(c) 2021 : Hans Welleman 1
1Unsymmetrical and/or inhomogeneous cross sections | CIE3109
CIE3109
Structural Mechanics 4
Hans Welleman
Module : Unsymmetrical and/or
inhomogeneous cross section
v2021
2Unsymmetrical and/or inhomogeneous cross sections | CIE3109
QuestionWhat is the stress (and strain) distribution in a crosssection due to extension and bending (shear) in case of aunsymmetrical and/or non-homogeneous cross section’ ?
z
z
z
y
y
y
(a) (b) (c)
concrete
steel
E1
E1
E2
unsymmetrical inhomogeneous both
new smart materials in composite structures(MSc)
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3Unsymmetrical and/or inhomogeneous cross sections | CIE3109
CIE3109 : Structural Mechanics 4
• 1-2 Inhomogeneous and/or unsymmetrical cross sections
• Introduction
• General theory for extension and bending, beam theory
• Unsymmetrical cross sections
• Example curvature and loading
• Example normalstress distribution
• Deformations
• 3 Inhomogeneous cross sections
• Refinement of the theory
• Examples
• 4-5 Stresses and the core of the cross section
• Normal stress in unsymmetrical cross section and the core
• Shear stresses in unsymmetrical cross sections
• Shear centre
Lectures
4Unsymmetrical and/or inhomogeneous cross sections | CIE3109
sectional definitions
Vz
Vy
BENDING and AXIAL LOADING
RELATION BETWEEN EXTERNAL LOADS AND DISPLACEMENTS (deformations)
• Structural level
• Cross sectional level
• Micro level (material science)
ux
uz
uy xz
y
ϕz
ϕx
ϕy
Fz
FxFy
F
global definitions
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5Unsymmetrical and/or inhomogeneous cross sections | CIE3109
Couples T and bending moments M• Formal approach, positive bending
moment M in a cross section acting in
plane from positive side towards negative
side.
• Formal approach external couple T,
following right-hand-rule or corkscrew rule
around an axis.
• Engineering practise with bending around
an axis similar to definition of T on positive
planes.
6Unsymmetrical and/or inhomogeneous cross sections | CIE3109
fibre in cross section x at position y,z
cross section
fibre
ASSUMPTIONS
• FIBRE MODEL
• SMALL ROTATIONS
OF THE CROSS SECTION
• UNIAXIAL STRESS SITUATION
IN THE FIBRES
• LINEAR ELASTIC MATERIAL
BEHAVIOUR
Hooke’s law : Eσ ε= ×
cross sections
fibres
beam axis
axis of symmetry
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SOLUTION PATH
• Determine the strain distribution due to the displacements of
the cross section?
• Determine the stress distribution caused by these strains?
• Determine the resulting forces in the cross section?
Load Sectional forces Stress Strain Deformations Displacements
(F en q) (N, V, M) (fibre) (fibre) (ε,κ) (u,ϕ)
Equilibrium Static Constitutive Compatibility Kinematische
relations relations relations relaties relaties
Constitutive relation (cross section)
• Use the constitutive relation on structural level (slide 28)
8Unsymmetrical and/or inhomogeneous cross sections | CIE3109
KINEMATIC RELATIONS
Relation between deformations and displacements
z
x-axis
z-axis
ϕy
uz
ux
ϕy
z
x
uzy
d
d−=ϕ
y
x-axis
y-axis
ϕzuy
ux
ϕzy
x
u y
zd
d=ϕ
fibre
cross section
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9Unsymmetrical and/or inhomogeneous cross sections | CIE3109
