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Lecture #7
Shear stresses
in thin-walled beam
THE CONCEPT OF SHEAR FLOW
Both of possible stresses – normal and shear –
usually act in the load-carrying structure.
Instead of consideration of the shear stress t, it is
convenient to introduce the shear flow q, using the
thickness d :
q  t d
2
SHEAR STRESSES RELATED QUESTIONS
- shear flows due to the shear force, with no torsion;
- shear center;
- torsion of closed contour;
- torsion of opened contour, restrained torsion and
deplanation;
- shear flows in the closed contour under combined action
of bending and torsion;
- twisting angles;
- shear flows in multiple-closed contours.
3
FORMULA FOR THE SHEAR FLOW
From the equilibrium of elementary piece of shell we
obtain (t is a tangential coordinate):
dq
dt
d 
d
dz
0
The formula for the shear flow could be derived by
integration:
t
q  t   q0 

0
d
dz
 d  dt
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FORMULA FOR THE SHEAR FLOW
If the normal stress is caused only by bending
moments, we get:
t
q  t   q0 

0
 z  x, y 
q  t   q0 
Nz
A
Qx
Iy

d
dz
M
Iy
y
 d  dt
x
 S y (t ) 
M
Ix
Qy
Ix
x
y
 S x (t )
5
FORMULA FOR THE SHEAR FLOW
Formula used for shear flows in Structural Analysis:
q  t   q0 
Qx
Iy
 S y (t ) 
Qy
Ix
 S x (t )
Compare to Zhuravsky formula used in Mechanics of
Materials:
t  y 
Q y  Sx( y)
b y Ix
6
CALCULATION OF STATIC MOMENTS
Way of calculation:
1) find principal axes;
2) choose tangential coordinate at the end of the
contour (or multiple coordinates for branching
contour);
3) make the calculation (integration for exact solution,
or sum for a discrete cross section), taking into
account the branching of contours;
4) check that static moment is zero at the end of
tangential coordinate.
7
CALCULATION OF STATIC MOMENTS
Exact formula:
t
Sx t 
 y  t   d  t  dt
0
For a discrete cross section:
Sx t 
n t 

Ai  y i
i 1
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CALCULATION OF STATIC MOMENTS - EXAMPLE
Vertical force of 100 kN is applied.
Given
cross section
Continuous
approach
Discrete
approach
9
CALCULATION OF STATIC MOMENTS - EXAMPLE
Discrete approach
S x , cm
3
q , kN m
t , M Pa
10
CALCULATION OF STATIC MOMENTS - EXAMPLE
Continuous approach
S x , cm
3
q , kN m
t , M Pa
11
CALCULATION OF STATIC MOMENTS - EXAMPLE
Comparison of continuous and discrete approach
used in Structural Analysis and the approach used in
Mechanics of Materials.
t , M Pa
12
CALCULATION OF STATIC MOMENTS
Symmetry of static moments:
1) If u is the axis of symmetry, the Su will be
symmetrical with respect to u.
2) If u is the axis of symmetry, and v is the axis
perpendicular to u, the Sv will be antisymmetrical with
respect to u.
13
WHERE TO FIND MORE INFORMATION?
Megson. An Introduction to Aircraft Structural Analysis. 2010
Chapter 16
… Internet is boundless …
14
TOPIC OF THE NEXT LECTURE
Shear center
All materials of our course are available
at department website k102.khai.edu
1. Go to the page “Библиотека”
2. Press “Structural Mechanics (lecturer Vakulenko S.V.)”
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