seismic analysis and design of cold-formed steel structural

Missouri University of Science and Technology Missouri University of Science and Technology

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International Specialty Conference on Cold-Formed Steel Structures

(1975) - 3rd International Specialty Conference on Cold-Formed Steel Structures

Nov 24th, 12:00 AM

Seismic Analysis and Design of Cold-formed Steel Structural Seismic Analysis and Design of Cold-formed Steel Structural

Members Members

Suresh G. Pinjarkar

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SEISMIC ANALYSIS AND DESIGN

OF

COLD-FORMED STEEL STRUCTURAL MEMBERS

by

Suresh G. Pinjarkar*, A.M. ASCE

Cold-formed steel structural members are widely used in nuclear

power plant installations. These members are primarily used for carrying

electrical cables and in HVAC duct work. In Seismic Category I Structures,

all electrical cables are often grouped together and carried through con

duits or cable buses in the case of power cables and in cable trays in

the case of control cables. These cable trays and ducts are supported by

the frames or other supporting systems from the slabs, beams, walls, etc.

The primary function of such a system is to support and ensure the safety

of the cables and ducts during the seismic event. lt is essential that

all systems which are vital for the control and safe shutdown of a nuclear

reactor remain functional under the most severe earthquake. The various

components of such a structural system are composed of cold-formed struc

tural steel and are classified as Category I items for seismic analysis

and design.

The seismic response of the secondary system such as cable trays and

ducts is greatly influenced by the nature and type of response of the

* Associate; Wiss, Janney, El:oltner and Associates, Northbrook, Illinois

865

866 THIRD SPECIALTY CONFERENCE

primary structure which supports it. A response spectrum method of

analysis is used to compute forces due to seismic excitations. Due to

an earthquake there is a differential motion between the various elements

and the supporting structure. Since this motion is relatively rapid, it

causes stresses and deformationsin various structural elements. In order

to survive a seismic event the various components must be strong as well

as ductile enough to resist forces and deformations imposed upon it.

The behavior of the secondary system under seismic conditions depends

not only on the earthquake motion to which it is subjected, but on the

properties of the various components of the system. These proper-

ties include member stiffnesses, types of supports, type of connec-

tions, damping characteristics and period of vibration. The purpose of

this paper is to provide a seismic resistant design taking into account

the various factors outlined above. The definition of terms used in this

paper is provided in the Appendix.

TYPES OF MEMBERS

The various types of cold-formed steel structural members more

widely used in Seismic Category I Systems are described below:

Cable Trays: Cable trays are continuous "U"-shaped members where cables

rest on the bottom of the tray and are held in place by two longitudinal

side walls. There are different types of trays according to their

function and trays from different manufacturers vary in size and shape

and in method of construction.

shown in Fig. 1.

Some of the important types of trays are

SEISMIC ANALYSIS AND DESIGN 867

Cable Buses: Cable buses are used to carry power cables and are of

various types. They are provided with spacers and separate holes for

each cable. See Fig. 2.

HVAC Ducts: These are continuous box type members as shown in Fig. 3.

Support Frames: Various types of channels or combinations of channels (see

Fig. 4) are used as supporting elements for cable trays and HVAC ducts.

These channels permit rigid metal construction without welding or drill-

ing. Standard components are used to create virtually unlimited variety

of support systems. The connections consist of a spring-loaded nut in-

serted anywhere along the continuous channel slot and then secured with

bolts to appropriate fittings. Serrations in the hardened nut engage

channel ridges to produce ~igid vise-like grip. Various support systems

are used to support the cable trays and the ducts as shown in Fig. 5.

ANALYSIS

The seismic analysis of cable tray and HVAC duct systems is

performed by use of the response spectrum or time-history concept of

analysis. The use of response spectrum method, however, provides the

most convenient and direct procedure for seismic analysis. It is there-

fore, necessary to obtain response spectra for various primary supporting

structural elements such as floor slabs and walls, etc. The response

spectra thus obtained is known as in-structure response spectra.