FIBRE MODEL
• Horizontal displacementz
x-axis
z-axis
ϕy
uz
ux
ϕy
z
x y
x z
( , , )
( , , )
u x y z u z
or
u x y z u z u
ϕ= + ×
′= − ×
z
y
y
y u
ϕ− ×
′− ×
y
x-axis
y-axis
ϕzuy
ϕzy
ux
x
u zy
d
d−=ϕ
x
u y
zd
d=ϕ
10Unsymmetrical and/or inhomogeneous cross sections | CIE3109
RELATIVE DISPLACEMENT-STRAIN
x y z( , )y z u y u z uε ′ ′′ ′′= − × − ×
x z y
x y z
( , , )
( , , )
u x y z u y z
or
u x y z u y u z u
ϕ ϕ= − × + ×
′ ′= − × − ×
( , , )( , )
u x y zy z
xε
∂=
∂
y z( , )y z y zε ε κ κ= + × + × with2
2
2
2
d
d;
d
d;
d
d
x
u
x
u
x
u z
z
y
y
x −=−== κκε
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11Unsymmetrical and/or inhomogeneous cross sections | CIE3109
STRAIN DISTRIBUTION
κz
κy
x-axis
z-axis
x-axis
y-axis
strain
strain
z
y
y z( , )y z y zε ε κ κ= + × + ×
Conclusion:
Strain distribution is fully described with three deformation parameters
12Unsymmetrical and/or inhomogeneous cross sections | CIE3109
STRAIN FIELD
Three parameters
• strain in fibre which coincides with
the x-axis
• slope of strain diagram in y-direction
• slope of strain diagram in z-direction
y z( , )y z y zε ε κ κ= + × + ×
neutral axis
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13Unsymmetrical and/or inhomogeneous cross sections | CIE3109
k
k
CURVATURE
• First order tensor
• Curvature in x-y-plane
• Curvature in x-z-plane
• Plane of curvature k
2
z
2
y κκκ +=
zk
y
tanκ
ακ
=
14Unsymmetrical and/or inhomogeneous cross sections | CIE3109
NEUTRAL AXIS
kk is perpendicular
to neutral axis
0),( zy =×+×+= κκεε zyzy
neutral axis
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15Unsymmetrical and/or inhomogeneous cross sections | CIE3109
FROM STRAIN TO STRESS
• Constitutive relation : link between deformations and stresses
y z
( , ) ( , ) ( , )
( , ) ( , )
y z E y z y z
y z E y z y z
σ ε
σ ε κ κ
= ×
= × + × + ×
16Unsymmetrical and/or inhomogeneous cross sections | CIE3109
STRESS AND SECTIONAL FORCES
• Static relations : link between stresses and
sectional forces
y
z
( , )
( , )
( , )
A
A
A
N y z dA
M y y z dA
M z y z dA
σ
σ
σ
=
=
=
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17Unsymmetrical and/or inhomogeneous cross sections | CIE3109
Elaborate …
( )
( )
( )
++×==
++×==
++×==
AA
AA
AA
zdAzyzyEdAzyzM
ydAzyzyEdAzyyM
dAzyzyEdAzyN
zyz
zyy
zy
),(),(
),(),(
),(),(
κκεσ
κκεσ
κκεσ
zzzyyzz
2
zyz
zyzyyyyz
2
yy
zzyyzy
),(),(),(
),(),(),(
),(),(),(
κκεκκε
κκεκκε
κκεκκε
EIEIESdAzzyEyzdAzyEzdAzyEM
EIEIESyzdAzyEdAyzyEydAzyEM
ESESEAzdAzyEydAzyEdAzyEN
A AA
A AA
A AA
++=++=
++=++=
++=++=
approach with “double-letter” symbols
18Unsymmetrical and/or inhomogeneous cross sections | CIE3109
CONSTITUTIVE RELATION
=
z
y
zzzyz
yzyyy
zy
z
y
κ
κ
ε
EIEIES
EIEIES
ESESEA
M
M
N
on cross sectional level
INDEPENDENT OF THE ORIGIN OF THE
COORDINATE SYSTEM
SPECIAL LOCATION OF THE ORIGIN OF THE COORDINATE
SYSTEM TO UNCOUPLE BENDING AND AXIAL LOADING
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19Unsymmetrical and/or inhomogeneous cross sections | CIE3109
NORMAL FORCE CENTRE
• Bending and axial loading are uncoupled:
=
z
y
zzzy
yzyy
z
y
0
0
00
κ
κ
ε
EIEI
EIEI
EA
M
M
N Axial loading
Bending
y z 0ES ES= =definition NC : if N = 0 then zero strain ε at the NC and the n.a.
