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MICHIGAN STATE UNIVERSITY LIBRARIES \\\\\\\\\\\\\ \\\\\\\\\\\\\\\ \\\\\\\\\\\\\\\\\\\\\\\ \\\\\\\ 3 1293 00906 9729 l This is to certify that the thesis entitled SEISMIC RESPONSE OF TIED ARCH BRIDGES presented by Aida Bellamine has been accepted towards fulfillment of the requirements for MS degree in Civil Engineering /2 (”man Major professor Date J24” 17/ [QQQ‘ ( 0-7639 MS U is an Affirmative Action/Equal Opportunity Institution ' LIBRARY MIchIgan State ‘ Unlverslty PLACE IN RETURN BOX to remove this checkout from your record. TO AVOID FINES return on or before date due. DATE DUE DATE DUE DATE DUE F l MSU Is An Affirmative Action/Equal Opportunity Institution c:\cIrc\ddedm.Dm3-D. 1 SEISMIC RESPONSE OF TIED ARCH BRIDGES By Aida Bellamine A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE Department of Civil and Environmental Engineering 1992 4’ \ \J (‘( ABSTRACT SEISMIC RESPONSE OF TIED ARCH BRIDGES By Aida Bellamine The bridges were classified into heavy rib-light deck, light rib-heavy deck and medium rib-medium deck types. Their seismic responses based on the AASHTO design spectrum were found to be very close to the response average to three time history ground motions. The seismic effect relative to live load effect was expressed in terms of a “structural zone factor”. An , “optimal” distribution of the structural material between the arch and the deck was considered. An “optimal” value of the cross-sectional area ratio of deck to rib was found to range between 0.75 and 0.97. It was observed that except for the tensile stress from the deck tie rod action, the seismic responses of the tied and deck types of arch bridges were quite similar. Copyright by AIDA BELLAMINE 1992 To my parents and brothers and sisters for their love and support. iv ACKNOWLEDGMENTS I would like to thank Dr. Robert K. L. Wen, Professor of Civil Engineering at Michigan State University. Without his support and assistance, this work would not have been accomplished. I would also like to thank the National Science Foundation for sponsoring the project. I would like to thank my committee members at Michigan State University: Dr. R. Harichandran, Dr. F. Hatfield and Dr. J. Lubkin. Chapter II TABLE OF CONTENTS LIST OF TABLES LIST OF FIGURES LIST OF SYMBOLS INTRODUCTION 1.1 Motivation 1.2 Object and Scope MODELING AND PARAMETERS 2.1 2.2 2.3 2.4 General Modeling 2.2.1 Arch Rib 2.2.2 Deck 2.2.3 Suspenders 2.2.4 Treatment of Mass and Damping Loading 2.3.1 Live and Dead Load 2.3.2 Dynamic Loading Parameters 2.4.1 Dimensional Parameters: Set A 2.4.2 Parameter Set B vi Page vi vii viii ...; OOOO\J\]\I\IO\O\O\&IIUILIIUIN TABLE OF CONTENTS (Continued) Chapter III 2.5 2.6 2.4.3 Sets C, D and E Response Quantities Samples of Existing Bridges 2.6.1 Properties of Actual Bridges 2.6.2 Definition of Three Bridge Types 2.6.3 Generated Bridges SEISMIC RESPONSE 3.1 3.2 3.3 3.4 3.5 General Method of Analysis 3.2.1 Equation of Motion 3.2.2 Time History Analysis 3.2.3 Spectral Analysis 3.2.4 Computer Program Used Natural Frequencies and Normal Modes 3.3.1 Natural Frequencies and Normal modes of a Tied Arch Bridge 3.3.2 Effect of Stiffness on the Fundamental Frequency Seismic Response 3.4.1 Ground Motions and Design Spectrum 3.4.2 Comparison of Results Relative Importance in Design 3.5.1 General 3.5.2 Dead and Live Load Response vii Page 10 13 13 13 14 14 25 25 25 25 26 27 28 29 29 3O 31 31 32 33 33 33 TABLE OF CONTENTS (Continued) Chapter IV VI 3.5.3 (DL+LL) Versus (DL+EQ), and the “Structural Zone Factor” “OPTIMAL DESIGN” 4.1 4.2 4.3 Introduction Parameters Results 4.3.1 Effect of Parametric Variations on the Fundamental Frequency 4.3.2 Effect on “Optimal ABAR” COMPARISON OF TIED AND DECK TYPE BRIDGES ' 5.1 General 5.2 Fundamental Natural Frequency 5.3 Dead Load Response 5.3.1 Stresses in the Deck 5.3.2 Stresses in the rib 5.3.3 Sum of Deck and Rib Stresses 5.4 Seismic Stresses 5.4.1 Stresses in the Deck 5.4.2 Stresses in the Rib 5.4.3 Sum of Deck and Rib Seismic Stresses CONCLUDING REMARKS 6.1 6.2 Recapitulation Concluding Remarks viii Page 34 49 49 49 52 52 53 67 67 68 .69 70 7O 70 71 71 71 72 78 78 80 TABLE OF CONTENTS (Continued) LIST OF REFERENCES 82 Table 2.1 2.2 2.3 2.4 2.5 2.6 3.1 3.2 3.3 3.4 4.1 4.2 5.1 5.2 5.3 LIST OF TABLES Parameters of Sample Real Bridges (Set A) Parameters of Sample Real Bridges (Set B) Alternative Parameters of Sample Real Bridges (Sets C, D and E) Classification of Bridge Types Representative Values of Representative Bridges Basic Parameters of Generated Bridges Comparison of Time History Solutions (Stress Amplification Factor) with Spectral Solutions for Three Bridge Types Comparison of Time History Solutions with Spectral Solution (Displacement/Span Length) Maximum Stresses and Structural Zone Factor Structural Zone Factor for Generated Bridges Fundamental Frequency in CPS (ABAR = 1.0) Values of “Optimal ABAR” and Corresponding Stress Fundamental Frequency of Tied and Deck Type Bridges Deck Maximum Stresses for Tied and Deck Type Arched Bridges Rib Maximum Stresses for Tied and Deck Arched Bridges Page 18 19 20 21 22 23 37 38 39 40 55 56 73 74 75 Figure 2.1 3.1 3.2 3.3 3.4 3.5 3.6 4.1 4.2 5.1 5.2 5.3 LIST OF FIGURES Tied Arch Bridge Model First Ten Modes of a Tied Arch Bridge Effect of RHO on Fundamental Frequency Distribution of Stress Resultants Due to Dead Load Distribution of Maximum Stress Due to Dead Load Distribution of Stress Resultants Due to Live Load Distribution of Maximum Stress Due to Live Load Fundamental Frequency as a Function of ABAR Maximum Dead Load and Earthquake Stresses Versus ABAR Deck Type Bridge Model Maximum Dead Load Stresses of Tied Bridges and Deck Bridges Maximum Seismic Stresses of Tied Bridges and Deck Bridges xi Page 24 41 46 47 47 48 48 57 62 76 77 77 LIST OF SYMBOLS A = “bedrock” acceleration AA = dead load compressive force divided by a “reference allowable Stress”, (see Equation 2.2) ABAR = ratio of the deck cross-sectional area to the rib cross-sectional area Ad = cross-sectional area of the deck ALPHA = dead load factor divided by the rib slenderness ratio, (see Equation 2.4) A, = cross-sectional area of the rib AS = cross-sectional area of the suspenders ASTAR = cross-sectional area of deck plus rib divided by “reference area AA”, (see Equation 2.1) BSMAXX = seismic