Lecture #7 Shear stresses in thin-walled beam. THE CONCEPT OF SHEAR FLOW 2 Both of possible stresses – normal and shear – usually act in the load-carrying.

Lecture #7Shear stressesin thin-walled beam

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 :

q

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.

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:

00

t dq t q dt

dz

0dq ddt dz

FORMULA FOR THE SHEAR FLOW

5

If the normal stress is caused only by bending moments, we get:

00

t dq t q dt

dz

, yz xz

y x

MN Mx y x y

A I I

0 ( ) ( )yxy x

y x

QQq t q S t S t

I I

FORMULA FOR THE SHEAR FLOW

6

Formula used for shear flows in Structural Analysis:

0 ( ) ( )yxy x

y x

QQq t q S t S t

I I

Compare to Zhuravsky formula used in Mechanics of Materials:

( )y x

x

Q S yy

b y I

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.

CALCULATION OF STATIC MOMENTS

8

Exact formula:

For a discrete cross section:

0

t

xS t y t t dt

1

n t

x i ii

S t A y

CALCULATION OF STATIC MOMENTS - EXAMPLE

9

Givencross section

Vertical force of 100 kN is applied.

Continuousapproach

Discreteapproach

CALCULATION OF STATIC MOMENTS - EXAMPLE

10

Discrete approach 3, cmxS

, kN mq , MPa

CALCULATION OF STATIC MOMENTS - EXAMPLE

11

Continuous approach 3, cmxS

, kN mq , MPa

CALCULATION OF STATIC MOMENTS - EXAMPLE

12

Comparison of continuous and discrete approach used in Structural Analysis and the approach used in Mechanics of Materials.

, MPa

CALCULATION OF STATIC MOMENTS

13

Symmetry of static moments:

1) If u is the axis of symmetry, the Su will be symmetrical with respect to u.

2) If u is the axis of symmetry, and v is the axis perpendicular to u, the Sv will be antisymmetrical with respect to u.

WHERE TO FIND MORE INFORMATION?

14

Megson. An Introduction to Aircraft Structural Analysis. 2010Chapter 16

… Internet is boundless …

TOPIC OF THE NEXT LECTURE

15

Shear center

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.)”