Recap of Important Concepts in Mechanics of Materials
Safety considerations of aircraft structures
Methodology for stress analysis
Strength, Stiffness, Stability, Vibration External loads internal loads stresses and deformations
Overview of stresses and deformation in aircraft
Fuselage (bending normal & shear stress; twisting shear stress; internal pressure normal stress) Wing (bending normal & shear stresses; twisting shear stress;)
Fundamentals of equilibrium
Force and moment equilibrium
Stress and strain
Normal stresses and strains of typical structures under in-plane force, bending, and internal pressure Shear stresses and strains of typical structures under bending and twisting Plane stress condition and Mohr’s circle method
Failure theories
Max. strength criterion for brittle materials Max. shear stress criterion (Tresca) Max. distortion energy criterion (Von Mises)
Sectional properties
(Polar) Second moment of area of various sections and parallel axial theorom Product of second moment of area of various sections
MECH3650 Aircraft Structures Torsion of thin-walled structures Prof Jinglei YANG Room 2553 (lift 27&28) 3469 2298
[email protected]
twisting of the fuselage
twisting of the wing
This set of notes and supplementary information can be found at
MECH3650 Aircraft Structures Torsion of thin-walled structures Prof Jinglei YANG Room 2553 (lift 27&28) 3469 2298
[email protected]
twisting of the fuselage
twisting of the wing
This set of notes and supplementary information can be found at
Contents of topic 4
Review of Torsion Development of theory for thin-walled closed section
Development of theory for thin-walled open section
Single cell and multi-cell Hybrid section
Development of theory for warping of thin-walled section
Shear Stresses and Deformations of Torsion
Thrust from engine located below wing
Sudden banking
Flexibility of wing tip
Asymmetric loads
Shear stresses in the skin of the wing.
Shear stresses in the skin of the fuselage.
Due to the thin-walled nature of aircraft structures, torsional deformation and stresses are of major concern.
Torsion of Solid Circular Shafts When a solid circular shaft is subjected to a torque about its twist center, the relationships between the angle of twist, shearing stresses, and the twisting moment are deduced from experiments. From these deductions, the basic assumptions used in the development of the torsion equations for solid circular section are: 1) plane section remains plane before and after deformations. 2) shear strains g vary linearly from the central axis reaching gmax at the periphery. 3) Within the elastic limit of the material, Hooke’s law can be used. 4) material is homogeneous. 5) deformations are small and hence small angle approximation is valid. 6) torque is applied through the twist centre, i.e the section will only twist and does not bend. 7) the effect of axial constraint is neglected.
Response of Solid Shafts subject to torque
Non-Circular Section
Circular Section
Response of Thin-Walled Shafts subject to torque
Thin-Walled Open Section
Thin-Walled Closed Section
Axial Constraint in Torsion of Thin-Walled Shafts walls develop both axial stresses and shear stresses
Constrained end develops axial stresses
St Venant free torsion
Torsion of Circular Shafts If plane section remains plane after deformation, then lines OC, BC (undeform) and OC’, BC’ (deformed) will always be straight. It can be shown from the deformed shaft, that shear strain g varies linearly from the central axis. MOMENT EQUILIBRIUM
DEFORMATION
T B g
C’
dz
g dz
g
C
rdq r
dz d q dz
dF
r O T
d q
Gg Gr
dq
dF r
dF dA
dF
dA
dT rdF r dA
T r dA r 2G dq
dF
d q dA dz
2
d q
Solid and Hollow Circular Shafts Combining the two equations, the equations of the elementary theory of torsion are derived: r
T J
G
d q dz
The polar moment of inertia, J , depends on the section of the shaft: 4 ro r i 4 2 J 2 rm3t
hollow shaft
J
thin tube
The corresponding shear stress in the thin tube can be expressed as:
Tr m 3
2 rm t
T 2 Ao t
Shear stress is constant over the thickness in the wall of the tube of constant t .
Torsion of Thin-Walled Single Cell Closed Sections In the absence of axial constraint, a closed tube subjected to a torque T at its twist center will result in a pure shear stress system. The shear stress varies linearly with the radius r in a thick-walled closed section. However when the thickness of the tube is small compared to other dimensions, the stress can be assumed to be uniform across the thickness. For non-circular thin-walled tube, the stress distribution across the wall thickness is also uniform. This observation greatly simplifies the analysis of thin-walled structures in torsion.
