Design Guide for Bridges for Service Life (2013)

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570 This appendix provides steps that could lead to development of design aid for piles subjected to axial load and lateral movement. The principal steps are explained and are customized for development of design aids for 50 ksi steel H-piles. C.1 eStimAtion oF mAximum ALLowAbLe StrAin Seasonal and daily temperature fluctuations subject steel H-piles in jointless bridges to cyclic loading that can result in fatigue failures of the H-piles. This possibility is espe- cially important as the magnitude of cyclic strain exceeds elastic limits. The seasonal and daily temperature fluctuations subject H-piles to one large annual cycle (due to seasonal temperature change) and a number of smaller load cycles (due to daily tem- perature loading) (Dicleli and Albhaisi 2004; Karalar and Dicleli 2010). Figure C.1 shows typical H-pile cyclic strain (Dicleli and Albhaisi 2004). This strain is the maximum longitudinal strain in steel H-piles, typically located at the point of fixity below the pile head. The following steps outline one alternative for predicting the fatigue life of steel elements subjected to variable amplitude cyclic loading. The steps involve concepts of cycle counting and use of damage models for keeping track of accumulated damages due to cycling loading (Gere and Goodno 2012). 1. Obtain the loading history to which the steel element is subjected. 2. Develop an S-N type curve for the material under consideration. In general, in the low-cycle regime (yearly seasonal cycle when the steel element is subjected to strain exceeding elastic limits), the data should be presented in terms of strain versus number of cycles to failure. C DESiGN OF piLES FOR FATiGUE AND STABiLiTy

571 Appendix C. DESiGN OF piLES FOR FATiGUE AND STABiLiTy 3. Use a cycle counting method, such as the rain flow method (ASTM 1049-85), to convert the variable amplitude loading into equivalent constant amplitude loading. 4. Use a damage model, such as Miner’s rule, to determine the time to failure. Dicleli and Albhaisi (2004) suggest using Equation C.1 for relating strain ampli- tude to fatigue life: M N2a f m ε ( )= (C.1) where ea = constant strain amplitude, Nf = fatigue life (number of cycles to failure) corresponding to ea, M = factor determined from experimental testing (0.0795), and m = exponent determined from experimental testing (–0.448). Dicleli and Albhaisi (2004) suggest using Miner’s rule as a damage model for steel H-piles. Equation C.2 expresses Miner’s rule: n N 1i ii n 1 ∑ ≤ = (C.2) where ni is the number of cycles associated with the loading number i, and Ni is the number of cycles to failure for the same case. Dicleli and Albhaisi (2004) assume that steel H-piles are subjected to two constant amplitude loadings, one corresponding to seasonal temperature changes and another representing daily temperature changes. Therefore, Miner’s rule can be expanded as shown by Equation C.3: n N n N 1s fs l fl + = (C.3) Figure C.1. Pile strain as a function of time. Source: Dicleli and Albhaisi 2004.

572 DESiGN GUiDE FOR BRiDGES FOR SERviCE LiFE In Equation C.3, ns and ni are the number of small and large amplitude strain cycles due to temperature changes during the service life of the bridge, respectively, and Nfs and Nfl are the total number of cycles to failure for the corresponding small and large ampli- tude strain cycles, respectively. According to Dicleli and Albhaisi (2004), for 100 years of service life, the number of small amplitude cycles is ns = 14,800, and the number of large amplitude cycles is nl = 100. These values were obtained by studying the field perfor- mance of several jointless bridges and developing the types of data shown in Figure C.1. For small and large amplitude loading, Equation C.1 can be customized, as shown by Equations C.4 and C.5, respectively (Dicleli and Albhaisi 2004): M N2as fs m ε ( )= (C.4) M N2al fl m ε ( )= (C.5) To facilitate development of an “allowable” strain to be used in selecting a steel pile capable of meeting the fatigue requirement, the small strain amplitude (eas) is related to the large strain amplitude (eal) by using β, a proportionality constant, result- ing in the relationship shown by Equation C.6: as alε βε= (C.6) In Equation C.6, β is estimated to be 0.25 (Karalar and Dicleli 2010). By substitut- ing Equation C.6 into Equation C.4 and solving for constant amplitude life to failure, Equation C.4 and C.5 could then be used to determine Equations C.7 and C.8 (Dicleli and Albhaisi 2004): N M 1 2fs al m 1 βε =     (C.7) N M 1 2fl al m 1 ε =     (C.8) By substituting Equations C.7 and C.8 into Equation C.3 and solving for eal , the maximum large amplitude strains that the pile can sustain without fatigue failure can be obtained as shown by Equation C.9 (Dicleli and Albhaisi 2004): n M n M 2 2 1 al s m l m 1 1ε β =     +                   (C.9) Substituting the previously stated values for the parameters in Equation C.9, which are ns = 14,800, nl = 100, b = 0.25, M = 0.0795, and m = –0.448, eal is then determined to be 0.002967.

