Master’s Thesis in Sustainable Structural Engineering An investigation of the design of cylindrical steel tanks modelled according to EN 14015 and according to the Eurocodes Author: Yonas Gebre Supervisor: Carmen Amaddeo Ext.Supervisor: Björn Mattsson Examiner: Björn Johannesson Course Code: 5BY31E, 30 credits Date: 2022-02-22, Linnaeus University, Faculty of Technology Department of Building Technology Abstract Storage tanks are above or below ground vessels for storing chemicals, petroleum and other liquid products. Above ground vertical cylindrical shells are typically thin walled structures prone to buckling and lose their stability especially when they are empty or have lower fluid level due to external loads. According to the Swedish National board of Housing, Building and Planning (Boverket), the Eurocodes and the Swedish national annex and building code for structural design, EKS (BFS 2011:10) should be used for verification of mechanical resistance of storage tanks. However the industry has been using a European design standard EN14015, for design of large site built steel tanks. The research question is if this design fulfils the requirements in the Swedish building code EKS and the Eurocodes. In order to investigate this, a parametric study of the buckling resistance of an empty tank has been performed, by comparing the design according to EN14015 With the requirements according to the Swedish building code and the Eurocodes. The finite element analysis was done with the finite element tool ABAQUS, The parametric study was carried out for three terrain categories0, I and II, for the six snow load zones and for six basic wind velocities according to the Swedish snow and wind maps in EKS. The buckling resistance also further investigated for three reliability classes, reliability class 1, 2 and 3 according to the Swedish national annex and for two fabrication classes, fabrication class A and B using EN1991-1-6 The finite element analysis result of linear elastic and nonlinear buckling analysis with imperfections showed that, design according to EN14015 can meet the requirements of Eurocodes and EKS at lower basic wind velocities, terrain category (I and II) for smaller imperfections . But it does not meet the requirements at terrain category-0, for all reliability classes and all imperfection classes. The tank shell showed in some cases an increase in the load proportionality factor in nonlinear analysis for the load combinations considered in this study. It is thus necessary to study further on the finite element modelling of thin walled large tanks on relations of local buckling effect due to highly stiffened regions and the effect of magnitudes and applications of imperfections for large tanks using EN1993-1-6 Keywords: Steel tank; Wind; Snow; FEM; Shell buckling; Non-linear analysis iii Acknowledgement This is a master thesis that has been written as the last project within the Master of Science in sustainable structural engineering program at Linnaeus University. I would like to thank my supervisor Carmen Amaddeo at Linnaeus University. My external supervisor Björn Mattson at Transport Styrelesen for the input on the thesis; you have supported me to keep the project going forward, giving me valuable input to complete this thesis work. And also many thanks to Osama Abdeljabar for your help in my thesis work. Last but not least I would like to thank those of you who helped me during my stay at Linnaeus University. Without the support from home the project would have been difficult to complete. Yours sincerely Yonas Gebre Växjö, Date, 02-03-2022. iv Table of content 1. Introduction.................................................................................................... 1 1.1 Background and problem description ................................................................. 1 1.2 Purpose and aim .................................................................................................. 2 1.3 Hypothesis and Limitations ................................................................................ 2 2. Literature review ........................................................................................... 3 3. Theoretical background ................................................................................ 5 3.1 Design according to EN 14015: 2005 ................................................................. 5 3.1.1 Design for fluid action ................................................................................ 5 3.1.2 Design against buckling .............................................................................. 5 3.2 Design according to the Swedish building code and the Eurocodes ................... 5 3.2.1 Reliability classes........................................................................................ 6 3.2.2 Self-weight................................................................................................... 6 3.2.3 Snow load .................................................................................................... 7 3.2.4 Wind load .................................................................................................... 8 3.3 Fabrication classes in EN1993-1-6 ................................................................... 10 3.3.1 4. Methodology and object description .......................................................... 12 4.1 Design according to EN 14015 ......................................................................... 12 4.2 Finite Element Analysis .................................................................................... 13 4.2.1 4.3 Different geographical locations .............................................................. 14 The studied tank ................................................................................................ 15 4.3.1 5. Geometrical imperfections ........................................................................ 10 Geometry of the tank ................................................................................. 16 Results and analysis ..................................................................................... 17 5.1 Design using EN14015 ..................................................................................... 17 5.1.1 Design for hydrostatic pressure ................................................................ 17 5.1.2 Design to prevent buckling ....................................................................... 18 5.2 Numerical simulations (FEM) .......................................................................... 18 6. Analyses and discussion .............................................................................. 25 7. Conclusions ................................................................................................... 27 v 1. Introduction Storage tanks are part of many industries for storing chemicals, petroleum products or any fluids or gases at ambient/elevated/low temperature. Tanks could be classified as either above or below ground [1]. Above ground storage tanks are constructed just above the ground or at higher elevation using structural supports. In an industrial set-up both above and below ground steel tanks are widely used to store water, petroleum and other chemical fluids. Above ground steel tanks can either be vertical cylindrical or horizontal but vertical cylindrical vessels are used mostly and built with flat bottoms that rest directly on prepared ground. Being in possession of a tank leads to the responsibility of ensuring that the tank doesn't affect the environment through leakage of chemicals or petroleum and other hazardous fluids. Tanks that contain flammable fluids have to be inspected regularly by an accredited control organization and have to follow the rules of several actors [2].These rules and regulations are all laws written by the parliament combined with regulations written by the government and regulations written by several authorities. They all have to be applied when designing of tanks as well as directives from EU [3]. Vertical above ground tanks are used in many industries to store water, oil, fuel, chemical and other fluids [4].The materials used varies depending on the fluid stored and the industry [4].Materials like Metals have been used almost exclusively in the oil industry and are most often short cantilever shells [4]. Silos and pressure vessels tend to be taller than storage tanks [4].oil storage tanks are constructed of curved steel sheets that are welded together to form a cylinder and are prone to fail by buckling e.g. due to wind pressure. Researches on storage tanks for containing oil and fuels have increased significantly in the last 20 years [4]. This is due to the huge economic, environmental and social losses caused by failures due to accidents or natural disasters [4]. 