BHP

THE DESIGN OF STRUCTURES SUBJECTED TODYNAMIC LOADS

by

Dr. Frank GattoBHP Engineering Pty Ltd

Date of presentation." 1 April, 1995

BHP Engin~.ering Pb’ Lid AC::N 008 630 500

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INTRODUCTION

This document is intended as a guide for structural engineers to analyseand design structures subjected to d.~nnamic loads. Design methods arepresented as well as criteria wKich specify an accep~ble level of vibration.

The effects of dynamic loads, that is, loads that vary with rime, can besignificantly greater than the effects of static loads with the samemagnitude. As well as increased stresses and deformations in a structure,other adverse effects include failure due to fatigue, ser~,iceability problemsand occupant discomforc

Tradi~iona!Iy, smact’,~res subjected to d)~amic loads have been designed bytrs~-,.g to ensure that the major natural frequencies of the structure are notclose to the frequency of the applied forces. V,’-i~e the calculation and studyof the structure’s natural frequencies present a guide to the beha~io ,ur of thestructure, it does not give the complete picture.

Some of the reasons why design by avoidance of natural frequencies should

be no.~t be relied upon exclusively include -

Structural engineers are interested in the vibration amplitudes.Acceptable levels of vibration specified by ~rious Codes of Practice areusually defined in terms of {-ibration amplitudes.

Unacceptable vibration amplitudes may be caused by -!. Vibrating loads applied at frequencies near the structure natural

frequency.2. Large vibratS~g loads applied at frquencies away from the structure

natmra! frequency.3. A combination of (1] and (2].

Different parts of the structure may tolerate different levels of ~Sbration.For example, kn a mine processin~ plant, the vibration amplitudes in theoccupant areas need to be less than in the remainder of the plant.

Typically, when calculating natural frequencies, only the majorstructure natural frequencies are considered. Most structures arecomplex and have many natural frequencies. Some of these naturalfrequencies are major structure natural frequencies and many arelocalised. Vibration at certain locations may be caused by Iocalisednatural frequencies.

The best method to use for the design of structures subject to dynamic loadsis to calculate the vibration response. In this document, some handcalculation methods are presented which are suitable for preliminary designand for simple structures. However more complex structures requirecomputer analysis. When underta_tdng a vibration analysis, it is alsoimportant to check the sensitivity of the modeIIing assumptions.

Structural designers should also consider other methods of reducing thevibration levels. Ways in which nuisance from vibration can be reducedinclude -

Reduction of vibration at its source.Reduction of the vibration transmitted to the immediate surroundings atthe source.

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¯ Placing equipment that causes vibration as far as possible from peopleand equipment likely to be affected.

¯ Protecting sensitive apparatus against external vibration.¯ Protecting the whole structure or part of it against external vibration.

For example, consideration should be given to using xibration isolationsystems or separating occupant areas from sources of vibration.

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2. DYNAMIC LOADS

Dynamic loads can be categorised into four main groups (see Figure 1).

Harmonic loads va~., according to a sine function with or without a phaseshift. They affect the-structure for a sufficiently tong time to permit asteady-state vibration response. They may be caused by machines withsS~nchronously rotating mases which are slightly out of balance or machineswith intentional out of balance forces.

Periodic loads exhibit a time ~riation which is repeated at regular intervals.Although the load ~ithin one period is arbitrary., its repetition allows it to bedecomposed into a series of harmopJc loads through a Fouriertra.nsformation. The duration of the load is long enough for a steady stateresponse to develop. They may a_rise arise from human motion such aswalking, ru_nning and machines ha~ing more than one unbalanced mass oroscillating parts.

Transient loads exhibit an arbitrary time ~riation without any periodicity.Such loads can be due to v~-’md, ~-ater waves, earthquakes, rail or trm~c andconstruction works.

Impulsive loads are als0 of a t~ansient narm-e. Their duration, however, isvery sho~ so that the structural member affected reacts in a differentma!-l!ler,

Loads

Dynamic

Dead load

Perrr~nent bad

~ SIoMy varying bads

Harmonic

Machine operation

Human motbn

Wind

PeriodicWater w~ves

Earthquakes

Transient

tmpuls;’ve

Rail and road traffic

Conarucibn works

~impact

Loss of support

Figure 1. Types of loading in structural engineering

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at

3.1.

