Rocking stiffness of mounting arrangements in electrical cabinets and control panels

Rocking stiffness of mounting arrangements in electricalcabinets and control panels

Jianfeng Yang, S.K. Rustogi, Abhinav Gupta *

Center for Nuclear Power Plant Structures, Equipment and Piping, North Carolina State University, Campus Box 7908, Raleigh, NC

27695-7908, USA

Received 18 June 2001; received in revised form 19 July 2002; accepted 24 July 2002

Abstract

Several studies have shown that the consideration of a rigid body-rocking mode in a cabinet is necessary to evaluate

accurate incabinet spectra. Observations from finite element analyses are used to study cabinet rocking behavior and to

show that accurate representation of the boundary conditions at the cabinet base is essential in the evaluation of correct

rocking mode. Simple formulations for evaluating the rocking stiffness are developed by conducting detailed analytical

studies for three different types of cabinet mounting arrangements. Availability of these formulations enables

incorporation of a cabinet rocking mode in the Ritz vector approach [Nucl. Eng. Des. 190 (1990) 225] for evaluating the

cabinet dynamic properties in significant mode and for generating the incabinet response spectra.

# 2002 Elsevier Science B.V. All rights reserved.

1. Introduction

Dynamic characteristics of the electrical control

panels and cabinets in a nuclear power plant are

often needed for seismic qualification of safety

related instruments such as relays that are

mounted on these cabinets. The cabinet dynamic

properties, in most cases, are calculated from a

finite element analysis. In some cases, experimen-

tal data obtained from either an in situ modal

testing or a shake table testing has also been used

to estimate the cabinet dynamic properties. The

calculated dynamic properties of cabinets are then

used to evaluate the earthquake input for safety

related instruments. The finite element analysis of

a cabinet is usually performed by assuming that

the cabinet is rigidly anchored at its base. This

assumption may or may not be valid depending

upon the particular arrangement used for mount-

ing the cabinet at its base. Several studies (Llam-

bias et al., 1989; Lee et al., 1990, 1991; Lee and

Abou-Jaoude, 1992; Gupta et al., 1998, 1999b)

have shown that the flexibility of the cabinet

mounting arrangement can significantly affect its

dynamic behavior. The flexibility of a mounting

arrangement is dependent upon not only the

nature of mounting such as welding or anchoring

but also on the structural details of a particular

mounting arrangement.* Corresponding author. Tel.: �/1-919-515-1385; fax: �/1-

919-515-5301

E-mail address: [email protected] (A. Gupta).

Nuclear Engineering and Design 219 (2002) 127�/141

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Llambias et al. (1989) have shown that modelingof structural details at the cabinet base can play a

significant role in response evaluation. They con-

ducted shake table tests using narrow band ran-

dom excitation on two cabinets connected

together. The results of their tests indicate that

as the excitation level increases, the natural

frequency of the cabinet decreases. This decrease

in the natural frequency occurs primarily due toyielding of anchor bolts and peeling of a base plate

during high excitation levels. It is found that the

first mode of vibration in each of the two

horizontal directions is purely a rocking mode

and that the non-linearities are associated with the

structural elements at the base of the cabinets.

These non-linearities due to the yielding of plinth

and gusset plates at the bottom corners of thecabinets results in a stiffness reduction at the base.

Sustained increase in the levels of excitation

eventually results in the tearing of gusset plates

and buckling of the plinth.

Lee et al. (1990) evaluates the effect of base

shimming on the seismic response of cabinets using

simple as well as detailed analytical approaches. It

is shown that the base support becomes discontin-uous due to the addition of shim plates in the in

situ condition, as opposed to the continuous

support which is often the configuration of a

cabinet mounting arrangement during shake table

testing. Therefore, the horizontal members at the

base can bend or twist freely in the in situ

condition due to the gap introduced by discrete

shim supports. On the other hand, these membersare restrained against rotations due to the con-

tinuous bearing during a shake table test. Conse-

quently, the stiffness imparted by the same

structural members is much higher in a shake

table test than that in the in situ condition.

