Accounting of earthquake effect assumes that the structure resists passively through the combination of strength, deformation and energy absorption. The damping level of such a system is usually very low and therefore the energy dissipation value of such a system in elastic behavior is also low. During the strong earthquake the deformations of such structure will exceed the limits of elasticity and the structure will not collapse only due to its ability to deform in an inelastic manner. Inelastic deformations take the form of localized plastic joints, which leads to increased pliability and energy absorption. Accounting this most of the earthquake energy is absorbed by the structure through local damages.
Let's consider the energy distribution in the structure. During the seismic effect a certain amount of energy is delivered to the structure. This energy is presented as kinetic and potential energy that must be absorbed or dissipated. If there is no damping in the structure, the oscillations will continue endlessly. However, the structure always has some damping characteristics and due to this the amplitude of oscillations decreases during the motion. It is possible to increase resistance of the structure to earthquake and reduce the number of damages by adding special damping elements to the structure. Such elements are integrated into the framework and absorb the energy that passes through them.
The law of conservation of energy of such system has the form:
where - total energy under earthquake action,
- total kinetic energy,
- elastic deformation energy,
- inelastic deformation energy,
- energy dissipated in a special damping element.
LIRA-SAPR allows you to take into account the work of special damping elements with the finite element of viscous damping (FE '62), whose scheme is shown in Fig. 1.
Fig. 1. Spring-damper system.
Let's consider the work of this element.
The energy loss in a single oscillation cycle in such an element can be defined as
where ' damping force.
In the linear mathematical model, the viscous damping force .
The equation of harmonic oscillations looks like
and the speed of movement is determined by the expression
With the fact that, we can write down
Then the energy loss in one cycle of oscillation is equal to
In resonance, and ,
Equation (4) can be written as
Expression (9) can be presented in the form:
The ellipse, defined by equation (10), can be represented graphically (Fig. 2).
Other mechanisms of energy dissipation can be represented as a viscous damper by equating the work per cycle, as it is done for a viscous damper.
Therefore, the equivalent damping factor is defined as
Let's solve a test example of oscillation of two frames. The geometrical and physical characteristics are identical. Length of span - 5m, height of the floor - 3 m, section of columns - I-section '35K1, section of beams - I-section '30B1. The same dynamic load is applied to both frames. In one frame between the floors we shall install the elements of viscous damper ('' 62) as it is shown on figure 3.
Characteristics of the viscous damper element are specified in the form (Fig. 4.),
Where ' element stiffness in axial direction (N/m), ' viscous damping coefficient (N*s/m).
To calculate the damping factor we will use the formula
Damping degree - dimensionless ratio of damping factor to critical damping:
The nature of the movement depending on the value of the degree of damping can be divided into three cases:
- < 1,0 oscillatory motion;
- > 1,0 non-oscillatory motion;
- = 1,0 critical damping motion.
Lets consider the problem of damping degree influence. For three frames with FE 62 (Fig. 5.) we will set different parameters of .
There are various mechanisms that can cause attenuation in the structure. LIRA-SAPR has a special FE, which simulates the operation of the linear element of the viscous damper. In this element, the damping force is proportional to the speed. In many cases such a simple expression for damping force is impossible. However, it is possible to obtain an equivalent viscous damper factor. For this purpose it is necessary to equate the loss of kinetic and potential energy with the dissipation energy.