Optimum design of longitudinally stiffened simply supported orthotropic bridge decks

Journal o f Sound and Vibration (1975) 40(2), 273-284

OPTIMUM DESIGN OF LONGITUDINALLY STIFFENED SIMPLY SUPPORTED ORTI-IOTROPIC BRIDGE DECKS S. F. NG AND G. G. KULKARNI Department of Civil Enghteering, University of Ottawa, Ottawa, Canada (Received 10 June 1974, and in revisedform 5 November 1974)

Design analysis of a stiffened bridge deck giving higher structural efficiencies in both static and dynamic behaviour is particularly difficult because of the stochastic nature of vehicle parameters entering into the mathematical formulation of the problem. The present paper deals with the optimum solution based on the maximization of mass ratio and frequencies, resulting in minimum weight design sections having higher strength to weight ratios under static lo~ds and smaller amplification factors under dynamic loads. Relationships between geometric dimensions of slab and stiffeners for two criteria of optimization are established by using the theory of orthotropic plate formulation and typical results are shown for practical ranges of design parameters. Comparative merits of the optimization criteria are discussed and conclusions are drawn based on applications to design problems. 1. INTRODUCTION The analysis of a stiffened simply supported bridge deck commences with the determination of geometric dimensions of the slab and stiffeners so as to satisfy the deflection and moment requirements under static loads. The dynamic analysis of such bridge decks includes interaction of bridge and vehicle parameters to determine the magnitude of impact factors under the action of moving vehicles. The important bridge parameters for dynamic analysis are frequencies and mass density per unit area. These quantities in turn are functions ofthe crosssectional dimensions of slab and stiffeners finalized from static considerations. Once the relative dimensions of slab and stiffeners are finalized to satisfy the requirements under static loads, the resulting constant values of frequencies and mass densities of the cross-section do not give any alternative to reduce the amplifications under dynamic loads. This is overcome in the proposed method of analysis which deals with obtaining higher structural efficiencies in both static and dynamic designs of the bridge deck. The research literature contains solutions dealing with separate static and dynamic analysis of these types of bridges idealized as orthotropic plates. Valetsos and Yamada [1 ] analysed the frequencies of these types of decks idealized as orthotropic plates. Heins and Looney [2] presented static analysis of orthotropie bridges using finite difference techniques. Iyengar [3] and others studied free vibration of beam slab type bridge decks. More recently, empirical solutions for frequencies of orthotropic plates were presented by the authors [4, 5, 6] and amplification factors were computed under multiple passage of vehicles. Kirk [7] studied the frequencies of stiffened plates for constant values of mass of plate and stiffener combinations. The research works cited above deal with either the static or the dynamic behaviour ofstiffened simply supported orthotropic bridge decks. Modern trends to slender aesthetic bridges, combined with materials of high strength characteristics, heavy vehicles with high speeds and methods of precise computations are yielding lighter bridges with little reserve strength. This in turn demands design based on optimum solutions which should have maximum structural efficiency both under static and dynamic loads. 273

