## 12507

Source: Reprinted from Ref. [1, p. 284], by courtesy of Marcel Dekker, Inc.

Source: Reprinted from Ref. [1, p. 284], by courtesy of Marcel Dekker, Inc.

higher modes. For the same reason, the lumped-mass method also yields larger rigidity in structural modeling. It is worthwhile to mention that for large structural systems, such as tall buildings, only the first several fundamental modes are essential for design. Thus, the eigenvalue inaccuracy resulting from the lumped-mass model can be ignored.

3.5.4 Axial, Torsional Vibration, and Flexural Vibration with Timoshenko Theory

Equations 3.86, 3.89, and 3.90 are basics for vibration of finite element systems and frameworks with Timoshenko theory in flexural vibration, axial as well as torsional vibrations. Equation 3.93 can be used for system response analysis. Detailed results are available [1, pp. 265-270, 285-317].

### 3.5.5 Dynamic Motion Equation with P-A Effect

When a member is subjected to a compressive force, P, the force times the member's deflection, A, yields an additional moment that is called second-order moment due to the P-A effect. For the dynamic-stiffness method, the P-A is implicitly expressed in the stiffness coefficient [1, pp. 238-240, 253-258]. For consistent-mass and lumped-mass methods, the P-A effect is formulated separately from stiffness in a geometric matrix. A geometric matrix is important in response analysis for tall buildings because heavy floor load transmitted to supporting columns can affect the vibrating frequencies significantly. The dynamic motion equation is similar to Equations 3.93 with additional term [Kg]{q} as follows:

where [Kg] is called geometric matrix in a consistent-mass model but string matrix in a lumped-mass model [1, pp. 303-306, 317-318]. The former involves both rotational and side sway d.o.f. and the latter involves side sway d.o.f. only. The negative sign corresponds to the axial force in compression. If the force in a member is in tension, then the [kg] should have a positive sign. Then, the system matrix is [Kg] = Y1 [kg]. It worthwhile to point out that compression reduces the stiffness that consequently reduces natural frequencies. It is also worthwhile to note that when the compressive force is a harmonic excitation, such as machinery vibration, then the structural response can be a dynamic instability problem. The instability behavior depends on the ratio of the structure's natural frequency to the forcing frequency of the axial force [4].

3.6 Elastic and Inelastic Response Analysis Methods Based on Nature of Exciting Forces

### 3.6.1 Nature of Exciting Forces

The nature of exciting forces and their categories were discussed in Section 3.1.1. Various response analysis methods are summarized herein for deterministic forces as well as nondeterministic forces. Earthquakes are treated for both deterministic and nondeterministic cases.

### 3.6.2 Modal Analysis

The method was presented in Sections 3.3.2 and 3.3.3 from which the response analysis can be obtained by using Duhamel's integral or response spectra. Note that this approach is only for an elastic system because it requires natural frequencies and mode shapes.

### 3.6.3 Direct Integration Methods

Direct integrations are of paramount importance in dynamic response analysis for many reasons including (1) they can be used for structural response at various deformation stages from elastic to inelastic and (2) they can be applied to motion equations for various irregular forcing functions such as earthquake accelerations in digital data. There are a number of numerical integration methods such as Newmark method, Wilson-6 method, Runge-Kutta fourth-order method, constant acceleration method, linear acceleration method, and average acceleration method. These methods have general basic characteristics: (1) determination of response involves computation of displacement, velocity, and acceleration; (2) at the beginning of integration, response parameter values must be given or have been calculated at one or more points proceeding with the specific time intervals of the integration; (3) truncation errors due to finite number of terms in Taylor series expansion for replacing the differential equation by a finite difference equivalent; and (4) propagation error resulting from tendency of error growth from the integration step to the next, wherein the solution can become unbounded and unstable. Two well-known methods are selected to outline integration procedures and their associated truncation as well as propagation errors.

3.6.3.1 Newmark Method (Linear Acceleration Version)

Newmark originally derived a general integration in which two parameters can be modified to change the integration to three cases:

1. Average acceleration method. The acceleration of the system remains constant over the time interval At and its value is equal to the average values of acceleration at the beginning and end of the interval.

2. Constant acceleration method. The acceleration of the system is constant and is equal to its value at the beginning of the time interval.

3. Linear acceleration method. The acceleration of the system varies linearly over the time interval.

Without detailed derivation, the end results of the linear acceleration method are expressed as h M+At [C] + [K]W + At)} = {F (t + At)} — [M]{A} — [C]{B} (3.95)

where and

{A} = "¿2 {*(')}" Dtt fX(t)} - 2{€(t)} (3.96a)

{B} = — 2{X (t)} — A {x(t)} — At {x(t)} (3.96b)

Note that x(t), X(t), and x(t) are supposed to be given or calculated at time t, thus the response parameters are unknown for the next step calculation at t + At. Thus, Equation 3.95 can be simplified to the following expression:

After {x(t + At)} is calculated, acceleration and velocity are then obtained as

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