Analysis and retrofitting of fixed steel platforms under low seismic loads

Chapter Introduction Chapter Introduction 1.1 Background Among the offshore structures, the steel fixed platform is the most common structure in oil exploration. In areas of low seismic activity, the platform design is usually controlled by storm or other environmental loading, such as wave and current, rather than earthquake. For areas where the peak ground acceleration is less than 0.05g, e.g., the Gulf of Mexico, no earthquake analysis is required (API RP 2A-WSD, 2000), since the design for environmental loading other than earthquake will provide sufficient resistance against potential earthquakes. However, past observations indicate that low seismicity areas are not totally free from the seismic hazard due to soil amplification of long period waves which travel long distances from the seismic faults. Seismic waves generated at the epicenter can be classified into two groups: high frequency waves, which have high intensity but damp out rapidly during propagation; low frequency waves, which have large displacement properties and damp out relatively slowly. Because the high frequency waves damp out quickly during propagation, the seismic waves at long distances are often rich in low frequency waves. Although the peak ground acceleration (PGA) of the low frequency waves may be very low, the induced motions may have disproportionally high displacement and possibly high velocity characteristics. In addition, such low frequency waves may be amplified several folds through the soft soil layers; if the natural period of the soil at the site is close to the predominant period of the ground motion at bedrock (this kind of amplification due to resonance is called site amplification effects). Furthermore, such amplification may be further enlarged, if the natural period of the -1- Chapter Introduction structures supported on such soil sites is close to the predominant period of the site. Therefore, due to the large displacement properties that low frequency waves possess and the amplification by the soil, the fixed steel platforms may be subjected to large displacements that may cause some concern. The platforms which are not designed for earthquake may still resist certain level of earthquake loading. This is due to reserve strength. The possible sources of reserve strength are (1) actual strength of the material used in construction is higher than the strength used in the design; (2) effects of structural elements that are not included in the prediction of lateral load capacity; (3) effects of minimum requirements on member sections in order to meet the stability and serviceability limits; and (4) redistribution of internal forces in the inelastic range (Rahgozar and Human 1998). Thus there is a need to evaluate the reserve strength in a fixed steel platform designed for wave load to see whether there is sufficient capacity to meet the demand due to low seismic loads and to investigate possible seismic retrofitting techniques if needed. 1.2 Literature review 1.2.1 Seismic hazard analysis The seismic hazard analysis refers to the estimation of some measure of the strong earthquake ground motion expected to occur at a selected site. This is necessary for the purpose of earthquake resistant design of a new structure or for estimating the safety of an existing structure of importance, like dams, nuclear power plants, longspan bridges, high-rise buildings, offshore structures, etc. at that site. By taking into account the entire available database on seismicity, tectonics, geology and attenuation characteristics of the seismic waves in an area of interest, the seismic hazard analysis provide an estimate of the site-specific design ground motion (Dravinski et al., 1980; -2- Chapter Introduction Westermo et al., 1980). One important application of hazard analysis is the preparation of seismic zoning maps for generalized applications (Lee and Trifunac, 1987; Trifunac, 1989a, 1990a; Anderson and Trifunac, 1977, 1978a, 1978b). By estimating the amplitudes of a parameter describing the ground motion at a closely spaced grid of sites