LIBRARY Michigan State University This is to certify that the dissertation entitled ANALYSIS AND STABILITY OF LARGE-SPAN FLEXIBLE CONDUITS presented by Benjamin Nduchekwe Okeagu has been accepted towards fulfillment of the requirements for Doctor of Philosophy degree in Civil Engineering 63% e 4 , Sgtm/ jor professor Date /\/0l/. /// (72 MS U is an Affirmative Action/Equal Opportunity Institution 0— 12771 MSU LIBRARIES m RETURNING MATERIALS: Place in book drop to remove this checkout from your record. FINES will be charged if book is returned after the date stamped below. ANALYSIS AND STABILITY OF LARGE-SPAN FLEXIBLE CONDUITS BY Benjamin Nduchekwe Okeagu AN ABSTRACT OF A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Civil and Sanitary Engineering 1982 ABSTRACT ANALYSIS AND STABILITY or LARGE-SPAN FLEXIBLE CONDUITS by .Benjamin Nduchekwe Okeagu The objectives of the present study are twofold: 1) To examine the characteristics of the coefficients of soil reaction for flexible conduits, and develop simple formulas for their evaluation. 2) To use such formulas in the prebuckling and buckling analyses of the conduits. Both circular and elliptical conduits are examined in order to investigate the effect of the shape of the conduit on its stability. Theneed for this study arises from the fact that existing studies employ physical idealizations that ignore certain salient parameters of the soil-structure interaction problem. The findings suggest the following: a) The coefficients of soil reaction vary considerably around the conduit, depending on the span of the conduit, the depth of enbedd- ment, and the direction of action. b) A good portion of the buckling strength of the conduit is derived from its interaction with the surrounding fill. c) The shape of the conduit has considerable influence on its stability. Theoretical results from the present study show reasonable agree- ment with ones from relevant buckling tests. ACKNOWLEDGEMENTS The present work has been carried out under the patient advice and supervision of Dr. George Abdel Sayed to whom the author is most deeply indebted. Dr. Sayed's contributions to this work are truly priceless and the author wishes to hope that he will continue to be his mentor. The writer is grateful to other members of his doctoral committee for their useful suggestions. Thanks are due to Ms. Nancy Hunt and Ms. Vicki Switzer for their skillful typing of the manuscript. Finally, the author is indebted to all members of his family for their support and prayers. TABLE OF CONTENTS ACKNOWLEDGEMENTS . . . . . . . . . . TABLE OF CONTENTS . . . . . . . . . . LIST OF TABLES LIST OF FIGURES . . . . . . . . . . . NOTATIONS . CHAPTER I. CHAPTER II. CHAPTER III. CHAPTER IV. CHAPTER V. REFERENCES TABLES . . FIGURES . . APPENDICES INTRODUCTION . . . . . REVIEW OF LITERATURE . DETERMINATION OF THE COEFFICIENT OF SOIL REACTION PRE-BUCKLING AND BUCKLING ANALYSIS OF ELASTICALLY SUPPORTED RING DISCUSSION AND CONCLUSIONS iii 0 O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 0 iii iv vii ix 24 38 67 76 80 132 167 3-16: 3-17: 3-18: 3-19: 3-20: 3-21: 3-22: LIST OF TABLES HYPERBOLIC PARAMETERS USED RESULTS RESULTS RESULTS RESULTS RESULTS RESULTS RESULTS RESULTS RESULTS RESULTS RESULTS RESULTS RESULTS RESULTS RESULTS RESULTS RESULTS RESULTS RESULTS RESULTS LOADING FOR FOR FOR FOR FOR FOR FOR FOR FOR FOR FOR FOR FOR FOR FOR FOR FOR FOR FOR P: P P=300#, P=300#, P=300#, P=100#, P=300#, P=SOO#, P=100#, P8300#, P=500#, P=300#, P=500#, P=100#, P=300#, P=500#, P=300#, P=500#, SCHEDULE c 33.3 , H = c H =4 C I # 4’ AND DIAMETER = 100 INCHES # 6.4’ AND DIAMETER.= 100 INCHES HC=6.4’ AND DIAMETER:200 INCHES HC=8’, AND DIANETER=200 INCHES Hc=11.73’, AND DIAMETERFZOO INCHES ’ AND DIAMETER=300 INCHES HC=6.39’, AND DIAMETER=300 INCHES Hc=6.39’ AND DIAMETERS300 INCHES HC=6.39’, AND DIAMETER=300 INCHES ’ Hc=8 , AND DIAMETER=300 INCHES Hc=8’, AND DIAMETER=300 INCHES Hc=8’, AND DIAMETERS300 INCHES HC=11.73’, AND DIAMETER=300 INCHES Hc=11.73’, AND DIANETER:300 INCHES Hc=12.64’, AND DIAMETER=300 INCHES Hc=12.64’, AND DIANETER:300 INCHES Hc-12.641 AND DIAMETER=300 INCHES HC=20.36’, AND DIAMETER:300 INCHES Hc=2o.36’, AND DIAMETER=3OO INCHES HC=20.36’, AND DIAMETER:300 INCHES iv PAGE 80' 81 82 83 84 as 86 87 88 89 9o 91 92 93 94 95 96 97 98 99 100 101 TABLE 3-23: 3-24: 3-25: 3-26: 3-27: 3-28 3-29: 3-30: 3-31: 3-32: 3-33: 3-34: 3-35: 3-36: 3-37: RECOMMENDED VALUES OF K1 AND nh (AFTER TERZAGHI 1955) RESULTS FOR HC = 4’, DIAMETER?200 INCHES, AND P=50 LBS RESULTS FOR Hc = 4’, DIAMETER = 200 INCHES AND P = 100 LBS RESULTS FOR HC = 5.4’, DIAMETER=200 INCHES, AND P=1OO LBS RESULTS FOR Hc = 8’, DIAMETER=200 INCHES, AND P=100 LBS RESULTS FOR Hc = 11.23’, DIAMETERezoo INCHES, AND P=100 LBS RESULTS FOR Hc = 4’, DIAMETERF3OO INCHES, AND P=100 LBS RESULTS FOR Hc = 6.4’, DIAMETERFZOO INCHES, AND P=50 LBS RESULTS FOR Hc = 8’, DIAMETER=3OO INCHES, AND P=SO LBS RESUUTS FOR HC = 8’, DIAMETER:3OO INCHES, AND P=1OO LBS RESULTS FOR Hc = 11.23’, DIAMETER=3OO INCHES, AND P=50 LBS RESULTS FOR H = 12.64’, DIAMETERFBOO INCHES, AND P=SO LBS RESULTS FOR H 12.64’, DIAMETER:300 INCHES, AND p=100 LBS COMPARISON OF PROPOSED AND FEM RESULTS FOR MEDIUM SOIL (CIRCULAR) COMPARISON OF PROPOSED AND FEM RESULTS FOR DENSE SOIL 7(CIRCULAR) THEORETICAL PRE-BUCKLING THRUSTS THEORETICAL PRE-BUCKLING DEFLECTIONS THEORETICAL PRE-BUCKLING MOMENTS VARIATION OF A WITH SPAN COMPARISON OF PROPOSED AND FEM RESULTS FOR DENSE SOIL (ELLIPTICAL) COMPARISON WITH TEST RESULTS COMPARISON WITH TEST RESULTS COMPARISON WITH TEST RESULTS COMPARISON OF THEORETICAL FORMULATIONS OF COEFFICIENT OF SOIL REACTION (SOIL SUPPORT MODULUS) PAGE 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 121 122 123 TABLE 4-10: 4-11: 4-12: 4-13: 4-14: 4-15: VALUES OF KS IN TONS?CU. FT FOR SQUARE PLATES, 1 1 FT X 1 FT, or BEAMS 1 FT WIDE, RESTING ON SAND (AFTER TERZAGHI, 1955) COMPARISON OF THEORETICAL CRITICAL PRESSURES (CIRCULAR) P VARIATION OF CRITICAL PRESSURE WITH Pl. 0 VARIATION OF CRITICAL PRESSURES WITH ASPECT RATIO (ELLIPTICAL SECTION) VARIATION OF CRITICAL PRESSURE WITH DEPTH VARIATION WITH DEPTH OF RELATIVE CROWN DEFLECTION DURING BUCKLING (CIRCULAR) DATA REDUCTION DATA REDUCTION vi PAGE 124 125 126 127 128 128 129 131 FIGURE 1-1: 2-2: 2-3a 2-3b 2-4: 2-6: 2-7: 3-1: 3-2a: 3-8: LIST OF FIGURES VARIATION OF THE CROWN MOMENT AS A FUNCTION OF SUBGRADE REACTION GEOMETRY OF CORRUGATION SOIL BEDDING AND ENGINEERED BACKFILL AROUND THE CONDUIT HYPERBOLIC STRESS-STRAIN RELATIONSHIP TRANSFORMED HYPERBOLIC STRESS-STRAIN RELATIONSHIP PRESSURE DISTRIBUTION ASSUMED IN THE MARSTON- SPANGLER THEORY SOIL PRESSURE DISTRIBUTION ACCORDING TO THE RING COMPRESSION THEORY: (a) CIRCULAR SECTION: (D) ELLIPTICAL SECTION; (c) PIPE-ARCH IDEALISATION OF THE STRUCTURE FOR ANALYSIS BY FRAME-ON- ELASTIC SUPPORTS METHOD DISPERSION OF LIVE LOAD THROUGH THE SOIL-FILL LINEAR STRAIN TRIANGULAR ELEMENT 9-NODE QUADRILATERAL ELEMENT 8-NODE QUADRILATERAL ELEMENT FINITE ELEMENT MESH FINITE ELEMENTS AROUND THE CONDUIT INTERFACE ELEMENT LOADING SCHEME VARIATION OF COEFFICIENT OF SOIL REACTION WITH DEPTH (300 INCH SPAN) VARIATION OF COEFFICIENT OF SOIL REACTION WITH DEPTH (200 INCH SPAN) vii PAGE 132 133 134 135 135 136 137 138 139 141 141 142 143 144 145 146 147 3-10: 3-11a: 3-11b: 4-6a: 4-6b: 4-7: 4-8: 4-10a: LOAD-DEFLECTION CURVE VARIATION OF 8* WITH % VARIATION OF 8* WITH 8 FOR 300 INCH SPAN VARIATION OF 8* WITH 8 FOR 200 INCH SPAN. SHEAR INTERACTION MODEL (AFTER KLOPPEL AND BLOCK, 1970) VARIATION OF THEORETICAL PRE-BUCKLING DEFLECTION WITH 6 VARIATION OF MOMENTS AROUND CONDUIT (TYPICAL) VARIATION OF A WITH SPAN VARIATION OF COEFFICIENT OF SOIL REACTION AROUND CONDUIT (CIRCULAR, DENSE) COMPARISON OF THEORETICAL CRITICAL PRESSURES (CIRCULAR) ELLIPTICAL CONDUITS VARIATION OF CRITICAL PRESSURE WITH Pl/Po VARIATION OF CRITICAL PRESSURE WITH DEPTH VARIATION OF CRITICAL PRESSURE WITH ASPECT RATIO (ELLIPTICAL SECTION) VARIATION OF RELATIVE CROWN DEFLECTION AT INITIATION OF BUCKLING WITH DEPTH APPROXIMATE BUCKLING CONFIGURATION (TYPICAL) MEYERHOFF AND BAIKE'S LOADING TESTS LOG N1 VERSUS LOG Hz N1 VERSUS N2 VARIATION OF CD WITH SPAN viii PAGE 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163a 163b 164 165 166 NOTATIONS Semi-major axis Parameters governing the virtual displacement components in the radial and tangential directions, respectively Semi-minor axis Load-dependent stability matrix Boussinesque coefficient Cohesion Pressure transfer coefficient Span of conduit The smaller of the span of conduit or width of load distribution Axial rigidity in 8-direction Modulus of elasticity of conduit material Initial tangent modulus Tangent modulus Modulus of soil reaction Poisson's ratio number Buckling stress Depth of cover Depth to the crown Coefficient of horizontal soil reaction Coefficient of normal soil reaction Coefficient of tangential soil reaction ix 1“s' \)s C, n Modulus parameter Modulus parameter Change of curvature of the centerline in the 6-direction Bending moment per unit length in the 8-direction Axial force per unit length acting in the 6-direction Atmospheric pressure Radius of curvature Radius of gyration Bending strain energy Strain energy of elastic supports Total potential energy Tangential component of deflections Normal component Of deflections Load-independent stability matrix H .2. D Reduction factor to account for the depth of cover k .ii k n Axial strains at the centerline in the 9-direction k s ni Poisson's ratio of soil Potential energy of external load Virtual displacement components during buckling in the normal and tangential directions, respectively Axial stresses in the 6-direction CHAPTER 1 INTRODUCTION Underground conduits are built using corrugated steel sheets and constructed to induce beneficial interaction between the conduit walls and the surrounding soil. The soil acts as an integral part of the structural system and the structure is referred to as a composite soil- steel structure. The benefits of such interaction have long been recognized by many researchers. As shown by Szechy (l), ascribing even a minimum amount of lateral support to the soil medium reduces the moments and stresses in the structure by a significant amount (Figure l-l). This reduction is enhanced (or in certain cases hindered) at an appropriate depth of filling by the arching effects of the soil (refer- ence 1)). For-a long time, underground conduits were limited to spans of less than 25 feet. Only in the last 15 years have soil-steel structures been built up through 54 feet spans and come to be regarded as economi- cal alternatives to conventional short span bridges. Construction of conventional bridges is labor intensive and much of this labor is highly