NCHRP FY 2004 Rotational Limits for Elastomeric Bearings. Final Report APPENDIX F. John F. Stanton Charles W. Roeder Peter Mackenzie-Helnwein

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1 NCHRP 1-68 FY 4 Rottionl Limits for Elstomeric Berings Finl Report APPENDIX F John F. Stnton Chrles W. Roeder Peter Mckenzie-Helnwein Deprtment of Civil nd Environmentl Engineering University of Wshington Settle, WA i -

2 TABLE OF CONTENTS APPENDIX F DEVELOPMENT OF DESIGN PROCEDURES. F-1 F.1 Derivtion of Sher Strins in the Elstomer. F-1 F.1.1 Response without Lift-off F- F Internl Stresses F- F.1.1. Bering Stiffnesses F-6 F Mximum Sher Strin F-8 F Combined Loding F-13 F Uplift nd Hydrosttic Tension F-17 F.1. Lift-off Permitted F- F. Sher Strin Cpcity F-6 F..1 Liner Model F-6 F.. Nonliner Model F-9 F...1 Theoreticl Bsis. F-9 F... Monotonic Lods F-31 F...3 Cyclic Lods F-33 F...4 Summry on Nonliner Model F-4 F.3 Axil nd Rottion Demnds from Ded nd Live Lods. F-4 F.3.1 Bckground nd Assumptions F-4 F.3. Methodology F-44 F.3.3 Computed Vlues F-44 F Upper Bound to Rottion. F-44 F.3.3. Criticl Combintion of Compression nd Rottion F-45 F.3.4 Therml Cmber F-49 F.4 Evlution of the Design Models F-5 F.4.1 Evlution Criteri F-5 F.4. Liner Model F-53 F.4.3 Nonliner Model F-53 F.4.4 Evlution Conclusions F-56 F.5 Detiled Development of Specifiction Provisions F-58 F.5.1 Method B F-58 F.5. Method A F-6 F.5.3 Discussion of Methods A nd B. F-69 F.6 Summry nd Conclusions F-7 F.6.1 Summry nd Conclusions on Computtion of Cyclic Sher Stress F-7 F.6. Summry nd Conclusions on Cyclic Sher Stress Cpcity F-71 - ii -

3 F.6.3 Summry nd Conclusions on Cyclic Sher Stress Demnd F-71 F.6.4 Summry nd Conclusions on Model Evlution F-7 LIST OF FIGURES Figure F.1. Displcement Fields for Component Lodcses....F- Figure F.. Stresses in the Elstomer due to Rottion nd Compression Lod...F-5 Figure F.3. Stiffness Coefficient B (from Stnton nd Lund)....F-7 Figure F.4. Stiffness Coefficient B r (from Stnton nd Lund)....F-7 Figure F.5. Sher Strin Coefficient C....F-11 Figure F.6. Sher Strin Coefficient C r....f-11 Figure F.7. Sher Strin Coefficient D....F-1 Figure F.8. Sher Strin Coefficient D r....f-1 Figure F.9. Hydrosttic Tension under Combined Compression nd Rottion...F-14 Figure F.1. Distribution of Verticl Strin cross the Bering....F-15 Figure F.11. Distribution of Verticl Stress cross the Bering....F-16 Figure F.1. Distribution of Sher Strin cross the Bering...F-16 Figure F.13. Hydrosttic Tension vs. Rottion for Vrious Compression Strins. S = 6....F-18 Figure F.14. Normlized Hydrosttic Stress s Function of α nd λ...f-19 Figure F.15. Lift-off: Assumed Bering Behvior...F-1 Figure F.16. Totl Sher Strin for Bering with Lift-off....F-4 Figure F.17. Post Lift-off Conditions: Rtio of Approximte nd True Totl Sher Strin...F-5 Figure F.18. Effective Strin vs Number of Cycles: 5% Debonding...F-8 Figure F.19. Correltion Coefficient vs Cyclic Fctor....F-8 Figure F.. Effective Strin vs Number of Cycles: 5% Debonding...F-9 Figure F.1. Tests CYC5-1 Debonding for Btch B1 Berings....F-34 Figure F.. Debonding vs. Number of Cycles. Test CYC5....F-36 Figure F.3. Debonding vs. Number of Cycles. Test CYC7....F-36 Figure F.4. Debonding vs. Number of Cycles. Test CYC9....F-37 Figure F.5. Debonding vs. Number of Cycles. Test CY11...F-37 Figure F.6. Debonding vs. Number of Cycles. Test CYC1....F-38 Figure F.7. Debonding vs. Number of Cycles. Test CYC15....F-38 Figure F.8. Ftigue Life for Cyclic Sher Strin due to Rottion: 5 % Debonding Criterion....F-39 Figure F.9. Ftigue Life for Cyclic Sher Strin due to Rottion: 5 % Debonding Criterion....F-39 Figure F.3. Ftigue Life for Cyclic Sher Strin due to Rottion: 75 % Debonding Criterion....F-4 Figure F.31. Ftigue Life for 5 % Debonding Criterion. Eqution (F-67) vs Mesured Dt...F-41 Figure F.3. Effect of Spn on Rottion Angle....F-46 Figure F.33. Effect of Girder Stiffness on Rottion Angle...F-47 Figure F.34. Effect of Lne Lod Fctor on Rottion Angle...F-47 - iii -

4 Figure F.35. Effect of Girder Stiffness on Totl Sher Strin....F-48 Figure F.36. Effect of Truck Loction on Mximum Sher Strin...F-48 Figure F.37. Effects of Spn nd Truck Loction on Mximum Sher Strin....F-49 Figure F.38. Girder nd AASHTO /Design Therml Grdient (Zone 1)...F-5 Figure F.39. Rottion due to Therml Grdient on Girder....F-51 Figure F.4. Strins due to Therml Grdient....F-51 Figure F.41. Predicted Debonding for Exmple Bering....F-54 Figure F.4. Compression Ftigue: Predicted vs. Observed Debonding...F-56 Figure F.43. Axil Stress Rottion Interction Digrm. 9 x Bering, 4 Lyers t.5...f-61 Figure F.44. Axil Stress Rottion Interction Digrm x 14.5 Bering, 3 Lyers t.5...f-6 Figure F.45. Axil Stress Rottion Interction Digrm x 14.5, 4 Lyers t f-6 Figure F.46 Allowble normlized xil stress s function of S /n....f-63 Figure F.47 Allowble xil stress s function of S /n....f-64 Figure F.48 Allowble xil lod s function of S /n...f-65 Figure F.49 Interction Digrm: Effect of Aspect Rtio. (n = 3, h ri =.5 for ll)...f-66 LIST OF TABLES Tble F-1. Computed Sher Strins t Initition of Debonding in Monotonic Tests...F-3 Tble F-. Multipliers for Rottion Angle due to Girder Continuity....F-45 - iv -