covering the complete area of a big city or an entire state, zoning maps are developed by contouring the sub-areas with equal hazard. Such maps find useful applications in the earthquake-resistant design of common types of structures, for which it is not possible to carry out the detailed site-specific studies. The zoning maps are also useful for land-use planning, assessing the needs for remedial measures, and estimation of possible economical losses during future earthquakes (Trifunac, 1989b; Trifunac and Todorovska, 1998). Seismic hazard analysis involves the quantitative estimation of ground-shaking hazards at a particular site. Seismic hazards may be obtained deterministically (deterministic seismic hazard analysis, DSHA), when a particular earthquake scenario is assumed, or probabilistically (probabilistic seismic hazard analysis, PSHA), in which uncertainties in earthquake size, location, and time of occurrence are explicitly considered. 1.2.1.1 Deterministic seismic hazard analysis The deterministic seismic hazard analysis aims at finding the maximum possible ground motion at a site by taking into account the seismotectonic setup of the area around the site and the available data on past earthquakes in the area (Krinitzsky, 1995; Romeo and Prestininzi, 2000). For this purpose, the magnitude of the largest possible earthquake is estimated for each of the seismic sources around the site of interest. The commonly used forms of seismic sources are the line, area, dipping plane, and the volume sources. The point source is also used sometimes when the epicenters -3- Chapter Introduction are concentrated in a very small area far away from the site of interest. The maximum magnitude in each of the sources is assumed to occur at the closest possible distance from the site. The DSHA involves the development of a particular seismic scenario on which the ground motion hazard evaluation is based. The scenario consists of the assumed occurrence of an earthquake of a specified size occurring at specified location. Earthquake sources may be identified from the records of historical seismicity. If sufficient data are available, the maximum intensity can be determined and used to estimate the location of the earthquake epicenter and magnitude of the event. A typical DSHA has four-step process (Reiter, 1990) as described below: 1. Identification and characterization of all earthquake sources that can produce significant ground motion at the site. 2. Selection of a source-to-site distance parameter for each source zone. In DSHA, the shortest distance between the source zone and site of interest is selected. 3. Selection of the controlling earthquake is generally expressed in terms of some ground motion parameter, at the site. The selection is made by comparing the levels of shaking produced by earthquakes assumed to occur at the distance identified in step 2. The controlling earthquake is described in terms of its size and distance from the site. 4. The hazard at the site is defined, usually according to the ground motions produced at the site by the controlling earthquake. Peak acceleration, peak velocity and response spectrum ordinates are used to characterize the seismic hazard. Although the deterministic seismic hazard analysis seems to be a very simple procedure and provides a direct framework for evaluation of worst-case ground motions, it cannot present information on the likelihood of occurrence of the -4- Chapter Introduction controlling earthquake, the likelihood of it occurring where it is assumed to occur and the level of shaking that might be expected during a finite period of time. The results of the deterministic approach based on a single earthquake at a fixed distance from a selected site are not always able to ensure the intended conservatism for all the structures covering a wide range of frequencies. This is because the ground motion in different frequency ranges may be dominated by earthquakes of different magnitudes and distances. Thus, to get a reliable estimation of the seismic hazard at a site, it is necessary to consider through PSHA the effects of all the earthquakes of different magnitudes with their proper spatial distribution around the site of interest (Cornell, 1968; Anderson and Trifunac, 1977, 1978a), and not just a single earthquake. Also, the uncertainties in specifying the input parameters should be taken into account (Lee and Trifunac, 1985). 