Y? a LIBRAR Y Michigan Saga University "- This is to certify that the thesis entitled An Experimental Study into The Nonlinear Behavior of A Parabolic Arch Bridge presented by Bruce Frazier Henley has been accepted towards fulfillment of the requirements for Department of Master of degreein Civil Engineering Science £\\ttalr\l -" / k l Major professor ./ ” 7 / - ,l' Date November 16, 1977 0-7639 AN EXPERIMENTAL STUDY INTO THE NONLINEAR BEHAVIOR OF A PARABOLIC ARCH BRIDGE By Bruce Frazier Henley A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE Department of Civil Engineering 1977 ABSTRACT AN EXPERIMENTAL STUDY INTO THE NONLINEAR BEHAVIOR OF A PARABOLIC ARCH BRIDGE By Bruce Frazier Henley An experimental study of the nonlinear behavior of a parabolic arch bridge model, up to the point of buckling, is reported. The bridge model is 96 inches long, and has a rise of 16% inches, and the ribs are 4 inches center to cen- ter and connected by lateral bracing beams at every 6 inches on the horizontal projection. Vertical and lateral loads were applied in various com- binations to simulate actual load conditions. By applying the load increments in a monotonic manner, the behavior of the bridge could be observed and measured. It was found that the symmetric and antisymmetric buck- ling modes are in close proximity of each other, but that the symmetric mode appears to be the failure mode. Several techniques were used to deduce the buckling load, and the load-displacement asymptote and Southwell plot methods worked reasonably well for both modes. The decrease of lateral stiffness due to vertical loads was also studied. However, the buckling load extrapolated from these results appears to be too low. ACKNOWLEDGMENTS The work reported herein has been supported by the Division of Engineering Research, and also by the Department of Civil Engineering. This work constitutes the author's thesis, which has been written under the direction of Dr. Robert K. Wen, to whom deepest gratitude is extended for his guidance and instruction. The author wishes to express his appreciation to his fellow graduate students, Tom Heck and Jose Lange for their assistance in the collection of the data. Special gratitude is extended to the Machine Shop of the College of Engineering for their assistance in fabricating the experimental apparatus. ii TABLE OF CONTENTS Chapter I INTRODUCTION ..................................... 1.1 Object and Scope ............................ 1.2 Notation .................................... II DESIGN AND CONSTRUCTION OF MODEL ................. 2.1 General ..................................... 2.2 Demensional Analysis ........................ 2.3 Properties of the Model ..................... 2.4 Computer Analysis ........................... 2.5 Test Apparatus and Construction ............. .1 Arch Ribs and Bracing ................ 2.5.2 End Supports ......................... 2.5.3 Panel Joints ......................... 2.5.4 Loads ................................ 2.5.5 Measurement Equipment ................ III TEST PROCEDURES .................................. 3.1 Vertical Load-Only Tests .................... 3.2 Combined Lateral and Vertical Load Tests.... IV RESULTS AND DISCUSSION ........................... 4.1 General ..................................... 4.2 Behavior Under Vertical Loads-Only .......... 4.2.1 Overall Behavior ..................... 4.2.2 Crown Behavior ....................... 4.2.3 Quarter Point Behavior ............... 4.3 Behavior Under Combined Lateral and Vertical loads ....................... 4.3.1 Overall Behavior ..................... 4.3.2 Crown Behavior ....................... 4.4 Buckling Load ............................... 4.4.1 Vertical Load-Only ................... 4.4.2 Combined Load Testing ................ 4.5 Discussion .................................. 4.5.1 Differences Among Test Results ....... 4.5.2 Comparisons with the Results of Others ............................... iii 21 21 21 21 22 32 32 32 Table of Contents (Continued) Chapter Page V CONCLUSION ....................................... 46 LIST OF REFERENCES ............................... 49 iv LIST OF TABLES Table Page 2-1 Dimensional Analysis Parameters and Values ...... 7 2-2 Properties of the Ribs and Bracing .............. 8 4-1 Tabulation of Experimental Buckling Loads ....... 41 Figure 2-1 2-3 2-4 2-5 4-1 4-2 4-3 4-4 4-5 4-6 4-7 4-8 LIST OF FIGURES Arch Bridge Test Model ........................... Front View of Model .............................. End View of Model ................................ Bracing .......................................... Hinges ........................................... Panel Joint ...................................... Displaced Shape of One Rib Under Typical Vertical Load-Only Test ....................... Load—Displacement Curve for the Crown Under Vertical Loads-Only ........................... Load-Displacement Plot for the Antisymmetric Component of A2 ............................... Load-Displacement Plot for the Symmetric Component of A 2 ............................... Displaced Shape of One Rib Under Typical Combined Load Tests ........................... Load-Displacement Plots for the Crown Under Combined Load Tests (q=0.0,1.0, 1.8,2.7,I3{7).. Load-Displacement Plots for the Crown Under Combined Load Tests (q=0.7. 1L3.2-2. 3-0) ...... Load-Displacement Plots for the Crown Under Combined Load Tests (q=0.3,31u5.2 5.3 3) ...... Lateral Stiffness versus Vertical Load ........... vi Page 10 12 12 13 13 15 23 25 26 27 28 29 30 31 36 List of Figures (Continued) Figure Page 4-10 Southwell Plot for the Crown .................... 37 4-11 Southwell Plot for the Antisymmetric Component of A2 ......................................... 38 4-12 Southwell Plot for the Symmetric Component of A2 ......................................... 39 vii I. INTRODUCTION 1.1 Obiect and Scope Arches are one of the oldest type of structures. In the days of the Roman Empire, arch bridges were built with masonry materials. Due to a lack of analytical knowledge, the design and construction of those structures were essen- tially based on experience. Because of their bulkiness, elastic stability usually was not a problem. The strength of masonry, however, limited the size and span length of those old structures. The introduction of steel and the development of struc- tural mechanics in the 18th Century greatly extended the capability of structural engineers. Arch bridges of increas- ingly long spans were built. For example, the Kill Van Kull bridge in New York has a 1,675 foot span. The more recent advent of the computer has further en- abled the engineer to make increasingly more precise calcu- lations. There is a tendency to use lighter and more slen- der construction. As the span lengths have increased, their relative widths have decreased. This produces a long slender arch bridge, which renders itself more susceptible to geometric instabilities. A study of the behavior of long slender arch bridges, up to or near the buckling load, could show the significance of the bracing upon the overall 1 stability of the arch bridge. To date, the buckling load of a single parabolic arch rib has been studied both analyti- cally and experimentally by a number of investigators (see, for example, References 5 and 7). Available studies of the entire arch bridge, a structural system that consists of two ribs with one of several commonly used bracing patterns, have been rather limited in their scope and numbers. It is this lack of information concerning the nonlinear behavior, to the point of buckling, that has stimulated this investigation, the ultimate goal being to aid in the design of the bracing system to provide a more stable bridge. The present study is limited to investigating the non- linear behavior of a through type parabolic arch bridge with Vierendeel bracing. The investigation is to be carried out by experimental study of a long slender arch bridge. This is accomplished in essentially two phases: First, the design and construction of a model basedcniprototype designs; sec- ondly, the execution and analysis of a series of loading tests. With a knowledge of the displacements at the crown and quarter points, the buckling load can be determined with- out damaging the model. Two loading conditions were employed in this study. First, a vertical load only test sequence, simulating live load plus dead load, was performed to obtain a vertical load— lateral displacement relationship, hereinafter to be called the vertical load-displacement relation. Secondly, a com- bined vertical and lateral load, simulating wind load, test series was performed to provide a lateral load-lateral displacement relation, similarly this will be termed the lateral load-displacement relation, with the vertical load as a parameter. Reduction of the data obtained provided reasonable agreement as to the buckling load when estimated by the asymptote estimation of the load-displacement plots and by the Southwell plots. The decrease of lateral stiffness due to vertical loads was also studied. However, the buckling load extrapolation from these results appears to be too low. In the following chapters, Chapter II contains a des- cription of the dimensional analysis, design, computer analy- sis, and construction of the laboratory model. The test pro- cedures are outlined in Chapter III. The results