ACCEPTANCE CRITERIA

"gm-ee major criteria need to be considered when evaJuating the effects of~ibration. Ernese are -. human body perception and response;¯ effects on equipment and machineo’; and¯ structural design and detailing with respect to fatigue life.Each of these criteffa will be discussed in detail in the follo~"mg section.

HUMAN BODY PERCEPTION

The human body can detect mag.~_itudes of ~ibra~ion lower than those whichwould norma!ly cause mechanical problems. The "discomfo~" or"annoyance" produced bv whole body ~ibration is a veO" influential factorand may be the one of the limiKng parame*,ers Ln l_he design of the structure.However, in some cases, it is d~cuh to eliminate the perception ofvibration. Therefore the objecNve is to idenKfy those components (i.e.frequency of ~dbraNon, directions of motion, etc.) which contribute most toadverse subjective reaction and concentrate on their reduction.

3.1.1. Vibration Exposure to AS2670.1

It is possible to predict the extent of subjec13ve reacNons (i.e. discomforx orannoyance) from measurements of vibration. The sensitivity of the humanbody to horizontal and vertical vibration has been incorporated into avibration evaluation guide published by Australian Standards. This isAS2670: Part 1 - Evaluation of human exT~osure to whole-body vibration- Part1: Generol requirements. Vibration exposure limits are given as a function ofthese quantities:¯ direction of motion, either horizontal or vei-dcal;¯ frequency of ~bration;¯ acceleration of the osci!lations; and¯ exposure time.

This relationship is sho~..-n in figures 2 and 3. For practical evaluationpurposes, three main human criteria are distinguished. These a_re :¯ the prese~,ation of health and safety ("ex~posure limit");¯ the preservation of working efficiency ("fatigue decreased proficiency

boundary"); and¯ the presen-ation of comfort ("reduced comfort boundary").

//L~os~limtt, The exposure limit recommended is set at approximately/~alf the~ level considered to be the threshold of pain (or limit of voluntary[~.l..erance) for healthy human subjects. These ~ not be exceeded

%~ithout special just_t+~cation and awareness of a~ )~,,

Fatigue decreased proficiency botmdary. This boundary species a limitbeyond which exposure to vibration can be regarded as carrying a significantrisk of impairing working efficiency in many types of tasks. The actualdegree of task interference in any situation depends on many factors,including individual characteristics as well as the nature and difficulty ofthe task. For example, a more stringent limit may be applied when the taskis of a particulayly demanding perceptual nature or ca!Is for an exercise offLne manual skill. By contrast, some relaxation of the limit might bepossible in circumstances in which the performance of the task (e.g. heavymanual work) is relatively insensitive to vibration.

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Reduced comfort boundary. The reduced ~ bounda-Dr is related todifficulties of cano.ing out such operations as em~_ reading and

~oo.oo0

1

1.ODD

0.100 ,

1Frequency[t~

Figure 2. Senstttvi~ of h~s to ~ ~ibratton for fatigue~n~y bm,~a~~F~ ~e ..lf~t ~ttpI~ ~

1 DO.ODD

10.000

1.000

0,100

1 10

Frequen¢’~ ~

.,

lm__16m

Ii lh2.5h

4h8h16h24h

100

Figure 3, Sensitivity of humax~s to ~ vibration for fatigueF~ ~l~sure limit multiply

accelerations by 2.0, For reduced comfa~2B~i~llvlde accelerations by3.15.

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3.1.2. Vibration Exposure to AS2670.2

AS2670:P~2 - Eva[uazion of human ex’posure to wh~le-body vibration- Part2: Continuous and shock-i~nduced uibrazton in bu~kiings (i ~o 80 Hz) isspeckfically for the assessment of human e_xposure to con ~u-nuous vibrationha buildings. Vibration is assessed on the basis of the ~ibra~on dose value(X~D\0. A base acceleration curve is provided (Figure 4) and a muldp!)~mgfactor is used to distinguish different classes of buil ~dings (Table t).

1.000 ___

Frequency (Hz)

Fig~e 4. Base vertical acceleration curve for 16 hour day or 8 hournight.

