Literature review
2.1 Dynamic analysis of structures
The inertial (mass), elastic (stiffness) and energy dissipation (damping) properties of an actual struc-ture result from the properties of all structural members in the strucstruc-ture and the interaction between the members [15]. For analysis purposes, however, a structure can be idealized as a system with a specific number of lumped masses and a massless supporting structure with a certain stiffness and a certain damping level. Relatively simple structures can be schematized with a single lumped mass, whereas multiple lumped masses are used in order to idealize more complex structures. Such sys-tems are called single degree of freedom (SDOF) and multiple degrees of freedom (MDOF) syssys-tems, respectively.
2.1.1 Equation of motion
Figure 2.1 shows the idealization of a simple, one-story structure, that is subjected to an external force p(t) in the lateral direction. Three separate components must be assigned to this SDOF system:
the mass component m1, the stiffness component k1 and the damping component c1.
a) idealized SDOF structure b) mass-spring-damper system Figure 2.1: Modelling of one-story structure [15].
In order to find the lateral displacement u(t) of the mass over time due to the external force p(t), the dynamic equilibrium equation, Eq. 1, must be solved. This differential equation, which is based on D’Alembert’s principle of dynamic equilibrium, is also called the equation of motion (EOM). The EOM equates the external force p(t) on a system to the resulting reaction forces within the system:
the inertia force fI, the damping force fD and the spring force fS.
fI+ fD+ fS = p(t) → m1u(t) + c¨ 1u(t) + k˙ 1u(t) = p(t) (1) where ¨u(t) is the acceleration, ˙u(t) is the velocity and u(t) is the displacement of the mass over time due to external force p(t).
Literature review page 7
Many structures, however, cannot be accurately idealized using a single lumped mass. Such struc-tures get schematized as systems with multiple lumped masses and, hence, multiple degrees of freedom. For MDOF systems, a mass matrix m, stiffness matrix k and damping matrix c must be determined which contain the masses, stiffnesses and damping coefficients corresponding to all de-grees of freedom for all eigenmodes of deformation. Figure 2.2 shows the idealization of a two-story structure, that is subjected to an external force p(t) in the lateral direction.
a) idealized 2DOF structure b) mass-spring-damper system Figure 2.2: Modelling of two-story structure [15].
In order to find the lateral displacement u(t) of the masses for all eigenmodes due to the external force p(t), the dynamic equilibrium equation, Eq. 2, must be solved. The EOM for MDOF systems is similar to the EOM of SDOF systems, despite of the fact that scalars are replaced by matrices and vectors, due to the higher number of degrees of freedom and eigenmodes.
m¨u + c ˙u + ku = p(t) (2)
For the 2DOF system that is shown in Figure 2.2, the EOM can be described as follows:
"
m1 0 0 m2
# (u¨1
¨ u2
) +
"
c1+ c2 −c2
−c2 c2
# (u˙1
˙ u2
) +
"
k1+ k2 −k2
−k2 k2
# (u1 u2
)
= (p1(t)
p2(t) )
(3)
In order to solve the EOM of a SDOF system, the scalars m1, k1 and c1 must be determined, whereas the matrices m, k and c must be determined for solving the EOM of a MDOF system. The determination of the three components is described in section 2.1.2. Using the derived scalars or matrices, the vibration properties of the idealized structure can be derived as described in section 2.1.3. Several methods can be used to solve the EOM. Some of these methods can only be used for analyzing linear structures, whereas others can also be applied for analyzing nonlinear struc-tures. In the current research, Newmark’s integration method is applied to solve the EOMs. This method, which can be applied to analyze both linear and nonlinear structures, is briefly introduced in Appendix A.3.
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2.1.2 Mass, stiffness and damping
The mass m1 can be determined by estimating which part of the structure will (most significantly) contribute to the inertia force of the system that arises as a result of earthquake loading. In case a structure contains a heavy roof and relatively light columns, it may be assumed that the corresponding mass m1is equal to the mass of the roof, while the mass of the columns is neglected due to their low inertial impact.
The spring stiffness k1represents the lateral stiffness of the structure. This value therefore depends on the material, the dimensions and the boundary conditions of the structural elements. In case both spring stiffness k1 and damping coefficient c1 do not change over time, independent of the load’s magnitude, the SDOF system can be categorized as a linear system. In reality, however, the stiffness k1generally changes due to a fluctuating and/or increasing load. Figure 2.3 shows the force-deformation relationship of a laterally in-plane (IP) loaded URM shear wall over several load cycles.