amplification factor for straigth beam stress Cd = deck half depth C, = rib half depth Dd = deck depth D, = rib depth E = Young’ smodulus f = stress Fa = reference allowable stress xii LIST OF SYMBOLS (continued) g = acceleration of gravity GAMMA = dead load factor in terms of the rib stiffness, (see Equation 2.14) GSTAR = dead load factor in terms of the total stiffness (that of the deck plus that of the rib), (see Equation 2.3) H = rise of arch Id = moment of inertia of deck I, = moment of inertia of rib K = Stiffness L = span length M = bending Moment M = total mass per unit length M, = rib mass per unit length N = number of panels P = axial force Rd = radius of gyration of deck section R, = radius of gyration of rib section RHO = Ratio of deck bending stiffness to rib bending stiffness = Id/I, = section modulus S = site (soil) coefficient SA = spectral acceleration SAM = ratio of axial stress to total stress SMAXX = seismic amplification factor for curved beam Stress Ti ith natural period site zone factor ZS xiii LIST OF SYMBOLS (continued) Zst = “Structural zone factor” g = critical damping coefficient xiv 1. INTRODUCTION 1.1 Motivation Major earthquakes in recent years have caused significant damage to some highway bridges. Such damage could pose a threat to the life line system of which bridges are links. This consideration motivated a number of researchers to conduct studies on the behavior and response of highway bridges under seismic excitations. As far as arch bridges are concerned, much of the past work was done on the “deck type arched bridges” (Figure 5.1), (which, in order to avoid confusion with the general term of the deck of a bridge, will sometimes be referred to in the present study as the “D- bridges”). In terms of existing arch bridges, the tied type bridges (Figure 2.1), (which may be in this work referred to as the “T-bridges”) are as frequently constructed as, if not more than, the D-bridges. A deck type arch~ bridge is usually built on rock foundations (e.g. crossing a gorge), while a tied arch bridge is built on softer soils (e.g. crossing a river). The purpose of this thesisis to investigate the seismic response of the tied type arch bridges. Research on the deck type arch bridges was reported by Dusseau and Wen [1] who studied the in-plane and out-of-plane dynamic response of three actual deck type arch bridges under unequal as well as equal support motions. Inelastic responses of D-bridges were reported by Wen and Lee [2]. R. J. Millies recently conducted a study on the effects of the lateral Stiffness of the end towers on the seismic response of the D-bridges in the three dimensional space [3]. Work on tied arch bridges has been scarce. In 1989, Lee and Torkamani [4] conducted a study of the seismic resistance of tied arch bridges in the three dimensional space. It compared two mathematical models for the bridge deck. In the first one the bridge deck was modelled by a series of Simple beam elements connected to the tension ties at panel points, in the second one the bridge deck system is represented by three dimensional super-elements, each treated as an independent substructure. The latter model can reflect the local bending of the individual deck stringers, while the former cannot. Phase-different wave ground motions and the flexibility of the soil surrounding the foundation were considered. It was found that the beam model of the deck system was not adequate to produce the seismic response in three dimensional space.The study was, however, conducted on a single existing bridge, which limits its generality. 1.2 Object and Scope The objective of this thesis is to study the general characteristics of the seismic responses of the tied arch bridges. Attempts were also made in the study to assess the significance of such responses in design practice, and to relate them to those of thedeck type bridges for which much more data are available. The work for this study was based on computer modelling. The model used is a two dimensional one, dealing with the in-plane elastic behavior of bridges under equal support ground motions consisting of horizontal and vertical components. No soil-Structure interaction was accounted for. The system is assumed to be elastic. After having established that geometric nonlinearity has little effect on the seismic behavior of T—bridges, the problems were solved as linear ones. Eight panels were used in the bridge model, with uniform mass distribution on the horizontal projection. The cross-sectional area and the bending stiffness were assumed to be constant for each strUctural member set: rib, deck and suspenders. Young’ s modulus was taken to be 29000 ksi. Three ground motion inputs were used for time history analysis, and the AASHTO [5] design spectrum was employed for spectral response analysis. In the following, Chapter two describes the bridge model used and the parameters of the problem. An examination of the range of values of these parameters for some existing bridges resulted in a classification of tied arch bridges into three types; heavy rib-light deck, light rib-heavy deck and medium rib-medium deck. Based on this classification, three bridges were generated from each of three prototype bridges for the study. Chapter three reports an investigation of the seismic responses to three ground motion inputs and to the AASHTO [5] design spectrum. It was found that the average of the time history responses to the three ground motion inputs (the 1940 El Centro and the CIT B1 and B2 accelerograms [6]) was very close to the spectral analysis response based on the AASHTO [5] design spectrum. This reinforces the validity of that spectrum for design purposes. Also, in chapterthree, a “structural zone factor” ZS, that compares the earthquake effects with the live load effects is developed. In Chapter four, an exploratory study on an “optimal” distribution of structural material between the deck and the rib was conducted. It was based on the assumption that the “optimal” ratio of the cross-sectional areas of the deck to the rib is that for which the maximum stresses in the rib and deck are equal. The value of that ratio was found to range from 0.75 to 0.97. Chapter five presents a comparative study of the T-bridges and the D-bridges, insofar as their dead load plus seismic responses are concerned. It was found that the responses of the two bridge types were quite close. Finally chapter six recapitulates the work done and presents the concluding remarks on this investigation. 