ce 579: structral stability and design
DESCRIPTION
CE 579: STRUCTRAL STABILITY AND DESIGN. Amit H. Varma Assistant Professor School of Civil Engineering Purdue University Ph. No. (765) 496 3419 Email: [email protected] Office hours: M-T-Th 10:30-11:30 a.m. Chapter 1. Introduction to Structural Stability. OUTLINE Definition of stability - PowerPoint PPT PresentationTRANSCRIPT
CE 579: STRUCTRAL STABILITY AND DESIGN
Amit H. Varma
Assistant Professor
School of Civil Engineering
Purdue University
Ph. No. (765) 496 3419
Email: [email protected]
Office hours: M-T-Th 10:30-11:30 a.m.
Chapter 1. Introduction to Structural Stability
OUTLINE
Definition of stability
Types of instability
Methods of stability analyses
Bifurcation analysis examples – small deflection analyses
Energy method Examples – small deflection analyses Examples – large deflection analyses Examples – imperfect systems
Design of steel structures
ENERGY METHOD
We will currently look at the use of the energy method for an elastic system subjected to conservative forces.
Total potential energy of the system – – depends on the work done by the external forces (We) and the strain energy stored in the system (U).
=U - We.
For the system to be in equilibrium, its total potential energy must be stationary. That is, the first derivative of must be equal to zero.
Investigate higher order derivatives of the total potential energy to examine the stability of the equilibrium state, i.e., whether the equilibrium is stable or unstable
ENERGY METHD The energy method is the best for establishing the equilibrium
equation and examining its stability The deformations can be small or large. The system can have imperfections. It provides information regarding the post-buckling path if large
deformations are assumed The major limitation is that it requires the assumption of the
deformation state, and it should include all possible degrees of freedom.
ENERGY METHOD
Example 1 – Rigid bar supported by rotational spring
Assume small deflection theory
Step 1 - Assume a deformed shape that activates all possible d.o.f.
Rigid bar subjected to axial force P
Rotationally restrained at end
Pk
L
L P
L cosL (1-cos)
k
ENERGY METHOD – SMALL DEFLECTIONS
Write the equation representing the total potential energy of systemL (1-cos)
L P
L cos
k L sin
L
kPTherefore
LPksdeflectionsmallForLPkTherefore
d
dmequilibriuFor
LPkd
d
LPk
LPW
kU
WU
cr
e
e
,
0;0sin,
0;
sin
)cos1(2
1)cos1(
2
1
2
2
ENERGY METHOD – SMALL DEFLECTIONS
The energy method predicts that buckling will occur at the same load Pcr as the bifurcation analysis method.
At Pcr, the system will be in equilibrium in the deformed.
Examine the stability by considering further derivatives of the total potential energy
This is a small deflection analysis. Hence will be zero. In this type of analysis, the further derivatives of examine the stability of
the initial state-1 (when =0)
PLkd
d
LPkLPkd
d
LPk
2
2
2
sin
)cos1(2
1
sureNotd
dPPWhen
mequilibriuUnstabled
dPPWhen
mequilibriuStabled
dPPWhen
cr
cr
cr
0
0
0
2
2
2
2
2
2
ENERGY METHOD – SMALL DEFLECTIONS
In state-1, stable when P<Pcr, unstable when P>Pcr
No idea about state during buckling.
No idea about post-buckling equilibrium path or its stability.
Pcr
P
Stable
Unstable
Indeterminate
ENERGY METHOD – LARGE DEFLECTIONS
Example 1 – Large deflection analysis (rigid bar with rotational spring)
L (1-cos)
L P
L cos
k L sin
abovegivenisiprelationshPbucklingpostThe
mequilibriuforL
kPTherefore
LPkTherefored
dmequilibriuFor
LPkd
d
LPk
LPW
kU
WU
e
e
sin,
0sin,
0;
sin
)cos1(2
1)cos1(
2
1
2
2
ENERGY METHOD – LARGE DEFLECTIONS
Large deflection analysis See the post-buckling load-displacement path shown below The load carrying capacity increases after buckling at Pcr
Pcr is where 0Rigid bar with rotational spring
0
0.2
0.4
0.6
0.8
1
1.2
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
End rotation
Lo
ad P
/Pcr
00
sin
sin
crP
P
mequilibriuforL
kP
ENERGY METHOD – LARGE DEFLECTIONS
Large deflection analysis – Examine the stability of equilibrium using higher order derivatives of
00,
).,.(0
)tan
1(
cossin
sin,
cos
sin
)cos1(2
1
2
2
2
2
2
2
2
2
2
2
2
ford
dBut
STABLEAlways
ofvaluesalleiAlwaysd
d
kd
d
LL
kk
d
d
L
kPBut
LPkd
d
LPkd
d
LPk
ENERGY METHOD – LARGE DEFLECTIONS
At =0, the second derivative of =0. Therefore, inconclusive.
