Advanced structures - wing section, beams, bending, shear flow and shear center

AE2302
AIRCRAFT STRUCTURES-II
INTRODUCTION

Course Objective
 The purpose of the course is to
teach the principles of solid and
structural mechanics that can be
used to design and analyze
aerospace structures, in
particular aircraft structures.

Function of Aircraft Structures
General
The structures of most flight vehicles are thin walled
structures (shells)
Resists applied loads (Aerodynamic loads acting on the wing
structure)
Provides the aerodynamic shape
Protects the contents from the environment

Definitions
Primary structure:
A critical load-bearing structure on an aircraft.
If this structure is severely damaged, the
aircraft cannot fly.
Secondary structure:
Structural elements mainly to provide enhanced
aerodynamics. Fairings, for instance, are found
where the wing meets the body or at various
locations on the leading or trailing edge of the
wing.

Definitions…
Monocoque structures:
Unstiffened shells. must be
relatively thick to resist bending,
compressive, and torsional loads.

Definitions…
Semi-monocoque Structures:
Constructions with stiffening members that may
also be required to diffuse concentrated loads into
the cover.
More efficient type of construction that permits
much thinner covering shell.

Function of Aircraft Structures:
Part specific
Skin
reacts the applied torsion and shear forces
transmits aerodynamic forces to the longitudinal
and transverse supporting members
acts with the longitudinal members in resisting the
applied bending and axial loads
acts with the transverse members in reacting the
hoop, or circumferential, load when the structure is
pressurized.

Part specific
Ribs and Frames
1. Structural integration of the wing and fuselage
2. Keep the wing in its aerodynamic profile

Part specific
Spar
1. resist bending and axial loads
2. form the wing box for stable torsion resistance

Part specific
 Stiffener or Stringers
1. resist bending and axial loads along with the skin
2. divide the skin into small panels and thereby
increase its buckling and failing stresses
3. act with the skin in resisting axial loads caused
by pressurization.

Simplifications
1. The behavior of these structural elements is often
idealized to simplify the analysis of the assembled
component
2. Several longitudinal may be lumped into a
single effective
3. longitudinal to shorten computations.
4. The webs (skin and spar webs) carry only shearing
stresses.
5. The longitudinal elements carry only axial stress.
6. The transverse frames and ribs are rigid within
their own planes, so that the cross section is
maintained unchanged during loading.

UNIT-I
Unsymmetric Bending of
Beams
The learning objectives of this chapter are:
•Understand the theory, its limitations, and
its application in design and analysis of
unsymmetric bending of beam.

UNIT-I
UNSYMMETRICAL BENDING
The general bending stress equation for elastic, homogeneous beams is given as
where Mx and My are the bending moments about the x and y centroidal axes,
respectively. Ix and Iy are the second moments of area (also known as
moments of inertia) about the x and y axes, respectively, and Ixy is the product
of inertia. Using this equation it would be possible to calculate the bending
stress at any point on the beam cross section regardless of moment orientation
or cross-sectional shape. Note that Mx, My, Ix, Iy, and Ixy are all unique for a
given section along the length of the beam. In other words, they will not
change from one point to another on the cross section. However, the x and y
variables shown in the equation correspond to the coordinates of a point on the
cross section at which the stress is to be determined.
(II.1)

Neutral Axis:
 When a homogeneous beam is subjected to elastic bending, the neutral axis
(NA) will pass through the centroid of its cross section, but the orientation of the
NA depends on the orientation of the moment vector and the cross sectional
shape of the beam.
 When the loading is unsymmetrical (at an angle) as seen in the figure below, the
NA will also be at some angle - NOT necessarily the same angle as the bending
moment.
 Realizing that at any point on the neutral axis, the bending strain and stress
are zero, we can use the general bending stress equation to find its
orientation. Setting the stress to zero and solving for the slope y/x gives
(

UNIT-II
SHEAR FLOW AND SHEAR CEN
Restrictions:
1. Shear stress at every point in the beam must be less than the
elastic limit of the material in shear.
2. Normal stress at every point in the beam must be less than the elastic
limit of the material in tension and in compression.
3. Beam's cross section must contain at least one axis of symmetry.
4. The applied transverse (or lateral) force(s) at every point on the beam
must pass through the elastic axis of the beam. Recall that elastic axis
is a line connecting cross-sectional shear centers of the beam. Since
shear center always falls on the cross-sectional axis of symmetry, to
assure the previous statement is satisfied, at every point the
transverse force is applied along the cross-sectional axis of symmetry.
5. The length of the beam must be much longer than its cross sectional
dimensions.
6. The beam's cross section must be uniform along its length.

Shear Center
If the line of action of the force passes through the
Shear Center of the beam section, then the beam
will only bend without any twist. Otherwise, twist will
accompany bending.
The shear center is in fact the centroid of the internal
shear force system. Depending on the beam's cross-
sectional shape along its length, the location of shear
center may vary from section to section. A line
connecting all the shear centers is called the elastic
axis of the beam. When a beam is under the action of
a more general lateral load system, then to prevent
the beam from twisting, the load must be centered
along the elastic axis of the beam.

Shear Center
 The two following points facilitate the determination of the shear center
location.
1. The shear center always falls on a cross-sectional axis of symmetry.
2. If the cross section contains two axes of symmetry, then the shear center is
located at their intersection. Notice that this is the only case where shear
center and centroid coincide.

