BASIC ELEMENTS OF EARTHQUAKE RESISTANT DESIGN
Structures on the earth are generally subjected to load of two types static and dynamic. Static loads are constant with time while dynamic loads are time varying. The majority of civil engineering structures are designed with assumptions that all applied loads are static. The effect of dynamic loads is not considered because the structure is rarely subjected to dynamic loads; more so, its consideration in analysis makes the solution more complicated and time consuming. This feature of neglecting the dynamic forces may some times become the cause of disaster, particularly in the case of earthquake. There is a growing interest in the process of designing civil engineering structures capable to withstand dynamic loads, particularly, earthquake induced load.
The dynamic force may be an earthquake force resulting from rapid movement along the plane of faults within earth’s crust. This sudden movement of fault releases great energy in the form of seismic waves, which are transmitted to the structures through their foundations, and cause to set the structure in motion. These motions are complex in nature and induce abrupt horizontal and vertical oscillations in structures, which result accelerations, velocities and displacements in the structure. The induced accelerations generate inertial forces in the structure, which are proportional to the acceleration of the mass and acting opposite to the ground motion.
The energy produced in the structure by the ground motion is dissipated through internal friction within the structural and non-structural members. This dissipation if energy is called damping. The structures always posses some intrinsic damping, which diminishes with time once the seismic excitation stops. These dissipative or damping forces are represented by viscous damping forces, which are proportional to the velocity induced in the structure. The constant of proportionality is called as linear viscous damping. The resisting force in the structures is proportional to the deformation induced in the structure during the seismic excitation. The constant of proportionality is referred to as stiffness of structure. Stiffness greatly affects the structure’s uptake of earthquake generated forces. On the basis of stiffness the structure may be classified as brittle or ductile.
Brittle structure having greater stiffness proves to be less durable during earthquake while ductile structure performs well in earthquakes.This behavior of structure evokes an additional desirable characteristic called ductility. Ductility is the ability of structure to undergo distortion or deformation without damage or failure.
The basic equation of static equilibrium under displacement method of analysis is given by
F(ext) = ky
Where, F(ext) is the external applied static force, k is the stiffness resistance, and y is the resulting displacement. The restoring force (ky) resists the applied force, F(ext).
Now, if the applied static force changes to dynamic force or time varying force the equation of static equilibrium becomes one of the dynamic equilibrium and has the form
F(t) = my(t) + cy(t) + k(t)y(t)
Where,
my(t) = inertia forces acting in a direction opposite to that of seismic motion applied to the base of the structure, whose magnitude is the mass of the structure times its acceleration, m is the mass (kg) and y(t) is the acceleration (m/sec2). Inertia forces are the most significant which depend upon the characteristic of the ground motion and the structural characteristics of structure. The basis characteristic of the structure and ground is its fundamental or natural period.
The fundamental periods of structures may range from 0.05 sec for a well anchored piece of equipment, 0.1 for a one storey frame, and 0.5 for a low structure up to 4 storeys and between 1 to 2 seconds for a tall building of 20 storeys.
Natural periods of ground are usually in the range of 0.5 to 1 sec so that it is possible for the building and ground to have the same fundamental period and therefore, there is high probability for the structure to approach a state of partial resonance called as quasi resonance. Hence, in developing a design strategy for a building, it is desirable to estimate the fundamental periods both of the structure and of the site so that a comparison can be made to see the existence of the probability of quasi resonance.
Cy(t) = damping force acting in a direction opposite to that of the seismic motion, c is the damping co-efficient (N sec/m) and y(t) the velocity (m/sec). The value of damping in a structure depends on its components. The damping effect is expressed as a percentage of the critical damping which is the greatest damping value that allows vibratory moment to develop. The degrees of damping in common types of structures are reinforced concrete 5 to 10%, metal frame 1 to 5%, and masonry 8 to 15%
k(t)y(t) = restoring force k(t) is the stiffness (N/m) or resistance is a function of the yield condition in the structure which is in turn a function of time. y(t) is the displacement in meters. F(t) is the externally applied force (N).
The equation above is a second order differential equation that needs to be solved for the displacement y(t). The number of displacement components required specifying the position of mass points is called the number of degrees of freedom to obtain an adequate solution. For some structures, single degree of freedom may be sufficient where as for others several hundred degrees of freedom may be required.