skilled. Major capital plant equipments, such as cranes and the like, are required and the conventional bridge components are usually made of high grade material. In contrast, the major component in soil-steel structures is soil which is widely available and one Of the cheapest building materials. Further, the high performance Of earth moving equipments make the construction of soil-steel structures highly pro- ductive and economical (2). Low costs and high productivity are the main incentives for using soil-steel bridge structures. A report by the United States Federal 2 Highway Administration estimates that using these low cost bridges results in savings Of 30% over other conventional short span bridges. Similar savings are reported in Canada (3), while the Australian exper- ience found the cost Of soil-steel bridges to be typically one-third that of the conventional bridges (4). Value analysis by a product designer (2) concludes that most conventional overpass structures do not represent optimum design. Alternative design using flexible metal arches and culverts was favored when considering all governing parameters. The design of flexible conduits is usually governed by the circum- ferential thrust in the conduit walls (5). If the depth of cover is equal to or more than a minimum specified depth of one-sixth of the span, moments in the wall are not required to be calculated. The justifica- tion for the neglect of moment lies in the manner in which the inter- face pressure between steel conduit wall and the surrounding soil mass changes with the movement of the wall. Even if bending moment occurs locally to cause partial yielding, the resulting movement of the wall would cause an increase in the interface pressure provided by the adja- cent soil mass, and this increase in pressure tends to inhibit any further movanent. In general, flexible conduits are designed to guard against the following primary modes of failure: 1) Wall crushing which occurs when the compressive stresses due only to the circumferential thrust exceed the axial strength of the wall. 2) Separation of seams when the thrust exceeds the seam strength. 3) Excessive deformation due to plastic yielding under combined compressive and bending stresses. 3 4) Bearing failure of soil (typically for small and shallow con- duits subjected to heavy live loads). 5) Soil failure above the conduit (stability of soil cover). 6) Buckling in the conduit walls both in the elastic and the elastic-plastic levels Of stresses. Simplified procedures are developed for the analysis and design Of soil-steel structures. These procedures proved to be adequate for the design of short and medium span conduits (up to span of 25 ft) and with covers of not less than 1/6 the span (6,7,8). Herein, the first four modes of failure are the dominant consideration for design. However, with the new trend in building conduits with larger spans and shallower covers, the latter two modes of failure tend to control the design. The behavior and design of long-span metal conduits under shallow cover have been examined by Duncan (9) who recommended that moments in conduits should be calculated when the height to span ratio, H/S, is less than one-fourth. Duncan did not include buckling as part of the design criteria but stated that "additional research is needed to define precisely the range of conditions for which buckling failure may occur.” The stability of soil-steel structures has been examined by many investigators. Leonards and Stetkar (10) presented a summary of the information and formulas available. Almost all stability studies deal with uniformly applied boundary pressure on circular cross-sections. The only stability study that is general in nature and accomodates sections other than circular was presented by Kloppel and Glock (11). However, this study neglects the bending deformations in the determi- nation of the critical load or pressure. Also, Kloppel and Glock considered the conduit to be supported by continuous elastic springs (similar to Winkler approach) with the assumption that the coefficient of soil reaction is constant with no variation with the depth or direc- tion of action. Recently, Hafez (12) examined the problem of soil failure above the conduit (failure mode number 5) and the author feels that attention should be paid to proper evaluation of the buckling criteria (failure mode number 6). The thrust of the present research is to develop a methodology which can deal with everyday analysis and stability problems of soil- steel structures under shallow or deep covers. Furthermore, the pro- posed methodology is applied to study the stability problems of Soil- steel structures. The problem is approached by employing a mathematical idealization of the soil response. This approach is similar to the analytical method proposed by Desai and Christian (13) for the design of footings, and the Spring method applied by Kloppel and Glock (1970) for the analysis and stability problems in soil-steel structures. However, unlike Kloppel's approach, it is recognized here that a large number of parameters influence the performance of soil-steel structures. Therefore, 1) A study is conducted in Chapter 3 to examine the parameters governing the coefficient of soil reaction, kn, normal to the surface of the conduit wall as well as the coefficient, ks, tangential to the wall surface, and to develop a simple formula for their evaluation. 2) The energy principles of mechanics and the associated varia- tional methods are used in Chapter 4 in the pre-buckling and buckling an"-"-‘~:L:yses of the conduit. The geometric non-linearity of the structure is incorporated in the formulation by using non-linear strain- dis£>lacement relations. Equilibrium is then based on the deformed 5 geometry of the structure and thus general instability is detected. 3) Both circular and elliptical cross-sections are examined in order to study the effect of the conduit shape on its stability. CHAPTER 2 REVIEW OF LITERATURE In order to provide some appreciation Of the complexity of the inter- action problem, and a motivation for the techniqueadopted in the present study, a review of current literature is presented in this chapter. 2-1 CONDUIT WALLS The conduit walls are usually made of cold formed corrugated steel plates. A typical corrugation profile is shown in Figure (2-1). The plates are usually shipped unassembled and bolted together at the site. 22-2 SOIL MATERIAL The structural integrity of soil-steel structures is dependent as much on the selection of adequate steel walls as it is on the soil materials used for bedding and backfill (Figure 2-2) . The bedding usually has a minimum thickness of 12 inches (30 cm) and is preshaped in the transverse direction to accomodate the conduit invert curvature. The backfill is placed and compacted in layers Of not more than 12 inches (30 cm). At no time does the difference in levels of backfill on the two sides of the conduit exceed twice the thickness of a com- PaCted layer. Granular materials are generally recomended for bedding and ha~cltfill. Such materials do not exhibit much change in their physical and engineering properties once they are constructed. Environmental Che-rages such as moisture do not affect these properties to the same degree as they affect those of cohesive soils. After placement of the bedding and backfill envelope, secondary Ina"terial can be used to achieve the desired grade above the conduit. 7 However, the behavior of such material should be taken into account especially with regard to the possible effects of arching above the conduit. 2—3 CONSTITUTIVE LAWS FOR SOIL MEDIA A major difficulty in the analysis of soil-structure interaction is an accurate description of the relations between stresses and strains in the soil media. In order to represent the interaction problem realis- tically, some form Of non-linear relation must be used. Numerous con- stitutive relations have been proposed over the years and are well documented. Typical among these are the Hardin model (14) and the hyper- bolic model. The later is proposed by Duncan and Chang (15) who related the tangent modulus E to the principal stresses (I1 and 03 by the t equation Rf(l-sin¢)(01-Oa) 2 Et = Ei l '- 2C cos¢ + 2 03 sino (2'1) where E i is the tangent modulus, R.f the failure ratio (that is the ratio between the measured compressive strength (01-03) f and the asymptotic Value of the stress difference for the hyperbolic stress-strain curve (Figure 2-3a) , C the cohesion, and d) the angle of internal friction. As suggested by Janbu (16) , the initial tangent modulus Bi is related to the confining pressure by (2.2) wFlex-e Pa is the atmospheric pressure, and k, n, constants to be determined experimentally . 8 substituting this into equation (2.1) gives the final expression for the constitutive relation 03 Rf(01-03)(1-sin¢) =kP — .— Et a (P ) 1 2C cos¢ + 2 U3 Sin¢ (2.3) In a similar manner, the expression for the tangent values of Poisson's ratio \J , and the shear modulus G at any stress level may be written as t t n 03 Ga-F log (3;) \) = t d(o -O) 2 1_ 1 3 (2.4) a n Rf(01-03)(1-sin¢) kP FHA [ - . J a Pa 2C cos¢ + 2 U3 Sin¢ aund 2 R (01-03)(l-sin¢)-1 Gt = Gi 1'- 2 03 sin¢ + 2C cos¢ (2'5) where G, F, and d are parameters to be determined experimentally, and G i is the initial value of the shear modulus. 2—4 FORCE ANALYSIS In this section, some of the existing techniques for calculating the force effects in the conduit walls of soil-steel structures are diSoussed. 