stability of structure having dynamic loadings

1. STRUCTURAL ENGINEERING
CV3602
Finite Element Analysis
Stability
1

2. General Introduction
• Build upon the principles of structural mechanics and analysis that you
learned in previous courses ➔study advanced topics in structural
engineering.
• To cover the basic principles of the Stability of Structures and its application
to the elastic analysis of structures.
Stability of structures
Aims
2
Structural Engineering (CV3602)

3. • Demonstrate comprehensive understanding of the background of the stability
of structures for linear elastic structures
• Know and understand the derivation of analytical solutions for calculating the
critical load of perfect columns
• Know and understand the use of stability functions to determine the critical
buckling load of frames
Learning outcomes
General Introduction
Stability of structures
3
Structural Engineering (CV3602)

4. General Introduction
Finite Element Analysis (FEA)
References
Structural Engineering (CV3602)
Reference
Timoshenko SP (1936) Theory of elastic stability. McGraw-Hill
Bleich, F (1952) Buckling load of metal structures, Mc Graw- Hill.
Chajes, A (1974) Principles of structural stability theory, Prentice-Hall
Bazant ZP, Cedolin L (1991) Stability of Structures, Dover
Galambos T, Surovek A (2008). Structural stability of steel. Concepts and
applications for structural engineers. Wiley & Sons
4
Chai HY, Sung Lee (2011). Stability of structures. Principles and applications.
Elsevier

5. General Introduction
• Slender columns are subject to buckling
• Small compressive axial loads result in axial shortening of the member.
• After exceeding critical load, the member deflects laterally (bending) giving
rise to large deformations which drive to collapse
General
P
Level of P
Critical
load
0
column
shortens
column
bends
Stability of structures
shortening
P
bending
5
Structural Engineering (CV3602)

6. General Introduction
• Members in axial tension fail when tensile stresses exceed material tensile strength
• Stocky members in axial compression fail when compressive stresses exceed material
compressive strength
• Slender members in compression fail in buckling
Modes of failure under axial loads
buckling
Stability of structures
Tension
Failure types
Compression
Material
Stocky
member
Slender
member
Material Buckling
6
Structural Engineering (CV3602)

7. General Introduction
• Buckling load cannot be determined solely by material strength.
• It depends on:
-Member dimensions
-Boundary conditions
-Material properties
The buckling problem
Complex problem
Stability of structures
7
Structural Engineering (CV3602)

8. General Introduction
-A direct answer does not exist
-We can only observe the buckling phenomenon and try to explain it
-One good explanation is given by Salvadori and Heller in the book
Structure in Architecture:
“The tendency of all weights to lower their position is a basic law of nature. It is another basic
law of nature that, whenever there is a choice between different paths, a physical
phenomenon will follow the easiest path. Confronted with the choice of bending out or
shortening, the column finds it easier to shorten for relatively small loads and to bend out for
relatively large loads. In other words, when the load reaches its buckling value the column
finds it easier to lower the axial load by bending than by shortening.”
Why compression members buckle?
Stability of structures
8
Structural Engineering (CV3602)

9. General Introduction
• Stable equilibrium: If the ball is slightly displaced from its original position then it will
return to this position after the removal of the disturbing force
• Unstable equilibrium: If the ball is slightly displaced from its rest position then it
doesn’t return after removal of disturbing force, but continues to move away from the
original position
• Neutral equilibrium: If the ball is slightly displaced from its original position neither
returns nor moves away from its original position after the removal of the disturbing
force. It stays in the position that the disturbing force has moved it.
Types of equilibrium
Equilibrium conditions of a ball in a gravity field
Stable Unstable Neutral
Stability of structures
9
Structural Engineering (CV3602)

10. General Introduction
• The ball is in equilibrium at any point along line ABC. In the region between AB the
equilibrium is stable and in the region between BC the equilibrium is unstable. At point
B (the transition of these two regions) the ball is in a state of neutral equilibrium.
• The behaviour of the column is similar. The column in straight configuration is stable at
small loads, but unstable at large loads (after buckling). Hence, it can be assumed that
a state of neutral equilibrium exists between these two states. The critical load
corresponds to this neutral equilibrium. This means that for the critical load the
column remains in equilibrium both in the straight and in slightly bent configuration.
This observation will be used for the determination of the critical buckling load.
Neutral Equilibrium
Stability surface (Chajes 1974)
Stability of structures
10
Structural Engineering (CV3602)

11. • Euler column is an idealized column for which it is assumed that:
1. It has a constant cross section
2. It is made of a homogeneous material
3. The material is linear elastic (Hooke’s law)
4. Member ends are simply supported. The lower end cannot move vertically and horizontally.
The upper end cannot move horizontally, but it can move vertically (roller support)
5. The column is perfectly straight and the load P is applied along its centroidal axis
(perfect column)
Definition of Euler column
P
L
Stability of structures Euler column
P
11
Structural Engineering (CV3602)

12. We study the equilibrium in the slightly bent configuration of the figure.
x is an arbitrary distance from the lower end of the column.
v is the deflection along the Y-axis
Critical load of the Euler column
P
P
P Y
L
X
P
P
x
v
M
Stability of structures
By applying moment equilibrium about the Z-axis at a section a distance x from the base we obtain
𝑀 + 𝑃 ∙ v = 0 → 𝐸𝐼 ∙ v′′ + 𝑃 ∙ v = 0 (1)
By assuming that
𝜇2
=
𝑃
𝐸𝐼
(2)
Euler column
P
12
Structural Engineering (CV3602)

13. We get that
v′′
+ μ2
∙ v = 0 (3)
This is a 2nd order linear and homogeneous ordinary differential equation with constant
coefficients.
The general solution for v is
v=A ∙ sin μx + B ∙ cos μx (4)
Where A and B are arbitrary constant coefficients.
Critical load of the Euler column
Stability of structures Euler column
13
Structural Engineering (CV3602)

