The document discusses shearing stress in beams. It provides basic relationships and conditions for solving shearing stress, including that shearing stress is constant in vertical cuts through thin-walled members. It examines shearing stress in specific cross sections like rectangular, I-profile, and U-profile beams. The maximum shearing stress occurs at the junction of the flange and wall for an I-profile. Shear centers are discussed for locating the line of action of shear forces in non-symmetric sections. Composite members made of different materials connected by welds or bolts are also addressed.

1. Department of Structural Mechanics
Faculty of Civil Engineering, VSB - Technical University Ostrava
Elasticity and plasticity
Theme 6
Shearing stress in
bending
• Basic relationships and conditions of solutions
• Shearing stress in chosen cross-sections
• Shear flux and shear centre
• Dimension of members in shear
• Composite beams

2. 2 / 74
Shearing stress in beams
Basic relations and conditions of solution
b
a
+
V
V
Rbz
Raz
F
-
V
V
In bending there come up usually together with bending moments also shear
forces, which causes the shearing stress.
0
, ≠
y
z M
V
0
=
=
=
= z
x
y M
M
V
N
In the plane xz:
Plane bending: external and internal forces are in planes xy or xz – the main
planes.
0
, ≠
z
y M
V
0
=
=
=
= y
x
z M
M
V
N
In the plane xy :

3. 3 / 74
Basic examples of shearing stress
Basic examples of shearing stress
Cracks in bridges after flood in 2002, South Czech, photo: Prof. Ing. Vladimír Tomica, CSc.

4. 4 / 74
Basic examples of shearing stress
Cracks in bridges after flood in 2002, South Czech, photo: Prof. Ing. Vladimír Tomica, CSc.
Basic examples of shearing stress

5. 5 / 74
Basic examples of shearing stress
Crack in the support of the concrete beam
photo: Prof. Ing. Radim Čajka, CSc.
Basic examples of shearing stress

6. 6 / 74
Basic examples of shearing stress
Základní vztahy a předpoklady řešení
Detail of the screw connection

7. 7 / 74
The formula of reciprocity of the shearing stresses
y
z
x
[ ]










=
z
yz
y
xz
xy
x
σ
τ
σ
τ
τ
σ
σ
sym.
Tenzor of stresses:
{ } { }T
xy
zx
yz
z
y
x τ
τ
τ
σ
σ
σ
σ =
Vector of stresses:
more topic n.8
Just 6 stresses components
S
z
d
y
d
x
d
xy
τ
yx
τ
yx
τ
xy
τ
yx
xy τ
τ = zy
yz τ
τ = xz
zx τ
τ =
similarly
Basic relations and conditions of solution

8. 8 / 74
Basic relations for derivation of shearing stress
Shearing stress in chosen cross-sections
+y
+z
A B
T
z
( )
z
b
y
S
A,
( )
z
y
y
z
zx
xz
b
I
S
V
.
.
=
=τ
τ
statical moment of separate part of cross section
...
y
S
z
V ... Shear force in section
y
I ... moment of inertia of the whole cross section
( )
z
b ... Width of cross-section in the assessing place
Grashof formula
Cross sectional characteristics for shear stress
in bending are:
- Sy
- Iy
[ ]
3
T
part
y m
z
A
S ⋅
=

9. 9 / 74
Shear stress in a rectangular cross section
Shearing stress in chosen cross-sections
z
y
b
h
Cross section
max
τ
o
2
Distribution of xz
τ
z
3
.
12
1
h
b
Iy =
( ) b
b z =
T
z
( ) A
V
bh
V
b
bh
h
bh
V
b
I
S
V Z
Z
z
z
y
y
z
ax
m
2
3
2
3
12
1
4
2
.
.
3
2
/
1 =
=
⋅
⋅
⋅
=
=
=τ
τ
[ ]
3
T
průr
části
y m
z
A
S ⋅
=
statical moment of separate part of cross section
...
y
S
z
V ... Shear force in section
y
I ... moment of inertia of the whole cross section
( )
z
b ... Width of cross-section in the assessing place

10. 10 / 74
Design and assessment of rectangular cross section in
shear
Dimensioning
Assessment after Ultimate Limit
State
Design
Realisation
Dimensioning
d
req
Ed f
A
V ,
,
Rd
Ed V
V ≤
Rd
V
To make bigger section
M
k
d
f
f
γ
=
3
.
2
3
max
d
z f
A
V
≤
=
τ
d
z
req
f
V
A
.
2
.
3
3
=
1
≤
Rd
Ed
V
V

11. 11 / 74
Shearing stress in thin-walled members
Shearing stress in chosen cross-sections
h
t
t
t w
f <<
,
,
t
h
Symmetric I
h
w
h
f
t
f
t
f
b
w
t
w (web)
f (flange)
Thin-walled beam
Open cross sections: I, U, T, C, Z Hollow box-beam sections (pipe):

