N. Munirudrappa, Professor of Civil Engineering (Retd), Bengaluru University, reveals the behavior of multi-storey buildings and structures of R.C.C and steel under Elastic Stability in his study on the stability of tall buildings.

Stability of Tall BuildingsPicture courtesy: Skidmore, Owings & Merrill (SOM)

Stability of tall buildings is of keen interest to structural engineers as these modern structures are very slender in nature and have different structural shapes, besides which, they are subjected to wind loads, leading to lateral loads acting at each storey, which results in stability of the structure. Stability directly affects the drift in the structure. Generally, one must consider two aspects: Elastic stability and Inelastic stability.

Defining Stability
The resistance offered by a structure to undesirable movement like sliding, collapsing, and overturning is called stability. In other words, the load which initiates lateral sway is also termed as Elastic Stability.

The stability of any structure depends on support conditions, arrangement of members, and the external loadings. Based on this, the structure can be classified as Stable or Unstable. The modern use of steel and high strength alloys in engineering structures, especially in bridges, ships and aircrafts has made elastic instability a very important consideration. In recent years there are extensive investigations - both experimental and theoretical - governing the stability of structural elements such as bars, plates and shells.

The first problem of elastic stability and instability concerning lateral buckling of compressed members was solved about 200 years ago by L. Euler (1933). At that time, the principal structural materials were wood and stone, which, being of low strength, necessitated stout structural members, for which the question of elastic stability was not of primary importance.

Earlier, the theory developed by Euler under elastic stability was not in vogue without any practical applications (slender bars). Only with the start of extensive construction of steel railway bridges during the latter half of the past century did the question of buckling of compression members become of practical importance. This in turn accounted for the study of stability of thin plates and thin shells. Experience revealed that structures may fail in some cases, not on account of high stresses surpassing the strength of materials, but owing to insufficient elastic stability of slender thin-walled members. Under the pressure of practical requirements, the problem of lateral buckling of columns originated by Euler has been extensively investigated; hence, buckling of compressed members is only a case of elastic instability.

The modern design of bridges, buildings of RCC/Steel, ships and aircrafts is confronted by a variety of stability issues. Though their sections may be solid in the form of struts, built-up, and lattice with columns and tabular members, where the possibility of local buckling as well as buckling as a whole can be observed. To visualize buckling aspects of structural systems here are a few examples:
  • The use of thin sheet material as in plate girders and airplane structures one should keep in mind that thin plates may prove unstable under the action of forces in their own planes and fail by buckling sideways (web buckling or web crippling).
  • Thin cylindrical slabs such as vacuum vessels, which have to withstand uniform external pressure, may exhibit instability and collapse at a relatively low stress if the thickness of the shell is too small in comparison with the diameters. The thin cylindrical shell may buckle also by axial compression, bending, torsion, or by a combination of these. To understand structural stability, we need to apply the concept of mathematics and the theory of materials’ strength, stiffness, and stability.
With reference to the study of tall buildings, the study related to beam-columns was necessary. Here, the structure at each floor slab is subjected to gravity loads and lateral loads (due to winds) (Ref Fig1).

As per the mathematics, the required load which initiates buckling is given by
P = π2 EI / L2 to be modified Pcr = π2 E Imin / l2e

The limiting value of compressive force P is called the critical load and is denoted by Pcr. Pcr was defined by Leonard Euler (1774). Pcr is responsible for the initiation of buckling (Elastic Stability). After removal of the load the buckling cannot be seen.

General Assumptions for Stability Analysis
  • The structure is perfectly straight without any initial deformation.
  • Initial eccentricities do not occur when vertical loads are applied.
  • The structure is considered as a whole, ignoring the possibility of local buckling.
  • Shear deformation, characteristic of small and medium structures, does not influence the buckling stability.
  • The simultaneous action of lateral and vertical loads is ignored (B.Taranath).
Theory of Elastic Stability (based on Euler’s theory) Under Axial Load
Generally, columns are subjected to axial loads, hence design of columns related to steel structures is based on an indirect method. The axial compressive stress developed is inversely proportional to slenderness ratio. Slenderness ratio decides the design of columns as short and long columns. For RCC columns, there is no definite relationship to say whether it is a long column or a short column. Generally, by the code it is stated that the length of the column to least lateral dimension is equal to 12 or greater than 12 and is termed as long column. This will be applied as correction factor to the stresses.

