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Rotation Limits for Elastomeric Bearings (2008)

Chapter: Chapter 3 - Interpretation, Appraisal, and Applications

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Suggested Citation:"Chapter 3 - Interpretation, Appraisal, and Applications." National Academies of Sciences, Engineering, and Medicine. 2008. Rotation Limits for Elastomeric Bearings. Washington, DC: The National Academies Press. doi: 10.17226/23131.
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Suggested Citation:"Chapter 3 - Interpretation, Appraisal, and Applications." National Academies of Sciences, Engineering, and Medicine. 2008. Rotation Limits for Elastomeric Bearings. Washington, DC: The National Academies Press. doi: 10.17226/23131.
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Suggested Citation:"Chapter 3 - Interpretation, Appraisal, and Applications." National Academies of Sciences, Engineering, and Medicine. 2008. Rotation Limits for Elastomeric Bearings. Washington, DC: The National Academies Press. doi: 10.17226/23131.
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Suggested Citation:"Chapter 3 - Interpretation, Appraisal, and Applications." National Academies of Sciences, Engineering, and Medicine. 2008. Rotation Limits for Elastomeric Bearings. Washington, DC: The National Academies Press. doi: 10.17226/23131.
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Suggested Citation:"Chapter 3 - Interpretation, Appraisal, and Applications." National Academies of Sciences, Engineering, and Medicine. 2008. Rotation Limits for Elastomeric Bearings. Washington, DC: The National Academies Press. doi: 10.17226/23131.
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Suggested Citation:"Chapter 3 - Interpretation, Appraisal, and Applications." National Academies of Sciences, Engineering, and Medicine. 2008. Rotation Limits for Elastomeric Bearings. Washington, DC: The National Academies Press. doi: 10.17226/23131.
×
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Suggested Citation:"Chapter 3 - Interpretation, Appraisal, and Applications." National Academies of Sciences, Engineering, and Medicine. 2008. Rotation Limits for Elastomeric Bearings. Washington, DC: The National Academies Press. doi: 10.17226/23131.
×
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Suggested Citation:"Chapter 3 - Interpretation, Appraisal, and Applications." National Academies of Sciences, Engineering, and Medicine. 2008. Rotation Limits for Elastomeric Bearings. Washington, DC: The National Academies Press. doi: 10.17226/23131.
×
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Suggested Citation:"Chapter 3 - Interpretation, Appraisal, and Applications." National Academies of Sciences, Engineering, and Medicine. 2008. Rotation Limits for Elastomeric Bearings. Washington, DC: The National Academies Press. doi: 10.17226/23131.
×
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Suggested Citation:"Chapter 3 - Interpretation, Appraisal, and Applications." National Academies of Sciences, Engineering, and Medicine. 2008. Rotation Limits for Elastomeric Bearings. Washington, DC: The National Academies Press. doi: 10.17226/23131.
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Suggested Citation:"Chapter 3 - Interpretation, Appraisal, and Applications." National Academies of Sciences, Engineering, and Medicine. 2008. Rotation Limits for Elastomeric Bearings. Washington, DC: The National Academies Press. doi: 10.17226/23131.
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Suggested Citation:"Chapter 3 - Interpretation, Appraisal, and Applications." National Academies of Sciences, Engineering, and Medicine. 2008. Rotation Limits for Elastomeric Bearings. Washington, DC: The National Academies Press. doi: 10.17226/23131.
×
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Suggested Citation:"Chapter 3 - Interpretation, Appraisal, and Applications." National Academies of Sciences, Engineering, and Medicine. 2008. Rotation Limits for Elastomeric Bearings. Washington, DC: The National Academies Press. doi: 10.17226/23131.
×
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Suggested Citation:"Chapter 3 - Interpretation, Appraisal, and Applications." National Academies of Sciences, Engineering, and Medicine. 2008. Rotation Limits for Elastomeric Bearings. Washington, DC: The National Academies Press. doi: 10.17226/23131.
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33 3.1 Proposed Design Rules Research showed that truck passage over a bridge causes shear strains in the bearing due to axial load effects that are much larger than the shear strains due to rotations. This fact causes difficulties in basing the design procedures directly on test results, because appropriate rotation data are available from the tests conducted for this study, but comparable com- pression fatigue data are not. The only ones available are those from NCHRP Report 298 (Roeder et al., 1987). The tests con- ducted in that study were on bearings without cover, which necessitated a method of measuring debonding different from the one used here, and renders very difficult the process of comparing the results of the two studies. Furthermore, most of the NCHRP 298 bearings were square with S = 5, whereas those used here were 22 in. × 9 in., with S = 6.2. Attempts to construct a design model were made with Non- linear and Linear Model for stress-strain relationships. In both cases equivalence of shear strains was assumed. The Nonlinear Model predicted behavior for a common bearing size that was in disagreement with field experience so that model was dis- carded, and the design methodology for Method B was based on the Linear Model. Suitable provisions for Method A then were derived from the proposed Method B architecture by de- veloping a maximum probable rotation value and computing axial stress allowed by Method B to accompany it. That axial stress forms the basis for the Method A design procedure in which explicit account of rotations is not required. The pro- posed design provisions for both methods are presented in Ap- pendix G. The development of these procedures is discussed in detail in Appendix F. One of the difficulties designers have experienced with the existing Method B AASHTO LRFD design rules is that large rotations combined with light axial loads often make design impossible because the no uplift provisions force the bearing to be very thick and causes problems with instability. The pro- posed provisions address this difficulty by allowing lift-off in bearings that have no external plates. The proposed Method B design provisions are based on the total shear strain approach, whereby shear strains from each type of loading are computed separately and then added. This concept has underlain the AASHTO design provisions for many years, but its use has not been transparent. The change to an explicit use of that approach is proposed now because it simplifies the specifications and is easier for designers to understand because of its transparency. It also is used by spec- ifications elsewhere (for example, AASHTO Seismic Isolation Specifications, EN 1337), although the numerical values pro- posed here differ slightly from those used in other specifica- tions. The difference with isolation bearings reflects that they are typically larger and thicker than conventional bridge bear- ings and are subject to different loading (that is, a very small number of large shear loadings during a lifetime). Differences with European specifications are based on the significant effect of cyclic loading seen during the tests for this project. Such ex- tensive test data were not available during development of EN 1337. The proposed specification provisions are slightly more conservative than those of EN 1337. The total allowable strain is higher than the one implicit in the 2004 AASHTO LRFD Specifications, but that fact is par- tially offset by the presence of a constant amplification factor that is applied to cyclic strains arising from traffic loading. The Method B design provisions in Appendix G, based on the Lin- ear Model, provide design rules for bearings that: • Are consistent with the debonding observed in the tests, • Are readily satisfied by bearings in common use today, thus passing a necessary and objective criterion of reasonability, • Penalize cyclic loads, in accordance with the findings of the testing program, which showed that cyclic loading led to much more debonding than did monotonic loading of the same amplitude, • Remove the previous restrictions on lift-off for bearings without external plates, and from which the girder can read- ily separate over part of the surface, C H A P T E R 3 Interpretation, Appraisal, and Applications

