BHP
THE DESIGN OF STRUCTURES SUBJECTED TO DYNAMIC LOADS
by Dr. Frank Gatto BHP Engineering Pty Ltd
Date of presentation."
1 April, 1995
BHP Engin~.ering Pb’ Lid AC::N 008 630 500
~ BHP
INTRODUCTION This document is intended as a guide for structural engineers to analyse and design structures subjected to d.~nnamic loads. Design methods are presented 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 be significantly greater than the effects of static loads with the same magnitude. As well as increased stresses and deformations in a structure, other adverse effects include failure due to fatigue, ser~,iceability problems and occupant discomforc Tradi~iona!Iy, smact’,~res subjected to d)~amic loads have been designed by trs~-,.g to ensure that the major natural frequencies of the structure are not close to the frequency of the applied forces. V,’-i~e the calculation and study of the structure’s natural frequencies present a guide to the beha~io ,ur of the structure, 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 are usually 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 the occupant areas need to be less than in the remainder of the plant. Typically, when calculating natural frequencies, only the major structure natural frequencies are considered. Most structures are complex and have many natural frequencies. Some of these natural frequencies are major structure natural frequencies and many are localised. Vibration at certain locations may be caused by Iocalised natural frequencies.
The best method to use for the design of structures subject to dynamic loads is to calculate the vibration response. In this document, some hand calculation methods are presented which are suitable for preliminary design and for simple structures. However more complex structures require computer analysis. When underta_tdng a vibration analysis, it is also important to check the sensitivity of the modeIIing assumptions. Structural designers should also consider other methods of reducing the vibration levels. Ways in which nuisance from vibration can be reduced include Reduction of vibration at its source. Reduction of the vibration transmitted to the immediate surroundings at the source.
Page 1
¯ ¯ ¯
Placing equipment that causes vibration as far as possible from people and 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 isolation systems 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 phase shift. They affect the-structure for a sufficiently tong time to permit a steady-state vibration response. They may be caused by machines with sS~nchronously rotating mases which are slightly out of balance or machines with 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 be decomposed into a series of harmopJc loads through a Fourier tra.nsformation. The duration of the load is long enough for a steady state response to develop. They may a_rise arise from human motion such as walking, ru_nning and machines ha~ing more than one unbalanced mass or oscillating 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 and construction works. Impulsive loads are als0 of a t~ansient narm-e. Their duration, however, is very sho~ so that the structural member affected reacts in a different ma!-l!ler,
Dead load Perrr~nent bad ~ SIoMy varying bads
Loads
Machine operation Human motbn Harmonic Wind Water w~ves Periodic Earthquakes
Dynamic
Rail and road traffic Transient Conarucibn works
tmpuls;’ve
~
impact
Loss of support
Figure 1. Types of loading in structural engineering
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ACCEPTANCE CRITERIA
at
"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
3.1.
The human body can detect mag.~_itudes of ~ibra~ion lower than those which would norma!ly cause mechanical problems. The "discomfo~" or "annoyance" produced bv whole body ~ibration is a veO" influential factor and 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 of vibration. Therefore the objecNve is to idenKfy those components (i.e. frequency of ~dbraNon, directions of motion, etc.) which contribute most to adverse 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 or annoyance) from measurements of vibration. The sensitivity of the human body to horizontal and vertical vibration has been incorporated into a vibration evaluation guide published by Australian Standards. This is AS2670: Part 1 - Evaluation of human exT~osure to whole-body vibration- Part 1: Generol requirements. Vibration exposure limits are given as a function of these 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 evaluation purposes, 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 limit beyond which exposure to vibration can be regarded as carrying a significant risk of impairing working efficiency in many types of tasks. The actual degree of task interference in any situation depends on many factors, including individual characteristics as well as the nature and difficulty of the task. For example, a more stringent limit may be applied when the task is of a particulayly demanding perceptual nature or ca!Is for an exercise of fLne manual skill. By contrast, some relaxation of the limit might be possible in circumstances in which the performance of the task (e.g. heavy manual work) is relatively insensitive to vibration.
