## 107073

Source: Reprinted from Ref. [1, p. 210], by courtesy of Marcel Dekker, Inc.

Source: Reprinted from Ref. [1, p. 210], by courtesy of Marcel Dekker, Inc.

Y22 = -26.28982 sin 12x + 26.26940 cos 12x + 26.28982 sinh 12x - 26.26940 cosh 12x Y31 = 23.87324 sin l3x Y32 = 23.87324 sin l3x

Y41 = -21.24624 sin l4x + 0.025582sinh l4x

Y42 = 15.02337 sin 14x - 15.02335 cos 14x - 15.02337 sinh 14x + 15.02335 cosh 14x

Then, from Equation 3.65

Eli Fki

in which the amplification factor of impulse load, for t < Z, Z = 0.1 s, is

The maximum displacement of the first mode occurs during the pulse at 0.0379 s, which is used for the response of other modes. The calculations are summarized in Table 3.2. Substituting the results from Table 3.2 into Equation 3.70 yields the moments at B for t = 0.0379 s associated with the first, second, and fourth modes as

M2 = -EIy"(L) = -EI[Yst1A1Yi1 (L) + Y^Y^H Yst4A4Y^L)] = -EI [0.034648(1.62093)(0) + (-0.017429(0.75101)(-0.036)) + 0.000279(1.070783)(0.06672)] = -4.912053(10-4)EI M3 = Ely" (0) = EI [Yst1A1Yi2(0) + Y!l2A2Y22(0) + Yst4A4 Y42(0)] = EI [0.034648(1.62093)(0) + (-0.017429(0.75101)(-0.036)) + 0.000279(1.070728)(0.06672)] = 4.912053(10-4)EI

Note that the third mode does not contribute to the structural response because Yst3 = 0. Note also that the equilibrium check on moments is satisfied at the joint because Y1M = 0.

3.4.5 Effects of Rotatory Inertia as well as Bending and Shear Deformation on Frequencies

Bernoulli-Euler theory is derived based on bending deformation only. In fact, a flexural vibration can include bending and shear deformation as well as rotatory inertia as derived in Timoshenko theory. Consider an element shown in Figure 3.20. Let z be the distance measured at any point from the neutral axis; then the displacement of a fiber located at z is ya = -z I (376)

FIGURE 3.21 Three-span beam.

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