Lee and Abou-Jaoude (1992) study the effect of

base uplift on cabinet response using an idealized

SDOF system representing the cabinet attached toa rigid base member that is supported at its ends

by two vertical parallel springs. The springs are

considered to be rigid in compression but flexible

in tension, thus allowing a rocking uplift motion at

the base. Such a system is non-linear and can

represent the various states of deformation that

include no uplift of the base as well as alternating

uplift of opposite edges. Lee and Abou-Jaoude

(1992) perform a time-history analysis of this

system assuming a linear behavior during each of

the three contact conditions. It is shown that the

modeling of base flexibility is critical in determin-

ing the correct dynamic characteristics of the

cabinet.In another study, Lee et al. (1991) corroborate

the above conclusions for an actual cabinet. They

perform a finite element analysis of the cabinet to

show that the presence of channels between the

base angle and the embedded plate lowers the

cabinet natural frequency. Such connections are

flexible because the base angle tends to deform

with respect to the bolts that connect the base

angle to the base channel. As for the SDOF

system, they use simple equivalent springs in their

finite element model and do not model the

structural elements at the base such as channels,

bolts, and base angles. Furthermore, no procedure

is given to evaluate the stiffness values of these

springs.

Gupta et al. (1999a) conduct detailed analytical

studies using several actual cabinets to illustrate

that the cabinet dynamic properties are needed for

only a few (often one) significant modes of

vibration. A Ritz vector approach is proposed to

calculate the dynamic properties in significant

cabinet modes (Gupta et al., 1999a). In this

approach, the significant cabinet mode is repre-

sented as a superposition of the global cabinet and

the local plate (or frame) mode. Unlike finite

element analysis or experimental testing both of

which are time and cost intensive, the Ritz vector

approach is an easy-to-use approach that also

gives accurate results. In another study (Gupta et

al., 1999a,b) the results obtained from detailed

finite element analyses are verified with the

corresponding results from in situ as well as shake

table testing of actual cabinets. These comparisons

between the analytical and experimental data are

used to conclude that the global cabinet mode can

either be a bending mode or a rocking mode. A

global bending mode can be directly considered in

the Ritz vector approach by incorporation of

appropriate mathematical functions. However, a

rocking cabinet mode cannot be directly consid-

J. Yang et al. / Nuclear Engineering and Design 219 (2002) 127�/141128

ered as no method exists to evaluate the flexibility

of a mounting arrangement in a cabinet.

In the present paper, a discussion is presented

on the stiffness imparted by different cabinet

mounting arrangements. Results obtained from

detailed finite element analyses are used to develop

simple formulations for evaluating the stiffness of

cabinet mounting arrangements in three different

types of widely used mounting configurations.

2. Cabinet mounting arrangements

Three different types of mounting arrangements

are considered in the present paper. Two of these

configurations correspond to cabinets DGLSB

and MS4716 that are also considered in Gupta et

al. (1999a) for the development of the Ritz vector

approach and in Rustogi and Gupta (2002) for the

verification of the finite element results by com-

parison with experimental data. The third type of

mounting arrangement that is considered in this

study has been used for mounting a variety of

cabinet types. It corresponds to cabinet PROT IV

described in Yang and Gupta (2000). The conclu-

sions of the present study can also be extended to

additional mounting arrangements that have not

been considered explicitly. The structural details of

the first configuration, corresponding to cabinet

DGLSB, are shown in Fig. 1. It consists of a 0.15-

in. thick base plate that is anchored to the floor by

twelve 0.5-in. diameter bolts. The base plate is

stiffened by an angle section and by a vertical

unistrut, as shown in Fig. 1. The structural details

Fig. 1. Structural details of the mounting arrangement in

cabinet DGLSB.

Fig. 2. Structural details of mounting arrangement in cabinet

MS4716.

J. Yang et al. / Nuclear Engineering and Design 219 (2002) 127�/141 129

of the second mounting arrangement considered in

this study, corresponding to cabinet MS4716, are

shown in Fig. 2. In this case, the 0.15-in. thick base

plate is anchored to the floor by only four 0.75-in.

diameter bolts. In addition, steel bars are used as

stiffeners for the base plate. It should be noted that

among these two configurations, the former does

not have any anchor bolts at the base plate corners

whereas the latter has anchor bolts only at the

corners. Fig. 3 shows the third type of mounting

arrangement considered in this study in which only

a triangular plate is used in each corner for

anchoring the cabinet to the floor instead of a

plate for the complete base. The triangular plate is

0.30-in. thick and the anchor bolts are 0.75-in. in

diameter. The triangular plate is welded to a base

frame that consists of two types of members. The

outer frame consists of channel sections whereas

tubular steel beams are used at intermediate

locations to provide continuity between adjacent

bays. A total of three bays exist in cabinet PROT

IV. However, different cabinet types can have

either more or fewer bays. Table 1 describes the

structural and geometrical details of variousmembers in the three types of mounting arrange-

ments as employed in cabinets DGLSB, MS4716,

and PROT IV.

