1. Torsion of opened cross sections.
Lecture #3
Torsion of opened cross sections.
Loads on frame due to fuselage bending

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. SHEAR CENTER AND TORSION - ILLUSTRATION3

4. TORSION IN MECHANICS OF MATERIALS
The measure of resistance to torsion is a polar moment of inertia Ir .
Specific twist angle
Maximal shear stress 4

5. TORSION IN THIN-WALLED CROSS SECTIONS
The polar moment of inertia Ir is calculated as a sum for rectangular portions of thin-walled cross section.
Specific twist angle Maximal shear stress 5

6. TORSION IN THIN-WALLED CROSS SECTION
Torsional moment is 1000 N·m.
Material is steel, G = 77 GPa.
Moment of inertia Ir = 4.14 cm4.
Shear stress tmax = MPa. 6

7. CALCULATION OF DEPLANATIONS (WARPING)
Since the hypothesis of planar cross section is not valid, the beam theory is not applicable.
Thus, specific theory developed by Vlasov is used.
Vlasov’s theory is based on two main hypotheses:
1) The cross section keeps its shape and rotates as a whole around the shear center.
2) There are no shear strains and stresses at the middle plane (gtz = 0). 7

8. u – displacement along t axis.
CALCULATION OF DEPLANATIONS
where w – displacement along z axis (deplanation);
u – displacement along t axis.
where f – angle of rotation of cross section along z axis;
r – lever from the shear center to the direction of t axis at the given point. 8

9. CALCULATION OF DEPLANATIONS
where w0 –displacement at the start point.
If start point is set on the axis of symmetry, we get
where w(t) – sectorial coordinate (doubled area covered by rotation of radius-vector): 9

10. CALCULATION OF DEPLANATIONS
Analytical values:
Max is 0.67 mm
At the corner is mm 10

11. B – bimoment (kind of scalar force factor):
NORMAL STRESSES AT RESCTRICTED TORSION
Normal stresses could be found using the formula
where Iw – sectorial moment of inertia:
B – bimoment (kind of scalar force factor): 11

12. NORMAL STRESSES AT RESCTRICTED TORSION
The distribution of normal stresses for real structure is usually quite complex, so it is usually wise to use FEA. 12

13. DISTORSIONAL BUCKLING
The open contour torsion theory is also used in the analysis of special buckling modes, called distorsional buckling:
lateral torsional buckling
torsional buckling (flexural-torsional instability)
The detailed analytical analysis of these phenomena is quite complicated, so it’s not explained in our course. However, it’s quite important for aerospace engineers. The proper reference for self-study is given on the last slide. 13

14. DISTORSIONAL BUCKLING
Lateral torsional buckling is widely happening in civil engineering structures 14

15. DISTORSIONAL BUCKLING
Flexural-torsional buckling is always analyzed for stringers in aerospace structures 15

16. COMPARISON OF OPENED AND CLOSED CONTOURS
For a tube of 25 mm diameter and thickness of 2 mm we get:
For the closed contour we get the polar moment of inertia of cm4, while for opened – cm4, which is 57 times smaller.
If we would increase the diameter, the difference will be increased dramatically.
Let’s take a thin-walled circle with radius 1 m and thickness of 2 mm.
For the closed contour we get 628,300 cm4, while for opened – only 1.67, which is 375 thousands times smaller. 16

17. Ix is moment of inertia for cross
LOADING OF FUSELAGE FRAME DUE TO BENDING OF FUSELAGE
If the cross section is subjected to bending with moment Mx , the specific normal force is equal to
Here d is effective thickness of
skin (includes stringers);
Ix is moment of inertia for cross
section: 17

18. LOADING OF FUSELAGE FRAME DUE TO BENDING OF FUSELAGE
The relative bending angle between two frames fx is equal to
Here a is a distance between
frames.
Normal stresses have a vertical projection due to the presence of bending angle. This projection tends to compress the frame as shown at the figure. 18

19. LOADING OF FUSELAGE FRAME DUE TO BENDING OF FUSELAGE
The distributed load on the frame could found as
By making few transformations, we get 19

20. … Internet is boundless …
WHERE TO FIND MORE INFORMATION?
Megson. An Introduction to Aircraft Structural Analysis. 2010
Chapter 17.2
For flexural-torsional buckling, refer to Chapter 8.6
… Internet is boundless … 20