5 APPENDIX F Development of Design Procedures. This ppendix describes the development of design procedures. Tht subject requires discussion of five primry topics, ech of which is treted in seprte subsection. The focus is on sher strins cused by the vrious lodings, becuse they re the criticl quntity tht determines whether debonding occurs. Section F.1 ddresses sher strin demnd, nd in prticulr it describes the reltionships between externl lods nd internl sher strins. Section F. ddresses sher strin cpcity. Two models re developed from the test results, nd they link the mplitude of the totl sher strin to the level of debonding dmge. This is done for both monotonic nd cyclic loding. Section F.3 ddresses loding demnd. The forces nd rottions tht the bering experiences must be known in order to determine the sher strins tht they cuse. These forces nd rottions should be evluted for ech bridge during its design, but estimtes of them re obtined in this section from typicl bridge geometries nd lodings to guide the development of the proposed bering design specifictions.. Section F.4 detils the evlution of the two different models for sher strin cpcity, nd Section F.5 describes the process of developing detiled design provisions from the foregoing sections. Section F.6 contins summry of the findings of the ppendix. F.1 Derivtion of Sher Strins in the Elstomer. Determintion of strins in the elstomer is complicted, becuse the mteril is nerly incompressible, is nonliner, nd experiences very lrge strins. An ccurte evlution is possible only with Finite Element Anlysis, s described in Appendix E, nd even then it presents significnt chllenges. A simpler, lbeit pproximte, pproch to nlysis is needed for design. Gent nd Lindley (e.g. Gent nd Lindley, 1959, Gent nd Meinecke, 197, Lindley nd Teo, 1978) pioneered the nlysis of lminted berings nd developed nd presented linerized nlysis procedure. Conversy (1967) extended it to llow for finite vlues of the bulk modulus, nd Stnton nd Lund (4) provided numericl vlues of ll the necessry coefficients, for different bulk modulus vlues. Tht pproch forms the bsis of the procedure used for the design method used in this reserch, nd is summrized here. It is pproximte, becuse it ssumes prbolic distribution of displcement through the thickness of the elstomer, but, s the Finite Element Anlyses show, tht pproximtion proves to be remrkbly good, nd, for the geometries nd stresses used in prcticl berings, the errors re smll compred with those rising from other sources, such s chrcteriztion of mteril properties. Its simplicity compred with ny other lterntives mkes it n ttrctive choice. Two types of behvior must be distinguished. The first is defined here s uplift n refers to ny sitution in which the loding surfce (i.e. the sole plte between the girder nd bering) remins in full contct with the bering t ll times. This lwys occurs when the bering hs bonded externl pltes. It my lso occur in the bsence of bonded externl pltes if the compression is lrge enough to prevent ny seprtion. The most - F-1 -

6 importnt consequence is tht hydrosttic tension stress my be induced in the elstomer if the rottion demnd is lrge reltive to the bering s rottion cpcity. In the second type of behvior, referred to here s lift-off, the rottion is lrge enough tht the loding surfce seprtes from the bering over prt of its re. By definition this cn only occur in the bsence of bonded externl pltes. From n nlyticl viewpoint, the first behvior involves constnt boundry conditions, while in the second the boundry (i.e. the region in contct with the loding surfce) chnges with the loding. Those ltter conditions constitute contct problem, for which nlysis is much more complex. The two types of behvior re ddressed in Sections F.1.1 nd F.1.. F.1.1 Response without Lift-off F Internl Stresses The internl stresses in the elstomer re developed here. They demonstrte fundmentl behviorl properties of the bering nd re used in subsequent sections. The nlysis is bsed on the following ssumptions: The rubber is perfectly bonded to the steel pltes. The steel pltes re rigid in both xil tension nd flexure. No edge cover exists. The rubber bulges lterlly in prbolic shpe. In the liner theory developed by Gent nd Lindley (1959) the internl stress stte cn be shown to consist of two sets of stresses superimposed on ech other. The displcement fields tht correspond to them re shown for single bonded lyer in Figure F.1. Lod cse 1 Lod cse Figure F.1. Displcement Fields for Component Lodcses. In the first lodcse, the elstomer is treted s being perfectly debonded from the steel pltes bove nd below it. Under pure xil lod, it shortens verticlly nd spreds lterlly, without ny resistnce from the steel pltes, due to the Poisson effect. For this loding, s for ll lodings, Poisson s rtio, ν, is very nerly equl to.5 becuse the elstomer is lmost incompressible. - F- -

7 The stresses nd strins re described in set of Crtesin coordintes, in which z is verticl, x is horizontl nd prllel to the bridge xis nd y is horizontl in the trnsverse direction. The verticl stress nd the locl sher strin t the edge re given by σ zz = Eε zz (F-1) γ = (F-) xz Here, positive stress nd strin re tensile. In the second lodcse, the verticl displcement is held constnt, nd horizontl stresses re pplied to the top nd bottom surfces of the elstomer, which re pulled inwrds to their originl horizontl positions. The nerly-constnt volume property cuses the lyer to bulge t the mid-thickness s its top nd bottom surfces re pulled inwrds. For this lodcse, the verticl stress nd the sher strin re controlled (Stnton nd Lund, 4) by t K 1G σ σ zz zz = Kε zz (F-3) dσ xx dτ + dx dz zx = (F-4) where G = sher modulus of the elstomer K = bulk modulus of the elstomer ε zz = verticl strin σ zz = verticl direct stress σ xx = horizontl direct stress τ zx = sher stress In Gent s theory, the stress t point is lso ssumed to be hydrosttic, so σ xx, σ yy nd σ zz re ll equl. For the generl cse of bering whose pln shpe is rectngle of rbitrry spect rtio, the problem is 3-D nd closed form solutions re not vilble. The equtions my be solved using series solution (e.g. Conversy 1967) or by numericl integrtion (e.g. Stnton nd Lund 4). 3-D Finite Element nlysis is lso chllenge becuse the nonlinerity of the problem requires n itertive solution, nd, given the mesh refinement needed, the computtionl demnds become enormous. However, for n infinite strip, the nlysis becomes -D, nd closed form solutions re possible. Furthermore, Finite Element Anlysis of the -D system is computtionlly fesible, nd my be used to explore behvior nd to verify the simpler liner theory. The primry results for n infinite strip re presented here. Eqution (F-3) my be simplified by using the Compressibility Index, λ, developed by Stnton nd Lund - F-3 -

