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Equations of Equilibrium
Structural Analysis: Equilibrium© 2021 Mayuresh Patil. Licensed under a Creative Commons Attribution 4.0 license https://creativecommons.org/licenses/by-nc-sa/4.0/ [email protected]
3D Elasticity Problem
• Variables:• 3 displacements:• 6 strains:
• 6 stresses:
• We have a total of 15 variables and so we need 15 equations
u(x, y, z) v(x, y, z) w(x, y, z)
✏xx(x, y, z) ✏yy(x, y, z) ✏zz(x, y, z)
✏xy(x, y, z) ✏yz(x, y, z) ✏xz(x, y, z)
�xx(x, y, z) �yy(x, y, z) �zz(x, y, z)
�xy(x, y, z) �yz(x, y, z) �xz(x, y, z)
Kinematics: 6 Strain-Displacement equations• Normal strain
• Shear strain
✏xx =@u
@x
✏yy =@v
@y
✏zz =@w
@z
�xy = 2✏xy =
✓@u
@y+
@v
@x
◆
�yz = 2✏yz =
✓@w
@y+
@v
@z
◆
�xz = 2✏xz =
✓@u
@z+
@w
@x
◆
Material Constitutive Law:6 Stress-Strain equations• (Homogenous) Isotropic Material
• where
• Invert the above matrix equations to get stress in terms of strain
G =E
2(1 + ⌫)
8>>>>>><
>>>>>>:
✏xx✏yy✏zz2✏yz2✏xz2✏xy
9>>>>>>=
>>>>>>;
=
2
6666664
1/E �⌫/E �⌫/E 0 0 0�⌫/E 1/E �⌫/E 0 0 0�⌫/E �⌫/E 1/E 0 0 0
0 0 0 1/G 0 00 0 0 0 1/G 00 0 0 0 0 1/G
3
7777775
8>>>>>><
>>>>>>:
�xx
�yy
�zz
�yz
�xz
�xy
9>>>>>>=
>>>>>>;
Equations of Equilibrium – linear momentum<latexit sha1_base64="UF4C9f31rkhA3f5TAuZzqEaAWjw=">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</latexit> �x = �xx �y = �yy �z = �zz
⌧xy = �xy ⌧yz = �yz ⌧xz = �xz
Notation:
Equations of Equilibrium – angular momentum<latexit sha1_base64="UF4C9f31rkhA3f5TAuZzqEaAWjw=">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</latexit> �x = �xx �y = �yy �z = �zz
⌧xy = �xy ⌧yz = �yz ⌧xz = �xz
Notation:
3 Equations of Equilibrium
• where, are known body forces per unit volume
• There are actually 6 equations of equilibrium but the three angular (moment) equations lead to the symmetry of the stress tensor (which we assumed without proof earlier)
@�xx
@x+
@�xy
@y+
@�xz
@z+ bx = 0
@�xy
@x+
@�yy
@y+
@�yz
@z+ by = 0
@�xz
@x+
@�yz
@y+
@�zz
@z+ bz = 0
bx(x, y, z) by(x, y, z) bz(x, y, z)
�xy = �yx
�xz = �zx