Fundamentals of Stability Theory (Galambose)
It is not necessary to be a structural engineer to have a sense of what it means for a structure to be stable. Most of us have an inherent understanding of the definition of instability that a small change in load will cause a large change of displacement. If this change in displacement is large enough or is in a critical member of a structure, a local or member instability may cause collapse of the entire structure.

An understanding of stability theory or the mechanics of why structures or structural members become unstable is a particular subject of engineering mechanics of importance for engineers whose job is to design safe structures.

Structural engineers are tasked by society to design and construct buildings, bridges and a multitude of other structures. These structures provide a load bearing skeleton that will sustain the ability of the constructed structure to perform its intended functions. For studies on structures under calamities, the role of the structural engineer is to keep them safe without getting damaged. This is governed by the important characteristics of the structure known as stiffness, strength and stability. The strength of structures, namely stability, examines how and what under loading conditions the structure will pass from stable to an unstable one.

In a well-defined structure, the user or occupant will never have to think of the structure’s existence. Absolutely safety of course is not an achievable goal; for example, the recent man-made disaster and collapse of the World Trade Center in USA, led to the conclusion that any structure may become unstable under extreme and unforeseeable circumstances.

Wind Effect on Structures
Wind effect on structures can be classified as static-static wind effect, which primarily causes elastic bending and twisting of the structure. For tall, long span and slender structures, a dynamic analysis is essential. Wind gusts cause fluctuating forces on the structure which induce large dynamic motions, including oscillations. Wind is essentially the large-scale horizontal movement of free air and is an important consideration when designing tall structures because since it exerts loads on building. It is a phenomenon of great complexity because of the many flow situations arising from the interaction of wind structures.

Stabilizing Structures Against Forces
The seismic performance of high-rise buildings is assessed in the present study through fragility relationships. Different approaches can be used to derive fragility functions (eg. Rossetto and Elnashai 2005). The approach of generating damage data through analytical simulations is the most realistic option, particularly for the UAE and hence is adopted in the present study (Mwafy 2012).

Several techniques for deriving vulnerability curves based on the numerically simulated structural damage statistics have been also proposed in the literature with diversity in structural idealizations, analysis methods, seismic hazards, and damage models. Most of these techniques require a lot of analyses to account for uncertainty. This is particularly true when adopting the multi-degree-of-freedom inelastic dynamic simulations for deriving the vulnerability relationships (which is the approach adopted herein).

The present study aims at investigating the relationship between seismic performance and cost-effectiveness of tall buildings through designing and developing detailed simulation models for high-rise buildings with various concrete and formwork. Over 1600 inelastic pushover analyses (IPAS) and incremental dynamic analyses (IDAS) are performed using 20 natural and artificial earthquake records to derive vulnerability relationships and to provide insights into the seismic response of the reference structures up to collapse.

Stability of Frames
A frame is said to be stable if the number of unknown reactions are greater than or equal to available equations of equilibrium. Stability is a field of mechanics that studies the behavior of structures under compression. When a structure is subjected to a sufficiently high compressive force or stress, it will have a tendency to lose its stiffness and display a noticeable change in geometry and become unstable. When instability occurs, the structure loses its capacity to carry the applied load and is incapable of maintaining a stable equilibrium configuration.

Stability Analysis: Applications on Framed Structures (RCC & Steel)

P-Delta Analysis (Case Study)
Mallikarjuna B.N. and Prof. Ranjith A focused on P-Delta analysis to be compared with linear static analysis. An 18-storey steel frame structure with 68.9m has been selected as a model for their research. The model is analysed by using STAAD Pro 2007 structural analysis software with consideration of P-Delta effect. At the same time, the influence of different bracing patterns has been investigated. The steel braces are usually placed in vertically aligned spans. This system allows obtaining a huge increase of stiffness with minimal added weight. So, it is very effective for the existing structure for which the poor lateral stiffness. The loads considered for the analysis are gravity load, live load, and wind load. The frame structure is analysed for wind load as per IS875 (part 3) 1987. After analysis, the comparative study is presented with respect to maximum storey displacement and axial force.