8 λ = S 3G (F-5) K where S = Shpe Fctor loded pln re LW = = perimeter re free to bulge t ( L + W ) (F-6) In Eqution (F-6), L is the bering length, W is the width nd t is the elstomer lyer thickness. The Compressibility Index indictes the extent to which bulk compressibility of the elstomer ffects the response, nd rises nturlly in the development of the equtions. x It is lso convenient to define the dimensionless coordinte, ξ =, where the origin is L t the center of the bering. Thus ξ = ±1 correspond to the edges. Then Eqution (F-3) becomes, for -D system in the x-z plne, 1 d λ σ zz dξ ( ξ ) σ zz ( ξ ) = Kε ( ξ ) zz (F-7) For pure xil loding, the verticl strin, ε zz (ξ), is constnt cross the cross-section nd therefore independent of ξ, so the verticl stress is obtined by tking ε zz (ξ) s constnt vlue, ε, in Eqution (F-7), which cn be solved to give ( λξ ) ( ) cosh σ zz ( ξ ) = K ε 1 (F-8) cosh λ The sher strin t the top or bottom surfce of the lyer is then ( λξ ) ( ) λ sinh γ zx ( ξ ) = 6S ε (F-9) λ cosh These solutions re for the generl cse of slightly compressible mteril, for which the compressibility is defined by λ. For completely incompressible mteril, λ =, nd the equtions become indeterminte. Equtions (F-8) nd (F-9) cn be solved by using binomil expnsion, nd the equtions reduce to nd σ γ zz xz ( ξ ) Eε ( 1 ξ ) = ( ξ ) 6Sε ξ S (F-1) = (F-11) The distribution of stress in the elstomer under compressive lod is illustrted on the right side of Figure F.. - F-4 -

9 Direct stress Compression Tension Sher stress Figure F.. Stresses in the Elstomer due to Rottion nd Compression Lod. For pure rottion, the verticl strin is distributed linerly cross the bering surfce, i.e. ε zz θ = L (F-1) t L ( ξ ) x = Sθ ξ where θ L = the rottion per lyer of the bering. For the generl, compressible, cse the verticl stress is derived from Eqution (F-7) s ( λξ ) ( ) sinh σ zz ( ξ ) = KS θ L ξ (F-13) sinh λ γ zx ( ξ ) ( λξ ) ( ) S λ cosh = 6 θ 1 L ( λ) sinh λ (F-14) These my lso be simplified for the specil cse of complete incompressibility (λ = ), to give nd 3 3 ( ξ ) θ ( ξ ξ ) σ zz = ES (F-15) L 3 - F-5 -

10 ( ξ ) θ ( 1 3ξ ) γ zx = S L (F-16) The distribution of stress in the elstomer due to rottion is illustrted in Figure F.. These equtions re used in Sections F.1.1. nd F to define the two most importnt bering properties: stiffness nd pek sher strin. F.1.1. Bering Stiffnesses The bering stiffnesses in response to xil lod nd rottion re needed for nlysis. The xil stiffness is needed for computing the xil deflection, which is in turn needed for evlution of hydrosttic tension. The rottionl stiffness is used for checking stbility of the bering, for verifying lterl-torsionl buckling of the girder (Mst 1993) nd, if the bering is equipped with slider, for determining whether the stinless steel will lift off from the PTFE. In most cses, under-estimting the stiffness leds to sfe prediction (i.e. it is conservtive choice). The mjor exception is hydrosttic tension, for which low estimte of stiffness is likely to led to n unsfe prediction of stress (i.e. internl frcture will be predicted not to occur when if fct it will). Stnton nd Lund (4) show tht these responses cn be expressed in terms of the shpe fctor, S, of the bering lyer. The xil nd rottionl stiffnesses of one lyer re: where K K ( A + B S ) P EA = = (F-17) Δ t ( A B S ) M EI = = r r (F-18) θ t r + L S = shpe fctor M = moment E = Young s modulus ( 3G) A, B, A r, B y = dimensionless constnts Δ = xil displcement θ L = rottion ngle pplied to ech lyer of the bering. For common S vlues (4 to 8), the A nd A r terms re smll compred with the B S nd B r S terms nd my be ignored in the interests of simplicity, with little error. Their vlues re given by Gent nd Lindley s A = A r = 1. for rectngulr shpes, nd for n infinite strip. Vlues for B nd B r re given in Figure F.3 nd Figure F.4, tken from Stnton nd Lund (4). They re functions of L/W, nd λ, the Compressibility Index. The xil stiffness coefficient, B, is shown only up to L/W = 1.. For lrger vlues of L/W, B cn be found by interchnging L nd W. - F-6 -

11 .5 B vs Aspect Rtio B true, lmbd =. B ppr, lmbd =. B true, lmbd =. B B ppr, lmbd =. 1.5 B true, lmbd =.4 B ppr, lmbd =.4 1 B true, lmbd =.6 B ppr, lmbd =.6.5 B true, lmbd =.8 B ppr, lmbd = Aspect Rtio B true, lmbd = 1. B ppr, lmbd = 1. Figure F.3. Stiffness Coefficient B (from Stnton nd Lund). B r B r vs. Aspect Rtio Br true, lmbd = Br true, lmbd =. Br true, lmbd =.4 Br true, lmbd =.6 Br true, lmbd =.8 Br true, lmbd = 1 Br ppr, lmbd = Br ppr, lmbd =. Br ppr, lmbd =.4 Br ppr, lmbd = Aspect Rtio Br ppr, lmbd =.8 Br ppr, lmbd = 1 Figure F.4. Stiffness Coefficient B r (from Stnton nd Lund). Empiricl pproximtions for these quntities in rectngulr berings, intended for use in the rnge < λ < 1., re: - F-7 -

12 B ( ) + ( λ) ( 1 min{ L W, W L} ) λ (F-19) (.4.4 ) + ( λ) ( 1 exp(.64l W )) B r λ / (F-) In figures F-3 nd F-4, the true vlues re shown s solid lines with solid symbols, nd the empiricl pproximtions re shown s dshed lines nd open symbols. For circulr berings, Gent nd Meinecke (197) give B =. nd B r = /3 for the incompressible cse. By using the pproximtion proposed for compression by Gent nd Lindley (1959), the effective modulus of elsticity my be expressed s 1 E eff 1 1 EB S + K (F-1) from which the stiffness coefficient B for the compressible cse becomes B (F-) 1+ λ Similrly, for rottion, 1 B r (F-3) λ For most elstomers used in bridge berings, K 45, psi, nd G 11 psi so, for S = 6, λ.167 nd the vlues of prmeters such s B differ little from those of completely incompressible mteril, for which λ =.. Becuse in this pproch the problem is chrcterized s liner, superposition is vlid nd the solutions from different lod-cses, such s compression nd rottion, cn be dded directly. This gretly simplifies the clcultions. F Mximum Sher Strin The mximum sher strin in the elstomer is of interest becuse it is often used s design criterion. It cn be obtined from Eqution (F-4). The results re, for xil lod nd rottion respectively, γ = ( C S) ε (F-4) γ r L = ( CrS) θ L (F-5) t where C, C r = dimensionless constnts γ = mximum sher strin cused by xil lod - F-8 -

