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1/14M.Chrzanowski: Strength of Materials
SM2-03: Bending
B E N D I N G
2/14M.Chrzanowski: Strength of Materials
SM2-03: Bending
N=0, Qy=0, Qz=0
My= M
y
z
x
MM
Mx=0, My ≠ 0, Mz=0
My(x)=M=const
My
Qz(x)=0
N=0, Qy=0, Qz=0
N=0, Qy=0, Qz=0
Mx=0, My =0, Mz ≠ 0
„Pure” bending
Formal definition: the case when set of internal forces reduces solely to the moment vector which is perpendicular to the bar axis
Example: a straight bar loaded by concentrated moments applied at its ends.
or
3/14M.Chrzanowski: Strength of Materials
SM2-03: Bending
Remarks on terminology
4/14M.Chrzanowski: Strength of Materials
SM2-03: Bending
NORMAL (proste) INCLINED (ukośne)
y
z
x
y
z
x
Mx=0, My ≠ 0, Mz ≠0Mx=0, My ≠ 0, Mz=0
M M
90o <90o
90o<90o
5/14M.Chrzanowski: Strength of Materials
SM2-03: Bending
PURE (czyste)
y
z
x
Mx=0, My ≠ 0, Mz=0
M
N=0, Qy=0, Qz=0
IMPURE („nie-czyste”)
y
z
x
Mx=0, My ≠ 0, Mz=0
M
N=0, Qy=0, Qz ≠ 0
Q
NON-UNIFORM (poprzeczne)
6/14M.Chrzanowski: Strength of Materials
SM2-03: Bending
End of remarks
7/14M.Chrzanowski: Strength of Materials
SM2-03: Bending
E.Mariotte (1620-1684)Galileo (1564-1642)
EXPERIMENTAL approach
8/14M.Chrzanowski: Strength of Materials
SM2-03: Bending
EXPERIMENTAL approach
Galileo (1564-1642)Jacob Bernoulli (1654-1705)
x
z
D
D’
P uD
wD
l
h
M=M(x)
Q=Q(x)
Mx=0, My ≠ 0, Mz=0
N=0, Qy=0, Qz ≠ 0
For h<<l shear forces can be neglected
N=0, Qy=0, Qz = 0
u is linear function of z !)(zuu
),(
,zx
x
zxux
is linear function of z
and does not depend
on x if M=const|x
zax
9/14M.Chrzanowski: Strength of Materials
SM2-03: Bending
zax tension
compressionazxy
azxz
Hooke law: ijkkijij G 2
zax
21)1/( Eazkk )21( )1(2/ EG
EazEE
azazGazx
11)21(2
zy
EEazazazG
011
)21(2
0 zxyzxy 0 zxyzxy
Continuum Mechanics application
y
z
x
10/14M.Chrzanowski: Strength of Materials
SM2-03: Bending
Eazx
zy 0azzy
zax
x y
zz
My
maxmax Eazx
maxz
?
Continuum Mechanics application
11/14M.Chrzanowski: Strength of Materials
SM2-03: Bending
x y
z
My
maxz
?
0 dANA
x 0 ySEa0dAzEaA
y-axis is the central inertia axis of cross-
section area
0 ydAMA
xz 0 yzJEa0 dAzyEaA
y-z axes are central principal inertia axesof cross-section area
0 zdAMA
xy dAzEaMA
y 2yy EaJM
yy EJMa
z
Equilibrium conditions
12/14M.Chrzanowski: Strength of Materials
SM2-03: Bending
azxzy
zax
Eazx
zEJ
M
y
yzy
zEJ
M
y
yx
zJ
M
y
yx
000
000
00zJ
M
Ty
yx
xz
xy
y
yx z
EJ
M
T
00
00
00
Axes
x – which coincides with bar axis
y,z – which are central principal inertia axes of the bar cross-section area are principal axes of strain and stress matrices
Continuum Mechanics application
13/14M.Chrzanowski: Strength of Materials
SM2-03: Bending
zJ
M
y
yx
x y
zz
My
xmax
maxz
y
y
y
yx W
Mz
J
M maxmax
maxz
JW y
y where Wy is called
section modulus For z=0 (i.e. along y-axis ) there is and0x
section of y-axis within bar cross-section is called neutral axis (for normal stress and strain)
Neutral axis
Neutral axis coincides with only non-zero bending moment component My
yM
Pure plane bending
14/14M.Chrzanowski: Strength of Materials
SM2-03: Bending
Important remarks
1. All above formulas are valid only for principal central axes of cross-section inertia
2. If moment vector coincides with any of two principal axes we have to deal with plane bending. If this is not the case – we have to deal with inclined bending and derived formulas cannot be used.
3. Bar axis (x-axis) is one of the principal axis of strain and stress matrices. As two remaining principal stress and strains are equal therefore any two perpendicular axes lying in the plane of bar cross-section are also principal axes.
4.The neutral axis for normal stress and strain coincides with bending moment vector.
15/14M.Chrzanowski: Strength of Materials
SM2-03: Bending
stop
