Transcript Title

SM1-08: Continuum Mechanics: Stress distribution
CONTINUUM MECHANICS
(STRESS DISTRIBUTION)
M.Chrzanowski: Strength of Materials
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SM1-08: Continuum Mechanics: Stress distribution
   


p

p
r
,
n
   
  
p  prn  rconst ; n   pn 
State of stress

rconst

n
Stress vector

nconst

p

r
Stress distribution
    
 
p  prr;n  nconst   pr 
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SM1-08: Continuum Mechanics: Stress distribution

PPi 

q qi 
Surface traction
(loading)
Volume V
Surface
Stress
vector

q
S
x3

p i  i   ijj
 


  0   PdV   p dS
V0
S0
i
i
 PdV    dS  0


V0
S0
 PdV     dS  0
i
V0
GGO
theorem
Volumetric
force
x1
M.Chrzanowski: Strength of Materials
Volume V0
x2
Surface
S0
ij  ij x1, x2 , x3,
ij
S0
j
 ij
 PdV   x
i
V0
V0
dV  0
j

 
 Pi  ij dV  0


x j 
V0 

 ij 
 Pi 
0



x
j


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SM1-08: Continuum Mechanics: Stress distribution
On the body surface stress vector has to
be balanced by the traction vector

q 

Stress on the body surface
qi  i   ijj

p
Coordinates of vector normal to the surface
qi   ijj
This equation states statics boundary conditions to comply with the solution of
the equation:

 ij 
 Pi 
0



x
j


M.Chrzanowski: Strength of Materials
This equation (Navier equation) reflects
internal equilibrium and has to be fulfilled
in any point of the body (structure).
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SM1-08: Continuum Mechanics: Stress distribution
Navier equation
in coordintes reads:

 ij 
 Pi 
0



x
j


 11  12  13
P1 


0
x1
x2
x3
 21  22  23
P2 


0
x1
x2
x3
 31  32  33
P3 


0
x1
x2
x3
We have to deal with the set of 3 linear
partial differential equations.
There are 6 unknown functions which have to
fulfil static boundary conditions (SBC):
qi   ijj
We need more equations to determine all 6 functions of stress distribution.
To attain it we have to consider deformation of the body.
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SM1-08: Continuum Mechanics: Stress distribution
Comments

 ij 

0
1. Equation Pi 

x j 

is derived from one of two
equilibrium equations, i.e. that the sum of forces acting over the body
has to vanish.
2.
The other equilibrium equation – sum of the moments equals zero – yield
already assumed symmetry of stress matrix, σij=
3.
σji
Navier equation is the special case of the motion equation i.e. uniform
motion (no inertia forces involved). The inertia effects can be included by
adding d’Alambert forces to the right hand side of Navier equation.
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SM1-08: Continuum Mechanics: Stress distribution
stop
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