13 γ r = mximum sher strin cused by rottion Equtions (F-17) nd (F-4) my lso be combined to give the sher strin due to xil lod directly in terms of the stress. This my be done most simply by ignoring the A term in Eqution (F-17), on the bsis tht it is much smller thn the B S term. Then σ γ = CSε CS = D 3GB S σ GS (F-6) where D C 3B = (F-7) Similrly, the sher strin cused by rottion my be expressed in terms of the L/t rtio lone (nd not the shpe fctor) s γ r L = Dr θ L t (F-8) where D r = Cr 4 + (F-9) ( 1 L / W ) For typicl berings, S is bout 6, B is bout, nd A is 1., nd the error resulting from ignoring the A nd A r terms is pproximtely 1.5%. Vlues for C, D, C r nd D r re shown in Figure F.5 through Figure F.8. Empiricl pproximtions for the sher strin coefficients, vlid in the rnge < λ <1., re where ( (.667 L / W ) )* ( 1.75λ +.14λ ) C (F-3) {( λ )( 1 + L W +.66λ ), ( λ) } C r min (F-31) D { d,( d d * L W )} mx 1 + / (F-3) - F-9 -

14 d d = λ * + (.1 *.413) 1 λ (.71 *.46) = λ * + λ ( * (.47) ) d 3 = λ * λ λ D r min, λ + L W (F-33) In Figure F.5 through Figure F.8, the true vlues re shown s solid lines with solid symbols, nd the empiricl pproximtions re shown s dshed lines nd open symbols. The vlue of D remins essentilly constnt for L/W > 3. To minimize possible confusion over the vlue S of W nd L, convention is needed. The one used here is tht W is lwys the length of the side prllel to the xis of rottion under considertion. This holds true for computing both stiffness nd sher strin coefficients. Usully, the bering will experience rottion bout its wek xis, so W will be the length of the long side nd L, the length of the short side. Thus, for 1 in. x in. bering, L = 1 in. nd W = in. for bending bout the wek xis, but L = in. nd W = 1 in. for bending bout the strong xis. The coefficients D nd D r given in Equtions (F-3) nd (F-33) compute the sher strin on the side of the bering, of length W, tht is the one prllel to the xis of rottion. Becuse both strins occur in the sme plce, they re dditive. Note tht, under xil lod lone, the lrgest sher strin occurs on the long side of the bering. For the common cse of rottion bout the wek xis, the lrgest sher strins due to both xil nd rottion loding individully occur in the sme plce, so sher strins need only be clculted there. If the primry loding is bout the strong xis, the lrgest totl sher strin my occur either t the long side (due to xil lod lone) or t the short side (due to xil plus rottion effects). Both must be clculted, nd the lrger controls. For circulr bering, D nd D r my be pproximted s D =1. (F-34) D =.375 (F-35) r The vlues for D nd D r in Equtions (F-34) nd re derived from Gent nd Meinecke (197). They re lso consistent with the equtions in the existing AASHTO LRFD Design Specifictions. They re lower thn the vlues for rectngulr berings, thereby demonstrting tht, for the sme rottion or verge xil stress, the circulr bering will experience lower sher strin. - F-1 -

15 C C vs Aspect Rtio C true, lmbd =. C true, lmbd =. C true, lmbd =.4 C true, lmbd =.6 C true, lmbd =.8 C true, lmbd = 1. C ppr, lmbd =. C ppr, lmbd =. C ppr, lmbd =.4 C ppr, lmbd = Aspect Rtio C ppr, lmbd =.8 C ppr, lmbd = 1. Figure F.5. Sher Strin Coefficient C. Cr C r vs. Aspect Rtio Aspect Rtio Cr true, lmbd = Cr true, lmbd =. Cr true, lmbd =.4 Cr true, lmbd =.6 Cr true, lmbd =.8 Cr true, lmbd = 1 Cr ppr, lmbd = Cr ppr, lmbd =. Cr ppr, lmbd =.4 Cr ppr, lmbd =.6 Cr ppr, lmbd =.8 Cr ppr, lmbd = 1 Figure F.6. Sher Strin Coefficient C r. - F-11 -

16 D D vs L/W true lmbd =. true lmbd =. true lmbd =.4 true lmbd =.6 true lmbd =.8 true lmbd = 1. ppr. lmbd =. ppr. lmbd =. ppr. lmbd =.4 ppr. lmbd =.6 ppr. lmbd =.8 ppr. lmbd = L/W Figure F.7. Sher Strin Coefficient D. Dr D r vs Aspect Rtio Aspect Rtio lmbd = lmbd =. lmbd =.4 lmbd =.6 lmbd =.8 lmbd = 1 lmbd = lmbd =. lmbd =.4 lmbd =.6 lmbd =.8 lmbd = 1 Figure F.8. Sher Strin Coefficient D r. Exmple. Consider bering with four internl lyers ech x 1 x ½ thick, mde from rubber with G =.135 ksi, K = 45 ksi, nd subjected to compressive lod of kips nd rottion of.1 rdins bout its wek xis. The shpe fctor is - F-1 -

17 LW S = = t The compressibility index is λ = S 1* ( L + W ) *.5* ( 1 + ) 3G K = * = 6.66 =. From Figure F.3 nd Figure F.4, B = 1.85 nd B r =.5 (exct vlues), so the xil stiffness of the lyer is ( A + BS ).45*( *6.66 ) = = 1348 kip / in P EA K = = Δ t The lod of kips cuses n xil strin of.5 kips ε =.5in =.97in / in =.97% 1348kips / in The rottionl stiffness of the lyer is K r M EI = = θ L ( ) ( A B S ) r + r.45* *6.66 = = 313,5 in kip / rd t so the rottion of θ L =.1 rd/4 lyers =.5 rd/lyer induces moment of M = 313,5*.5 = 784 in kips For λ =. nd L/W = AR =.5, the coefficient vlues re C = 7.6 nd C r =.9. The pek sher strins due to combined xil force nd rottion re: γ, mx C Sε = 7.6*6.66*.97 = 1.5 = L θ L 1.5 γ r, mx = CrS =.9*6.66* * =.48 t.5 Note tht, for these clcultions, the gross dimensions of the berings were used. This ignores the fct tht the cover rubber behves slightly differently thn the rubber in the core of the bering. The specifictions proposed in Appendix G include slight refinement of the definition tht ccounts pproximtely for the difference. F Combined Loding A bering subjected to combined compression nd rottion my be nlyzed using the liner theory outlined in Sections F through F There re two mjor questions of interest: whether bering without externl pltes experiences lift-off t the tension edge, nd consequently higher sher strins on the compressive side, nd whether bering with externl bonded pltes experiences internl hydrosttic tension. Figure F.9 illustrtes the development of hydrosttic tension. Note tht the mximum hydrosttic tension typiclly does not occur t the edge. Recll tht the liner theory.5 - F-13 -

