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1/10M.Chrzanowski: Strength of Materials
SM2-04: Inclined bending
INCLINED BENDING
2/10M.Chrzanowski: Strength of Materials
SM2-04: Inclined bending
If bending moment does not coincide with any of principal central axis of cross-section inertia we have to deal with inclined bending.
y
z
M
Plane of load acting
yM
zM
yy MM
Direction of bending moment vector
cosMM z zz MM
MM
sinMM y
y
z
MyM
zM
y
z
yyz cot)90tan(
+
3/10M.Chrzanowski: Strength of Materials
SM2-04: Inclined bending
y
z
zM
yM
y
z
yJ
M
z
zx y
J
Mz
J
M
z
z
y
yx
We use superposition principle
For neutral axis:
cosMM z sinMM y
yJ
zJ
Mzy
x
cossin
0x yJ
Jz
z
y cot
y
z
zJ
M
y
yx
Neutral axis
M
Jy>Jz
Jy<Jz
y
z
y
z
4/10M.Chrzanowski: Strength of Materials
SM2-04: Inclined bending
yJ
Mz
J
M
z
z
y
yx Stress distribution:
0x
yJ
Jz
z
y cot
y
z
M
Neutral axis
is represented by the equation of the plane
max
minTension
zone
Compression zone
max
This plane cuts thorough cross-section plane at the neutral axis
5/10M.Chrzanowski: Strength of Materials
SM2-04: Inclined bending
Animation of neutral axis movement dependent on bending moment
direction
6/10M.Chrzanowski: Strength of Materials
SM2-04: Inclined bending
y
z
80o
70o
60o
50o
40o
30o
20
10o
00o
h/b=1,41
Jy/Jz =2 Loading plane
Neutral axis
Bending moment vector
Angle
7/10M.Chrzanowski: Strength of Materials
SM2-04: Inclined bending
y
z
h/b=1,41
Jy/Jz =2
Loading plane
Neutral axis
Bending moment vector
80o
70o
60o
50o
40o
30o
20
10o
00o
8/10M.Chrzanowski: Strength of Materials
SM2-04: Inclined bending
y
z
b/h=1,41
Jy/Jz =1/2
Loading plane
Neutral axis
Bending moment vector
80o
70o
60o
50o
40o
30o
20
10o
00o
9/10M.Chrzanowski: Strength of Materials
SM2-04: Inclined bending
y
z
Płaszczyzna obciążeniaOś obojętna
Wektor momentu
h=b=a
Jy/Jz =1
y
zZmax =a/2=0,5a
Zmax
=a/2½=0,7a
aJ
M5,0max a
J
M7,0max
Ideal shape: independent of plane of loading angle maximum stress is the same!
y
z
10/10M.Chrzanowski: Strength of Materials
SM2-04: Inclined bending
stop
11/10M.Chrzanowski: Strength of Materials
SM2-04: Inclined bending
y
z
h/b=1,41
Jy/Jz =2 Loading plane
Neutral axis
Bending moment vector
Angle
80o
70o
60o
50o
40o
30o
20
10o
00o
12/10M.Chrzanowski: Strength of Materials
SM2-04: Inclined bending
y
z
h/b=1,41
Jy/Jz =2 Loading plane
Neutral axis
Bending moment vector
Angle 80o
70o
60o
50o
40o
30o
20
10o
00o
13/10M.Chrzanowski: Strength of Materials
SM2-04: Inclined bending
y
z
Płaszczyzna obciążeniaOś obojętna
Wektor momentu
h=b=a
Jy/Jz =1
y
z
α=45°
Zmax =a/2=0,5a
Zmax
=a/2½=0,7a
aJ
M5,0max a
J
M7,0max
Ideal shape: independent of plane of loading angle maximum stress is the same!