2. MODELING AND PARAMETERS 2.1 General This chapter describes the modeling and the parameters of the present study. In the first section, the bridge model used is described. In the second section, the static and dynamic loading are described. In the third section, the parameters of the Study are enumerated. Finally, in the last section, seven existing bridges were considered. They were classified into three categories according to their design characteristics. 2.2 Modeling The bridge model used in the present study (Figure 2.1) is a two dimensional elastic finite element model. The bridge is divided into eight panels of equal length. It has a roller support at the left end and a hinged support at the right end. The arch rib and deck are rigidly connected at the supports providing rotational as well as translational continuity. All loads and masses are lumped at the panel points. 2.2.1 Arch Rib The rib is a parabolic arch represented by a series of curved beam elements [7] connected at the panel points. It is worth mentioning here that the larger the number of panels and hence the number of nodal points, the closer the (lumped) static load distribution approaches a uniform one. That would decrease the bending stresses. It is well known that bending stresses vanish in a parabolic arch when the load distribution is uniform on its horizontal projection. The cross-sectional area and the moment of inertia are assumed to be the same for all arch rib curved beam elements, and constant along their lengths. 2.2.2 Deck The deck is represented by a series of Straight beam elements connected at panel points. It also acts as a tie which eliminates the need for thrust abutments. The cross-sectional area and the moment of inertia are assumed to be the same for all deck straight beam elements, and constant along their lengths. 2.2.3 Suspenders The arch rib and the deck are connected at the panel points by suspenders. These suspenders are modelled as essentially rigid truss elements. 2.2.4 Treatment of Dead Load, Mass and Damping The dead loads on the deck and the rib are assumed to be uniformly distributed on the horizontal projection. As mentioned previously, they are, as the mass, lumped at the panel points. Damping is assumed to be of the Rayleigh type, having critical damping ratios of 0.02 for the first two modes. 2.3 Loading 2.3.1 Live Load In this study, when live load is considered, it is assumed to be uniformly distributed over the left half of the bridge deck. Its magnitude is computed according to the AASHTO specifications [5]. It consists of a uniform load of 0.64 kips/ft/lane plus a concentrated load of 18 kips to be placed at the most critical section. For convenience in using the computer program, the concentrated load was converted to a uniform load distributed over one half of the span on which the 0.64 kips/ft/lane was also applied. For the model, the number of lanes used in the load computation was equal to one half of the number of lanes for the bridge concerned. 2.3.2 Seismic Loading Three ground motion inputs are used in the present study. They include the 1940 El Centro earthquake, and the B1 and B2 accelerograms which are two artificial earthquakes [6] of similar magnitude to the El Centro one. For dynamic analysis by the spectral method, the design response spectrum specified by AASHTO [5] is used. 2.4 Parameters In this study, five sets of parameters are introduced: A, B, C, D and E, each complete by itself, so that the different sets can be used alternately. The first set, labeled A, is the basic dimensional parameters set. Second set B is mostly dimensionless. It is introduced to make the results more meaningful. Sets C, D and E are minor variations (involving two dimensionless parameters) of set B. They are used at different stages of the investigation. 2.4.1 Dimensional Parameters: set A 1. 3" n O a. I 2 3 4 5 6. 7 8 9 The dimensional parameters are enumerated as follows: L = length of bridge span . H = rise of bridge arch cross-sectional area of arch rib . C, = half depth of arch rib I, = moment of inertia of arch cross—section Ad = cross-sectional area of bridge deck - Half depth of deck Id = moment of inertia of deck cross-section . N = number of panels 10. §= damping ratio 11. M = total mass per unit length 12. M, = rib mass per unit length 13. E = Young’ s modulus 14. AS = cross-sectional area of bridge suspenders 2.4.2 Parameter set B dimensionless ones in order to provide greater insight into the behavior of the system. Set B is introduced to replace set A. It consists of the following Some of the above listed parameters may be combined parameters: 1. 2. L = length of bridge Span ASTAR = total cross-sectional area of deck plus rib scaled by a “reference area”, AA d+Ar ASTAR = AA (2.1) where AA _ 14ny (2.2) 8x (I) xFa is equal to the rib dead load compressive force at the crown divided by a “reference allowable stress”, Fa (set equal to 22 ksi herein). 3. ABAR = Ad/A, = ratio of the cross-sectional area of the deck to the cross-sectional area of the rib MngL3 4. GSTAR = Ex (Id+lr) (2.3) This parameter is proportional to the dead load displacement divided by the Span length L. 5. RHO = Id/I, = ratio of the moment of inertia of the deck to the moment of inertia of the rib 6. H/L = rise to span length ratio. 7. M,/M = rib mass to total mass ratio 8. C,/R, = ratio of rib half depth to R,, rib section radius of gyration 9. Cd/Rdl= ratio of deck half depth to Rd, deck section radius of gyration 10. N = number of panels 11. §= Damping ratio 10 12. M = total mass per unit length 13. E = Young’ s modulus (E = 29000 ksi) 14.As = cross-sectional area of the bridge suspenders As mentioned earlier, the parameters of set B are mostly dimensionless. The reason for replacing set A by set B is because the results presented in terms of dimensionless parameters are generally more meaningful. Also, according to the theory of dimensional analysis, the total number of dimensionless parameters for the problem may be reduced from the dimensional set, i.e, 14, by three; three being the basic dimensions of the problem, i.e., length, force (or mass) and time. In the set presented here, effectively, the number of parameters can be regarded as reduced by two. The “time” dimension is not made dimensionless because it seems unwieldy to scale the seismic motion in terms of dimensionless time. In set B, one may eliminate M and E as parameters when dimensionless responses, such as stress amplification factors and displacement ratios, are considered. The cross-sectional area of the suspenders A, may also be eliminated as a parameter by making the suspenders essentially rigid members. 