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CIE3109 Structural Mechanics 4 Hans Welleman
Module : Unsymmetrical and/or inhomogeneous cross section Unsymmetrical and/or inhomogeneous cross sections 1 | CIE3109
CIE3109 : Structural Mechanics 4
Lectures • 1-2 Inhomogeneous and/or unsymmetrical cross sections • Introduction • • General theory for extension and bending, beam theory Unsymmetrical cross sections • 3 • 4-5 • Example curvature and loading • Deformations Inhomogeneous cross sections • Refinement of the theory • Examples Stresses and the core of the cross section • Normal stress in unsymmetrical cross section and the core • • • Example normalstress distribution Shear stresses in unsymmetrical cross sections Shear centre 2 | CIE3109 Unsymmetrical and/or inhomogeneous cross sections
Question
What is the stress and strain distribution in a cross section due to extension and bending in case of a unsymmetrical and/or non-homogeneous cross section’ ?
concrete
y E
1
y
steel
y E
2
z
(a) unsymmetrical
E
1
z
(b) inhomogeneous (c) both Unsymmetrical and/or inhomogeneous cross sections
z
3 | CIE3109
BENDING and AXIAL LOADING
y
V y z
F
z
F
x y
F
y
u
x
F
x V z
z u
y
u
z global definitions
x
sectional definitions RELATION BETWEEN EXTERNAL LOADS AND DISPLACEMENTS • • Structural level Cross sectional level Unsymmetrical and/or inhomogeneous cross sections 4 | CIE3109
ASSUMPTIONS
• FIBRE MODEL • SMALL ROTATIONS OF THE CROSS SECTION • UNIAXIAL STRESS SITUATION IN THE FIBRES • LINEAR ELASTIC MATERIAL BEHAVIOUR Hooke’s law : fibre in cross section
x
at position
y,z
5 | CIE3109 Unsymmetrical and/or inhomogeneous cross sections
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 (
F
en
q
) Sectional forces (
N, V, M
) Equilibrium relations Static relations Stress (fibre) Strain (fibre) Deformations (
,
) Constitutive Compatibility relations relaties Kinematische relaties Displacements (
u,
)
Constitutive relation (cross section)
Unsymmetrical and/or inhomogeneous cross sections 6 | CIE3109
KINEMATIC RELATIONS
Relation between deformations and displacements
y z u x z-as
y
d
u z
d
x z u z u x x-as
z y
y y y-as
z
d
u y
d
x
Unsymmetrical and/or inhomogeneous cross sections
u y x-as
z
7 | CIE3109
FIBRE MODEL
• Horizontal displacement
y
d
u z
d
x
z
d
u y
d
x
z
y y z
u
x y
y-as z-as
z
or
u u x x y z x-as x-as u u z y
y
z
u
x z y Unsymmetrical and/or inhomogeneous cross sections 8 | CIE3109
RELATIVE DISPLACEMENT-STRAIN
u
x
y
z
y
or
u
x y
z u
z
u x y z
x
u
x
y u
y y z z with d
u x
d
x
;
y
d 2
u y
d
x
2 ;
z
d 2
u z
d
x
2 Unsymmetrical and/or inhomogeneous cross sections 9 | CIE3109
STRAIN DISTRIBUTION
Conclusion:
Strain distribution is fully described with three deformation parameters y
z-axis z
z
y-axis y
strain
x-axis
z
x-axis
