Welcome to Blogs @ Andrew Qu
Blog Index
All blogs
Search results

Triangular Element for Plate Bending - Part 2

Summary

Describes strain and stress calculation for a simple triangular element for thin plate bending.

Strain, Stress and Moment

Continue from Part 1, we can now define strains as

{ε} =    ∂2w/∂x2
 ∂2w/∂y2
2∂2w/∂x∂y

Remember from Area Coordinates that

L1
L2
L3
= a1
a2
a3
b1
b2
b3
c1
c2
c3
1
x
y
Let
        Lcy2 = ∂2(½L1L2L3)/∂y2 = c1c2L3 + c1L2c3 + L1c2c3

   ∂2w/∂y2 = β4(2c1c1L2 + 4L1c1c2 + Lcy2) + β5(2c1c1L3 + 4L1c1c3 + Lcy2)
               + β6(2c2c2L3 + 4L2c2c3 + Lcy2) + β7(2c2c2L1 + 4L2c2c1 + Lcy2)
               + β8(2c3c3L1 + 4L3c3c1 + Lcy2) + β9(2c3c3L2 + 4L3c3c2 + Lcy2)

Similarly, let

       Lcx2 = ∂2(½L1L2L3)/∂x2 = b1b2L3 + b1L2b3 + L1b2b3

   ∂2w/∂x2 = β4(2b1b1L2 + 4L1b1b2 + Lcx2) + β5(2b1b1L3 + 4L1b1b3 + Lcx2)
               + β6(2b2b2L3 + 4L2b2b3 + Lcx2) + β7(2b2b2L1 + 4L2b2b1 + Lcx2)
               + β8(2b3b3L1 + 4L3b3b1 + Lcx2) + β9(2b3b3L2 + 4L3b3b2 + Lcx2)

Let,

   Lcxy = ∂2(½L1L2L3/∂x∂y = ½(b1c2L3 + b1L2c3 + c1b2L3 + L1b2c3 + c1L2b3 + L1c2b3)

   ∂2w/∂x∂y = β4(2c1b1L2 + 2L1b1c2 + 2L1c1b2 + Lcxy) +
       β5(2c1b1L3 + 2L1b1c3 + 2L1c1b3 + Lcxy) + β6(2c2b2L3 + 2L2b2c3 + 2L2c2b3 + Lcxy) +
       β7(2c2b2L1 + 2L2b2c1 + 2L2c2b1 + Lcxy) + β8(2c3b3L1 + 2L3b3c1 + 2L3c3b1 + Lcxy) +
       β9(2c3b3L2 + 2L3b3c2 + 2L3c3b2 + Lcxy)

Putting these into matrix form:
ε =
000XX4 XX5 XX6 XX7 XX8 XX9
000YY4 YY5 YY6 YY7 YY8 YY9
000XY4 XY5 XY6 XY7 XY8 XY9
CI d

XX4 = (2b1b1L2 + 4L1b1b2 + Lcx2)
XX5 = (2b1b1L3 + 4L1b1b3 + Lcx2)
XX6 = (2b2b2L3 + 4L2b2b3 + Lcx2)
XX7 = (2b2b2L1 + 4L2b2b1 + Lcx2)
XX8 = (2b3b3L1 + 4L3b3b1 + Lcx2)
XX9 = (2b3b3L2 + 4L3b3b2 + Lcx2)

YY4 = (2c1c1L2 + 4L1c1c2 + Lcy2)
YY5 = (2c1c1L3 + 4L1c1c3 + Lcy2)
YY6 = (2c2c2L3 + 4L2c2c3 + Lcy2)
YY7 = (2c2c2L1 + 4L2c2c1 + Lcy2)
YY8 = (2c3c3L1 + 4L3c3c1 + Lcy2)
YY9 = (2c3c3L2 + 4L3c3c2 + Lcy2)

XY4 = 2(2c1b1L2 + 2L1b1c2 + 2L1c1b2 + Lcxy)
XY5 = 2(2c1b1L3 + 2L1b1c3 + 2L1c1b3 + Lcxy)
XY6 = 2(2c2b2L3 + 2L2b2c3 + 2L2c2b3 + Lcxy)
XY7 = 2(2c2b2L1 + 2L2b2c1 + 2L2c2b1 + Lcxy)
XY8 = 2(2c3b3L1 + 2L3b3c1 + 2L3c3b1 + Lcxy)
XY9 = 2(2c3b3L2 + 2L3b3c2 + 2L3c3b2 + Lcxy)

 Let,
 
ε = B d
B =
000XX4 XX5 XX6 XX7 XX8 XX9
000YY4 YY5 YY6 YY7 YY8 YY9
000XY4 XY5 XY6 XY7 XY8 XY9
CI

Stresses are defined as moment per unit length.

   {M} = [D]{ε} = [D] [B] {d}

Mx
My
Mxy
= Et3

12(1-v2)
1  v 0
v  1 0 
0  0(1-v)/2
 ∂2w/∂x2
 ∂2w/∂y2
2∂2w/∂x∂y

Stiffness matrix is then,

   [K] = [D][B]

where,

D
= Et3

12(1-v2)
1  v 0
v  1 0 
0  0(1-v)/2

Part 1 - Displacement, shape functions

Ads from Google
Dr Li Anchor Profi
www.anchorprofi.de
Engineering anchorage plate design system
©Andrew Qu, 2015. All rights reserved. Code snippets may be used "AS IS" without any kind of warranty. DIY tips may be followed at your own risk.