2.4.3 Sets C, D and E Sets C, D and E are each obtained by replacing two of the dimensionless parameters of set Bby two different ones. 1. Set C is obtained by replacing ASTAR and C,/R,, of set B, by ALPHA and Dd/D,, where MngL2 Ex ,/l,xA, ALPHA = (2.4) 11 or, in terms of set B parameters, J8xFax é) xGSTARx (1+ABAR) x (1+RHO) ALPHA = “ASTAR (2.5) and Si Dd R110 R d b: ' Jm" T (2'6) 79—, Set C parameters were used to classify the samples of existing tied arch bridges into three types, as discussed in section 2.6.2, and in Chapter 3, when examining the seismic response of the three bridge types. 2. Set D is obtained from set B by replacing ASTAR and GSTAR by (L/R), and GAMMA respectively, where L L (7,) = -————-——-—— (2.7) c (Id+1,) Ar or L ASTAR GSTAR E (E) = H" X (2.8) C 8xFax(Z)x(1+ABAR) and GAMMA = GSTARX (1+RHO) (2.9) 12 Set D parameters are introduced in Chapter 5, and used to compare the seismic response of the tied type bridges with the deck type bridges. 3. Set E is obtained from set B by replacing ASTAR and GSTAR by V and ALPHA respectively, where V is the volume of deck and rib members. It may be expressed as a function of set B parameters as: MngszASTARx (ABAR+LTR) v = H (2.10) 8xFax (Z) x (1+ABAR) where LR, the curved length of the arch rib, is a function of H/L and L. ALPHA is as given by Equation 2.4. Set E was used to generate three types of bridges to be discussed in section 2.6.3 in order to investigate whether three bridges, with the same length and amount of structural material, but of different types, would behave differently, especially under seismic loading. It is emphasized again that each of the five sets is complete in the sense that it uniquely defines the problem; i.e., a given bridge may be uniquely defined by any one of the sets. 2.5 Response Quantities The response quantities considered in the present study are: 1. The maximum live load stress,fLL . The maximum dead load stress,fDL . The maximum earthquake stress,fEQ . The maximum combined dead load and earthquake stresses: fDL+fEQ .U‘AUJN The maximum combined dead load and live load StressesszL+fLL 13 6. The stress amplification factor, which is the ratio of seismic plus dead load stress to dead load stress: BSMAXX, for the deck straight beam elements, and SMAXX for the rib curved beam elements 7. The nodal vertical displacements, scaled by the span length. 2.6 Samples of Existing Bridges 2.6.1 Properties of Actual Bridges Seven actual bridges are considered: the North Fork Stillaguamish River Bridge, the Leavenworth Centennial Bridge, the Fort Duquesne Bridge, the Fort Henry Bridge, the Glenfield Bridge, the Fort Pitt Bridge, and the West End-North Side Bridge. The parameters of concern for these bridges were computed from data obtained from the Steel Design Handbook [8]. The dimensional properties (set A) of the above bridges are presented in Table 2.1. The parameters of set B are presented in Table 2.2. Table 2.3 lists the parameters of sets C, D and E that replace the parameters of set B. 2.6.2 Definition of Three Bridge Types When examining the numerical values of the parameters of set C, it was noticed that the bridges may be classified into three groups: heavy rib- light deck, designated as type A, light rib-heavy deck, designated as type B, and medium rib-medium deck, designated as type C. In Table 2.4 are shown the classification of the seven sample bridges and the associated values of the parameters. Table 2.5 lists the definitions of the “representative values” for each bridge type. The three sets of representative values thus defined are used as reference parameter sets to generate three bridge sets with the same length, amount of structural l4 material, and total weight, corresponding to each type. This is described in the next section. 2.6.3 Generated Bridges In the previous section, three relatively distinct types of tied arch bridges are defined. It is natural to investigate for a given bridge belonging to a given type, whether it would behave differently, particularly with reference to seismic response, if it were “designed” as the other types. For this consideration, the following procedure was followed. 1. A representative bridge of each type is selected: Fort Henry (type A), Glennfield (type B), and Leavenworth (type C). 2. The length, volume (of structural material) and mass of each of these real bridges, along with the representative values chosen for each bridge type, are used to generate three bridges, with the same length, volume and mass, but with the other parameters pertaining to the different types in accordance with the definition. The question to be answered is whether the three bridges of different types, would respond differently under live load, dead load, and particularly seismic excitation. This is investigated in chapter three. In the remainder of this section, it is shown how three bridge types are generated. For each representative or prototype bridge the volume of the structural material is computed as: v=LxAd+L,xA, (2.11) 15 From the fixed volume of each representative bridge, two additional bridges are generated, using their representative values of the parameters. The parameters that define each bridge that is generated are computed as follows: A = V (2.12) r L Lx (ABAR* +—L—’) where ABAR* is taken from the representative values defining each bridge type. (In the expressions given in this section, quantities with a star superscript represent representative values as listed in Table 2.5; those without, pertain to properties of the prototype bridge.) 1 _ (ngflxL4 r - (2.13) 52 x A, x ALPHA“2 where Mg is the total dead load per unit length for the prototype bridge, A, is the rib cross-sectional area as computed above and ALPHA.