14. To evaluate the arbitrary parameters A and B we apply the boundary conditions.
1) At x=0 → v=0
0 = A ∙ sin 0 + B ∙ cos 0 → B = 0 (5)
2) At x=L → v=0
0 = A ∙ sin μL (6)
Critical load of the Euler column
P
P Y
X
Equation (6) is valid when:
a) A=0 (7)
This is called the trivial solution and it valid for all P. It represents the fact that when the shape of
the column remains perfectly straight then equilibrium holds for all loads.
b) sin(μL)=0 (8)
Equation (8) yields
μ·L=n·π for (n=1,2,3…) → μ=π/L (for n=1 which is the critical case) (9)
Stability of structures Euler column
L
14
Structural Engineering (CV3602)

15. Critical load of the Euler column
By substituting Eq. (2) into Eq. (9) and solving for P we obtain
Pcr =
EI∙π2
L2 (10)
Pcr is called the Euler critical load.
• The Euler load is the smallest load for which neutral equilibrium in a slightly displaced position
is feasible. Hence, it is the load at which the column ceases to be in stable equilibrium and it
marks the transition from stable to unstable equilibrium.
• The greater values of P for higher n values are mathematical solutions of the problem but
without any significance for the stability problem.
For n=1→ μ=π/L Eq. (4) gives that
v = A ∙ sin
π
L
x (11)
Hence, v is a sinusoidal function of arbitrary amplitude A.
Stability of structures Euler column
15
Structural Engineering (CV3602)

16. Eigenvalue problems
The previous theory is based on the assumption of small deflections, which drives to a linear
differential equation. This is why it is called linear column theory
The large-deflection theory drives to a nonlinear differential equation ➔ nonlinear column theory
The linear column theory should not be confused with the linear beam bending theory according
to which displacements are proportional to loads. This is not the case for the stability problem
Linear column theory belongs to the class of eigenvalue problems. In these problems, non-zero
values of the dependent variable v(x) exist only for certain discrete values of other parameters.
The values of the other parameters are known as eigenvalues and the solutions as eigenvectors.
In our case, the critical loads P =
n∙EI∙π2
L2 (n>=1) are the eigenvalues and the solutions v = A ∙
sin
n∙π
L
x are the eigenvectors. For n=1 we get the critical load and the eigenvector is called
fundamental buckling mode
Stability of structures Euler column
16
Structural Engineering (CV3602)

17. Critical load of the Euler column
Stability of structures Euler column
Lateral deflection amplitude
Load P
Pcr
• Up to Euler load he column remains straight (zero lateral amplitude).
• At the Euler load the column is deflected laterally with indeterminate amplitude
• This response is called bifurcation and occurs only to perfect columns
Lateral deflection – compression axial force P response of the Euler Column
Bifurcation
17
Structural Engineering (CV3602)

18. Effective Leff or critical Lcr buckling length of a column, with different boundary conditions at its
ends, is the length of an equivalent Euler column that has the same Pcr as the examined column.
Hence, we can use Euler formula with Leff to calculate Pcr.
𝑃𝑐𝑟 =
𝜋2
𝐿𝑒𝑓𝑓
2 𝐸𝐼
With the concept of effective length we can use the same formula for all boundary conditions.
Stability of structures Other boundary conditions
The critical or effective buckling length concept
Boundary
conditions
Hinged-hinged Fixed-fixed Hinged-fixed Fixed-free
Effective
length
Critical load 𝑃
𝑐𝑟 =
𝜋2
L2
𝐸𝐼 𝑃𝑐𝑟
4𝜋2
L2
𝐸𝐼 𝑃
𝑐𝑟
2.04𝜋2
L2
𝐸𝐼 𝑃𝑐𝑟 =
𝜋2
4L2
𝐸𝐼
L
eff
=
L
L
L
L
L
L
eff
=
0.5L
L
eff
=
0.7L
L
eff
=
2L
Structural Engineering (CV3602) 18

19. The Euler formula can be written for design purposes as:
𝑃𝑐𝑟 =
𝜋2
𝐿𝑒𝑓𝑓
2 𝐸𝐼 → 𝜎𝑐𝑟 =
𝜋2
𝐴𝐿𝑒𝑓𝑓
2 𝐸𝐼 (1)
Where A is the area of the cross-section and σcr is the critical buckling stress .
We define r as the radius of gyration given by the following equation:
𝑟 =
𝐼
𝐴
(2)
r is clearly a section property. By combining Eqs (1) and (2) we obtain
𝜎𝑐𝑟 =
𝜋2
λ2 𝐸 (3)
In Eq. (3), λ = Leff/r is the slenderness which is a non-dimensional geometric parameter. Eq (3)
shows that the critical buckling stress is simply a function of material (elastic modulus) and the
slenderness ratio (member property).
For design purposes a safety factor (γ>1) is usually applied as:
𝜎𝑐𝑟,𝑑𝑒𝑠𝑖𝑔𝑛 =
𝜋2
𝛾⋅λ2 𝐸 (4)
Stability of structures Buckling design curve
Buckling design curve
Structural Engineering (CV3602) 19

20. If we plot the relationship of σcr versus the slenderness ratio by Eq. (3) we get a single buckling
critical curve for a given material (e.g. steel, timber)
Area 1: No failure
Area 2: Failure by buckling prior to yielding
Area 3: Failure by yielding prior to buckling
Area 4: Failure by yielding and buckling
Stability of structures
Buckling design curve
Slenderness λ
Critical
stress
σ
cr
Column design curves for specific material
Yielding
Buckling
1
2
4
3
fy
σcr
Buckling design curve
Structural Engineering (CV3602) 20