• Introduce a new check for hydrostatic tension stress to guard against internal rupture of the elastomer in bearings that have external plates and are subjected to light axial load and large rotations, and • Eliminate the absolute compressive stress limit (presently 1.6 ksi or 1.75 ksi) to encourage the use of bearings with higher shape factors for high load applications. Such bear- ings performed extremely well in testing program. A stress limit related to GS remains. A cyclic amplification factor of 2.0 is proposed. This is higher and more conservative, than the European value of 1.0 or 1.5 (which value is to be chosen by the bridge’s owner). However, the cyclic test program conducted for this research showed that cyclic loading is significantly more damaging than static loading. Thus the European minimum value of 1.0 appears to be unrealistically low. The European Specification (EN 1337) uses the same total strain capacity of 5.0 proposed here, so the existence of a higher cyclic amplification factor in the present proposals makes them inherently more conservative than those of EN 1337. Despite that, the proposed rules are still sim- pler, more versatile, and more liberal than those in the 2004 AASHTO LRFD Specifications. Their greater simplicity, com- bined with the potential for higher allowable stresses when rotation is low and is explicitly taken into account, may result in designers preferring them to Method A for the majority of steel-reinforced elastomeric bearings. The total shear strain caused by compression, rotation, and shear displacements is calculated. For each component, strain is divided into a static and a cyclic component. Strains arising from truck loading, of which several million cycles must be expected during the life of the bridge, are treated as cyclic. All others, including shear from thermal effects, are regarded as static. The total shear strain must satisfy: where = shear strain caused by axial load, = shear strain caused by rotation, and = shearstraincaused by shear displacement. Subscripts “st” and “cy” indicate static and cyclic loading respectively. This single provision replaces the four equations in the ex- isting Article 14.7.5.3.2 (“Compressive Stress”) and the six equations in the existing 14.7.5.3.5. (“Combined Compres- sion and Rotation”). The provisions that presently limit the live load and total load separately are addressed by the pres- ence in Equation (3-1) of the amplification factor of 2.0 on γ s s rth = Δ γ θr r ri iD L h = ⎛ ⎝⎜ ⎞ ⎠⎟ 2 γ σa a sD GS = γ γ γ γ γ γa st r st s st a cy r cy s cy, , , , , ,.+ +( ) + + +( ) ≤2 0 5 0. (3-1) cyclic load effects. The existing restrictions on lift-off (present AASHTO Equation 14.7.5.3.5-1) have been removed for bear- ings without external bonded plates, because the tests showed no evidence of fatigue failure on the tension side of such bear- ings subjected to combined axial load and rotation. The po- tential for such fatigue was the basis for the provisions presently in force, which were based partly on research conducted on very small laboratory specimens (Caldwell et al., 1940). That study showed reversal of strain during cycling was particularly damaging, and, because at the time lift-off was taken to imply strain reversal, design provisions were developed to restrict lift-off. The removal of this restriction allows lift-off of the girder or sole plate from the bearing, which solves most of the existing problems with design for simultaneous light axial load and large rotation. In the 2004 AASHTO LRFD Specifications, the total axial stress is restricted to 1.60 ksi or 1.75 ksi (in the presence or absence of shear displacements), regardless of shape factor. Study of the mechanics of bearings shows that load-carrying ability is more logically related to the product GS (shear mod- ulus of the rubber times the shape factor) than to absolute stress, because GS largely controls the shear strain due to com- pression. The previous 1.75 ksi limit was imposed because, at the time, only a limited number of tests had been conducted on higher shape factor bearings. However, that situation has now changed. Several bearings with higher shape factors were tested in this program, and many isolation bearings now have been built and tested, almost all of which have shape factors higher than 6, which lies near the top of the range commonly used for conventional bridge bear- ings. (Many isolation bearings have shape factors of 20 or more.) Isolation bearings also are typically stressed more highly than conventional bearings, and bearings tested in this program with shape factors of 9 and 12 were loaded repeatedly to a stress of 12 ksi (the test machine capacity), and showed no debonding at all (see tests SHF1-C2 and SHF2-C2 in Appendix A). The com- pressive capacity of such bearings is well demonstrated. The pro- posed limit of 3.0 on the static component of γa implies a per- manent axial stress no higher than approximately 2.0GS, which is still conservative. For example, in the axial load tests, debond- ing initiated at average axial stresses between 4.7 GS and 7.8 GS, and specimens SHF1-C2 and SHF2-C2 did not debond at all under cycles of axial load to 12GS. That limiting stress of 2.0GS is not expected to be used often. The size of many bearings is controlled by the desire to make them as wide as the girder flange in order to pro- mote lateral stability during erection of the girders. The re- sulting compressive stresses are then relatively low. How- ever, the need occasionally arises for a bearing with a very large capacity, and use of higher allowable compressive stresses permits an elastomeric bearing to be considered for those cases. 34

Restriction against uplift is necessary if the bearing has ex- ternal plates bonded to it, because large rotations combined with light axial load could lead to hydrostatic tension and brit- tle internal rupture of the elastomer. Previously, this provision was unnecessary, because the no uplift provisions prevented such behavior. Gent and Lindley (1959b) studied the problem and found that rupture occurred at a hydrostatic tension stress of approximately 0.9E,or about 2.7G.The problem is expected to arise only rarely, but it must be addressed nonetheless. It is achieved by requiring where σhyd is the peak hydrostatic tension, computed by and where the axial strain, εa, is given by and is to be taken as positive for compression in Equation (3-6). Constant Ba is given approximately by where L/W is the aspect ratio of the bearing in plan. For val- ues of α greater than 0.333, the hydrostatic stress is compres- sive, and no limit is required. The values of εa and θL used in Equation (3-5) consist of the static components plus 2.0 times the cyclic components. Although Equations (3-2) to (3-7) appear somewhat com- plicated, they are not difficult to evaluate, especially if pro- grammed into a spreadsheet. The apparent complexity arises because the location of the maximum hydrostatic tension stress varies with the relative magnitudes of rotation and axial load. The equations were obtained for an infinite strip bear- ing by the linear theory and validated by FEA. Agreement was very good. They are conservative for other shapes. The previous restriction on shear displacement (limited to half the rubber thickness) is retained without change, as are the provisions for compressive deflection, stability, steel re- inforcement, and seismic conditions. B L W W L a = −( ) + − +( ) − ⎧⎨⎩ ⎫ 2 31 1 86 0 90 0 96 1 . . . . min , λ λ ⎬⎭ ⎛⎝⎜ ⎞⎠⎟ 2 (3-7) ε σ a a aB GS = 3 2 (3-6) α ε θ = a LS (3-5) f α α α α( ) = +⎛⎝⎜ ⎞⎠⎟ − −( ) ⎧⎨⎪⎩⎪ ⎫⎬⎪⎭⎪ 4 3 1 3 12 1 5 2 . (3-4) σ θ αhyd L G S f 3 3 = ( ) (3-3) σhyd G3 0 75≤ . (3-2) The foregoing concepts define Method B. Method A was formulated with the goal to be consistent with Method B to the greatest extent possible. That was done by develop- ing a design rotation that represents the maximum value likely to occur in practice and using Method B to determine the corresponding allowable axial stress. Two restrictions on the use of Method A were found to be necessary. First, it should not be used if external bonded plates are present. This was done to avoid having to address the problems asso- ciated with hydrostatic tension. Most bearings are fabricated without external plates so the restriction is not expected to be serious. Second, Method A may not be used if S2/n > 16. This re- striction is necessary to avoid excessive shear strains at the bearing’s edge when the rotation is large. Bearings with few, thin layers do not accommodate rotations well and experi- ence relatively large shear strains due to the rotation. They also have large values of S2/n. The restriction is part of the price of ignoring rotations during design. The allowable axial stress is related to the limiting value of S2/n: the higher the stress used, the lower must be the largest S2/n permissible. Appendix F describes the process of selecting a suitable com- bination of allowable stress and maximum S2/n. Care was taken to ensure that bearings suitable for a typical freeway overpass could still be designed using Method A, which is the common practice today. One difficulty arose. Method A covers several bearing types other than steel-reinforced elastomeric bearings, and they are designed without any amplification factors to repre- sent the effects of cyclic loading. Because this research pro- gram did not address those bearing types, no information was generated on their response to cyclic load, and there was no basis for altering the provisions for their design. It was de- cided to leave Method A designs in terms of nonamplified stresses, and to alter the allowable stresses developed from Method B accordingly. In that regard, Methods A and B are not fully compati- ble. However, in other ways, they are more compatible than the case today. Under the 2004 Specifications, it is possible to design a bearing for a set of loads under Method A, and to find that, under Method B, same bearing with the same loads fails to meet all the design criteria. Under the proposed specifications, such occurrences will be much less common. (Review of the development of Method A, verified by trial and error, showed that this can only happen if the rotations are larger than the design rotations used in the development of Method A, and this is most improbable.) Apart from the addition of these two restrictions, Method A follows the same structure as in the existing specifications. The allowable stresses in it are raised by 25%, in parallel with the increases in Method B. 35