are pre- sented in Chapter IV along with a discussion of their mean- ing. A summary of this report is then contained in Chapter V. 1.2 Notation The symbols used herein are listed below with their def- initions: A Cross sectional area (in.2) C Torsional constant (in.4) E Young's modulus (lbs./in.2) H Rise of the crown above the hinges (in.) h Width of bridge (c. to c. of arch ribs) (in.) Ixx Major axis moment of inertia (in.4) Iyy Minor axis moment of inertia (in.4) J Polar moment of inertia (in.4) Kh Lateral stiffness of bridge (1bs./in.) L Span length (in.) S w A A1,A2,A3,A 2 * A2 A** 2 4 Vertical load (lbs./inch of bridge) Panel width (in.) Lateral load (lbs./inch of bridge) Lateral displacement (in.) Dial gages readings along bridge Asymmetric component of recorded A2 Symmetric component of recorded A2 Terms which appear with a bar (') over them represent quantities associated with the bracing, and unbarred terms refer to quantities associated with the arch ribs. II. DESIGN AND CONSTRUCTION OF MODEL 2.1 General Before a realistic model of an arch bridge can be constructed, it is important to first obtain the values of certain parameters of existing "prototype" bridges. Then with the aid of dimensional analysis, the ranges of values of the properties of the model can be formulated. After the shapes and other properties of the model have been estab- lished, a numerical solution can be obtained to get a feel for the forces and displacements produced by the applied loads. After the numerical solution showed what appeared to be acceptable behavior, the physical model was then construc- ted. The following three sections will deal more specifi- cally with the above mentioned topics, along with the test set-up. 2.2 Dimensional Analysis The fundamental objective of dimensional analysis is to reduce the number of independent variables, and establish a set of dimensionless variables that will ensure proper simili- tude between the physical systems (1). The dimensionless parameters chosen for this investigation are listed in Column (1) of Table 2-1. The next step is to compute the "practical" range of values that would be used to create a model. This was 5 6 accomplished by computing the values of the dimensionless parameters corresponding to four real arch bridges: the Cold Spring Canyon bridge near Santa Barbara, California, the Ohio State Route 8 bridge near Cleveland, Ohio, the South Street bridge over I-84 near Middlebury, Connecticut, and the Colorado River bridge on Utah State Route 95. The ranges of such values for the above mentioned bridges are listed in column (2) of Table 2-1. 2.3 Properties of the Model The dimensionless parameters and the ”practical" ranges of values presented above would allow the creation of an in- finite variety of model arch bridges. But if one or more of the independent variables can be solved for by fixing the value of say just one independent variable, the number of correct solutions for the remaining unsolved variables will be greatly reduced. Since the width of a testing frame in the Structures Laboratory and the length of a sheet of alumi- num are both 96.0 inches, this was chosen for the length (L) of the bridge. Then by using Columns (1) and (2) in Table 2-1, direct substitution of L will yield values for the rise (H), width (h), bracing spacing (S), and the areas (A) and (A). The remaining values are dependent upon the choice of the cross sections of the rib and bracing. This turns out to be an iterative process to obtain the most acceptable solu- tion. Presented in Table 2-2 is the final choice for the sec- tional properties for both the ribs and the bracing. The corresponding values of the dimensionless parameters are listed in Column (3) of Table 2-1. It can be seen that not TABLE 2-1 Dimensional Analysis Parameters and Values Dimensionless Desired Values of Tokarsz ‘ Parameters Range Model Values 0 0.30 0.32 x 0.32 x H/L 0.13 4 0.17 0.17 x 0.2 S/L 0.05 4 0.15 0.063 x 0.067 x h/L 0.037 4 0.13 0.042 x 0.14 IXX/Ixx 0.0095 4 0.035 0.026 x 0.032 x Iyy/Iyy 0.0015 4 0.014 0.13 0.00052 A/A 0.10 4 0.25 0.24 x 0.067 6/0 = 3/3 (0.35 4 3.5)x10'4 0.087 0.0011 x Iyy/Ixx 0.10 4 0.65 0.12 x 0.016 J/Iyy 2.5 4 10.5 9.39 x 1.02 xxx/A2 1.7 4 5.1 0.74 x 0.74 x Ll/K’ 197.1 4 429.5 217.8 x 111.8 GC/EIyy 1.79 4 7.50 3.56 x 0.75 w/Eyfir‘ (1.4 4 2.6)x10"7 N.A. q/E,gr' (1.3 4 2.2));10-6 2 * ---(x) indicates that the value is within the range of existing bridges. 