“ is the ratio of dead load factor to slenderness ratio of the arch rib, taken from the representative value parameter sets defining each bridge type. The value of the parameter GAMMA may be computed from GAMMA = ALPHA* x% (2.14) in which If R = _ (2.15) 16 using the central value of (M,/M), the rib mass is equal to * M _ M, = (7') >2. mm L3 84 e: 8.24 3: N3 32 SE .95 so? #3 8A : 2A 2A N3 one 2: Rd 3m 35 0:8 £82 _ 31o mom 2 83. SA as 93 9: $5 SE SE. £83553 _ e a A e A 5 Be: Bee 2 9.5 2 at UD m {b ._ EC .0 m at 3 EC 4 SEE E3 3 < < 2 SB momgem Gem 295m mo €038.23 fim DEE. l8 28:95 _ e2 3.? com sum 3% 50: Em SN 25 23:56 Ed 32 BE :8 . 582 GA SE .95 so? 36 Eng Seem 582 omfi odmv 5.535234 Am Homv Swarm Bum oEEam mo SBDESE N.N DEE. Seem l9 33:95 <2 3% 3% AS a .o eon 3.88 52 SS: N3 85 Sum eon. 8.45m e22 3.2: 23 as 20556 <2 232 NBA 2% £5 a: eon £52 <22 4.0 sec., where T,- is the ith natural period, A is the (“bedrock”) acceleration in g and S is a site (soil) coefficient. In this analysis, A and S are takento correspond to the “strongest” ground motion and soil, i.e., A = 0.4, and S = 1.0. 31 3.4.2 Comparison of results In this section, the response computed by the spectral solution is compared with the responses computed by the time history solutions when the structure is subjected to the ground motions mentioned above. The response quantities considered here are the stress amplification factor: maximum dead load plus seismic stress/maximum dead load stress, BSMAXX for the straight beam elements; SMAXX for the curved beam elements; and the ratio of nodal displacement to span length. These quantities are computed for the representative bridge of each type: a/a (heavy rib, light tie), b/b (light rib, heavy tie), and c/c (medium rib, medium tie). The values of the time history response quantities are averaged and compared to the spectral values. Table 3.1 shows the stress amplification factors.The spectral value compares very well with the average of the three ground motions. Table 3.2 shows the displacement ratios for the three time history ground motions, their average value, and the spectral value. The spectral value of the displacement ratio is also close to the average of the three ground motions. This suggests that the response spectra, recommended by the AASHTO [5], predicts well the response of this bridge type to seismic excitations. 3.5 Relative Importance in Design 3.5.1 General When studying the response of a bridge structure to earthquake excitations, for a proper perspective, it is important to first investigate its behavior under static loading. In this section, the distribution of the stresses and of the stress resultants under dead and live” loads are first 32 examined. Then the stresses under the load combinations normally considered in design, i.e., dead load plus live load and dead load plus earthquake load, are compared by introducing a structural zone factor, denoted by Z”, at which the effect of live load plus dead load and that of dead load plus earthquake load are equal. 3.5.2 Dead and Live Load Response The bridge considered in this section is the representative bridge of Type A (bridge a/a: Heavy Rib-Light Deck). Figure 3.3 shows the distribution of the stress resultants in the bridge members under dead load. Figure 3.4 shows the stress distribution in the bridge members under dead load. The internal forces (stress resultants) are much higher in the arch rib than in the deck. This is understandable since, in the case of this bridge (heavy rib, light tie), the rib is much stiffer than the deck (RHO = 0.01548). However, the stresses in the arch and in the deck are close. The rib has higher axial forces and bending moments than the tie. It also has larger area and moment of inertia. The differences tend to vanish when the stress is computed. From structural theory, it is known that the bending moment on a parabolic arch vanishes when the loading on the horizontal projection is uniform. In the case of this bridge, eight panels are used for a length of 577.5 ft, and the mass is lumped at the panel points. The load distribution is not exactly uniform, which explains the relatively large bending moments in the arch rib. Figure 3.5 shows the stress resultants distribution in the arch rib and the bridge deck under live load. The anti-symmetry of the bending moment inthe arch is due to the fact that only the left half of the bridge is loaded. 33 The axial forces in the rib and tie are close, although the moments in the arch are much higher than in the tie (compared to the dead load case). That explains why the difference in the stresses is bigger than in the case of the dead load. At the midspan, where the internal forces are very close in the arch and in the tie, the stresses are equal. Figure 3.6 shows the distribution of the live load stresses in the deck and the rib. 3.5.3 (fDL+fLL) versus (fDL-i-fEQ), and the “Structural Zone Factor” In this section, a structural zone factor is computed and discussed in order to investigate which load combination: dead load plus live load or dead load plus seismic load governs the design of each one of the bridge types. The zone factor is computed by making the dead load plus live load stress equal to the dead load plus a fraction (= Z“) of the earthquake stress, the dead load and earthquake stresses being reduced by 33% to account for the usual increase in allowable stress of 1/3 in the case of seismic design. Thus, one may write: fDL+ZSI XfEQ fDL +fLL = 1.33 (3.7) from which: 0.33f + 1.33f ZS, = DL LL (3. 8) fEQ 34 where fLL is the live load stress, fDL is the dead load stress, fEQ is the earthquake stress and Z5, is the “structural zone factor”. When Z3, is equal to unity, the live load effect is just equal to the effect of an earthquake of the intensity considered here (i.e., AASHTO A = 0.4, S = 1.0). Table 3.3 shows the maximum live load, dead load, and seismic load stresses and the structural zone factor as computed above for the prototype bridge of each type (a/a, b/b, and c/c bridges), both for the tie and for the arch rib. For more exploratory data, Table 3.4 shows the “structural zone factor” for the three bridge types and for the generated bridges of the other two types corresponding to them. It is noticed, from Table 3.4, that for bridge types b and c, Z5, is always greater than 1.0 for the curved beam elements of the arch rib. This would mean that if these bridges are built at a site with smaller seismicity than considered here, live load effects would govern the design. It is also noticed that for type a, ZS, is always less than 1.0, for the straight beam elements of the bridge tie. The “structural zone factor” can be compared to the site zone factor 25 which may be computed as follows: (A) site x Ssile zs — (A X5) (3.9) used in which Ame and SS,“ are, respectively, the ground acceleration and soil factor for the site under consideration and (A x S)used = (0.4 x 1.0) as used in obtaining the results presented. If the “structural zone factor” is smaller than the site zone factor, then earthquake loading would govern the design. 