Table I. Multiplying factors used to specify satisfactory magnitudes ofbulldin~ vibration.

Time Multiplying factor

Critical working areas (e.g. Day 1hospital operating theatres, Night 1

precision laboratories)Residential Day 2to4

Night 1.4Office Day 4

Night 4Workshops Day 8

Night 8

Table 1 leads to magnitudes of rib.ration below which the probability ofadverse comments is low. Doubling of the suggested ~ibration magnitudesmay result in adverse comment and this may increase significantly ff themagnitudes are quadr~pled.

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3,2. EFFECTS OF VIBRATION ON MACHINE BEHAVIOUR

With regard to machine behavior_r, the design of a structure may beconsidered satisfactory ff the follo~’Lng conditions are satisfied:¯ no damage is done to the roach.the;¯ the performance of the machine or adjacent machines is not impaired;

and¯ ~cessive maintenance and repair costs for the macl~ine and De

adjacent macl-Knes are not generated.

This criteria for acceptance must be based on the particular ~pe of reactS.heand other mact~_nes and equipment operating in the vicir~i~. For example,more stringent criteria may have to be applied ff the operation of precisionins~r’m-nents in an area subject to vibration is involved. EquipmentmanKfacm_rers may provide inYorma~on on acceptable ambient ~bra~ionlevels for their sensi~ve equipment.

Vibration tolerances for heavy macl-zines are given in figure 5.

1.00E+O0

1.00E-01

0olO

1.00E-02

1.00E-03

Frequency (Hz)

Figure 5. General machinery vibration severity chart.

lOO

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3.3. EFFECTS OF FATIGUE

Many structures suffer a reduction in strenguh after they have beensubjected to cyclic loading. This phenomenon is kno~-n as fatigue and isessentially a process, of crack mitigation and subsequent propagation. T’neeffect of any form of dynamic response is to reduce the fatigue life because:¯ the amplitude of the stress variation at any point is increased;¯ the frequency of the stress reversal is associated with a combination of

the natural frequency of the structure and the forcing frequency.

The actual mode of failure and time to failure are dominated by the localphysical features of the struct-ure as well as external factors such ascorrosion, temperature, pre-treatment, etc. The failure of a structure istherefore dominated by the weakest links in the failure chain and mucheffort is required to identify, and eliminate them.

For high cycle fatigue, AS4100-t990, Section 11 has a good ~-eatment of thedesign of structures and structural elements subject to loadings which couldlead to fatigue. Within the limitations described in AS4100, Section 1 t.4states that fatigue assessment is not required for a member, connection ordeta_il ff the normal or shear design stress ~satisfy

f" < 26MPa

or ff the number of stress cycles, nso satisfies

Even t~hough mechanical failure due to material fatigue is by far t_he mostcommortly known deteriorating effect of vibrations, a ~bra~ng mechanicalconstruction may fail in practice for other reasons as well. Failure may, forinstance, be caused by the occurrence of one, or a few, excessive "4brationamplitudes [brittle materials, contact - failures in relays and switches,collisions between two vibraKng systems, etc).

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4. SENSITIVITY

Vibration analysis is a function of mass, damping, stiffness and the appliedforces. A change in any one of these factors may be sufficient to make thebeha~iou.r of the structure or certain elements of the structure unacceptable.

Some of the modelling assumptions which can affect rise accuracy of theresults include:¯ simplistic modelling of cormec~on details Ipirmed or f~xed joints, s1~p at

joints, etc);¯ change of stiffness due to floors and secondary members not considered

in model;¯ difficulty in calculating the exact mass;¯ changing live loads during structure operation (e.g. spillage due so poor

housekeeping;¯ difficulty in modelling the interaction bet~veen structure and mach_ine;¯ difficulty in modelling the interaction between structure and soil; and° difficulty in correctly specifying the s~racm_re damping.

In general, it has been found that it is very difficult to predict the naturalfrequency of a structure to better than ±10% of the actual structure naturalfrequency.

It is very importar~t for the de’signer to undertake a series of sensiti~itystudies and evaluate the importance of various factors. Figure 6 shows theresults of a se~.si~i~iry analysis on a simple structure. As can be seen, theactual dsmamic displacements may be much greater than the calculatedresults.