8 TABLE 2-2 Properties of the Ribs and Bracing Span Length, L = 96.0 in. Rise, H = 16.25 in. Bracing Spacing, S = 6.0 in. Width, h = 4.0 in. (Cross Section) rib, = 0.75 deep x 0.26 in. wide Area, A = 0.1943 in2 Torsional Constant, C = 3.40 x 10‘3 in4 Moments of Inertia, Ixx = 9.10 x 10"3 in4 ryy = 1.08 x 10-3 in (Cross Section) brace, = 0.25 deep x 0.188 in. wide Area, A = 0.0469 in2 Torsional Constant, C = 2.97 x 10'4 in 4 Moments of Inertia, ixx = 2.44 x 10'4 in4 Iyy - 1.37 x 10‘4 in4 all of the dimensionless values fall in the desired ranges.' This is due to the necessity of having to use solid cross- sections for both the ribs and the bracing, while the proto- types used box sections for the ribs and box sections or stan- dard rolled sections for the bracing. From Column (1) of Table 2-1, it can be seen that the loads on the structure are directly related to the strength of the structure. Of the three basic construction materials to choose from, steel, aluminum, and plastic, aluminum was cho- sen. It provided more linear and less creep behavior than plastics, and required smaller loads than did steel. Shown in Figure 2-1 is a schematic drawing of the model used. 2.4 Computer Analysis Before construction of the actual model, a numerical so— lution for the linear elastic behavior of the arch bridge was obtained. The SAP IV (3) finite element program was used. The computer model simulated in every respect the physical model, except that between each pair of panel joints the arch ribs were approximated by two straight equal length beam ele- ments, since SAP IV has no curved beam elements in its library. By considering single fold symmetry about the crown, only one half of the structure need be studied. The boundary condi- tions consisted of allowing the crown to displace vertically, laterally, and to twist about its longitudinal axis. At the support of each rib, only translations and axial twist were restrained. The axial twist was provided by adding a tor- tional boundary element at the end of each rib, and tangent 10 .Hmpoz ummH mwpwum nou< HuN MMDUHh :wuquo ® mamcmm wv swam 14 m. w. Zdflfiwuwmvflhé ’ y... 11 to the rib slope at that point. The computer solution indicated that when the model was subjected to a lateral loading (w) of 0.71 lbs./in., corres- ponding to a 100 m.p.h. wind, the lateral displacement at the crown would be 2.00 inches, giving a lateral stiffness (Kh) of 0.353 lbs./in.. It should be noted that since SAP IV is limited to linearly elastic structures, Kh is then indepen- dent of any vertical load applied to the model. Indeed, come puter solutions by SAP IV cannot reflect nor predict any of the nonlinear behavior exhibited by the physical model. In this sense, the computer solution has only limited applica- tion to this study, which as will be shown later, was domi- nated by nonlinear behavior. 2.5 Test Apparatus and Construction 2.5.1 Arch Ribs and Bracing Figures 2-2 & 3 show the arch bridge model. The ribs were cut from a single sheet of 2024-T3 aluminum with a re- 7 psi, and a Poisson's ratio of ported Young's modulus of 10 0.32. The value of Young's modulus was confirmed by a ten- sion test performed on a sample coupon. The ribs were cut to shape, instead of being bent, to avoid the stresses of bend- ing. The bracing, item (1) in Figure 2—4, were machined from a single bar of 2024—T3 aluminum. This figure also shows the connections used in attaching the bracing to the ribs. Holes were drilled through the ribs so that the panel "joint col- lars", item (B) in Figure 2-4, could be securely affixed with the bracing attached to the collar. 12 FIGURE 2-2 Side View of Bridge Model FIGURE 2-3 End View of Bridge Model l3 FIGURE 2-4 Bracing FIGURE 2—5 Hinges 14 2.5.2 End Support Shown in Figure 2—5 is the end support assembly. The hinge assembly must satisfy the previously mentioned bound— ary conditions, and be able to provide a stable platform that will not displace during the course of testing. The hinges were designed similar to a universal joint that is commonly used in automobile transmissions. The only displacements permitted were rotations about the two pins. The base plates that held the two sets of hinges were leveled to within 1 mm of each other with the aid of a sur- veying theodolite. 2.5.3 Panel Joints On the arch bridge, the panel joint is where the bracing and the deck hangers frame into the rib. Figure 2-6 shows a typical panel joint from below, the rib (A), bracing (B), and the load hangers (c). The load hangers were designed to orient the vertical loads through the center line of the rib before and after displacements. This is important since any misalignment of the loads could create an eccentricity which would give rise to moments that would alter the behavior of the bridge. 