Consider the Taylor series expansion of at =0
Determine the first non-zero term of ,
Since the first non-zero term is > 0, the state is stable at P=Pcr and =0
nn
n
d
d
nd
d
d
d
d
d
d
d
0
4
04
43
03
32
02
2
00 !
1.....
!4
1
!3
1
!2
1
cos
sin
cos
sin
)cos1(2
1
4
4
3
3
2
2
2
LPd
d
LPd
d
LPkd
d
LPkd
d
LPk
kPLLPd
d
LPd
d
d
d
d
d
cos
0sin
0
0
0
04
40
3
30
2
20
00
24
1
!4
1 44
04
4
kd
d
ENERGY METHOD – LARGE DEFLECTIONS
Rigid bar with rotational spring
0
0.2
0.4
0.6
0.8
1
1.2
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
End rotation
Lo
ad P
/Pcr
00
STABLE
STABLESTABL
E
ENERGY METHOD – IMPERFECT SYSTEMS
Consider example 1 – but as a system with imperfections The initial imperfection given by the angle 0 as shown below
The free body diagram of the deformed system is shown below
Pk L0
L cos(0)
L (cos0-cos)
L P
L cos
k(0 L sin
0
ENERGY METHOD – IMPERFECT SYSTEMS
abovegivenisiprelationshPmequilibriuThe
mequilibriuforL
kPTherefore
LPkTherefored
dmequilibriuFor
LPkd
d
LPk
LPW
kU
WU
e
e
sin
)(,
0sin)(,
0;
sin)(
)cos(cos)(2
1
)cos(cos
)(2
1
0
0
0
02
0
0
20
L (cos0-cos)
L P
L cos
k(0 L sin
0
Rigid bar with rotational spring
0
0.2
0.4
0.6
0.8
1
1.2
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
End rotation
Lo
ad P
/Pcr
00 00.05 00.1 00.2 00.3
ENERGY METHOD – IMPERFECT SYSTEMS
:
sinsin
)(
0
00
belowshownofvaluesdifferentforipsrelationshP
P
P
L
kP
cr
ENERGY METHODS – IMPERFECT SYSTEMS
As shown in the figure, deflection starts as soon as loads are applied. There is no bifurcation of load-deformation path for imperfect systems. The load-deformation path remains in the same state through-out.
The smaller the imperfection magnitude, the close the load-deformation paths to the perfect system load –deformation path
The magnitude of load, is influenced significantly by the imperfection magnitude.
All real systems have imperfections. They may be very small but will be there
The magnitude of imperfection is not easy to know or guess. Hence if a perfect system analysis is done, the results will be close for an imperfect system with small imperfections
ENERGY METHODS – IMPERFECT SYSTEMS
Examine the stability of the imperfect system using higher order derivatives of
Which is always true, hence always in STABLE EQUILIBRIUM
tan.,.cossin
)(.,.
cos.,.
0cos.,.
0
cos
sin)(
)cos(cos)(2
1
0
0
2
2
2
2
0
02
0
eiL
k
L
kifei
L
kPifei
LPkifeid
dif
stablebewillpathmEquilibriu
LPkd
d
LPkd
d
LPk
ENERGY METHOD – SMALL DEFLECTIONS
P
k
L
PL
L (1-cos)
L cos
L sin
k L sinO
Example 2 - Rigid bar supported by translational spring at end
Assume deformed state that activates all possible d.o.f.Draw FBD in the deformed state
ENERGY METHOD – SMALL DEFLECTIONS
Write the equation representing the total potential energy of system
PL
L (1-cos)
L cos
L sin
k L sinO
LkPTherefore
LPLksdeflectionsmallFor
LPLkTherefore
d
dmequilibriuFor
LPLkd
d
LPLk
LPW
LkLkU
WU
cr
e
e
,
0;
0sin,
0;
sin
)cos1(2
1
)cos1(2
1)sin(
2
1
2
2
2
22
222
ENERGY METHOD – SMALL DEFLECTIONS
The energy method predicts that buckling will occur at the same load Pcr as the bifurcation analysis method.