20.00

16,00

14.00

12.00

10.00

8.00

6.00

4.00

2.00

0.00

i i i"’W’e"/I I I\1

I I I ! /! ~ ! \1 I

I i i I ~ j / \ I\,~ ~ I !/I Ill ~\~ \ ~ I

10 15 20 25

Forcing frequency (Hz)

Figure 6. Results from sensitivity arm.lysis.

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5. PRELIMINARY DESIGN

T’ne following rules provide acceptable levels of sela-iceabflity and ares~itable for preliminary sizing. Table 2 presents span to depth ratios w~chcan be used for preliminary, design purposes.

Table 2. Spa.~ to depf.~ ratios.ColumnsMaximum slenderness ratio

BeamsMaximum depth ratiosConveyor decks and walkwaysFloor beams other than minor beamsMajor beams supporting drives less than 200 kWMajor beams supporting drives greater than 200 kWMajor beams supporting screens, crushers, etcGirtsPurlins not lappedPurlins lapped

1:80

1:251:201:151:121:101:501:301:50

K a more detailed preliminary design is required, equ3valent static loadfactors may be used. The supporting structure is designed for a static loadcalculated by muldplFLng the weight 6f the structure times the equivalentstatic load factor. These factors are presented in Table 3.

Table 3. Equlvalent static load factors.Screens

]

5Centrifuges 2.5C~shers 3Bucket elevator supports

125.2Reciprocating machine supports (pumps and compressors)Rotation machinery supports

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6. FINAL DESIGN

The table below presents the vibration design procedure.

Obtainfj, the forcing frequency and the

amplitude of the force Fd

Calculate the vib,-a~on response, ~d and ~d

Iffy is ~-ithin 30% of peak

Then there may be excessive ~-ibration

Set factor of s~e~, 4~ = 4

Iffy is not ~’ithin 30% of peak

Set factor of safeD,, ¢ = !. 5 to 2

Is ¢ x 8~ > allo~ablc for people?X

Is ¢ x 8a > allowable for machiner~9 ./

6.1. REGULAR STRUCTURES

The calculation of the vibration response of "regular stractures" can becarried out using some simplified calculations. The range of applicability ofthese simplified calculations is sho~-n in Table 4.

Ta5IeApplicable structures:Squat structures - height to minimumwidth is less than 3

applicability fo~- s, tz~ple analysis.Non applicable structures:Tall structures

Structure is well braced

Individual members are all checked

Irregular structures

More than one source of vibration

Large dynamic forces.

The calculation of the vibration response is based on the following generalapproach:

l. Determine the natural frequencfles of the stracture.2. Determine dynamic amplification factor.

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3. Calculate the pseudo-static deflection and pseudo-static stress.4, Calculate the dynan-dc deflection and dynarrdc stress,

~nese steps are e_\-plained in more detail hn the fo!]o~mg sections.

6,1,1. Natural Frequency of Structure

natural frequency can be determined from -standard formulae;static deflection method; orcomputer analysis.

Fable 5 shows some S~d~d fo~a~ for uhe #.rst natural frequency.

Table 5, Standard formulae for natural frequencies.Description I Natural frequency (Hz)

Uniform beam, cantilever, lumped_ 1 [3Elmass at end f~ - 2r~ "~M/3

Uniform beam, simply suppoded,lumped mass at centre

Uniform beam, simply supported,uniformly distributed mass

Uniform beam, simply supported,lumped mass at general position

Uniform beam, simply supported,uniformly distributed mass andlumped ,mass at general position

Frame structure, N levels, uniformcolumns, equal mass at each level,fixed at base

f~ - 2~ ~Ma~ (L- ai

wherefn = natural frequency of &rst mode

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= elastic modulussecond moment of ~rea

= sum of second moment of area of all columns between each ]evellengthdistance of mass from suppor~

b=[-a= concentrated mass= mass per unit lenK~h ~

The ~tati~ d~fiection met_hodcan be used to obtain an estimate of thestructure natural frequency. The static deflection of a structure under itsown weight and the natural frequency of the structure are related. Thisrelationship can be used to estimate the f~rst natural frequency of comp]e_xstructures whose s~aNc deflection is k~nown, i’n_is relationship is given by -

where

The accuracy of the above equation depends on the degree to which thestatic deflection conforms to the mode shape of vibration. The staticdeflection method generally uh’derestimates the natur~l frequency of morecomplex structures. For example, for a ur~orm simply supported beamwith a uniformly distributed mass, the approx~nadon of t_he static deflectionmethod underestimates the in-st na~u_ral frequency of the beam by about11%.