As can be seen, the unloading and reloading curves differ from the initial loading curve. Therefore, the value of the stiffness k1 is not a constant factor: its value depends on the past deformations u of the element and on whether the deformation is increasing or decreasing. Hence, for nonlinear systems, the component k1u(t) in the EOM is replaced by k1(u(t), ˙u(t))u(t).
Figure 2.3: Force-deformation relationship laterally IP loaded URM shear wall [16].
The damping coefficient c1 cannot be determined using the damping properties of the individual structural elements, since such properties are not well established [15]. Also, several factors would then not be included, such as energy that is dissipated in friction at connections in steel structures, in friction between structural and non-structural elements, and in the opening and closing of micro-cracks in concrete structures. Instead, the damping coefficient c1 is determined using the damping ratio ζ and the critical damping coefficient ccr,1 of the structure. The damping ratio ζ must be based on available data on similar structures, whereas the latter coefficient ccr,1 often is determined using the values of the mass m1and the spring stiffness k1of the structure. This regularly applied type of damping, depending on the mass and the stiffness of a system, is called Rayleigh damping.
Usually, the damping ratio ζ of a structure is assumed to be 0.05 (i.e. 5%), resulting in the following damping coefficient c1:
c1= ccr,1ζ = ccr,10.05 = (2p
k1m1)0.05 (4)
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In case the damping ratio ζ equals one (i.e. 100%), the damping coefficient c1 equals the critical damping coefficient ccr,1. In this situation, the mass m1 loses all its restoring force, that is the force in the spring, by the time it reaches its equilibrium state. Therefore, the mass m1 does not overshoot this state and oscillation is completely inhibited. A corresponding system is called a critically damped system. Systems with a damping ratio ζ smaller than 1 are called underdamped systems, as are all existing structures [15]. For such systems, the mass m1overshoots its equilibrium state due to its non-zero restoring force and then oscillates around this state. However, damping causes a decrease in the amplitude of the oscillation over time. Systems with a damping ratio ζ greater than 1 are called overdamped systems. Due to high damping strengths, such systems absorb most of the restoring force even before the mass m1reaches its equilibrium state. As a result of this early omittance of the restoring force, it takes longer for the system to reach the equilibrium state than for the critically damped systems [17]. The dynamic behavior of the three types of damped systems is illustrated in Figure 2.4.
Figure 2.4: Free vibration of critically damped, overdamped and underdamped system [15].
2.1.3 Vibration properties
The dynamic behavior of an undamped (c1= 0) idealized system can be described by its vibration properties: the natural circular frequency ωn[rad/s], the natural period Tn[s] and the natural cyclic frequency fn [Hz]. For a damped (c1 > 0) idealized system, the vibration properties do not only depend on the mass and stiffness components, but also on the damping ratio ζ of the examined system. The subscript n of the undamped properties is then replaced by subscript d for the damped properties. For both damped and undamped SDOF systems, the equations for the determination of the vibration properties are described in Eq. 5. These equations are the resulting equations of an eigenvalue analysis on the analyzed system. The basis of deriving vibration properties using eigenvalue analyses is introduced in Appendix A.2.
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ωn=r k1
m1 Tn= 2π ωn fn= 1
Tn
= ωn 2π
↔
ωd = ωn
p1 − ζ2 Td= 2π
ωd
= Tn p1 − ζ2 fd= 1
Td
= ωn
p1 − ζ2 2π
(5)
The dynamic response of a system as a result of specific dynamic loading depends on the system’s vibration properties. In case a system is subjected to a sinusoidal load with a vibration period that is similar to the natural period Tn of the system, the system will show a high amplified dynamic response. This phenomenon is called resonance. In static analysis methods, resonance is taken into account by means of dynamic amplification factors (DAFs), which are defined as the ratio between the system’s response as a result of a dynamic load and the system’s response as a result of a static application of the maximum amplitude of the same load [18]. Figure 2.5 shows the DAF that applies to the acceleration of a SDOF system with different damping ratios ζ. Indeed, the highest DAFs are obtained for the scenario in which the ratio between the excitation frequency and the natural circular frequency is equal to 1.
Figure 2.5: Dynamic amplification factor for different frequency ratios and damping ratios [15].
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