1.2.1.2 Probabilistic seismic hazard analysis During the past 50 years, the use of probabilistic concept has made the uncertainties in the location, size and rate of recurrence of earthquakes and ground motion characteristics to be explicitly considered in the evaluation of seismic hazards. Cornell (1964) studied the probability distribution of a dependent variable and derived its relationship with other independent variables whose probability distributions are known or can be assumed. Esteva (1966) and Rosenblueth (1968) studied the earthquake ground motions, their dependence on magnitude and distance, and the relationship between the frequency of occurrence of earthquakes and the frequency of occurrence of ground motions at a site. The PSHA can be described as a procedure involving four steps (Reiter, 1990): -5- Chapter Introduction 1. Identification and characterization of earthquake sources. It is like the first step of DSHA, except that the probability distribution of potential rupture locations within the source must be characterized. 2. Characterization of the seismicity or temporal distribution of earthquake recurrence. In this step, a recurrence relationship that represents the average rate at which an earthquake of some size will be exceeded is used to characterize the seismicity of each source zone. 3. The ground motion produced at the site by earthquakes of any possible size occurring at any possible point in each source zone must be determined with the use of predictive relationship. 4. The uncertainties in earthquake size, earthquake location and ground motion parameter prediction are combined to obtain the probability that the ground motion parameter will be exceeded during a particular time period. In the PSHA, the “logic-tree” formulation is often used to incorporate the effects of both aleatory and epistemic uncertainties (Coppersmith and Youngs, 1986; SSHAC, 1997; Savy et al., 2002). The logic-tree methodology considers a large number of different probabilistic models and model parameters, and computes the hazard for all the combinations of parameter values defined by the end branches of the logic-tree. Each of the input parameters is assigned an appropriate weight to define a discrete probability density function for the frequency of exceeding a value of a strong motion parameter. One can then obtain the various statistical estimates of the frequency of exceeding. 1.2.1.3 Disaggregation It has become common to display the relative contributions to the hazard by different random components, specifically, the magnitude, M, the source-to-site -6- Chapter Introduction distance, R and ε , a measure of the deviation of the ground motion from the predicted value. The above results, which are obtained separately for each fault and successively combined for all the faults in the region, are called the disaggregation of the PSHA. The disaggregation has two major objectives: get the contributions to a fixed hazard level in terms of fundamental quantities and provide seismological parameters describing the earthquake that contribute most to a fixed hazard value. The first application of disaggregation was introduced by McGuire and Shedlock (1981) and the relevance of disaggregation was pointed out by the National Research Council (NRC) (1988) and recognized lately by U.S. Regulatory Commission (NRC) and the U.S. Department of Energy (DOE). McGuire (1995) published a disaggregation method that finds an earthquake representative of the disaggregated uniform hazard-response spectrum. The calculation involves: 1. execution of the PSHA analysis in terms of magnitude, distance and ε for each ground motion prediction equation considered. 2. analysis of the contributions of each seismic source to check which source dominates the hazard. 3. selection of the representative magnitude-distance combination. 