The methods for generating in-structure response spectra are well

documented in the literature. Briefly, a time-history analysis of the

entire structure is performed to obtain the time-history response at the

selected mass points. The next step is to subject

Coble :1 [OOQbOc

Variable rung spacing

(a) Solid Bottom (b) Ladder or Trough Type

Fig. l - Cable Trays

~ Fig. 2 - Cable Buses Fig. 3 - HVAC Ducts

u =c: ~ ~: Fig. 4 - Typical Channel Supports

CP 0' CP

-l :5 ::0 0 C/l "1:1 tr1 (j

:; tl ~ (j

0 z ., tr1 ::0 tr1 z (j tr1

/Fixed or hinged I support

[ 0 0 0 0 J

[ 0 0 0 0 J

·.-Wall

Fig. 5 - Support Systems for Cable Trays, Conduits and HVAC Ducts

[ J

,-Floor slab

Vl

t:l Vl

3:: -(')

> z > t"' -<: ~ Vl

> z 0 0 tTl Vl

Ci z

(7J 0'< \0


a single-degree-freedom system,with the natural frequency range of

interest and various damping ratios,to this time-history motion. The

response spectrum for a particular mass point (supporting structural

element) is obtained by plotting the maximum acceleration against its

frequency. The same procedure is repeated for various damping values.

There are a number of uncertainties involved in computing the fre-

quencies and amplitude calculations. These uncertainties are due to

variation in elastic properties of both structure and foundation, ideal-

ization of structure with lumped masses, and elastic properties of dis-

crete parts. These uncertainties are usually accounted for by broadening

of the peak responses as shown in Fig. 6. The resulting smoothed spe·ctra

are used for the purpose of design.

Variation in damping characteristics has a significant effect on the

amplitude of the peak response, as seen in Fig. 7. The damping values

are, therefore, determined conservatively. The recommended values are

given in Table 1. Higher damping values may be used if supported by test

data.

TABLE l

DAMPING VALUES* (Percent of Critical Damping)

Structural Component Operating Earthquake

Welded Connec-tions 2

Bolted Connections 4

*AEC Regulatory Guide 1.61

Basis Safe Shutdown (OBE) Earthquake (SSE)

4

7

3.0

2.5 en

(.!)

2.0 c: 0 - 1.5 c .... Q)

Q)

0 0 1.0 <(

0.5

0.1

-For design use-

----- Actually

\..~ ......... ' , ... ' ...... ,.,""

, , ,,

0.2 0.3 0.4 0.5

', /

I I

I I

I I

I

Period - seconds

Fig. 6 - Typical In-structure Response Spectra

0.7 0.8 0.9

Vl

[:] Vl

~ -n ;I> z ;I> r ~ Vl -Vl

;I> z 0 0 tT1 Vl -0 z

'7J ---1 ......

872

en -c:: ::1

tn

c:: 0

0 ... Q)

Q) (.} (.}

<t

THIRD SPECIALTY CONFERENCE

50 33 20 20.0

10.0 8.0

6.0 5.0 4.0

3.0

2.0

1.0 0.8 I.

0.6 0.5

1--1--

0.4

0.3

0.2

0.1

.05 0.02 0.03 0.05

Frequency {CPS)

10 5.0 3.3 2.0

7£~1% Damping 2% Damping 5% Damping

/h ~0% Damping

r;:l.'l X~ /; ~ v '< \\

\\ I .\ I \ ~ '/ v-··---~~1 '\ \\ ~ v--1'\. \.\

'\ '\ r--~

'\ .\ 1'\

0.1 0.2 0.3 0.5

Period - seconds

Fig. 7 - Typical Response Spectra for Various Damping Values

1.0 0.5

~ \ ~ ~ " '\ ~ ~

""-.. ~

1.0 2.0


Recordings of seismic events indicate that the earthquake motions

occur simultaneously in all three directions without consistent relations

among the motions in various directions. Hence, it is necessary to

compute the total effect of all components by taking the square root of

the sum of the squares of the three components of motion (two orthogonal

horizontal and one vertical motion). For the purpose of design, therefore,

the in-structure response spectra are developed for all three components

of earthquake motion. These spectra are developed both for Safe Shutdown

Earthquake (SSE) and Operating Basis Earthquake (OBE).