18 ssumes tht the direct stress t point is the sme in ll directions, so it is lso equl to the hydrosttic stress. The totl verticl stress in the elstomer is obtined for n infinite strip by dding Equtions (F-8) nd (F-13) nd including the terms from the first lodcse. The result, given in Eqution (F-36) is lso written in terms of the coefficients A, etc., so tht the eqution my be extended for use with spect rtios other thn n infinite strip. + = Stress Compression. Tension Figure F.9. Hydrosttic Tension under Combined Compression nd Rottion. σ zz ( ξ ) ( λξ ) ( λ) = A E + BK 1 ε + Ar Eξ Br K ξ 4 cosh sinh 3 cosh sinh (F-36) ( λξ ) Sθ L ( λ) The chrcteristics of the totl response depend on the reltive mgnitudes of ε nd Sθ L, which represent the mgnitudes of the individul lodings. For this purpose it is convenient to introduce the vrible θ c, defined s ε θc = (F-37) S The strin ε is negtive for compression, in which cse θ c >. Physiclly, θ c represents the rottion t which the verticl displcement on the tension side of the bering strts to become net upwrds. It is clled the chrcteristic rottion. To find the loction of the mximum verticl stress, Eqution (F-36) must be differentited nd the result set to zero. Doing this nd using the fct tht leds to cosh ( ) = sinh ( λξ ) + 1 λξ (F-38) - F-14 -

19 [ H H ] sinh ( ) + [ F H ] sinh( λξ ) + [ F H ] = r λξ (F-39) r r r Where H λ =.75B K ε (F-4) cosh ( λ) H F r r λ = 3.75Br K θls (F-41) sinh ( λ) ( EA KB ) θ S = (F-4) r r L Eqution (F-39) is qudrtic in sinh(λξ), which cn be solved in closed form. The vlue of ξ, the loction of the pek hydrosttic stress, cn be extrcted from it nd substituted bck into Eqution (F-36) to find the pek hydrosttic stress. If ξ is less thn 1., the mximum verticl stress occurs within the bering nd the elstomer experiences verticl tension nd, by impliction, hydrosttic tension. If ξ > 1., the pek stress occurs outside the bering, so the result is of no prcticl interest from the viewpoint of hydrosttic tension, since the verticl stress everywhere within the bering is compressive..8.6 Verticl Strin vs x/l.4 Verticl Strin vert strin xil vert strin rot vert strin totl x/l Figure F.1. Distribution of Verticl Strin cross the Bering. - F-15 -

20 Direct stress/e Direct stress/e vs x/l direct stress/e xil direct stress/e rot direct stress/e totl x/l.5 1 Figure F.11. Distribution of Verticl Stress cross the Bering. 3 Sher Strin vs x/l 1 Sher Strin sher strin xil sher strin rot sher strin totl x/l Figure F.1. Distribution of Sher Strin cross the Bering. Figure F.1 through Figure F.1 show the verticl stress, the verticl strin nd the sher strin cross the elstomer lyer for the cse θ = θ c. In the figures, negtive stress nd strin indicte compression. (ε = -.3 in/in ws used in this cse, but tht fct does not ffect the figures). For these conditions the tension side of the bering experiences net upwrd displcement, or verticl tensile strin. However, the totl verticl stress is everywhere compressive. This implies tht the sole plte does not lift off from the elstomer lyer. The fct tht verticl tension strin does not necessrily led to verticl tension stress represents behvior tht differs from tht of conventionl mterils. It is explined by the - F-16 -

21 fct the elstomer s ner incompressibility cuses the mteril to shift lterlly s well s verticlly. Some rubber from the compressed side is indeed forced outwrds into the bulge on tht side. However, some of it is lso pushed cross the bering to the tension side, where it occupies the spce left by the upwrd movement of the top plte on tht side. This behvior is confirmed by the fct tht the signs of the sher strins t the two edges re opposite (see Figure F.1), showing tht both fces re bulging outwrds. If the tension fce bulges outwrd, the hydrosttic stress just inside it must be compressive. Tht result is confirmed in Figure F.11. It is shown in Section F tht, in the bering with externl pltes, the hydrosttic stress first becomes tensile only when the rottion reches vlue of pproximtely 3θ c. At tht rottion, the sher strin on the tension fces chnges sign, so the edge bulges inwrds, confirming the existence of hydrosttic tension. Furthermore, in berings with no externl pltes, lift-off lso strts t rottion of θ = 3θ c, since the interfce cn crry no tension stress. Thus the tension side must rise due to rottion through three times the distnce required to rech its originl elevtion, in order to cuse sher strin reversl t the edge or hydrosttic tension in the interior of the elstomer. F Uplift nd Hydrosttic Tension If externl pltes re bonded to the bering, rottion my cuse hydrosttic tension in the interior of the rubber. Excessive hydrosttic tension my cuse the rubber to rupture (Gent nd Lindley, 1959b). A comprehensive nlysis of the internl stresses is complex, becuse it must ddress both 3-D effects nd finite compressibility. However, simplified solution cn be obtined by pplying the liner theory to n infinite strip lyer, s ws done in Section F An infinite strip llows the nlysis to be -D, which simplifies the closed form nlysis nd lso mkes Finite Element Anlysis fesible. The nlysis described below is confirmed by the Finite Element studies described in Appendix E. Equtions (F-36) through (F-4) presented wy of finding the pek verticl stress in the elstomer, including the effects of mteril compressibility. However, the equtions re little complex. They cn be simplified by specilizing them for the geometry of the infinite strip nd by ssuming tht the mteril is completely incompressible. Those ssumptions reduce Eqution (F-36) to ( ξ ) = E( ξ 3α )( 1 ξ ) σ hyd (F-43) 3 where ξ = x/l is mesured from the center of the bering lyer nd ε θ α = c = (F-44) Sθ L θl Note tht ε is negtive for compressive strin t the center of the bering, so in most cses, α will be positive. Eqution (F-43) shows tht the distribution of hydrosttic stress is governed by the single vrible α. This observtion simplifies the clcultions significntly. - F-17 -

