## 1

FIGURE 3.16 Prismatic beam.

by courtesy of Marcel Dekker, Inc.)

separation of variable technique should be employed for solution. Thus y(x, t ) = Y (x)-g (t) (3.53)

where Y(x) is called the shape function, which is expressed in terms of x along a member and g(t) is the time function, which is related to the variable time of motion. The shape function is

Y = A sin Ix + B cos Ix + C sinh Ix + D cosh 1x (3.54)

where

p is a constant independent of time and has unit of rad/s; A, B, C, and D are arbitrary constants and can be determined by using the boundary conditions of a member. The derivation is based on Bernoulli-Euler theory with consideration of bending deformation.

### 3.4.2.2 Dynamic-Stiffness Coefficients

For the arbitrary member, ''ij," of a framework shown in Figure 3.17, let the end moments, M{, Mj, the end shears, Vi, Vj, and their associated end deflections, Yi, Yj, as well as the end slopes, di, dj, be considered positive. According to the shape function derived in Table 3.1, the following boundary conditions can be established. Let |Qe}=[M; Mj V Vj]T and {qe} = [0, 8j Y{ Y,]T, then the flexural dynamic-stiffness coefficients can be expressed as

sinh f cosfâ€”cosh f sinf

0 0