1.1 Background and problem description The Swedish National Board of Housing, Building and Planning (Boverket) incorporated in May 2011 the Eurocodes in their building code for structural design of construction works. This building code, Boverket mandatory provisions amending the board’s mandatory provisions and general recommendations (2011:10) on the application of European design standards (Eurocodes), EKS, covers also the design of large tanks. This means that the Eurocodes now have to be followed. Thus, the Eurocodes and EKS shall now be used for the verification of the mechanical strength of steel tanks. However, the industry have been using the design methods in the European standard EN 14015:2005, Specification for the design and manufacture of site built, vertical cylindrical, flat bottomed, above ground, welded, steel tanks for the storage of liquid at ambient temperature and above. One problem with the design models in EN 14015 is that it does not have any explicit models to account for different snow loads or wind loads. According to EKS and the Eurocodes it is necessary to verify that the load bearing capacity is larger than the actions effects from permanent and variable actions [5]. Depending on the geographical position the wind load and snow load may vary considerably in Sweden. Another problem with the model in EN 14015 regarding buckling is that, it has limitations. It does not consider explicitly the wind load, or the weight of the roof and any snow load on it. The structural design models of the shell courses and the distances 1 between stiffening rings is thus quite simplified. More over the models are based on the allowable or working stress method, whereas the Swedish building code and the Eurocodes are based on the partial factor method. In the allowable stress method all reliability of the resistance of the structure is placed on the strength of the material. In the EKS and the Eurocodes the reliability is distributed to both the actions and the resistances. It is also possible to differentiate between different levels of reliability in this system, depending on the consequences of failure of a construction work or a building. 1.2 Purpose and aim The purpose of the thesis is to clarify under which circumstances, the design according to EN 14015:2005 comply with the Swedish building code. In order to do this, the design of a large tank according to EN 14015 is compared with the design using finite element analysis according to the Eurocodes and the Swedish national choices in EKS. The comparison is made for different geographical locations by using possible wind and snow load combinations. The result of this thesis can also be used to provide information or feedback to the industry and the designer on the use and limitation of EN14015 design method. 1.3 Hypothesis and Limitations The hypothesis of this study is that, steel tanks designed according to EN14015 design method do not fulfil the requirements of the Swedish building code, EKS, at some geographical locations and terrain types in Sweden. The investigation is limited to tank shell buckling, and only one tank with specific size is studied in three terrain categories. It is also assumed that the tank is situated close to the cost line of Sweden. The different combinations of wind and snow load are thus limited to a certain number of combinations of wind speeds and snow loads. Moreover it is also assumed that the self-weight of the roof of the tank is the same, irrespective of the snow load at the different geographical locations. 2 2. Literature review A study on numerical evaluation on the shell buckling of empty thin-walled tanks under wind load according to current American and European design codes was performed by Chrysanthos at et. al. [6]. They made a comparison between the current design codes API650 and EN1993-1-6 by performing linear bifurcation analysis, LBA, geometrically nonlinear elastic analysis of the perfect tank shell, GNA and geometrically nonlinear buckling analysis of an imperfect tank shell, GINA, recommended by Eurocodes. These analyses were done in order to evaluate the buckling resistance of two existing above ground, vertical, thin-walled, cylindrical steel tanks, with large diameters and variable thickness. Both tanks are self-supported, flat bottoms, having nine courses with variable shell thickness and considered empty. This study also indicates the efficiency of current design specifications in addressing structural stability of empty large tanks when subjected to wind actions. They also stated that EN1993-1-6 and EN1993-4-2 have not yet seen many field applications and their results may raise doubts. Linear bifurcation analyses of the FE-models of both tanks were carried out using ABAQUS. The tanks were subjected to wind pressure distribution as proposed by EN1993-1-6 and uniform wind pressure adopted by API 650. The results were found by Comparing boundary condition discrepancies between fully fixed and compression-only boundary conditions. Based on the result of the analysis, shell buckling was initiated at the thinner shell courses for both tanks, the buckling mode of tank T-761 is located below both wind girders, since the ring stiffeners provide greater stiffness to the upper thin shell courses due to smaller distance between wind girders for fully fixed boundary conditions and EN1993-1-6. Wind pressure might overestimate buckling capacity for T-761. From Linear bifurcation analysis of the two models, they found that tank T-776 and T761 showed lower Critical buckling load capacity when the tanks’ bottom boundary conditions are partially fixed, or compression-only springs are used and when the wind load acting on the shell is not uniform. But under uniform wind velocity pressure distribution on the tank shell, both tanks showed similar critical elastic buckling capacity when considering both partially fixed and fully fixed bottom boundary conditions. They finally concluded that; • • • • Linear bifurcation analysis, LBA, is a good indicator for critical buckling load capacity and it also provide a bifurcation point and buckling modes that can be used as imperfection shapes in nonlinear analysis. Fully fixed boundary conditions may overestimate the buckling capacity of tanks. Uniform wind pressure distribution is evaluated as the more unfavorable compared to the EN1993-1-6, distribution, which is experimentally confirmed. It also provides a different behavior to the shell, allowing smaller displacements and stresses. The imperfection amplitudes proposed by EN1993-1-6 decrease considerably the nonlinear buckling resistance of the cylindrical tanks but also cause a progressively stiffening response with rapidly growing displacements of the tank shells. The largest deformable criteria for estimating the critical load is arbitrary and does not provide reasonable and satisfactory results. 3 API650 propose empirical design methods, which do not quantify the buckling critical state and do not account for imperfections so that the theoretical back ground of the method should be investigated and improved in the future. In a study by Lei Chen and J .Michael Rotter [10] on ‘‘Buckling of anchored cylindrical shells of uniform thickness under wind load’’ they conclude that cylindrical shells subjected to non-uniform wind pressure display different buckling behaviours from those of cylinders under uniform external pressure. In this study, different size silos and anchored tanks of all typical geometries against buckling under wind loads. Considering different aspect ratios (H/D), like stocky, intermediate and slender cylinders. The buckling behaviour of silos and anchored tanks of commonly used geometry under wind pressure was studied using finite element analysis, ABAQUS. Silos aspect ratio is in the range of (0.5< L/r < 8 with 200< r/t<1000) where t, is the wall thickness. And tanks are often squatter and thinner with ratio of (0.1< L/r <4 with 500 < r/t< 2000). Linear elastic bifurcation and geometrical imperfect nonlinear buckling analysis were performed for cylinders with (r/t =200) and for different length to radius ratios such 0.5,1 and 4. The results showed that the critical buckling is similar to the classical circumferential buckling mode, but when the length to radius ratios changes to higher values, the critical buckling mode also changes, axial compressive stresses are induced by the unsymmetrical pressure on the shell and this produces a main long buckle on the windward meridian extending over whole height. When considering geometrical nonlinearity stocky cylinders with (L/r <3) pre-buckling deformations are small before limit load is reached and its nonlinear buckling pressure is close to the linear elastic critical pressure, this indicate that geometrical nonlinearity has little effect on the buckling strength of stocky cylinders. Finally, they conclude that cylindrical shells subjected to non-uniform wind pressure displays different buckling behaviours than those subjected to uniform external pressures. In addition different aspect ratios of cylinders also results a complex buckling patterns on the shell and yield quite different results of linear and nonlinear buckling analysis. Tanks under uniform external pressure always experience circumferential buckling and little affected by change in geometry. For intermediate cylinders, pre-buckling ovalization of the cross-section has an important effect on the buckling strength. 