3. Finite element analysis

The global rocking behavior of the cabinet can

be represented by a rigid body rotation of the

cabinet that has a rotational spring of stiffness Ku

at the base, as shown in Fig. 4. The rocking

stiffness Ku can be evaluated from a finite element

analysis by applying lateral static load on the top

of cabinet. The lateral load causes a moment M at

the cabinet base and results in a rigid body

rotation u . The rocking stiffness Ku can then be

calculated as:

Ku�M

u(1)

For cabinets DGLSB, MS4716, and PROT IV,

Ku obtained from the above procedure are 0.5�/

108, 0.35�/108 and 8.5�/108 lb-in. rad�1, respec-

tively. For the purpose of developing a formula-tion to evaluate the rocking stiffness of the cabinet

mounting arrangement, we performed detailed

finite element analyses. These analytical studies

are used to evaluate the various parameters that

affect the value of Ku in the cabinets. Three finite

element models corresponding to the three differ-

ent mounting configurations are created using

ANSYS. A correct finite element model must beable to represent the base plate uplift and the

corresponding global rocking mode of the cabinet.

Therefore, we model the base plate using the shell

element. Triangular shell elements are used in the

region of high principal stresses to avoid shear

locking that is observed to be present in rectan-

gular shell elements. At each bolt location, the

base plate is considered to be fixed in five degreesof freedom, the three translations and the two

rotations. The sixth degree of freedom, rotation

about the bolt axis, is considered to be free. The

fundamental mode of vibration for each cabinet

evaluated from such a finite element model is

found to be a global rocking mode. However, this

Fig. 3. Structural details of mounting arrangement in cabinet

PROT IV.


Table 1

Structural and geometrical details of cabinets

Cabinet name Height H (in.) Depth D (in.) Length W (in.) Base plate thickness t (in.) Weight Mg (lb) Anchor bolt diameter (in.) Bolt distance Db (in.)

DGLSB 74 16 72 0.15 742 0.5 1.5

MS4716 72 24 48 0.15 676 0.75 1.5

PROT IV 74 30 80 0.30 1017 0.75 2.75

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model is still incorrect for several reasons. Since

the base plate is considered to be fixed at the bolt

locations only, a rocking cabinet mode results in

an incorrect base plate deformation such that a

particular region of the base plate deflects down-

wards below the floor level, as shown in Fig. 5. It

is incorrect to simply constrain the vertical degrees

of freedom in the base plate where a violation of

the boundary condition occurs. The boundary

conditions are violated only for a downward

deflection but not for an upward deflection, which

can occur in higher modes of vibration. Therefore,

it is a non-linear system. However, the conclusion

reached by Gupta et al. (1999a) that only one

mode (fundamental mode in this case) is sufficient

to calculate accurate incabinet spectra is used to

avoid a non-linear analysis. Since the objective is

to accurately evaluate the rocking stiffness of

cabinet mounting arrangement in the fundamental

rocking mode, we use an iterative procedure for

selecting the nodes at which the vertical degrees of

freedom requires constraint against downward

deflection. To begin with, we constrain the vertical

degrees of freedom at all the nodes along the

cabinet edge about which the rocking takes place.

Next, we perform a static analysis by applying a

lateral load on the top of cabinet and identify the

nodes at which the boundary conditions are still

violated. These degrees of freedom are constrained

in this step and the process repeated until no more

violations occur. The dynamic properties and the

amplifications in the resulting finite element mod-

Fig. 4. Global rocking mode of a cabinet.

Fig. 5. Deformation of base plate in the cabinet rocking mode.

Fig. 6. Deflected shape for base plate uplifting in cabinet

DGLSB.


els are found to be in good agreement with the

corresponding values observed from in situ testdata for cabinet DGLSB and shake-table test data

for cabinet MS4716. The details of these compar-

isons between the finite element analysis results

and the experimental data are given in Gupta et al.

(1998), Rustogi and Gupta (2002).

