STRUCTRAL STABILITY AND DESIGN

OUTLINE
 Definition of stability
 Types of instability
 Methods of stability analyses
 Examples – small deflection analyses
 Examples – large deflection analyses
 Examples – imperfect systems
 Design of steel structures

STABILITY DEFINITION
 Change in geometry of a structure or structural component under
compression – resulting in loss of ability to resist loading is
defined as instability in the book.
 Instability can lead to catastrophic failure  must be accounted in
design. Instability is a strength-related limit state.
 Why did we define instability instead of stability? Seem strange!
 Stability is not easy to define.
*Every structure is in equilibrium – static or dynamic. If it is not
in equilibrium, the body will be in motion or a mechanism.
*A mechanism cannot resist loads and is of no use to the civil
engineer.
*Stability qualifies the state of equilibrium of a structure.
Whether it is in stable or unstable equilibrium.
 Structure is in stable equilibrium when small perturbations do not
cause large movements like a mechanism. Structure vibrates
about its equilibrium position.
 Structure is in unstable equilibrium when small perturbations
produce large movements – and the structure never returns to its
original equilibrium position.
 Structure is in neutral equilibrium when we can’t decide whether
it is in stable or unstable equilibrium. Small perturbation cause
large movements – but the structure can be brought back to its
original equilibrium position with no work.
 Thus, stability talks about the equilibrium state of the structure.
The definition of stability had nothing to do with a change in
the geometry of the structure under compression – seems
strange
 BUCKLING Vs. STABILITY
 Change in geometry of structure under compression – that results in
change in it’s ability to resist loads – called instability.
 Not true – this is called buckling.
 Buckling is a phenomenon that can occur for structures under
compressive loads.
 The structure deforms and is in stable equilibrium in state-1.
 As the load increases, the structure suddenly changes to
deformation state-2 at some critical load Pcr.
 The structure buckles from state-1 to state-2, where state-2
is orthogonal (has nothing to do, or independent) with state-1.
 What has buckling to do with stability?
 The question is - Is the equilibrium in state-2 stable or
unstable?
 Usually, state-2 after buckling is either neutral or unstable
equilibrium

P<Pc P=Pcr P>Pcr

P

P P P

 Thus, there are two topics we will be interested in this course
 Buckling – Sudden change in deformation from state-1 to
state-2
 Stability of equilibrium – As the loads acting on the
structure are increased, when does the equilibrium state
become unstable?
 The equilibrium state becomes unstable due to:
 Large deformations of the structure
 Inelasticity of the structural materials
 We will look at both of these topics for
 Columns
 Beams
 Beam-Columns
 Structural Frames

Structure subjected to compressive forces can undergo:

1. Buckling – bifurcation of equilibrium from deformation state-1 to
state-2.
 Bifurcation buckling occurs for columns, beams, and
symmetric frames under gravity loads only
2. Failure due to instability of equilibrium state-1 due to large
deformations or material inelasticity
 Elastic instability occurs for beam-columns, and frames
subjected to gravity and lateral loads.
 Inelastic instability can occur for all members and the
frame.
 We will study all of this in this course because we don’t want our
designed structure to buckle or fail by instability – both of which
are strength limit states.
 BIFURCATION BUCKLING
 Member or structure subjected to loads. As the load is increased,
it reaches a critical value where:
 The deformation changes suddenly from state-1 to state-2.
 And, the equilibrium load-deformation path bifurcates.
 Critical buckling load when the load-deformation path bifurcates
 Primary load-deformation path before buckling
 Secondary load-deformation path post buckling
 Is the post-buckling path stable or unstable?

SYMMETRIC BIFURCATION

 Post-buckling load-deform. paths are symmetric about load axis.

 If the load capacity increases after buckling then stable
symmetric bifurcation.
 If the load capacity decreases after buckling then unstable
symmetric bifurcation.
 ASYMMETRIC BIFURCATION

INSTABILITY FAILURE

 There is no bifurcation of the load-deformation path. The

deformation stays in state-1 throughout
 The structure stiffness decreases as the loads are
increased. The change in stiffness is due to large
deformations and / or material inelasticity.
 The structure stiffness decreases to zero and
becomes negative.
 The load capacity is reached when the
stiffness becomes zero.
 Neutral equilibrium when stiffness becomes
zero and unstable equilibrium when stiffness
is negative.
 Structural stability failure – when stiffness
becomes negative.
 FAILURE OF BEAM-COLUMNS

Snap-through buckling
 Introduction to Structural Stability

OUTLINE
 Definition of stability
 Types of instability
 Methods of stability analyses
 Examples – small deflection analyses
 Examples – large deflection analyses
 Examples – imperfect systems
 METHODS OF STABILITY ANALYSES
 Bifurcation approach – consists of writing the equation of
equilibrium and solving it to determine the onset of buckling.
 Energy approach – consists of writing the equation expressing
the complete potential energy of the system. Analyzing this
total potential energy to establish equilibrium and examine
stability of the equilibrium state.
 Dynamic approach – consists of writing the equation of
dynamic equilibrium of the system. Solving the equation to
determine the natural frequency () of the system. Instability
corresponds to the reduction of  to zero.