4. modification of ε until the computed spectral ordinates reproduce the values of the target uniform hazard spectrum. The disaggregation describes the conditional probability distribution of M, R and ε of an event for which spectral acceleration exceeds a specified level at the site. In probability theory, the different representations of the disaggregated hazard are called, respectively, the marginal probability mass function (PMF) of M, and the joint PMF of M-R and M-R- ε , based on condition that the spectral acceleration is greater than a specified level at the site. -7- Chapter Introduction The disaggregation of the hazard in terms of the joint M-R- ε distribution was introduced and recommended by Stepp et al. (1993), Chapman (1995), McGuire (1995), Abrahamson (1996), Kramer (1996), Abrahamson and Silva (1997). Recently, Spudich (1997), Harmsen and Frankel (2001) used four dimensions: latitude, longitude, M, and ε for the joint probability distribution. This disaggregation scheme permits the display of the hazard on a typical map of the faults surrounding the site, allowing an immediate identification of the locations on the faults dominating the hazard. Practically speaking, this formulation, along with the knowledge of the most likely magnitude, may be very helpful in establishing the specific earthquakes that present the greatest hazard to the site. 1.2.2 Capacity Assessment It is important to answer the question how to determine the capacity of the offshore platform by means of advanced analyses. Usually, in a design situation the approach is to obtain a characteristic capacity higher than the characteristic environmental loads with a return period of typically 100 years multiplied by some partial safety factor for loads and resistance. Using conventional design procedures for ultimate limit state design (ULS), the characteristic capacity is normally taken as first yield or first component buckling decided on the basis of linear analysis. Obviously this method is conservative. Figure 1.1 (Skallerud and Amdahl, 2002) illustrates a typical offshore steel platform’s deformation curve with environmental loads. The structural response is typically such that after some plasticity has developed, a critical compression member becomes unstable. Although the load at member failure could be said to be the capacity of the structure, the structure often regains resistance because the reduction in load-carrying capacity is compensated by load redistribution to still intact members. When more members fail by buckling or -8- Chapter Introduction yield, the structure at last reaches a limit load. With the increasing interest in taking advantage of the system strength beyond first component failure to resist extreme loading events, it is important to understand how the various parameters (number of legs, bracing configuration, numer of bays, and so on) are critical to strength and how variations in these parameters may affect strength. In order to correctly assess the ultimate strength after the first component failure and post-ultimate behavior, static pushover analysis (Kallaby and Millman, 1975) is used to identify the reserve and residual strength of the platform. Usually, the reserve strength is referred to preultimate performance measures and the residual strength is referred to post-ultimate performance measures. However, there is no consensus regarding which performance measures should be used in deciding the above parameters. Indeed, many performance measures are used in the industry. Figure 1.1 Global load versus global displacement for an offshore platform 1.2.2.1 Reserve strength Reserve strength exists at the component level to allow for uncertainties in both the resistance of the component and the loading to which it is subjected. Based on -9- Chapter Introduction statistical data, characteristic values are adopted to ensure that the probability of failure is acceptable. Beyond that, safety factors are applied to improve the certainty of survival and to allow for which no statistical data are available. At the system level, however, there are additional sources of reserve strength. The