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Figure 2.60. Seven-story building; third-mode forces and displacements.

(b = 0.05), it is determined that the second and third mode spectral accelerations (0.500g) are 80% greater than the first mode spectral acceleration (0.276g). On the basis of mode shapes and modal participation factors, modal story forces, shears, overturning moments, acceleration, and displacements are determined.

Figure 2.61a shows story forces obtained by multiplying the story acceleration by the story mass. The shapes of story force curves (Fig. 2.61a) are quite similar to the shapes of the acceleration curves (Fig. 2.61d), because the building mass is essentially uniform.

Figure 2.61b shows story shears that are a summation of the modal story forces in Fig. 2.61a. The higher modes become less significant in relation to the first mode because the forces tend to cancel each other due to the reversal of direction. The SRSS values do not differ substantially from the first mode values.

Figure 2.61c shows the building overturning moments. Again, the higher modes become somewhat less significant because of the reversal of force direction. The SRSS curve is essentially equal to the first mode curve.

Figure 2.61d shows story accelerations. Observe that the second and third modes do play a significant role in the structure's maximum response. While the shape of an individual mode is the same for displacements and accelerations, accelerations are proportional to displacements divided by the squared value of the modal period, which accounts for the greater accelerations in the higher modes. The shape of the SRSS combination of the accelerations is substantially different from the shapes of any of the individual modes because it accounts for the predominance of the various modes at different story levels.

Figure 2.61e shows the modal displacements. Observe that the fundamental mode predominates, while the second and third mode displacements are relatively insignificant. The SRSS combination does not differ greatly from the fundamental mode. It should be noted that for taller and irregular buildings, the influence of the higher modes becomes larger.

Figure 2.61. Seven-story building. Modal analysis summary: (a) modal story forces (kips); (b) modal story shears (kips); (c) modal story overturning moments (kip-ft); (d) modal story acceleration (g's); (e) modal lateral displacements (in.).
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