stresses around a hole: setup - soest · stresses around a hole: setup i main topics a introduction...
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GG303 Lecture 21 10/26/09 1
Stephen Martel (Fall 2009: Conrad) 21-1 University of Hawaii
STRESSES AROUND A HOLE: SETUP
I Main Topics
A Introduction to stress fields and stress concentrations B Stresses in a polar (cylindrical) reference frame C Equations of equilibrium D Solution of boundary value problem for a pressurized hole
II Introduction to stress fields and stress concentrations A Importance
1 Stress (and strain and displacement) vary in space 2 Stress concentrations can be huge and have a large effect
B Common causes of stress concentrations 1 A force acts on a small area (e.g., beneath a nail being hammered) 2 Geometric effects (e.g., corners on doors and windows) 3 Material heterogeneities (e.g., mineral heterogeneities and voids)
III Stresses in a polar (cylindrical) reference frame (on-in convention)
GG303 Lecture 21 10/26/09 2
Stephen Martel (Fall 2009: Conrad) 21-2 University of Hawaii
IV Equations of equilibrium (force balance)
A In Cartesian (x,y) reference frame
1 Force = (stress)(area) 2 Force balance in the x-direction
a
!
Fx" = 0= [#$ x x](dydz) +[#$ y x](dxdz) +[$ x x+%$ x x%x
dx](dydz)+[$y x+%$ y x
%ydy](dxdz)
b
!
Fx" = 0= [#$ x x
#xdx](dydz) +[
#$ y x
#ydy](dxdz)
c
!
Fx" = 0= [#$ x x
#x+#$y x
#y](dxdydz)
d
!
"#x x"x
+"#y x
"y= 0
The rate of σxx increase is balanced by the rate of σyx decrease 3 Force balance in the y-direction
a
!
Fy" = 0 =[#$yy](dxdz) +[#$x y](dydz) +[$ y y+%$y y
%ydy](dxdz)+[$ x y+
%$ x y
%ydx](dydz)
b
!
Fy" = 0 =[#$y y
#ydy](dxdz) +[
#$ x y
#xdx](dydz)
c
!
Fx" = 0= [#$y y
#y+#$ x y
#x](dxdydz)
d
!
"#y y
"y+"#x y
"x= 0
The rate of σyy increase is balanced by the rate of σxy decrease
GG303 Lecture 21 10/26/09 3
Stephen Martel (Fall 2009: Conrad) 21-3 University of Hawaii
B In polar (r,θ) reference frame (apply chain rule; see supplement) 1
!
Fr" = 0=#$
rr
#r+1
r
#$r%
#%+$rr&$%%r
2
!
F"# = 0=$%""$"
+$%
r"
$r+2%
r"
r
V Solution of boundary value problem for a pressurized hole
A Governing equation for an axisymmetric problem
1 Displacements are purely radial for a uniform suction (pressure) in hole; the displacements are toward (away from) the hole
2 For constant values of r, stresses and displacements in a polar reference frame do not change with θ, so ∂(ur, θrr, etc.)/∂θ = 0
3 By symmetry the shear stress σrθ = 0 4 Equation B2 of section IV is identically equal to zero 5 Equation B1 becomes:
!
"#rr
"r+#
rr$#%%
r= 0, and then
!
d"rr
dr+"
rr#"$$
r= 0
This is our governing equation. It is a differential equation. B One General Solution Method
1 First solve for the stresses in terms of the strains using Hooke’s law, and then solve for the strains in terms of the displacements. This leaves a differential equation to solve in terms of the displacements.
GG303 Lecture 21 10/26/09 4
Stephen Martel (Fall 2009: Conrad) 21-4 University of Hawaii
Second, after solving for the displacements, take the derivatives of the displacements to find the strains. Third, solve for the stresses in terms of the strains using Hooke’s law. The solution is not hard, but it is somewhat lengthy.
2 The general solution will contain constants. Their values are found in terms of the stresses or displacements on the boundaries of our body (i.e., the wall of the hole and any external boundary), that is in terms of the boundary conditions for our problem.
C Strain-displacement relationships Cartesian coordinates Polar coordinates a
!
"xx
=#u
#x
!
"rr
=#u
r
#r
b
!
"yy =#v
#y
!
"## =ur
r
c
!
"xy =1
2
#u
#y+#v
#x
$
% & '
(
!
"r# = 0
D Strain- stress relationships 1 Plane stress (σzz = 0)
Cartesian coordinates Polar coordinates
a
!
"xx =1
E# xx $%# yy[ ]
!
"rr
=1
E#
rr$%#&&[ ]
b
!
"yy =1
E# yy $%# xx[ ]
!
"## =1
E$## %&$ rr[ ]
c
!
"xy =1
2G#xy
!
"xy =1
2G#xy
2 Plane strain (εzz = 0) Cartesian coordinates Polar coordinates a
!
"xx =1#$ 2
E% xx #
$
1#$
& '
( ) % yy
*
+ , -
. /
!
"rr
=1#$ 2
E%
rr#
$
1#$
& '
( ) %**
+
, - .
/ 0
b
!
"yy =1#$ 2
E% yy #
$
1#$
& '
( ) % xx
*
+ , -
. /
!
"## =1$% 2
E&## $
%
1$%
' (
) * &rr
+
, - .
/ 0
c
!
er" =
1
2G#
r"
!
er" =
1
2G#
r"
GG303 Lecture 21 10/26/09 5
Stephen Martel (Fall 2009: Conrad) 21-5 University of Hawaii
E Stress-strain relationships 1 Plane stress (σzz = 0)
Cartesian coordinates Polar coordinates
a
!
"xx =E
1#$ 2( )%xx + $%yy[ ]
!
"rr
=E
1#$ 2( )%rr
+ $%&&[ ]
b
!
"yy =E
1#$ 2( )%yy + $%xx[ ]
!
"## =E
1$% 2( )&## + %&
rr[ ]
c
!
"xy = 2G#xy
!
"r# = 2G$
r# 2 Plane strain (εzz = 0)
Cartesian coordinates Polar coordinates a
!
"xx =E
(1+ #)$xx +
#
(1% 2# )$xx + $yy( )
&
' ( )
* +
!
"rr
=E
(1+ #)$rr
+#
(1%2#)$rr
+$&&( )'
( ) *
+ ,
b
!
"yy =E
(1+ #)$yy +
#
(1% 2# )$yy + $xx( )
&
' ( )
* +
!
"## =E
(1+ $)%## +
$
(1&2$ )%## + %
rr( )
'
( ) *
+ ,
c
!
"xy = 2G#xy
!
"r# = 2G$
r# F Axisymmetric Equilibrium (Governing) Equations
1 Plane stress (σzz = 0) In terms of stress In terms of displacement
!
"#rr
"r+#
rr$#%%
r= 0
!
d2ur
dr2
+1
r
dur
dr"ur
r2
= 0
2 Plane strain (εzz = 0) In terms of stress In terms of displacement
!
"#rr
"r+#
rr$#%%
r= 0
!
d2ur
dr2
+1
r
dur
dr"ur
r2
= 0