2‘4 - 1 Marston-Spangler Theory The theory of loads on buried conduits developed by Marston (17) Bung: later modified by Spangler (18) represents one of the earliest fonnal investigations conducted on this subject. Marston based his theory on an assumed column of soil transferring load directly on the conduit and derived the following expression W= CyB2 (2.6) Behere C = a calculation coefficient B = span Of the conduit Y = unit weight Of soil W = load on the conduit. Spangler later extended Marston's theory to flexible conduits. Whereas Marston considered only a single concentrated load, Spangler assumed the pressure distribution shown in Figure (2-4) . Soil pressures at the top and bottom of the conduit are assumed uniform while a parabolic lateral pressure is assumed at the sides with a maximum at the spring- 1ine. The vertical pressures are assumed to extend over the span of the conduit while the lateral pressures subtend an angle of 100° at the center of the conduit. The uniform pressure is taken as the sum of the over-burden pressure and any distributed live load, P , at the top Of L the conduit PC = ysh + PL (2.7) where PC = the uniform vertical pressure, h = the depth of cover above the crown of the conduit, P = the equivalent live load pressure includ- L 1119 impact. For a conduit uniformly supported by a well compacted soil, the ma~3‘=:imum horizontal pressures at the sides are taken to be 1.35 the veli‘tical pressure on the t0p of the structure. Based on the assumed 10 pressure distribution, the thrust in the wall of the conduit is found to be 0.7 PCR at the top and bottom of the wall and PCR at the sides, with a maximum of about 1.1 PCR at the haunches. Similarly the moment in the wall is taken as 0.02 PCR2 at the top, sides and bottom, and —0.02 P¢R2 at the haunches. Based on the model shown in Figure (2-4) , Spangler derived what has now come to be known as the IOWA FORMULA for calculating the crown deflection where D13 K1= We rs EIB E'= D11<1Wr3 Ax = EI+ 0.061 E'r3 (2'8) Deflection lag factor of compensate for the volume change of the soil with time. Bedding constant which varies with the angle Of bedding. Load on the conduit per unit length. Radius of the conduit. Conduit wall stiffness per inch. Modulus of soil reaction. The Iowa formula had been used extensively in culvert design with a 5% decrease in the vertical diameter of the culvert generally con- sidered the safe limiting value for the control of deflection. The conduit was considered to be in a state of incipient failure if the de'12::ease in the vertical diameter reached 20%, prompting the use of a factor ofsafety of 4.0 against instability. In view of its empirical nature, Spangler's theory applies with lincited success only to small-span conduits under deep fillings. With rec=ent trends toward larger spans, the theory is grossly inadequate for 11 the following reasons: 1) The 5% limit for control of deflection is too generous since for large spans -- culverts spanning as much as 54 feet -- a 5% decrease in the vertical diameter can be quite excessive. 2) Watkins (1960) has found that under certain conditions, the conduit wall can fail by ring buckling long before the 20% limit on the vertical crown deflection is attained. 3) For culverts under shallow cover, subjected to live loads, the assumption of a pressure distribution extending over the full span of the conduit can be over-conservative (Bakt, 1980) . 4) The assumed pressure distribution is arbitrary and so is the parameter defined as the modulus Of soil reaction. 2—4.2 Ring Compression Theory White and Layer (1960) assume a uniform pressure around a circular conduit buried to a depth of at least one-eight its span in a well- compacted fill. The uniform pressure, P, consists of the overburden pressure), yh, and a uniform live and impact load pressure, P L That is , P = yh + PL (2.9) where Y is the unit weight of soil, h the depth Of cover, and PL the equivalent live and impact load pressure. The circumferential thrust, Tr is expressed as T = PD/2 (2.10) whelIe D is the span of the conduit. The ring compression theory is also extended to non-circular conduits 12 and thus implies that the soil pressure is greatest at the point of mini- ruum radius as illustrated in Figure (2-5) . 2—4.3 Method of Watkins Watkins (19) gives the following expression for the thrust, TL, in the conduit wall due to live loads TL = 0.5 Cp 0L (1+1) Dh (2.11) C = a pressure transfer coefficient 0' = the equivalent uniformly distributed pressure at the level of the crown I = the impact factor. The pressure transfer coefficient, cp, accounts for the arching action of dead loads. (IL is computed from Boussinesq's theory Of force effects on an elastic half space, and is expressed as PCB a: L H (2.12) 2 C Where P is the concentrated load applied at the level of the embank- ment, He the depth of cover to the crown, and C the Boussinesque b coefficient. In using Boussinesq‘s theory, it is assumed that it applies even to large cavities in the elastic half space. This assumption is found to be invalid and the load dispersion differs in the longitudinal and transverse directions of the conduit (20). The use of cp presumes that the phenomenon Of arching applies to live and dead loads in like manner. There is no evidence to support these assumptions. 13 2—4.4 Kaiser Aluminum Method This method is based on finite element analysis and provides enxlpressions for the thrusts and bending moments due to live loads. lieerice, T = k LL 2. p ( 13) sviaerre T = thrust due to live load L H k = 1.0 for --< 0.25 P D '— h H H = 1.23 - -—-for 0.25 <'-— < 0.75 (2.14, a-c) Dh -D '— = 0.5 for 31-> 0.75 Dh -' D = Span of the conduit LL = the equivalent line load corresponding to applied concentrated forces. me bending moments in the metal arch, due to live load is given as ML = KmDhLL (2 . 15) ML a bending moments due to live loads km = 0.018 - 0.004 Log” Nf for Nf _<_ 5000 = 0.0032 for Nf _>_ 5000 0.265 - 0.053 Log1° Nf R = < 1.0 (2.15, 3,13) H 0.75 - (I? 14 3 ES (Dh) H H II fluxural rigidity per unit length of conduit [*1 II secant modulus of the fill material. Tire equivalent line loads in equations (2.13) and (2.15) are obtained from Boussinesq's theory in much the same way as Watkin's method. The Kaiser Aluminum method leads to conclusions that do not agree with test data. For example, it predicts that for depths of cover between 0.3 and 0.5 meters (1.0 and 17.0 feet), live load effects remain constant. In contrast, tests by Bakt (20) show that live load effects decrease quite rapidly with the depth Of cover. 2-4.5 Frame on Elastic Supports This procedure employs the Winkler model, replacing a unit length Of the culvert wall by a two-dimensional frame and the supporting soil by discreet elastic springs (Figure 2-6). Two interacting zones Of earth pressure are identified -- a zone of active and a zone Of passive earth pressure. The active pressure is due to the movement of the soil toward the conduit and consists of a radial and a tangential component. me tangential component is the result of frictional forces developed bet‘veen the soil and the conduit as the conduit deflects downwards. It is considered negligible in the upper portion of the culvert ( 2E:‘-'~<'>;ppel and Glock (1970) . The radial active pressure is taken in the £611“ of a cosine function P - P cos (—5‘) (2 17) where P8 is the vertical compression at the crown and 0° the spreading 15 angle (Figure 2-6). The spreading angle (11° depends on the ratio, A, of the horizontal active pressure to the vertical compression in the soil, and is expressed as: 'IT 2 A = cos (m) (2.18) where : 0.5 for depths of cover exceeding the span >2 II 0.0 for depths of cover less than the span. The high value Of A for deep filling accounts for the reduction of the vertical compression of the soil by arching. For shallow culverts, A is taken to be zero to reflect the fact that the vertical compression at the level of the Crown, Ps (due mainly to live loads), is much The vertical compression larger than the horizontal active pressure. at the level of the crown, PS, is given as y}! + PC) for PC < yH (deep cover) '0 II (2.19, a-b) 1.1 (7H + Po) for PC > yI-I (shallow cover) where Po is the live load pressure, H the depth of cover, and Y the unit weight of the soil. le 10% increase in the case of shallow fillings accounts for concentra- t - 1011 of live loads on top of the crown. In the zone of passive pressure, the walls of the culvert move o “Wards against the supporting fill. The passive pressure is assumed t Q aCt in the form of spring supports, each having a tangential reaction, '1? ' For a typical location, n, on the culvert and a normal reaction, F. 16 wall, these components are given as: F = P + s w (2.20) n n n n where Pn is the active part of applied loading, 8n the spring constant, and Wn the radial displacement. Similarly, (2.21) where U is the coefficient Of friction. 2-4.6 The Finite Element Method The geometric and material non-linearities encountered in soil- steel structures render a complete analytical solution intractable The finite element method (21) is clearly the only technique that is able to simulate most of the aspects of the interaction problem with a mini- It is capable of modelling the mum of over-simplifying assumptions. Presence or lack of friction between the soil and the conduit walls as well as the non-linear behavior of the soil and conduit walls. First a finite element mesh is drawn to simulate the soil mass and A two-dimensional analysis is then performed to tl'le culvert wall. cmpute the nodal stresses, displacements and other quantities Of Interest. Clearly the complexity, accuracy and therefore degree of rigor of the finite element method depend on the type of elements and the refinement of the discretization. 2‘4-7 Theory of Elasticity A circular soil-steel structure has been analyzed as an elastic C:§?.‘.l.indrical shell embedded in an isotropic elastic medium of infinite The problem is considerably simplified by introducing Q3“":1ent (22) . 17 some physical idealizations. The complete solution process, as might be expected, is very rigorous in detail and the final expressions are equally involved. Burns' solution (22) has received a lot of attention in spite of being quite restrictive. The culvert is considered to be embedded to a depth of at least 1‘: times its diameter in a weightless, homogenous, isotropic and linearly elastic medium of infinite extent. Stresses and deformations are determined for two limiting cases: (i) full slip (that is zero shear stress between the soil and the conduit wall), and ( ii) no slip. Assumptions such as the ones mentioned above over-simplify the problem, severly limiting the range of applications of elasticity so lutions . 2-4-8 Practical Code Provisions on Force Analysis The Ontario Highway Bridge Design Code gives live load thrusts, TL , as TL = 0.5 OLDHmf(I+l) (2.22) o = the equivalent uniformly distributed load at the level of the crown. m = modification factor for multi-lane loading. I = dynamic load factor. D = the smaller of the span Of the conduit or width Of load distribution . ‘3 equivalent distributed load UL is calculated on the basis of a 2:1 afLs£>ersion (Figure 2-7a) -- that is, the lines of dispersion slope down 18 to the crown at the ratio Of 2 vertically to l horizontally. The modification factor mf is taken as 1 for a single vehicle, and 0.95 for two vehicles. The impact factor, I, for a single lane is given as 0.4 for H 0 H II 2.0 for H ?