274

S . F . NG AND G. G. KULKARNI

Attempts are being made in this investigation to obtain cross-sections of bridge decks which have smaller amplifications under dynamic loads and at the same time satisfy the code requirements of deflections and stress levels under static loads. The method includes simultaneous treatment of parameters involved in static and dynamic analysis, and aims at achieving minimum weight solution to the design problem. The assumptions inherent in the method are as follows. 1. The deck system comprising slab and longitudinal stiffeners can be replaced by an equivalent orthotropic plate of constant thickness having elastic properties coinciding with the axis of symmetry. 2. Only the first mode of vibration of the deck system is considered. 3. Damping in the system is neglected. 4. The fundamental frequency and mass of both the bridge and the vehicle are assumed to have predominant effect in computing amplification factors. Other parameters entering into the dynamic analysis are not treated in obtaining the optimum solution. 5. The amplification factors refer to the deflections and strains at the centre of the span of the bridge. 6. Both bridge and the vehicle are assumed to have a single degree of freedom and the vehicle is assumed to travel with uniform velocity and on a predetermined path. 2. PARAMETERS OF THE PROBLEM The important bridge parameters in dynamic analysis are bridge frequency, f B, and total mass of the bridge, MB. The frequency, f B, of the bridge combines with the frequency, f~, of the vehicle to form a common parameter frequency r a t i o , fR, given byfsbrv. Similarly the total mass, Mn, of the bridge combines with the total mass, My, of the vehicle to form the mass ratio MR, given by MdMB. The numerical values offB and MB can be finalized from static analysis and desirable ranges of the ratiosfa and MR can be obtained so as to give smaller amplification factors, for a given range of frequency and mass of vehicles. 3. BASIS FOR OPTIMUM SOLUTION The effect of the frequency ratio, fB/f~, on amplifications is shown in Figure 1. These are calculated for a mass ratio MdMB = 0"01 and six typical values of bridge and crossing frequencies. Calculations were based on beam idealizations and amplifications are plotted for the passing of one and three vehicles, shown as (i) and (iii), respectively. The effect of the mass ratio, MJMn, on amplifications is shown in Figure 2. These are calculated for six typical cases of bridge frequencies and crossing frequencyf~ = 0-5rr. Results shown are again for passing of one and three vehicles as denoted by (i) and (iii) respectively. From the figures presented, the following observations are made: (i) the amplification factor, At, increases rapidly as the frequency r a t i o , fR, increases from 0 to 1 ; (ii) peak amplifications are observed atfR = 1 ; (iii) the amplification has a steady and decreasing trend betweenfR = 1 to 6 and more; (iv) the amplification decreases with an increase in mass ratio and the trend is predominant whenfR is close to unity; (v) the amplification is independent of mass ratios forfR = 3 and more. From the above observations, it is noted that higher values of frequency and mass ratios are associated with decreasing magnitudes of amplifications. Higher bridge frequencies tend

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OPTIMUM DESIGN OF BRIDGE DECKS (0;

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to yield a frequency ratio greater than unity thus avoiding the critical resonance range (fR = 1) and thus locating the resulting amplifications in a zone of steady variation with decreasing trend. Higher mass ratios tend to decrease amplifications when the frequency ratio is in the critical range o f resonance. Further, higher frequencies and mass ratios can both be attained by minimizing the mass density of the bridge which in turn tends to minimize the weight ofthe bridge deck. Higher values of frequencies give higher stiffnesses with minimum cross-section area thus giving economical sections of higher strength to weight ratios. These two parameters, namelyfB and MB, can be obtained during the static design process. The dimensions of slab and stiffeners for a given span thus obtained can be checked for requirements under static loads, and maximized forfB and MR for minimum weight sections. The assumptions governing the interdependencies of amplification factors on frequency and mass ratios were investigated by dynamic tests on nine modes. These models were investigated in two different series: namely, Series A (4 models) and Series B (5 models). The remaining parameters such as velocity of the vehicle and its initial conditions of vibration before entering the bridge were the same for all tests. The results o f these investigations are presented in Table 1 for a typical case. It can be seen that for both Series A and B an increase in the frequency and mass ratio causes a reduction in the amplification factors. The same can also be observed for all other Combinations of vehicle parameters. This trend of experimental

276

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TABLE I

Effect of frequency and mass ratioson anaplification factors Series model Al A, A4 As B~ B2 Ba B~

Bs

Frequency ratio fR

Mass ratio -~IR

Amplification factor Ar

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1"25 0'975 0"70 0"58 0"715 0"625 0 "515 0"415 0"39

1"45585 1"49151 1"52526 1"56373 1"45589 I "50549 I "52949 1-54930 1 "59505

277

OPTIMUM DESIGN OF BRIDGE DECKS

results further supports the analytical assumption that design based on maximum frequencies results in a higher mass ratio (lighter structures) and hence smaller amplification factors giving higher strength under the dynamic action of vehicles. 4. OPTIMUM SECTION FOR MAXIMUM FREQUENCIES The fundamental frequency (m = 1) of a bridge of span a, rigidity D~ and mass density Po is given by fs,, =

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a

(I)

--,

~/Po

and is assumed independent of the aspect ratio, b]a. As higher frequencies are multiples of fundamental frequencies, it is sufficient to maximize the fundamental frequencyfn in equation (1). This is done by adjusting relative dimensions of slab and stiffeners to give maximum values of the ratio Dx[pofor a given span. With plate and stiffeners assumed to be made of the same material, having a modulus of elasticity E a n d density l~er unit volume p, the rigidity, D~, and mass density, Po, of the orthotropic plate are calculated in terms of dimensionless parameters ~,, fl, and ~s (a list of nomenclature is given in the Appendix). :,1

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(2)

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(4)

278

s . F . NG AND G. G. KULKARNI

The mass density, Po, is given by Po =

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(5)

By using equations (2) to (5), the relationship among the parameters ~,, fl, and ~, can be found for a maximum value of Vt-ff~/po. This is done by means of a numerical technique. 5. oPTIMUM SECTIONS--MAXIMUM MASS RATIO The deflection, 6~, of a bridge of span a, aspect ratio b/a and under a point load P at the centre, can be expressed in the form pa 4

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Dv"~-~D~KD.