3.2 Design Examples A spreadsheet was prepared for designing bearings, based on the proposed design provisions in Appendix G. For any set of loadings, the spreadsheet computes the total shear strain and the hydrostatic tension stress. It also makes the checks re- quired by the existing AASHTO LRFD Specifications, so that the proposed and existing designs can be compared. The spreadsheet was used to prepare the design examples presented in this section. In the examples, the AASHTO nota- tion of θi is used in place of θL for the rotation per layer, and ref- erence is made to the number of each source equation from Ap- pendix G. Numerical values were taken from the spreadsheet. Because of rounding, they may in some cases differ slightly from those obtained by following the calculations with a hand calculator. Numerical values are given to four digit accuracy to help minimize this problem. Such accuracy is not warranted in practice by the reliability of the material properties or the design equations. Six examples are intended to illustrate different features of the design process and are summarized in Table 3.1. Relevant features in each are highlighted in the table. Most are self- explanatory. The “Rotation Sum” feature concerns the man- ner in which rotations are added. If, for example, initial girder camber causes a positive rotation, not all of which is removed by the application of full dead load, then the effect of adding live load might be to reduce the relative rotation of the top and bottom surfaces of the bearing. How then should the rotations be added in applying the proposed design equations? Equations in the following examples numbered with G (such as G-10) refer to equations in Appendix G, the proposed spe- cifications. Appendix G is located on the web at http://trb.org/ news/blurb_detail.asp?id=8556. Example 1 Common bearing, lift-off permitted, AASHTO Type V girders. Design Criteria A prestressed concrete girder bridge is supported on elas- tomeric bearings. The AASHTO Type V girders span 120 ft. and have 28 in. wide bottom flanges. Under full dead load, the load is 105 kips and the girder end is horizontal. The live load causes an axial load of 48 kips and a rotation of 0.0025 radians. Shear displacement due to thermal effects is ± 0.75 inches, all to be taken at one end of the girder because the other end of the bridge is fixed against longitudinal movement. Concrete shear keys prevent transverse movement at the girder ends. Design by Method A, if possible. Solution For shear displacements, Try W = 25 in. (to fit the flange) and two internal layers of 0.75 in. each. Select an elastomer with G = 0.110 ksi (approx. 50 durometer). Trial and error (with a spreadsheet) shows that a bearing with gross plan dimensions of 25 in. × 10 in. will work. Calculations are as follows: For acceptance under Method A, There are no external bonded plates OK The total axial stress is The total axial stress/GS is The bearing is thus satisfactory under Method A. It also proves to be satisfactory under both the existing and the σa GS = = < 0 6340 0 110 4 663 1 236 1 25 . . . . .  OK σa = + = < 105 48 241 3 0 6340 1 25 . . .ksi ksi OK S n 2 24 663 2 10 872 16= = < . . OK L W eff eff = + = = 0 5 10 0 9 5 9 75 0 5 25 . ( . . ) . . ( .   inches 0 24 5 24 75 2 + = = + . ) . inches S W L h W L eff eff ri eff eff( ) = +( ) = = 24 75 9 75 2 0 75 24 75 9 75 4 663 . . . . . .   A Weff effL = 241 3 2. in hrt s≥ = =2 2 0 75 1 5Δ  . . inches (G-10) 36 No. Bridge type Bearing type Rotn. Sum Hydrostatic Skew 1 AASHTO psc "Method A" Simple 2 PSC girder Standard omplex No No 3 teel girder Standard imple Yes 4 S Steel girder Standard imple No Yes 5 PSC box Large Simple 6 Special High S C S S Simple No No No No No No No Table 3.1. Bearing design examples.

proposed Method B approaches. It does not satisfy the ex- isting Method A because the axial stress/GS exceeds the limit of 1.0. Example 2 Common bearing, lift-off permitted. Design Criteria A prestressed concrete girder bridge is supported on elas- tomeric bearings. The girders span 125 ft. and have 25 in. wide bottom flanges. Under construction conditions, the load is 50 kips and the rotation is 0.008 radians, due to cam- ber. Under full dead load, the load is 120 kips and the girder end is horizontal. The live load causes a load of 70 kips and a rotation of 0.002 radians. Shear displacement due to thermal effects is ± 0.75 in., all to be taken at one end of the girder, be- cause the other end of the bridge is fixed against longitudinal movement. Concrete shear keys prevent transverse move- ment at the girder ends. Thermal camber causes a rotation of 0.0015 radians. Solution For shear displacements, Try W = 23 in. (to fit the flange) and three internal layers of 0.5 in. each. Select an elastomer with G = 0.110 ksi (approx. 50 durometer). Assume K = 450 ksi (default value). Trial and error shows that the minimum possible gross length is 7.013 in. Use 8 in. to avoid the absolute minimum. Edge cover is 0.25 in. The shims are therefore 22.5 in. × 7.5 in. Calculations are as follows. The subscripts para and perp indicate conditions relevant to rotation about axes parallel and perpendicular to the support face. Thus, in a bridge without skew, bending of the girder causes a rotation θpara, while torsion of the girder would cause an end rotation θperp. Here the long side of the bearing is placed parallel to the support face. L W eff eff = + = = 0 5 8 0 7 5 7 75 0 5 23 0 . ( . . ) . . ( .   inches + = = +( 22 5 22 75 2 . ) . inches S W L h W L eff eff ri eff eff ) = +( ) = = 22 75 7 75 2 0 5 22 75 7 75 5 781 . . . . . .   A W Leff eff S G K = = = = 176 3 3 5 781 3 0 11 450 0 1565 2. . . . in λ  hrt s≥ = =2 2 0 75 1 5Δ  . . inches (G-10) Coefficients needed for shear strains are Under service conditions: Axial load = 120 kips DL + 70 kips LL, Shear deformation = 0.75 in., Rotation (rad.) = 0.005 misalignment + 0.0015 thermal + 0.002 LL, and Because the rotation occurs about the weak axis, the shear strains on the long side are critical. Shear strain there due to (amplified) axial load is The components of the total amplified rotation are illus- trated in Figure 3.1. Under full DL, the girder is horizontal. If thermal camber is ignored, the rotation under amplified (DL + LL) is (0.0 + 2  0.002) = 0.004 rad. downwards. However, the thermal camber is always upwards so, if it is included, the amplified sum of the DL + thermal + 2  LL gives a smaller total of only 0.0015 rad. downwards, and is not the critical load case. Alternatively, in the interests of simplic- ity, a very conservative estimate always can be obtained by tak- ing the sum of the absolute values of the components or, in this case, 0.0 (DL) + 0.0015(thermal) + 2  0.002(LL) = 0.0055 rad. γ σa a para aD GS = = =, . . . . .1 4075 1 475 0 110 5 781 3 264  (G-3) σa = + = 120 2 0 70 176 3 1 475 . . .  ksi D L W r perp, min . . . . , .= − + + ⎧ 1 552 0 627 2 233 0 156 0 5 λ λ⎨⎩ ⎫⎬⎭ = − + min . . . . . . 1 552 0 627 0 1565 2 233 0 156 0 1   565 2 9355 0 5 0 2800 + ⎧⎨⎩ ⎫⎬⎭ = . , . . (G-7) D L W r para, min . . . . , .= − + + ⎧ 1 552 0 627 2 233 0 156 0 5 λ λ⎨⎩ ⎫⎬⎭ = − + min . . . . . . 1 552 0 627 0 1565 2 233 0 156 0 1   565 0 3407 0 5 0 5 + ⎧⎨⎩ ⎫⎬⎭ =. , . . (G-7) D D a a 1 2 2 1 060 0 210 0 413 1 1030 1 506 0 = + + = = − . . . . . . λ λ 071 0 406 1 5048 0 315 0 195 0 047 2 3 λ λ λ + = = − + − . . . . .Da λ2 1 2 3 0 2856= − = + ⎛⎝⎜ ⎞⎠⎟⎧⎨ . max ,,D D D D L W a para a a a⎩ ⎫⎬⎭ = − ⎛⎝⎜ ⎞⎠max . , . . . . 1 1030 1 5048 0 2856 7 75 22 75⎟ ⎧⎨⎩ ⎫⎬⎭ = = − 1 4075 1 1030 1 5048 0 2 . max . , . .,Da perp 856 22 75 7 75 1 1030 . . . ⎛⎝⎜ ⎞⎠⎟⎧⎨⎩ ⎫⎬⎭ = (G-4) 37

