituting ) y ) x lim + ( ( ) ( ( ) u + x x, y + - x u x, y v x, y + y + y v - x, y = xy x 0 x y y 0 u v y x = + xy u u + + 1 1 x y u 1 < < x v v u = + xy 1 < < x y y
FIGURE 2.9 Cartesian coordinates in 3-D
1 0 0 0 - - E E E 1 0 0 0 - - E E E x x y 1 0 0 0 y - - E E E z z = 0 0 0 1 0 0 yz yz G zx zx 0 0 0 0 1 0 xy xy G 0 0 0 0 0 1 G
( ) E 1 - E )( E )( 0 0 0 ( )( ) ( ) ( ) 1 - 2 1 + 1 - 2 1 + 1 - 2 1 + x x E )( E(1 - ) E )( 0 0 0 y y ( ) ( )( ) ( ) 1 - 2 1 + 1 - 2 1 + 1 - 2 1 + z z = E )( E )( E(1 - ) 0 0 0 yz yz ( ) ( ) ( )( ) 1 - 2 1 + 1 - 2 1 + 1 - 2 1 + zx zx 0 0 0 G 0 0 xy xy 0 0 0 0 G 0 0 0 0 0 0 G Where: E G = + 1 ( ) 2
1 ( ) W = x + y + z + + + x y z xy yz zx xy yz zx 2