runs through the NC
20Unsymmetrical and/or inhomogeneous cross sections | CIE3109
LOCATION NCy
z
NC NCy
NCz
y
z
dAzyE ),(
NCNCzzzyyy +=+=
NCzNC
z
NCyNC
y
),(),(
),(
),(),(
),(
zEAESdAzyEzzdAzyE
dAzzyEES
yEAESdAzyEyydAzyE
dAyzyEES
AA
A
AA
A
×+=+×
=×=
×+=+×
=×=
y
NC
zNC
ESy
EA
ESz
EA
=
=
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21Unsymmetrical and/or inhomogeneous cross sections | CIE3109
RESULT
• Bending and axial loading are
uncoupled
• Moment is first order tensor
2
z
2
y MMM +=
y
zmtan
M
M=α
22Unsymmetrical and/or inhomogeneous cross sections | CIE3109
MOMENT
M
m
x-axis
z-axis
y-axis
Mz
My
ααααm
MOMENT
M
TENSION
COMPRESSION
N.C.
SUMMARY of formulas
TENSION
COMPRESSION
CURVATURE
κκκκ
k
n.a.
x-axis
n.a.
z-axis
y-axis
κκκκz
κκκκy
ααααk
N.C.
×
=
×=
×+×+=
z
y
zzyz
yzyy
z
y
zy
EIEI
EIEI
EA
M
M
N
zyzyEzy
zyzy
κ
κ
ε
εσ
κκεε
),(),(),(
),(
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23Unsymmetrical and/or inhomogeneous cross sections | CIE3109
PLANE OF LOADING SAME
AS PLANE OF CURVATURE ?
y y
z z
yy yz y y
zy zz z z
M
M
M
EI EI
EI EI
λκ
κλ
κ
κ κλ
κ κ
=
= ⇔
× = ⇔
0 eigenvalue problemyy yz
zy zz
EI EI
EI EI
λ
λ
− = −
24Unsymmetrical and/or inhomogeneous cross sections | CIE3109
PRINCIPAL
DIRECTIONS
( ) ( )
y yy y
z zz z
221 1
yy zz 2 2
12
0
0
, ( )
tan 2 ,( )
yy zz yy zz yz
yz
y z
yy zz
M EI
M EI
EI EI EI EI EI EI
EI
EI EI
κ
κ
α
=
= + ± − +
=−
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25Unsymmetrical and/or inhomogeneous cross sections | CIE3109
ROTATE TO PRINCIPAL
DIRECTIONS
y yy y y yy y
z zz z z zz z
0
0
M EI M EI
M EI M EI
κ κ
κ κ
= = ⇔ =
direction (1)
direction (2)
z
y
=
=
m k
m k
yy zz
1) 0 principal direction
2) / 2 other principal direction
3) EI EI
α α
α α π
= = =
= = =
=
z zz z zzm k
y yy y yy
tan( ) tan( )M EI EI
M EI EI
κα α
κ= = =
m k ??α α=
26Unsymmetrical and/or inhomogeneous cross sections | CIE3109
EXAMPLE 1
Determine the position of the plane loading m-m for the specified position of the neutral line which makes an angle of 30o degrees with the y-axis. The rectangular cross section is homogeneous.
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27Unsymmetrical and/or inhomogeneous cross sections | CIE3109
EXAMPLE 2
A triangular cross section is loaded by pure bending only. The cross section is homogeneous and the Youngs modulus is E. The normal stress in point A and C is equal to 10 N/mm2.