.... Ni~ ~i~;~ the option to calculate na~oa-al frequencies. ~gnis is thesimplest optLon but care must be take when using these programs. Sometips when using these programs a_re - "

. A hand check of the first mode frequency using an approxSmate orempirical method is s~ongly advisable to ensure that the results arerealistic.

There is even more need than with static analysis to ~iew computeranalysis results as approximate. It is very difficult to predict naturalfrequencies of real structures with a high degree of precision.

Programs such as MicroStran and SpaceGass assume that the mass islumped at the node points. This ~Ii give a significantly different answerthen ff the mass is assumed to be uniformly distributed. Add extranodes along the member ff modelling distributed mass.

6.1.2. Dynamic Amplification Factor

Once the structure natural frequency is known, the equation below is usedto calculate the dynamic amplification factor -

where

1

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Blip

6,2,

dynarrflc amplification factorforcing frequency

natural frequencydamping ratio

Representative values of the damping ratio are given in Table 6.

Table 6. T’ypical values of damping. ,,.Damping ratio

Continuous steel structures 0.02 - 0.04Bolted steel structures O.O4 - 0.07Prestressed concrete structures 0.02 - 0.05Reinforced concrete structures 0.04 - 0.07Small diameter piping systems 0.01 - 0.02Equipment and large diameter piping 0.02 - 0.03

6.1.3. Pseudo-Static Amplitudes

Tine pseudo-static amplitudes of vibration of the structure are the deflectionsand stresses due to the dynamic loads applied as though they are staticloads.

6.1.4. Dynamic Amplitudes of Vibration

The d~-namic amplitudes of ~ibration are calculated by multiplying thepseudo-static amplitudes of vibration by the dynamic amplilication factor,i.e. for deflection -

and for stress -

O"d -=-- ~) X O"s

whered)mamic deflection amplitude

pseudo-static deflection

dynamic stress amplitude

pseudo-static stress amplitudedynamic amplification factor

COMPLETE ANALYSIS

Many simple dynamic problems can be solved quickly and adequately by themethods outlined in the previous section. However there are situationswhere more detailed numerical analysis may be required and finite elementanalysis is a versatile technique widely available for this purpose.

Progra~-ns such as Strand and Nastran can calculate the d}nnarnic responsefor a ~-iety of problem ~b~pes. However, dynamic analysis is more complexthan static analysis and care is required so that results of appropriateaccuracy are obtained at a reasonable cost when using f’mite elementprograms. It is often ad%sable to investigate simple idealisations initiallybefore embarking upon detailed models.

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There are different analysis methods for sol~q_ng vibration problems~include -¯ Modal analysis.¯ Direct integration.¯ Modal frequency analysis,¯ Direct frequency analysis.

These

The designer needs to decide whether to solve in the time domain or thefrequencT domain. T32oically ff the excitation is periodic then it is preferableto solve in the frequency domain, If the problem involves a transient or hassome nonlineaxities then it is necessary to solve in the time domain.

Some tips when using these programs are -¯ A hand check of the first mode frequency using an approximate or

empirical method is s~-ongly advisable to ensure that the results a_rerealistic.

¯ There is even more need than ~,ith star.ic analysis to x~ew computeranalysis results as approximate. It is very difficult to predict naturalfrequencies of real structures with a high degree of precision. It isimportant to ,-----------------undertake sensiti~dry analyses. At best, the answer is +10%of the true walue.

¯ Most computer programs assume that the mass is lumped at the nodepoints. This x~ill give a significan~Jy different answer then if the mass isassumed to be uniformly, distributed. Add exT_ra nodes along themember ff modelling distributed mass.

¯ For modal mnalysis, it is necessary to decide on the number of sigltificantmodes to use. Typically for a well-modelled structure, only the firsttaventy major modes should be significant.