DESIGN PROCEDURES

The methods and procedures for seismic calculations depend upon the

type of structural component to be analyzed, and accordingly, the analysis

Is divided into two parts:

Rigid Component: In this ··ase the component is rigidly attached to the

supporting element a11d has a 11atural period of vihratio11 eqtial to or less

than 0.02 (or frequency greater than 50). Ir1 tl1is case the componcJlt

simply "rides" along with the support and the acceleration of the compo

nent is the same as that of the support without amplification. The

acceleration to which the component is subjected is assumed to be equal to

the acceleration corresponding to zero period from the in-structure re-

sponse spectrum.

Flexible Component: When the component is flexible there is a resonant


effect between the component and structure. If the component has a

natural frequency close to the resonant frequency of the structure, the

component motion is greatly amplified. Near the point of resonance the

acceleration of the component may be several times that of the supporting

point. The response of flexible components depends upon the stiffness of

various members, type of connections, type of supports and material

characteristics and loads.

The complex flexible systems are analyzed by performing dynamic

analysis. For the purpose of a dynamic analysis the system may be repre

sented by a lumped-mass multi-degree-of-freedom system consisting of a

discrete number of masses connected by a set of mass-free elastic members.

The masses are chosen so that all significant modes are included. Also a

mass is lumped at points where significant concentrated weight is located.

The resulting system is analyzed using response spectrum modal analysis

technicque or time-history analysis. All significant modes are included

in the analysis. A square-root of the sum of the square method criterion

is used i11 combini11g model responses.

The structural system is subjected to the seismic excitations in

three orthogonal directions; namely, two horizontal and one vertical. The

individual responses are combined according to the square-root of the sum

of the square criterion to determine the complete response of the compo

nent structure to an earthquake. An example of a flexible system is given

in Fig. 8 which shows a cantilever type of hanger carrying three levels

of cable trays. This system is represented by 3 lumped masses. The indi

vidua 1 mode shapes are shown in Fig. 8 (d). (e). and (f). The combined

response due to all three modes is shown in Fig. 8(c).

Support

Motion

~

c.___)

t:____1 II [.__J

L...J~I (_1

(a)

Lumped mass model

tbA

~

~

(b)

u

L_j

~

(c)

:!

!....1

l..___..\

1st Mode

(d)

Fig. 8 - Dynamic Analysis of Flexible Systems

+ n +

w

~

2nd Mode

(e)

~ f-J

~

3rd Mode

(f)

Vl tTl v; 3:: I)

> z > r -< Vl

Vl

> z tJ tJ tTl Vl

C1 z

~ ---1 (fo


Pseudo-Dynamic or Equivalent Static Analysis

Since the complete dynamic analysis of the system is quite cumber

some, static analysis based on the dynamic response, also known as pseudo-

dynamic analysis, may be performed under certain conditions. In pseudo-

dynamic analysis the structure is represented by a single-degree-of

freedom system and the period of vibration is computed by simple analytical

methods. The seismic forces are then obtained by multiplying the mass by

the appropriate acceleration and applying it at the center of gravity of

the mass. A detai 1 description of Lhis procedure is given later. The analy

sis could be further simplified by using the peak acceleration value

obtained from the in-structure spectra. This value is multiplied by a

factor of 1.5 to obtain equivalent static loads to account for all modes.

Any simplified analysis which results in equally conservative response is

a] so acceptable.

The various <·ompor1et1ts of tl1e ,·able tray or HVAC du<·t system are de

signed to withstand the effects of dead load, live load, and seismic forces.

The following load conditions are investigated and checked to determine the

most severe cot1dition.

l. Dead Load

2. Dead Load plus Live Load

3. Dead Load plus Safe Shutdown Earthquake (SSE)

4. Dead Load plus Operating Bases Earthquake (OBE)


Stress Combination

Earthquake motions occur simultaneously in various directions. Two

orthogonal and one vertical direction is considered in the analysis. The

stresses resulting from the two horizontal and one vertical motio11 are

~ombined as follows:

0 dL

where,

(\

0dL

0 X

(j y

n z

2 2 + n + o y z

design stress

stresses due to dead load only

stresses due to horizontal earthquake motion along "X" direction

stresses due to horizontal earthquake motion along "Y"

direction

stresses due to vertica1 eartbquake motion

Allowable Stresses

The allowable stresses for loading condition l (d.L) shall be as

per AISI Specifications for the Design of Cold-Famed Steel Structural

Members. The allowable stresses for loading 2 (d.L +L.L.) are assumed

to be l. 33 times those given in the AISI specifications. For loading

3 (d.L +SSE) the allowable stresses given in the AISl specifications

are multiplied by a factor of 1.70. For this case the design stress

should always be less than 90% of the yield stress. The allowable

stresses for loading 4 (d.L + OBE) should be the same as that for dead

load alone.