strain y Unsymmetrical and/or inhomogeneous cross sections 10 | 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 Unsymmetrical and/or inhomogeneous cross sections 11 | CIE3109
CURVATURE
• • • • First order tensor Curvature in x-y-plane Curvature in x-z-plane Plane of curvature
k
y 2 z 2 tan k z y
k k
Unsymmetrical and/or inhomogeneous cross sections 12 | CIE3109
NEUTRAL AXIS
k
is perpendicular to neutral axis
k neutral axis
(
y
,
z
)
y
y
z
z 0 Unsymmetrical and/or inhomogeneous cross sections 13 | CIE3109
FROM STRAIN TO STRESS
• Constitutive relation : link between deformations and stresses y z Unsymmetrical and/or inhomogeneous cross sections 14 | CIE3109
STRESS AND SECTIONAL FORCES
• Static relations : link between stresses and sectional forces
N M
y
A
A
y
M
z
A
z
Unsymmetrical and/or inhomogeneous cross sections 15 | CIE3109
Elaborate …
N M
y
M
z
A
(
y
,
z
)
dA
A E
(
y
,
z
)
A
A y
(
y
,
z
)
dA
z
(
y
,
z
)
dA
A E
(
y
,
A E
(
y
,
z
)
z
)
y
y
y
y
y
y
z
z
dA
z
z
z
z
ydA
zdA N
M
y
M
z
A A
E A
E
( (
E y
(
y
, ,
z y
,
z
) )
z
)
dA ydA zdA
y y y
A A
A
E E E
( ( (
y y y
, , ,
z z z
) ) )
y ydA
2
dA yzdA
z z z
A
A A
E E E
( ( (
y
,
y y
, ,
z z z
) ) )
zdA z yzdA
2
dA
EA
ES
y
ES
z
ES
y y
EI
yy y
EI
yz y approach with “double-letter” symbols
ES
z z
EI
yz z
EI
zz z Unsymmetrical and/or inhomogeneous cross sections 16 | CIE3109
CONSTITUTIVE RELATION
on cross sectional level
N M M
y z
EA ES ES
y z
ES
y
EI
yy
EI
zy
ES EI
yz
EI
z zz
κ κ ε
y z INDEPENDENT OF THE ORIGIN OF THE COORDINATE SYSTEM SPECIAL LOCATION OF THE ORIGIN OF THE COORDINATE SYSTEM TO UNCOUPLE BENDING AND AXIAL LOADING Unsymmetrical and/or inhomogeneous cross sections 17 | CIE3109
NORMAL FORCE CENTRE
• Bending and axial loading are uncoupled:
N M M
y z
EA
0 0 0
EI
yy
EI
zy 0
EI
yz
EI
zz
κ κ ε
y z Axial loading Bending definition NC :
ES
y
ES
z 0 if
N
= 0 then zero strain at the NC and the
n.a.
runs through the NC Unsymmetrical and/or inhomogeneous cross sections 18 | CIE3109
LOCATION NC
y
y
y
NC
z
z
z
NC
y y
NC
z
NC
y
NC
E
(
y
,
z
)
dA ES
y
ES
z
A
E
(
y
,
z
)
A
E
(
y
,
z
)
y dA
ydA
A
E
(
y
,
z
)
z dA
A
E
(
y
,
z
)
zdA
y
NC
A
E
(
y
,
z
)
dA z
NC
A
E
(
y
,
z
)
dA
ES
y
ES
z
EA
z y
NC
EA
z
NC
y
NC
ES
y
EA z
NC
ES
z
EA z
19 | CIE3109 Unsymmetrical and/or inhomogeneous cross sections
RESULT
• • Bending and axial loading are uncoupled Moment is first order tensor
M
M
y 2
M
z 2 tan m
M M
y z Unsymmetrical and/or inhomogeneous cross sections 20 | CIE3109
y-axis
SUMMARY of formulas
COMPRESSION MOMENT M
n.a.
COMPRESSION TENSION
y N.C.
k
CURVATURE
k
n.a.
x-axis y-axis
M
y N.C.
TENSION
m
MOMENT M M
z
z z-axis
(
y
,
z
) (
y
,
z
)
y E
(
y
,
y
z z
) (
y
,
z
)
z
N M M y z
EA EI yy EI yz EI yz EI zz
y z
m
z-axis
Unsymmetrical and/or inhomogeneous cross sections
x-axis
21 | CIE3109
PLANE OF LOADING SAME AS PLANE OF CURVATURE ?