4. Stiffness of configurations 1 and 2

Typical uplifting shapes of the cabinet base

plates are shown in Figs. 6 and 7 for both

configurations, respectively. As seen in these

figures, the rigid body rotation of the base plate

about the front edge of cabinet is restrained at bolt

locations leading to localized deformations in the

vicinity of each anchor bolt. The localized defor-

mation around a bolt occurs in a cup-like forma-

tion, shown in Fig. 8, and represents a region of

high base plate curvatures. The base plate curva-

tures in regions away from a bolt location arerelatively negligible and hence, do not contribute

significantly to Ku . Furthermore, the resistance

offered by each anchor bolt to the base can be

represented by a discrete spring whose stiffness is

equal to the stiffness offered by the cup-like

deformation around that bolt. Fig. 9 shows such

an equivalent representation.

In Eq. (1), M represents the overturning mo-ment about the cabinet base and u the rigid-body

rotation. The base plate also uplifts with a rotation

u about the front edge (edge O�/O? in Fig. 9). This

rotation of base plate is resisted by an equivalent

discrete spring at each bolt location. Let the

rotation u result in a vertical deflection dvi and

force Fvi in the ith spring located at distance di

from the edge of rotation. Spring forces in each ofthe discrete springs offer resistance to rotation,

which is represented by the resisting moment M

about the edge of rotation. We can then write:

M�XN

i�1

Fvidi (2)

where N is the total number of anchor bolts. If the

spring stiffness of ith spring is Kvi , Eq. (2)

becomes:

M�XN

i�l

Kvidvidi�uXN

i�1

Kvid2i (3)

Since all anchor bolts are identical and the cup-

Fig. 7. Deflected shape for base plate uplifting in cabinet

MS4716.

Fig. 8. ‘Cup-like’ deformation of base plate around an anchor

bolt.

Fig. 9. Equivalent model with vertical springs at anchor bolt

locations.


like deformation around each anchor bolt issimilar, we can assume Kvi �/Kv for all i . Eq. (3)

then becomes:

M��XN

i�1

d2i

�Kvu (4)

and the rocking stiffness Ku is obtained as:

Ku�M

u�

�XN

i�1

d2i

�Kv (5)

Next, we develop an expression to evaluate Kv

for a cup-like deformation. A typical cup-likedeformation around a bolt is shown in Fig. 8. As

explained earlier, this cup-like deformation repre-

sents a region of high base plate curvatures along

both X and Z directions. Dimensions of this cup-

like deformation are proportional to the distance

of anchor bolt from the nearest base plate edge,

Db. Therefore, we represent the dimensions of this

region along X and Z direction as axDb and azDb,respectively, where ax and az are constants. The

strain energy of this rectangular plate region is

given by:

U �Et3

24(1 � n2)

� gaxDb

0

gazDb

0

��@2u

@x2

�2

�2n@2u

@x2

@2u

@z2�2(1�n)

��

@2u

@x@z

�2

��@2u

@z2

�2�dxdz (6)

where u is the transverse plate deflection; E is theYoung’s modulus of elasticity for base plate

material; n is the Poisson’s ratio; and t is the

base plate thickness. The transverse plate deflec-

tion u may be expressed as:

u�u0f

�x

axDb

�f

�z

azDb

�(7)

where f is a function that represents the deformed

shape, and u0 is the generalized displacement.

Substituting Eq. (7) in Eq. (6) and differentiating

U with respect to u0 would yield an expression of

the form:

@U

@u0

�Kvu0 (8)

in which Kv can be expressed as:

Kv�CEt3

12(1 � n2)

1

D2b

(9)

where C is a constant whose value is governed by

the choice of the function f for deformed shape.

Eqs. (5) and (9) together give the followingexpression for Ku :

Ku�C

�XN

i�1

d2i

�Et3

12(1 � n2)

1

D2b

(10)

If the distance di is expressed as ciD where ci is a

constant for each i , Eq. (10) can be re-written as:

Ku�C

�XN

i�1

c2i

�D2 Et3

12(1 � n2)

1

D2b

(11)

5. Stiffness of configuration 3

As discussed earlier, this mounting configura-

tion consists of primarily three structural sys-

Fig. 10. Outer frame structure having rotational stiffness Ku 1

in cabinet PROT IV. (b) Intermediate tubular beams having

rotational stiffness Ku 2 in cabinet PROT IV.


