Damping Formulation

4. 7 Mass Formulation

4.8 Damping Formulation

Rayleigh damping is a common model used in practice. Modal damping is also frequently used for modal analysis, but is not practical for nonlinear time history analysis. Classical or Caughey damping is sometimes used to get a better fit over several modes than Rayleigh damping can provide. In nonlinear time history anal- yses, excessively large and unrealistic damping forces can be observed with linear damping since the resulting damping forces can easily exceed the physical means by which damping would actually occur. The damping forces have been observed to be large relative to the forces carried by a yielding frame. Damping remains a poorly understood or measurable quantity in building structures, so another type of damping is used in this program which is more physically intuitive. Nonlinear damping called story damping has been chosen to limit the viscous forces to specified values so that unrealistically large damping forces are not generated when structural yielding causes high relative velocities. These nonlinear inter-story dashpots resist

relative horizontal motions between floors. Forces in the dashpots will be capped at a specified inter-story velocity. The dashpots can be placed at the same location as shear springs and assembled in the same way. Dashpot "stiffness" can be chosen to provide a desired percentage of critical damping.

F

---~~--~~---~x

.

Figure 4.11 Nonlinear inter-story damping force.

Nonlinear, inter-story damping is appealing because it makes sense physically.

Initially, when a structure has only been mildly shaken by an earthquake, many non-structural components can absorb energy and dampen the response. For large amplitude excitation, these elements will most likely be damaged and no longer able to dissipate more energy. When this source of damping is limited, the inelastic be- havior of the structural frame will provide the remaining damping. A bilinear model (figure 4.11) defines the force-velocity relationship by providing a cutoff velocity

v{J

that corresponds to a yield force

F{j.

The yield force is chosen to maintain reason- able damping force levels. The cutoff velocity is chosen for the given yield damping force to provide acceptable elastic damping in the lowest mode of the building (1%

to 5%). The velocity required to achieve a desired level of damping in the first mode can be determined experimentally, and this will be discussed in chapter 7. As the structure undergoes nonlinear response, the damping will level off when inter-story velocities exceed the cutoff velocity.

Additional mass and stiffness proportional damping can be used in the formu- lation, but they are set by the original mass and stiffness matrices and do not vary as the stiffness changes in nonlinear response of the structure. A small amount of this Rayleigh damping is used for numerical purposes. The overall damping is non- classical, so even in modal form the equations of motion are coupled by off diagonal damping terms. The effective modal damping resulting from the story damping will be discussed in chapter 7.

Three types of damping can be provided: mass proportional, stiffness propor- tional, and story damping resulting in the following equation for damping:

(4.57)

where esT is the nonlinear story damping. Higher order terms of a Caughey Se- ries for classical damping are not used, but Rayleigh damping can be achieved by choosing coefficients ao ^{and a1 so that}

(4.58)

is satisfied (Clough and Penzien 1993, p. 236). For equal damping in two modes, this simplifies to

(4.59)

The story damping can be assembled the same way the shear spring stiffness was by replacing k~,ss with c~,sr, ~x with

x[tN,

^and ~P{tN with

qfN

in equation

(4.25). Here, the 3 x 3 sub-matrix is

( 4.60)

i,ST

cj

=

0

0

0 0

0 0

where the damping force yield level can be different in the two orthogonal horizontal directions. The effect of one story dashpot i becomes

(4.61)

which can be written as (4.62)

for story j. Nate that this relationship is not linearized as in the case of shear build- ing stiffness. Linearization for velocity nonlinearities is not as straightforward as for displacement nonlinearities. Because the story damping is calculated based on initial elastic properties, the total damping matrix can be expressed as C from equa- tion 4.57 assuming the damping is linear. This is why the damping force equation can be written above for total motion, not incremental motion. The nonlinearities are accounted for by corrections to the residual force vector. This will be discussed in chapter 6.

The story damping is summed for the

n1

dashpots on story j as

( 4.63) where

C~T =

n~

a [ci_,ST] .

J i=l J

The story damping for the entire structure is assembled over the total number of

stories n8 as

(4.64)

where n8 is the number of stories.

The total damping forces using the complete damping matrix (4.57) are

( 4.65)

and the resulting forces are linear. The portion of the forces assumed to be linear is

(4.66) ^qsT,L

=

^esT*·

As in the case of shear springs, the transformation

T}

from equation 4.19 pro- vides the exact relation between the nonlinear forces at dashpot location i and the master node at level j:

(4.67) ^{ST,NL _} [Ti]T ^i,NL

qj - j qj .

The forces can be summed for the

nj

dashpot locations at each floor j and assembled into the total story damping force vector as

(4.68)

This allows the nonlinear damping forces to be written as a linear portion and a nonlinear correction:

(4.69)

total

~ q(x)

=

linear correction

~

qL

+

(qST,NL _ qST,L).

The effect of nonlinear damping on the equations of motion will be discussed in chapter 6 and quantification of the damping level will be discussed in chapter 7.