Transcript Quadrilateral Elements

EMA 405
Quadrilateral Elements
Introduction


Quadrilateral elements are 4-sided, planar
elements
They can address plane stress, plane strain,
or axisymmetric analyses
◦ Plane stress - z=0 (for thin structures)
◦ Plane strain - z=0 (for thick structures)
◦ Axisymmetric will be addressed in the next
chapter

2-D triangular elements are related and can
be thought of as quad elements where two
nodes are coincident
Element Types in ANSYS
Constant Strain Triangle
 Option for Plane42 in ANSYS

u ( x, y )  c1  c2 x  c3 y
3
v( x, y )  c4  c5 x  c6 y
 x  c2
 y  c6
2
1
 xy  c3  c5
What if triangles have midside nodes
(linear strain triangle)?
u ( x, y )  c1  c2 x  c3 y  c4 x 2  c5 xy  c6 y 2
v( x, y )  c7  c8 x  c9 y  c10 x 2  c11 xy  c12 y 2
3
4
5
1
6
 x  c2  2c4 x  c5 y
 y  c9  c11 x  2c12 y
 xy  c3  c5 x  2c6 y  c8  2c10 x  c11 y
2
Plane2 element
in ANSYS
Bilinear quadrilateral element?
u ( x, y )  c1  c2 x  c3 y  c4 xy
4
v( x, y )  c5  c6 x  c7 y  c8 xy
3
1
2
 x  c2  c4 y
 y  c7  c8 x
 xy  c3  c4 x  c6  c8 y
Option in Plane42
element in ANSYS
Bilinear Quadrilateral (cont.)

This element does not permit curvature in bending
this
not
this
Quadratic quadrilateral element?
u ( x, y )  c1  c2 x  c3 y  c4 x 2  c5 xy  c6 y 2  c7 x 2 y  c8 xy 2
v( x, y )  c9  c10 x  c11 y  c12 x 2  c13 xy  c14 y 2  c15 x 2 y  c16 xy 2
 x  c2 x  2c4 x  c5 y  2c7 xy  c8 y 2
4
8
3
5
7
1
6
2
Plane82
element in
ANSYS
Improved bilinear quadrilateral
element
This is the default behavior of Plane42 in
ANSYS
 It adds curvature terms to the assumed
displacements to permit curvature in
bending

Von Mises Stress
Also known as equivalent stress
 A common model for the onset of yielding in
multiaxial stress states

1
e 
2


2
2
2
















6





x
y
x
z
y
z
xy
xz
yz
2
2
2