�yz = �zy<latexit sha1_base64="F7M94HZ3+SHUwQCejcgpYU7tN3I=">AAACPHicbZBLS8NAFIUnPmt8RV26GSyCq5KIoBuhIILLivYBTQiT6aQdOnkwM5GmIT/MjT/CnSs3LhRx69ppG6UPLwycOd+9zNzjxYwKaZov2tLyyuraemlD39za3tk19vYbIko4JnUcsYi3PCQIoyGpSyoZacWcoMBjpOn1r0a8+UC4oFF4L9OYOAHqhtSnGEllucadLWg3QG42SHN4CX9v6SCHtq3/weE0HM7CdBamuWuUzYo5LrgorEKUQVE113i2OxFOAhJKzJAQbcuMpZMhLilmJNftRJAY4T7qkraSIQqIcLLx8jk8Vk4H+hFXJ5Rw7E5PZCgQIg081Rkg2RPzbGT+x9qJ9C+cjIZxIkmIJw/5CYMygqMkYYdygiVLlUCYU/VXiHuIIyxV3roKwZpfeVE0TiuW0rdn5ep1EUcJHIIjcAIscA6q4AbUQB1g8AhewTv40J60N+1T+5q0LmnFzAGYKe37B2Fzr2s=</latexit><latexit sha1_base64="F7M94HZ3+SHUwQCejcgpYU7tN3I=">AAACPHicbZBLS8NAFIUnPmt8RV26GSyCq5KIoBuhIILLivYBTQiT6aQdOnkwM5GmIT/MjT/CnSs3LhRx69ppG6UPLwycOd+9zNzjxYwKaZov2tLyyuraemlD39za3tk19vYbIko4JnUcsYi3PCQIoyGpSyoZacWcoMBjpOn1r0a8+UC4oFF4L9OYOAHqhtSnGEllucadLWg3QG42SHN4CX9v6SCHtq3/weE0HM7CdBamuWuUzYo5LrgorEKUQVE113i2OxFOAhJKzJAQbcuMpZMhLilmJNftRJAY4T7qkraSIQqIcLLx8jk8Vk4H+hFXJ5Rw7E5PZCgQIg081Rkg2RPzbGT+x9qJ9C+cjIZxIkmIJw/5CYMygqMkYYdygiVLlUCYU/VXiHuIIyxV3roKwZpfeVE0TiuW0rdn5ep1EUcJHIIjcAIscA6q4AbUQB1g8AhewTv40J60N+1T+5q0LmnFzAGYKe37B2Fzr2s=</latexit><latexit sha1_base64="F7M94HZ3+SHUwQCejcgpYU7tN3I=">AAACPHicbZBLS8NAFIUnPmt8RV26GSyCq5KIoBuhIILLivYBTQiT6aQdOnkwM5GmIT/MjT/CnSs3LhRx69ppG6UPLwycOd+9zNzjxYwKaZov2tLyyuraemlD39za3tk19vYbIko4JnUcsYi3PCQIoyGpSyoZacWcoMBjpOn1r0a8+UC4oFF4L9OYOAHqhtSnGEllucadLWg3QG42SHN4CX9v6SCHtq3/weE0HM7CdBamuWuUzYo5LrgorEKUQVE113i2OxFOAhJKzJAQbcuMpZMhLilmJNftRJAY4T7qkraSIQqIcLLx8jk8Vk4H+hFXJ5Rw7E5PZCgQIg081Rkg2RPzbGT+x9qJ9C+cjIZxIkmIJw/5CYMygqMkYYdygiVLlUCYU/VXiHuIIyxV3roKwZpfeVE0TiuW0rdn5ep1EUcJHIIjcAIscA6q4AbUQB1g8AhewTv40J60N+1T+5q0LmnFzAGYKe37B2Fzr2s=</latexit><latexit sha1_base64="F7M94HZ3+SHUwQCejcgpYU7tN3I=">AAACPHicbZBLS8NAFIUnPmt8RV26GSyCq5KIoBuhIILLivYBTQiT6aQdOnkwM5GmIT/MjT/CnSs3LhRx69ppG6UPLwycOd+9zNzjxYwKaZov2tLyyuraemlD39za3tk19vYbIko4JnUcsYi3PCQIoyGpSyoZacWcoMBjpOn1r0a8+UC4oFF4L9OYOAHqhtSnGEllucadLWg3QG42SHN4CX9v6SCHtq3/weE0HM7CdBamuWuUzYo5LrgorEKUQVE113i2OxFOAhJKzJAQbcuMpZMhLilmJNftRJAY4T7qkraSIQqIcLLx8jk8Vk4H+hFXJ5Rw7E5PZCgQIg081Rkg2RPzbGT+x9qJ9C+cjIZxIkmIJw/5CYMygqMkYYdygiVLlUCYU/VXiHuIIyxV3roKwZpfeVE0TiuW0rdn5ep1EUcJHIIjcAIscA6q4AbUQB1g8AhewTv40J60N+1T+5q0LmnFzAGYKe37B2Fzr2s=</latexit>
3D Elasticity Equations
• 6 strain-displacement equations • 6 stress-strain equations• 3 equations of equilibrium• Total 15 equations for 15 variables• 3 displacements• 6 strains• 6 stresses
Boundary conditions
• Displacement (Dirichlet)
• Traction (Neumann)
2D Elasticity (Plane Stress) Problem