To any of these totals must be added the allowance for mis- alignment of 0.005 rad., which should always be taken in the sense that is most disadvantageous. The misalignment allowance is small (approximately 1⁄8 in. elevation difference between the two ends of the 23 in. long bearing). Despite that, it still constitutes about half of the total design rotation, and demonstrates the importance of the ac- curacy of the initial setting. Here the conservative estimate of 0.0055 rad. is used for purposes of illustration. Including the misalignment allowance, the total rotation is Shear strain due to rotation about the weak axis is Shear strain due to shear displacement is The total shear strain check is given by The shear strain due to the static component of the axial load is The bearing has no external plates, so a check on hydro- static tension is not necessary. Stability criteria are satisfied. Under initial conditions: Load = 50 kips Total rotation = 0.005 + 0.008 = 0.013 rad. Shear displacement = 0.0 These loadings are static, so no amplification factor is needed. Shear strain due to axial load is γ σa a para aD GS = = =, . . . . .1 4075 0 284 0 110 5 781 0 628  (G-3) σa = = 50 176 3 0 284 . . ksi γ σa a para aD GS = = ( ) =, . . . . 1 4075 120 176 3 0 110 5 781 1 507 3 0. .< (G-2) γ γ γ γtot a r s= + + = + + = ≤3 264 0 420 0 500 4 185 5 0. . . . . (G-1) γ s s rth = = = Δ 0 75 1 5 0 5 . . . inches inches (G-10) γ θr r para ri iD L h = ⎛ ⎝⎜ ⎞ ⎠⎟ = ⎛⎝⎜ ⎞⎠⎟, . . . 2 0 5000 7 75 0 5 2 0 0105 3 0 420 . . ⎛⎝⎜ ⎞⎠⎟ = (G-6) θtot = + + ( ) =0 005 0 0015 2 0 0 002 0 0105. . . . . rad. Shear strain due to rotation is Total shear strain for rotation about the weak axis is For rotation about the strong axis, the service level shear strain due to axial load is The total amplified rotation is due to misalignment alone Shear strain due to rotation is Shear strain due to shear displacement is zero, so the total shear strain is Under initial conditions, the strains are The total amplified rotation is Shear strain due to rotation is Shear strain due to shear displacement is zero, so the total shear strain is According to the existing LRFD Method B specifications, this bearing would just fail the compressive stress limit of 1.66GS. It also would just fail the combined stress requirement. However, it does satisfy all the performance checks with the proposed specifications. γ γ γ γtot a r s= + + = + + = ≤0 492 0 966 0 000 1 458 5 0. . . . . (G-1) γ θr r perp ri iD L h = ⎛ ⎝⎜ ⎞ ⎠⎟ = ⎛⎝⎜ ⎞⎠, . . . 2 0 2800 22 75 0 5 ⎟ ⎛⎝⎜ ⎞⎠⎟ = 2 0 005 3 0 966 . . (G-6) θtot = 0 005. rad. γ σa a perp aD GS = = =, . . . . .1 1030 0 284 0 110 5 781 0 492  (G-3) γ γ γ γtot a r s= + + = + + = ≤2 558 0 966 0 000 3 524 5 0. . . . . (G-1) γ θr r perp ri iD L h = ⎛ ⎝⎜ ⎞ ⎠⎟ = ⎛⎝⎜ ⎞⎠, . . . 2 0 2800 22 75 0 5 ⎟ ⎛⎝⎜ ⎞⎠⎟ = 2 0 005 3 0 966 . . (G-6) θtot = 0 005. rad. γ σa a perp aD GS = = =, . . . . .1 1030 1 475 0 110 5 781 2 55  8 (G-3) γ γ γ γtot a r s= + + = + + = <0 628 0 521 0 000 1 148 5 0. . . . . (G-1) γ θr r para ri iD L h = ⎛ ⎝⎜ ⎞ ⎠⎟ = ⎛⎝⎜ ⎞⎠⎟, . . . 2 0 5000 7 75 0 5 2 0 013 3 0 521 . . ⎛⎝⎜ ⎞⎠⎟ = (G-6) 38 Initial Initial DL DL DL+LL DL+thermal+LL DL+thermal Figure 3.1. Girder end rotation components.

The bearing may also be evaluated under the proposed Method A. Results are as follows: No bonded external play exist OK The total (nonamplified) compressive stress is The total (nonamplified) compressive stress/GS is The bearing therefore does not satisfy the proposed Method A requirements. By adding an extra shim, and mak- ing four layers at 3⁄8 in. each rather than 3 layers at 1⁄2 in., the design satisfies Method A. Under Method B, the bearing is quite highly stressed in compression and the rotations are quite small, so it is not surprising that, with three layers of rubber, it does not quite satisfy Method A criteria. Example 3 Steel bridge, common bearing size, large camber, but lift-off prevented. Hydrostatic tension is possible. Design Criteria A steel plate girder bridge is supported on elastomeric bear- ings. The girders span 150 ft. and have 22 in. wide flanges. Under construction conditions, the load is 32 kips and the ro- tation is 0.04 radians (camber). (This camber is extreme but was selected to create conditions in which hydrostatic tension might pose problems. Such large cambers are likely only with very slender steel girders.) Under full dead load, the end of the girder is exactly horizontal and the total load is 115 kips. The live load causes a load of 70 kips and a rotation of 0.002 radi- ans. Shear displacement due to thermal effects is ± 1.0 in., all to be taken at one end. The bearing is to have external plates bonded to the elastomer, and is to be bolted to the girder directly after erection. Use a 50 durometer elastomer with G = 0.110 ksi. Solution Try a bearing 20 in. wide, with a total rubber thickness of 2 in. Note that, under initial conditions, hydrostatic ten- sion may cause a problem. Check it first. Trial and error (with a spreadsheet) shows that a bearing 20 in. × 8 in., with four 1⁄2 in. thick rubber layers, will work. Calculations are shown below. σa GS = = > 1 078 0 110 5 781 1 695 1 25 . . . . . ksi OK  σa = + = < 120 70 176 3 1 078 1 25 . . .ksi ksi OK S n 2 25 781 3 11 139 16= = < . . OK The bearing properties (for rotation about the weak axis) are: Coefficients needed for shear strains are The axial stiffness coefficient, Ba, is given by Eq. G-33 as Under initial conditions, the axial stress is σa eff P A = = = 32 153 1 0 209 . . ksi B L W W L a ≈ −( ) + − +( ) − ⎧⎨⎩ ⎫ 2 31 1 86 0 90 0 96 1. . . . min ,λ λ ⎬⎭ ⎛⎝⎜ ⎞⎠⎟ = −( ) + − + 2 2 31 1 86 0 1507 0 90 0 96 0. . . . . .  1507 1 0 3924 1 7508 2( ) −( ) = . . (G-16) D L W r perp, min . . . . , .= − + + ⎧ 1 552 0 627 2 233 0 156 0 5 λ λ⎨⎩ ⎫⎬⎭ = − + min . . . . . . 1 552 0 627 0 1507 2 233 0 156 0 1   507 2 5484 0 5 0 3033 + ⎧⎨⎩ ⎫⎬⎭ = . , . . ( )strong axis (G-6) D L W r para, min . . . . , .= − + + ⎧ 1 552 0 627 2 233 0 156 0 5 λ λ⎨⎩ ⎫⎬⎭ = − + min . . . . . . 1 552 0 627 0 1507 2 233 0 156 0 1   507 0 3924 0 5 0 5 + ⎧⎨⎩ ⎫⎬⎭ = . , . . (weak axis) (G-6) D D a a 1 2 2 1 060 0 210 0 413 1 1010 1 506 0 = + + = = − . . . . . . λ λ 071 0 406 1 5045 0 315 0 195 0 047 2 3 λ λ λ + = = − + − . . . . .Da λ2 1 2 3 0 2867= − = + ⎛⎝⎜ ⎞⎠⎟⎧⎨ . max ,,D D D D L W a para a a a⎩ ⎫⎬⎭ = − ⎛⎝⎜ ⎞⎠max . , . . . . 1 1010 1 5045 0 2867 7 75 19 75⎟ ⎧⎨⎩ ⎫⎬⎭ = = − 1 3920 1 1010 1 5045 0 2 . max . , . .,Da perp 867 19 75 7 75 1 1010 . . . ⎛⎝⎜ ⎞⎠⎟⎧⎨⎩ ⎫⎬⎭ = (G-4) A W L in S G K eff eff= = = = = 153 1 3 5 566 3 0 11 450 0 2. . * . .λ 1507 L W eff eff = + = = 0 5 8 0 7 5 7 75 0 5 20 0 . ( . . ) . . ( .   inches + = = +( 19 5 19 75 2 . ) . inches S W L h W L eff eff ri eff eff ) = +( ) = 19 75 7 75 2 0 5 19 75 7 75 5 566 . . . . . .   39