tems*/the anchoring through the triangle plates,

the outer frame comprising of channel sectionmembers, and the tubular steel beams at the

junction of two adjacent bays. From a detailed

finite element analysis of a cabinet with this

mounting arrangement, it was found that the total

rotational stiffness imparted by this mounting

arrangement can be represented by two discrete

rotational springs with stiffnesses Ku1 and Ku2,

respectively, connected in parallel, i.e.:

Ku�Ku1�Ku2 (12)

where Ku1 corresponds to the rotational stiffness

imparted solely by the outer frame of channel

section members and the four triangular plates

with anchor bolts in the four corners as shown in

Fig. 10(a). Further, Ku2 corresponds to the rota-

tional stiffness imparted by tubular steel beams

and the triangular plates with anchor bolts

attached to these beams, as shown in Fig. 10(b).Expressions developed in the previous section

for the first two mounting configurations with

base plates can be used directly to calculate the

rotational stiffness Ku1. This is so because the

resistance offered by each anchor bolt to the base

can be represented by vertical spring whose value

can be evaluated using Eq. (9). In addition, thechannel section members in the outer frame at the

base do not impart any meaningful resistance to

the rigid body rocking of the cabinet. These frame

members do not exhibit any bending due to the

high in-plane stiffness of the side plates. Therefore,

we can write:

Ku1�Kv

X4

i�1

d2i (13)

where di is the distance of anchor bolts from the

cabinet rocking edge as shown in Fig. 11, and the

summation is performed for the four anchor bolts.

Eq. (13) can be simplified further by using

symmetry and performing the summation for

only two anchor bolt locations, i.e.:

Ku1�2Kv

X2

i�1

d2i (14)

Next, we present a detailed discussion on the

stiffness imparted by a tubular beam and the

triangular plates attached to it for anchoring the

cabinet. Observations made in the detailed finite

element analysis show that each tubular steel beam

Fig. 11. Simplified model for evaluating Ku 1 in cabinet PROT

IV.Fig. 12. Simplified model for evaluating Ku 2 in cabinet PROT

IV.


deforms like a fixed-end cantilever during theuplifting of cabinet base. Since each beam is

connected to four anchor bolts, we represent it

by a simplified beam-spring model shown in Fig.

12. It should be noted that the spring shown in

Fig. 12 represents the effect of two anchor bolts

that are located near the cabinet edge at which

uplifting occurs, and the length of the cantilever

beam is equal to the distance of the uplifting edgefrom the anchor bolts near the opposite edge

about which rocking occurs. The stiffness im-

parted by the two anchor bolts located near the

cabinet edge about which the rocking occurs is

negligible and, therefore, ignored. The expression

for the rotational stiffness imparted by the beam-

spring system shown in Fig. 12 can be developed

using the method of superposition as describedbelow.

For a load P applied at the tip of the cantilever,

point A in Fig. 12, the resisting moment is given

by:

M�PD�Ku2u2�Ku2

�dA

D

�(15)

in which dA is the vertical displacement at point A,

and D is the cabinet depth. This gives:

Ku2�PD2

dA

(16)

The total vertical deflection dA can be expressed

as a superposition of the displacement dAP due to a

load P applied at point A of a simple cantileverand the displacement dA

K due to a load 2KvdB

applied at the spring location B of a simple

cantilever beam, as shown in Fig. 12, i.e.:

dA�dPA�dK

A (17)

where

dPA�

PL3

3EI(18)

dKA �

2KvdBl2(3L � l)

6EI(19)

in which dB is the vertical displacement at point

B*/the location of spring, in the beam-spring

system. E and I are the Young’s modulus and

the moment of inertia for the tubular beam,respectively.

Once again, the vertical displacement dB can be

expressed as a superposition of the displacements

dBP and dB

K , the displacements at point B of a

cantilever beam due to load P at the tip and due to

load 2KvdB at point B, respectively, i.e.:

dB�dPB�dK

B (20)

in which:

dPB�

Pl3

6EI(3L� l) (21)

dKB �

2KvdBl3

3EI(22)

Substituting Eqs. (21) and (22) into Eq. (20), we

get:

dB�Pl2(3L � l)

4Kvl3 � 6EI(23)

Eqs. (18)�/(20) give:

dA�P

6EI

�2L3�

Kvl4(3L � l)2

2Kvl3 � 3EI

�(24)

Furthermore, Eqs. (16) and (24) give:

Ku2�6EID2

�2L3�

Kvl4(3L � l)2

2Kvl3 � 3EI

��1

(25)

As the location of the anchor bolt (springlocation at point B) is close to the cabinet edge

(point A), Eq. (25) can be simplified further by

assuming L :/l , i.e.:

Ku2�2KvD2�3EID2

L3(26)

It should be noted that the expression of Ku2 in

the above equation gives the rotational stiffnessfor only one tubular beam and the corresponding

anchor bolts. For two such beam-anchor bolt

systems encountered in cabinet PROT IV and

shown in Fig. 3, the total rotational stiffness Ku2

will be twice of that given by Eq. (26). Therefore,

Eqs. (12), (13) and (27) give:


Ku�2

�Kv

X2

i�1

d2i �2KvD2�

3EID2

L3

�(27)

The above expression can be generalized further

for cabinet types with ‘m ’ bays instead of three

bays considered in PROT IV. Such cabinets will

have (m�/1) tubular beams separating the ‘m ’

bays. Therefore, we can write:

Ku�2Kv

X2

i�1

d2i �(m�1)

�2KvD2�

3EID2

L3

�(28)

6. Parametric study

Eqs. (11) and (27) give two simple formulations

for evaluating the rocking stiffness of cabinetmounting arrangements. However, these expres-

sions are based on Eq. (9) that contains the

constant C whose value changes from one cabinet

type to another. We conduct a parametric study to

evaluate the constant C empirically. To begin

with, let us consider the mounting configurations

1 and 2. Several static analyses are performed

using each of the two models, shown in Figs. 1 and2, by varying the various parameters over a range

of values observed in the actual cabinets. The

parameters varied in the model include the base

plate thickness t ; the bolt distance from the nearest

base plate edge Db; cabinet depth D ; cabinet height

H ; and cabinet width W .

Detailed finite element analyses reveal that Ku

values in these two configurations depend only on

the values of t , Db and D . The variation of Ku with

H , and W is negligible. Figs. 13�/18 show the

Fig. 13. Variation of rocking stiffness Ku with base plate

thickness t in cabinet DGLSB.

Fig. 14. Variation of rocking stiffness Ku with cabinet depth D

in cabinet DGLSB.

Fig. 15. Variation of rocking stiffness Ku with bolt distance Db

in cabinet DGLSB.

Fig. 16. Variation of rocking stiffness Ku with base plate

thickness t in cabinet MS4716.


variation of numerically evaluated Ku with t , Db

and D for both the mounting configurations. For

configuration 1, corresponding to cabinet

DGLSB, Ku varies as t2.5 to t2.9 with base plate

thickness; as (1/Db)1.6 to (1/Db)1.7 with bolt dis-

tance; and as D2.0 to D2.1 with the cabinet depth.

In Eq. (11), variations of Ku with t , Db and D are

close to those obtained from the parametric study.

Similar observations can be noted in Figs. 16�/18

for the configuration 2 corresponding to cabinet

MS4716 and Figs. 19�/21 for configuration 3

corresponding to cabinet PROT IV. These varia-

tions are not exactly equal to t3, (1/Db)2, and D2

due to the complex nature of actual curvatures

near the anchor bolts. It should be noted that Eqs.

(11) and (28) for evaluating Ku are still valid

because the differences between the actual and the

observed variations of Ku with t , Db and D can be

accommodated in the value of C that is obtained

empirically. The value of C is obtained by least-


in cabinet MS4716.


in cabinet MS4716.

Fig. 19. Variation of rocking stiffness Ku with triangular plate

thickness t in cabinet PROT IV.


in cabinet PROT IV.


in cabinet PROT IV.


Table 2

Rocking stiffness Ku for the three types of mounting configurations

Cabinet base type Base plate thickness t (in.) Cabinet depth D (in.) Bolt distance Db (in.) Ku (108 lb-in. rad�1) Percent difference