2.5.4 Lgads The loads that are applied to the structure are done so by using combinations of solid lead cylinders, and canisters containing various amounts of lead shots. The solid lead cyl- inders, which were used only for the vertical loads were formed in 16 oz. beer cans, and came in weights of about 3.50, 6.75, and 1.75 lbs. each. These cylinders had screw hooks FIGURE 2-6 16 attached at both ends so that several cylinders in combina— tion could be connected to any given panel joint. The lead shot, 0.14 inch diameter balls, was held in separate canis- ters with a wire hook attached to the lid so that it too could be added in tandem with the solid cylinders. For ease of construction, the solid lead cylinders were not formed to an exact weight. Thus, the lead shot canisters were also used to equalize the vertical loads on the panel joints. The lead shot canisters were oriented so that their ring tab open- ings remained on top. This facilitated the addition of the lead shot load increments into the canister without having to take the canister off of the structure for loading which would disturb the system. The lateral loads which simulated wind loads, were at- tached to the "leeward" rib of the bridge. Using monofila- ment nylon fishing line, the line was passed through a nylon pulleywhere the lateral load is transformed from the lead weights. Figures 2-2 & 3 show the vertial loads (A), and the lateral loads (B) in place. 2.5.5 Measurement Equipment Four dial gages, whose least count were 0.0005 inches, were stationed along the bridge. From one support, the gages were located 9, 27, 48, and 69 inches, and the deflections measured were labeled, respectively, A1, A2, A3, and A'z. where A3 corresponded to the crown, and A2 and A'2 were at about the quarter points. Hereinafter, they will be re- ferred to as the quarter point displacements. With this ar- rangement of dial gages, the deformed behavior of the model could be monitored. l7 III. TEST PROCEDURES Two test programs were conducted: First, using the vertical loads only, and secondly, using both vertical and lateral loads. These tests are described in the following sections. 3.1 Vertical Load-Only Tests As mentioned previously, the vertical loads which repre- sent the dead load plus the live load were suspended from the panel joints. The panel loads were equal and the arches should essentially be in a state of uniform compression. In this test, at each vertical load level, the lateral displace- Imaus were, of course, due to the unavoidable imperfection in the system. Initial tests indicated that in order to obtain meaning- ful results thechanges in vertical loads must be monotonic. That is, alternate loading and unloading should be avoided. This required a programming of the loading procedure. Since the lead shot canisters were not large enough to hold thexmdr ume of lead needed for the maximum load required, it became necessary to use a "piece-wise-continuous" approach to the vertical loading. This approach consisted of applying a base load set to be equal to several load increments below the max- imum load of the preceding series of tests, excepting, of course, the initial load segment for which the base load is 18 l9 zero. Increments of lead shot were then added to the base load monotonically. With an overlapping of data points, it is then possible to coalesce the data so that it would re- present the structural behavior under a monotonic loading. In the course of developing the above test technique, it was found that the behavior of the model bridge was very sensitive to the way in which the vertical loads were applied to it. Care must be exercised to align the load to avoid ec- centricities, to smoothly transfer the loads onto the model to avoid any impact loading, and it also became clear that the base loads could not usually be applied in a single step. The practice of adding the base loads in two or more steps was then adapted to correct the latter problem. The vertical base loads had to be applied in syStematic manner to prevent the model from behaving erratically. These loads were applied in pairs, with two panel joints per station and then symmet- rically about the crown, starting at the supports and working towards the crown. The vertical load increments, consisted of 0.35 lbs. per canister of lead shot, were added in a similar manner as were the base loads, except that only one canister was incremented per side at a time. To avoid an unbalanced loading situation during the load incrementation process, the loads were simul- taneously added to the panel joints that are symmetric with respect to the crown, but on different ribs. 3.2 Combined Lateral and Vertical-Load Tests This set of tests were designed to evaluate the effects of the vertical loads upon the lateral stiffness of the 20 structure when subjected to simulated wind loads. This