Lecture #7 Shear stresses in thin-walled beam. THE CONCEPT OF SHEAR FLOW 2 Both of possible stresses – normal and shear – usually act in the load-carrying.
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Lecture #7Shear stressesin thin-walled beam
THE CONCEPT OF SHEAR FLOW
2
Both of possible stresses – normal and shear – usually act in the load-carrying structure.
Instead of consideration of the shear stress , it is convenient to introduce the shear flow q, using the thickness :
q
SHEAR STRESSES RELATED QUESTIONS
3
- 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.
FORMULA FOR THE SHEAR FLOW
4
From the equilibrium of elementary piece of shell we obtain (t is a tangential coordinate):
The formula for the shear flow could be derived by integration:
00
t dq t q dt
dz
0dq ddt dz
FORMULA FOR THE SHEAR FLOW
5
If the normal stress is caused only by bending moments, we get:
00
t dq t q dt
dz
, yz xz
y x
MN Mx y x y
A I I
0 ( ) ( )yxy x
y x
QQq t q S t S t
I I
FORMULA FOR THE SHEAR FLOW
6
Formula used for shear flows in Structural Analysis:
0 ( ) ( )yxy x
y x
QQq t q S t S t
I I
Compare to Zhuravsky formula used in Mechanics of Materials:
( )y x
x
Q S yy
b y I
CALCULATION OF STATIC MOMENTS
7
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.
CALCULATION OF STATIC MOMENTS
8
Exact formula:
For a discrete cross section:
0
t
xS t y t t dt
1
n t
x i ii
S t A y
CALCULATION OF STATIC MOMENTS - EXAMPLE
9
Givencross section
Vertical force of 100 kN is applied.
Continuousapproach
Discreteapproach
CALCULATION OF STATIC MOMENTS - EXAMPLE
10
Discrete approach 3, cmxS
, kN mq , MPa
CALCULATION OF STATIC MOMENTS - EXAMPLE
11
Continuous approach 3, cmxS
, kN mq , MPa
CALCULATION OF STATIC MOMENTS - EXAMPLE
12
Comparison of continuous and discrete approach used in Structural Analysis and the approach used in Mechanics of Materials.
, MPa
CALCULATION OF STATIC MOMENTS
13
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.
WHERE TO FIND MORE INFORMATION?
14
Megson. An Introduction to Aircraft Structural Analysis. 2010Chapter 16
… Internet is boundless …
TOPIC OF THE NEXT LECTURE
15
Shear center
All materials of our course are availableat department website k102.khai.edu
1. Go to the page “Библиотека”2. Press “Structural Mechanics (lecturer Vakulenko S.V.)”
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