_ 2 meters (80 inches). For depths of filling, H, between 0.16 D and 2 meters, a linear inter- polation is permitted. The OHBDC method avoids the assumption of a load dispersion extend- ing over the full span of the conduit which may be conservative for shallow conduits. The method of the American Iron and Steel Institute (A181) is similar to Watkin's method except that the arching effects (of both live and dead loads) are completely ignored. The thrust, T , due to L live loads is given simply as TL = 0.5 O'L (I-t-1)Dh . (2.23) wit}: identical notations as in equation (2.11) . The American Association of State Highway and Transportation of ficials' method (AASHTO) is virtually identical to AISI method. The same expression is used for the thrusts due to live loads, with identi- c: 31 notations. That is, = . 2.24 TL 0 5 UL (I+l)Dh ( ) 1e only difference is that for a depth Of cover, H, exceeding 0.61 meters (about 2 feet) , the live load is assumed dispersed in such a way as to be uniformly distributed over a square of sides 1.75 H. In the 19 case of multiple concentrated loads with over-lapping square areas, the effective area is defined by the outside limits of the over-lapping squares. The total width Of dispersion in this case is confined to the span of the conduit (Figure 2-7b) . 2—5 STRENGTH ANALYSIS The existing techniques for calculating the distribution of force effects on soil-steel structures were the subjects of the preceeding sections on force analysis. The present section examines the ability of the structures to sustain the force effects. Rather detailed study of the literature on this subject is given in the report by Leonards and Stetkar (1977) . With the exception of Kloppel and Glock (1970) , all theories deal with uniform radial boundary pressures . 2—5- 1 Practical Code Provisions on Strength Analysis For strength analysis, the OHBDC considers the conduit wall to be divided into two zones -- an upper zone in which the radial displace- ments are directed toward the inside of the conduit, and a lower zone wit’li radial displacements outward towards the soil. , of the wall in both zones is The elastic buckling stress, fb 3 E B = 2.25 fb (mm/r)! ( ) w here r = the radius of gyration, R the radius of curvature of the wall, 6- B a reduction factor to account for the depth of cover. For depths o _ fi cover exceeding twice the radius Of curvature at the crown, B is t 3‘an as 1.0, and for other cases as: 20 B = (:5) (2.26) The factor K is a function Of the relative stiffness Of the conduit wall with respect to the adjacent soil, and is expressed as (2.27) where 131 is the flexural rigidity of the conduit wall. The demarcation between the two buckling zones (that is the upper and lower zones) is accounted for through the factor A. For buckling in the lower zone, A is taken as 1.22, and for the upper zone as H 0.25 1 = 1.22 [1.0 + 2(_ 3) ] (2.28) E R Where E = E' [1 - (3:1?) ] (2.29) H = depth of cover above the crown of the conduit E' = modulus of soil reaction. Both the AASHTO specifications (American Association of State Highway and Transportation Officials, 1973) and the A151 (American Itch and Steel Institute, 1971) use equation (2.25) to calculate the en~a.stic buckling stress in the wall Of the conduit. The latter assumes a Value of K of 0.27 for corrugated steel pipes with backfill compacted t9 85% standard density. 21 2-6 COEFFICIENT 0F SOIL REACTION The concept of coefficient of soil reaction, K, was first intro- duced by Winkler (1867) and has since been applied by a number of investigators. It had previously been erroneously thought that this coefficient was an exclusive soil parameter which‘could be expressed purely in terms of the elastic constants Of the soil medium. This misconception was first pointed out by Terzaghi (23) . Attempts to incorporate other salient properties of the soil-structure system have since been made by Mayerhof and Baike (24), Kloppel and Block (1970) , and Luscher (25) . For culverts embedded in sand backfill, Meyerhoff and Baike gave the following expression for the coefficient of soil reaction (2.30) where E8 is the modulus of soil, R the condUit radius, and K the coeffi- Cient of soil reaétion. me authors Offer no rationale for their expression other than that "the res istance of fills in the horizontal direction will usually govern in tlie case Of sands and gravels." Kloppel and Glock derived their expression by considering a plane State of strain of an elastic plate with a circular Opening (repre- Bel'l‘ting the conduit). The plate (representing the soil medium medium) is considered to have a constant modulus of elasticity Es, and the o petting a radius R. The opening is subjected to a radial compressive E Q3=‘<=e Po and the authors show through plate theory that K is given by: “r“ ‘ 22 E S k = W (2.31) where vs is the Poisson's ratio of soil. Herein it should be noted that the plane strain analysis of isotropic media resulted in the existence of tensile stresses of equal magnitude and perpendicular to the radial stresses. The expression for the coefficient of soil I‘éactio'n due to Luscher is: R. 2 1 Es [1 - (R0) ] k = Ri 2 (2.32) (l + vs) {1 + (F) (l - 2 vs)} R o where: Es = the soil modulus vs = the poisson's ratio of soil R. = inside radius of elastic ring of soil support R 8 outside radius of elastic ring of soil support R = conduit radius. The expression was based on empirical results derived from small scale model tests and includes the effects of the dgpth of filling. For a fairly deep cover, the ratio (2;) in equation (2.32) is nearly zero, and if the poisson's ratio,rv:, of the soil medium is taken as 0.5, both expressions (Luscher, Kloppel and Block) simplify to that given in equation (2.30). In all the above expressions, the coeffi- cient of soil reaction is assumed to be constant all around a given 23 conduit, and the surrounding soil medium is represented by an isotropic, homogenous, linear continuum. CHAPTER 3 DETERMINATION OF THE COEFFICIENT OF SOIL REACTION INTRODUCTION Many investigators have attempted to examine_the behavior of soil- steel structure systems by employing diverse empirical and analytical techniques. These range from empirical estimation of the ring compres- sion stresses to highly sophisticated finite element analyses incorpor- ating non-linear and stress-dependent properties of the soil media. In between these two approaches, exist the idealized models of soil-structure interaction analysis (26,27) which attempt to strike a balance between them. This approach utilizes the physical idealization or analog modelling of the soil-structure interaction problems in terms of Winkler element (Kloppel and Glock (1970)). Herein it should be noted that the above range of interaction analyses and the idealized models are not unique to soil-steel structures but also exist in the analysis of soil-supported footings and rafts. Analytically, the problems in soil-steel structures are consider- ably simplified by the introduction of a physical idealization of the soil-structure interaction. By using such idealization, a number of problems can be examined relatively conveniently, such as analyses of live load effects, stability problems and three dimensional behavior of the structures. Admittedly, the difficulty in this approach is that the spring constants and shear stiffness are not unique soil properties independent of the problem under consideration. They are related to the soil properties, as well as the geometric and stiffness parameters of the structure. However, despite the complexity and the approximate nature of the analog modelling schemes, they present very 24 25 useful tools to analysts and designers. They provide the facility to readily investigate the influence of soil support as well as the conduit geometry and stiffness properties on the failure characteristics of the structure. The objective of this chapter is to improve this approach and make it more attractive to engineers. Herein explicit results are obtained incorporating the different parameters governing the soil effects, and more accurate idealization is achieved for the coefficient of soil reaction. They are obtained by relating the results of internal force components and deformations calculated with rigorous finite element analysis (12) to equivalent results obtained from the system modelling of the problem. The finite element analysis developed in (12) also forms the basis for verifying the results of the system modelling. 3-1 FINITE ELEMENT FORMULATION The composite system of the conduit walls and the supporting fill is discretized by'a number of finite elements (Figures (3-1) - (3-4)). Higher order elements are used around the culvert walls to reflect the steeper variations in soil stresses. Further away from the conduit 'walls, constant strain triangular elements are used to model the soil imass, while the conduit wall itself is discretized into twenty loeam.elements. The constitutive relations for the soil media are Ibased on the stress-dependent hyperbolic model as shown in Chapter 2. The development of the finite element model and computer program is ;Emesented in detail in reference (12) and briefly outlined here. An analytical model is used to reflect the normal and shear stresses :resulting from the interaction between the conduit and the surrounding Soil. Such interaction results from the relative movement of the soil Viith respect to the conduit wall at the soil-conduit interface, and the 26 relative movement of the soil particles with respect to each other. The interface element is a two-node spring type element (Figure (3-5)) possessing no physical dimensions, enabling it to be placed between the conduit and the soil without distorting the conduit geometry. Each interface element is assigned a normal and a tangential stiffness, kn and ks, respectively. Both are taken as zero in case of tension between the soil and the conduit wall. In order to minimize the overlap Ibetween nodes on either side of the interface in compression, kn is assigned a very large number, while a non-linear relationship is used for the shear stiffness On nS Rfs TS 2 Ts KI Yw (Pa) L 1 OntanA" (3 ’ 1) vehere Ts is the applied shear stress, Rfs the failure ratio, on the xiormal stress, Yw the unit weight of water, A the angle of friction loetween the soil and the conduit wall, KI a dimensionless stiffness rrumber, and n8 the stiffness exponent. An analytical incremental procedure is used to simulate construc- . / tion processes. The filling is applied in ten successive increments and L'" ii ssequence of linear analyses are carried out using the stress-strain 1?€33Lationships of the form '{A0} = [c]{A€} (3.2) “filters {A0} is the incremental stress vector, {As} the incremental strain Vector, and [c] the constitutive matrix. The effect of soil compaction is included in the form of equivalent 27 nodal loads applied on top of each construction layer. Before proceed- ing to the next layer, these are removed by applying equal and opposite forces. 