Plane Strain/Stress Elements

Question 1

Plane stress

Answer 1

applied to solids that are loaded i plane so the out of plane stresses are negligible
- sigma_z = tau_xz = tau_yz = 0
- displacements are only a function of x and y

Question 2

Plane strain

Answer 2

solids loaded in plane and restrained in the out of plane direction, such as a slice of a 3D body
- w=0 therefore epsilon_z = gamma_xz = gamma_yz = 0

Question 3

Plane stress/strain similarities

Answer 3

both are 2D problems with direct strains and stresses in x/y and the xy all non zero
-affects [D] with that for isotropic materials used in the notes
-for a poissons ratio greater than zero plain strain is usually stiffer than plane strain

Question 4

Continuity

Answer 4

Required of the displacement field for it in the stresses/strains
-C0 continuity refers to the 0th derivative
-achieved by having shape functions that are C0 continuous through out the element domain and at the element boundaries

C0 continuity
-external nodes shouldnt cause displacements of the opposite edge
- internal nodes shouldnt cause displacement of element edges

Question 5

Constant strain triangles shape functions

Answer 5

• local axis reference system should have x-axis going through 2 nodes
• shape functions are linear since there are 3 conditions on u(x,y) and v(x,y), linearity from pascals triangle

u(x,y) = a0 + a1x + a2y

• solve for ai in the form of u1,u2,u3 so that u(x,y) can be written as a linear summation of the u’s

Question 6

Constant strain triangle limitations

Answer 6

{epsilon} = [B]{d}
-{d} is the nodal disp. while {u} is the displacement field
-[B] is independent of x/y hence the name constant stress
* Bad for large change in strain which comes with irregular geometry and non-uniform loading. Can still be used with large refinement but must check the smoothness of the strain solution

Question 7

Applying constant strain triangle equations at an element level

Answer 7

-t and A can be functions of x/y
*applying a unit t avoids it specification for plane stress/strain and constant thickness
*applying the principle of virtual work the integrands become constant for constant t and D

-switch to global axis using [T]
{d}=[T]{u}
{R}e=[T] transposed {t}

-often used to change from a fine to a coarser mesh

Question 8

Rectangular elements

Answer 8

Now 4 conditions so an additional term from Pascals triangle is used

u(x,y) =