4 3. Theoretical background In this section a brief review of the basic theories and regulations related to shell buckling analysis is presented. Stability of a structure can be analysed by computing its critical load, i.e., the load corresponding to the situation in which a perturbation of the deformation state does not disturb the equilibrium between the external and internal forces [7].Buckling instability occurs in elastic and plastic modes and can occur locally or globally or both locally and globally, overall or take place as a combination of local and overall buckling [7]. Shell buckling due to wind is one of the most common damages caused by the impact of a strong wind load [8]. Different design methods are used to check the stability of a vertical, above ground cylindrical tanks. The mostly used standards, the Eurocodes, EN14015 and the Swedish national annex EKS, are adopted for this thesis work. 3.1 Design according to EN 14015: 2005 EN14015 design standard uses empirical methods for selecting the thickness of the shell courses, depending on the geometry of the tank, the operational liquid level, the material used, the density of the liquid to be contained and allowance for corrosion. This method is based on the concept of limiting the tensile stresses of the shell due to hydrostatic pressure while it does not consider tank shell buckling explicitly. 3.1.1 Design for fluid action During operation, the load due to the liquid content shall be the design weight of the product to be stored from the maximum design level to empty. Also, the internal pressure load shall be the load due to the specified test pressure and test internal negative pressure [8]. For a full tank, the design hydrostatic pressure is the maximum liquid pressure acting at the bottom of the tank shell course, which is the product of the specific weight of the liquid, maximum height measured from the top of the tank shell. This fluid action is resisted by providing thicker bottom shell courses. In this project, the tank is assumed to be empty and the internal negative pressure is considered for the calculation. Empty tanks are highly susceptible to buckling since the effect of the external wind load and the internal negative pressure on the tank shell becomes more pronounced. As a result, the tank shell may buckle at lower snow load. 3.1.2 Design against buckling In this design standard, shell buckling is considered indirectly through an empirical design method that ensure the stiffening of the shell by providing it with circumferential ring stiffeners at specific heights, depending on the thickness of the shell courses, the height of the shell and a three second gust wind velocity [6], which is considered to be the design wind speed acting on the tank shell. If the gust wind speed is greater than 45m/sec, another wind speed should be considered after reaching an agreement between the client and the designer [6]. 3.2 Design according to the Swedish building code and the Eurocodes Eurocodes design standards provides theoretical background and methodologies for evaluating explicitly the buckling resistance of the shell structures [8]. The provisions include analytical formulation for calculating the buckling capacity in terms of stresses. 5 The Eurocodes also proposes numerical analysis methods like linear bifurcation analysis and nonlinear buckling analysis. In this project, the effect of the wind velocity pressure and snow load on mechanical strength of the tank shell considering different terrain categories and snow load zones, since wind velocity pressure is dependent on the reference wind load, which varies with terrain category. The snow load also varies with the geographical locations. Wind loads on cylindrical tanks are simulated as pressure distribution acting on the circumferential shell [8]. According to EN1993-4-2, wind simulations through equivalent uniform pressure throughout the circumference of the tank shell, are adopted for this study [9]. Figure 1 Wind pressure distribution on cylindrical tanks (left) and equivalent uniform pressure (right) 3.2.1 Reliability classes The reliability of a structure can be classified based on the extent of the risk of human injuries caused due to the failure of the structure [13].The Eurocode system uses for this purpose, consequence classes [8]. The consequence class, classifications of structures based on effect of the consequence of failure of the structures considers also social and environmental risks. In this study three reliability classes, RC1-RC3 are used according to the Swedish national choice in EKS. Reliability class-1 is a lower reliability with a coefficient of 0.83, and should be used for example for a steel tank for storing water at agricultural farm area. Steel tanks for containing liquids or gases with explosive nature can cause a high risk of injuries to human if failure occurs. In such case the steel tank can be classified as Reliability class-3, a higher reliability class with a coefficient of 1.And reliability class-2 with a coefficient 0.91 is considered when a structural failure causes medium risk of injuries. This study uses all the three reliability classes to compute the buckling load capacity of the steel tank under study. The implementation of these coefficients is shown in the Equation (1), where the coefficient is denoted γd 3.2.2 Self-weight The design self-weight of permanent actions on a structure can be determined by using Equation 6.10 b from the Swedish national annex, is given by; πΊπΊπ·π· = πΎπΎππ . ππ. πΎπΎπΊπΊ_π π π π π π . πΊπΊππ (1) Where, πΎπΎππ is the reliability coefficient, ππ is Self-weight reduction factor, πΎπΎπΊπΊ_π π π π π π is the load factor for safety and πΊπΊππ self-weight of the structure. 6 In tank shell design, the shell wall constitutes its self-weight and the weight of the roof. The self-weight of the shell is balanced or supported by the tank bottom support and the tank shell is assumed in its’ equilibrium state before the tank roofing weight is applied. The assumed roof self-weight is 58 ton, which is based on the calculated data obtained from similar tank roof design of a Mid-Rock tank, T-501. The same self-weight of the roof is assumed for all snow load zones and terrain categories considered under this study. The weight of the roof (πΊπΊππ ) is calculated by adding the total weight of all the roofing components per circumferential unit. The area of the roof is calculated in the Mathcad document in Appendix A, as described in Eurocode 1 [11].The design value of the self-weight of the roof include a reliability coefficient,( πΎπΎππ ), a self-weight reduction factor,( ππ) and a partial factor for permanent load, (πΎπΎπΊπΊ_π π π π π π ), see equation (1).Using the Swedish building code, the reduction factor is a constant which was set to 0.89 and the partial factor, which works as a safety factor, is 1.35[10].The design self-weight of the roof is calculated for the three reliability classes at each terrain category. 