At Pcr, the system will be in equilibrium in the deformed. Examine the stability by considering further derivatives of the total potential energy
This is a small deflection analysis. Hence will be zero. In this type of analysis, the further derivatives of examine the
stability of the initial state-1 (when =0)
LPLkd
d
andsdeflectionsmallFor
LPLkd
d
LPLkd
d
LPLk
22
2
22
2
2
22
0
cos
sin
)cos1(2
1
ATEINDETERMINd
dkLPWhen
UNSTABLEd
dLkPWhen
STABLEd
dLkPWhen
0
0,
0,
2
2
2
2
2
2
ENERGY METHOD – LARGE DEFLECTIONS
abovegivenisiprelationshPbucklingpostThe
mequilibriuforLkPTherefore
LPLkTherefore
d
dmequilibriuFor
LPLkd
d
LPLk
LPW
LkU
WU
e
e
cos,
0sincossin,
0;
sincossin
)cos1(sin2
1
)cos1(
)sin(2
1
2
2
22
2
PL
L (1-cos)
L cos
L sin
O
Write the equation representing the total potential energy of system
ENERGY METHOD – LARGE DEFLECTIONS
Large deflection analysis See the post-buckling load-displacement path shown below The load carrying capacity decreases after buckling at Pcr
Pcr is where 0Rigid bar with translational spring
0
0.2
0.4
0.6
0.8
1
1.2
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
End rotation
Lo
ad P
/Pcr
cos
cos
crP
P
mequilibriuforLkP
ENERGY METHOD – LARGE DEFLECTIONS
Large deflection analysis – Examine the stability of equilibrium using higher order derivatives of
UNSTABLEHENCEALWAYSd
d
Lkd
d
LkLkd
d
LkLkd
d
LkPmequilibriuFor
LPLkd
d
LPLkd
d
LPLk
.0
sin
cos)sin(cos
cos2cos
cos
cos2cos
sincossin
)cos1(sin2
1
2
2
222
2
222222
2
2222
2
22
2
2
22
ENERGY METHOD – LARGE DEFLECTIONS
At =0, the second derivative of =0. Therefore, inconclusive.
Consider the Taylor series expansion of at =0
Determine the first non-zero term of ,
Since the first non-zero term is < 0, the state is unstable at P=Pcr and =0
nn
n
d
d
nd
d
d
d
d
d
d
d
0
4
04
43
03
32
02
2
00 !
1.....
!4
1
!3
1
!2
1
0sin2sin2
0cos2cos
0sin2sin2
1
0)cos1(sin2
1
23
3
22
2
2
22
LPLkd
d
LPLkd
d
LPLkd
d
LPLk
occursbucklingwhenatUNSTABLEd
d
LkLkLkd
d
LPLkd
d
0
0
34
cos2cos4
4
4
2224
4
24
4
ENERGY METHOD – LARGE DEFLECTIONS
Rigid bar with translational spring
0
0.2
0.4
0.6
0.8
1
1.2
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
End rotation
Lo
ad P
/Pcr
UNSTABLE
UNSTABLE
UNSTABLE
ENERGY METHOD - IMPERFECTIONS
Consider example 2 – but as a system with imperfections The initial imperfection given by the angle 0 as shown below
The free body diagram of the deformed system is shown below
P
k
L cos(0)
L0
PL
L (cos0-cos)
L cos
L sin
O0
L sin0
ENERGY METHOD - IMPERFECTIONS
abovegivenisiprelationshPmequilibriuThe
mequilibriuforLkPTherefore
LPLkTherefore
d
dmequilibriuFor
LPLkd
d
LPLk
LPW
LkU
WU
e
e
)sin
sin1(cos,
0sincos)sin(sin,
0;
sincos)sin(sin
)cos(cos)sin(sin2
1
)cos(cos
)sin(sin2
1
0
02
02
02
02
0
20
2
PL
L (cos0-cos)
L cos
L sin
O0
L sin0
3max
32
0max
00
cos
sinsin0)sin
sinsin(0
)sin
sin1(cos)
sin
sin1(cos
LkP
Lkd
dPP
P
PLkP
cr
ENERGY METHOD - IMPERFECTIONS
0
0.2
0.4
0.6
0.8
1
1.2
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
End rotation
Lo
ad P
/Pc
r
00 00.05 00.1 00.2 00.3
Envelope of peak loads Pmax
ENERGY METHOD - IMPERFECTIONS
As shown in the figure, deflection starts as soon as loads are applied. There is no bifurcation of load-deformation path for imperfect systems. The load-deformation path remains in the same state through-out.
The smaller the imperfection magnitude, the close the load-deformation paths to the perfect system load –deformation path.
The magnitude of load, is influenced significantly by the imperfection magnitude.
All real systems have imperfections. They may be very small but will be there
The magnitude of imperfection is not easy to know or guess. Hence if a perfect system analysis is done, the results will be close for an imperfect system with small imperfections.
However, for an unstable system – the effects of imperfections may be too large.