 Each method has its advantages and disadvantages. In fact, you

can use different methods to answer different questions
 The bifurcation approach is appropriate for determining the
critical buckling load for a (perfect) system subjected to loads.
 The deformations are usually assumed to be small.
 The system must not have any imperfections.
 It cannot provide any information regarding the post-
buckling load-deformation path.
 The energy approach is the best when 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.
 FOR ANY KIND OF BUCKLING OR STABILITY ANALYSIS –
NEED TO DRAW THE FREE BODY DIAGRAM OF THE DEFORMED
STRUCTURE.
 WRITE THE EQUATION OF STATIC EQUILIBRIUM IN THE DEFORMED
STATE
 WRITE THE ENERGY EQUATION IN THE DEFORMED STATE TOO.
 THIS IS CENTRAL TO THE TOPIC OF STABILITY ANALYSIS
 NO STABILITY ANALYSIS CAN BE PERFORMED IF THE FREE BODY
DIAGRAM IS IN THE
Example 1 – Rigid bar supported by rotational spring
k P
Rigid bar subjected to axial force P
Rotationally restrained at end
L
Step 1 - Assume a deformed shape that activates all possible d.o.f.

k L P

L sin


L (1-cos)
L cos

Write the equation of static equilibrium in the deformed state

M o 0   k  P L sin   0
k
P 
L sin 
For small deformations sin   
k k
 Pcr  
L L

Thus, the structure will be in static equilibrium in the deformed state

when P = Pcr = k/L
When P<Pcr, the structure will not be in the deformed state. The
structure will buckle into the deformed state when P=Pcr

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
P
L
L sin


k L sin

L cos
L (1-cos)
 M o 0  (k L sin  )  L  P L sin   0
k L2 sin 
P 
L sin 
For small deformations sin   
k L2
 Pcr  kL
L
Thus, the structure will be in static equilibrium in the deformed state when P = Pcr
= k L. When P<Pcr, the structure will not be in the deformed state. The structure
will buckle into the deformed state when P=Pcr.

Example 3 – Three rigid bar system with two rotational springs

P k k P
A D
B C
L L L

Assume deformed state that activates all possible d.o.f. Draw FBD in the deformed
state.

P k k P
1 2
L sin 2 D
A L sin 1 1 – 2) L
L
C
B 1 – 2)

Assume small deformations. Therefore, sin=

k P
P  D
1 – 2) L sin 2 L
A  C
L sin 1
L k(22-1)

B 1+(1-2)
k(21-2)
 M B 0  k (21   2 )  P L sin1  0  k (21   2 )  P L 1  0

M C 0  k (2 2  1 )  P L sin 2  0  k (2 2  1 )  P L  2  0

 Equations of Static Equilibrium

k (21   2 )  P L 1  0 2k  PL k  1  0

   
 k (2 2  1 )  P L  2  0  k 2k  PL 
  2  0

Therefore either  and  are equal to zero or the determinant of the

1 2
coefficient matrix is equal to zero.
When  and  are not equal to zero – that is when buckling occurs – the
1 2
coefficient matrix determinant has to be equal to zero for equil.
Take a look at the matrix equation. It is of the form [A] {x}={0}. It can also be
rewritten as [K]-[I]){x}={0}

  2k k 
    1  0
  L L  P1 0
2k  0 1    0
  k     2   

 L L  
This is the classical eigenvalue problem. ([K]-l[I]){x}={0}.
We are searching for the eigenvalues (l) of the stiffness matrix [K]. These
eigenvalues cause the stiffness matrix to become singular. Singular stiffness matrix
means that it has a zero value, which means that the determinant of the matrix is
equal to zero.
2k  PL k
0
k 2k  PL
 (2k  PL ) 2  k 2  0
 (2k  PL  k )  (2k  PL  k )  0
 (3k  PL )  (k  PL )  0
3k k
 Pcr  or
L L

Smallest value of Pcr will govern. Therefore, Pcr=k/L

Each eigenvalue or critical buckling load (Pcr) corresponds to a buckling shape that
can be determined as follows
Pcr=k/L. Therefore substitute in the equations to determine ϴ1 and ϴ 2
 k (21   2 )  P L 1  0  k (2 2  1 )  P L  2  0
Let P  Pcr  k Let P  Pcr  k
L L
 k (21   2 )  k1  0  k (2 2  1 )  k 2  0
 k1  k 2  0  k1  k 2  0
1   2 1   2
All we could find is the relationship between q1 and q2. Not their specific values.
Remember that this is a small deflection analysis. So, the values are negligible. What
we have found is the buckling shape – not its magnitude.
The buckling mode is such that ϴ1 and ϴ 2 Antisymmetric buckling mode