3.2.3 Snow load The snow loads were given by the snow standard as described by equation. The snow load shape coefficient (ππππ ) is given in EN1991-1-3[12] for the slope of the roof. The exposure coefficient (Ce ) is determined according to the topography; such as windswept, normal or sheltered, where the sheltered topography leads to a larger exposure coefficient [12].A normal topography was chosen for this study. The thermal coefficient (Ct) is only used for heated construction and where the roof has a poor thermal insulation [13]. The thermal coefficient depends on the heat transfer coefficient, temperature in the surroundings and whether or not the roof has a snow guard that is the probability of the snow staying on the roof. The characteristic value of snow load on the ground (π π π π ), is determined by the National Annex for the geographical location of the tank [13]. In this study all the characteristic snow loads acting on the tank shell at different geographical locations in Sweden were considered. A more description of the calculations can be seen in Appendix A. The snow load on the roof is assumed to be vertically on horizontal plane. The design snow load on the tank roof acting on the top of the tank shell perimeter is determined by using equation 6.10.b of the Swedish building code. Compute the design snow load, considering each selected snow load zone when snow load is accompanying variable action and also considering the reliability classes, reliability class A and B. πππ·π· = πΎπΎππ . πΎπΎππ . ππo . ππππ (2) where: πΎπΎππ - reliability coefficient, πΎπΎππ - variable action load factor, ππo - combination factor for accompanying variable load ππππ - characteristics snow load on the tank roof 7 ππππ = ππππ . Ce . Ct . π π π π (3) The characteristics snow load on the tank shell is the product of snow load shape coefficient (ππππ ), exposure coefficient (Ce ), thermal coefficient (Ct), and the basic ground snow load (π π π π ), See equation (3). In this study the characteristics snow load on the tank shell is calculated for the selected snow load zones in Sweden. 3.2.4 Wind load The Eurocodes and the Swedish national annex are used to calculate the design wind load [15].The main factor that influences the design wind pressure acting on the tank shell is the terrain category for which the tank is modelled. The external wind pressure consists of some constants factors like terrain factor, (ππππ ) wind velocity peak factor, (ππππ ), wind turbulence intensity, (πΌπΌππ ), and the wind turbulence intensity is also dependent on the orographic coefficient, (ππππ ), the height of each shell course from the bottom of the tank and the external pressure buckling coefficient (Cb), the peak velocity pressure, (qp), which is given by the national annex. The peak velocity pressure is determined through several factors which are described in the Mathcad document in the Appendix A, but can be shortly described as factors determined for the terrain types and the basic wind velocity. In this study three terrain categories were chosen. The terrain types are, terrain category-0 (at sea or coastal area exposed to the open sea), terrain category-I (Lakes or flat and horizontal area with negligible vegetation and without obstacles) and terrain category-II (area with regular cover of vegetation or buildings). The basic wind velocity depends on the geographical location where the coastal areas generally have a higher basic wind velocity than further in the country [13].The location was set in the case of Sweden, for all the six different basic wind velocities in the three selected terrain categories. The equivalent external wind pressure on the tank shell at each shell course is calculated by using equation 11.28, EN1993-4-2[9]. According to the standard the peak wind velocity pressure shall be multiplied by a maximum wind pressure coefficient (kw), which is also a function of the radius of the tank, distance between stiffening rings and the thickness of the shell plates between the stiffening rings [15].Similar to the design snow load and selfweight of the tank roof, the design wind pressures acting on the shell are calculated at the selected terrain types for reliability class 1, 2 and 3. The common design parameters or constants are being the same for all the three terrain types but the terrain roughness length, (π§π§ππ ), wind pressure turbulence intensity on the shell (πΌπΌππ ), and the terrain factor, (ππππ ), and varies for each terrain type. The turbulence intensity of the wind pressure at a height (z) is given by: πΌπΌππ = 1 π§π§ οΏ½ππππ ln οΏ½ οΏ½οΏ½ π§π§ππ where: ππππ is the orographic coefficient, π§π§ππ is the reference height in [m], and π§π§ is the height in [m] measured along the shell height, starting from the tank bottom. The terrain factor is given ππππ is given by; 8 (4) ππππ = 0.19 οΏ½ π§π§ππ οΏ½ π§π§πππΌπΌπΌπΌ 0.07 (5) The peak wind velocity pressure, acting on each shell course along the height of the tank is given by equation (6) below: ππππ (π§π§), where z is measured along the shell height and zo is the reference height. 2 π§π§ π§π§ ππππ (z) = οΏ½οΏ½ππππ . ln οΏ½ οΏ½ . ππππ οΏ½ + 2. ππππ . οΏ½ππππ2 . ππππ . ln οΏ½ οΏ½οΏ½οΏ½ . ππππππ π§π§ππ π§π§ππ The equivalent wind velocity pressure acting on the tank shell is given by: ππππππ_πΈπΈπΈπΈ ππππππ_πΈπΈπΈπΈ = πππ€π€ πππ€π€_max _πΈπΈπΈπΈ (6) (7) where: course. πππ€π€ is the maximum wind pressure coefficient, πππ€π€_max _πΈπΈπΈπΈ is the maximum wind pressure [kN/m2], acting at the top of each shell The internal suction pressure acting on the tank shell at each terrain category is taken as 40 percent of the equivalent wind pressure acting at the top of the tank shell, πππ π _πΈπΈπΈπΈ . πππ π _πΈπΈπΈπΈ = 0.4πππππππΈπΈπΈπΈ ,π‘π‘π‘π‘π‘π‘ (8) The characteristic equivalent pressure acting on the tank shell is the sum of equivalent external pressure, acting along its height at the top of each shell course, and the internal negative pressure acting on the tank shell, which is given by equation (9). πππΈπΈπΈπΈ = ππππππ_πΈπΈπΈπΈ + πππ π _πΈπΈπ·π· (9) where: ππππππ_πΈπΈπΈπΈ is the characteristic external wind load [kN/m2], πππ π _πΈπΈπΈπΈ is equivalent internal negative pressure [kN/m2]. In terrain category II, the tank was located at a place, where some isolated obstacles, like buildings, trees and low vegetation, exists with separations of at least 20 times the obstacle height [15]. The turbulence intensity of the wind at a height (π§π§), the peak velocity pressure, (ππππ ) and the corresponding characteristic equivalent pressure (πππΈπΈπΈπΈ ) acting on the shell were lower when compared with the results in terrain category 0 and terrain category I In terrain category I, the tank was located at a place, where lakes or flat and horizontal area with negligible vegetation and without obstacles [15]. The turbulence intensity at a 9 height (π§π§), the peak velocity pressure, (ππππ ) and the corresponding characteristics equivalent pressure acting on the shell were lower when compared with terrain category 0 but higher than the results in terrain category II. In terrain category 0, the tank was located near to the sea or coastal area exposed to open sea [16]. The turbulence intensity at a height (π§π§), the peak velocity pressure, (ππππ ) and the corresponding characteristic equivalent pressure acting on the shell were higher when compared with the results in terrain category I and terrain category II. The design wind pressures are calculated using the reliability classes (1, 2 and 3).The imperfections classes (A and B) were used in the selected terrain categories. For additional calculation parts see the in appendix A. 3.3 Fabrication classes in EN1993-1-6 3.3.1 Geometrical imperfections There are three fabrication classes in EN1993-1-6. Larger or smaller imperfections are allowed depending on the fabrication class. Imperfections may have a large impact on the buckling capacity of thin walled structures. For thin walled structures, geometric imperfections refer to deviation from the shape of perfect geometry. Initial geometric imperfections are the main attribute that dominate the buckling and post-buckling behaviour of thin walled structures by creating large differences between theoretical and experimental prediction of collapse loads [15]. In this project, imperfection classification, as described by the Eurocodes buckling relevant geometrical tolerance, specifically fabrication tolerances, are considered for non-linear buckling analysis. Fabrication tolerance class A and B were used in the analysis. The dimples formed should be measured at every position, in both meridional and circumferential directions. The depth of initial dimples should be assessed in terms of the dimple parameters, U0X, U0θ and U0W and these values should satisfy the conditions stated in EN1993-1-6. The depth of initial dimples created on the shell wall, calculated by using gauge of length (lg) for meridional compressive stressed region, (ππππππ ) and for circumferential compressive stressed or shear stressed region, (lgθ). The gauge length, (ππππππ ), for across welds in both circumferential and meridional directions. The amplitude of initial imperfection is the maximum value of the product of the gauge lengths calculated in Equations (10), (11) and (12). See appendix B for calculation of initial dimple tolerances. ππππππ = 4√ππππ (10) where, ππ is the radius of the tank shell[m], π‘π‘ is the minimum thickness of shell courses[mm], and ππππππ is gauge length along meridional direction[mm] ππππππ = 2.3(ππ 2 ππππ)0.25, ππππππ ≤ ππ (11) where: 10 ππ is length of the shell segment between two ring stiffeners[m], and ππππππ is gauge length[m] along circumferential direction ππππππ = 25π‘π‘ or ππππππ = 25π‘π‘ππππππ , ππππππ ≤ 500ππππ where: ππππππ is gauge length due to additional across welds in meridional and circumferential direction. 