struc stbility-2

Structural Stability

General Principles for Structural Stability

1940 Tacoma Narrows Bridge, Washington – torsional vibration under 40 mph (64 km/h) winds

1970 Milford Haven Bridge, Wales – errors in the box girder design

1968 Ronan Point, Newham, East London – gas explosion demolished a load bearing wall, causing progressive collapse of floor slabs

2013 Menara UMNO, Penang– lightning arrester toppled

Tensile fracture

Compression buckling in web

Compression buckling in flange

Buckling rail lines

What is BUCKLING ??

• A physical phenomenon of a reasonably straight, slender member (or body), bending laterally (usually abruptly) from its longitudinal position due to compression.

• When a slender structure is loaded in compression, for small loads, it deforms with hardly any noticeable change in geometry and load-carrying ability.

• On reaching a critical load value, the structure suddenly experiences a large deformation and it may lose its ability to carry the load. At this stage, the structure is considered to have buckled.

What is BUCKLING ??

• For example, when a rod is subjected to an axial compressive force, it first shortens slightly but at critical load, the rod bows out, and it is said that the rod has buckled.

• In the case of a thin circular ring under radial pressure, the ring decreases in size slightly before buckling into a number of a circumferential waves.

• For a cruciform column under axial compression, it shortens and buckles in torsion.

• Buckling is considered as a failure mode as it reduces the compressive resistance of structures to a level lower than the ultimate value its material can withstand.

Classification of buckling

•Overall / global buckling

•Local buckling

• Interactive buckling (local & overall)

Classification of buckling

Overall / global buckling

Overall / global buckling (torsional buckling)

Local buckling in a column

Local buckling in a beam flange

Local buckling in the compression flange of a box girder

Local buckling in the flange and web of a plate girder

Local buckling in the web and flange of a composite plate girder

Local buckling in the web and flange of a curved plate girder

Local buckling in the web of a tapered plate girder

Concepts of stability..

• Primary aim of structural calculations is to produce safe and strong structures.

• Structures fail due to:

▫ Yielding (plastic failure)

▫ Buckling (compressive failure)

▫ Fracture (tensile failure)

▫ Fatigue (long term effect)

Concepts of stability

• The deformation of a structure under loads is governed by the following necessary and sufficient conditions: ▫ Equilibrium conditions (Stress)

▫ Stress-strain relationship (Stress & Strain)

▫ Strain-displacement relationship (Strain & Displacement)

together with the boundary conditions. • These are referred to as Governing Equations. • These conditions govern the equilibrium position of the

loaded structure.


• When a change in geometry of structure or structural component under compression results in loss of its ability to resist loadings, this condition is called instability.

• Instability can lead to catastrophic failure in a structure; it must be taken into account in designing a structure.

• In Ultimate Limit State (ULS) design, the structure or structural component is designed against all pertinent limit states that may affect the safety or performance of structures at their maximum load-carrying capacities.

• Strength limit states, deal with performance of structures at their maximum load-carrying capacities. ▫ Examples: structural failure due to either formation of plastic

collapse mechanism, or to member or frame instability.


• Serviceability Limit State (SLS) are concerned with structural performance under normal service conditions. They pertain to appearance, durability, maintainability of structure. ▫ Examples: deflections, drift, vibration and corrosion.

• Importance of considering stability in design is recognised by most practicing engineers; however, the subject remains perplexing to some. This is because the use of first order structural analysis (which is familiar to most engineers) is not permissible in stability analysis.

• In stability analysis, the change in geometry of the structure must be taken into account. Consequently, equilibrium equations must be written based on geometry of structure that becomes deformed under load. This is known as second-order analysis.


• The concept of stability is best illustrated by the well-known example of a ball on a curved surface. For a ball initially in equilibrium, a slight disturbing force applied to the ball on a concave surface will displace the ball by small amount, but ball will return to its initial equilibrium position once it is no longer being disturbed. In this case, the ball is in a stable equilibrium.


• If the disturbing force is applied to a ball on convex surface and then removed, the ball will displace continuously from, and never return to, its initial equilibrium position, even if the disturbance was infinitesimal. The ball is now in unstable equilibrium.


• If the disturbing force is applied to the ball on a flat surface, the ball will attain a new equilibrium position to which the disturbance has moved it and will stay there when the disturbance is removed. The ball is in a neutral equilibrium.



• The definitions of stable and unstable equilibrium apply only to cases in which the disturbing force is very small.

• It is possible for a ball, under certain conditions to go from one equilibrium position to another. For example, a ball that is “stable” under small disturbance may go to an unstable equilibrium under large disturbance (Fig. a) or vice versa (Fig. b).

• This is known as effect of finite disturbance.

Fig. (a) Fig. (b)

Types of instability..

• Buckling (instability) of structures can be grouped into 2 categories:

▫ Bifurcation buckling

▫ Limit load buckling


• Bifurcation buckling ▫ The deflection under compressive load changes from one

direction to an alternative (different) direction. ▫ For examples, from axial shortening to lateral deflection (buckling of

columns loaded axially and buckling of thin plates subjected to in-plane compressive forces) or buckling of rings subjected to radial compressive forces.

▫ The load at which the bifurcation occurs is called the critical load.

▫ The deflection path that exists prior to bifurcation is known as the primary path, and the deflection path after bifurcation is called the secondary/post-buckling path.


• Limit load buckling ▫ This type of instability is characterized by the fact

that there is only a single mode of deflection from start of loading to the limit or maximum load.

▫ The structure attains a maximum load without any previous bifurcation.

▫ For examples,

buckling of shallow arches and spherical caps under uniform external pressure


• Limit load buckling

Methods of buckling analysis

1. Vector approach

2. Energy approach

to obtain the GOVERNING EQUATIONS in the form of eigenvalue problems

eigenvalue represents the buckling load whilst eigenvector the buckling mode

Why do we do buckling analysis ?

• Buckling load provides the basis for commonly used formulas in the design codes.

• An essential step towards understanding the buckling behaviour of complex structures, initial imperfections, residual stresses, etc.

• To obtain more accurate results especially when there are no standard solutions available.

The history

• Leonhard Euler (1707-1783) ▫ The first study on elastic stability on equilibrium equation and

buckling load of a compressed elastic column. • Joseph-Louis Lagrange (1736-1813)

▫ Developed the energy approach for the study of mechanics problems.

• Jules Henry Poincare (1854-1912) ▫ Founder of the bifurcation theory and the classification of

singularities. • Aleksandr Mikhailovich Liapunov (1857-1918)

▫ Introduced the generalised energy functions (Liapunov functions) • Theodore von Karman (1881-1963)

▫ Worked on inelastic buckling of columns, hysteresis loops and plastic deformation of beams.

• Warner Tjardus Koiter (1914-1997) ▫ Initiated the classical non-linear bifurcation theory.

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