Thin-Walled Single Cell Closed Section Consider a tube of an arbitrary shape with varying wall thickness across its section but constant along its length, and subjected to a torque T through its twist center.
Examine the force equilibrium of an element cut from the tube.
1t 1dz 2t 2 dz
1t 1 2t 2 q constant The symbol q is defined as the shear flow in the tube wall.
z dz s ds
Bredt-Batho Theory In the cross-section of the tube, the shear force acting on the element ds is q ds. The applied torque T is resisted by shear force q ds times distance r about the twist center :
T
q ds r q r ds T 2 Ao q
ds
r
O
q
T
2 Ao
where Ao is the total area enclosed by the mid-line of the tube.
q ds
In this derivation, the sweeping of radius about point O is counterclockwise (same direction as applied torque) and this action of sweep yields a “+r” . The theory of the torsion of closed tubes is known as the BredtBatho theory.
For any thin-walled tube, the shear stress at any point of the tube where the wall thickness is t : zs
q
T 2 A
Deformation of Thin Walled Closed Sections The angle of twist per unit length of the tube is defined as f =d q /dz . The elastic shear strain energy per unit length is,
1
U g V U 2
2
2G dV
vol
U dz
2
T
8 A Gt ds 2 o
T
2 2
8 Ao G
dz t
ds
ds
t
dV=tds dz
The external work done per unit length of the tube is T f /2 , Hence equating this work to the internal strain energy yields :
f
dq dz
T
4A G 2 o
ds t
or f
1 2 Ao
q
Gt ds
Equating this to the torsion formula f =T/JG, J can be defined as : J
For thin-walled circular tube of constant t ,
4 r
t 2 r t
2 2
J
2 r
3
4 Ao2 ds t
Example: Thin-Walled Tube of constant (segment) thickness in Torsion (Benham, Crawford & Armstrong, “Mechanics of Engineering Materials”)
The light-alloy stabilizing strut of a high-wing monoplane is 2m long and has the cross section shown. Determine the torque that can be sustained and the angle of twist if the max shear stress is limited to 28MPa. Take G=27GPa. The enclosed area, Ao=(25)2 + 50x50=4460mm2 The maximum shear stress depends on the minimum wall thickness,
zs
q t
T
T 2 Aot zs 2 4460 2 28 500kNmm
2 Aot
The angle of twist,
q
dq dz
T
4 Ao2G
ds t
500000 2000 4 44602 27000 0.0476
d
where
102.4
2.73o
25 50 2 2 102.4 t 3 2
ds
Example: Tube of varying wall thickness in Torsion An aircraft wing consists of the cross section shown in the figure. The inclined webs have linearly varying thicknesses. The length of one half of the wing measured from the fuselage connection to its tip is 6 m. Assuming that fuselage connection is rigid and the wing can be treated as a cantilever. Calculate the maximum shear stress in the wing, and the angle of twist at the free end when the wing is subjected to a uniformly distributed torque of 25 kN-m per metre length. Shear modulus G=28.5 GPa throughout and assumes shear flow is constant in the walls at any section.