SHEAR STRESS DISTRIBUTION
RECTANGLE T-SECTION

EXAMPLES
 For the beam and loading shown, determine:
(a) the location and magnitude of the maximum transverse shear force 'Vmax',
(b) the shear flow 'q' distribution due the 'Vmax',
(c) the 'x' coordinate of the shear center measured from the centroid,
(d) the maximun shear stress and its location on the cross section.
Stresses induced by the load do not exceed the elastic limits of the material. NOTE:In this problem
the applied transverse shear force passes through the centroid of the cross section, and not its
shear center.
FOR ANSWER REFER
http://www.ae.msstate.edu/~masoud/Teaching/exp/A14.7_ex3.html

Shear Flow Analysis for
Unsymmetric Beams
 SHEAR FOR EQUATION FOR UNSUMMETRIC SECTION IS

SHEAR FLOW DISTRIBUTION
 For the beam and loading shown, determine:
 (a) the location and magnitude of the maximum
transverse shear force,
 (b) the shear flow 'q' distribution due to 'Vmax',
 (c) the 'x' coordinate of the shear center measured
from the centroid of the cross section.
 Stresses induced by the load do not exceed the
elastic limits of the material. The transverse shear
force is applied through the shear center at every
section of the beam. Also, the length of each member
is measured to the middle of the adjacent member.
 ANSWER REFER
 http://www.ae.msstate.edu/~masoud/Tea
ching/exp/A14.8_ex1.html

Beams with Constant Shear Flow
Webs
Assumptions:
1. Calculations of centroid, symmetry, moments of
area and moments of inertia are based totally on
the areas and distribution of beam stiffeners.
2. A web does not change the shear flow between two
adjacent stiffeners and as such would be in the state
of constant shear flow.
3. The stiffeners carry the entire bending-induced
normal stresses, while the web(s) carry the entire
shear flow and corresponding shear stresses.

Analysis
 Let's begin with a simplest thin-walled stiffened beam. This means a beam with
two stiffeners and a web. Such a beam can only support a transverse force that
is parallel to a straight line drawn through the centroids of two stiffeners.
Examples of such a beam are shown below. In these three beams, the value of
shear flow would be equal although the webs have different shapes.
 The reason the shear flows are equal is that the distance between two adjacent
stiffeners is shown to be 'd' in all cases, and the applied force is shown to be
equal to 'R' in all cases. The shear flow along the web can be determined by the
following relationship

Important Features of
Two-Stiffener, Single-Web Beams:
1. Shear flow between two adjacent stiffeners is constant.
2. The magnitude of the resultant shear force is only a function of the
straight line between the two adjacent stiffeners, and is absolutely
independent of the web shape.
3. The direction of the resultant shear force is parallel to the straight line
connecting the adjacent stiffeners.
4. The location of the resultant shear force is a function of the enclosed
area (between the web, the stringers at each end and the arbitrary
point 'O'), and the straight distance between the adjacent stiffeners.
This is the only quantity that depends on the shape of the web
connecting the stiffeners.
5. The line of action of the resultant force passes through the shear
center of the section.

EXAMPLE
 For the multi-web, multi-stringer open-section beam shown, determine
(a) the shear flow distribution,
(b) the location of the shear center
 Answer

UNIT-III
Torsion of Thin - Wall Closed
Sections
 Derivation
Consider a thin-walled member with a closed cross section subjected to pure torsion.

Examining the equilibrium of a small
cutout of the skin reveals that

Angle of Twist
By applying strain energy equation due to shear and
Castigliano's Theorem the angle of twist for a thin-
walled closed section can be shown to be
Since T = 2qA, we have
If the wall thickness is constant along each segment of
the cross section, the integral can be replaced by a
simple summation

Torsion - Shear Flow Relations in Multiple-
Cell Thin- Wall Closed Sections
 The torsional moment in terms of the internal
shear flow is simply

Derivation
For equilibrium to be maintained at a exterior-interior wall (or web)
junction point (point m in the figure) the shear flows entering
should be equal to those leaving the junction
Summing the moments about an arbitrary point O, and assuming clockwise
direction to be positive, we obtain
The moment equation above can be simplified to

Shear Stress Distribution and Angle of
Twist for Two-Cell Thin-Walled Closed
Sections
 The equation relating the shear flow along the exterior
wall of each cell to the resultant torque at the section is given as
This is a statically indeterminate problem. In order
to find the shear flows q1 and q2, the compatibility
relation between the angle of twist in cells 1 and 2 must be used. The compatibility
requirement can be stated as
where

 The shear stress at a point of interest is found according to the
equation
 To find the angle of twist, we could use either of the two twist formulas
given above. It is also possible to express the angle of twist equation
similar to that for a circular section

Shear Stress Distribution and Angle of Twist for
Multiple-Cell Thin-Wall Closed Sections
 In the figure above the area outside of the cross section will be designated as
cell (0). Thus to designate the exterior walls of cell (1), we use the notation 1-
0. Similarly for cell (2) we use 2-0 and for cell (3) we use 3-0. The interior walls
will be designated by the names of adjacent cells.
 the torque of this multi-cell member can be related to the shear flows in exterior
walls as follows

 For elastic continuity, the angles of twist in all
cells must be equal
 The direction of twist chosen to be positive is clockwise.

TRANSVERSE SHEAR LOADING OF BEAMS WITH CLOSED
CROSS SECTIONS

EXAMPLE
 For the thin-walled single-cell rectangular beam and loading shown, determine
(a) the shear center location (ex and ey),
(b) the resisting shear flow distribution at the root section due to the applied load
of 1000 lb,
(c) the location and magnitude of the maximum shear stress
ANSWER REFER
http://www.ae.msstate.edu/~masoud/Teaching/exp/A15.2_ex1.html

Advanced structures - wing section, beams, bending, shear flow and shear center

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