M
M M y z
EI EI yy zy
EI yz EI zz
EI yy EI
zy
EI EI yz
zz
0 eigenvalue problem Unsymmetrical and/or inhomogeneous cross sections 22 | CIE3109
PRINCIPAL DIRECTIONS
M M
y z
EI
yy 0 0
EI
zz
EI
, yy zz 1 2
EI yy
EI zz
1 2 (
EI yy
EI zz
) 2 2
EI yz
tan 2 ,
y z
1 2 (
EI EI yz
yy EI zz
) Unsymmetrical and/or inhomogeneous cross sections 23 | CIE3109
ROTATE TO PRINCIPAL DIRECTIONS
z
direction (1)
y
direction (2)
M M
y z
EI
yy 0 tan( 1) 2) 3) m )
M
z
M
y m m
EI
yy k k
EI
zz 0 0
EI
zz
EI EI
zz yy z y
EI
zz
EI
yy
M
y
M
z tan( k ) / 2
EI
yy
EI
principal direction zz m y z k ??
other principal direction Unsymmetrical and/or inhomogeneous cross sections 24 | CIE3109
EXAMPLE 1
Determine the position of the plane loading specified position of the neutral line which makes an angle of 30 o degrees with the homogeneous.
y m-m for the -axis. The rectangular cross section is Unsymmetrical and/or inhomogeneous cross sections 25 | 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/mm 2 .
a) Calculate the magnitude and direction of the resulting bending moment b) Compute the stress in point B Unsymmetrical and/or inhomogeneous cross sections 26 | CIE3109
F1 = HELP
I
zz
1 36
bh
3
I
yy
1 48
b
3
h
1
bh
3 36
(tan
)
2
I
yz
1 36
bh
3
tan
Unsymmetrical and/or inhomogeneous cross sections 27 | CIE3109
Equillibrium conditions
d
N
q x
0 d
x
d
V y
q y
0 d
x
d
V z
d
x
q z
0 and and d
M y
V y
d
x
0 d
M z
d
x
V z
0 d
N
q x
d
x
d 2
M y
d
x
2
q y
d 2
M z
d
x
2
q z
Unsymmetrical and/or inhomogeneous cross sections 28 | CIE3109
Differential equations (structural level)
Kinematics: d
u x
d
x
u x
'
y
d 2
u y
d
x
2
u y
"
z
d 2
u z
d
x
2
u z
" Constitutive relations:
EAu
M M
y z
x
"
EA
0
EI u
0
yy EI EI
'''' yy
y
'''' zy
yz y
Equilibrium relations: 0 zz
κ κ EI u zz ε
y z
z z
'''' '''' d
N
q x
d
x
d 2
M y
d
x
2
q y
q x
q q z y
d 2
M z
d
x
2
q z
axial loading bending Unsymmetrical and/or inhomogeneous cross sections 29 | CIE3109
Result
u x
"
q x EA Q y yy
u y y
"''
zz
EI q yy yy y y
y EI EI zz
zz
EI q
yz yz yz EI yz z
2
yz
2
yz EI z Q z zz
u z z
"''
EI q z y z EI EI
zz
EI q
EI
2
y
2
yz y zz EI z zz yy EI u yy EI u zz z y
"'' "''
Q y
Q z
modify the load in all forget-me-not’s OR solve differential equation 30 | CIE3109 Unsymmetrical and/or inhomogeneous cross sections
EXAMPLE (see lecture notes)
EI
1 9
Ea
4 4 2 2 4
x
0 : (
u y
0;
u z
0;
y
0;
z
0)
x u u y y
"''
u u z z
"''
l
: (
u y
0;
u z
0;
M y
0;
M z
0)
EI q C zz
y EI EI yy
C x C x zz
2
EI
2
yz z
2
EI q yz EI EI yy
4
zz EI
2
q x yz
16
z
4 3
q z Ea
4 4
EI q yy D yy
z
zz
2
EI q yz D x
2
yz y D x
3
EI q yy
2
EI EI yy D x zz
4
z
3
EI
2
yz
8
q x q z Ea Ea z
4 4 4 Unsymmetrical and/or inhomogeneous cross sections 31 | CIE3109
Result
Unsymmetrical and/or inhomogeneous cross sections 32 | CIE3109
SUMMARY displacements
Original y-z coordinate system
• • • 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)
Principal 1-2 coordinate system
• 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 Unsymmetrical and/or inhomogeneous cross sections 33 | CIE3109