400 mm
3.5 mm
500 mm
375 mm
3.5 mm
250 mm
125 mm 25 kNm/m
3 mm 6 mm 4.5 mm
Example: Tube of varying wall thickness in Torsion The cantilever wing is subjected to a uniformly distributed torque of 25 kN-m per metre length, 6m
25 kNm/m T R
T(z) Nm
T(z)
T z
q
The shear flow q is :
2 Ao
M
z
0,
T R
25 6 150 kNm
Maximum shear stress occurs at fixed end (at z=0),
max
max
qmax tmin
z
25 kNm/m
T max
T R
T(z)
2 Aot min
150 103 2 Ao 0.003
2
N/m
M 0, z
T z 150 25z kNm
z
Example: Tube of varying wall thickness in Torsion Analyzing the enclosed area, 484.12 mm
395.09 mm
375 mm
250 mm
125 mm
A2
250 375 2
A1 395.09
Ao
123443.75 mm 2
max
244473.75 mm 2 0.2445 m 2 150 103
125 375 484.12 2 121030 mm 2
102.25 MN/m2 MPa
Example: Tube of varying wall thickness in Torsion Analyzing the rate of twist,
d q dz
q
2 Ao G
ds t
Analyzing the integral of ds/t ,
ds
t
250 125 3.5
3.5
2
1
ds t
2
2
ds t 250 mm
t
3.5mm
3.5mm
t1 dt
3.0mm
t2
ds
0
L
s
6.0mm 4.5mm
ds t
ds dt
ds
t 2
125 mm
dt
dt t dt t 1 t ds ds ln t2 ln t 1 t dt
ds t
1
ds
ds
2
t
0 400 6.0 4.5
ln 4.5 ln 6.0 76.72
0 500 4.5 3.0
ln 3.0 ln 4.5 135.16
250 125
Example: Tube of varying wall thickness in Torsion Analyzing the rate of twist,
d q dz
q
2 Ao G
The angle of twist at the free end is :
q
therefore q
ds t L
0
The shear flow q is :
q ds dz A G t 2 o 1
4 Ao2G
530.90
ds t
q
T z 2 Ao
T z dz L
0
6
150 25 z dz 4 A G 2 o
0
6
z 2 150 z 25 2 6 4 0.2445 28.5 10 2 0 7.7902 105 150 6 12.5 62 0.03506 rad 530.9
o
=2.01
Torsion of Thin-Walled Multi-cell Tubes The fact that shear flow in a single cell section is constant is used here. The torque applied at a section with multiple cells will be distributed as :
q1
q2
2 Cell 1
Cell 2
q2 q1
1
T2 T1
3
q3
T2
T1
T3
T
unknowns = q1, q2 .... qn
Torsion of Thin-Walled Multi-cell Tubes Let’s use a 2-cell thin walled closed section as a basis for discussion. B
Equilibrium along length : q2
qw
q1 C
A 1
Cell 1
qw
A w
L O
Cell 2
q1 L q2 L qW L 0
q1 q2
A 2
qW q1 q2 q12
D
A
Taking moment equilibrium about arbitrary point O, the torque is: T
OBCA
q1r ds
OADB
q2 r ds
OAB
qw r ds
2 A1 Aw q1 2 A2 Aw q2 2 Awq w 2 A1q1 2 A2 q2 With an additional cell, there exists an additional independent shear flow associated with this cell, which cannot be determined by statics. The degree of indeterminacy of a tube having n cells is (n-1).
Thin-Walled Multi-cell Tubes For a n -cell closed section of general shape. The n independent shear flows are taken as unknowns. The equation of equilibrium for multi-cell becomes : n
Aq
T 2
r
r
r 1
where q r is the shear flow in cell r and A r is the enclosed area of cell r . Hence (n-1) equations of compatibility (or deformation) must be used to supplement the single equation of equilibrium that is available. Saint-Venant torsion theory states that a cross section does not distort in its own plane, hence each cell has the same rate of twist. For the i -th cell : 1 q d q f ds dz 2 Ai G t i
Saint Venant’s Torsion Theory Saint Venant’s Torsion theory states that : 1) The cross section does not distort during deformation, the deformation at any point on the cross section can be described by translations and rotation. 2) Even with warping, plane sections rotate as a rigid body. This implies that the in-plane displacement components of u and v follows those of a rigid body rotation. 3) If the cross section is free to warp, there will be no axial stresses arising from the twisting. This is called a state of Saint-Venant free torsion.
q1q q q2q O O
Example: Multi-cell Thin-Walled Tube in Torsion (R.D Cook, W.C Young, “Advanced Mechanics of Materials”)
Find the rate of twist of the section and shear stresses in the walls of the 3-cell section shown. Given that the T=2kNm, G=70GPa and dimensions shown in the figure. 4mm
Apply the rate of twist for each cell:
q2
q1 2
35
3mm
3
45 q3
4mm
40mm
1 Ai
5mm
1 4mm
2Gf
40mm
q
t
45 35 40 80 40 q q q 1 13 12 40 80 4 3 3 1
2Gf
40 35 40 35 q2 q23 q21 40(35) 4 5 3 3
2Gf
45 40 45 40 q32 q3 q31 40(45) 5 4 3 3
1
1
q12 q1 q2 q21 q2 q1
Solving simultaneously yields: q1 165Gf
Substitute these into the torque eqn : n
T
i
ds
2 Arq r
q2 168.3Gf q3 175.76Gf
r 1
2 106
2Gf 40(80)(165) 40(35)(168.3) 40(45)(175.76)
f 13 2 106 rad / mm
Knowing f , qi can be obtained and hence the shear stresses in the wall.