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Reduced comfort boundary. The reduced ~ bounda-Dr is related to difficulties of cano.ing out such operations as em~_ reading and ~oo.oo0
1
1.ODD
0.100 , 1
Frequency[t~
Figure 2. Senstttvi~ of h~s to ~ ~ibratton for fatigue ~n~y bm,~a~~F~ ~e ..lf~t ~ttpI~ ~
1 DO.ODD lm
__16m Ii lh 2.5h 4h
10.000
8h 16h 24h
1.000
.,
0,100
1
10
100
Frequen¢’~ ~
Figure 3, Sensitivity of humax~s to ~ vibration for fatigue F~ ~l~sure limit multiply accelerations by 2.0, For reduced comfa~2B~i~llvlde accelerations by 3.15.
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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- Part 2: Continuous and shock-i~nduced uibrazton in bu~kiings (i ~o 80 Hz) is speckfically for the assessment of human e_xposure to con ~u-nuous vibration ha 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!)~mg factor 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 hour night. Table I. Multiplying factors used to specify satisfactory magnitudes of bulldin~ vibration. Time Multiplying factor
Critical working areas (e.g. hospital operating theatres, precision laboratories) Residential Office Workshops
Day Night
1 1
Day Night Day Night Day Night
2to4 1.4 4 4 8 8
Table 1 leads to magnitudes of rib.ration below which the probability of adverse comments is low. Doubling of the suggested ~ibration magnitudes may result in adverse comment and this may increase significantly ff the magnitudes 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 be considered 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.he and other mact~_nes and equipment operating in the vicir~i~. For example, more stringent criteria may have to be applied ff the operation of precision ins~r’m-nents in an area subject to vibration is involved. Equipment manKfacm_rers may provide inYorma~on on acceptable ambient ~bra~ion levels 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
lOO
Frequency (Hz)
Figure 5. General machinery vibration severity chart.
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~ BHP 3.3.
EFFECTS OF FATIGUE Many structures suffer a reduction in strenguh after they have been subjected to cyclic loading. This phenomenon is kno~-n as fatigue and is essentially a process, of crack mitigation and subsequent propagation. T’ne effect 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 local physical features of the struct-ure as well as external factors such as corrosion, temperature, pre-treatment, etc. The failure of a structure is therefore dominated by the weakest links in the failure chain and much effort is required to identify, and eliminate them. For high cycle fatigue, AS4100-t990, Section 11 has a good ~-eatment of the design of structures and structural elements subject to loadings which could lead to fatigue. Within the limitations described in AS4100, Section 1 t.4 states that fatigue assessment is not required for a member, connection or deta_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 most commortly known deteriorating effect of vibrations, a ~bra~ng mechanical construction may fail in practice for other reasons as well. Failure may, for instance, be caused by the occurrence of one, or a few, excessive "4bration amplitudes [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 applied forces. A change in any one of these factors may be sufficient to make the beha~iou.r of the structure or certain elements of the structure unacceptable. Some of the modelling assumptions which can affect rise accuracy of the results 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 natural frequency of a structure to better than ±10% of the actual structure natural frequency. It is very importar~t for the de’signer to undertake a series of sensiti~ity studies and evaluate the importance of various factors. Figure 6 shows the results of a se~.si~i~iry analysis on a simple structure. As can be seen, the actual dsmamic displacements may be much greater than the calculated results. 20.00
16,00
14.00 12.00
10.00
i
i
i"’W’e"/I I
I\1
8.00
I
I
I ! /! ~
! \1
I~j/
\ I\,
I
6.00
I i i
4.00
~ ~I
!/I Ill
~\~ \
~I
2.00
0.00 10
15
20
25
Forcing frequency (Hz)
Figure 6. Results from sensitivity arm.lysis.
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~ BHP
5.
PRELIMINARY DESIGN T’ne following rules provide acceptable levels of sela-iceabflity and are s~itable for preliminary sizing. Table 2 presents span to depth ratios w~ch can be used for preliminary, design purposes.
