1. Torsion rigidity and shear center of closed contour
Lecture #1
Torsion rigidity and shear center
of 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. CALCULATION OF TWIST ANGLE FOR CLOSED CROSS SECTION
To find the twist angle for a portion of thin-walled beam subjected to external loads we use Mohr’s formula under virtual displacement method:
At the left side we have real twist angle Df (from external loads) multiplied by ̅M (unity moment). The right side represents the energy stored in the volume V and calculated using unity shear stresses ̅t and real strains g.
The relation between unit moment ̅M and unit shear flow ̅q could be found using Bredt’s formula: 3

4. CALCULATION OF TWIST ANGLE FOR CLOSED CROSS SECTION
From the formula from previous slide we can derive that
Here Dz is the length of portion of thin-walled beam having the twist angle Df ; t is tangential coordinate. Taking ̅M=1, we get the relative twist angle
The absolute angle of rotation relation f is calculated by integration and
using the starting value f0 : 4

5. EXAMPLE OF TWIST ANGLE CALCULATIONS – GIVEN DATA
EQUIVALENT DISCRETE CROSS SECTION 5

6. EXAMPLE – SHEAR FLOW DIAGRAMS AND TWIST ANGLE
CONTINUOUS APPROACH
q = °/m
DISCRETE APPROACH
q = °/m
(0.1% error) 6

7. EXAMPLE – TWIST ANGLES FOR THREE DIFFERENT SHEAR FORCE POSITIONS
q = °/m
q = °/m
q = °/m 7

8. TORSION RIGIDITY
In Mechanics of Materials, the formula for relative twist angle was
The denominator G ∙ Ir is referred as torsion rigidity. Taking the formula for q for thin-walled beam and using Bredt’s formula, we can derive
Example: Taking the hollow circular cross section we get Ir = 2pR3d which is same with Mechanics of Materials. 8

9. PROPERTIES OF SHEAR CENTER
1. The transverse force applied at shear center does not lead to the torsion of thin-walled beam.
2. The shear center is a center of rotation for a section of thin-walled beam subjected to pure torsion.
3. The shear center is a position of shear flows resultant force, if the thin-walled beam is subjected to pure shear. 9

10. CALCULATION OF SHEAR CENTER POSITION FOR CLOSED CROSS SECTION
For closed cross section, we use equilibrium equations for moments to find q0. Thus, moment of shear flows MC (q) would depend on the position of force Qy and would not be useful to find XSC.
We find the shear center position from the condition that the twist angle should correspond to the torsional moment which is calculated as Qy multiplied by the distance to shear center: 10

11. CALCULATION OF SHEAR CENTER POSITION FOR CLOSED CROSS SECTION
Thus, we need to calculate the torsion rigidity of wingbox G·Ir and relative twist angle q :
Finally, we can get shear center position XSC by formula 11

12. CALCULATION OF SHEAR CENTER POSITION
FOR CLOSED CROSS SECTION - EXAMPLE
For the abovementioned example, we get:
- torsion rigidity G·Ir = 1.1·106 N·m2 ;
- relative twist angle q = °/m ;
- torsional moment MT = kN·m ;
- shear center coordinate XSC = m .
shear center 12

13. … Internet is boundless …
WHERE TO FIND MORE INFORMATION?
Megson. An Introduction to Aircraft Structural Analysis. 2010
Chapter 17.1 for torsion of closed cross sections
Paragraph for shear center in closed cross sections
… Internet is boundless … 13

14. Calculation of shear flows in multiple closed contours
TOPIC OF THE NEXT LECTURE
Calculation of shear flows in multiple closed contours
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.)” 14