• The out-of-plane stresses are zero or negligible
• Variables:• 2 displacements:• 3 strains:
• 3 stresses:
• We have a total of 8 variables and so we need 8 equations
u(x, y, z) v(x, y, z) w(x, y, z)
✏xx(x, y, z) ✏yy(x, y, z) ✏zz(x, y, z)
✏xy(x, y, z) ✏yz(x, y, z) ✏xz(x, y, z)
�xx(x, y, z) �yy(x, y, z) �zz(x, y, z)
�xy(x, y, z) �yz(x, y, z) �xz(x, y, z)
�zz ⇡ 0 �xz ⇡ 0 �yz ⇡ 0
2D Kinematics: 3 Strain-Displacement equations• Normal strain
• Shear strain
✏xx =@u
@x
✏yy =@v
@y
✏zz =@w
@z
�xy = 2✏xy =
✓@u
@y+
@v
@x
◆
�yz = 2✏yz =
✓@w
@y+
@v
@z
◆
�xz = 2✏xz =
✓@u
@z+
@w
@x
◆
2D Material Constitutive Law:3 Stress-Strain equations• (Homogenous) Isotropic Material
• Or invert the above equation
• where G =E
2(1 + ⌫)
8<
:
✏xx✏yy2✏xy
9=
; =
2
41/E �⌫/E 0�⌫/E 1/E 0
0 0 1/G
3
5
8<
:
�xx
�yy
�xy
9=
;
8<
:
�xx
�yy
�xy
9=
; =E
1� ⌫2
2
41 ⌫ 0⌫ 1 00 0 1�⌫
2
3
5
8<
:
✏xx✏yy2✏xy
9=
;
2 2D Equations of Equilibrium
• where, are known body forces per unit volume
@�xx
@x+
@�xy
@y+
@�xz
@z+ bx = 0
@�xy
@x+
@�yy
@y+
@�yz
@z+ by = 0
@�xz
@x+
@�yz
@y+
@�zz
@z+ bz = 0
bx(x, y, z) by(x, y, z) bz(x, y, z)
@�xx
@x+
@�xy
@y+
@�xz
@z+ bx = 0
@�xy
@x+
@�yy
@y+
@�yz
@z+ by = 0
@�xz
@x+
@�yz
@y+
@�zz
@z+ bz = 0
2D Elasticity Equations - 8 equations/variables
• 3 strain-displacement equations • 3 stress-strain equations• 2 equations of equilibrium• Total 8 equations for 8 variables• 2 displacements• 3 strains• 3 stresses
Equations of Equilibrium in terms of Strain
2D Elasticity Equations - 5 equations/variables
• 3 strain-displacement equations • 2 equations of equilibrium (in terms of strain)• Total 5 equations for 5 variables• 2 displacements• 3 strains
Equations of Equilibrium in terms of Displacements
2D Elasticity Equations - 2 equations/variables
• 2 equations of equilibrium (in terms of displacment)• Total 2 equations for 2 variables• 2 displacements
Equations of Equilibrium in terms of Displacements • Two equations and two unknowns
• For isotropic material• Similar to Navier’s Equations of 3D Elasticity
• Next class we will try to write the equations in terms of one new variable –Airy stress function
@2u
@x2+
1� ⌫
2
@2u
@y2+
1 + ⌫
2
@2v
@x@y+
1� ⌫2
Ebx = 0
@2v
@y2+
1� ⌫
2
@2v
@x2+
1 + ⌫
2
@2u
@x@y+
1� ⌫2
Eby = 0
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