(6)

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(7)

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(9)

By using Dy = [EHa/12(1 - v2)]~] and D= from equation (2), equation (9) is expressed in dimensionless form and then solved for maximum MR for constant values of a, p and &c. The term &da can be replaced to meet the code requirements of ratio deflection to the span. The coefficient of deflection, Ko, is obtained by writing a separate programme which includes superposition of individual solutions of an infinite orthotropic strip and a finite plate. 6. NUMERICAL SOLUTION--MAXIMUM FREQUENCY AND MAXIMUM MASS RATIO Numerical solutions to equations (I) and (9) were obtained for the following practical ranges of a,, fl, and ~ : a, = 0.15 to 0.50 in 0.005 increments, fl, = 0.1 to 0.9 in 0.10 increments, ~, = 1-0 to 3.5 in 0-5 increments. The frequency and mass coefficients, KF and KM, are obtained as multipliers to H ~ m 2z~2/a2 and (6c/a 4p) x 101~ These are shown in Figure 4 and Figure 5 respectively. Numerical values were obtained by varying fl, through an interval of 0-01 for constant 0~.

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It is seen from Figure 4 and Figure 5 that both frequency and mass coefficients first increase with increase in fl,, and that these same quantities attain peaks and then decrease with further increase in fl,. The magnitude offl, where the peak occurs depends on the magnitude ofcq. In both cases, absolute peak values of the coefficients decrease with increase in czsand fl=. The physical interpretation of increase, peak and decrease of these coefficients is a measure of the rate of variation in strength/weight ratio with increase in fl~. Increase in values of the coefficients Kr and KM with an increase in fls indicates that the rate of increase in stiffness is more than that in the cross-sectional area. Peak indicates optimum combination of stiffness and area while decrease in coefficients indicates that the rate of increase in dead weight is more than that in stiffness. I

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280

s.F. NG AND G. G. KULKARNI

7. RELATIONSHIPS BETWEEN ~s, fl~ FOR MAXIMUM FREQUENCY AND MASS RATIO COEFFICIENTS The peaks in Figures 4 and 5 appear to form a smooth curve indicating that the relationship o f cq and fls follow a definite pattern in the region of optimum magnitudes. The values of fl~ which correspond to peaks of frequency and mass coefficients are plotted against ct~ in Figures 6(a) and 6(b). It is seen that the relationships follow a linear pattern and can be represented by straight lines, the equations of which are fls = 2~s,

(10a)

for maximum frequency

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.In; (b) max..M~.

8. LIMITATIONS The two criteria for optimum frequency and mass ratios given by equation (10) cannot be simultaneously satisfied by a given optimum section. Frequency coefficients obtained by using equation (10) have been calculated and are given in Table 2. The magnitudes of frequency coefficients calculated by using mass ratio maximization (equation 10(b)) are TABLE 2

Comparison offrequency coeJ~cients of opthnum sections by methods offrequency and mass-ratio maximizations

=~ 0.1 0.15 0.20 0.25 0.30 0.35

Frequency by maximum frequency approach 0.3218 0.3153 0"3095 0"3045 0"3005 0-2973

Frequencies by maximum mass ratio approach ,9 ~~j = 0-1 1.5 2 2-5 3 3.5 0.3194 0 . 3 2 0 1 0-3207 0 - 3 2 1 1 0 " 3 2 1 4 0.3216 0"3129 0.3137 0 " 3 1 4 2 0-3147 0.3150 0-3152 0.3072 0.3079 0"3085 0"3089 0.3092 0.3094 0.3022 0-3030 0.3035 0"3040 0.3043 0.3044 0.2980 0.2988 0"2994 0.2999 0.3002 0-3004 0.2946 0.2955 0 . 2 9 6 1 0.2966 0.2970 0.2972