11 (12) 4. Methodology and object description 4.1 Design according to EN 14015 In this section, a Mathcad tool is used for design calculations according to EN14015. This design is then used in order to study if the design fulfils the requirements in the Swedish building code for mechanical strength of construction works. In EN14015 the design of the tank can be divided in two steps. The first step is computation of the design hydrostatic pressure acting on the shell, which is maximum at the bottom of the shell course. Hydrostatic pressure is the product of the specific weight of the liquid to be contained and the height of the tank shell. The second step is design for the stability of the tank and this is also done after determining the shell thickness so that stresses in the tank shell from the action of hydrostatic pressure do not exceed the least of two third of material yield strength or 260MPa. Computation of the shell thickness is influenced by factors like geometry of the tank, material to be contained, material strength, corrosion allowance, design pressure and height of shell courses from top of the tank shell. And then design of the shell for external wind loads and internal negative pressure, which is buckling resistance of the tank shell, is prevented by providing ring stiffeners and determines the minimum spacing of secondary rings using an equivalent minimum shell thickness. The required minimum design thickness of the shell plates is derived from equation (13) ππππ = π·π· [98ππ(π»π»ππ − 0,3) + ππ] + ππ 20ππ (13) where: c is corrosion allowance, for the shell assumed fully painted (c=0), D is tank diameter in [m], π»π»ππ is the distance from the bottom of the considered course to the design liquid height [m], ππ is the design pressure [mbar], S is the allowable design strength [MPa]. According to EN14015, 9.1.1 a, the allowable design stress, S is minimum of 2/3 of the yield strength of the shell material or 260 MPa, P is the design pressure [MPa], assumed to be zero, since an empty tank was considered, and W the density of the liquid to be contained, [Kg/l], for diesel a density of 780Kg/l was used. The required minimum number of stiffening rings and their spacing on the shell π»π»ππππ 5 ππππππππ = πΎπΎ οΏ½ 3 οΏ½ π·π· 0.5 (14) 12 where: π»π»ππππ is the minimum spacing between the stiffeners, ππππππππ is the minimum shell thickness, and K is a factor determined by an empirical formula in equation (15). πΎπΎ = 95000 2 (3.563ππππ + 580ππππ ) where: (15) ππππ is a 3 second gust wind speed [45m/s], and ππππ is design internal negative pressure. Taking the ratio of the total equivalent stable height (π»π»π¬π¬ ) of the shell to the minimum spacing between the stiffeners (π»π»ππππ ). And comparing their values [6] using the relations, 2π»π»ππππ <π»π»π¬π¬ < π»π»ππππ and also their location on the tank shell is determined by 1/3(π»π»π¬π¬ ) and 2/3(π»π»π¬π¬ ) from the top of the tank shell. 4.2 Finite Element Analysis The stability analysis of a finite element model of the shell was done by using ABAQUS [14], based on Eurocodes recommendations [15].The cylindrical tank shell was discretized by using 4-node curved elements identified as S4R, with reduced integration. The mesh (some 88000 elements) was defined by means of convergence studies [14]. In this project the structural response of the tank shell has been computed using linear bifurcation analysis (LBA), to obtain the critical elastic buckling load factor (CBLF) and geometrical nonlinear analysis with imperfections (GMNIA). Linear bifurcation analysis determines the elastic critical buckling resistance, which is a key reference value for all shell buckling analyses. This analysis provides the first estimate of the elastic buckling strength, which is unique and computable when compressive stresses occur in the structure. It can be represented by a relatively simple equation following a parametric numerical study. As result, linear bifurcation analysis can be adopted as the elastic reference failure strength in EN-1993-1-6 to be modified approximately to account for the more complex effects of imperfections [10]. In this project the linear bifurcation analysis was done for different design buckling load combinations of wind and snow for the three terrain categories. Also the three reliability classes were considered. After running the linear buckling analysis the Eigen value for the first Eigen mode was obtained and it was used as the applied load multiplying factor in the load module to get the critical elastic buckling strength of the shell element. In general, the result of linear bifurcation analysis (LBA) cannot be justified as a reliable strength assessment for practical shell structures because the true strength may be quite sensitive to geometric nonlinearity and geometric imperfections, which are both ignored in linear bifurcation analyses. Non-Linear buckling analysis for numerical post buckling analysis is based on incremental and iterative Newton-Raphson method to obtain solutions for nonlinear 13 problems. Riks algorithm was chosen for the different variations of Newton-Raphson method, which is an arc-length technique that can provide solutions in nonlinear problems [16], and this method, is generally used to predict unstable geometrically nonlinear collapse of a structure. Geometrically nonlinear static problems sometimes involve buckling or collapse behaviour. Where the load-displacement response shows negative stiffness and the structure must release strain energy to remain in equilibrium. The Riks method uses the load magnitude as an additional unknown, it solves simultaneously for loads and displacements. That is, during periods of response, the load and /or the displacement may decrease as the solution evolves. In this analysis the result from the linear elastic buckling analysis, Eigen mode shape 1, was used as the initial imperfect shape for nonlinear analysis. Also geometrical imperfections were considered according to the allowed limits for the different fabrication classes in EN1993-1-6. Imperfection class A and B were primarily used in this study. 4.2.1 Different geographical locations In this study three selected terrain categories, together with snow load and basic wind velocities at different geographical locations were investigated. For, on ground snow load and basic wind velocity map of Sweden, see Figure 2 and Figure 3 [13]. Figure 2. On ground snow load map of Sweden according to the national annex EKS (BSF 2011:10). 14 Figure 3. Basic wind velocity map of Sweden according to the national annex EKS (BSF 2011:10). Table 1.Basic Wind and ground snow load combinations vb (m/s) Snow load on the ground (kN/m2) 21 3.0 22 3.5 23 2.5 3.0 24 2.5 3.0 25 26 4.5 3.5 1.5 1.0 1.5 Table 1, shows the basic wind velocity and ground snow load combinations using the basic wind velocity and ground snow load map of Sweden. This load combinations were used in the three categories selected in this study. 4.3 The studied tank The studied tank is dome shaped, above ground, vertical, cylindrical tank, painted shell surfaces to prevent corrosion and normal steel is used. The tank is assumed to be placed at different selected geographical locations in Sweden and used to contain petroleum. The tank has fixed bottom and pinned top shell boundary conditions and also stiffened by ring stiffeners at two positions along its height. For the material properties of the shell and the stiffener [6] see Table 2. 15 Table 2.Material properties Designation E ν fyk 4.3.1 Value 210GPa 0.3 420MPa Description Stiffness, Young’s modulus Transverse contraction, Poisson’s ratio Yield strength Geometry of the tank The studied tank was provided with the following dimensions: • • • • • the diameter of the tank, D=26m the height of the tank shell, H=21m the tank shell type - variable thickness tank shell tank roof height above the shell wall, h=2.5m the tank roof type - fixed type and dome-shaped See also Figure 4. The thickness of the shell courses varied from 6mm at the top to 9mm at the bottom. Figure 4. Geometry of the tank (left) and shell wall thickness(right), with permission from EN 1993-1-6 16 5. Results and analysis This chapter has been divided in to two sections, the first section, presents the result of the Mathcad calculation for the design of the tank according to EN14015.The second one presents the results from the finite element analysis of the tank designed according to EN14015. 