The axial strain [Eq. (G-16)] is then The rotation is 0.04 (camber) + 0.005 (misalignment), so the dimensionless variable α is The hydrostatic tension can be obtained from Eq. (G-30) as Hydrostatic tension exists, but its magnitude is acceptable. If hydrostatic tension had posed a problem it might have been solved by specifying a construction procedure that requires the bearing not to be bolted to the girder until after the slab has been poured and the rotation due to camber had been elimi- nated. However, such a provision is undesirable because it could be forgotten on site. Making the bearing thicker would be a better alternative. Under initial load, for rotation about the weak axis, the shear strains due to compression are Shear strain due to rotation is Total shear strain is Under full service load, the calculations are similar to those of Example 1 and are not presented here. It is instructive to check the bearing against the existing LRFD design specifications under initial conditions, given the large rotation and light axial load. Eq. 14.7.5.3.5-1 of the LRFD Specifications requires (Note that B and σs in the existing specifications are the same as the quantities L and σa in this report. Also, gross, rather σ θ s ri GS n B h > ⎛⎝⎜ ⎞⎠⎟ ⎛ ⎝⎜ ⎞ ⎠⎟ =1 0 1 0 0 110 5 566 2 . . . .  0 045 4 8 0 5 1 763 2 . . . ⎛⎝⎜ ⎞⎠⎟ ⎛⎝⎜ ⎞⎠⎟ = ksi γ γ γ γtot a r s= + + = + + =0 475 1 351 0 000 1 827. . . . (G-1) γ θr r para ri iD L h = ⎛ ⎝⎜ ⎞ ⎠⎟ = ⎛⎝⎜ ⎞⎠⎟, . . . 2 0 5000 7 75 0 5 2 0 045 4 1 351 . . ⎛⎝⎜ ⎞⎠⎟ = (G-6) γ σa a para aD GS = = =, . . . . .1 3920 0 209 0 110 5 566 0 47  5 (G-3) σ θ αhyd i G S f= ( ) = ⎛⎝⎜ ⎞⎠⎟3 3 5 566 0 045 4 0 0603 3  . . . 3 0 3508 2 25= <. . (G-12) f α α α α( ) = +⎛⎝⎜ ⎞⎠⎟ − −( ) ⎧⎨⎪⎩⎪ ⎫⎬⎪⎭⎪ = 4 3 1 3 1 0 02 1 5 2 . . 603 (G-13) α ε θ = = = a iS 0 01147 5 566 0 045 4 0 1832 . . . / .  (G-15) ε σ a a a aG A B S = +( ) = +3 0 209 0 330 1 0 1 7508 5 5662 . . . . . 2 0 01147 ( ) = . (G-15) than effective, bearing dimensions are used for all calcula- tions in the existing Specifications). However Thus, the bearing fails by a significant margin to satisfy the uplift provisions of the existing Specifications because of the combination of high rotation and low axial load. The fact that the initial conditions prove acceptable under the pro- posed Method B provisions is an important change. A girder with such a large camber also poses potential prob- lems during construction. The bottom flange will elongate, and the bearing will move longitudinally, as the girder is stressed by the addition of the dead weight of the deck and other components. An approximate value for the magnitude of the movement can be obtained by considering the two major sources of it. In the interests of simplicity, the girder is assumed here to be symmetric. The girder is assumed to be 5 ft. deep (span/depth = 30). End rotation of the girder gives a movement at each end of The girder also flattens out from its cambered shape. The neutral axis by definition does not change length, but its hori- zontal projected length does, because its initial curved shape changes to a straight line by the time the camber has been com- pletely eliminated. The change in length is given by If the cambered shape is a parabola, this gives If it is sinusoidal where a0 = the initial camber height. The exact cambered shape is seen to have little effect on the magnitude of the longitudinal movement. Using the parabolic shape, and noting that The movement must occur at the nonfixed end of the bridge, where Δtotal = + =2 1 2 0 48 2 88 . . . in. θ θ camber a L a L L = = = ⎛⎝⎜ ⎞⎠⎟ = 4 2 667 2 667 4 2 6 0 0 2 2 Δ . . . 67 0 04 4 1800 0 48 2 . . ⎛⎝⎜ ⎞⎠⎟ = in. Δ = ≈π 2 0 2 0 2 4 2 467 a L a L . Δ = 2 667 0 2 . a L Δ = ⎛⎝⎜ ⎞⎠⎟∫ 12 2 0 dy dx dx L Δ = = =d 2 60 2 0 04 1 2θ . . in. σ s = = 32 20 8 0 200 kips inch ksi  . 40