281

OPTIMUM DESIGN OF BRIDGE DECKS I

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consistently less. The same trend was observed when mass ratios were calculated by using both equations (10a) and (10b). This shows that a given section cannot be optimized to obtain simultaneous peaks o f frequencies and mass ratios. However, these two criteria in equation (10) can be considered as giving two boundaries enclosing an area which contains sections close to both. This is given in Figure 7 where mass ratios for optimum sections are calculated by using both the maximum frequency and the maximum mass ratio approaches and these are plotted against the ratio b[a for ~, = 0.I to 0.35 and ~ = 1 to 3.5. The curves shown by dotted lines are mass ratios obtained by using the maximum frequency approach, which therefore gave consistently smaller values for all combinations.

282

S.F. NG AND G. G. K.ULKARNI 9. R E L A T I V E

MERITS OF THE TWO METHODS

OF OPTIMUM

APPROACH

Deflection coefficients (P~/abV"DT~y)x 10-2 are calculated by using the maximum frequency and maximum mass ratio methods and are given in Figures 8(a) to 8(0 for b/a = I'0 to 0.5. Deflection coefficients are plotted against ~:, for czs= 0.I to 0.35. It is seen that deflections calculated by the maximum frequency approach are consistently lower than those calculated by means of the maximum mass ratio approach. This shows that relatively stronger sections are obtained by using the maximum frequency approach and are therefore preferable for design considerations.

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OPTIMUM DESIGN OF BRIDGE DECKS

283

10. CONCLUSIONS 1. Miiaimum weight design can be achie~,ed by maximizing the frequency or mass ratios. 2. Sections designed to give maximum frequency and m a x i m u m mass tend to yield smaller amplification factors under dynamic loads. 3. The relationship between relative geometric dimensions o f slab and stiffener take a definite linear pattern for obtaining maximum frequency and maximum mass ratio. 4. Sections designed by using maximum frequency approach give relatively smaller deflections and are preferable from design considerations.

REFERENCES 1. A. S. VAL~SOS and Y. YAMADA1958 8th Progress Report, Highway Bridge Impact Investigations, University of Illinois, Urbana, Illinois. Free vibration of simple span I-beam highway bridges, Part B. 92. C. P. I-IEINSand C. T. G. LOONEY1968 Proceedings of the American Society of Civil Engineers, Structures Division 565-592; Bridge analysis using orthotropic plate theory. 3. K. T. I~NGAR and K. S. JAOADLSH1964 Bulletin of the Indian Society of Earthquake Technology 1, 15-25. On the vibration of beam and slab bridges. 4. S. F. NG and G. G. KULKARNI1972 Journal of Sound and Vibration 21,249-261. On the transverse free vibration of beam-slab type highway bridges. 5. S. F. NG and G. G. KU~-KARNI1974 Proceedings of the American Society of Civil Engineers, Structures Division, 100, No. ST4, 842-848. Empirical frequencies of vibration of slab type bridges. 6. S.F. NG and G. G. KULKARNI1972 InternalResearch Report, University of Ottawa, CivilEng#zeering Department. On forced vibrations of highway bridges. 7. C. L. KInK 1970 Journal of Sound and Vibration 13, 375-388. Natural frequencies of stiffened rectangular plates. APPENDIX: NOMENCLATURE

f , frequency in Hz of bridge M~ total mass of bridge deck f, vehicle frequency (Hz) Mo total mass of vehicle frequency ratio, fall, MR mass ratio, Mo/Ms AF amplification factors, ratio dynamic deflection to static deflection f~ crossing frequency, nv[a V velocity of vehicle b width of bridge a span of bridge b/a aspect ratio, A~ D.,D, flexural rigidities of orthotropic plate in directions of span and width, respectively Po mass density of plate per unit area number of half sine waves in direction of x m p density per unit volume of slab material hlH Ols A t/s 4, t/h

h thickness of slab H total thickness of section t thickness of stiffeners $ chordwise (width) spacing of stiffeners E modulus of elasticity of the material of slab and stiffeners Poisson's ratio

284 e~, g P 6c A, Kr Ku KD

S. F. NG AND G. G. KULKARNI

depth of centre of gravity of repeating T section acceleration due to gravity load at centre deflection of bridge under load P at centre cross-section area per unit width of deck, H[ct, + ,8,(1 -- cq)] coefficient of frequency coefficient of mass ratio coefficient of deflection