Table 2. Spa.~ to depf.~ ratios. Columns Maximum slenderness ratio
1:80
Beams Maximum depth ratios Conveyor decks and walkways Floor beams other than minor beams Major beams supporting drives less than 200 kW Major beams supporting drives greater than 200 kW Major beams supporting screens, crushers, etc Girts Purlins not lapped Purlins lapped
1:25 1:20 1:15 1:12 1:10 1:50 1:30 1:50
K a more detailed preliminary design is required, equ3valent static load factors may be used. The supporting structure is designed for a static load calculated by muldplFLng the weight 6f the structure times the equivalent static load factor. These factors are presented in Table 3.
Table 3. Equlvalent static load factors. Screens Centrifuges ] C~shers Bucket elevator supports Reciprocating machine supports (pumps and compressors) Rotation machinery supports
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5 2.5 3 125.2
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 be carried out using some simplified calculations. The range of applicability of these simplified calculations is sho~-n in Table 4.
Ta5Ie applicability fo~- s, tz~ple analysis. Applicable structures: Non applicable structures: Squat structures - height to minimum Tall structures width is less than 3 Structure is well braced
Irregular structures
Individual members are all checked
More than one source of vibration Large dynamic forces.
The calculation of the vibration response is based on the following general approach: l. Determine the natural frequencfles of the stracture. 2. Determine dynamic amplification factor.
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~ BHP 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; or computer 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 [3El mass 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
f~ - 2~ ~Ma~ (L- ai
Uniform beam, simply supported, uniformly distributed mass and lumped ,mass at general position
Frame structure, N levels, uniform columns, equal mass at each level, fixed at base
where fn = natural frequency of &rst mode
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~ BHP = elastic modulus second moment of ~rea = sum of second moment of area of all columns between each ]evel
length distance 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 the structure natural frequency. The static deflection of a structure under its own weight and the natural frequency of the structure are related. This relationship can be used to estimate the f~rst natural frequency of comp]e_x structures 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 the static deflection conforms to the mode shape of vibration. The static deflection method generally uh’derestimates the natur~l frequency of more complex structures. For example, for a ur~orm simply supported beam with a uniformly distributed mass, the approx~nadon of t_he static deflection method underestimates the in-st na~u_ral frequency of the beam by about 11%. .... Ni~ ~i~;~ the option to calculate na~oa-al frequencies. ~gnis is the simplest optLon but care must be take when using these programs. Some tips when using these programs a_re " .
A hand check of the first mode frequency using an approxSmate or empirical method is s~ongly advisable to ensure that the results are realistic. There is even more need than with static analysis to ~iew computer analysis results as approximate. It is very difficult to predict natural frequencies of real structures with a high degree of precision. Programs such as MicroStran and SpaceGass assume that the mass is lumped at the node points. This ~Ii give a significantly different answer then ff the mass is assumed to be uniformly distributed. Add extra nodes along the member ff modelling distributed mass.
6.1.2. Dynamic Amplification Factor Once the structure natural frequency is known, the equation below is used to calculate the dynamic amplification factor -
1 where Page 13
Blip dynarrflc amplification factor forcing frequency natural frequency damping ratio
Representative values of the damping ratio are given in Table 6. Table 6. T’ypical values of damping. ,,. Damping ratio Continuous steel structures Bolted steel structures Prestressed concrete structures Reinforced concrete structures Small diameter piping systems Equipment and large diameter piping
0.02 - 0.04 O.O4 - 0.07 0.02 - 0.05 0.04 - 0.07 0.01 - 0.02 0.02 - 0.03
6.1.3. Pseudo-Static Amplitudes Tine pseudo-static amplitudes of vibration of the structure are the deflections and stresses due to the dynamic loads applied as though they are static loads.
6.1.4. Dynamic Amplitudes of Vibration The d~-namic amplitudes of ~ibration are calculated by multiplying the pseudo-static amplitudes of vibration by the dynamic amplilication factor, i.e. for deflection -
and for stress O"d -=-- ~) X O"s
where d)mamic deflection amplitude pseudo-static deflection dynamic stress amplitude pseudo-static stress amplitude dynamic amplification factor
COMPLETE ANALYSIS
6,2,
Many simple dynamic problems can be solved quickly and adequately by the methods outlined in the previous section. However there are situations where more detailed numerical analysis may be required and finite element analysis is a versatile technique widely available for this purpose. Progra~-ns such as Strand and Nastran can calculate the d}nnarnic response for a ~-iety of problem ~b~pes. However, dynamic analysis is more complex than static analysis and care is required so that results of appropriate accuracy are obtained at a reasonable cost when using f’mite element programs. It is often ad%sable to investigate simple idealisations initially before embarking upon detailed models.