failure of one component may not limit the capacity of the structure as a whole, provided there is adequate ductility and redundancy such that loads can be redistributed. For highly redundant structures, a sequence of component failures occur before the ultimate strength is reached. The above factors are implicit sources of reserve strength which are not controlled or quantified in design. Similarly, conservatisms embodied within codes, material yield strengths exceeding the minimum criteria specified, component limit states less onerous than ultimate strength and overdesign of non-structural requirements, may be considered as implicit sources of reserve strength. Reserve strength is more commonly defined as the ability of a structure to sustain loads in excess of the design value. The Reserve Strength Ratio (RSR) (Tius and Banon, 1988) may be defined as: RSR= Ultimate Platform Resistance Design Load (1.1) According to Health & Safety Executive (HSE) study (OTO 1997), an approximate value for reserve strength may be obtained by examining the proportion of design load that is environmental in nature (wave and current load, wind load, buoyancy). As a global safety factor is applied to the combined dead and live load, an estimate of reserve strength can be obtained by removing this safety factor and assigning the associated margin for increased environmental load. This simplified method was initially proposed by Bea and Mortazavi (1995) and later adopted in HSE study. The ISO guide (ISO 1998) defines the design load as the unfactored 100 year global environmental action, while the ultimate platform resistance is determined in - 10 - References 122(3), pp.110-117. 117. Robert, A. (1975). Evaluation of Seismicity and Earthquake Shaking at Offshore Sites. Offshore Technology Conference, OTC 2354, pp. 179-190. 118. Romeo, R. and Prestininzi, A. (2000). Probabilistic versus Deterministic Seismic Hazard Analysis: An Integrated Approach for Siting Problems, Soil Dyn. Earthq. Eng., Vol. 20, pp.75-84. 119. Rosenblueth, E. and Esteva, L. (1966). On seismicity. Seminar on Applications of Statistics in Structural Mechanics. 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Lett., Vol. 68(1), pp. 58-73. - 230 - Appendix Appendix A Earthquake Catalog from USGS United States Geological Survey (USGS) Branch of Earthquake and Geomagnetic Information http:// www.usgs.gov/ ZONE 1A Geographic Grid Search Earthquakes= 34 Latitude: 11.000N 4.000N Longitude: 122.000E 117.000E Catalog Used: PDE Magnitude Range: 5.0 - 10.0 Data Selection: Historical & Preliminary Data CAT YEAR MO DA ORIG TIME PDE PDE PDE PDE PDE PDE PDE PDE PDE PDE PDE PDE PDE PDE PDE PDE PDE PDE PDE PDE PDE PDE PDE PDE PDE PDE PDE PDE PDE PDE PDE PDE PDE PDE 1973 1975 1976 1976 1976 1976 1976 1976 1976 1976 1976 1977 1980 1981 1982 1984 1987 1988 1992 1993 1993 1993 1994 1994 1996 1997 1997 1997 1999 1999 1999 2000 2001 2002 04 10 07 07 07 07 07 07 07 08 09 07 10 12 06 03 03 12 02 02 04 10 11 11 11 02 02 05 03 03 04 02 07 04 28 17 25 26 26 26 26 26 26 14 18 13 23 25 14 14 07 14 07 28 08 13 02 27 09 12 13 12 04 06 23 27 31 07 203943.90 140408 140317.80 025639.30 030315.10 053510.30 083612.20 084934.60 094350.60 111028 075444.90 230807.80 140021.40 002815.79 192934.53 003918.15 172055.71 170628.07 223408.76 004528.70 201721.35 005233.63 014355.54 182708.01 061114.55 172123.47 220317.62 134526.39 085201.90 131533.67 143414.14 120359.35 164130.93 010357.81 LAT LONG 6.39 10.26 5.09 4.96 5.06 4.99 4.90 4.89 4.99 4.71 4.64 7.80 6.52 4.76 8.09 5.20 7.30 5.75 7.71 8.25 6.22 7.58 5.10 5.77 10.25 10.26 10.24 10.18 5.40 5.37 5.48 10.65 8.02 7.22 117.70 121.80 118.29 118.31 118.39 118.59 118.05 118.34 118.55 118.42 118.03 122.00 117.96 118.48 121.47 118.39 121.78 117.86 121.96 121.73 120.23 121.48 118.64 119.32 121.71 121.56 121.62 121.66 121.94 121.91 121.98 121.97 117.47 117.05 DEP 33 38 33 33 33 33 33 33 33 36 33 46 51 39 26 50 34 79 38 32 46 43 55 27 33 33 33 33 33 32 33 33 33 33 MAGNITUDE 5.40 5.20 5.30 6.20 5.30 5.20 5.30 5.30 5.10 5.10 5.00 5.20 5.10 5.40 5.00 5.60 5.20 5.10 5.00 5.30 5.20 5.50 5.70 5.50 5.40 5.20 5.00 5.90 7.10 5.20 5.00 5.20 5.50 5.10 