PSEUDO-DYNAMIC ANALYSIS OF TYPICAL COMPONENTS

I. Cable Trays, Cable Buses and HVAC Ducts

The analysis and design of cable trays, cable buses, and HVAC ducts

is very similar and therefore, only the analysis and design of cable

trays is described here. A typical cable tray support system is

~hown in Fig. 9(A). For computing forces in vertical plane, the

cable tray is assumed to be ~ontinuous over two supports as shown in

Fig. 9(B). This assumption gives results which are consistent with

the comprehensive dynamic analysis of the entire system. The forces

induced due to various loads are computed as follows:

1. Dead Load Only

Referring to Fig. 9(B), the maximum design moment will be

wL2/l0

where

w dead load per unit length of pan cable tray

L support spacing

2. Dead Load Plus Live Load

Referring to Fig. 9(B) and 9(C) maximum design moment will be

~L wL2/l0 + 0.08PL

where,

P = concentrated live load, 200 lbs. placed anywhere on the tray.

3. Safe Shutdown Earthquake (SSE)

The computations described here are based on a study of analyti-

cal models by the author of entire cable tray systems including

trays, hangers, etc. The objective of the study was to develop a

Hanger

frame


,..----Floor slab

Longitudinal bracing

Cable tray or HVAC duct

J! 1-

£ 1-

Fig. 9(A) Cable Tray or HVAC Duct Support System

* * i f *%* i * f .<;£ i + + * * * %l L .\~ L ~I~ L .I

Fig. 9(B) Dead Load Condition

! 200 lb

L 7h I 7Er .. \~3 ~ • 23 L .. \. L

Fig. 9(C) Live Load Condition

y

X - -i I y

Fig. q(D) Reference Axes for Member Properties

X


"implified approach for the design of various components of the cable

tray system. It was determined that the effect of the first mode was

more than 90% of the total dynamic response due to all significant

modes. Therefore, equivalent static analysis is performed in lieu of

dynamic analysis.

The forces caused due to dead load and seismic excitations in vertical

and horizontal directions are computed as follows:

(a) Moment due to vertical excitation:

M (wL 2

/10) v

where,

a v

a acceleration, as obtained from the response spectrum curve of the v

supporting slab for vertical component of SSE, corresponding to

the vertical period of vibration, of the cable tray.

The vertical period of vibration Tv of the rable tray for a three span

continuous beam shown in Fig. 9(B) is computed as follows:

v J WLJ = ~- ----~---

27.76 X 106

xI T

X

where,

I moment of inertia of the tray about horizontal axis, (in.4

)

W wL (kips)

L = span in inches

(h) Maximum moment due to horizontal excitation in the direction

perpendicular to longitudinal axis of cable trays:

2 ~ ~ (wL /I 0) . ah


where,

ah peak acceleration from the response spectrum of the supporting

slab due to horizontal component of SSE.

The peak acceleration value of ah is used due to uncertainty in

determining the period of vibration in the horizontal direction.

This is due to the fact that period is a function of relative lateral

stiffnesses of cable tray and the supporting hangers and can not be

determined by simple methods. However, the model studies indicate

that the total response in this case is primarily due to the lg mode

and therefore, the factor of 1.5 is not required.

(c) Forces due to horizontal excitation in direction parallel to the

longitudinal axis of the cable trays:

Axial stress due to horizontal excitation in the direction of the

longitudinal pan axis is very small compared to the forces caused due

to excitation in other directions. Therefore, they are not included

in the design. However, it is imperative that adequate bracing

should be provided to prevent any longitudinal motion during the

earthquake.

4. Operating Basis Earth~<l_ke __ (.OBE)

The forces induced due to OBE are computed in a similar way by sub

stituting the floor response spectra for OBE earthquake.