¯ For direct integration it is necessary to decide on the integration schemeand to select a suitable ~t!rne step. A larger time step will decreaseaccuracy and not count the contribution of higher modes. A smallertime step ~,ill require excessive computer time.

¯ Secondary members, floor plate, cladding, etc, may be neglected in astatic analysis but can alter the dy-namic characteristics of the sr~dcturesignificantly.

¯ A given finite element model witl represent higher modes ~th decreasingaccuracy, v, rnen using finite element programs, the accuracy of t~hefrequencies deteriorates for the higher modes of a particalar model.They underestimate the exact frequency.

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7. EXAMPLE

i-~.is section presents an example describing the procedure for the design ofs.~-uctures subject to ~ibrar_ion.

PROBLEM DESCRIPTION

Consider a structure required to support a vibrating screen. A plan of thesu-ucture is shox~-n in Figure 7. T’ne screen weighs 7 tormes and itsoperating frequency is 900 rpm. The screen is supported on four isolationsprings and the force transmitted to the structure at the .operating frequencyis 800 N per support. Each beam carries an ex-u-a mass of 100 kg/m due to~_!k~,ays and floor plate. "Fne beams are restrained at the top flange and itis assumed T_hat morion does not occur in the tra_nsverse direction, i.e.morion only in the Z direction.

Ln uKis example the units shall be N, mm and sec.

3 8 10 15

4 ~ Screen Screen ~ 11~ support support ~

. ~, Screen Screen ~, ..e ~ support support ~ ~ "

1500

3000

1500

1500 2500 1500

T~X

Figure 7. Plan of structure supporting vibrating screen.

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PRELIMXNARY DESIGN

Beam depLh = Span= 5500mm10 I0

Choose 460UB67 [for demonstration o~y]

m = 67.1 kg/m

I = 294x106 mm4

A = 8540 mm2

= 550 mm

DYNAMIC LOADS

From ~ibrating screen manufacturer -

NATURAL FREgUENCY

The first natural frequency from a computer analysis usLng Nastran isshown below. Note that bounciary condidons have been set such that onlymotion i~ithe Z direction can occur.

From the analysis

fn = !7.6 Hz

DYNAMIC AMPLI~’ICATION FACTOR

Damping ~. = 0.02 for a continuous steel structure

15r=--=0.85

17.6

1Q = ~(l- r~ )’ + (2~r)~

1

0.85 ). +(2x0.02x0.85)a

FACTOR OF SAFETY

The forcing frequency or operating frequency is ~-ith.in 30% of the naturalfrequency. Therefore

¢=4

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PSEUDO-STATIC DEFLECTION AND STRF, SS

Apply the dynamic loads as static loads and calculate t_he displacement ands~ess. Results are sho~-n in We figure below.

Point

123456789

10111213141516

Applied force(N)

SupportSupport

0

Displacement(ram)

0I oI 0.054

800 0.1150 0.138

800 0.1150 ! 0.0540 I 0.070

t 0.070! 0.054

0.1150.1380.1150.054

0

0

8000

8o0oSupportSupport

Bending stress(MPa)

.06

.06.06.06.06.06.06.O6.O6.06.06.06

The maximum pseudo-static deflection and pseudo-static stress

0.138 mm1.06 NfPa

DYNAMIC/LM~PLITUDES OF VIBRATION

The dynamic amplitudes of ~ibration are given by

~ =Qx~,6~ = 3.63x0.138 = 0.50 ram

O’� = Q X

o’~ = 3.63x1.06 = 3.85

ACCEPTABLE VIBRATION LEVELS

f~ = 15 Hz, 6d = 0.50 mm

Acceleration = fo =

J~o = 4 X/I;2 X152

X~S = ~X0

4.44

s~

f~s- ~ -3.14m/

!!

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COMPLETE ANALYSIS

A forced vibration mnalysis was done using Nastran. The equations ofmoNon were solved in the frequency domain and the vibration amplitude atNode 5 is presented in the fig ,ure below.

Displacement at Node 5

3,5

o

1o 12 14 16 18 20Forcing frequency (Hz)

The maximum displacement at 15 l-Lz is 0.5 mm. The forcing frequency oroperating frequency is wit_bin 15 % of a major peak. Therefore the factor ofsafety to be used is 4.

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