3-2 FACTORS AFFECTING THE COEFFICIENT OF SOIL REACTION The coefficient of soil reaction, k, is the unit pressure developed as the sides of the conduit move outward a unit distance against the fill. As noted earlier, this coefficient is not a unique soil property, depending instead on a variety of parameters pertaining to the soil- conduit system. The parameters selected for investigation in this study are l) The degree of compaction, 9; 2) the depth of cover, H (ft, in); 3) the span of the culvert, D (ft, in); 4) the flexural rigidity of the culvert wall, 31 (inZ/in); 5) soil modulus, ES (psi); 6) Poisson's ratio of soil, VS; 7) magnitude and direction of soil displacement As, 6 respectively: 8) the unit weight of soil, Y (pcf); 9) the relative density of soil (defined as dense and medium). Mathematically these are expressed as K = f(9, H, D, E1, E3, vs, AS, a, y) (3.3) Depending on the constitutive model used, expressions for the soil ‘“K3<111lus, Es' and the Poisson's ratio, vs, are very complicated in general, as well as non-linear and stress-dependent. Consequently, the develop- ment of an analytical model incorporating the wide variability of these Paranaters is virtually impossible. For purposes of computational 28 convenience, this study is restricted to two particular class of soils, namely a well-compacted dense, and a medium dense granular backfill. This limitation is valid since it is required in practice to use only well-compacted, granular soil. For such type of soils, the hyperbolic parameters, on the basis of studies by Duncan et al (1977) may be taken equal, or as close as possible to those in Table (3-1). Therefore Es and vs are considered to be prescribed quantities and their effects on the coefficient of soil reaction are accounted for. A stronger case for the elimination of E5 and vs from extensive consideration comes from the fact that the subsequent analysis utilizes the theory of dimensional analysis which requires that the significant parameters be dimensionless as well as independent of each other. It was noted previously that Es and Vs are dependent upon stress levels, which in turn vary with the depths of cover, H. Therefore to satisfy the limitation of independence as required by the theory of dimensional analysis, 33, vs and H cannot be considered separately. It has been found convenient to eliminate ES and vs in favor of the more readily amenable parameter, H. With ES and vs eliminated from further consideration, the theory of dimensional analysis (28) is applied to furnish the following non- dimensional form of equation (3.3) A 5: E! 37.1. .2 Y f (D. Ym' D. e, m (3.4) (A brief discussion of the theory of dimensional analysis is presented in the appendix . ) The advantage of equation (3.4) lies in the reduction of the number (bf independent terms. Rather than conduct a parametric study involving Iline separate terms as required by equation (3.3), the number of terms 29 is reduced to only five in equation (3.4). This represents substantial savings in time and expense in the present study. Furthermore, each non-dimensional term is considered varied if at least one of the para- ‘meters it consists of is varied. Therefore the choice of which para- 'meters to vary is often dictated by convenience and economy. The method of Kloppel and Glock presented in Chapter 2 identifies two interacting zones of earth pressure -- a zone of active and a zone of passive pressure. In order to compute the coefficient of soil 'reaction,it is desirable to devise a technique for separating the effects of one from the other. (Throughout the rest of this chapter, emphasis is placed on the normal component, kn' of the coefficient (of soil reaction. In the following chapter, shear interaction is emamined.) To achieve the desired separation, equal normal concentrated :Eorces are applied at the nodes of the beam elements to induce outward Ciisplacements all around the conduit as shown below. ‘V5L1:h such a device, the influence of active pressure is eliminated (311:1 the coefficient of soil reaction normal to the conduit wall is given simply as: 30 (3.5) 3‘ ll .14.? ni where Ai is the normal displacement at the ith interface node, Oi the normal stress at i in the direction of A1, and kni the desired coeffi- cient of soil reaction. 3-3.1 The Effects of Compaction and Flexural Rigidity, EI Soil stabilization is probably the single most important factor in most culvert installations. Rather than compute the response to a {range of values of the degree of compaction, this study accounts for compaction by specifying a dense, granular backfill compacted to 'the recommended AASHTO standards. (Later, the case of medium «dense soil is examined.) In this way, the degree of compaction is elimi- xaated from further consideration as a separate, independent entity. IPurthermore, the primary effect of compaction is usually to improve the <1ua1ity of the soil, notably the unit weight. Since the unit weight, ‘Yn is retained in equation (3.4), the influence of compaction is in effect reflected. There is evidence in the literature (29) that the effect of the flexural rigidity, EI, of the conduit wall on the coefficient of soil reaction is quite negligible. This conclusion is presumed accurate and titles term.EI/YD“ therefore dropped from equation (3.4). Hence no Separate examination of this term is conducted herein. 3-23.2 The Effects of the Depth of Cover, and the Span of the Conduit The effects of the depth of cover, H, and the span of the conduit, ‘Dor are examined in this study by computing values of the coefficient of 3°11 reaction corresponding to a range of values of H and D. Results 31 for H of 4.0, 6.4, 8.0, 11.73, 12.64, and 20.36 feet and for D of 300, 200 and 100 inches are presented in Tables (3-2) - (3-21). They are also presented in a non-dimensional form by plotting k/Y versus H/D in Figures (3-7) and (3-8). The plots are nonlinear, and can be des— cribed with sufficient accuracy as suggested by Bowles (30), by the fol- lowing relationship H (3.6) -<|3" ll 3’ + O J. with AS = 0 for sand filling. Equation (3.6) is developed in detail later. 3-3.3 The Effects of Magnitude and Direction of Soil Displacement (As, 8) To study the effects of the magnitude of soil displacements, a range of uniform normal forces are applied according to the loading schedule summarized in Table (3-22). Corresponding values of the coefficient of soil reaction are shown in Tables (3-2) - (3-21) for diameters of 100, 200 and 300 inches. These clearly show that the coefficient of soil reaction is practically independent of the magnitude of soil dis- placements, for displacements not exceeding 0.1 inches. Furthermore, the load-displacement relationship (Figure 3-9) is linear in the practical range of displacements. Evidence that subgrade reaction may be related to the direction of scfii displacement comes from Terzaghi (1955) and Vesic (31). Terzaghi proposed expressions for the coefficients of vertical and horizontal subgrade reaction, kv and kn respectively, based on the results of Plate load tests. According to him, the coefficient of vertical sub- grade reaction, kv, for beams on elastic foundation may be expressed as: 32 2 , _ 3+1 29 kv — K1 (—-B ) (1 + —B) (3.7a) For piles under lateral loads, a similar expression is given for the coefficient of horizontal subgrade reaction D kh - n B (3.7b) where in equations (3.7), B = the width of the beam or pile D = the depth of embeddment nh and K1 8 constants based on results of plate load tests. Recommended values of K1 and nh for sand filling are given in Table (3-23) . By extending the results of laboratory triaxial tests to footings, Vesic (31) proposed that the coefficient of vertical subgrade reaction, kv, may be expressed as k = 0.65 12 B8B Es (3 8) V' B EbI 1-\)2 ° where: U! l width of footing I I moment of inertia of footing O‘ modulus of elasticity of footing up M I modulus of elasticity of soil Poisson's ratio. C II Though empirical in nature, equations (3.7) and (3.8) clearly show tfluit.kv and Rh are significantly different for identical sets of soil 33 parameters and beam, footing or pile dimensions. Since the results of the present study indicate that the coefficient of soil reaction is independent of the magnitude of soil displacements, the only variable responsible for this difference must be the direction of soil displace- ments, 9. The effects of variations in 8 is accounted for in the sub- sequent discussion. 3-4 DEVELOPMENT OF THE EXPRESSION FOR THE NORMAL COMPONENT OF COEFFICIENT OF SOIL REACTION In the preceeding sections, the effects of a number of parameters on the coefficient of soil reaction were discussed and after deleting those factors considered negligible, the coefficient of soil reaction was shown to be given by the following equation 1T1 = fUTZI 1TB) (3.9) where 1T1 =k/Y 7T2 = H/D 7T3 = 6 Equation (3.9) is referred to as a prediction equation, and represents an unknown function which must be established by a suitable analytical procedure. A rational procedure for achieving this is discussed by Murphy (Reference (28)) and adopted here without proof. The method involves plotting the dependent variable as successive functions of each cxf the independent variables with all but one of the later held constant each time. As an illustration, consider the case described in equation C3.9). First H1 is plotted as a function of Hz, with W3 held constant at 753, say. Fran such a graph, a suitable curve-fitting technique is enmiloyed to develop a relationship between H} and Hz (for the constant 34 value of N3). Suppose this function is designated as 7r; = f('rr2, '53) (3.10a) Next, the procedure is repeated for W3 with W2 held constant (at E} say) resulting in a similar expression in Hz. Suppose this is written as 1r; = £52, 1T3) (3.10b) An equation of the form of (3.10a) or (3.10b) is called a component equation and the choice of the constant values 3} and E} are completely arbitrary. It is shown in (28) that the component equations may be combined into a prediction equation as f('nz, Fahffiz. NJ) (3 11) "1(“21 1T3) = __ _ _ F(W2' TF3)s 2 where: S = the number of dimensionless, independent parameters (three in the present case). F(§}, 33) = equation (3.19a) evaluated at E}, or equation (3.10b) evaluated at W3 Hence, the prediction equation can be expressed as a product of its component equations combined in some appropriate manner. Obviously, since this technique is semi-empirical, the chances of error increase with the number of variables involved. The technique is now applied to the present study. The sign con- (rention for this purpOse is that 6 is positive in the clockwise direction, increasing from zero at the crown. Furthermore, only one- half of the conduit (0 _<_ 6 5_ 180°) is considered since the coefficient of? soil reaction is virtually symmetrical about the vertical axis of the conduit (Tables 3-2 to 3-35) . Hence, any expressions developed 35 for one half of the conduit, automatically satisfy the other. Figures (3.7) and (3.8) show plots of k/y versus H/D with 6 held constant at suitable values. From these graphs and using the method described herein, in conjunction with the method of least squares, the expression for the coefficient of soil reaction for the dense soil is found to be (3.12) Ulm d 9 - 1 S b + 6. 