Anil K. Chopra - Earthquake Engineering for Concrete Dams

Under the approximation of Eq. (2.2.1), the equation of motion for a dam supported on rigid foundation with an empty reservoir is

(2.2.2)

in which the generalized mass

(2.2.3)

where the integration extends over the cross‐sectional area of the dam monolith; the mass density of the dam concrete m k( x , y ) = m ( x , y ), k = x and y is considered separately for the horizontal and vertical components of dam motion for convenience later in expressing the hydrodynamic effects in terms of an added mass and added damping; ; ω 1, and ζ 1are the fundamental natural frequency and the viscous damping ratio of the dam alone;

(2.2.4)

Equation (2.2.2)can be rewritten as

(2.2.5)

where

(2.2.6)

For harmonic free‐field ground acceleration , where ω is the exciting frequency, the modal coordinate can be expressed in terms of its complex‐valued frequency response function, . Upon substitution into Eq. (2.2.2)and canceling e iωton both sides gives

(2.2.7)

We will later extend Eq. (2.2.7)to include dam–water interaction ( Section 2.4) and dam–foundation interaction ( Section 3.2.4).

In preparation for response spectrum analysis of the dam including dam–water–foundation interaction subjected only to horizontal ground motion (to be developed in Chapters 3and 4), such analysis for the dam alone is presented first.

The response history of the modal coordinate due to arbitrary ground acceleration in the x ‐direction can be computed from dam response to harmonic ground motion, characterized by the frequency response function ( Eq. (2.2.7)), using standard Fourier synthesis techniques. Alternatively, it can be expressed in terms of D 1( t ), the deformation response of the first‐mode single‐degree‐of‐freedom (SDF) system, an SDF system with vibration properties – natural frequency ω 1and damping ratio ζ 1– of the first vibration mode of the dam. The equation of motion of this SDF system subjected to ground acceleration is given by

(2.2.8)

Having temporarily limited the earthquake response analysis to the x ‐component of ground motion, the superscript x may be dropped from , , and . Comparing Eq. (2.2.5)to Eq. (2.2.8)gives the relation between q 1and D 1:

(2.2.9)

where D 1( t ) can be determined by numerically solving Eq. (2.2.8). Substituting Eq. (2.2.9)in Eq. (2.2.1)gives the displacement history of the dam

(2.2.10)

We will be especially interested in the peak value of response, or for brevity, peak response , defined as the maximum over time of the absolute value of the response quantity:

(2.2.11)

where the subscript “ o ” attached to a response quantity denotes its peak value. The peak displacements can then be expressed as

(2.2.12)

where is the ordinate of the deformation response (or design) spectrum for the x‐ component of ground motion evaluated at period T 1= 2 π / ω 1, and damping ratio ζ 1; the subscript “ o ” that denotes peak value will subsequently be dropped to simplify notation.

The equivalent static forces associated with the peak displacements [ Eq. (2.2.12)] are given by (Chopra 2017: Section 17.7)

(2.2.13)

in which is the ordinate of the pseudo‐acceleration response (or design) spectrum, and w k( x , y ) = g m k( x , y ). Because the vertical ( k = y ) component of displacements in the fundamental vibration mode, , is much smaller than their horizontal ( k = x ) component, the associated vertical forces may be dropped, leaving only the horizontal ( k = x ) component of forces in Eq. (2.2.13):