22 3.5 Hydrosttic tension S = 6 Tension stress/e eps = eps = -.15 eps = Rottion/lyer (rd) Figure F.13. Hydrosttic Tension vs. Rottion for Vrious Compression Strins. S = 6. For specific berings subjected to specific lodings, ε nd θ L, figures such s Figure F.13 cn be prepred. It shows the pek tensile stress, normlized with respect to E, for vrious ε nd θ L vlues, for bering with S = 6 nd incompressible mteril. Negtive xil strin indictes compression. The lower regions of the curves for ε re curviliner becuse the loction of the pek hydrosttic stress moves inwrds towrds the center of the bering s the rottion increses while the xil strin remins constnt. Figure F.13 shows the results for single bering, but results for rnge of berings cn be shown on single dimensionless plot. This is chieved by finding the loction, ξ, of the mximum hydrosttic stress from Eqution (F-43), nd substituting bck to find the pek stress. The vlue of ξ tht mximizes σ hyd (ξ) is given by 1 ξ mx = α + α + (F-45) 3 If ξ mx > 1, the direct stress is compressive everywhere within the bering nd no hydrosttic tension exists. This is the cse when α > 1/3, which implies tht the rottion per lyer lies below 3θ c. For design, the problem is likely to be to determine the mximum hydrosttic tension stress, if ny, tht exists under prticulr combintion of xil lod nd rottion. It is obtined by using in Eqution (F-43) the vlue of ξ found from Eqution (F-45). The result, fter normliztion, is - F-18 -

23 1.5 σ hyd,mx 4 1 σ hyd, norm = = f ( α ) = α( 1 α ) + α + 3 (F-46) ES θ L 3 3 This representtion is convenient becuse it llows ll combintions of shpe fctor, xil strin nd rottion to be shown on single plot. The constnt of normliztion in Eqution (F-46) contins the term S 3 so, for given α, the hydrosttic stress rises rpidly with incresing S. The configurtion most likely to led to problems with hydrosttic stress is therefore bering with externl bonded pltes, high shpe fctor, light xil lod, nd lrge rottion. This combintion of events my occur in steel bridges during construction. If bolting or welding the bering to the girder relly is essentil for the service lod condition, the possibility of temporry hydrosttic tension during construction my be voided by leving the berings unttched until fter the deck hs been cst. By then, the initil rottion due to cmber, nd the potentil for internl rupture in the bering, will lrgely hve been eliminted. Similr clcultions, but including compressibility, were conducted using Equtions (F-36) through (F-43). The hydrosttic stress ws gin normlized using with respect to ES 3 θ L. It ws found tht ll berings with single vlue of λ, regrdless of the individul vlues of S, E nd K, ll ly on single curve. Curves for different λ vlues re shown in Figure F.14, with the pproximte curve for complete incompressibility, mrked pprox nd computed from Eqution (F-46). Common berings hve S 6 nd λ.. In tht rnge, the pproximte incompressible curve in Figure F.14 gives vlues tht re very close to the exct ones. Given the other pproximtions in the clcultions, it is proposed tht Eqution (F-46) be ccepted s stisfctory design eqution for computing the pek hydrosttic tension stress, regrdless of the compressibility of the elstomer. It ws developed for n infinite strip, but is conservtive for other geometries..6.5 σhy d/(e*s 3 *θl) vs α σ hyd/(e*s 3 *θl) lmbd = lmbd =.4 lmbd =.8 lmbd = 1. pprox lph Figure F.14. Normlized Hydrosttic Stress s Function of α nd λ. - F-19 -

24 To illustrte the use of Eqution (F-46) nd Figure F.14, consider bering with bonded externl pltes nd S = 1. It is subjected to loding tht cuses ε = -.1 in /in nd θ L =.1 rdins per lyer. Then α = f σ ε.1 = =.833 Sθ 1 *.1 L ( α ) = α( 1 α ) + α + =. 154 = ES 3 θ f α 1.5 ( ) = E( 1 3 *.1*.154). E hyd, mx L = 66 Gent nd Lindley (1959b) nlyzed rupture under hydrosttic tension using frcture mechnics nd, to confirm their findings, they conducted experiments in which they pplied pure tension to smll circulr rubber lyers bonded to externl pltes. They found tht sudden rupture occurs when the hydrosttic tension stress reches stress of σ rupture. 9E (F-47) The internl rupture is consequence of the locl stress stte in the rubber nd not the overll loding. Thus, the fct tht Gent nd Lindley s work ws bsed on pure tension, but the problem here is cused by combined compression nd rottion, is immteril nd use of Eqution (F-47) s rupture criterion is vlid. F.1. Lift-off Permitted If lift-off is free to occur becuse no externl pltes exist, the problem becomes geometriclly nonliner, nd therefore more difficult, even if the mteril nonlinerity is ignored on the bsis tht the displcements re smll. An pproximte nlyticl pproch cn be developed for the -D problem of n infinite strip, nd is presented here. Consider n infinite strip bering lyer of length L. A rigid sole plte rests on it, inclined t n ngle θ, nd supports lod per unit width, p, smll enough tht the plte mkes only prtil contct with the elstomer lyer, s shown in Figure F.15. The following ssumption is mde: The bering my be divided into two prts: the one on the unloded side experiences no verticl lod nd no internl stress or strin, while the one on the loded side behves s though the verticl edge t the neutrl xis were free surfce. The pproximtions implied by these ssumptions re: The verticl (xil) stress on the unloded region is zero. This is not quite true becuse the steel plte will be bent t the neutrl xis nd the out-of-plne sher stresses in it will cuse some verticl stress on the elstomer. - F- -

25 No interction exists between the two regions of the bering. This is lso not quite true becuse the profile of the left edge of the loded prt (s shown) will not be completely free from sher strin. L/ L/ ηl Figure F.15. Lift-off: Assumed Bering Behvior. The cross-section of one lyer of the bering is shown in Figure F.15. The instntneous shpe fctor, S i, of the right hnd, compressed, region is given by S ηl = i (F-48) t where the distnce ηl is the length of the compressed region. The dividing line between the two prts of the bering is tken to be the point t which contct is lost. The length, ηl, of the compressed region is relted to the xil strin nd rottion by the fct tht α, s defined in Eqution (F-44), hs the vlue 1/3 for the right hnd region. (This is consequence of its being t incipient lift-off). The xil strin in the middle of the instntneous compressed region, ε i, nd the rottion ngle re then relted by S i 3ε θ i = (F-49) L where ε i is negtive if the strin is compressive, s it is expected to be. The subscript i indictes the instntneous vlue, corresponding to the compressed region s covering only prt of the totl re of the bering. The lod per unit width on the bering is p, (negtive if compressive) nd it is relted to the verge strin in the compressed region by ( A + B S ) ε EB S p = σ i = ε ie i i i (F-5) ηl This cn be combined with Equtions (F-48) nd (F-49) to give - F-1 -