5.1 Design using EN14015 The calculation results of the tank shell using the design models in EN14015 regard the hydrostatic pressure and the stability of the shell to prevent buckling. For an empty tank it is the buckling due to the loads from the roof, snow and wind that the tank is designed for with respect to instability and buckling of the shell. But the design hydrostatic pressure is calculated for determination of the tank shell thickness. The hydrostatic pressure effect is higher towards the bottom shell courses. The design hydrostatic pressure acting on the tank shell when the tank is at full level is calculated by equation (16). PH = γQ . γ. H (16) The result of the design hydrostatic pressure is 25MPa, which is smaller than the allowable tensile strength of the shell material (260MPa) and also the thickness of the shell course increases from top to the bottom of the shell. 5.1.1 Design for hydrostatic pressure Based on this standard and some assumptions, like length of each shell course, since this study is focus on an empty tank and the shell surface is painted to prevent corrosion, allowable design strength of the shell material calculated from characteristics yield strength of the shell material. A total of elven shell courses are assumed, out of which nine of them have equal height of 2m and for the last two top shell courses 1.5m height is used. For additional calculation parts see in the Appendix B. The shell thickness is computed by equation (13). According to EN14015 design standard, the minimum thickness provision for the shell courses having minimum calculated thickness, using normal steel material was used for the tank shell is 6mm [6]. Design for wind it has been done by first computing equivalent stable height of the shell courses, which are the product of each shell course height with the ratio of the minimum shell thickness to the shell course under consideration. The total equivalent stable height of the tank is 17.34m, which is the sum of equivalent stable height of each shell course. 17 Table 3.Shell course thickness and equivalent height Course number 11_top 10 9 8 7 6 5 4 3 2 1_Bottom 5.1.2 h(m) 1.5 1.5 2 2 2 2 2 2 2 2 2 H(m) 21 19.5 18 16 14 12 10 8 6 4 2 He(m) 1.5 1.5 2 2 2 2 2 1.58 1.14 0.89 0.73 eci (mm), calculated 0.72 1.56 2.4 3.24 4.09 4.93 5.77 6.62 7.46 8.09 8.72 eci(mm), provided 6 6 6 6 6 6 6 6.6 7.5 8.3 9 Design to prevent buckling The maximum permitted spacing between secondary stiffening rings on shells of minimum thickness and the required number of stiffeners were calculated using equation (14) and (15). From the equation (15), the maximum spacing between the ring stiffeners, (Hp1), was calculated to be 5.78m. The ratio of the total equivalent stable height of the tank shell to the maximum permitted spacing between the stiffeners is 2.77 m, and comparing these values yields, two secondary ring stiffeners are required. According to EN14015, Table 17, provide two angular shaped secondary stiffeners with minimum size 120x80x10 [mm] and the ring stiffeners are positioned at 5.78m and 11.6m from the top of the tank shell. 5.2 Numerical simulations (FEM) The result of static structural finite element analysis, using ABAQUS for the different combinations of the investigated terrain types, reliability classes, snow loads, wind loads and fabrication classes are shown below. The results using the load combinations in this study, the elastic critical buckling load factor (CBLF), the Load proportionality factor (LPF) in nonlinear buckling analysis are listed in tables and also presented some graphs of LPF verses lateral displacement of the shell, (U3). In terrain category II the Linear bifurcation and nonlinear buckling analysis results for this terrain category using the given load combinations, considering reliability class1 and reliability class 2 and using fabrication class A and B 18 Figure.5. Linear elastic buckling (left) and Nonlinear buckling (right). Figure 6. von Mises equivalent stress(left) and nonlinear buckling top view(right). Note that the deformation is exaggerated in, Figure.5 and Figure 6, which are scaled 20 times the original deformation. The linear critical buckling load factor for Eigen mode 1 is 1.32, the load proportionality factor, LPF is 1.24, obtained from nonlinear analysis, see Figure.5. The shell is stiffer at the top region due to the two closely spaced ring stiffeners but it shows a little reduction in the buckling load factor in nonlinear analysis and the corresponding lateral maximum displacement along the wind direction is reached 49.64mm at the stiffened region. The equivalent von Mises stresses, see Figure 6, increases from bottom of the tank shell to its top and it reaches a maximum where local buckling or dimples formation occur. The critical elastic buckling load factors and Load proportionality factors for nonlinear buckling are presented in Table 4 and Table 5. 19 Table 4. Critical buckling load factor(CBLF) and Load proportionality factor(LPF) for Terrain CategoryII, reliability class-1 and Fabrication class-A. vb(m/s) sk,(kN/m2) Critical buckling load factor, CBLF Ultimate buckling load factor, LPF 21 3 1,68 1,54 22 4.5 1,49 1,36 23 3.5 1,41 1,31 24 3 1,32 1,24 25 1.5 1,26 1,67 26 1.5 1,17 1,24 Table 5. Critical buckling load factor( CBLF) and Load proportionality factor(LPF) for Terrain CategoryII ,Reliability class-2 and Fabrication class-A. vb(m/sec) Sk,(kN/m2) Critical buckling load factor Ultimate buckling load factor, LPF 21 3 1.53 1,42 22 4.5 1.36 1,24 23 3.5 1.29 1,20 24 3 1.20 1,14 25 1.5 1.15 1,22 26 1.5 1.07 1,43 The load proportionality factor (LPF) verses horizontal displacement of the tank shell (U3) for a basic wind velocity of 24m/s and a ground snow load of 3.0kN/m2 according to Figure.7. For the six basic wind velocities, see Figure 8. Additional results are presented in the Appendix A. 20 Figure.7. LPF verses displacement for terrain category-II, reliability class 1, fabrication class-A Figure 8. LPF verses displacement for terrain category-II , reliability class-1 and fabrication class-A The results for terrain category I, the Linear bifurcation and nonlinear buckling analysis results for this terrain category using the given load combinations, considering reliability class1 and 2 and using Fabrication class A and B are shown in Table 6 and Table 7. The critical elastic buckling load factors and load proportionality factors for nonlinear buckling are presented; see Table 6 and Table 7. The result in Figure 10 shows some increment in the buckling load capacity in nonlinear analysis. 21 Table 6 Critical buckling load factor( CBLF) and Load proportionality factor(LPF) for Terrain Category-1 ,Reliability class-1 and Fabrication class-A. vb (m/s) Sk(kN/m2) Critical buckling load factor, CBLF Ultimate buckling load factor, LPF 21 3 1.48 1,35 22 4.5 1.32 1,18 23 3.5 1.24 1,13 24 3 1.07 1,07 25 1.5 1.09 1,11 26 1.5 1.02 1,05 Table 7.Critical buckling load factor( CBLF) and Load proportionality factor(LPF) for Terrain Category-1 ,Reliability class-2 and Fabrication class-B vb (m/s) Sk(kN/m2) Critical buckling load factor, CBLF Ultimate buckling load factor, LPF 21 3 1.35 1,52 22 4.5 1.13 1,20 23 3.5 1.15 1,35 24 3 0.97 1,27 25 1.5 0.99 1,45 26 1.5 0.93 1,40 The following graphs shows, the load proportionality factor (LPF) verses horizontal displacement of the tank shell (U3) for basic wind velocity of 23m/s and a ground snow load of 3.5kN/m2 see Figure 9. And also for the six basic wind velocities, see Figure 10. Figure 9. LPF verses displacement (U: U3) for basic wind speed 23m/s at terrain category-1, reliability class-1, fabrication class-A: 22 Figure 10. LPF verses displacement (U: U3) for all basic wind speeds at terrain category-1 , reliability class1 and fabrication class-A In terrain category 0, the FE-model analysis results obtained in this category under the given load combinations, considering reliability class1 and 2 and using fabrication class A and B. The critical elastic buckling load factors and load proportionality factors for nonlinear analysis is given below, see Table. 8. Table. 8. Critical buckling load factor (CBLF) and Load proportionality factor(LPF) for Terrain Category0, Reliability class-2 and Fabrication class-A vb(m/sec) Sk(kN/m2) Critical buckling load factor, CBLF Ultimate buckling load factor, LPF 21 3 1,27 1,42 22 4.5 1,13 1,18 23 3.5 1,06 0,96 24 3 0,99 0,91 25 1.5 0,93 0,94 26 1.5 0,87 0,89 A graph of the load proportionality factor verses horizontal displacement of the tank shell for basic wind velocity 26m/s and ground snow load 1.5kN/m2 see Figure 11 and, for the six basic wind velocities for terrain category 0, see Figure 12. 