12. 12 / 74
Shear stress in I profile
Shearing stress in chosen cross-sections
Section
Wall Distribution of
z
y
Flange Distribution of
xz
τ
xy
τ
max
,
xz
τ
o
2
h
w
h
f
t
f
t
f
b
w
t
o
1
Condition of solutions:
• shearing stress is constant in the vertical
cut to the fragment of the wall (see Detail)
• is parallel to circumference of the section
xy
τ
Detail
Detail

13. 13 / 74
Shear stress in the wall of the I-profile
Shearing stress in chosen cross-sections
Distribution
z
y
xz
τ
max
,
xz
τ
o
2
h
w
h
f
t
f
t
f
b
w
t
z
t
I
S
V
y
y
z
x
.
.
=
τ
Basic formula:
t ... Thickness in the assessing
fragment
y
S ...statical moment of a
separate partn of the section
x
τ ... Shearing stress in the plane
perpendicular to the axis x
xz
τ
xy
τ
vertical part
horizontal part
Shearing stress in the wall xz
τ
(quadratic function)
[ ]
3
T
part
y m
z
A
S ⋅
=

14. 14 / 74
Shear stress in the flange of I-profile
Shearing stress in chosen cross-sections
z
y h
w
h
f
t
f
t
f
b
w
t
Distribution of xy
τ
o
1
t
I
S
V
y
y
z
x
.
.
=
τ
Basic formula
Shearing stress in the flange xy
τ
(Linear function)
[ ]
3
T
part
y m
z
A
S ⋅
=
T
z

15. 15 / 74
Maximum stress in the I-profile
Shearing stress in chosen cross-sections
Distribution of
z
y
Distribution of
xz
τ
xy
τ
max
,
xz
τ
o
2
h
w
h
f
t
f
t
f
b
w
t
o
1
( )( )
[ ]
2
max .
.
.
.
4
.
.
.
8
h
t
t
h
t
b
t
t
I
V
w
f
w
f
w
y
z
+
−
−
=
τ
0
=
z

16. 16 / 74
Shear centre
Shear flux and shear centre
At double symmetric cross sections the resulting force goes throught the centre of
gravity, at non symmetric sections it is different – if a plane of the load is not a plane
of symmetry, than the load must go throught the shear centre A, otherwise the
member would be twisted (stress from torsion).
We obtain resulting shear forces Qf in the profile by the integration of shearing
stresses along the single walls of the open profile. They are equivalent to shear force
Vz.
+y
+z
T
A
0
h
0
h
t
t
z
V
o
45
f
Q
f
Q
x
τ
x
τ
z
V
f
Q
f
Q
2
z
f
V
Q =

17. 17 / 74
Shear centre of the U-profile
Shear flux and shear centre
z
y T
0
h
A
z
V
f
Q
f
Q
0
b
a
w
Q
xy
τ
xy
τ
xz
τ
s
h
t
h
s
t
S f
f
y .
.
.
2
1
2
.
. 0
0
=
=
s
y
z
f
y
y
z
xy
I
s
h
V
t
I
S
V
.
2
.
.
.
. 0
=
=
τ
Flange:
Profiles U, UE, UPE - size a in tables

18. 18 / 74
Shear centre of U-profile
Shear flux and shear centre
z
y T
0
h
A
z
V
f
Q
f
Q
0
b
a
w
Q
xy
τ
xy
τ
xz
τ
s
M
( )
[ ]
2
2
0
0
0 4
.
.
.
.
4
.
.
.
8
z
h
t
h
b
t
t
I
V
w
f
w
y
z
xz −
+
=
τ
Web
... likewise I profile
z
w V
Q =
0
.
. h
Q
a
V f
z =
Statical moments to point M → y
f
z
f
I
h
b
t
V
h
Q
a
.
4
.
.
. 2
0
2
0
0
=
=

19. 19 / 74
Composite members
Composite members
a b
b
a b
0
≠
V
welds, screws, bolts
a
a
( ) ( )
( ) y
y
z
z
y
y
z
z
zx
z
x
I
S
V
b
I
S
V
b
b
Q
.
.
.
.
.
*
=
=
= τ
[kN/m]
a
I
S
V
a
Q
Q
y
y
z
x
x .
.
.
*
=
=
[kN]
Shear force for one
connecting member
↓

20. 20 / 74
Questions for the exam
1. Shearing stress in bending of the rectangular
cross section
2. Shearing stress in bending of thin-walled
members, shear centre
3. Assessment of the members under shear
stress in bending
4. Composite members
Questions for the exam