26 S i 3p B teθ = 4 (F-51) L Eqution (F-51) reltes ηl, the length of the compressed region, to the loding prmeters p nd θ L. (It does so indirectly, since S i is function of ηl). The rottion per lyer, θ L, is expressed s multiple, ρ, of the chrcteristic rottion for the full bering section, θ c, θ L = ρθ c (F-5) p θ c = ε = (F-53) 4 S S EB t Note tht the coefficient ρ is the inverse of α, used in Section F In Equtions (F-5) nd (F-53), the subscript refers to the properties of the complete bering, rther thn the instntneous loded portion. The physicl mening of θ c is tht, when it is pplied t the sme time s the lod p, the verticl displcement on the tension edge of the bering is zero. As explined in SectionF.1.1.4, this is not the sme s the initition of lift-off. These two equtions cn be substituted into Eqution (F-44) to give η = 3 S = i 4 (F-54) S ρ Coefficient η defines the proportion of the bering re tht is subjected to compressive stress, nd ρ defines the mplitude of the rottion. For ρ >3, η < 1, lift-off occurs nd the compressed region is smller thn the bering surfce. The components of sher strin my now be computed s γ p p ρ = CSiε i = C = C (F-55) teb S teb S 3 i p γ r = CrSi θl = Cr 3ρ (F-56) teb S These cn be dded to give γ tot p ρ = ( C 3 ) + Cr (F-57) teb S 3 Eqution (F-57) my lso be expressed in terms of the verge stress, σ, referred to the entire bering surfce. The stress is not rel, becuse the lod is pplied only to prt of the bering surfce, but it is still convenient mesure of the lod. The lod p cuses n verge stress on the whole bering of - F- -

27 p σ = = cσ GS (F-58) L where c σ is dimensionless coefficient whose vlue my be expected to lie in the rnge. < c σ < 3.. Then, using E 3G, nd ignoring the negtive sign, γ tot cσ GSL ρ cσ ρ = ( C 3 ) ( 3 ) + Cr = C + Cr (F-59) teb S 3 3B 3 For n infinite strip nd incompressible conditions, C = 6 nd C r =, so this becomes γ tot 3cσ ρ = (F-6) B 3 Prior to lift-off, the totl sher strin my be clculted by conventionl mens, becuse the compliction of the vrying contct re does not exist. It cn be shown tht, under those conditions, the totl sher strin is cσ γ tot = ( C + ρcr ) (F-61) 3 B Eqution (F-61) is liner in θ L (or ρ), s might be expected, while Eqution (F-59) is nonliner. At incipient lift-off, when θ L = 3θ c, nd ρ = 3., they give the sme vlue, s they should. The individul curves tht relte γ tot to ρ re vlid before nd fter lift-off respectively, nd both the curves represented by the two equtions nd their slopes re continuous where they meet, t ρ = Bering with lift-off: totl sher strin vs rottion γ totl(in/in) sig/gs =.5 sig/gs =.5 sig/gs =.75 sig/gs = 1. Lift-off ρ =θ/θ c Figure F.16 shows the reltionship between totl sher strin nd rottion ngle, for fixed lod per unit width, p. Severl curves, ech representing different vlue of p, re - F-3 -

28 shown. They re mde dimensionless by expressing the loding, p, s σ /GS, so tht berings of different sizes nd mterils my be represented on the sme plot. The totl sher strin is the sum of the sher strins due to compression nd rottion, nd is shown for the compression edge of the bering lyer. Ech curve is composite of Equtions (F-59) nd (F-61). They meet t ρ = 3., which represents incipient lift-off for ll cses. As cn be seen, the curves fltten out slightly fter lift-off. This suggests tht the totl sher strin my be pproximted conservtively by using the simple expedient of ignoring the fct tht lift-off occurs, nd computing the components of sher strin using the liner, pre-lift-off equtions (e.g. Eqution (F-61)). The rtio of the correct sher strin, computed using Eqution (F-59) nd the pproximte one, obtined using Eqution (F-61), is shown in Figure F.17. It shows tht the rtio lies in the rnge.8 to 1. for ρ vlues up to 1, nd is therefore conservtive. This finding ws confirmed using Finite Element Anlysis. A physicl explntion cn be given for this finding. Consider bering in which lift-off hs occurred. Now freeze the position of the sole plte, nd pull the seprted elstomer upwrds so tht it is once gin ttched to the sole plte. The elstomer in tht region will experience verticl tension stress. Now relese the verticl displcement of the sole plte but keep the sme rottion. The sole plte will move downwrds in order to reestblish equilibrium, becuse of the new tension stress in prt of the elstomer. The bulge on the compression side will increse nd the sher strins there will be lrger when the elstomer is everywhere ttched to the externl pltes. The converse is therefore true; lift-off will reduce the bulging nd the sher strin on the compression side, if the rottion remins the sme. γ totl(in/in) Bering with lift-off: totl sher strin vs rottion sig/gs =.5 sig/gs =.5 sig/gs =.75 sig/gs = 1. Lift-off ρ =θ/θ c Figure F.16. Totl Sher Strin for Bering with Lift-off. - F-4 -

29 γ true/γ.pprox AR = AR =.5 AR =.333 AR =.5 AR = AR = 1 Sher strin rtio vs ρ ρ=θ/θ c Figure F.17. Post Lift-off Conditions: Rtio of Approximte nd True Totl Sher Strin. γ totl(in/in) Bering with lift-off: totl sher strin vs rottion sig/gs =.5 sig/gs =.5 sig/gs =.75 sig/gs = 1. Lift-off ρ =θ/θ c Figure F.16 leds to two observtions. First, lift-off is unlikely to occur under service conditions if the bering support nd the underside of the girder re prllel under full ded lod becuse the service rottions re likely to be too smll. In tht cse the postlift-off clcultions re not needed. As n exmple, consider bering with S = 6, three lyers, loded to n verge stress of 1. GS (pprox 6 psi). Lift-off strts when cσ 1. θ > 3θ = 3 = =.8 rd lyer c 3S B 6 *1.333 / (F-6) or.65 rdins totl in the three lyers. This rottion is most unlikely to be reched during the service life of the bridge. During construction, the xil lod is lighter, thereby - F-5 -

