Lecture #8
Shear stresses
in closed contour
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.
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FORMULA FOR THE SHEAR FLOW
The formula for shear flow has a constant term q0:
Qy
Qx
q  t   q0 
 S y (t ) 
 S x (t )
Iy
Ix
constant
term
variable
term
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DETERMINATION OF CONSTANT SHEAR FLOW
Variable term qf satisfies two equations of equilibrium
for sums of forces.
Constant term q0 satisfies the equation of equilibrium
for sum of moments.
q  t   q0  q f  t 
Sum of moments equation:
M q 0  M qf  M Q  0
where Mq0 , Mqf - moments from constant and
variable parts of shear flow, respectively;
MQ – moment from external shear force.
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BREDT’S FORMULA FOR MOMENT
FROM CONSTANT SHEAR FLOW
Bredt’s formula (put minus if tangential coordinate
direction does not correspond to positive direction of
moment)
where W is an area of closed cross section multiplied
by 2.
Finally, constant part of shear flow could be found:
The signs in this formula should be used according to
directions of moments.
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EXAMPLE – GIVEN DATA
EQUIVALENT DISCRETE CROSS SECTION
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q f  t  , kN m
EXAMPLE –
DISCRETE
APPROACH
q0 , kN m
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EXAMPLE – DISCRETE APPROACH
q  t  , kN m
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q f  t  , kN m
EXAMPLE –
DISTRIBUTED
APPROACH
q0 , kN m
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EXAMPLE – DISTRIBUTED APPROACH
q  t  , kN m
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EXAMPLE –
COMPARISON OF
DISTRIBUTED AND
DISCRETE
APPROACH
(discrete diagrams
are highlighted in
green)
qf t 
qt 
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EXAMPLE –
SHEAR FLOWS
DEPENDING ON
SHEAR FORCE
POSITION
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 xz , MPa
EXAMPLE –
EFFECT OF
SHEAR
STRESSES ON
EQUIVALENT
STRESSES
 z , MPa
 Mises , MPa
 Mises    3  
2
z
2
xz
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WHERE TO FIND MORE INFORMATION?
Megson. An Introduction to Aircraft Structural Analysis. 2010
Chapter 16.3
… Internet is boundless …
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TOPIC OF THE NEXT LECTURE
Shear center
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.)”
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