mb GS mb GS mb GS Ms GS mb GS mb GS mb GS mb GS mb GS mb GS mb GS mb GS mb GS mb GS mb GS mb GS mb GS mb GS mb GS MwHRV mb GS MwHRV mb GS MwHRV MwHRV mb GS mb GS MwGS MwHRV MwHRV mb GS MwHRV MwHRV MwHRV IEFM DTSVNWG NFPO TFS .F .F.G .F.G .G .G .G .F.G .G .G .G .G 5D.G .G .G .G .G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . - 231 - Appendix Appendix B Members Size of 40m X-brace Platform (Figure 3.8) Member Outside Diameter D (m) Thickness t (m) A*A 0.85 0.022 B*B 0.85 0.022 AB 0.24 0.012 AC 0.85 0.022 BD 0.85 0.022 AN 0.28 0.016 BN 0.28 0.016 CN 0.28 0.016 DN 0.28 0.016 CD 0.24 0.012 CE 0.85 0.020 DF 0.85 0.020 CO 0.26 0.016 DO 0.26 0.016 EO 0.26 0.016 FO 0.26 0.016 EF 0.26 0.014 EG 0.80 0.020 FI 0.80 0.020 EP 0.28 0.012 FP 0.28 0.012 GP 0.28 0.012 IP 0.28 0.012 GH 0.26 0.012 HI 0.26 0.012 - 232 - Appendix Appendix C Members Size of 40m K-brace Platform (Figure 3.8) Member Outside Diameter D (m) Thickness t (m) A1A’ 0.90 0.020 B1B’ 0.90 0.020 A’B’ 0.45 0.018 A’C’ 0.90 0.020 B’D’ 0.90 0.020 A’N’ 0.36 0.014 B’N’ 0.36 0.014 C’N’ 0.50 0.016 D’N’ 0.50 0.016 C’E’ 0.90 0.020 D’F’ 0.90 0.020 C’O’ 0.32 0.012 D’O’ 0.32 0.012 E’O’ 0.30 0.012 F’O’ 0.30 0.012 E’G’ 0.90 0.020 F’I’ 0.90 0.020 E’H’ 0.30 0.014 F’H’ 0.30 0.014 G’H’ 0.30 0.012 I’H’ 0.30 0.012 - 233 - Appendix Appendix D Members Size of 80m X-brace Platform (Figure 3.20) Member Outside Diameter D (m) Thickness t (m) A*A 1.10 0.024 B*B 1.10 0.024 AB 0.35 0.016 AC 1.10 0.024 BD 1.10 0.024 AN 0.36 0.016 BN 0.36 0.016 CN 0.36 0.016 DN 0.36 0.016 CD 0.38 0.018 CE 0.85 0.020 DF 0.85 0.020 CO 0.40 0.018 DO 0.40 0.018 EO 0.40 0.018 FO 0.40 0.018 EF 0.40 0.020 EG 0.85 0.020 FI 0.85 0.020 EP 0.36 0.014 FP 0.36 0.014 GP 0.36 0.014 IP 0.36 0.014 GH 0.35 0.014 HI 0.35 0.014 - 234 - Appendix Appendix E Members Size of 80m X-brace Platform without Horizontal Members (Figure 3.20) Member Outside Diameter D (m) Thickness t (m) A*A 1.10 0.026 B*B 1.10 0.026 AB 0.40 0.018 AC 1.10 0.026 BD 1.10 0.026 AN 0.40 0.018 BN 0.40 0.018 CN 0.40 0.018 DN 0.40 0.018 CE 0.90 0.020 DF 0.90 0.020 CO 0.40 0.018 DO 0.40 0.018 EO 0.40 0.018 FO 0.40 0.018 EG 0.90 0.020 FI 0.90 0.020 EP 0.50 0.020 FP 0.50 0.020 GP 0.50 0.020 IP 0.50 0.020 GH 0.40 0.014 HI 0.40 0.014 - 235 - Appendix Appendix F Input File for SHAKE 91 Option - dynamic soil properties #1 modulus for clay 0.001 0.01 0.05 0.995 0.952 0.800 0.10 0.20 0.667 0.40 0.500 0.60 0.333 1.0 0.250 0.167 Damping for clay 0.001 0.01 0.05 0.10 0.20 0.40 0.60 1.0 2.55 3.02 4.70 6.17 8.00 9.83 10.75 11.67 0.40 0.60 1.0 #2 modulus reduction for sand 0.001 0.01 0.05 0.962 0.714 0.333 0.10 0.20 0.200 0.111 0.040 0.024 Damping for sand 0.001 0.01 0.05 0.10 0.20 1.62 5.57 11.67 13.80 15.22 0.40 0.60 16.06 1.0 16.36 16.61 #3 modulus for rock half space 0.0001 0.0003 1.000 1.000 0.001 0.003 0.01 0.03 0.9875 0.9525 0.900 0.810 0.1 0.725 1.0 0.550 Damping in Rock 0.0001 0.4 0.059 0.001 0.8 0.01 1.5 0.1 3.0 1.0 4.6 Option -- Soil Profile Bor Site 5.77 873 35.76 1050 .050 .127 37.63 1584 .050 .137 44.39 1373 .050 .138 55.97 1675 .050 .131 13.52 1260 .050 .136 .050 .131 - 236 - Appendix 29.92 2318 .050 .135 39.37 5508 .050 .133 .010 .141 11475 Option -- input motion: 2623 4096 .02 25. 01.prn (8f10.6) Option -- sublayer for input motion {within (1) or outcropping (0): Option -- number of iterations & ratio of avg strain to max strain 0.7 Option -- sublayers for which accn time histories are computed & saved: Option -- compute & save response spectrum: 9 9.81 0.05 Option -- compute & save response spectrum: 9.81 0.05 Option 10 -- compute & save amplification spectrum: 10 0.05 - surface/rock outcrop execution will stop when program encounters 0 - 237 - Appendix Appendix G Buckling Load of a Column with Hinged Ends The differential equation for a column with pinned ends subjected to an axial force equal to the critical buckling load Pcr (see Figure G.1) is given by the equation: d2y EI = − M = − Pcr y dx (G1) L Figure G.1 Buckling load for pin-ended column When EI is constant, this equation is satisfied when y varies as a sine curve, and the solution for Pcr is the Euler buckling load π EI / L2 , where L is the length of the member. In