Finite element Simple formulation

Configuration 1 (DGLSB) 0.10 16 1.0 0.22 0.22 1

0.15 20 1.5 0.50 0.51 2

0.20 24 2.0 0.88 0.97 11

0.10 16 2.0 0.07 0.05 �/32

0.20 24 1.0 2.68 4.19 56

Configuration 2 (MS4716) 0.10 16 1.0 0.12 0.16 25

0.15 20 1.5 0.35 0.36 2

0.20 24 2.0 0.70 0.68 �/2

0.10 24 2.0 0.13 0.09 �/35

0.20 24 1.0 1.89 2.99 58

Configuration 3 (PROT IV) 0.20 25 2.25 3.70 3.02 �/18

0.30 30 2.75 8.53 8.54 0

0.40 35 3.25 14.33 14.23 �/1

0.20 35 3.25 2.97 2.04 �/31

0.40 35 2.25 28.70 42.50 48

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squares fit procedure. It is 6.71 for the mountingconfiguration 1, and 13.30 for the mounting

configuration 2. Therefore, for mounting config-

uration 1 we can write:

Ku�6:71

�XN

i�1

c2i

�D2 Et3

12(1 � n2)

1

D2b

(29)

where 0.1 in5/t 5/0.2 in; 1.0 in5/Db5/2.0 in; 16

in5/D 5/24 in and, for mounting configuration 2,

we get:

Ku�13:30

�XN

i�1

c2i

�D2 Et3

12(1 � n2)

1

D2b

(30)


in5/D 5/24 in.

A similar parametric study was conducted for

mounting configuration 3 and the value of con-

stant C is found to be 16.44. Consequently, the

expression for Kv in Eq. (27) becomes:

Ku�16:44Et3

12(1 � n2)

1

D2b

(31)


in5/D 5/35 in.In mounting configuration 3, the variation of

Ku with thickness is close to t2 and not t3 due to

relatively thick triangular plate and significant

contribution from the tubular beams used in this

configuration. However, one can still use t3 based

on the formulations presented earlier. Any error

due to this difference is accommodated in the

value of C that is obtained empirically in Eq. (31).The base stiffness Ku calculated using the above

formulations and the corresponding finite element

analyses for a few cases are compared in Table 2.

These cases are representative of the complete

range of differences in the proposed formulations

and the finite element analyses results. Many more

cases, not included in Table 2, were considered in

the finite element analysis for comparison. As seenin this table, the proposed formulations give

accurate results for a majority of cases. On the

other hand, significant error is present when the

base plate thickness and bolt distance are close to

their extreme values. It should be noted that the

large differences occur when the base plate thick-

ness is close to its upper bound value and the boltdistance is close to its lower bound value or vice

versa. These extreme cases are mostly academic in

nature and do not have any practical significance.

For most conditions in the actual practice, the

results obtained from the proposed formulations

are in good agreement with the corresponding

results obtained from the finite element analyses.

7. Summary and conclusions

In the evaluation of the dynamic properties for

electrical cabinets and control panels by a finite

element analysis, the cabinets are, often, consid-

ered to be rigidly fixed at base. A finite element

analysis using these boundary conditions at thecabinet base cannot consider the rigid body rock-

ing motion of a cabinet. Several existing studies

have shown that the evaluation of a rigid body

rocking mode in a cabinet is necessary to evaluate

accurate incabinet spectra. In the present paper,

detailed finite element analyses are used to study

the rocking behavior of cabinets and show that

accurate representation of the boundary condi-tions at the cabinet base is essential in the

evaluation of correct rocking mode. Often, simple

methods are used for the evaluation of cabinet

dynamic properties and incabinet response spec-

tra. The Ritz vector approach presented by Gupta

et al. (1999a) is one such method that calculates

accurate dynamic properties of cabinets in signifi-

cant vibration modes. The significant mode can beeither a global mode or a local mode. It can also be

a superposition of the two modes. A global

bending mode can be included directly in the

Ritz vector approach by considering appropriate

Ritz vector. However, consideration of global

rocking mode requires knowledge of the rocking

stiffness imparted by the cabinet mounting ar-

rangement. In the present paper, simple formula-tions for evaluating the rocking stiffness are

developed by conducting detailed analytical stu-

dies for three different types of cabinet mounting

arrangements. Availability of these formulations

enables incorporation of a cabinet rocking mode

in the Ritz vector approach.


Acknowledgements

This research was partially supported by the

Center for Nuclear Power Plant Structures, Equip-

ment and Piping at North Carolina State Uni-

versity. Resources for the Center come from the

dues paid by member organizations, Department

of Energy through a sub-award to Duke Engineer-

ing and Services under the Nuclear Energy Re-search Initiative grant, and from the Civil

Engineering Department and College of Engineer-

ing in the University.

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Rocking stiffness of mounting arrangements in electrical cabinets and control panels