Torsion of Thin-Walled Open Sections – Background Prior to developing a simple theory for thin-walled open sections. Examine the response of a thin-walled flat bar in torsion.
Axial Stresses,
sz
Shear Stresses, zx
Torsion of Thin-Walled Open Sections – Background Away from the axial contraints, within St. Venant’s free torsion area Examine the shear stresses in the cross-section.
Shear Stresses, zx
Shear Stresses, zy
Torsion of Thin-Walled Open Sections – Shear Stresses Away from the axial contraints, within St. Venant’s free torsion area Examine the shear stresses in the cross-section in details. b
t
T
h ~b 2h dT
dh
Torsion of Thin-Walled Open Sections - Theory An approximate solution for thin-walled open section start with torsion of a strip of rectangular cross-section whose t<
b
Consider the strip to be built up of a series of thin walled concentric tubes which all twist by the same amount. Neglecting the small edge regions, the enclosed area of one of these tube is:
Ao 2bh
t
The shear stress of this tube is:
h
T
2 Aot
dT
4bh dh
The angle of twist of this tube,
2h dh
f
T
4 Ao2G
Combining the two eqns yields:
ds t
2b dT 2 2 42bh G dh 8bh G dh
dT
4bh dh
dT
f 8bh2 G 4bh
2Ghf
Torsion of Thin-Walled Open Sections – Theory The applied torque can be found from :
t / 2 1 T 4bh dh 4bh 2Ghf dh Gf bt 3 Gf J R 0 4bh dh 3
dT
f
T GJ R
The maximum shear stress occurs at h=+t/2 and h=-t/2 :
2Ghf
2Th
max
J R
Imagine that the rectangular cross section is distorted into a C, T or L shape. The sum of the J R of each part of the cross section contributes to the total torque, i.e: b2 t
t2 t3
b3
J R
1
3
b t 3
3 i i
i 1
t1 b1
J R = ?
b
Tt J R
Torsion of Thin-Walled I-Beam
zx
zy
Example: Low Torsional Stiffness of Thin-Walled Open Sections Compare the torsional stiffnesses of a closed circular thin-walled section with that of an equivalent circular thin-walled section with a small slit. t
The ratio of the GJ :
t
R
R
GJ R
c
GJ
2 R c t 3 3 2 R t 3
2 R c t 2 6 R
0.001
3
c=0.1t
0.0008
(a) closed GJ G
GJ G
(b) open 2 o
4 A ds
4 R
3
GJ R G
t 2
d e0.0006 s o l c / n e p0.0004 o
2
2 R t
G 2 R3t
GJ R G
bt
3
2 R c t 3 3
0.0002
0 0
50
100
R/t
150
200
Torsion of Hybrid Section It is usual to find aircraft components having combinations of open and closed section beams. ti
f
bi
T GJ
4 A02 1 J ds 3 t
A o
bi
A o
N
b t
3 i i
i 1
ti
In general, the torsional stiffness (GJ) of the closed portion is dominant. Thus the torsional stiffness of the open portion is usually ignored. Shear stresses in the open portion should always be checked. Closed portion
f
q 2 A0 G
ds
t
q t
2 A0 q
GJ
Open portion
ma x
Tt J
GTt GJ
Example: Torsion of an hybrid section (THG Megson, “Aircraft Structures for Engineering Students”)
(a) Find the angle of twist per unit length in the wing whose cross-section is as shown in the figure when it is subjected to a torque of 10 kNm. (b) Find also the maximum shear stress in the section. G=25000 N/mm2. Wall 12 (outer)=900 mm. Nose cell area=20000mm2.
1
3
2 mm 300 mm
1.5 mm
4
2 600 mm