re- quired, first, the establishment of a vertical load, as de- scribed previously, and secondly, a monotonically increasing series of lateral loads. Using the dial gage readings of the horizontal displacements measured from the tests, a fam- ily of load-displacement curves can be constructed to show the effects of the vertical load on the lateral stiffness. The lateral load increments consisted of measured amounts of lead shot, weighing either 0.22, 0.11 or 0.06 lbs., added in a fashion, symmetric with respect to the crown, pro- gressing from the support towards to the crown. The differ- ences in the increments used was brought about by the change in responsiveness toward lateral displacements as.the vertical loads were increased. When the vertical loads were small, the larger increments provided a good linear data spread. But with the larger vertical loads, smaller lateral load in- crements were required to produce similar results. When the lateral displacement reached about 0.75 inches, these tests were stopped in order to prevent any damage from occuring to the structure. IV. RESULTS AND DISCUSSION 4.1 General The results of this investigation will be presented in terms of both the overall behavior of the model and the be- havior of the crown and for a point close to the quarter point for both loading systems employed. With the informa- tion obtained from the behaviors of the crown, estimates can be made of the buckling load of the model. Comparisons will then be made between the results obtained from this investi- gation 11) those of certain previous investigations. 4.2 Behavior Under Vertical Loads-Only 4.2.1 Overall Behavior If one considers a simply supported beam-column, it is well-known that its first mode of buckling will appear as a half sine wave; this may be called the symmetric buckling mode. At an axial load four times larger than that required to cause the beam-column to buckle in the symmetric first mode, the second mode will appear. This mode appears as a full sine wave, and may be referred to as the antisymmetric buckling mode. Thus, in the case of a beam-column, the buck- ling loads corresponding to the symmetric and antisymmetric modes are quite distinct in the sense that the latter is four times the former. Indeed, for this reason, it has little en- gineering significance. 21 22 However, for arch bridges, this is not necessarily true. Due to the curvature of the structure and possibly the brac- ing,the antisymmetric mode may correspond to a lower load level than that for the symmetric mode. There also exists the possibility that the buckling loads corresponding to the two modes could be quite close to each other so that some form of interaction nugr occur prior to entry into either buckling mode as the loads are increased. Shown in Figure 4-1 is the displaced shape of one rib of the bridge model for four typical vertical load tests. The dial gages were labeled A1, A2, A3, and A'z, and their locations have been given in Chapter II. It is shown that the model appears to be deforming in a manner that is a com- bination of a symmetric and antisymmetric modes. If the de- formation was antisymmetric, there should be no displacement at the crown, and if it were symmetric, A'z, should not have 2 (at mirror point of A'z, and A3 should be greater than A'Z). Since the model was not tested to the opposite sign of A actual collapse, the exact nature of the final failure mode is not known. It can be speculated that the appearance of the combina- tion of the two modes could be the result of (A) that the ini- tial imperfection (eccentricities) contains substantial com- ponents in both modes and (B) that the critical loads for the modes are close to each other. 4.2.2 Crown Behavior If the symmetric mode was the mode of failure for the model, a study of the behavior of the crown could enable one 23 .05- .04—4 . q = 18.204 (lbs./in.) .03wn .02- .01'- DISPLACEMENTS (in. ) -.Ol«— —.02-4 - -.03- I l l r A1 A2 A3 A 2 DIAL GAGE STATIONS ALONG THE MODEL FIGURE 4—1 Displaced Shape of the One Rib under Typical Vertical Load-Only Tests. 