a) Calculate the magnitude and direction of the resulting bending moment
b) Compute the stress in point B
28Unsymmetrical and/or inhomogeneous cross sections | CIE3109
F1 = HELP
α
α
tan
)(tan
3
36
1
2
3
36
13
48
1
3
36
1
bhI
bhhbI
bhI
yz
yy
zz
=
+=
=
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29Unsymmetrical and/or inhomogeneous cross sections | CIE3109
EXAMPLE 3
a) Find the location of the NCb) Compute all cross sectional propertiesc) Find the location of the centre of gravity
30Unsymmetrical and/or inhomogeneous cross sections | CIE3109
STRUCTURAL LEVEL
Equillibrium
conditions
2
2
2
2
d d0
d d
d d d0 and 0
d d d
d d d0 and 0
d d d
x x
y y y
y y y
z z zz z z
N Nq q
x x
V M Mq V q
x x x
V M Mq V q
x x x
+ = = −
+ = − = = −
+ = − = = −
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31Unsymmetrical and/or inhomogeneous cross sections | CIE3109
STRUCTURAL LEVEL
Differential Equations
d'
d
xx
uu
xε = =
2
2
d"
d
y
y y
uu
xκ = − = −
2
2
d"
d
zz z
uu
xκ = − = −
Kinematics:
Constitutive relations:
=
z
y
zzzy
yzyy
z
y
0
0
00
κ
κ
ε
EIEI
EIEI
EA
M
M
N
d
dx
Nq
x= −
2
2
d
d
y
y
Mq
x= −
2
2
d
d
zz
Mq
x= −
Equilibrium relations:
subst
ituteaxial loading
bending
'''' ''''
'''' ''''
"x x
yy y yz z y
yz y zz z z
EAu q
EI u EI u q
EI u EI u q
− =
+ =
+ =
32Unsymmetrical and/or inhomogeneous cross sections | CIE3109
Result
2
2
"
"''
"''
xx
zz y yz z
y
yy zz yz
yy z yz y
z
yy zz yz
qu
EA
EI q EI qu
EI EI EI
EI q EI qu
EI EI EI
= −
−=
−
−=
−
2
2
"''
"''
zz yy y yz yy z
yy y
yy zz yz
yy zz z yz zz y
zz z
yy zz yz
EI EI q EI EI qEI u
EI EI EI
EI EI q EI EI qEI u
EI EI EI
−=
−
−=
−
"''
"''
yy y y
zz z z
EI u Q
EI u Q
=
=2
2
zz yy y yz yy z
y
yy zz yz
yz zz y yy zz z
z
yy zz yz
EI EI q EI EI qQ
EI EI EI
EI EI q EI EI qQ
EI EI EI
−=
−
− +=
−
modify the load in all forget-me-not’sOR solve differential equation
"x xEAu q= −
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33Unsymmetrical and/or inhomogeneous cross sections | CIE3109
EXAMPLE (see lecture notes)41
, 9
4 2
2 4i jEI Ea
=
2 2 4
2 2 4
3"''
2
3"''
zz y yz z yz z zy
yy zz yz yy zz yz
yy z yz y yy z zz
yy zz yz yy zz yz
EI q EI q EI q qu
EI EI EI EI EI EI Ea
EI q EI q EI q qu
EI EI EI EI EI EI Ea
− − −= = =
− −
−= = =
− −
42 3
1 2 3 4 4
42 3
1 2 3 4 4
16
8
zy
zz
q xu C C x C x C x
Ea
q xu D D x D x D x
Ea
= + + + −
= + + + +
0 : ( 0; 0; 0; 0)
: ( 0; 0; 0; 0)
y z y z
y z y z
x u u
x l u u M M
ϕ ϕ= = = = =
= = = = =
34Unsymmetrical and/or inhomogeneous cross sections | CIE3109
Result
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35Unsymmetrical and/or inhomogeneous cross sections | CIE3109
SUMMARY displacements
• solve differential equations
• Use “pseudo” load for y - and z -direction and use this load in
standard engineering equations
• Use curvature distribution in combination with moment area
theoremes to obtain displacements and rotations (see notes)
• Use principal directions, decompose load in (1) and (2) direction,
and compute displacements in (1) and (2) direction with standard
engineering equations. Finally transformate the displacements
back to y - and z -direction
Original y-z coordinate system
Principal 1-2 coordinate system
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