This is significantly larger than the ±1.0 inches of thermal expansion, so provision should be made for resetting the girder on the bearings during construction. This problem is more likely to occur in steel bridges because they are typically more flexible than concrete ones. Example 4 Common bearing, but lift-off allowed. The bridge also has 55 degrees skew. No thermal effects. The complexities in this example pertain to the skew. Design Criteria A steel plate girder bridge is supported on elastomeric bearings. The girders span 150 ft. and have 22 in. wide flanges. Under construction conditions, the load is 32 kips and the rotation is 0.04 radians (bending camber), and there is no torsional rotation. Under full dead load, the girder is ex- actly horizontal in bending, has a torsional rotation of 0.01 radians, and the total load is 115 kips. The live load causes a load of 70 kips and rotations of 0.002 radians (bending of girder) and 0.003 radians (torsion of girder). Shear dis- placement due to thermal effects is ± 1.0 in., all to be taken at one end, in the longitudinal direction. Use an elastomer with G = 0.110 ksi. The bearings are to be oriented with their long edges par- allel to the support. Guides at the support prevent move- ment perpendicular to the girder axes, but allow longitudi- nal displacement. Solution Try a rectangular bearing that will fit under the girder flange, with a total rubber thickness of 2 in. Trial and error (with a spreadsheet) shows that a bearing 18 in. × 9 in. will carry the loads and will fit in the space available. Calculations for the service loading are shown below. The orientation and labeling of the axes are shown in Figure 3.2. The bearing properties (for rotation about the weak axis) are: Coefficients needed for shear strains are Under service conditions: The local rotation and shear demands on the bearing for the strong (perp) and weak (para) axis rotation directions must be computed from the global coordinate system, defined by the longitudinal and transverse directions. In each case, the ro- tations due to loading are computed first, then the allowance for misalignment is added. The bearings are oriented at a skew angle of β = 55 degrees. With the positive directions as shown in Figure 3.2, the rotations in the global coordinate system are: Girder bending: rθtrans DL LL= ( ) − ( )0 000 0 002. . ad. Girder torsion: θlong DL LL= ( ) +0 010 0 003. . ( ) rad. Load kips DL kips LL= +115 70 D L W r = − + + ⎧⎨⎩ ⎫⎬⎭min . . . . , . 1 552 0 627 2 233 0 156 0 5 λ λ = − + + min . . . . . . 1 552 0 627 0 1587 2 233 0 156 0 1587 2   . , . . 0286 0 5 0 3389 ⎧⎨⎩ ⎫⎬⎭ = ( )strong axis (G-7) D L W r = − + + ⎧⎨⎩ ⎫⎬⎭min . . . . , . 1 552 0 627 2 233 0 156 0 5 λ λ = − + + min . . . . . . 1 552 0 627 0 1587 2 233 0 156 0 1587 0   . , . . 4930 0 5 0 5 ⎧⎨⎩ ⎫⎬⎭ = ( )weak axis (G-7) D D D D L W a para a a a, max , max . = + ⎛⎝⎜ ⎞⎠⎟⎧⎨⎩ ⎫⎬⎭ = 1 2 3 1 1037 1 5050 0 2852 8 75 17 75 1 3, . . . . .− ⎛⎝⎜ ⎞⎠⎟⎧⎨⎩ ⎫⎬⎭ = 643 1 1037 1 5050 0 2852 17 75 8 7 Da perp, max . , . . . . = − 5 1 1037 ⎛⎝⎜ ⎞⎠⎟⎧⎨⎩ ⎫⎬⎭ = . (G-4) D D a a 1 2 2 1 060 0 210 0 413 1 1037 1 506 0 = + + = = − . . . . . . λ λ 071 0 406 1 5050 0 315 0 195 0 047 2 3 λ λ λ + = = − + − . . . . .Da λ2 0 2852= − . S W L h W L eff eff ri eff eff = +( ) =2 17 75 8 75 2 0 5 17 . . .   . . . . 75 8 75 5 861 155 3 3 2 +( ) = = = = = A W L S G K eff eff in λ 5 861 3 0 11 450 0 1587. . .  = L W eff eff = +( ) = = + 0 5 9 0 8 5 8 75 0 5 18 0 . . . . . .   inches 17 5 17 75. .( ) = inches 41 trans perp long para β girder axis Figure 3.2. Orientation of axes for skew bridge.

The rotations in the local bearing axes (perp and para) are obtained by using the transformation matrix: The total amplified rotations, including the misalignment allowance, are therefore The shear deformations are obtained using the same trans- formation matrix The amplified axial stress is For shear strains on the long side (rotation about the weak or para axis), the shear strain due to (amplified) axial load is Shear strain due to rotation about the weak (para) axis is Shear strain due to shear deformation is The total shear strain check is given by The shear strain due to the static component of the axial load also is acceptable For rotation about the strong, or perp, axis, the service level shear strain due to axial load is γ σa a perp aD GS = = = =, . * . . * . .1 1037 1 642 0 110 5 861 2 811 (G-3) γ σa a para aD GS = = ( ) =, . . . * . 1 3643 115 155 3 0 110 5 861 1 567 3 0. .< (G-3) γ γ γ γtot a r s= + + = + + = ≤ 3 475 0 781 0 287 4 542 5 0 . . . . . (G-1) γ s s rth = = = Δ 0 5736 2 0 0 287 . . . inches inches (G-10) γ θr r para ri iD L h = ⎛ ⎝⎜ ⎞ ⎠⎟ = ⎛⎝⎜ ⎞⎠⎟, . . . 2 0 5000 8 75 0 5 2 0 02040 4 0 781 . . ⎛⎝⎜ ⎞⎠⎟ = (G-6) γ σa a para aD GS = = =, . . . . .1 3643 1 642 0 110 5 861 3 47  5 (G-3) σa = + = 115 2 0 70 155 3 1 642 . . .  ksi Δ Δ Δ Δ perp para long t { } = − ⎡ ⎣⎢ ⎤ ⎦⎥ cos sin sin cos β β β β rans { } = − { }0 57360 8192.. in. θ θ perp para { } = + + − 0 005736 2 0 0000824 0 005 0 0 . . . .  08192 2 0 003605 0 005 0 01090 0 02040− −{ } = +−{ . . .. }rad. θ θ β β β β θ θ perp para long t { } = − ⎡ ⎣⎢ ⎤ ⎦⎥ cos sin sin cos rans DL LL { } = + − 0 005736 0 0000824 0 008192 . ( ) . ( ) . ( ) . ( ) . DL LL−{ }0 003605 rad Shear strain due to rotation is The total shear strain is According to the existing LRFD Method B specifications, this bearing would just fail the compressive stress limit of 1.66GS, and would fail by a substantial margin of the uplift provision (about the strong axis) and the combined stress limit. Failure to meet the latter two criteria is not surprising, because they are very conservative and were a large part of the reason for conducting the research. The bearing also fails to satisfy the proposed Method A re- quirements. The total axial stress/GS is The shear strains due to combined loading about the strong (perp) axis also prove to be excessive. Example 5 Long-span box girder bridge, large bearing, lift-off permitted. Design Criteria A prestressed concrete box girder bridge is supported on elastomeric bearings. There is no skew. The girders span 300 ft. Under full dead load, the end rotation is 0.003 (cambered up- wards) and the load is 1,200 kips. The live load causes a load of 120 kips and a rotation of 0.0018 radians (downwards). Shear displacement due to thermal effects is −2.75, +1.00 in. Allow for 0.0015 radians of thermal rotation in the final condition. Solution For shear displacements, Axial forces appear to dominate the design. Try a high shape factor bearing (generally good for axial capacity but bad for rotation capacity.) Make the bearing approximately 2 to 1 aspect ratio, to minimize the rotational effects. Assume 1⁄4 in. edge cover. Use G = 0.135 ksi (approx. 55 durometer) to increase the load capacity. hrt s≥ = =2 2 2 75 5 5Δ * . . inches (G-10) σ s GS = +( ) ( ) = > 115 70 17 75 8 75 0 110 5 861 1 848 . * . . * . . 1 25. ksi γ γ γ γtot a r s= + + = + + = ≤ 2 811 1 164 0 410 4 385 5 0 . . . . . (G-1) γ θr r perp ri iD L h = ⎛ ⎝⎜ ⎞ ⎠⎟ = ⎛⎝⎜ ⎞⎠, . . . 2 0 3389 17 75 0 5 ⎟ ⎛⎝⎜ ⎞⎠⎟ = 2 0 01090 4 1 164 . . (G-6) 42