more complicated cases such as columns with variable cross section or with intermediate axial loads, it is difficult to get an analytical solution to the above equation. In such cases the, the finite-difference form of G1 (Ghali and Nevill, 1989), using the equivalent normalized concentrated elastic load may be used. Discretizing the column into n internal nods with equal spacing λ , then: - 238 - Appendix ⎧ y i −1 ⎫ [− − 1]⎪⎨ yi ⎪⎬ = wi λ ⎪y ⎪ ⎩ i +1 ⎭ (G2) where: y i : the lateral deflection of the i node The normalized elastic load wi can be expressed in terms of the deflection at three consecutive points by: P λ wi = cr 12 ⎡ ⎢ ⎣ EI i −1 10 EI i ⎧ y i −1 ⎫ ⎤⎪ ⎪ ⎥⎨ yi ⎬ EI i +1 ⎦ ⎪ ⎪ ⎩ y i +1 ⎭ (G3) and substituting for wi in Equation G2, we obtain: ⎧ y i −1 ⎫ [− − 1]⎪⎨ yi ⎪⎬ = Pcr λ λ ⎪ y ⎪ 12 ⎩ i +1 ⎭ ⎡ ⎢ ⎣ EI i −1 10 EI i ⎧ y i −1 ⎫ ⎤⎪ ⎪ ⎥ ⎨ yi ⎬ EI i +1 ⎦ ⎪ ⎪ ⎩ y i +1 ⎭ (G4) If the above equation is applied at each of the n internal nodes, the following simultaneous equations can be written: [A]n×n {y}n×1 = Pcr λ [ B]n×n [C ]n×n { y}n×1 12 (G5) Where ⎤ ⎤ ⎡10 ⎡ −1 ⎥ ⎥ ⎢ 10 ⎢− − ⎥ ⎥ ⎢ ⎢ L ⎥ ⎥ ; [B] = ⎢ λ= ; [ A] = ⎢ . . . . . . n +1 λ⎢ ⎥ ⎥ ⎢ 10 ⎥ − − 1⎥ ⎢ ⎢ ⎢⎣ ⎢⎣ 10⎥⎦ − ⎥⎦ ⎡1 / EI ⎢ / EI ⎢ [C ] = ⎢ ⎢ ⎢ ⎢⎣ . / EI n −1 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ / EI n ⎥⎦ - 239 - Appendix Multiplying both sides of above equation by A-1, we obtain: [ H ]{ y} = γ {y} (G6) where [ H ] = [ A] −1 [ B][C ] (G7) γ = 12 /( Pcr λ ) (G8) and The solution of equation G6 is an eigenvalue problem. The buckling load can be calculated from the largest eigenvalue γ by the equation: Pcr = 12 /(γλ ) (G9) - 240 - Appendix Appendix H Variation of Moment of Inertia along the Steel Brace When the steel brace is subjected to axial compression load, first yielding typically occurs in the inner side of the buckled brace at mid-height. As the axial load and corresponding lateral deflection increase, yielding spreads within the cross section and also in the longitudinal direction of the steel brace. This indicates that the moment of inertia varies at the cross sectional level with the applied load and also longitudinally within the yielded region. In order to find this region for steel brace with circular cross section, a nonlinear finite element analysis was carried out. The procedure is presented in detail next. H.1 Finite element modeling The 2.4m long circular steel brace (88.9mm X 4mm) with hinged ends shown in Figure G.1 is divided into 50 elements in finite element software ABAQUS. The initial out-of-straightness of sinusoidal shape with amplitude of e ' = 2.4mm (L/1000) at the mid-height is adopted in order to get the buckling load. The first-order, sheardeformable 2D beam element type B21 is adopted in ABAQUS to simulate material nonlinearity (elastic-plastic) and the nonlinear effect of large deflection. H.2 Results Figure H.1 shows the stress distribution along the steel brace when the buckling load is reached. From this figure, it can be seen that the yield region is between 0.72m (0.3L) and 1.68m (0.7L) from the top. - 241 - Appendix 370 Stress (MPa) 350 330 310 290 270 A B C D 250 0.4 0.8 1.2 1.6 2.4 2.8 Length (m) Figure H.1 Stress distribution along the steel brace The load-axial displacement response of the steel brace using the numerical model in chapter and ABAQUS is compared in Figure H.2 in order to validate the assumption of parabolic variation of the moment of inertia over a region of ± 0.2L from the mid-height. 350 Axial Force (kN) 300 250 200 Numerical Model 150 ABAQUS 100 50 0 Axial Compression (mm) Figure H.2 Load vs. axial compression of steel brace - 242 - [...]