35 Table 3.1 Comparison of Time History Solutions (Stress Amplification Fac- tor) with Spectral Solutions for Three Bridge Types _ r BSMAXX M, I SMAXX M, I Type A bridge El Centro 1.78 8, 1 1.85 5, 1 I B1 1.80 5, 1 2.11 4, 2 il BZ 1.83 8, 1 2.31 7, 2 Average 1.80 2.09 Spectral 1.83 8, 1 2.19 7, 2 ' Type B bridge El Centro 1.73 8, 2 1.30 7, 2 Bl 1.56 8, 2 1.35 5, 1 B2 1.98 5, 1 1.71 6, 2 Average 1.76 1.45 l 8, 2 Type C bridge El Centro 2.16 8, 1 1.47 6, 2 B1 3.04 1, 2 2.07 3, 1 B2 3.20 8, 1 1.86 3, 1 Average 2.80 1.80 Spectral 2.99 8, 1 1.68 7, 2 Table 3.2 Comparison of Time History Solutions with Spectral Solution 36 (Displacement/Span length) 1:? Deck 22:11: Arch left Arch fig: quarter midspan quarter qua?“ crown quarter point point pomt point Type A bridge El Centro 0.001060 0.001650 0.001010 0.000973 0.001510 0.000925 B1 0.001150 0.002030 0.001080 0.001070 0.001880 0.001010 B2 0.001230 0.001590 0.001360 0.001120 0.001460 0.001250 Average 0.001150 0.001760 0.001150 0.001050 0.001620 0.001060 Spectral 0.001168 0.001659 0.001146 0.001077 0.001512 0.001054 Type B bridge El Centro 0.001630 0.002300 0.001720 0.001410 0.002090 0.001510 Bl 0.001770 0.002710 0.001840 0.001560 0.002420 0.001630 B2 0.002190 0.003320 0.002340 0.001900 0.002980 0.002040 Average 0.001860 0.002780 0.001970 0.001620 0.002500 0.001730 Spectral 0.001839 0.002743 0.001946 0.001613 0.002450 0.001721 Type C bridge El Centro 0.000752 0.001000 0.000821 0.000696 0.000936 0.000759 B 1 0.001260 0.001060 0.001020 0.001180 0.001000 0.000949 B2 0.001150 0.001090 0.001090 0.001070 0.001010 0.001020 Average 0.001050 0.001050 0.000977 0.000982 0.000982 0.000909 Spectral 0.000936 0.001139 0.000921 0.000872 0.001064 0.000858 37 Table 3.3 Maximum Stresses and Structural Zone Factor Bridgedeck 51:12:38.1?) 3535521333 52323;) 2.. Bridge deck Type a 2.030 16.060 12.180 0.660 Type b 5.130 30.820 17.060 0.996 Type c 6.430 10.070 14.520 0.820 Bridge arch rib Type a 5.301 15.600 17.700 0.690 Type b 3.620 43.200 16.650 1.140 Type c 4.420 19.410 11.820 1.040 38 Table 3.4 Structural Zone Factor for Generated Bridges Deck 0.80 1.00 1.04 J Arch 1.05 1.14 1.21 I Type C c/a c/b c/c Deck 0.62 0.74 0.82 I] Arch 0.89 1.05 1.04 39 (a) First Mode, Natural Frequency = 0.52041 (HZ) (b) second Mode, Natural Frequency = 1.12356 (HZ) Figure 3.1 First Ten Modes of A Tied Arch Bridge (continued) 40 (c) Third Mode, Natural Frequency = 2.09616 (HZ) (d) Fourth Mode, Natural Frequency = 2.15025 (HZ) Figure 3.1 First Ten Modes of A Tied Arch Bridge (continued) 41 (e) Fifth Mode, Natural Frequency = 3.26269 (HZ) (f) sixth Mode, Natural Frequency = 4.44230 (HZ) Figure 3.1 First Ten Modes of A Tied Arch Bridge (continued) 42 (g) Seventh Mode, Natural Frequency = 5.47949 (HZ) (h) Eight Mode, Natural Frequency = 5.76301 (HZ) Figure 3.1 First Ten Modes of A Tied Arch Bridge (continued) 43 (i) Ninth Mode, Natural Frequency = 11.51545 (HZ) (j) Tenth Mode, Natural Frequency = 15.99208 (HZ) Figure 3.1 First Ten Modes of A Tied Arch Bridge (continued) Fundamental Frequency (CPS) Fundamental Frequency (CPS) 0.5 ~ - 0.4 - ~ 0.2 ~ ‘ I 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 RHO = I‘ll, (a) II = constant 0.8 ~ . 0.7 - - 0.5 — - 0.3 - . 0.2 r _ A I n J A o 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 RHO=I¢/I. (b) I, + Id = constant Figure 3.2 Effect of RHO on Fundamental Frequency 45 E 8000* ' EL- , :4 5000 _ _ g): IK'RRib Bending Moment 3 2 m ' ............................. ' g 2000 _ \Rib Anal Force _ $3 a: m 0 ‘ ‘ a \Deck Bending Moment :1: [— ‘0 40(1)" — fi a: '\Deck Axial Force 5 .4000 >- _ E 0 l 2 3 4 5 6 7 8 9 10 PANEL POINT NUMBER Figure 3.3 Distribution of Stress Resultants Due to Dead Load 1 —Deck Stresses _ o RibStresses MEMBER STRESSES (KSl) 5 A A 1 I l 41* 1 A l 0 l 2 3 4 5 6 7 8 9 10 PANEL POINT NUMBER Figure 3.4 Distribution of Maximum Stress Due to Dead Load 46 E 8000 - " é =6 5000 - - E V 4000 » V‘ . E -{—— Rib Bending Moment 1— 2000 Rib Axial Force _ _1 _ D Deck Bending Moment . if 0 _ .............. _ m {(3 KDeck Axial Force :2 «‘7; -2000 . — a: U.) § In .4000 ’- 2 0 1 2 3 4 5 6 7 8 9 10 PANEL POINT NUMBER Figure 3.5 Distribution of Stress Resultants Due to Live Load v 1 18 _ - Deck Stresses o RibStresses MEMBER STRESS ES (KSI) 5 PANEL POINT NUMBER Figure 3.6 Distribution of Maximum Stress Due to Live Load 4. “OPTIMAL DESIGN” 4.1 Introduction As noted in the sample real bridges presented in Chapter 2, there are widely different material distributions between rib and deck structural members. It is interesting to inquire whether there is an “optimal” distribution of structural material between the rib and the deck, so far as earthquake effects are concerned. An “optimal value” may be defined, based on ABAR (the ratio of the cross-sectional area of the rib to that of the deck) as follows: with other parameters fixed, the “optimal value” of ABAR is taken to be that value at which the maximum dead load plus earthquake load stress in the rib and that in the deck are equal. It will be shown that a deviation from that value would lead to a higher stress in either the rib or the deck. 4.2 Parameters For the purposes of this chapter, parameter set B is used. The parameters that are varied are: 1. L, span length. Three values: 200.0 ft, 400.0 ft and 800.0 ft are considered. 2. ASTAR, as defined previously, the sum of the cross-sectional areas of the rib and the deck scaled by a reference area, (see equation 2.1). Three 47 48 values: 1.00, 2.00, and 3.00 are considered. 