Trial and error shows that a bearing of 30 in. × 15 in., with 11 layers at 0.5 in. each will suffice. Final calculations are as follows: The bearing properties (for rotation about the weak axis) are: Coefficients needed for shear strains are The axial stiffness coefficient, Ba, is given by Eq. G-33 as B L W W L a ≈ −( ) + − +( ) − ⎧⎨⎩ ⎫ 2 31 1 86 0 90 0 96 1. . . . min ,λ λ ⎬⎭ ⎛⎝⎜ ⎞⎠⎟ = −( ) + − +( ) 2 2 31 1 86 0 90 0 96 0 2958. . . . * .λ 1 0 4958 1 6032 2 −( ) = . . (G-16) D L W r perp, min . . . . , .= − + + ⎧ 1 552 0 627 2 233 0 156 0 5 λ λ⎨⎩ ⎫⎬⎭ = − + min . . * . . . * . 1 552 0 627 0 2958 2 233 0 156 0 2958 2 0169 0 5 0 3181 + ⎧⎨⎩ ⎫⎬⎭ = . , . . (strong axis) (G-7) D L W r para, min . . . . , .= − + + ⎧ 1 552 0 627 2 233 0 156 0 5 λ λ⎨⎩ ⎫⎬⎭ = − + min . . . . . . 1 552 0 627 0 2958 2 233 0 156 0 2   958 0 4958 0 5 0 4924 + ⎧⎨⎩ ⎫⎬⎭ = . , . . (weak axis) (G-7) Da perp, max . , . . . . = − ⎛⎝1 1583 1 5205 0 2614 29 75 14 75⎜ ⎞⎠⎟⎧⎨⎩ ⎫⎬⎭ = 1 1583. (G-4) D D D D L W a para a a a, max , max . = + ⎛⎝⎜ ⎞⎠⎟⎧⎨⎩ ⎫⎬⎭ = 1 2 3 1 1583 1 5205 0 2614 14 75 29 75 1, . . . . .− ⎛⎝⎜ ⎞⎠⎟⎧⎨⎩ ⎫⎬⎭ = 3909 Da3 20 315 0 195 0 047 0 2614= − + − = −. . . .λ λ Da2 21 506 0 071 0 406 1 5205= − + =. . . .λ λ Da1 21 060 0 210 0 413 1 1583= + + =. . . .λ λ λ = = =S G K 3 9 861 3 0 135 450 0 2958. * . . A W L ineff eff= = 438 8 2. S W L h W L eff eff ri eff eff = +( ) =2 29 75 14 75 2 0 5 2 . * . * . 9 75 14 75 9 861 . . . +( ) = Weff = + =0 5 30 0 29 5 29 75. * ( . . ) . inches Leff = + =0 5 15 0 14 5 14 75. * ( . . ) . inches Under service conditions (amplified loads): For rotation about the critical parallel axis, shear strain due to axial load is shown in Figure 3.3. The rotation consists of 0.003 rad. (DL) and −0.0018 rad. (LL). These two oppose each other, as shown in Figure 3.3, so the peak rotation theoretically occurs under dead load alone. Therefore the rotations should be treated as a static load of the DL + LL [0.0030 + (−0.0018) = 0.0012 rad.] plus a live load of –(−0.0018) = +0.0018 rad. 1.0 times the static rotation plus 2.0 times the cyclic then gives 1.0  0.0012 + 2.0  0.0018 = 0.0048 rad., to which must be added the misalignment al- lowance. The total amplified rotation is then 0.0048 + 0.005 = 0.0098 rad. Shear strain due to rotation is Shear strain due to shear displacement is The total shear strain is The static axial stress is and the corresponding shear strain is Similar calculations for rotation about the strong axis, with only the construction misalignment rotation of 0.005, give γ σa a perp aD GS = = =, . * . . * . .1 1583 3 282 0 135 9 861 2 855 (G-3) γ σa a para aD GS = = =, . * . . * . .1 3909 2 735 0 135 9 861 2 857 3 0< . (G-3) σa = = 1200 438 8 2 735 . . ksi γ γ γ γtot a r s= + + = + + = <3 429 0 382 0 500 4 311 5 0. . . . . (G-1) γ s s rth = = = Δ 2 75 5 5 0 5 . . . inches inches (G-10) γ θr r para ri iD L h = ⎛ ⎝⎜ ⎞ ⎠⎟ = ⎛⎝⎜ ⎞⎠, . . . 2 0 4924 14 75 0 5 ⎟ ⎛⎝⎜ ⎞⎠⎟ = 2 0 0098 11 0 382 . . (G-6) γ σa a para aD GS = = =, . * . . * . .1 3909 3 282 0 135 9 861 3 429 (G-3) σa = + = 1200 2 0 120 438 8 3 282 . * . . ksi 43 DL DL+LL Figure 3.3. DL and LL rotations for bearing design.

The shim thickness, based on Fy = 36 ksi, must satisfy This thickness requires 10 gage sheet steel, which is slightly thicker than the 11 gage commonly used for convenience in manufacturing. It illustrates the need for thicker shims when the axial stress on the bearing is high. The bearing has no external plates, so hydrostatic tension does not need to be checked. Stability criteria are satisfied. Initial conditions also are satisfactory. Because the bearing thickness is less than 8 in. and its plan area is less than 1000 in2, special testing is not required. Note that, according to the existing LRFD Specifications, this bearing would fail the compressive stress limits and the combined stress limits under final conditions. However, it has a shape factor (approx 10) that is higher than in most bridge bearings. It also would fail to satisfy the proposed Method A requirements, because the nonamplified axial stress is 3.01 ksi, which significantly exceeds the limit of 1.25 ksi. While it is quite large, it is still well within the fabrication capabilities of most manufacturers. For example, vibration isolation bear- ings are often twice as thick as this. Example 6 Heavily loaded special purpose bearing, lift-off permitted. Design Criteria An elastomeric bearing is used to support a steel beam that is intended to yield cyclically under seismic loading in a large bridge. Special installation procedures ensure that there is no misalignment whatsoever. The dead load is the self-weight of the beam, which is negligible. The seismic load consists of 12 cycles of load to 2200 kips, accompanied by a rotation of 0.00167 rad. and a shear displacement of 0.25 in. The space available is 20 in. × 24 in. Solution Because the number of cycles of load is much smaller than under traffic load, treat the seismic load as static. The axial load appears to control the design, so use a high shape fac- tor and a relatively hard elastomer with G = 175 psi, (approx 60 durometer). Assume 1⁄4 in. edge cover. h h F s a y ≥ = = 3 3 0 5 3 008 36 0 1253max * . * . . σ inches AASHTO Eq. (14.7.5.3.7-1) γ γ γ γtot a r s= + + = + + = <2 855 0 512 0 000 3 367 5 0. . . . . (G-1) γ θr r perp ri iD L h = ⎛ ⎝⎜ ⎞ ⎠⎟ = ⎛⎝⎜ ⎞⎠, . . . 2 0 3181 29 75 0 5 ⎟ ⎛⎝⎜ ⎞⎠⎟ = 2 0 005 11 0 512 . . (G-6) Trial and error show that a bearing of 24 in. × 20 in., with 2 layers at 0.25 in. each will suffice. Final calculations are as follows: The bearing properties (for rotation about the weak axis) are: Coefficients needed for shear strains are Under service conditions (amplified loads): Shear strain due to axial load is γ σa a para aD GS = = =, . * . . * . .1 5103 4 690 0 175 21 566 1 877 (G-3) σa = = 2200 469 1 4 690 . . ksi D L W r perp, min . . . . , .= − + + ⎧ 1 552 0 627 2 233 0 156 0 5 λ λ⎨⎩ ⎫⎬⎭ = − + min . . * . . . * . 1 552 0 627 0 7366 2 233 0 156 0 7366 1 2025 0 5 0 3070 + ⎧⎨⎩ ⎫⎬⎭ =. , . . (strong axis) (G-7) D L W r para, min . . . . , .= − + + ⎧ 1 552 0 627 2 233 0 156 0 5 λ λ⎨⎩ ⎫⎬⎭ = − + min . . * . . . * . 1 552 0 627 0 7366 2 233 0 156 0 7366 0 8316 0 5 0 3429 + ⎧⎨⎩ ⎫⎬⎭ =. , . . (weak axis) (G-7) Da perp, max . , . . . . = − ⎛⎝1 4338 1 6740 0 1969 23 75 19 75⎜ ⎞⎠⎟⎧⎨⎩ ⎫⎬⎭ = 1 4338. (G-4) D D D D L W a para a a a, max , max . = + ⎛⎝⎜ ⎞⎠⎟⎧⎨⎩ ⎫⎬⎭ = 1 2 3 1 4338 1 6740 0 1969 19 75 23 75 1, . . . . .− ⎛⎝⎜ ⎞⎠⎟⎧⎨⎩ ⎫⎬⎭ = 5103 Da3 20 315 0 195 0 047 0 1969= − + − = −. . . .λ λ Da2 21 506 0 071 0 406 1 6740= − + =. . . .λ λ Da1 21 060 0 210 0 413 1 4338= + + =. . . .λ λ λ = = =S G K 3 21 566 3 0 175 450 0 7366. * . . A W Leff eff= = 469 1 2. in S W L h W L eff eff ri eff eff = +( ) =2 23 75 19 75 2 0 25 . * . * . 23 75 19 75 21 566 . . . +( ) = Weff = + =0 5 24 0 23 5 23 75. *( . . ) . inches Leff = + =0 5 20 0 19 5 19 75. *( . . ) . inches 44

