1. Lecture #7 Shear stresses in thin-walled beam

2. 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  :

3. 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.

4. 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:

5. FORMULA FOR THE SHEAR FLOW 5 If the normal stress is caused only by bending moments, we get:

6. FORMULA FOR THE SHEAR FLOW 6 Formula used for shear flows in Structural Analysis: Compare to Zhuravsky formula used in Mechanics of Materials:

7. 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.

8. CALCULATION OF STATIC MOMENTS 8 Exact formula: For a discrete cross section:

9. CALCULATION OF STATIC MOMENTS - EXAMPLE 9 Given cross section Vertical force of 100 kN is applied. Continuous approach Discrete approach

10. CALCULATION OF STATIC MOMENTS - EXAMPLE 10 Discrete approach

11. CALCULATION OF STATIC MOMENTS - EXAMPLE 11 Continuous approach

12. CALCULATION OF STATIC MOMENTS - EXAMPLE 12 Comparison of continuous and discrete approach used in Structural Analysis and the approach used in Mechanics of Materials.

13. CALCULATION OF STATIC MOMENTS 13 Symmetry of static moments: 1) If u is the axis of symmetry, the S u will be symmetrical with respect to u. 2) If u is the axis of symmetry, and v is the axis perpendicular to u, the S v will be antisymmetrical with respect to u.

14. WHERE TO FIND MORE INFORMATION? 14 Megson. An Introduction to Aircraft Structural Analysis. 2010 Chapter 16 … Internet is boundless …

15. TOPIC OF THE NEXT LECTURE 15 Shear center All materials of our course are available at department website k102.khai.eduk102.khai.edu 1. Go to the page “Библиотека”Библиотека 2. Press “Structural Mechanics (lecturer Vakulenko S.V.)”Structural Mechanics (lecturer Vakulenko S.V.)