4/o+12 Wm (- -l)m lm=1 (4.16a) —) + 6. 4/a+ 12 Wm(-1)m 1m=1 (4.16b) 44 4-3 SOME OBSERVATIONS ON THE SOLUTION SCHEME A computer program is written to solve equations (4-16) for various soil-conduit parameters. In addition to the unknown displacement coefficients, Wn, the point of inflection, 60, and the constant coef- ficient of soil reaction, kg, in the tangential direction to the wall of the conduit are also unknown. The program is designed to iterate over both of the later quantities (that is Go and ks) until an acceptable convergence is achieved. In the absence of experimental data, solutions from the finite element method C12) are used as the sole basis for verifying the pre-buckling displacements and stress-resultants. Literature on the coefficient of soil reaction, ks, in the tangen- tial direction to the wall of the conduit, is very scarce. Kloppel and Glock (1970) have proposed a model of shear interaction that argues for a total exclusion of the tangential component of the coefficient of soil reaction. According to this model, a set of shear stresses is induced around the upper section of the conduit as a result of the direct influence of live load as shown in Figure (4-Ja). Subsequent deformation of the conduit induces a similar set of shear stresses acting in an opposite sense to those due to loading (Figure 4-1b). Both sets of stresses counteract each other to an extent that is not pre- cisely determinate. However, it seems reasonable, according to this model, to ignore any resultant shear stresses since they are adjudged too small to make a significant contribution to the overall soil- structure interaction. The provision in this computer solution, of a tangential (in addi- tion to a normal) component of the coefficient of soil reaction is believed to be a more realistic modelling of the interaction phenomenon. The results of the computer solution indicate that for any set of conduit dimensions and live loads, the soil-structure interaction is 4S modelled with sufficient accuracy by taking the coefficient of tangen- tial reaction, kg, to be constant. In particular, it has been found convenient for the purposes of developing an appropriate expression for kg, to take this constant coefficient as some multiple of the normal component, kn , at the invert, expressed by the following equation 1 k = Ak (4.17) s ni where, >2 II a constant less than 1.0 Ofor036ieo The computer program, as noted earlier, performs an iterative routine. Starting with very small values of RS and 60, subsequent solutions are sought with small increments of these till a reasonable convergence is achieved. The displacement at the crown of the conduit is negative (inwards). Between the crown and the springline, the dis- placement reverses and becomes positive (outwards) just beyond the point of inflection, 60. In other words, the test for convergence is the angle 90 (incremented from zero) beyond which w(60) just reverses directions. A typical set of results is presented in Table (4-1) - (4-3) and results from finite element analysis are also presented for comparison. These results are also plotted in Figures (4-2) and (4-3). In addition, values of the constant coefficient, A, corresponding to these are presented in Table (4-4) and Figure (4-4). Of interest is the indication that A is practically independent of the span, D, of the conduit. Using the method of least squares, A is found to be approximately equal to 0.2. If this value is substituted into Equa- tion (4-17) the expression for kS may finally be written as 46 ks = 0.32 (4.25 - 9i%§2)¢o + 1 (4.18) 4-4 BUCKLING ANALYSIS The ring compression theory of White and Layer (1960) suggests that the flexural rigidity of underground flexible conduits governs mainly in the installation stages while the compressive strength of the conduit material and joints governs the behavior under load, provided there is an adequate backfill. The theory, however, disregards the actual properties of soils. Furthermore, buckling may in fact govern the behavior, under load, of flexible conduits whose spans are much larger than those considered in the ring compression theory. Therefore an adequate examination of the buckling limits of large-span flexible conduits is necessary. The second variation of potential energy is used to establish the criterion of elastic stability. The theory was developed with specific reference to elastic stability by E. Trefftz and has since been employed extensively (33, 34). It is based on the concept that a stationary mechanical system is in stable equilibrium if, and only if, the poten- tial energy, V, of the system attains a relative minimum; hence the change, AV, of potential energy is such that AV > O for any small vir- tual displacement of the system that is consistent with the constraints. The potential energy for an elastic system, is given in Equation (4-2). Consequently, the change, AV, in potential energy due to an infinite- simal (virtual) displacement from an equilibrium configuration is AV = AU + A9 (4.19) For an elastic system, AU may be written as —l- 620 + 3— 630 + + i- an (4.20) AU = 5U + 2! 3! ... n! in which GnU (the nth variation of U) is the volume integral of a homo- genous polynomial of nth degree in the components of the virtual dis- placement vector and its first derivatives (35). Similarly, the change, A9, in the potential energy of the external load is 1 69+ooo+—6Q (4.21) The principle of virtual work requires that the first variation (5U + 69) of the potential energy vanish for any equilibrium configura- tion. Thus if the virtual displacements are small, the sign of AV is controlled by the sign of 52H + 629. Therefore the equilibrium is stable if, and only if, 620 + 629 > 0 for all virtual displacements, and the criterion of stability, is that the second variation of poten- tial energy be positive-definite. The critical load for a structure is the limiting load at which the structure first loses its stability -- that is, the load at which 62V ceases to be positive-definite as the load is increased from zero. Accordingly, the question of stability resolves into a mathematical study of the nature of the second varia- tion of the potential energy. More importantly, the theory is readily generalized for multiple-degree-of-freedom systems. For a structure whose potential energy is a function of say, two variables A and B, and for arbitrary small virtual displacements A1 and B; from some equili- brium configuration (Ao, Bo), the change, AV in potential energy may be written in a Taylor's series expansion as 48 AV = g—X' (A0! B0) A1 + %% (A0! BO) El 1 32 v 2v + ‘27 3A2 (A0, B0) A2 + 2 g—-—ABB (A0, Bo) A181 + 32v 2 553-(A0, Bo) Bl (4.22) Hence, _ 1 2 1 2 Av - 6v + 37-6 v + 37-6 v + ... where: OV = g—X (A0, B0) A1 + % (A0, Bo) Bl (4.231)) and 32 v 232v 32v 52V: 3T2 (A0, Bo) A2 + m (A0180) A131+382 The appropriate condition for the limit of positive-definiteness of a quadratic form is that the determinant of the coefficients equal zero. Hence, in the present example, the condition for the initial loss of stability may be written as 32v 32v a? ‘on 30’ ma ”‘0' 130’ = 0 (4.24) 2V 2 37% (Aw 30’ '33} ‘on 30’ The use of the vanishing of the second variation of potential energy as a criterion of stability raises the question whether the (4.23a) (A0, Bo) B2 (4.23C) equilibrium is stable at the critical load itself -- that is, whether the equilibrium is stable for P fi-Pcr' or merely for P < Pcr' 49 Fortunately, this distinction is of little consequence in the determi- nation of the critical load (34) and the use of the vanishing of the second variation of potential energy is a sufficient criterion. 4-5 ENERGY EXPRESSIONS IN BUCKLING ANALYSIS In the pre-buckling analysis, it was assumed that linear theory was sufficient to ensure an accurate determination of the deflections and stress-resultants. In contrast, the development of expressions for the second variation of potential energy requires consideration of non-linearity (33), and equilibrium is based on the deformed geometry of the conduit. Recognizing this necessity, non-linear terms are retained in the geometric relationships. It is realized however, that retention of all non-linear terms is not practical and certain simplifying assumptions therefore become imperative. For example, the ring is assumed to buckle with- out any incremental membrane strains, and it is further assumed that the prebuckling membrane strains may be neglected without any loss of accuracy (36). The expressions for the strains and changes of curvature may be written as — — ‘1— — - l — .— 66 - R {(ve w) + 2 (v + we) } (4.25a) and, E - i- (Cz' + G) (4 25b) 66 R2 66 ° where Kee = change in curvature of the centerline in the 6-direction, 2% = axial strain of the centerline in the 6-direction, the bar denotes the sum of the pre-buckling equilibrium configuration and the corres- ponding virtual component. If w and v represent the displacement vector defining an equilibrium configuration, and C, n the respective components of the incremental virtual displacement vector during buckling, 50 then, neglecting the pre—buckling rotation of the ring element, v + which is very small, equations (4.25) may be rewritten as E' = 2-{v - w + - + l-( + )2} (4 e R e “a C 2 “ C8 ' and, E. - 1' (w + w + g + c) (4 88 if 88 88 ° The bending and membrane strain energy U13 and Um respectively, given by 2N1 ._ Ub = R Io 3 (Maxeeme (4. and 2N .. 