Shear strain due to rotation is Shear strain due to shear displacement is The total shear strain is The bearing has no external plates, so hydrostatic tension does not need to be checked. Stability criteria are satisfied. Initial conditions also are satisfactory. Because the bearing thickness is less than 8 in. and its plan area is less than 1000 in2, special testing is not required. Note that, according to the existing Method B LRFD Speci- fications, this bearing would not be permitted, largely because the applied axial stress exceeds the limit of 1.60 ksi, in the pres- ence of shear deformations. However, during the testing phase of the research, some bearings with shape factor 9 were loaded to 12 ksi (that is, to a stress of 12 GS) with no signs of damage. The bearing in this example is required to carry only 1.876 GS. Note also that the initial assumption that the axial force dominated proved incorrect. The shear strain due to total (amplified) axial load is 1.877, whereas the shear strain due to rotation is 1.783, so the magnitudes of the axial and rotation effects are similar. If the total shear strain of 4.160 were to be regarded as un- acceptably high, the simplest way to reduce it would be to use more rubber layers and lower the rotation component of the shear strain. However, there is no need to design for lower stresses because the shear strains are already low enough to prevent any debonding under such a small number of cycles. In addition, the loading is caused by the extreme seismic event, so some debonding damage may be tolerable. 3.3 QC/QA Procedures Chapter 2 summarized some issues related to QA/QC testing and evaluation procedures. Evaluation of QA/QC procedures was not the primary goal of this research, so less effort was spent on it than was given to testing and FEA. Nevertheless, several recommendations are appropriate and are described here. • It is clear that today’s elastomeric bearings are of higher qual- ity than those made in the past. Two of the reasons are the concentration of manufacturing into four major companies and the effectiveness of the QA/QC requirements imposed by the specifications. Each of the four major companies is large enough to support an effective QA/QC operation, and γ γ γ γtot a r s= + + = + + = <1 877 1 783 0 500 4 160 5 0. . . . . (G-1) γ s s rth = = = Δ 0 25 0 5 0 500 . . . inches inches (G-10) γ θr r para ri iD L h = ⎛ ⎝⎜ ⎞ ⎠⎟ = ⎛⎝⎜ ⎞, . . . 2 0 3429 19 75 0 25 ⎠⎟ ⎛⎝⎜ ⎞⎠⎟ = 2 0 00167 2 1 783 . . (G-6) each takes pride in producing quality bearings. This perhaps is made possible by the relatively stringent specifications, which tend to inhibit low-cost, low-quality fabricators from entering the market. The presence of such operators would create intense price competition, to the likely detriment of quality with little if any real reduction in bridge costs. This observation should be taken into account when considering any QA/QC tests for modification or elimination. The cost advantages that might be gained through elimination of some of these tests are small, whereas the potential losses as- sociated with lower quality standards are high. The cost of re- placing a bearing in a bridge, including any possible litigation costs, is several orders of magnitude higher than the cost of a material test in a manufacturer’s laboratory. As a result, it is suggested that major changes should be undertaken only after very careful consideration of the costs and benefits. • The test procedures previously associated with the current Method B were fairly complex and expensive, and they ap- pear to have deterred some engineers from using the design method. Yet the compressive stresses allowed by Method B may be only slightly higher than those permitted under Method A. (The increase depends on the amplitude of the rotation, which is not accounted for explicitly by Method A.) The long-duration load test was particularly troublesome, but has now been eliminated. Therefore, it is proposed that any additional testing be required for large bearings rather than those designed using Method B. Large bearings are tentatively defined as thicker than 8 in. or with a plan area larger than 1000 in2. There remains the question of what additional testing to apply to such bearings. The regimes in M251-06 have some serious drawbacks, but the AASHTO T-2 Committee has chosen to eliminate the long term test. It is recommended that the choice of required additional testing be a matter for discussion between the T-2 Com- mittee members, a representative group of manufacturers, and the researchers. Each group is a stakeholder in the issue and brings specialist knowledge to the table that is neces- sary to reach a good decision. • Very large and unusual bearings pose problems. Their size may make testing them difficult, because the loads needed might exceed the capacity of most test machinery, espe- cially if the test load is to be significantly higher than the service load. However, they are the bearings most in need of testing, because of the difficulties involved in curing large bodies of rubber and the consequences of a failure are more serious in a large bearing than in a small one. Three possible alternatives are envisaged: − First is to require that the manufacturer produce an extra full sized bearing for testing. It then should be cut up as needed, and destructive shear and compressive tests should be performed on the parts to evaluate the material properties throughout the bearing. 45

− Second is to core the center of one of the large bearings. Portions of the bearing core could be tested in shear and compression to provide the information needed through the thickness of the bearing. The cored bearing could then be refilled with precured elastomer and put into service. If the core is taken in the center of the bearing and the effect of the coring on the strength of the reinforce- ment is considered during design, the refilled bearing may provide good service if the bearing proves satisfac- tory in all other ways. − Third is to test the bearing with a tapered plate. The taper angle should be selected to create a combination of rotation and compression on the most heavily stressed edge that leads to the same local shear strain as would be caused by the desired test load applied through parallel load plates. Calculations should be done using the linear theory strain coefficients, Da and Dr. • Current definitions of the number of test specimens re- quired by the AASHTO M251-06 Specifications are less clear than they should be. For example, the definition of a “lot” of bearings may lead to excessive testing. There is lit- tle reason for duplicating tests on bearings of similar bear- ing sizes and geometries made from the same compound. It is suggested that a set of bearings made from one com- pound and whose dimensions differ by no more than 10% be grouped into a single lot for the purposes of establishing test requirements. Inconsistencies between various specifi- cations and within specifications should also be resolved. • The definition of the word “batch” also should be reviewed for consistency with practice. The process of mixing the in- gredients for a rubber compound is called batching. Some manufacturers batch their own rubber while others buy it prebatched, from a separate supplier. Thus the word “batch” could mean the bearings made from a single delivery of pre- mixed rubber, which might be enough for only some of the bearings for a whole bridge, or it could mean a group of bear- ings for the same job. The different interpretations may lead to different numbers of bearings being tested. 46

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Rotation Limits for Elastomeric Bearings Get This Book
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 Rotation Limits for Elastomeric Bearings
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TRB's National Cooperative Highway Research Program (NCHRP) Report 596: Rotation Limits for Elastomeric Bearings explores the elastomeric bearing design procedures suitable for adoption in the American Association of State Highway and Transportation Officials' load and resistance factor design (LRFD) bridge design specifications.

The appendixes to the report include the following:

Appendix A Test Data

Appendix B Survey of Current Practice

Appendix C Test Apparatus and Procedures

Appendix D Test Results Overview

Appendix E Finite Element Analysis

Appendix F Development of Design Procedures

Appendix G Proposed Design Specifications

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