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~ BHP There are different analysis methods for sol~q_ng vibration problems~ These include ¯ Modal analysis. ¯ Direct integration. ¯ Modal frequency analysis, ¯ Direct frequency analysis. The designer needs to decide whether to solve in the time domain or the frequencT domain. T32oically ff the excitation is periodic then it is preferable to solve in the frequency domain, If the problem involves a transient or has some 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_re realistic. ¯ There is even more need than ~,ith star.ic analysis to x~ew computer analysis results as approximate. It is very difficult to predict natural frequencies of real structures with a high degree of precision. It is important 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 node points. This x~ill give a significan~Jy different answer then if the mass is assumed to be uniformly, distributed. Add exT_ra nodes along the member ff modelling distributed mass. ¯ For modal mnalysis, it is necessary to decide on the number of sigltificant modes to use. Typically for a well-modelled structure, only the first taventy major modes should be significant. ¯ For direct integration it is necessary to decide on the integration scheme and to select a suitable ~t!rne step. A larger time step will decrease accuracy and not count the contribution of higher modes. A smaller time step ~,ill require excessive computer time. ¯ Secondary members, floor plate, cladding, etc, may be neglected in a static analysis but can alter the dy-namic characteristics of the sr~dcture significantly. ¯ A given finite element model witl represent higher modes ~th decreasing accuracy, v, rnen using finite element programs, the accuracy of t~he frequencies 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 of s.~-uctures subject to ~ibrar_ion.
PROBLEM DESCRIPTION Consider a structure required to support a vibrating screen. A plan of the su-ucture is shox~-n in Figure 7. T’ne screen weighs 7 tormes and its operating frequency is 900 rpm. The screen is supported on four isolation springs and the force transmitted to the structure at the .operating frequency is 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 it is 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
1500 Screen ~ 11 support ~
4 ~~ Screen support
3000
. ~, Screen e ~ support
Screen ~, .. support ~ ~
" 1500
1500
2500
1500
T~X Figure 7. Plan of structure supporting vibrating screen.
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PRELIMXNARY DESIGN Beam depLh = Span= 5500mm = 550 mm 10 I0 Choose 460UB67 [for demonstration o~y] m = 67.1 kg/m I = 294x106 mm4 A = 8540 mm2
DYNAMIC LOADS From ~ibrating screen manufacturer -
NATURAL FREgUENCY The first natural frequency from a computer analysis usLng Nastran is shown below. Note that bounciary condidons have been set such that only motion 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
15 r=--=0.85 17.6
1
Q = ~(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 natural frequency. Therefore
¢=4
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~ BHP PSEUDO-STATIC DEFLECTION AND STRF, SS
Apply the dynamic loads as static loads and calculate t_he displacement and s~ess. Results are sho~-n in We figure below. Point
Applied force
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Support Support 0 800 0 800 0 0 0
(N)
800 0
Displacement (ram) 0
Bending stress (MPa)
0.054 0.115 0.138 0.115 0.054 0.070 0.070 0.054 0.115 0.138 0.115 0.054 0
.06 .06 .06 .06 .06 .06 .06 .O6 .O6 .06 .06 .06
I I
! I t !
8o0
o
Support
o
Support
The maximum pseudo-static deflection and pseudo-static stress
0.138 mm 1.06 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 f~s- ~ -3.14m/
s~ Page 18
!!
COMPLETE ANALYSIS A forced vibration mnalysis was done using Nastran. The equations of moNon were solved in the frequency domain and the vibration amplitude at Node 5 is presented in the fig ,ure below.
Displacement at Node 5 3,5
o 1o
12
14 16 Forcing frequency (Hz)
18
20
The maximum displacement at 15 l-Lz is 0.5 mm. The forcing frequency or operating frequency is wit_bin 15 % of a major peak. Therefore the factor of safety to be used is 4.
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