3 ,- tJm - 2 I NeeedB (4. o where N6 = the axial force per unit length acting at the centerline the 6-direction. It is shown in reference (35) that NO and Me are given by Et ;. EI _. °' = — - - + 0 Ne R (V W) 137-(W W) (4 and, E1 .2 ._ = 4. Me E;- (w + w) ( Therefore using equations (4.26) and (4.28) in equation (4.27), the strain energy components may finally be written as: W 6’ 26a) 26b) are 27a) 27b) in 28a) 28b) 51 1 21‘s: 2 Ub ='§§- o:§3’(w + w66 + C + C66) d9 (4.29a) and, C II 2w R Et '2' IOWE‘Ve'WWe‘ 5’ El '1 1 F‘w+‘;ee+§+"88)-‘ {E[Ve'w+”8'§ + 2: % (n ‘+ :9) ]}de (4-29b) At this juncture, the strain energy, U , of the elastic supports k is included. Then by expanding what is left of the total strain energy (Ub + Um + Uk) and applying the fundamental definition of the second variation stated earlier, the second variation of the strain energy may be written, retaining no higher than quadratic terms, as 62D = R [Imk §2+ ii; (2; + m2] d6 (4 30) o n R 66 ' The problem of buckling of rings subjected to nonuniform pressures is much more involved than that of uniform pressure. The first attempt to consider nonuniform loads was apparently due to Almroth (37) who considered a pressure load of the form P = Po(l+cosfi). If the load remains normal to the conduit wall as the conduit deforms, the second variation of potential energy of external forces is shown in reference (35) to be given by: 52 2“ 629 = f p (g2 + 2C6” + n2) d6 (4.31) o It is of interest to note that 529, unlike the first variation, depends only on the loading function and the virtual displacement components. The problem of determining the limits of elastic stability is now reduced to one of seeking the appropriate expressions for the second variation, 62V, of the total potential energy. In this particular case, equations (4.30) and (4.31) combine to give: 2 2” RI 2 2 2 2 6 v = ID [E3'(C88 + c) + RKnC + p(; + 2C6” + n )]88 (4.32) Equation (4.32) may be solved by any number of suitable methods. In one such method, the appropriate Euler equations of variational calculus (see appendix) are found and together with the associated boundary condition, these yield a boundary value problem. The Euler equations for an integral of the form of equation (4.32) are (4033' a-b) a? d 3F + d2 3F _ —-_ 2 - 8; d6 8:6 de 3:66 0 where F.is the integrand in equation (4.32) Solving the final set of differential equations can sometimes, as in the present case, present difficulties. A simpler approach is the "direct-energy" method, so-called because the second variation of potential energy is minimized direct- ly without resort to Euler equations. This is done by evaluating 53 the integrals in equation (4.32) term-by-tenm (after assuming suitable admissible displacement functions), and then applying the criterion for the limit of positive-definiteness of quadratic form discussed earlier. Because of the assumption of admissible displacement functions, this gives an upper-bound solution. 4-6 SYMMETRIC BUCKL ING If the buckling waves occur in a symmetric mode, the virtual dis- ,placement components c;and n may be taken in the form of infinite Fourier series C(G) = Z A cosnG n=2 n 00 (4034' a-b) n(e) = X B sinnG n=2 n where rigid body displacements have been neglected by deleting terms corresponding to n = 1. Equations (4.34) automatically satisfy the boundary requirements that the admissible displacement functions be periodic in 6. Further, the coefficient of soil reaction, kn, may be conveniently expressed as an infinite series k (X) G) o . . k = — k. k 0 n 2 -+ .Z Jc0536 + z bSian (4 35) 3—1 b—l where, 2 fl k0 = F f kn(6)d6 o (4.36, a-C) 2 w = - .6d6 kj w I kn(6)cosJ O 54 kb = 0 (kn is symmetric about the vertical axis of the ring) Using equations (4.34) and (4.36) in (4.32), we get: {I N[% IE 2 (l-nm)2AnAmcosn6cosm6 Rn=2 m=2 co co so + R (7§-+ Z k.cos.6) Z Z A A cosnecosme . j j n m j=l n=2 m=2 co co co co + P(6) { Z Z An Am cosnGcosmG - 2 Z Z nAn B msinn63inm6 n=2 m=2 n=2 m=2 (D co + Z X B nB msinnGsintHde (4.37) n=2 m=2 where, P(6) = P0 + Plcose It is seen (Figure (4-5)) that a good approximation is obtained by taking only two terms of the soil coefficient function - that is, k 0 ' = = kn(9) = 2 + klcose and also by taking P1 Po' so that P(6) P (l+cosB). 0 Now, N w n I cosnBcosmGdG = I sinnesinmede = 3» n = m o o = O, n # m 2“ f cosjecosn6cosm6d6 = O, n # j + m or j # m + n or m f j + n o = g; n = j+ m or j = m + n or m = j + n 2W I cosjesinnesinmede = 0, n # j + m or j ¢ n + m or m # j + n o = %y n = j + m or j = n + m or m = j + n 2W f sinjecosnecosme = O o 55 Using the above relations in equation (4.37) and performing the integra- tion, the second variation of potential energy becomes 62v=—§— Z (1-n2)2A2+ o A2 R n 2 n n=2 n=2 oo oo oo +PTrEZB-2znAB+ZA2],n=m o n n n . n=2 =2 =2 1T 00 00 + Rk1 z Z POW m m 2 AA+ Z ZAA n=2 m=2 n m 2 n m n=2 m=2 CD 00 (X) 00 +2X ZnAB+Z ZBB],n=m+1 n m n m n=2 m=2 n=2 m=2 or m = n + l n, m # l (4.38) Equation (4.38) represents a quadratic form in the displacement parameters. Differentiating this quadratic form with respect to each of the parameters, a set of homogenous linear equations in the para- meters is obtained. The matrix containing these parameters is called the stability matrix. Clearly, the stability matrix contains two sets of terms, namely a submatrix of load-independent terms Xnm' and one of load-dependent berms 33m. The critical pressure is represented by the value of PC for which the determinant of the stability matrix vanishes. Because of the coupling of terms in An, Bn and Am, Bm (mfn), the indi- cated differentiation is accomplished in two parts -- first with n=m, and then with n#m. Withi’as defined earlier, differentiation for n=m gives 3F ZWEI 2 2 -I_ = - + - - = . 3An '—§y— (l n ) An fiKbRAn Pow(2an 2An) 0 (4 39a) 3F ‘3— = 2P TTB - ZnTTA P _ (4.391)) B o n n o 56 Solving for Bn in equation (4—39b) and substituting in equation (4.39a) gives 2EI [... R3 (1-n2)2 +1<0R + 2Po(n2-l)] An = o (4.39c) Equation (4.39c) constitutes the diagonal elements, xhn and Ban of the stability matrix. That is, 2E1 2 2 Xnn 'Eg— (l n ) KOR ( a) 3* = 2(n2-l) (4.40b) nn Similarly for n # m, differentiation yields: 53;'= TTk1R(An+1 + An-l) + PonEAn+1 + An-l + n(Bn+1 + Ian-1)]=0 . (4.41, a-b) 3F _ - _ 55—D- Pofl[-'{Bn+l - (n+1) An+1} {En-l (n 1) A‘n-lfl -0 A special class of problems is obtained by letting the off-diagonal elements (An Bn-l) of the stability matrix vanish. In +1' Bn+1' An-l' this case, equations (4.41) are identically zero, and equation (4.39c) then simplifies to K R _ EI(n2-l) o Pcr - R + 2(nz-l) (4’42) which is the classical solution for a uniformly supported circular ring under uniform boundary pressure. For the non-uniformly supported and non-uniformly loaded ring, equation (4.41b) is satisfieid in one of two ways: 57 (i) (n+1) An+ and Bn Bn+1 l -1 (n-l) An- 1 (ii) B (n-l) An_ and an (n+1) An+ n+1 l -l 1 In either case, substitution into equation (4.41a) yields A ) l + n-l 3F___ 8A.n - 0 — 1Tk1R (An+ + Po" [{n(n-l)+l} An_1 + {n(n+l) + 1}]An+1 (4.43) Equation (4.43) constitutes the off-diagonal elements of the stability matrix. (The fact that cases (ii) and (iii) give identical results is due to the symmetry of the stability matrix.) Hence ' X(n,n+1) = X(n,n-l) = klR B*(n,n-l) = n(n-l) + l (4.44, a-c) B*(n,n+1) = n(n+l) + 1 As stated earlier, the stability condition is X + P°B* = O (4.45) This is the standard eigenvalue problem, and the lowest eigenvalue represents the buckling load. Several techniques are available for solving matrix eigenvalue problems. This study employs the iterative Jacobi method, as outlined in the appendix, because of its ability to furnish the eigenvectors along with the eigenvalues without requiring a separate set of procedures as in most other method. This is parti- cularly useful if an approximate geometric configuration of the ring during buckling, is desired. 58 4-7 NONSYMMETRIC BUCKLING For this case, the boundary requirements are still that the virtual displacement components be periodic in 6. Hence the following displace- ment functions are admissible CD g = n£2(Ancosn6 + aninne) (4.46, a-b) n = n22(Cnsinn6 + DncosnG) Using these equations in equation (4.32) and carrying out the integra- tion yields 2 RI“ m 2 k‘oTrR w 6 V = O = i;— Z (l-nz) (A121+ B51) 4" 2 z (A:1 + Bi) n=2 ' n=2 co co co on + P n( 2 A2 + 2 B2 + Z c2 + X D2 ° n=2 n n=2 “ n=2 ” n=2 “ co co -ZZnAC+2ZnBD),n=m n=2 n n n=2 n n kl‘n'R 0000 0000 P017 0000 coco + 2 ‘5 ZAnAm ' 2 3.3m) + '2— ‘Z 2%. ' Z 5‘3an n m n m n m n m m co co co co co CD 00 " Z ZCnCm + 2 £13an + 2 Z ZnIAnCm + 22 ZanDm) , n=m+l n m n m n m n m or m=n+l n,m#l (4.47) Proceeding in a manner similar to the symmetric case, the quadratic form is differentiated with respect to the displacement parameters to obtain the elements of the stability matrix. Hence, for n=m, differentiation gives: 59 3F ZWEI 2 2 3An 2 R3 (l n ) An TIRkOAn ZWPO (A.n nCn) 3F ZflEI 2 2 _= =——— - + — 33m 0 R3 (1 n ) Bn TrRkan + 2‘”?0 (En nDn) §§_.= 0 = zflp (C _nA ) (4.48, a-d) 3C 0 n n n 3? __ _ BDn - O — 21TPO (Dn an) Solving for Cn and Dn in the last two equations and substituting in the first two gives 8? ZWEI 2 2 2 BAn O R3 (l n ) An TTkoRAn ZHPO (n l)An 2 (4.49, a-b) 3F ZWEI 2 2 3Bn 0 R3 (1 n ) Bn flkoRBn 21r1>o (n 1)Bn Each of equations (4.49) is a function of only one type of displace- ment.parameters. Therefore only one set of these equations is necessary to generate the elements of the stability matrix. Hence 2 X(n,n) = fig; (l-nz) + koR (4.50a) B*(n,n) = 2(n2-l) (4.50b) Similarly for n f m, differentiation gives 3? -——-= = ' + + Ban 0 1TRkl‘An-fl + An-l) + Pofl(ncn+l + An+1 nCn-l An-l) (4.51, a-b) 8F _ _ _ _ -——-= o - Pon[(n+1)An+1 + (n 1)An_1 cn+1 cn_l] 60 Equations (4.51) are seen to be identical to equation (4.41) obtained for the symmetric case. Therefore the elements of the stability matrix can be written directly as x(n,n+l) = X(n,n-l) = klR B*(n,n-l) n(n-l)+-1 (4.52, a-c) B*(n,n+l) n(n+l)+-l 4-8.l Elliptical Cross-Section So far, the discussion has centered exclusively on circular cross- section, resulting in the simplest possible solution, but lacking generality. In what follows, the theory is extended to rings of ellip- tical cross-section. As suggested by Brush and Almroth (33), shells of a general shape subjected to axisymmetric load can be expected to fail through the passing of a limit point. Therefore the same criterion is used to define the limit of elastic stability of the elliptical ring as for the circular ring -- that is the load at which 62V ceases to be positive-definite as the load is increased from zero. In the previous chapter, the coefficient of soil reaction in the normal direction to the wall of the conduit, was shown to be related to the span, D, of the conduit, the depth of cover, H, and the direction of action, 6, by equation (3.16). In order to show that this expression applies equally well to general shapes, it is necessary to extend it to an elliptical section. For this purpose the following expression for the depth, H, to any point on the elliptical conduit is quite useful (Figure (4-6a)): 61 H = H + Z (4.53a) where: Hc = depth to the crown bcose } Z = b {l ' 2611/2 (4.53b) [azsin26+b2cos Like the circular ring, the results are compared to the corresponding solutions from finite element analysis. The properties of the ellipse chosen for this purpose are Span, D = 286 inches 80.5 inches Semi-minor axis, b Semi-major axis, a 143 inches The results for depths of cover to the crown of 4, 6 and 8 feet are shown in Table (4-5) and agreement with results from finite element analyses is seen to be reasonable. Therefore the expressions for the coefficients of soil reaction proposed herein are believed.to apply reasonably well to conduits of arbitrary geometry. (In practice of course, geometry is prescribed.) The pre-buckling analysis for the elliptical conduit is identical in outline to that of the circular section. Hence, a separate elaborate presentation is not considered necessary here. 4-8.2 Stability Analysis The radius of curvature of the elliptical section varies around the conduit, and the expressions for the second variation of potential energy (equation (4.32)) must be modified to reflect this. The radius of curvature, R, at any point on the conduit is given by (39) 62 l R(6) = azb2 [azsin26+ b2c0526]3/2 = 2'2" l (37) (4 54) [l--€2cosze]3/2 . where 2 _ _ 2_2 E - 1 (a) a = semidmajor axis b = semi-minor axis radius of curvature. 