3. GSTAR, the “dead load factor” (see Equation 2.3). It is a measure of the total structural stiffness. Three values: 2.0, 5.0 and 10.0 are considered. In addition, in this chapter RHO (the deck bending stiffness to rib bending stiffness ratio) is assumed to be equal to the squared value of ABAR. The assumption is based on the following derivation: A box cross-sectional area (of depth D and thickness t) is assumed for both the rib and the deck (the depth and thickness are not necessarily equal for the deck and the rib). The ratio RHO can be written as: RHO = (4.1) The radius of gyration of the rib may be reasonably assumed to be proportional to the depth of its cross-section: Rr = arXDr (4.2) Also, the radius of gyration of the deck is assumed to be proportional to the depth of its cross-section: R — ct >~ 8 3 E l 15— — '5 5 E 1. _ 13 200.08 5 IL 40008 0..) 800.08 ‘64 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 ABAR (c) ASTAR=1.0, GSTAR=10.0 ‘1 2.5— — 83 B 2- - >6 8 3 a 200.08 ,3; 154/? - E 5 400.08 E s 1/ _ E 1.1. . W 800.08 0.5— _ ‘64 0.5 06 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 ABAR (d) ASTAR=2.0, GSTAR=2.0 Figure 4.1 Fundamental Frequency as a Function of ABAR (continued) 57 Lb) 2.5- - a D- 9, 2- - 5‘ E g g is- - E fl 0 200.08 E 1_ _ E ”:3 400.08 0'5 800.08 6.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 ABAR (e) ASTAR=5.0, GSTAR=5.0 3 T I ‘7 I 2.5- - ‘53 8 2- - >\ 8 8 8 _ E 15- .73 E ”:3 200.08 400.0ft 0.5 _ 800.08 6.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 ABAR (f) ASTAR=10.0, GSTAR=10.0 Figure 4.1 Fundamental Frequency as a Function of ABAR (continued) 58 h) 2.5- - a O- 8 2» 1 5‘ 5 200.08 8 i 1.5- - .3 400.08 0 e g 1- _ t: 12 800.0ft 0.5- - 6.4 05 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 ABAR (g) ASTAR=3.0, GSTAR=2.0 3 . . . . 2.5- - 2.“? 8 2— _ >\ 8 g 12’ u. 15- — E E E 20008 g 1— . 5: 400.08 0" 800.08 6.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 ABAR (h) ASTAR=3.0, GSTAR=5.0 Figure 4.1 Fundamental Frequency as a Function of ABAR (continued) 59 3 4 . . 4 fii. . 2.5- — a D- 8 2- - >8 8 3 3 - u- 15- E E 5 1- - £1 200.08 400.08 0.3 - 800.08 ‘64 05 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 ABAR (i) ASTAR=3.0, GSTAR=10.0 Figure 4.1 Fundamental Frequency as a Function of ABAR (continued) 7 . . . . Deck: 200.08 6 /— _. Deck: 400.08 Deck: 800.08 a 5~ - § / "’ I U 1 63 4- .......... 9‘ ............................ ...... _ 6 > , ———Rib: 200.08 E 3 y '-———— Rib:400.0ft _ g ”4— Rib: 800.0ft 2 2- _ 1— - 6.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 ABAR (a) ASTAR=1.0, GSTAR=2.0 7 . . . . . . Deck: 400.0 ft Deck: 800.0 ft N & E U! 6 1.1.1 + 5‘ :1 \ ‘———Rib:400.08 {U E 2_ Rib: 800.08 - 1— - 6.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 ABAR (b) ASTAR=1.0, GSTAR=5.0 Figure 4.2 Maximum Dead Load and Earthquake Stresses Versus ABAR Max(DL+EQ) Stress/Fa Max(DL+EQ) Stress/Fa 61 7 4 4 4 . 6 /——Deck:200.0ft _ /——Deck:400.0ft 5_ Deck:800.0ft 4 >. ‘P- ' 3_ \P—Rib: 200.08 q ‘\ \——4— Rib: 400.08 2 \————Rib:800.08 1— . 6.4 05 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 ABAR (c) ASTAR=1.0, GSTAR=10.0 7 . 4 6- _ 5— _ 4- ... l ' \\ l— Rib: 200.08 \ Rib: 400.0 8 - Rib: 800.0 8 1 p<> A 0.5 0.6 0.7 0.8 0.9 l 1.1 1.2 1.3 1.4 ABAR (d) ASTAR=2.0, GSTAR=2.0 Figure 4.4.2 Maximum Dead Load and Earthquake Stresses Versus ABAR (continued) Max(DL+EQ) Stress/Fa Max(DL+EQ) Stress/Fa 7 r 6 . _ 5 ~ _ 4 _ Deck: 200.0 ft _ —— Deck: 400.0 ft 4 F 3 Kick: 800.08 _ 62 \ l _ \ \.________. Rib: 400.0 8 _ Rib: 800.0 8 {0.4 0:5 0:6 0:7 0:8 0:9 1 1:1 1:2 1:3 1.4 ABAR (e) ASTAR=5.0, GSTAR=5.0 7 6- .. 5" .1 4 _ /— Deck: 200.0 8 _ Deck: 400.0 8 Deck: 800.0 ft ’1 ’ Rib: 200.08 \ \————Rib:400.0_ft Rib: 800.0 ft QC) 0.7 0.8 0.9 l 1.1 1.2 1.3 1.4 A O 111 ABAR (f) ASTAR=10.0, GSTAR=10.0 Figure 4.2 Maximum Dead Load and Earthquake Stresses Versus ABAR (continued) Max(DL+EQ) Stress/Fa Max(DL+EQ) Stress/Fa 63 /r-— Deck: 200.0 ft _ 7' y/—— Deck: 400.0 8 / r———- Deck: 800.0 8 *. 7 \—Rib: 200.08 \ ‘— Rib: 400.08 \_—— Rib: 800.08 h-I l . 0.7 0.8 0.9 l 1.1 1.2 1.3 1.4 oCD A 0 111 o O\ ABAR (g) ASTAR=3.0, GSTAR=2.0 7 . . . . . 6 - - 5 ._ _. 4 - _ 3 . /— Deck: 200.0 8 _ ,./ /.»——— Deck: 400.08 2 Deck: 800.0 ft 1 _ b ")4 Rib: 200.0 8 _ 4— Rib: 400.0 8 Rib: 800.0 8 64 0.5 06 07 0.8 0.9 1 1.1 1.2 1.3 1.4 ABAR (h) ASTAR=3.0, GSTAR=5.0 Figure 4.2 Maximum Dead Load and Earthquake Stresses versus ABAR (continued) Max(DL+EQ) Stress/Fa /¥ Deck: 200.0 8 _ / /—— Deck: 400.0 ft 90 0.5 0.6 0.7 . 0.8 0.9 1 1.1 1.2 1.3 1.4 (i) ASTAR=3.0, GSTAR=10.0 Figure 4.2 Maximum Dead Load and Earthquake Stresses versus ABAR (continued) 5. COMPARISON OF TIED AND DECK TYPE BRIDGES 5.1 General In this Chapter, the response to seismic excitations of T-bridges is compared to that of D-bridges. The motivation to conduct this comparison is that most of the past work has been done on D-bridges. Relatively little research has been done on T-bridges. It would be of interest to compare the two types. The D-bridge model is illustrated in Figure 5.1. The: in-plane structure has a parabolic arch represented by a series of curved beam elements. The deck is represented by a series of straight beam elements. The columns connecting arch and deck are represented by truss elements. The bridge has eight panels of equal length. The arch is hinged at the supports, and the deck has roller supports. The cross-sectional area and the moment of inertia are assumed to be the same for each set of elements. The D—bridge model (Figure 5.1) differs from the tied bridge model (Figure 2.1) in the following ways: 1. Most conspicuously, they differ in the location of the roadway relative to the arch. The deck of the D-bridge is above the arch, that of the T-bridge is beneath the arch. The arch and deck are connected at the arch crown in the D-bridge case. They are connected at the supports in the T-bridge case. 2. The D-bridge has a hinged rib, i.e., there is zero moment at either rib 65 66 supports. In the T-bridge case, in general, the rib has zero moment at the supports only if the deck beam has zero bending stiffness. That is due to the continuous connections between the rib and deck at the supports. 