... is a simple and quick method to obtain the ultimate strength capacity of an offshore structure under the earthquake load However, if wave load and earthquake load are considered simultaneously then one has to resort to nonlinear time domain analysis 1.2.4 Retrofit of steel structure using fiber reinforced polymers (FRP) If the seismic adequacy of steel platform is insufficient, seismic retrofitting may... wave loads, when they are subjected to far field effects of earthquakes and to propose retrofitting schemes for platforms which are vulnerable More specifically, the study sought to find: 1 the seismic hazard curve in Northern Borneo 2 the performance of fixed steel platforms at this site and at another nearby site at Northeastern Kalimantan, designed for wave loads, under simultaneous action of -... initial out -of- straightness imperfection, and the contribution of CFRP sheets The model was verified using experimental results and showed good agreement: the percentage increase in strength ranged between 11 and 39% Axial stiffness is also increased by up to 46%, regardless of the value of out -of straightness; the effectiveness of CFRP retrofitting increases as the values of out -of- straightness of the... The seismic analysis methods fall into three main groups of ascending sophistication: 1 Simple methods: static coefficient method This can often be carried out by hand calculation It provides an order of magnitude estimate of the response and is usually conservative; 2 Linear methods: response spectrum analysis and time domain analysis (normal modes) These are the main methods for dynamic seismic analysis. .. for retrofitting offshore structural members is a viable procedure, but to date no work has been reported on what extent the buckling strength of tubular members with circular cross section would be enhanced when retrofitted by CFRP 1.3 Objectives and Scope The main objective of the research work reported herein is to evaluate the seismic vulnerability of offshore steel platforms in Northern Borneo and. .. retrofitting may be required Seismic retrofitting refers to wise modification of the structural properties of an existing structure, in order to enhance its strength and ductility for future earthquake (Aoyama and Yamamoto 1984) For steel structures, current methods of retrofitting typically utilize steel plates that are bolted or welded to the structure However, constructability and durability drawbacks... accelerograms at the surface of the selected site are obtained The dynamic analyses of 40m and 80m deep platforms with different bracing configurations at Northern Borneo and Northeastern Kalimantan under the wave load and earthquake load were then carried out Based on the results obtained, two retrofitting techniques were applied on the platforms: viz retrofitting using high strength grout and CFRP In order... surface of the selected site are then obtained and used in a 3D seismic analysis of offshore platforms at Northern Borneo and Northeastern Kalimantan Chapter 5 describes the experimental study on steel brace retrofitted with CFRP An analytical model is developed to predict the behavior of axially loaded slender braces strengthened with CFRP sheets Chapter 6 describes the modeling of grouted members and. .. due to the effects of soil loads, traffic loads, and the internal pressure The results of the analysis showed that carbon fiber sheets provide better performance than glass or aramid in improving the internal pressure capacity of pipes at ultimate stress Figure 1.4 Schematic of the repairing (a) and section of the wound pipe (b) Vatovec et al (2002) tested rectangular steel tubes retrofitted with different... met Krawinkler and Seneviratna (1997) found that the pushover analysis could provide a good estimate of global and local inelastic deformation demand The technique would detect the weakness in the structure such as the formation failure mechanisms, excessive deformation demands, strength and stiffness irregularity, overloads on structural elements and connections, and global instability of the structural . the demand due to low seismic loads and to investigate possible seismic retrofitting techniques if needed. 1.2 Literature review 1.2.1 Seismic hazard analysis The seismic hazard analysis. domain analysis. 1.2.4 Retrofit of steel structure using fiber reinforced polymers (FRP) If the seismic adequacy of steel platform is insufficient, seismic retrofitting may be required. Seismic. involves: 1. execution of the PSHA analysis in terms of magnitude, distance and ε for each ground motion prediction equation considered. 2. analysis of the contributions of each seismic source to