23 Figure 11. LPF verses displacement (U: U3) for basic wind speed 26m/s at terrain category-0, reliability class-2, fabrication class-A. Figure 12. LPF verses displacement (U: U3) for all basic wind speeds at terrain category-0, reliability class2 and fabrication class-A. From the results presented in tables and graphs shown above, for the three terrain categories, the shell buckled in all cases before reaching the yield strength of the steel. The stress level is about 170 MPa. The critical elastic buckling load factors and the load proportionality factors indicate the buckling load capacity of the tank shell in linear and nonlinear cases. 24 6. Analyses and discussion The results obtained from the calculation document, design of the thank shell according to EN14015 and the designed tank shell was investigated if it fulfils the requirements in the Swedish building code. The thickness of shell courses were determined by designing the tank against hydrostatic pressure. The design hydrostatic pressure is 25MPa, maximum at the bottom shell course and it is lower than the allowable strength (260MPa) of the shell material. According to EN14015 minimum thickness criteria for the tank shell using normal steel and using the hydrostatic design pressure, the thank shell was provided with a minimum thickness of 6mm at the top six courses and a maximum thickness of 9mm at the bottom course. Design stability of the tank shell against the wind is influenced by the external wind and internal negative pressure, the minimum shell thickness and the tank diameter. Using EN14015, the wind action is resisted by providing the required number of stiffeners and their position on the tank shell. As a result, the tank shell under this study was provided with two ring stiffeners (L-shaped, 120x80x10mm) located at 5.78m and 11.6m from the top of the tank shell. The finite element analysis for checking the designed tank shell by using EN14015 was investigated by creating the model and using Eurocodes and EKS to apply the design loads on the model. In the finite element analysis linear and nonlinear buckling analyses were carried out considering the variation in terrain category, reliability classes and fabrication tolerances. Critical elastic buckling load and nonlinear buckling load proportionality factors were obtained using the six basic wind velocities and snow load combinations. The change in terrain category affects the linear and nonlinear buckling capacity of the tank shell. Table. 9 shows the buckling capacity of the shell at higher reliability and lower fabrication classes in terrain category 0 and II see Table. 9. For the wind and snow load combinations, the linear and nonlinear buckling capacity of the tank decreases, when the terrain changes from lower terrain category (II) to higher (0).One exception showed for basic wind velocity (21m/s), in nonlinear buckling, the change in terrain category from lower to higher class, CBLF and LPF showed the same value in both terrain categories. Table. 9. Reliability class-2 and Fabrication class-A. Terrain Category II 0 vb(m/sec) Sk(kN/m2) CBLF LPF CBLF LPF 21 3 1.53 1,42 1,27 1,42 22 4.5 1.36 1,24 1,13 1,18 23 3.5 1.29 1,20 1,06 0,96 24 3 1.20 1,14 0,99 0,91 25 1.5 1.15 1,22 0,93 0,94 26 1.5 1.07 1,43 0,87 0,89 The buckling behavior of the shell in the same higher terrain category and lower fabrication class see Table.10. The reliability class of the tank changes from lower to higher class, the tank shell buckling capacity decreases in linear and nonlinear analysis. Both the CBLF and LPF showed reduced values at reliability class 2. 25 Table.10.Terrain category-0 and Fabrication class-A Reliability class 1 2 vb(m/sec) Sk(kN/m2) CBLF LPF CBLF LPF 21 3 1.53 1,42 1,27 1,42 22 4.5 1.36 1,24 1,13 1,18 23 3.5 1.29 1,20 1,06 0,96 24 3 1.20 1,14 0,99 0,91 25 1.5 1.15 1,22 0,93 0,94 26 1.5 1.07 1,43 0,87 0,89 The nonlinear buckling behavior of the shell showed an increase in buckling capacity with in the same higher terrain category and higher reliability class (2), see Table.11, When fabrication tolerance class of the tank changed from lower to higher class, the result showed that linear buckling load factor is the same since it is independent of imperfections but there was an increase in the load proportionality factor (LPF) values at terrain category 0. Table.11. Terrain Category-0 and Reliability class-2 Fabrication class A B vb(m/sec) Sk(kN/m2) CBLF LPF CBLF LPF 21 3 1,27 1,42 1.28 1,52 22 4.5 1,13 1,18 1.13 1,19 23 3.5 1,06 0,96 1.06 1,25 24 3 0,99 0,91 0.99 1,24 25 1.5 0,93 0,94 0.93 1,36 26 1.5 0,87 0,89 0.87 1,27 26 7. Conclusions The results of finite element analysis of the tank shell, showed that design according to EN14015 complies with the Eurocodes and the Swedish building code at lower basic wind velocities and possibly also for higher basic wind velocities for terrain category II. The design using 14015, at terrain category I for lower reliability class, RC-1 and for lower fabrication class, fabrication class-A and at lower basic wind speeds also complies with the Eurocodes and the Swedish national annex. Design of steel tanks using EN14015 standards may not fulfill the requirements of Eurocodes and EKS at higher terrain categories, higher reliability and imperfection classes, Like terrain category 0, reliability class 2 and for fabrication class A. For example such petroleum tanks are mostly large and located near the sea or coastal areas open to the sea for easy shipment process (loading and unloading) of petroleum. And they are located at higher terrain category, stability design of the tanks should follow the Eurocodes and EKS design standards. Consideration of equivalent uniform pressure along the shell circumference, according to EN1993-4-2 for the design of large tanks at higher terrain categories and reliability classes may not give satisfactory results in nonlinear buckling analysis. Some wind-snow load combinations at higher terrain categories, at higher reliability and imperfections classes, the nonlinear finite element analysis of the tank showed an increase in buckling capacity, load proportionality factor for the load combinations considered in this study. Finally this requires also further studies on the relations of local buckling effect due to highly stiffened regions and the effect of the magnitudes and applications of imperfections for large tanks using EN1993-1-6 design standard. 27 Reference list [1] Storage Tanks URL:https://en.wikipedia.org/wiki/Storage_tank (visited on 01-02 2022) [2] Swedac. Cisterner.URL: https://www.swedac.se/amnesomraden/cisterner/ (visited on 01-02- 2022) [3] The Swedish National Board of Housing, Building and Planning (Boverket) https://www.boverket.se/en/start/ (visited in feb.2022) [4] Luis A.Godoy. “Buckling of vertical oil storage steel tanks: Review of static buckling studies”. In: Thin-walled structures volume103 (June.2016), pages.1-21. [5] Chrysanthos Maraveas, Georgios A. Balokas and Konstantinos D. Tsavdaridis “Numerical evaluation on shell buckling of empty thin-walled steel tanks under wind load according to current American and European design codes” volume 95 (October 2015) pages 152-160. [6] SS-EN 14015:2005 “ Specification for the design and manufacture of site built, vertical, cylindrical, flat-bottomed, above ground, welded, steel tanks for the storage of liquids at ambient temperature and above.” [7] Silva, V.D.Mechanics and Strength of Materials. Springer, Netherland, 2006. [8] Eurocode 3. Design of steel structures – Part 1-6, strength and stability of shell structures. European Standard EN 1993-1-6; 2007. [9] Eurocode 3. Design of steel structures – Part 4-2, tanks. European Standard EN19934-2; 2007. [10] Lei Chen, Michael Rotter “Buckling of anchored cylindrical shells of uniform thickness under wind load”. In: Engineering structures 41 (2012), 199-208 [11] Eurocode 1. Actions on structures - Part 1-4, general actions - wind actions. European Standard EN 1991-1-4; 2005. [12] Eurocode 1. Actions on structures - Part 1-3, general actions - snow load. European Standard EN 1991-1-3; 2003. [13] The Swedish National Board of Housing, Building and Planning. Boverket. “Boverkets författningssamling BFS 2019:1 EKS 11”. In: (2019). [14] ABAQUS. Documentation. Dassault Systems Simulia Corp. Providence RI; 2020 [15] Eurocode 1. Actions on structures – Part 1-4, general actions - wind actions. European Standard EN 1991-1-4; 2005, Table 4.1. 