24 to predict the critical buckling load. Should a structure be perfectly built and loaded such that there are no eccentrici- ties, it could be deformed without the occurrence of buckling. There could be no lateral displacement due to vertical loads alone. But every structure contains some degree of initial imperfection. This could be in the form of members not being straight, the structure not being plumb, play in the connec- tions, or the loads may be applied with eccentricities. As the loads are gradually increased, a linear relation- ship will develop between the applied loads and the lateral displacements. After a certain level of loading has been reached, all subsequent loads would have a nonlinear relation- ship to the lateral displacement, with each equal load increment creating successively larger displacement increments. It is this nonlinear behavior that reduces the overall stiffness of the structure. When the stiffness is reduced to zero, the structure is said to have buckled. Shown in Figure 4-2 is a plot of the vertical load ver— sus lateral displacement at the crown. The data points form a smooth curve, with the exception of the first four points. The deviation of these four points from the expected path, is probably due to the fact that the structure was in the pro— cess of adjusting to the initial slacks existing in the vari- ous joints and hinges. After the structure has "removed the slacks", it can be seen that a linear region extends to about 2.0 lbs./in.. Beyond 2.0 lbs./in., the structure progresses into a nonlinear range. The distribution of data points is seen to be quite ‘1, VERTICAL LOAD (lbs./in.) 8.0 9.0 llLlJll 7.0 5.0 6.0 4.0 2.0 3.0 1.0 25 {-estimated asymptote IlnLll LILLIJLJ I I I l I I r_ I I I I I .01 .02 .03 .04 .05 .06 .07 .08 .09 .10 .ll .12 A3, LATERAL DISPLACEMENT (in.) FIGURE 4-2 Load-Displacement Curve For The Crown Under Vertical Loads-Only 26 estimated asymptote 9 ‘ from Figure 4-2 d 8-————— ———_—-— 1 q 0 71 ’7 4 .fi 6.. ‘7 4 CD :3 d y! 5 Q . < . c5 *3 4 .- 23 8 E3 4 E? 3 ‘- gfl d 6* 2 1d IIIIiITTi'UU'l'I‘rTIT r -2 0 5 10 15 20 A: LATERAL DISPLACEMENT (in.) x 10'3 ’ FIGURE 4-3 Load Displacement Plot For The Antisymmetric Component of A2 27 ‘ estimated asymptote from ——_—— q, VERTICAL LOAD (lbs./in.) LJ I l l l I l T’ I I l 2 3 4 5 6 7 8 9 Af*, LATERAL DISPLACEMENT (in.) x 10"2 L FIGURE 4-4 Load-Displacement Plot For The Symmetric Component of A2 LATERAL DISPLACEMENTS (in.) .ch 28 .63-1 . . af0.000 (lbs./ina) w=l.984 (lbs./in.) DIAL GAGE STATIONS ALONG THE MODEL FIGURE 4-5 Displaced Shape of One Rib Under Typical Combined Load Tests w, LATERAL LOAD (lbs./in.) 29 ; v q V " V 1.5'- . q=0.o (lbs./in.) I v A 1 91:1.0 . V fi II V A 1.01— v o q d v ‘ . (i=1 8 . q :1 V O I o . «i=2 7 a /‘9 I . u / I . 4:3 7 i// _ 1 I l I 1 0.1 0.2 0.3 0.4 0,5 0,5 A3, LATERAL DISPLACEMENT (in.) FIGURE 4-6 Load-Displacement Plots For The Crown Under Combined Load Tests (91=0.O, 1.0, 1.8, 2.7, 3.7) w, LATERAL LOAD (lbs./in.) 30 l A L m n A l j Q=0.7(Ibs./In.) H U1 l A lLlelll [—4 O [AAJLIAJLLLI 0 U1 > 0’ L A l A U I I I I I 0.1 0.2 0.3 0.4 0.5 0.6 A3, LATERAL DISPLACEMENT (in.) FIGURE 4-7 Load-Displacement Plots For The Crown Under Combined Load Tests (‘1=0.7, 1.3, 2.2, 3.0) 31 1_5._ ‘ <1=0.8(lbs./in.) } .. j ‘1 A 1 ‘ d . -H d o A \ U) :3 1.0 - A - 8 : ... “‘7” .4 . / #4 A I . Ii 4 a ' \I 0 (1:3'3 3 ‘ .‘ 3. 1 Au/ 0 05- / q ‘ 1 I I I 0.1 0.2 0.3 0.4 0.5 0.6 A3, LATERAL DISPLACEMENT (in.) FIGURE 4-8 Load-Displacement Plots For the Crown Under Combined Loads (01=0.8, 1.5, 2.5, 3.3) 32 smooth which seems to justify the use of the segmented test- ing procedure that was discussed in the preceding Chapter. An extrapolation of this curve could lead to an estimate of the buckling load which will be discussed later. 4.2.3 Quarter Point Behavior Since the displaced form of the arch bridge ribs appears to be a combination of the symmetric and antisymmetric modes, the behavior of the quarter point should be investigated. If the displacements are a combination of the two predominant modes, their components could be separated. The following formulas were used to extract the two mode components: A** — A Sin (iii) = (0.773) A3 é Symmetric Component 2 3 ‘k _ 7W: ' . . A2 - 42 - 42 = Antisymmetric Component Presented in Figures 4-3 & 4 are the load-displacement plots of the extracted antisymmetric and symmetric components of A2 respectively. The general behavior can be seen to re— semble that of Figure 4-2, Figure 4-4 more so than Figure 4-3. With these plots, additional estimates of the buckling loads can be made. 4.3 Behavior Under Combined Lateral and Vertical Loads 4.3.1 Overall Behavior If one was to observe the horizontal projection of the deformed bridge subjected to lateral loads, it could be seen that the model would behave in a manner similar to that of a simply supported beam carrying a uniformly distributed load. In Figure 4-5, the generally symmetric shape of the deformed 33 bridge can be seen for typical tests using a combination of vertical and lateral loads. It was noted in testing, that there also existed a need for an initializing lateral load for the model to adjust to the slack in the system in addition to the vertical initial- izing load mentioned previously. Beyond this initializing load, the structure would behave linearly. The