SM2-07: Shear
SHEAR
&
NON-UNIFORM BENDING
M.Chrzanowski: Strength of Materials
1/11
SM2-07: Shear
Mx = 0, My = 0, Mz = 0
Qx=N= 0, Qy ≠ 0, Qz = 0
Formal definition: the case when set of
internal forces reduces solely to the
shear force vector perpendicular to the
bar axis
y
Qy
Mx = 0, My ≠ 0, Mz = 0
or
Qx= N=0, Qy= 0, Qz ≠ 0
Shear is associated with bending !
x
z
Qx=N=0, Qy= 0, Qz ≠ 0
y
y
x
Qz
z
My
NON-UNOFRM BENDING
M.Chrzanowski: Strength of Materials
x
Qz
z
PURE SHEAR
2/11
SM2-07: Shear
Pure shear
M
Q
Reactions
Mu0
?
M.Chrzanowski: Strength of Materials
Q=const
3/11
SM2-07: Shear
Pure shear
P
P
t
t
P
t
t
A
A
A
A
4At = P
M.Chrzanowski: Strength of Materials
t = P/4A
P
Mean shear stress
4/11
SM2-07: Shear
Non-uniform bending
Z
Z
X
X
h
l
Bernoulli hypothesis of plane
cross-sections does not hold!!
For h/l<<1 distorsion is small and we
will use the formula for normal stress
derived from this assumption :
M.Chrzanowski: Strength of Materials
x 
My
Jy
z
5/11
SM2-07: Shear
Non-uniform bending
Z
Z
P
P
X
X
tzx
Z
txz
txz= txz(z,y)
X
?
„point” image
txz
txz
tzx

Qz = P = t xz dA
A
M.Chrzanowski: Strength of Materials
6/11
SM2-07: Shear
Non-uniform bending
Z
A*
 x x  dx 
 x x 
D,F
B,C
z
B,D
y
M y x 
b (z)
C,F
M y  x  dx 
t xz x 
t xz x  dx 
Qz x 
Qz x  dx 
x
Prismatic bar!
D
N *  x     x  x dA
A*
x 
My
Jy
z
A*
dx
 x x  dx  
M y  x  dx 

Jy
M.Chrzanowski: Strength of Materials
b (z) N
B
~
t~
C
*
x  dx     x x  dx dA
A*
zx
z
~
lim N * x   N * x  dx   t~zx  b( z )  dx
dx0
F

t~zx 
Qz x   S *y z 
J y  bz 
 t~xz
7/11
SM2-07: Shear
Non-uniform bending



1
S
z 
~
Qz x 
Formula t xz 



Jy
b
z


holds for prismatic bars only!
and is given in main principal axes of cross-section inertia.
Distribution along z-axis
A
z
A*
Also:
z
A



S
z 
~
max t xz  max x Qz x  max z 



b
z


zmax
qi   ijj
y
zmin
qx  0   x  0  t xy  0  t~xz 1  t~xz  0
B
For A: S*(zmax)=0 since A*=0
For B: S*(zmax)=0 since A*=A
M.Chrzanowski: Strength of Materials
Kinematic Boundary Conditions in A and B:
t~xz  0
t~xz  0
qy  0  t yx  0   y  0  t yz 1  t yz  0
q  0  t~  0  t  0   1    q
z
zx
zy
z
z
z
8/11
SM2-07: Shear
Non-uniform bending
Distribution along z-axis; special cases
b(z) = b =const|z
b(z) – linear function of z
z
z
Parabola 2o
tmax= 3Q/2bh
h/2
Parabola 3o
2h/3
tmax= ?
y
y
h/2
h/3
b
b(z) – step-wise change
h/2
h/2
c
tb
tmax
y
y
h/2
b
Parabola 2o
z
z
tc
tb/ tc=c/b
h/2
b
M.Chrzanowski: Strength of Materials
b
9/11
SM2-07: Shear
Non-uniform bending
Stress distribution in beams – trajectories of main principal stresses
x 
M y x 
t zx 
 1, 2
Jy
txz
z
Qz x   S z 
*
y
x
x
 t xz
tzx
J y  bz 

1
 x 
 x2  4t xz2
2
2
t xz
tg1,2  
 z  1,2
 z  0 tg1, 2  
tzx
z
t xz
 1, 2
M.Chrzanowski: Strength of Materials
txz
x
z
x
For z=0:
x  0
tg1, 2  1
 1, 2  t xz
1, 2  45o
10/11
SM2-07: Shear
Non-uniform bending
l/2
2=t
1=t
Principal
stress
trajectories
2=45o
2=t
1=45o
z
1=t
x
My  0
Qz  0
M.Chrzanowski: Strength of Materials
11/11
SM2-07: Shear
stop
M.Chrzanowski: Strength of Materials
12/11