Stability Analysis of Tall Buildings
Tall buildings of ever-growing heights are being constructed worldwide. Consequently, structural systems have emerged. In the last few decades, one can see a tendency towards high-rise structures which are slender in nature.

Tall buildings and structural efficiency improvements lead to increased reserves of stiffness, resulting in reduced stability. The more slender the structure, more will be the appearance of bending instability. Due to these uncertainties in the building configuration, stability analysis of tall buildings represents an important issue for structural design.

Another uncertainty has been observed from earthquakes; it has been observed that a building with discontinuity in the stiffness and mass subjected to concentration of forces and deformation at the point of discontinuity may lead to the failure of member at the junction and collapse of the building.

Seismic Effects on High-Rise Buildings
When earthquakes occur, a building undergoes dynamic motion. This is because the building is subjected to inertia forces that act in opposite directions to the acceleration of the earthquake excitations. These inertia forces called seismic loads are usually dealt with by assuming forces external to the building.

Time histories of earthquake motions are also used to analyse high-rise buildings and their elements and contents for seismic design. The earthquake motions for dynamic design are called design earthquake motions. In the previous recommendations, only the equivalent static seismic loads were considered to be seismic loads. In ISO/TC98 which deals with “Bases for design of structures”, the term “action” is used instead of “load” and action includes not only load as external force but various influences that may cause deformations to the structures. In the future “action” may take the place of “load”.

Failure of Structures
It can be a total collapse or more unstable such as a beam that suffers excessive deflection causing floors to crack. If the structure crosses the various limit states it fails. The various limit states are excessive deflection, large rotations at joints, and cracking of metal or concrete by corrosion or excessive vibration under dynamic loads.

Stability failures are often catastrophic and occur more often during erection. This may be due to carelessness and inexperience of the structural engineer. Structural stability is generally associated with the presence of compressive axial force or axial strain in any structural element is part of the crosssection of beam or column. Local stability occurs in a single portion of a column-web buckling in a steel column.

Conclusion
The most important factor in the designing of high-rise buildings is the buildings need to withstand the lateral forces imposed by the winds and potential earthquakes. Most high-rises have frames made of steel or steel and concrete. Their frames are constructed of columns (vertical support members) and beams (horizontal support members), Cross-bracing or shear walls may be used to provide a structural frame with greater rigidity in order to withstand wind stresses.

The shear force and bending moments are higher for ground storey columns with respect to first storey columns. Behaviour of square column is better than rectangular column in terms of storey drift, base shear, and roof displacement. The shear walls used to eliminate the lateral load and soft storey effects, when the shear walls are kept centrally, they are not affected much by the behaviour of structures. The effect of masonry infills in the structures increases the stiffness of the structural element, thus minimizing the stability of the structure.

Bibliography
  • Bazant, Z.P., Cedolin, L. and Hutchinson, J.W., 1993. Stability of structures: elastic, inelastic, fracture, and damage theories.
  • Chen, W.F. and Lui, E.M., 1987. Structural stability: theory and implementation (No. 624.12 C5).
  • Euler, L., 1744. Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes. apud Marcum-Michaelem Bousquet.
  • Taranath, B.S., Taylor and Francse group, 2009. Reinforced concrete design of tall buildings.
  • Timoshenko, S.P. and Gere, J.M., 1963. Elastic stability theory. Warsaw: Arkady.
  • Smith, J., Coull, A. and Structures, T.B., 1991. Analysis & Design. Wiley and sons, New York.
  • Zalka, K.A, 2002. Global structural analysis of buildings. Taylor and Francse.
  • Galambos, T.V. ed., 1998, 4th Edition, Guide to stability design criteria for metal structures. John Wiley & Sons.
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