3. The deck of the D-bridge is on rollers. Under dead load, it has zero axial force. The following is a comparison, first of the dead load response, then of the seismic response of the two bridge types, with the same set of reference parameters (set D): L=750.0 ft, H/L=0.175, M=4.80 kips/ft, Mr/M = 0.265, GAMMA=5.25, (L/R)C = 200.0, ABAR=0.688, C,/R,=1.275, Cd/Rd = 1.275. The varying parameter is RHO; the ratio of deck bending stiffness to rib bending stiffness. The range over which RHO is varied is: 0.0, 0.1, 0.2, 0.5, and 1.0, although, any values of RHO larger than 0.5, are probably of little practical interest in the case of D-bridges. At this point, it is important to mention which dimensional parameters change with RHO, and which remain fixed. For the data presented, the rib bending stiffness remains constant. With increasing RHO, the deck bending stiffness increases, and therefore, the total stiffness increases. Also, for the data presented the area of the rib increases. Since ABAR is constant, the area of the deck also increases, and therefore, the total area increases. Summarizing, in the results presented in this chapter, with increasing RHO, 1d, Ad and A, all increase, and 1, remains constant. 5.2 Fundamental Natural Frequency Table 5.1 is a listing of the fundamental frequency of tied and deck type arched bridges for different values of RHO. For both bridge types the fundamental frequency increases with RHO. This is expected since the total structural stiffness increases with RHO. What seems unexpected is that the 67 fundamental frequency of the tied bridge is higher than that of the deck bridge. Initially, it is expected that the deck type bridge has higher fundamental frequency, because it corresponds to a tie-bridge with an infinite axial stiffness of the deck (tie). It seems superficially to be a stiffer structure, and thus should have a higher frequency. However, the results show that the deck bridge has a lower fundamental frequency than the tie bridge. The reason for this lies in the support conditions. For the horizontal degrees of freedom at the deck level, the D-bridge is more flexible than the T-bridge and hence has lower frequencies. In an attempt to prove this, the mass is removed from the deck. With mass on the rib only, the two bridges are similar for the horizontal degrees of freedom. Actually, the D-bridge is a little stiffer (both rib supports are hinged). As expected, the fundamental frequencies of the two bridges are very close. The frequency of the D- bridge is slightly higher (0.505 hetrz as opposed to 0.504 hertz for the T- bridge). 5.3 Dead Load Response Figure 5.2 is a plot of the maximum dead load stresses among all point on the arch rib, similarly for the deck, for both tied and deck type bridges, versus RHO,i.e., the deck to rib bending stiffness ratio. The location of the various maximum stresses are noted in Tables 5.2 and 5.3 for the deck and the rib, respectively. For both bridge types, the maximum dead load stresses gradually decrease with increasing RHO values. This can be explained by the fact that both the total bending stiffness and the total area increase with increasing RHO, as mentioned earlier. Consequently, the stresses decrease. Thus, these trends seem obvious. But the relative responses for the various cases at given values of RHO are informative. 68 5.3.1 Stresses in the Deck In the D-bridge case, the dead load stresses in the deck are small because the deck has roller supports, i.e., there is no axial load. All the stresses are due to bending. At zero value of RHO, there is no moment in the deck, the stresses are zero. For a positive value of RHO, there are some bending stresses, though small. As RHO increased from 0.0 to 0.1, the deck stresses increase for both types. Further increase in RHO results in a decrease in deck stress for the T-bridge. For the D-bridge, the stress remains essentially constant. The deck dead load stresses in the T-bridge are much larger than those in the D-bridge. Those large stresses are mostly due to the axial tensile force in the T-bridge deck. The percentages of the T-bridge deck dead load stresses in bending are generally twice those of the D-bridge, as shown in Table 5.2. However, their actual magnitudes are much smaller than those due to axial force. Of course, the equivalent of that axial force is provided by the rib support horizontal reaction in the case of the D-bridge. 5.3.2 Stresses in the Rib It is noted from Figure 5.2 that the stresses in the rib for the T-bridge and the D-bridge are quite close. This is because the rib behavior for the two bridges are typically of thearch kind. Although the sources of the horizontal thrust differ, the magnitude and effect are similar. 5.3.3 Sum of Deck and Rib Stresses Also in Figure 5.2 is plotted the sum of the maximum dead load stresses among all points in the deck and among all points in the rib. For 69 both bridge types, as RHO increased from 0.0 to 0.1, rib stresses decreased by approximately the same amount as the deck stress increase, on the order of 5 percent. Since the rib stresses are similar for both bridges, the difference of the sum is similar to the difference in the deck stresses, discussed in section 5.3.1. The difference lies in the axial tensile force in the deck of the tie bridge. 5.4 Seismic Stresses The maximum seismic stresses of both T-bridge and D-bridge are computed by use of the AASHTO spectrum. The maximum stresses and their location are listed in Tables 5.2 and 5.3. The results are plotted in Figure 5.3. The dead load stresses are not included in the seismic stresses discussed herein. 5.4.1 Stresses in the Deck The seismic deck stresses in the D-bridge are not greatly different from those in the T-bridge, albeit smaller (Figure 5.3). As RHO increases from 0.0 to 0.1, the deck stresses increases for both bridge types. For RHO bigger than 0.1, the rate of increase drops rapidly. It is noted that for larger values of RHO, the values of the deck stresses for the two bridge types get closer. 5.4.2 Stresses in the Rib As the deck stresses in the T-bridge are larger than the deck stresses in the D-bridge, the rib stresses in the T-bridge are smaller than the rib stresses in the D-bridge. For both bridge types, as RHO was increased from 0.0 to 0.1, the rib stresses decreased by approximately 10 percent. For RHO 70 bigger than 1.0, the rate of decrease drops. The stresses also get moderately closer with larger RHO values. It is interesting to look at the sum of the rib and deck stresses. 5.4.3 Sum of Deck and Rib Seismic Stresses The sum of the seismic stresses in the deck and the rib for the two bridge types is close, with the D-bridge total stress moderately larger. 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