Lecture #1Torsion rigidity and shear centerof closed contour

SHEAR STRESSES RELATED QUESTIONS

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- 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.

The relation between unit moment M and unit shear flow

q could be found using Bredt’s formula:

CALCULATION OF TWIST ANGLEFOR CLOSED CROSS SECTION

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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 (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 and real strains .

V

M dV

M q

The absolute angle of rotation relation is calculated by integration and

using the starting value 0 :

CALCULATION OF TWIST ANGLEFOR CLOSED CROSS SECTION

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From the formula from previous slide we can derive that

Here z is the length of portion of thin-walled beam

having the twist angle ; t is tangential coordinate.

Taking M=1, we get the relative twist angle

t

z q qdt

M G

00

z

z z dz

1; .t

q qdt q

G

EXAMPLE OF TWIST ANGLE CALCULATIONS – GIVEN DATA

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EQUIVALENT DISCRETE CROSS SECTION 27GPaG

EXAMPLE – SHEAR FLOW DIAGRAMS AND TWIST ANGLE

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, kN mq t

DISCRETE APPROACH

CONTINUOUS APPROACH

= - 0.473 °/m

= - 0.474 °/m(0.1% error)

1 i i

i i i

q LG

EXAMPLE – TWIST ANGLES FOR THREE DIFFERENT SHEAR FORCE POSITIONS

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= - 0.473 °/m

= + 1.327 °/m

= - 0.988 °/m

Example: Taking the hollow circular cross section we get

I = 2R3 which is same with Mechanics of

Materials.

TORSION RIGIDITY

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In Mechanics of Materials, the formula for relative twist angle was

The denominator G ∙ I is referred as torsion rigidity.

Taking the formula for for thin-walled beam and using Bredt’s formula, we can derive

TMG I

2

t

dtG I

G

PROPERTIES OF SHEAR CENTER

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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.

CALCULATION OF SHEAR CENTER POSITIONFOR 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:

T y Q SC

dM Q X X G I

dz

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CALCULATION OF SHEAR CENTER POSITIONFOR CLOSED CROSS SECTION

Thus, we need to calculate the torsion rigidity of wingbox G·I and relative twist angle :

2

t

dtG I

G

1; .t

q qdt q

G

Finally, we can get shear center position XSC by

formula

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SC Qy

G IX X

Q

, kN mq t

shear center

For the abovementioned example, we get:- torsion rigidity G·I = 1.1·106 N·m2 ;

- relative twist angle = - 0.473 °/m ;- torsional moment MT = - 9.2 kN·m ;- shear center coordinate XSC = 0.192 m .

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CALCULATION OF SHEAR CENTER POSITIONFOR CLOSED CROSS SECTION - EXAMPLE

WHERE TO FIND MORE INFORMATION?

Megson. An Introduction to Aircraft Structural Analysis. 2010Chapter 17.1 for torsion of closed cross sectionsParagraph 16.3.2 for shear center in closed cross sections

… Internet is boundless …13

TOPIC OF THE NEXT LECTURE

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Calculation of shear flows in multiple closed contours

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.)”