Torsion (cont.) - Faculty of Aerospace Engineering

Non-Homogeneous Composite Beams
under Tip Loading
M. Grebshtein, M. Kazar, V. Rovenski and O.Rand
Faculty of Aerospace Engineering
Technion – Israel Institute of Technology
Haifa 32000, Israel.
Presented at the
The 7th Annual Israeli Symposium On Composite Materials and Structures,
Technion - IIT, June 30, 2003.
Faculty of Aerospace Eng., Technion - I.I.T.
1

Outline
• Motivation and Background
> Notation
> Material properties
> Non-homogeneous cross-section definition
> Interlaminar conditions
• The proposed analytical methodology
> St. Venant’s Semi-Inverse methodology
> The three auxiliary problems of plane deformation
> The bending and torsion functions
• Illustrative examples
> Torsion
> Bending
• Concluding remarks
Faculty of Aerospace Eng., Technion - I.I.T.
2

Motivation
• Fiber-reinforced composite materials are extensively used in many engineering
applications (civil and military aircraft, space, automotive, commercial,
etc.), from replacement pieces to all-composite design.
• Composite structures exhibit some profound advantages:
> Specific strength and stiffness
> Improved fatigue characteristics
> Elastic coupling (for beams: “bending-twist”, “bending-extension”
and “extension-twist”).
Faculty of Aerospace Eng., Technion - I.I.T.
3

Existing Methods
• Analytic formulations (linear theory) [Lechnitskii (50), Sokolnikoff (56),
Muskhelishvili (53), Rukhadze].
• Semi-analytic methods [Kosmatka (91), Kim and Dugundje (93),
Rand (93,94,97), Yamane and Friedmann (93),
Pai and Nyfeh (94), Robbins and Reddy (93-95),
Laulusa (94), Zapfe and Lesieutre (95),
Francescatti and Kaiser (96), Betti and Gjelsvik (96),
Song and Librescu (97), Soldatos and Watson (97)].
• High order theories [Robbins and Reddy (93-95), Suresh and Nagaraj (96),
McCarthy and Chattopadhyay (96)].
• Numerical (approximate) methods [Sankar (93), Floros and Smith (97),
Chopra et al (95-97), Makeev and Armanios (99)].
Faculty of Aerospace Eng., Technion - I.I.T.
4

Notation
Tip loads
Surface loads
Body loads
“root”
“outer
surface”
v(x,y,z)
“tip”
u(x,y,z)
w (x,y,z)
Faculty of Aerospace Eng., Technion - I.I.T.
5

Material Properties
Orthotropic
‘x’,y’,z
Faculty of Aerospace Eng., Technion - I.I.T.
6

Non-Homogeneous Cross-Section
Definition
Generic Laminated
Faculty of Aerospace Eng., Technion - I.I.T.
7

The Interlaminar Conditions
Stress resultants (equilibrium):
Displacements (compatibility):
Faculty of Aerospace Eng., Technion - I.I.T.
8

The proposed analytical methodology
St. Venant’s Semi-Inverse methodology (“solution hypothesis”).
The three auxiliary problems of plane deformation (biharmonic BVP).
The bending and torsion functions (harmonic BVP).
Faculty of Aerospace Eng., Technion - I.I.T.
9

Tip Axial Force Effect
Homogeneous
Non-Homogeneous
Faculty of Aerospace Eng., Technion - I.I.T.
10

Transverse Bending Force Effect
bending function of the first kind
Faculty of Aerospace Eng., Technion - I.I.T.
11

The three auxiliary problems of plane deformation
No external surface loads:
Internal loads equilibrium:
{
{
Faculty of Aerospace Eng., Technion - I.I.T.
12

The three auxiliary problems of plane deformation (cont.)
Prescribed deformation discontinuity:
The first auxiliary problem:
{
Faculty of Aerospace Eng., Technion - I.I.T.
13

The three auxiliary problems of plane deformation (cont.)
Biharmonic BVP:
{
Faculty of Aerospace Eng., Technion - I.I.T.
14

Source of the Three Auxiliary Plane-Strain Problems
a) The First Auxiliary Problem:
Extension by a tip force.
b) The Second Auxiliary Problem:
Bending about the x direction by a tip
force or a tip moment.
c) The Third Auxiliary Problem:
Bending about the y direction by a tip
force or a tip moment.
Faculty of Aerospace Eng., Technion - I.I.T.
15

The Bending Function of the First Kind
Harmonic BVP:
Field Equation:
B.C.:
:Continuity
Faculty of Aerospace Eng., Technion - I.I.T.
16

Displacement:
Strain:
Stress:
Field Equation:
Stress:
Torsional Moment Effect
Displacement:
Faculty of Aerospace Eng., Technion - I.I.T.
17

Combining all Tip Loads Effects
Faculty of Aerospace Eng., Technion - I.I.T.
18

Example: Torsion of Non-Homogenous Beam
Faculty of Aerospace Eng., Technion - I.I.T.
19

Torsion (cont.)
Torsion function ϕ
Faculty of Aerospace Eng., Technion - I.I.T.
20

Torsion (cont.)
Stress component τ yz
Faculty of Aerospace Eng., Technion - I.I.T.
21

Torsion (cont.)
τ
Stress component xz
Faculty of Aerospace Eng., Technion - I.I.T.
22

Bending of Non-Homogenous Beam
y
x
Faculty of Aerospace Eng., Technion - I.I.T.
23

Bending of Non-Homogenous Beam
(cont.)
Solution of the third auxiliary problem
Faculty of Aerospace Eng., Technion - I.I.T.
y
x
24

Bending of Non-Homogenous Beam
(cont.)
Solution of the third auxiliary problem (cont.)
x
x
y
y
σ xx
σ yy
Faculty of Aerospace Eng., Technion - I.I.T.
y
x
25

Bending of Non-Homogenous Beam
(cont.)
Solution of the third auxiliary problem (cont.)
x y
σ zz
x
τ xy
Faculty of Aerospace Eng., Technion - I.I.T.
y
y
x
26

Bending of Non-Homogenous Beam (cont.)
y
The bending function χ 1
x
x
The bending function χ 2
Faculty of Aerospace Eng., Technion - I.I.T.
y
27

The Torsion Function
x
The torsion function ϕ
y
Faculty of Aerospace Eng., Technion - I.I.T.
28

Solution for Py
x
σ xx
y
x
Py
σ yy
Faculty of Aerospace Eng., Technion - I.I.T.
y
y
x
29

Solution for Py (cont.)
x
σ zz
y
Py
τ xy
Faculty of Aerospace Eng., Technion - I.I.T.
x
y
y
x
30

Solution for Py (cont.)
x
τ yz
y
x
τ xz
Py
Faculty of Aerospace Eng., Technion - I.I.T.
y
y
x
31

Concluding Remarks
• The paper presents a derivation and closed-form exact analytic
solutions for the Saint-Venant's problem of beams of nonhomogenous
cross-sections.
• The proposed methodology is composed of six sub-solutions
for the tip loads effect.
• The derivation is founded on three auxiliary biharmonic BVPs
and exploits Saint-Venant's semi-inverse methodology and known
solutions for homogeneous beams. In addition, the derivation
employs Neumann-type BVPs for the bending and the torsion
functions.
• Illustrative examples of solutions were carried out via polynomial
expansion of the involved stress functions.
Faculty of Aerospace Eng., Technion - I.I.T.
32