573 Appendix C. DESIGN OF PILES FOR FATIGUE AND STABILITY Based on the calculated maximum large strain amplitude of eal = 0.002967, the maximum cyclic curvature amplitude Yf at fatigue failure of the pile is expressed by Equation C.10: ε ψ = d 2 f al p (C.10) where dp is the width of the pile in the direction of the cyclic displacement. Knowing the cross section of the steel pile to be used, complete nonlinear moment curvature characteristics of the pile can be developed. From this relationship, the max- imum moment that a steel pile can sustain without failure can be estimated using the maximum “allowable” curvature, as obtained from Equation C.10. The maximum moment that a steel pile can sustain can then be used to obtain the maximum lat- eral displacement that the steel pile can accommodate without fatigue failure. The maximum lateral displacement is obtained through a nonlinear pushover analysis as described in the next section. C.2 Pushover AnAlysis exAmPle The development of the design aids required conducting a static pushover analysis. Static nonlinear pushover analysis using the finite element software SAP2000 can be used to estimate the maximum lateral displacement capacity of steel H-piles based on fatigue consideration. Only two sections (HP10x57 and HP12x84) meet the compact ductility requirements for A36 and A50 steels, as described in Section 8.6.2.4.2. These two cross sections were used for pushover analysis. C.2.1 Soil–Pile Interaction Model For the purpose of a pushover analysis, the p-y curve for piles driven in clay can be simplified as a bilinear curve, as shown in Figure C.2. Figure C.2. Actual and modeled p-y curves for clay. Source: Dicleli and Albhaisi 2004. L oa d pe r un it le ng th , ( P)

574 DESiGN GUiDE FOR BRiDGES FOR SERviCE LiFE In this figure, the ultimate response Pu is estimated as shown by Equation C.11: P C d9u u p= (C.11) where Cu is the undrained shear strength of the clay, and dp is the pile width. The elastic modulus of the clay soil can be estimated as shown by Equation C.12: E C9 5s u 50ε = (C.12) where ε50 is the soil strain at 50% of ultimate soil resistance. Table C.1 lists the corresponding values of Cu and e50 for different consistencies of clay soil. tABLE C.1. rePreSentAtive vALueS oF Cu And e50 Consistency of Clay Cu (psi) e50 Soft 2.9 0.020 Medium 5.8 0.010 Stiff 17.4 0.005 C.2.2 Description of the model To conduct a pushover analysis, the pile was modeled using SAP2000 and divided into small beam elements, each 1 ft in length. For the purpose of the analysis, a 40-ft length of pile was modeled for soft- and medium-density clays. The models show that this length is sufficient to provide a relative fixed condition in the lower portion of the pile. The pile tip is restrained from movements in all directions. The soil response to lateral deflection was modeled using nonlinear link elements placed every foot. The load deflection properties of the link elements were defined based on the p-y curve described in Chapter 8. Material properties were assumed to be 36-ksi steel for the pile section. Nonlinear beam elements with the capability of developing hinges at both ends were used in the pushover analysis. The properties of these hinges were defined on the basis of the ori- entation and the level of axial load on the pile. For a given axial load in the pile, soil condition, and steel section, a pushover anal- ysis was then performed to obtain the maximum lateral displacement capable of meet- ing the fatigue limit. On the basis of the assumptions made, the maximum moment that a pile can sustain without fatigue failure was established. This maximum moment is used in pushover analysis to establish the maximum lateral displacement. Results of the pushover analyses for various axial loads are shown in Figure C.3 and Figure C.4.

575 Appendix C. DESiGN OF piLES FOR FATiGUE AND STABiLiTy C.2.3 Results of the Analyses Using the described method and by performing pushover analyses, the maximum dis- placement that steel H-piles with a specified minimum yield strength of 50 ksi can ac- commodate has been estimated and is shown in the Figures C.3 and C.4. These figures can be used to determine the maximum lateral displacement that a pile can sustain, based on fatigue considerations. An interesting aspect of the data shown in these figures is that piles oriented with bending about the strong axis provide larger displacement (up to four times). Many departments of transportation orient the steel piles about the weak axis of bending, based on the logic that it could provide larger lateral displacement. The results shown in the figures contradict this belief. Figure C.3. Lateral displacement capacity of compact HP sections in soft clay (c = 2.9 ksi) (a) HP10x57 (b) HP12x84. Figure C.4. Lateral displacement capacity of compact HP sections in medium clay (c = 5.8 ksi) (a) HP10x57 (b) HP12x84. (a) (a) (b) (b)