Post on 30-Dec-2015
description
1/23M.Chrzanowski: Strength of Materials
SM2-05: Composed bending
COMPOSED BENDING
(Eccentric tension/compression)
2/23M.Chrzanowski: Strength of Materials
SM2-05: Composed bending
zJ
M
y
yx
x y
zz
My
)(zx
maxz
Neutral axis for bending
yM
000
000
00x
T
Plane bending
3/23M.Chrzanowski: Strength of Materials
SM2-05: Composed bending
A
Nx
x y
zz )(zx Neutral axis for
bending
yM
000
000
00x
T
N
Neutral axis for tension
+
-
Tension
4/23M.Chrzanowski: Strength of Materials
SM2-05: Composed bending
000
000
00x
T
zJ
M
y
yx
y
zz
My
)(zx Neutral axis
yM
N + =
+A
N
0xNeutral axis equation:
zi
zz
N
A
J
M
A
N
A
Nz
J
M
yy
y
y
y20110
0zN
M y
2y
y iA
J
„Eccentric”
Squared inertia radius
Plane bendingCombined bending
z0
5/23M.Chrzanowski: Strength of Materials
SM2-05: Composed bending
yJ
Mz
J
M
A
N
z
z
y
yx Normal stress
0zNM y 0yNM z
yJ
yNz
J
zN
A
N
zyx
00
20
201/
zy
x
i
yy
i
zzAN
0xNormal stress at neutral axis
20
2010
zy i
yy
i
zz
AN/
Eccentrics
Normal stress in terms of normal force and eccentrics y0 ,z0
Non-dimensional normal stress
Neutral axis equation
Bi-plane combined loading
6/23M.Chrzanowski: Strength of Materials
SM2-05: Composed bending
NEUTRAL AXIS MOVEMENTin (y,z) plane
7/23M.Chrzanowski: Strength of Materials
SM2-05: Composed bending
Side view Stress distribution Cross-section view
8/23M.Chrzanowski: Strength of Materials
SM2-05: Composed bending
A+
N/A
Neutral axis in an „infinity”
NN
Side view Stress distribution Cross-section view
9/23M.Chrzanowski: Strength of Materials
SM2-05: Composed bending
NA+
N/A
Side view Stress distribution Cross-section view
Neutral axis already being „seen”
10/23M.Chrzanowski: Strength of Materials
SM2-05: Composed bending
NA
N/A
+
Side view Stress distribution Cross-section view
Neutral axis outside of cross-section
11/23M.Chrzanowski: Strength of Materials
SM2-05: Composed bending
NA+
N/A
NN
Side view Stress distribution Cross-section view
Neutral axis touching cross-section contour
12/23M.Chrzanowski: Strength of Materials
SM2-05: Composed bending
NA+
-
N/A
Side view Stress distribution Cross-section view
Neutral axis within cross-section
13/23M.Chrzanowski: Strength of Materials
SM2-05: Composed bending
20
2010
zy i
yy
i
zz
Dual interpretation of neutral axis equation
Neutral axis co-ordinates
Eccentric co-ordinates
14/23M.Chrzanowski: Strength of Materials
SM2-05: Composed bending
20
2010
zy i
yy
i
zz
1. If neutral axis coincides with cross-section contour edge given by the equation:
yz then from the transformed equation of neutral axis:
yi
y
z
i
z
iz
z
yy
20
0
2
0
2
one can find co-ordinates of normal force position (eccentricity):
0
2
z
iy
2
0yiz
20
0
2
z
y
i
y
z
i
2
0ziy
Dual interpretation of neutral axis equation
15/23M.Chrzanowski: Strength of Materials
SM2-05: Composed bending
Kzz Kyy
02
22
0 yi
y
z
i
z
iz
yz
K
K
y
K
y
then by inserting these co-ordinates into neutral axis equation one can obtain eqaution of a straigth line defining the position of a normal force:
20
2010
zy i
yy
i
zz
2. If a number of neutral axis touches cross-section corner of given co-ordinates:
Dual interpretation of neutral axis equation
16/23M.Chrzanowski: Strength of Materials
SM2-05: Composed bending
4 cm
14 cm
2 cm
8 cm
2 cm3 cm
8 cm
3 cm
Example of cross-section kernel finding
17/23M.Chrzanowski: Strength of Materials
SM2-05: Composed bending
7,22 cm
y
z
a
b
c
d
D
B
A
C
Example of cross-section kernel finding
Mode 1: Finding eccentrties
18/23M.Chrzanowski: Strength of Materials
SM2-05: Composed bending
7,22 cm
y
z
e
E
Example of cross-section kernel finding
Mode 2: Finding normal force acting lines
19/23M.Chrzanowski: Strength of Materials
SM2-05: Composed bending
7,22 cm
y
z
f
F
Example of cross-section kernel finding
Mode 2: Finding normal force acting lines
20/23M.Chrzanowski: Strength of Materials
SM2-05: Composed bending
7,22 cm
y
z
G
g
Example of cross-section kernel finding
Mode 2: Finding normal force acting lines
21/23M.Chrzanowski: Strength of Materials
SM2-05: Composed bending
7,22 cm
y
z
Example of cross-section kernel finding
22/23M.Chrzanowski: Strength of Materials
SM2-05: Composed bending
7,22 cm
y
z
Example of cross-section kernel finding
23/23M.Chrzanowski: Strength of Materials
SM2-05: Composed bending
Cross-section kernel defines normal force positions (eccentrities) such that resulting normal stress in the whole cross-section area has the same sign (plus for N>0 and minus dla N<0).
Cross-section kernel has always a convex form.
Cross-section kernel