21 II Hence, the second variation of potential energy for the elliptical ring may be written as {I EIa3 9/2 bzknc2 1T{-—-g—-(l- E 2cos29) (C +C)2+ 66 a[l-ezcosze]a/2 + P(C2+2Cen+n2)}d6 (4.55) (As a check, it is seen that for a = b = R, equation (4.55) reduces to equation (4.32) obtained for the circular conduit of radius, R.) As in the circular cross-section, the only boundary requirement for the elliptical ring is still that the virtual displacement functions be periodic in 6. Therefore the same set of functions are admissible for the elliptical ring as for the circular. Hence, no §(9) = ng Ahcosne m (4.56, a-b) 0(9) = Z B nsinnG 63 where rigid body displacements have been automatically deleted as ex— plained for the circular section. Introducing these into equation (4.55) gives 2'” 3 °° 2 EIa 9 2 62V = I {—Efi—-(l-€2cosze) / Z[Ancosn6-n2Ancosn6] o n bzkn 0000 + 3/2 X 2A Amcosnecosme a[l-ezcosze] n m n coco coco + P[Z XAhAmcosnecosmG - 2 Z ZnAthsinnasinme n m n m coco + g anBmSinnGSinm6]}d8 (4.57) Evidently, equation (4.57) is difficult to integrate in terms of elementary functions, hence recourse is sought to numerical integration schemes. In particular, the trapezoid rule (38) is used. Briefly, for a function f(6) defined on some interval O-fl say, integration by trapezoid rule furnishes the following n A6 n-l I f(6)d6 = 1;-[f(o) + f(fl) + 2 .2 fi] + Error terms (4.58) 0 i=1 If the uniform interval A6 is kept sufficiently small, the error terms are relatively negligible. (The trapezoid rule is developed in detail in the appendix.) Then using equations (4.53) and (3.16) in (4.57) numerical integration yields the following expression for the elliptical conduit embedded in dense fill: EIa3 9/2 a a 62V - 15?— [(1—52) E E1 (1-n2-m2+n2m2)anam Q” _29/2 _2 2 22 _m-I-n + (1:) £5(ln~m+nm)AnAm( 1) 9/2 c0321 cosm_1r_ + (1-0.75£2) 6 6 (l-nz-mz-t-nzmz)A A n m 5M8 3MB 9 / 2 c0391 cosm_‘n_ + (14.2522) 3 3 (l-nz-m2+n2m2)A A n m 5M8 5M8 0081le 008217- N + E (l-nz-m2+n2m2)AnAm 3- 2 ”M8 Q Q +(1;,2552)9/2 Z Z (1—n2-m2m2m2mnam “3311‘“- ”51273 3 n m 9/2 0° 0° cosS‘nn cosSmn + (p.758) 2 Z (l-nz-m2+n2m2)AnAm 6 5 n m C W 2 m a 2 .m on d a 3.24: _ m+n 2b 4» 12 {213/5 ZZAnAm+——b (1) “Oi—D ZAnAm n m n m G O 22%% 2 . co 1r comm + “0.47st 0.4» 3%{1- 8661" 1/2} “ m 3/2 '96— -é- [.25a2+.75b2] [ 25+ 752:] O 0 a2 .Q Q . 5 i “.3. + 0.72— 0142- {1 - 'Sb ‘ n m ”891 ”3!"- a D 3/2 3 3 [.75a’+.255’]1/2J 2 [49.75%] 2 4r“ a + 0.93-b—Va+% X 2A A “‘21 “‘35"- a n m n m 2 2 fl Q 2 A A cosZ'nn cosZ‘mn 2 b .51: n m n m 3 3 + 1.15%- a+5 {1 + 2 2 3,2} 2 3/2 .75a +.25b [ ] [.75+.2s%] G O 2V 8661) X {13an 5 co 51m b b . n m cos_1_r_n_ 3 44.38:!- a+B-{ZH- 3/2 6 } .25 2 .75152 1/2 2 E a + 3 [39.75%] m m m + Pon [2 B: -2 Z nAan + Z A:]' n=m n n n P" mm coco mm +—:—[ZZAnAm+ 2 Z Z nAan- EZBan], n=m-+1 nm um nm or m=n+l n,m#l (4.59) The quadratic form is reduced to a stability matrix and the resulting eigenvalue problem solved in much the same manner as the circular cross- section. 4-9 COMPARISON WITH TEST RESULTS The present study is compared in Tables (4-6) to (4-8) with results of buckling tests by Meyerhof and Baike (1963), Watkins and Moser (40), and Luscher (1963). While agreements with the results of Watkins and Moser is quite good, there is considerable discrepancy with Luscher's results. This is the special case of uniform boundary pressures and uniform elastic support (coefficient of soil reaction) discussed in the preceding chapter. The discrepancy may be due in part to the fact that Luscher's results in themselves exhibit a great deal of scatter due probably to the rather small dimensions of the test parameters (0.815 inch for the radius of the conduit and less than 0.75 inch for the depth of cover). For such small scale tests, the effects of imper- fections may be considerable. Meyerhof and Baike's tests do not really belong to the standard class of buried conduits. Quarter sections of a circular conduit rest- ing on compacted sand backfill, were loaded to failure by loads applied directly to the ends of the sheets (figure 4-11). This is a more severe condition of loading than the case encountered in practice in which the load is applied on the buried conduit through the fill. 66 (The present study falls in the later category.) Consequently, the later case should yield higher buckling pressures than the former. Table (4-7) shows that this is the case, with results from the present study (based on a minimum cover of 3 inches, and medium soil) giving consistently higher critical pressures than those of Meyerhof and Baike. However, in all but one case, the two results agree to within 20%. CHAPTER 5 DISCUSSION The present study may be broadly divided into two parts: 1) An examination of the characteristics of the coefficient of soil reaction. 2) Pre-buckling analysis and the determination of the limits of elastic stability. 5-1 COEFFICIENT OF SOIL REACTION The concept of coefficient of soil reaction is used to describe the restraint offered against the outward movement of the conduit by the supporting fill. The significant work in this area belongs to Meyerhof and Baike (1963), Kloppel and Glock (1970), and Luscher (1963) as reviewed in Chapter 2. A comparison of the theoretical formulation is presented in Table (4-9). As noted earlier, the above studies considered the soil medium to be represented by an isotropic, homogeneous, linear continuum. Further, for a Poisson's ratio of soil, VS' of 0.5 the expressions simplify to a constant value of coefficient of soil reaction given by k = S (5.1) where E8 is the soil modulus considered to be constant, and R the con- duit radius. This author believes that the assumptions inherent in equation (5.1) are quite conservative and suggests the expressions developed in Chapter 3. These expressions result from a numerical modelling of the interaction 67 68 process by means of non-linear finite element stress analysis. A non- linear constitutive relationship (the hyperbolic model) is used to represent the mechanical behavior of the granular backfill. Any realis- tic modelling of the interaction problem must include some form of non-linear stress-strain relationship for the soil medium. Furthermore, it should be possible to determine the material parameters by making use of test results from conventional soil tests. The finite element analysis on which this study is based satisfies these requirements in that the hyperbolic parameters used in Table (3-1) are taken from the results of triaxial tests by Duncan and associates (1977). For this reason, the hyperbolic constitutive relationship is clearly an improve- ment over the linear-elastic model upon which equation (5.1) is based. Furthermore, an analytical modelling of compaction and construction processes is an important feature of the finite element analysis used in the present study. Special elements are adopted to represent the behavior at the interface between the backfill and the structure. The objective of developing a simple methodology applicable to everyday problems in soil-steel structures, is accomplished by means of the concept of dimensional analysis. By such procedure, the problem sim- plifies to parameters that are more readily amenable. More importantly, such simplification in no way precludes a thorough and comprehensive treatment of the problem. While the original finite element analysis incorporates the mechanical non-linearities of the soil-structure system, the final expressions for the coefficient of soil reaction contain only such simple parameters as the depth of filling, the span of the conduit, and the direction of action - all of which can be readily defined without any ambiguity. Experimentalwo+w where the arrow means "replaced by". (Both wb and wl must be contin- uous and twice differentiable in the interval 0 STX s L). For convenience, let w1(x) E e g (x) where e is an arbitrary small constant and C (x) is an admissible but arbitrary function (i.e. g (x) satisfies the necessary geometric boundary conditions). Then, 197 L BI n n V+AV=I°[—2—(w°+82;)2+q(w°+€r;)]dx (D.3) where AV is the change in potential energy resulting from a variation in w. By expanding Equation (D.3) and subtracting Equation (D.2) from the result, the change in potential energy AV can be written as, 2&1. L 2 I (z; )2 dx (0.4) L AV = e )0 (E1 w "C" + qg) dx + E The sum of the first~order terms in the expression for AV is called the "first variation" of V and denoted by the symbol 6V. The sum of the second-order terms is called the "second variation" of V and de- noted by the symbol i-GZV. For our example, L 5v = a I. (Exwo";" + q);) dx (D.5 a-b) L 1 E1 n 362v=627f° (z; )zdx B) THE EULER EQQATIONS Consider a structure for which the integrand F is a function of one independent variable x, and one dependent variable w and its de- rivatives such that: x V = le F (x, W, W’, W") dx (D06) 198 Suppose as before, we permit an infinitesimal variation in w such that w -—-)>-wo + w with wo being an equilibrium.position. If 1' w1(x) - €§(x), 8 being an arbitrary small constant and C(x) any arbi- trary admissible function, then: x AV = [XI [F(x, wo + 6;, w: + g;’, w: + 6;") (D.7) - F(x,wo,wo',wo")dx‘ Expansion of Equation (D.7) in Taylor's series gives, for the first variation: x 1 BF 3F , as .. 5V=€Ix (fi°§+fi;§+-fi:§)dx (D.8a) The criterion for equilibrium is that the first variation of V be equal to zero, and because 8 is arbitrary we then have: x 1 BF 3E 8F , n Ix°(3—w §+'3-;.oz; +W§)dx—O (D.8b) O O 0 Repeated integration by parts yields: "1 3r d3F d2 as- Ix°(§5°“?a§37;+af§$g)§dx=° (13.9.) 199 which ultimately simplifies to: _ag.-.E-aF ...iap so aw dxaw: dx23_wf ' x