1. Lecture #9
Shear stresses
in closed contour

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. 2

3. FORMULA FOR THE SHEAR FLOW
The formula for shear flow has a constant term q0:
constant
term
variable
term 3

4. MQ – moment of internal shear force.
DETERMINATION OF CONSTANT SHEAR FLOW
Variable term qf satisfies two equations of equilibrium for sums of forces.
Constant term q0 satisfies the equation of equilibrium for sum of moments.
Sum of moments equation:
where Mq0 , Mqf - moments from constant and variable parts of shear flow, respectively;
MQ – moment of internal shear force. 4

5. BREDT’S FORMULA FOR MOMENT FROM CONSTANT SHEAR FLOW
Bredt’s formula (put minus if tangential coordinate direction does not correspond to positive direction of moment)
where W is an area of closed cross section multiplied by 2.
Finally, constant part of shear flow could be found:
The signs in this formula should be used according to directions of moments. 5

6. EQUIVALENT DISCRETE CROSS SECTION
EXAMPLE – GIVEN DATA
EQUIVALENT DISCRETE CROSS SECTION 6

7. EXAMPLE – DISCRETE
APPROACH 7

8. EXAMPLE – DISCRETE APPROACH8

9. EXAMPLE – DISTRIBUTED
APPROACH 9

10. EXAMPLE – DISTRIBUTED APPROACH10

11. EXAMPLE – COMPARISON OF DISTRIBUTED AND DISCRETE APPROACH
(discrete diagrams are highlighted in green) 11

12. SHEAR FLOWS DEPENDING ON SHEAR FORCE POSITION
EXAMPLE –
SHEAR FLOWS DEPENDING ON SHEAR FORCE POSITION 12

13. EFFECT OF SHEAR STRESSES ON EQUIVALENT STRESSES
EXAMPLE –
EFFECT OF SHEAR STRESSES ON EQUIVALENT STRESSES 13

14. … Internet is boundless …
WHERE TO FIND MORE INFORMATION?
Megson. An Introduction to Aircraft Structural Analysis. 2010
Chapter 16.3
… Internet is boundless … 14

15. 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.)” 15