Stress Combination

The stresses caused due to dead load and excitation in horizontal

a11d vertical directions are combined as follows:


0 = (M /S) +' /(M /S ) 2 + (M /S')z dL X \1 V X -n y

where design

o =;stress at the point under consideration

S section modulus about 'X' (horizontal) axis, see Fig. 9 (D) X

S' section modulus about 'Y' (vertical) axis,modified to reflect y

different allowable stresses for bending of the tray in

vertical and horizontal planes. The allowable stress for

bending in horizontal plane is considerably smaller

compared to that in the vertical plane which is used as a basis

for design.

S section modulus about 'Y' (vertical) axis y

fv allowable stress for bending in vertical plane. This stress

is different for top and for the bottom of the tray in

compression.

fh allowable stress for bending in horizontal plane

s' y

s y

The stress "o" is computed for the top of the tray to be in

compression at the center of the span and bottom of the tray to be

in compression at the continuous support. This stress is then

checked against the corresponding allowable design stress fv.

Allowable Stresses

The top flange or the lip of the cable trays is often laterally

unbraced and can buckle separately by a deflection of the compression

flange relative to the tension flange accompanied by out-of-plane


bending of the web and the rest of the section. Accurate analysis

of such a situation is extremely complex. The allowable bending

stress for such cases may be determined according to Section 3 of

the Supplementary Information on AISI Specifications.

Allowable stresses in bending when the bottom flange is in

compression may be computed by determining the effective width as

per Sec. 2.3 of the Specifications. Allowable bending stress for

bending in the plane of the web may be computed as per Sec. 3.4.2 of

the Specifications.

In case of cable trays without a solid bottom or other members

where theoretical determination of properties is impossible, the

properties of sections should be determined by actual tests.

II. Surport Frames

The cable trays, cable buses, and HVAC ducts are supported by means

of various types of support frames as shown in Fig. 5. Selection

of a type of frame depends on the available support conditions,

clearances, type of connections, etc. Only trapeze type of hanger

frame shown in Fig. lO(A) are considered herein. Since the lateral

stiffness of trays or ducts is very small, the hanger frames act

independently during excitation in the horizontal direction. The

frame, therefore, could be represented by a single-degree-of-freedom

system with the mass of the trays lumped at the tray level, as shown

in Fig. lO(B). The forces induced in the trapeze frame due to

various loadsare computed as follows:

884

\ \ \ \ \ \ \ \


Iv Iv

I. b

Fig. IO(A) Fig. IO(B)

Seismic Seismic Excitation

-4 ...

I I I I I I I I I \ \

l Excitoti~n

Fig. !O(C)

\ \ \ \ \ \

---~.J --- ---Fig. lO(D)

Fig. 10 - Typical Support Frame

\ I I I I I I I I

h


1. Dead Load Only

The forces due to dead load W are calculated from simple frame

analysis. W is the reaction of the hanger from cable trays.

W = w. L

where,

w dead load of tray per linear foot

L hanger spacing, ft.

2. Forces Due to Horizontal Excitation

F =F.w.ah

where,

F forces in the frame due to unit horizontal load

ah horizontal acceleration corresponding to the horizontal

period of vibration Th computed from the response spectra

of the supporting floor slab.

considering Fig. 10(C)1 Th is given by

= 2TI !W JKi,

where,

K

T h

(\

2TI

lateral frame stiffness

deflection

l b IV + -.-.-J

2 h Ih

due to unit

3 l b IV Wl1 (l + -.-.-)

2 h Ih

load

Ih moment of inertia of horizontal member

l moment of inertia of the vertical member v

h height of the frame

885


b = width of the frame

If the horiz~member

2rr j6E'Qi is rigid (Ih oo), then

3. Forces Due to Vertical Excitation

F = F dL

where,

a v

FdL forces in the frame due to dead load

av vertical acceleration corresponding to the vertical period

T v

6 v

0 v

of vibration Tv computed from the response spectra of

supporting floor slab.

2rr /i-vertical static deflection at the center of the horizontal member

5 Wb3

[ 1 -4

~~t. 1v l 384 EI 3. v

1 + 3 IJ. Ih

The forces due to dead load plus excitation in horizontal and vertical

directions are combined as shown earlier to determine the maximum stresses

in each member. Since th?• forces caused due to seismic excitations are

reversible in sign, the forces are combined so as to give absolute maximum

forces in the members. The horizontal members are designed for bending

only. The vertical members are designed for bending plus axial tension,and

bending plus axial compression,if it exists.