30 reducing the lift-off rottion, θ c, nd the rottion is lrger, in which cse lift-off is possible. Second, even if lift-off does occur, use of the no-lift-off eqution (Eqution (F-61)) cuses only smll error in the predicted totl sher strin on the compression side, nd tht error is conservtive. This discussion suggests tht the liner equtions (Eqution (F-61) my sfely be used to predict totl sher strins even fter lift-off. This finding simplifies design by llowing one set of equtions to be used for computing sher strins under ll circumstnces. F. Sher Strin Cpcity Two models were developed to determine sher strin cpcity nd to form the bsis for design. One, referred to here s the Liner Model, uses Gent s liner theory to relte lods to nominl sher strins. For design, it limits to fixed number the totl nominl sher strin due xil lod, rottion nd sher displcements. In this model, the nominl strins due to cyclic effects re multiplied by constnt numericl fctor to reflect their greter potentil to inflict debonding dmge on the bering. Tht model is described in Section F..1 The second model, clled the Nonliner Model, represents n ttempt to obtin closer fit with the dt thn is possible with the Liner Model. It differs in three mjor respects from the Liner Model. First, it uses nonliner reltionship between the lod nd nominl strins. Second, the components of strin tht re cused by cyclic lods re multiplied by coefficient tht is function of the number of cycles, rther thn constnt. Third, the strin cpcity is not fixed number, but is function of the mount of debonding deemed cceptble. The nonliner model is more complex thn the liner one. It is described in Section F.. Both models re described here. The Nonliner Model ws developed first. However, for resons discussed in Section F.4, it is considered less desirble s design procedure t this time, nd the design procedure defined in Appendix G, nd bsed on the Liner Model, is recommended for doption by AASHTO. The Nonliner Model is described here so tht, when the necessry dt become vilble, it my be developed into complete nd fully clibrted design procedure. F..1 Liner Model In the Liner Model, the nominl sher strins re obtined from the lods by using Gent s equtions (Equtions (F-4) nd (F-5). The totl sher strin tht represents the demnd on the bering is obtined by ( γ st + γ r, st + γ s, st ) + cn ( γ, cy + γ r, cy + γ s, cy ) γ cp, (F-63) In Eqution (F-63), the subscripts, r nd s refer to xil, rottion nd sher responses, nd st nd cy refer to sttic nd cyclic lod respectively. The left hnd side of the eqution defines the totl sher strin demnd. It is expressed s sttic component plus cyclic component tht is multiplied by n mplifiction coefficient, c N, to ccount for the dmging effects of cyclic loding. The right hnd side represents the sher strin - F-6 -

31 cpcity, γ cp, nd is tken to be constnt. A similr pproch hs been dopted by others (e.g. BS 54, EN1337). Numericl vlues re needed for the constnts c N nd γ cp. They were obtined from the cyclic test dt. For prticulr level of debonding, the totl nominl strin from ech test ws plotted ginst the number of cycles needed to rech tht debonding level. An exmple is shown in Figure F.18. The verticl xis, lbeled Effective Strin, consists of the left hnd side of Eqution (F-56), in which the cyclic strins hve been multiplied by constnt fctor. The vlue used in Figure F.18 is.. To determine the best vlue of c N, different vlues were tried nd the correltion coefficient (R ) of the dt ws found nd plotted ginst the cyclic fctor, c N. The results re shown in Figure F.19, for both 5% nd 5% debonding. These vlues of debonding, rther thn the initition of debonding, were used becuse of the sctter in the ltter. Even these dt show considerble sctter, s demonstrted by the rther low R vlues. However, t both levels of debonding, there is cler trend tht indictes tht the lrgest correltion coefficient, nd therefore the best fit, occurs with c N.. Tht vlue ws therefore selected. The effective strin vs. number of cycles for 5% debonding, using c N =., is shown in Figure F.. A vlue for γ cp, the strin cpcity, is lso needed. It is rgued in Section F.3.1 tht up to 5 million cycles of rottion, due to fully lden trucks, my be imposed on bering during the lifetime of the bridge. The best fit line in Figure F.18 crosses the 5 million cycle point t n effective strin of 4.7. The corresponding vlue for 5% debonding, shown in Figure F., is 5.6. The vlue dopted here ws 5.. The reson for doing so is tht some of the tests (such s SHF5-C, with shpe fctor of 9) never reched 5% debonding, nd so re not included in Figure F.18. There is no simple wy to include them in the dt nlysis, but their effect would certinly be to rise the best fit line nd to increse the strin corresponding to 5 million cycles bove 4.7. If the best fit line is tken s is, the effective strin of 5. corresponds to 17 million cycles. In both plots, the exct slope of the best fit line is dominted by the high-cycle dt of tests CYC1 nd CYC15 (log(n) 5.5 nd 6.5 respectively). Also R is not sensitive to the slope of the line, so predictions to 5 million cycles involve some uncertinty. - F-7 -

32 1 1 5% Debonding y = x R = log(n cycles) Figure F.18. Effective Strin vs Number of Cycles: 5% Debonding Correltion Coefficient vs. Cyclic Fctor.5. R % 5% Cyclic Amplifiction Fctor Figure F.19. Correltion Coefficient vs Cyclic Fctor. - F-8 -

33 1 1 5% Debonding y = -.546x R = log(n cycles) Figure F.. Effective Strin vs Number of Cycles: 5% Debonding. F.. Nonliner Model F...1 Theoreticl Bsis. The sher strin cpcity of berings subjected to monotonic nd cyclic loding ws investigted using experiments described in Appendices A, C nd D. However, physicl limittions prevent sher strin from being mesured during the experiments. The primry difficulty is the existence of the rubber cover, but, even in the bsence of cover, ccess to the criticl loction is nywy difficult becuse of the loding pltes tht enclose the bering in the test rig. Accurte mesurements were mde with micrometer depth gge of the height of the bulge in the rubber, with the intention of using them s proxies for the sher deformtion. However, they proved difficult to correlte with the sher strin. Furthermore, the Finite Element studies showed tht the cover provided significnt smoothing effect over the bulging of the internl lyers, nd tht the correltion between the two ws wek. Physicl mesurements of bulging therefore provided little useful numericl verifiction of the mplitude of the sher strin. It is thus necessry to use theoreticl mesures to relte sher strin to lod. This is done in two steps. The first is to relte the compressive lod to the globl, or verge, compressive displcement, nd this process requires considertion of bering stiffness. The step is necessry for xil force, for which the loding is pplied s force, but is in generl unnecessry for rottion, becuse the loding is pplied s displcement (rottion). This is so becuse the girder is typiclly so much stiffer in bending thn the bering tht the rottion of both cn simply be tken s the free rottion of the girder. The second step is to relte the locl sher strin in the elstomer t the edge of the shims to the globl deformtion mesures of compressive displcement nd rottion ngle. The Finite Element studies demonstrted tht, for smll strins, the vlues for sher strin in - F-9 -