magnitudes of the displacement measured, as shown in Figure 4-5, are an order-of—magnitude larger than those that appear in Figure 4-1 for vertical loads alone. This would mean that factors such as imperfections and internal friction would have a smaller influence on the behavior under horizontal loads than that on the behavior under vertical loads only. 4.3.2 Crown Behavior Presented in Figures 4-6, 7 & 8 are plots of the lateral displacement versus lateral loads for given levels of vertical loads. It can be seen that for each of the vertical loads, there existed an essentially linear relationship between lat- eral displacement and the load. However, such results were unobtainable for vertical loads oflh()lbs./in., and greater, since upon application of the lateral load required to remove the initial slack, the structure seemed to be already in a state of impending collapse; i.e., no linear range of the load-displacement relation could be obtained. In Figures 4-6, 7 & 8, the slopes of the lines represent the lateral stiffness of the bridge at different levels of vertical loading. Plotted in Figure 4-9 are these stiffnesses versus vertical load level. It can be seen that as the 34 vertical load was increased, the lateral stiffness of the structure decreased. When the stiffness decreases to zero, buckling is said to take place. This will be discussed in the following section. 4.4 Buckling Load The results presented in this section are separated into two parts: Firstly, the results derived from the vertical load only testing and, secondly, the results obtained from the combined loading. 4.4.1 Vertical Load-Only As mentioned previously, buckling occurs when the stiff- ness of the structure is reduced to zero. When this happens, no further load increment is required to provide additional displacements. From Figure 4-2, the point of zero stiffness can be located by finding the asymptote of the load—displace- ment curve. The asymptote was estimated, by eye, to be about 8.0 lbs/in. . This approach, however, only holds if the buck- ling mode is of the symmetric type. In Figures 4-3 & 4, the plots of vertical load versus lateral displacements for A: and AS* can be seen. Since these displacements occur close to the one-quarter point, they could be used to evaluate the antisymmetric mode. The asymptote for the extracted symmetric mode is about 8.0 lbs./in.. No estimate is possible for the extracted antisymmetric mode since the curve has not leveled off noticeably. By using the data from Figures 4-2, 3 & 4, it is possible to construct the "Southwell plots" (6) for another estimation of the buckling load. The Southwell plot consists of plotting 35 A/q versus A. Figures 4-10, 11 & 12 present the Southwell PIOtS for A3» A: and 43* respectively. Normally, the South- well plot would yield a single straight line, the inverse of whose slope would be equal to the buckling load. However, in Figure 4-10 two distinctly different straight line segments are seen. The corresponding values of the estimated buckling loads from Figure 4-10 are, 11.80 and 9.17 lbs./in.. From Figures 4-11 & 12, the estimated buckling loads of the sym- metric and antisymmetric modes are 11.14 and 11.34 lbs./in., respectively. 4.4.2 Combined Load Testing Another way to determine the buckling load is to deter- mine the vertical load at which the stiffness under the ap- plication of lateral loads is zero;that is, where the curve representing the data points of Figure 4-9 will intersect the axis. The nature of the distribution of the data points is such that they seem to allow for more than one curve to fit. Therefore, two curves will be given here; first, a linear re- gression line, Line A, then a quadratic regression line, Line B. These can be seen in Figure 4-9 along with the SAP IV solution for the lateral stiffness that was described in Chap— ter 2. The latter solution agrees with the experimental data reasonably well at zero vertical load. But such analytical result based on a linear model is meaningless in the presence of substantial vertical loads. The values of the buckling loads yielded by lines A and B are 5.59 and 6.73 1bs./in., respectively. dA , BRIDGE STIFFNESS (le./in.) 36 0.50 \‘\ ISAP IV Solution q \\o o.\ o\\\ 0.333.; oR\ ‘Q\ \0 \\ 9N 3.\ 0167- ‘o\ \\ .. \\. \ Aj‘\\(:3 \ \, l E '3 Z '5 2 9., VERTICAL LOAD (lbs./in.) FIGURE 4-9 Lateral Stiffness versus Vertical Load 37 d B . A 4 1 d 3.: d m I O r-l :4 IN 2 — £1 '14 ‘7 . ,0 H \ "‘ ‘ C' H V u ‘U" \ < 1,- 0 q ' I I ' U 1 ' i I I 1 1 23 4 5 6 7 8 91011 A3. LATERAL DISPLACEMENT (in.) x 10-2 FIGURE 4-10 Southwell Plot For the Crown 38 N< mo qufiomEou ownumsfihmfiuc< 05H Mom uoam HHm3£uuom Hand mmDUHm m-oH x A.c4v Hzm2m0