The periods of vibration for the case with fixed support could be

determined similarly. However, the support frame configurations are rarely


simple as mentioned above. Most frames have multilevel trays and are

provided with diagonal bracing. A complete dynamic analysis of such

frames is required to compute the forces induced due to earthquake

motions.

SYNOPSIS

The use of cold-formed steel structural members in nuclear power

plants is discussed. A design criteria for a seismic resistant design

of such members is presented. Simplified methods of seismic analysis

are illustrated. However, the design information provided by the

manufacturers is not adequate for complete dynamic analysis and more

research should be undertaken to provide it.


APPENDIX I - DEFINITIONS

Operating Basis Earthquake (OBE): The earthquake which produces the

vibratory ground motion for which structures, systems, and components,

necessary for power generation and safety of the plant, are designed to

perform their intended function. This earthquake is usually assumed to

be equivalent of the 50 percent of the Safe Shutdown Earthquake. OBE

could occur several times at the site during the life of the plant.

Response Spectrum: A plot of the maximum response (acceleration, velo

city, or displacement) of a family of idealized single-degree-freedom

damped oscillators as a function of natural frequencies (or periods) of

the oscillators to a specified vibratory motion input at their support.

Safe Shutdown Earthquake (SSE): The earthquake which produces the maxi

mum vibratory ground motion that the nuclear power plant is designed to

withstand without functional impairment of those compone11ts 11ecessary to

shut down the reactor and maintain the plant in safe condition. This

earthquake is expected to be the largest earthquake which could occur at

the site during the life of the plant.

Seismic Category I Components: Those structural components which are

essential to the safe shutdown and control of the reactor if earthquake

occurs. These components must perform their intended function during

and after the earthquake.

a v

b

E


APPENDIX II - NOTATION

vertical acceleration

horizontal acceleration

width of the support frame

modulus of elasticity

F forces in support frame due to horizontal excitation

F forces in support frame due to unit horizontal load

FdL forces in support frame due to dead load

f v

g

h

l v

l X

K

L

M v

p

s X

s y

allowable stress in cable tray for bending in horizontal plane

allowable stress in cable tray for bending in vertical plane

acceleration due to gravity

height of the support frame

moment of inertia of horizontal member of support frame

moment of inertia of vertical member of support frame

moment of inertia of cable tray about horizontal axis

lateral frame stiffness

cable tray support spacing

moment in cable tray due to dead load only

moment in cable tray due to horizontal excitation

moment in cable tray due to dead load plus live load

moment in cable tray due to vertical excitation

concentrated live load

section modulus of cable tray about 'X' (horizontal) axis

section modulus of cable tray about 'Y' (vertical) axis

890

s' y

v

w

CJ

CJ X

0 y

0 z

6 v


modified section modulus of cable tray about 'Y' (vertical) axis

horizontal period of vibration

vertical period of vibration

total load over length 'L' of the cable tray

dead load per unit length of cable tray or HVAC duct

design stress

stresses due to dead load only

stresses due horizontal earthquake motion in 'X' direction

stresses due horizontal earthquake motion in 'Y' direction

stresses due to vertical earthquake motion

lateral deflection of support frame due to unit load

vertical static deflection at the center of the horizontal member of the support frame


APPENDIX III - REFERENCES

1. IEEE Std. 344-1971, "IEEE Guide for Seismic Qualification of Class I Electric Equipment for Nuclear Power Generating Stations", Institute of Electrical and Electronics Engineers, Inc.

891

2. M. Stoykovich, "Seismic Design and Analysis of Nuclear Power Plant Components", Specialty Conference on Structural Design of Nuclear Power Plants, Dec. 17-18, 1973, Chicago.

3. Timoshenko, S.P., and Young, D.H., Vibration Problems in Engineering, 3rd Edition, D. Van Nostrand Co., New York, 1955.

4. USAEC, Dictorate of Regulatory Standards, Regulatory Guide 1.61, "Damping Values for Seismic Design of Nuclear Power Plants", Oct. 1973.

5. USAEC, Dictorate of Licensing, Regulatory Standard Review Plan, Sec. 3.8, "Design of Seismic Category I Structures".

seismic analysis and design of cold-formed steel structural

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