```Lecture #6
Normal stresses
in thin-walled beam
FOREWORD
The calculation of stresses in the wing is generally a
statically indeterminate problem, involving three basic
mechanical equations: equilibrium, compatibility,
constitutive.
2
BASIC EQUATIONS OF SOLID MECHANICS
Equilibrium
equations
This is not only the sum of
forces or moments, but applies
for elementary volume as well
Constitutive
equations
Physical law, expresses
the relation between
stress and strain
Compatibility
equations
Solid body should
remain continuous
while being deformed
3
WAYS TO FIND NORMAL STRESSES IN THE WING
There are two major possible ways:
1. General energy-based methods for statically
indeterminate problems (effective only for some
specific problems). E.g.: Papkovich theorem (will
study later).
2. Beam bending theory, accordingly modified to take
account of the physical law.
4
WAYS TO SOLVE A SOLID MECHANICS PROBLEM
Displacements
are set as
unknowns
Strains are
derived
Stresses are
derived
Equilibrium
equations are
solved
Compatibility
equations
Constitutive
equations
Stresses are
set as
unknowns
Constitutive
equations
Strains are
derived
Equilibrium
equations
Compatibility
equations
Equilibrium
equations
Compatibility
equations are
solved
5
BASIC EQUATIONS IN BEAM THEORY
Equilibrium
equations
Internal force factors are equal to
appropriate integrals of moments
Constitutive
equations
Taken into account using the
reduction coefficients (a.k.a.
effectiveness factors)
Compatibility
equations
Hypotesis of planar cross sections
6
LIMITATIONS OF BEAM THEORY
Beam theory gives unsatisfactory results in
following cases:
1. Low aspect ratio of the wing.
2. Zone of sweptback angle change.
3. Cut-outs or any other irregularities.
7
NORMAL STRESSES IN THIN-WALLED BEAMS
The distribution of normal stresses obeys the
hypothesis of planar cross sections:
w  x, y  a  b  x  c  y
For the case of uniform linear material, it comes to be
 z  x, y 
Nz
A

M
Iy
y
x
M
Ix
x
y
8
CROSS SECTION DISCRETIZATION
(CE – conservative estimation, OE – optimistic estimation)
The discretization of real cross section is usually used
to possess the calculations of moments of inertia and
other geometrical properties:
- small but complex elements like stringers are
substituted by point areas positioned at stringer
center of gravity (CE) or at the skin surface (OE);
- skins are substituted by center lines (CE);
- complex center line is substituted by polygonal
curve;
- portions of skins are substituted by point areas (CE).
9
CROSS SECTION DISCRETIZATION
The problem is to find the moment of inertia.
Dimensions:
a = 60 mm; h = 22 mm; d1 = 4 mm; d2 = 6 mm;
H = 120 mm.
10
CROSS SECTION DISCRETIZATION – variant #1
One option is to
substitute the real
cross section by
center lines with
appropriate thicknesses – usually
used in FEA.
Ix, cm4 Iy, cm4
Exact
975
2478
Discrete
1000
2494
Error,%
+2.5
+0.6
11
CROSS SECTION DISCRETIZATION – variant #2
Another option is to
use concentrated
stiffeners and webs
– very good
discretization.
Ix, cm4 Iy, cm4
Exact
975
2478
Discrete
972
2477
Error,%
-0.3
-0.0
12
CROSS SECTION DISCRETIZATION – variant #3
Finally, all thin
members could be
substituted by point
areas – ideal for
manual
calculations.
Ix, cm4 Iy, cm4
Exact
975
2478
Discrete
972
2419
Error,%
-0.3
-2.4
13
CROSS SECTION DISCRETIZATION – variant #4
If point areas are
positioned at the
skin surface, the
moment of inertia is
overestimated
which is not
recommended.
Ix, cm4 Iy, cm4
Exact
975
2478
Discrete
1092
2419
Error,%
+12
-2.4
14
WAYS TO TAKE THE ACCOUNT OF PHYSICAL LAW
There are many possible ways, some of them are:
2. Method of variable secant modulus.
3. Method of reduction coefficients (a.k.a.
effectiveness factors).
The last one is widely used since it is convenient for
cross sections with multiple materials, and to study a
post-buckling behavior.
15
METHOD OF REDUCTION COEFFICIENTS
The reduction coefficient is the ratio between the real
stress in the member and the fictitious stress obeying
the Hook law  i , f  E f   i , f :
i 
i
 i, f
The reduction coefficient is introduced to maintain the
condition that fictitious strain is equal to real one:
 i   i, f
16
METHOD OF REDUCTION COEFFICIENTS
The method of reduction
coefficient is iterative one. At
the first iteration, we set the
initial reduction coefficient as
the ratio between Young
moduli:
i 
Ei
E i, f
Then we calculate fictitious
geometrical properties:
Ai , f   i  Ai
17
METHOD OF REDUCTION COEFFICIENTS
Next we calculate the
fictitious stress and find the
real stress, forming a new
iteration of reduction
coefficient:
i 
i
 i, f
The process stops when
values of reduction
coefficients are converged for
all points in the section.
18
METHOD OF REDUCTION COEFFICIENTS
Calculation of
red.coef. 
Fictitious geom.
properties
Fictitious and real
stresses
New reduction
coefficients 
Resultant real
stresses
Initial value is the ratio between
Young moduli
Area and axial moments of
inertia are calculated using
fictitious areas of members
Real stresses are derived from
material physical law or
compression diagram for panel
The process is stopped upon
convergence of red.coefficients
19
Megson. An Introduction to Aircraft Structural Analysis. 2010
Chapter 15 – beam bending theory;
Chapter 19 - discretization of the cross section.
Method of effectiveness factors (reduction coefficients) is
explained in
Bruhn. Airplane Design Handbook
Chapter A19.11
… Internet is boundless …
20
TOPIC OF THE NEXT LECTURE
Shear stresses in thin-walled beam
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.)”
21
```