28 Appendix A. Mathcad calculation document Steel Tank Design 1. General Data 1.1 Geometrical dimensions :used from Mid-Rock Tank T-501 Diameter : Dt β 26 m Height : Ht β 21 m Roof type: Dome shaped roof Rafters on the inside and must not be welded to the roof plates hr Hight of the roof: Perimeter/Grith of the tank shell: Us Us β Dt ⋅ π = 81.68 m Section areas: At π ⋅ Dt 2 At β ββ = 530.93 m 2 4 Volume/ Tank capacity : V V β At ⋅ Ht = 11149.51 m 3 Used Volume/ tank capacity: Vu Vu β 10600 m 3 1.2 Requirements Design negative pressure: Pu 5 mbar N Pu β 500 ββ m2 Design Over pressure Po : to be neglected since empty tank is assumed for this project. Tank type: Closed tank Tank with low pressure, (acc. SS-EN14015; Table 3) Design temperature: TD =25 ° c Stored Medium: Diesel Density of stored medium : kN γdiesel β 0.86 ββ m3 1.3 Standards used Appendix Page 1 of 55 Appendix A. Mathcad calculation document 1.3 Standards used SS-EN14015:2005 : used for design of the tank shell on Mathcad Eurocodes and EKS 11: used for FE-Modeling of the tank shell on ABAQUS 1.4 Material Material : Carbon and Carbon Manganese steels/ Mild steel: S420ML, EN14015,Table 7 Temperature : Maximum design metal temprature; T=100 ° c 2. Loads: 2.1 Main Loads 2.1.1 Dead Loads weight of roof: steel plate Wroof Steel: Plates(shell, roof) Insulation: No insulation used 2.1.2 Live loads Stored medium : Max.density: Diesel kN ββ m3 γdiesel β 0.86 Snow load : All snow zones are used in this project Wind Load : All basic wind velocity pressures are also used. 3. Design of the tank shell according to EN 14015:2005 3.1 Design for internal loads cβ0 Assume the shell is protected from corrosion,by paints D β 26 m p Neglect the design pressure N fyk β 420 ββ mm 2 Characterstic yield strength of the tank shell material S :Allowable design stress,MPa : According to EN 14015,9.1.1 a Appendix Page 2 of 55 Appendix A. Mathcad calculation document β fyk N β , 260 ββ S β min βββ β mm 2 β β 1.5 kg m 860 ββ ⋅ 10 ββ ⋅ 21 m = 0.18 MPa 3 m sec 2 kN W β 0.860 ββ m3 Hc : The distance from the considered course to the design liquid level/ hieght Assuming a constant course height , h=2m except at the top two courses of the tank h10 =1.5m and h11 =1.5m The height of the tank is divided into 11 rounds Elevation of the lower edge of the round measured from the top edge of the cylinderical height of the tank: β‘ 1.5 β€ β’ 1.5 β₯ β’ β₯ β’ 2 β₯ β’ 2 β₯ β’ 2 β₯ hββ’ 2 β₯ m β’ β₯ β’ 2 β₯ β’ 2 β₯ β’ 2 β₯ β’ 2 β₯ β’ β₯ β£ 2 β¦ β‘ 2 β€ β’ 4 β₯ β’ β₯ β’ 6 β₯ β’ 8 β₯ β’ 10 β₯ Hc β β’ 12 β₯ m β’ β₯ β’ 14 β₯ β’ 16 β₯ β’ 18 β₯ β’ 19.5 β₯ β’ β₯ β£ 21 β¦ z11 :Top shell course z1 :Bottom shell course z1 =Bottom shell course β‘2β€ Hc1_2 β β’ β₯ m β£4β¦ β‘ 6 β€ β’ 8 β₯ β’ β₯ β’ 10 β₯ β’ 12 β₯ Hc3_11 β β’ 14 β₯ m β’ 16 β₯ β’ β₯ β’ 18 β₯ β’ 19.5 β₯ β’β£ 21 β₯β¦ z11 =Top shell course The required Minimum design thickness of the Shell Plates, ec required Shell surface is painted to prevent corrosion cβ0 Neglect design pressure pβ0 D eci β ββ⋅ ββ98 ⋅ W ⋅ ββHc - 0.3 mββ + pββ + c 20 ⋅ S ............................eq(1) Appendix Page 3 of 55 Appendix A. Mathcad calculation document β‘ 0.72 β€ z11 =Top shell course β’ 1.56 β₯ β’ β₯ β’ 2.4 β₯ β’ 3.24 β₯ β’ 4.09 β₯ eci = β’ 4.93 β₯ mm β’ β₯ β’ 5.77 β₯ β’ 6.62 β₯ β’ 7.46 β₯ β’ 8.09 β₯ β’ β₯ z1 =Bottom shell course β£ 8.72 β¦ β‘ 6 β€ β’ 6 β₯ β’ β₯ β’ 6 β₯ β’ 6 β₯ β’ 6 β₯ e β β’ 6 β₯ mm β’ β₯ β’ 6 β₯ β’ 6.6 β₯ β’ 7.5 β₯ β’ 8.3 β₯ β’ β₯ β£ 9 β¦ Specified plate thickness ,e (mm) according to EN14015 emin β 6.0 mm The minimum Shell plate thickness emin : β‘ 6.0 β€ β’ 6.0 β₯ β’ β₯ β’ 6.0 β₯ β’ 6.0 β₯ β‘ 6.0 β€ ec1_9 β β’ 6.0 β₯ mm ec10_11 β β’ mm β£ 6.0 β₯β¦ β’ 6.6 β₯ β’ β₯ β’ 7.5 β₯ β’ 8.3 β₯ β’β£ 9 β₯β¦ 3.2 Design for Wind Primary stiffening ring : not required since the tank is fixed roof type, the shell is adequaetly stiffened by the roof structure: EN 14015, 9.3.1.2 The Equivalent Stable height of a course is given by: 5 β 2 β emin β He β h βββ β β e β h1_9 β 2 m He1 β h1_9 ..............................................................eq(2) h10_11 β 1.5 m β‘2 β€ β’2 β₯ β’ β₯ 2 β₯ 5 β’ β β emin β 2 β’ 2 β₯ βββ β =β’2 β₯ m β’ 1.58 β₯ β ec1_9 β β’ β₯ β’ 1.14 β₯ β’ 0.89 β₯ β’β£ 0.73 β₯β¦ emin β 6 mm 5 β He2 β h10_11 β emin β 2 β‘ 1.5 β€ m ββββ = β’ β£ 1.5 β₯β¦ β ec10_11 β The sum of the equivalent stable heights, HE for the full shell height is : HE β ∑ ββHe1ββ + ∑ ββHe2ββ = 17.34 m Appendix Page 4 of 55 Appendix A. Mathcad calculation document HE β ∑ ββHe1ββ + ∑ ββHe2ββ = 17.34 m The maximum permitted spacing of secondary stiffening rings on Shells of minimum thickness, Hp is calculated PV β 5 mbar Design internal suction pressure.[5mbar], maximum value for non-pressure closed tank m ββ sec VW β 45 95000 K β βββββββ= 9.39 β3.563 ⋅ V 2 + 580 ⋅ P β β W Vβ Assume emin β 6.0 ...................................................................eq(3) D β 26 mm m 1 β 2 βe 5 β min β ⋅ m = 6.25 m Hp1 β K ⋅ βββ ββ D 3 ββ ......................................................................eq(4) HE = 2.77 ββ Hp1 Since 2 Hp1 ∠ HE ∠ 3 Hp1 : Two seconadry stiffners are required These are ideally located at HE Hp1 β β= 5.78 m 3 and 2 ⋅ HE Hp2 β ββ= 11.6 m 3 Two secondary stiffening rings are needed to stablize the shell,which are located at 5.78m and 11.6m below the primary ring on the shell of equivalent height, HE Since both the upper and the lower stiffners are located with in the height of shell courses of minimum plate thickness. No height adjustment is needed. Provide two angular shape secondary stiffners with minimum size 120x80x10 [mm] : according to EN14015: 2005 Table 17. 4. Design of the tank shell according to Eurocodes and EKS-11 4.1. Actions on the steel Tank 4.1.1. Dead Loads Roof: Self weight of the roof data used from Midrock tank ,T-501 , designed and kN constructed for terrain type-0 and snow load zone,where - sk β 2.5 ββ m2 In this document this data is used for terrain type-II,terrain type-I terrain type-0 and all snow zones. Appendix Page 5 of 55 Appendix A. Mathcad calculation document Dome-shaped roof with assumed maximal rafter spacing Lβ2 m Us = 81.68 m Grith /perimeter of the tank shell Us Rafters : steel girders IPE140 Number of the girders: nR Us nR β β= 40.84 L Assumed number of girders nR β 52 Roof shell or plate thickness, minimum value: troof troof β 6 mm Radius of the roof: according to EN14015:2005 rr β 1.5 ⋅ Dt = 39 m kg Density of the roof plate: γroof β 7850 ββ m3 kN γroof β 78.50 ββ m3 hroof β 2.548 m The roof height above the tank shell:h Area of roofing surface Aroof β 2 π ⋅ rr ⋅ hroof = 624.37 m 2 Roofing weight that includes the weight of the roof plate, Girders, purlins ,bracings and weight of welds(1.8% of total weight) Tank roof self-weight data obtained from MID-ROCK Tank-T501, data obtained from Swedish National Board of Housing,Building and Planning (BOVERKET) Roof key stone or central ring wkeyst β 4618.01 kg Roof Girder wGD β 23739.63 kg purlins and bracings wpu_bra β 2855.92 kg Supporting ring/primary ring wsup_ring β 8241.08 kg Roofing steel plate, 4mm thickness wplate β 18240.58 kg Wroof β wkeyst + wGD + wpu_bra + wsup_ring + wplate = 57695.22 kg Diameter of the shell D β 26 m The roofing Dead load acting on the shell top pereimeter Wroof ⋅ g kN G β βββ = 6.93 β π⋅D m Appendix Page 6 of 55 Appendix Wroof ⋅ g kN G β βββ = 6.93 β π⋅D m A. Mathcad calculation document 4. Tank Shell D β 26 m kg ρsteel_plate β 7800 ββ m3 H β 21 m Ashell β π ⋅ D ⋅ H = 1715.31 m 2 Area of the tank shell 4.1.2. Characterstic variable Loads Snow load: Snow Load is calculated according to the ''50 year'' high For the three terrain catagory types-0,I and II: Thermal coefficient: Ct Ct β 1.0 Exposure cofficient, Ce Ce β 1.0 Snow load shape cofficient: μi β 0.8 The characterstic snow Load on the ground: sk Region: Selecting the eight different snow load zones in Sweden and thier corresponding characterstic snow load on the ground, sk Normal topograhpy and swedish snow zones :1,2, ...8 are assumed β‘ 5.5 β€ β’ 4.5 β₯ β’ β₯ β’ 3.5 β₯ 3.0 β₯ kN sk β β’ ⋅ ββ β’ 2.5 β₯ m 2 β’ 2.0 β₯ β’ β₯ β’ 1.5 β₯ β£ 1.0 β¦ Snow loads with designations pursuant according to SS-EN 1991-1-3 Sn β μ i ⋅ C e ⋅ C t ⋅ s k kN For sk ≥ 3 ββ m2 kN For 2 ≤ sk < 3 ββ m2 ψ0_1 β 0.8 ψ0_2 β 0.8 ψ0_3 β 0.8 ψ0_4 β 0.8 ψ0_5 β 0.7 ψ0_6 β 0.7 Appendix Page 7 of 55 : snow zone- 3.0 to 5.5 : snow zone-2.0 to 2.5 Appendix A. Mathcad calculation document kN For 1 ≤ sk < 2 ββ m2 ψ0_7 β 0.6 ψ0_8 β 0.6 : snow zone-1.0 to 1.5 Snow load :Unfavourable action snow Load in these regions β‘ ψ0_1 0 0 0 0 0 0 0 β€ β‘ 0.8 β’ β₯ 0 0 0 0 0 β₯ β’ 0.0 β’ 0 ψ0_2 0 0 ψ0_3 0 0 0 0 0 β₯ β’ 0.0 β’ 0 β’ β’ 0 0 0 ψ0_4 0 0 0 0 β₯ β’ 0.0 β₯= ψ0_i β β’ 0 0 0 ψ0_5 0 0 0 β₯ β’ 0.0 β’ 0 β’ β’ 0 0 0 0 0 ψ0_6 0 0 β₯ β’ 0.0 β’ β₯ 0.0 0 0 0 0 0 0 ψ0_7 0 β’ β₯ β’ 0.0 0 0 0 0 0 0 ψ0_8 β₯β¦ β£ β’β£ 0 0.0 0.8 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.8 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.8 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.7 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.7 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.6 0.0 0.0 β€ 0.0 β₯ β₯ 0.0 β₯ 0.0 β₯ 0.0 β₯ 0.0 β₯ β₯ 0.0 β₯ 0.6 β¦ β‘ 4.40 β€ β’ 3.60 β₯ β’ β₯ β’ 2.80 β₯ The chracterstics snow load on the tank shell is the product 2.40 β₯ kN of the shape,exposure , themal cofficients of snow and Sn β μ i ⋅ C e ⋅ C t ⋅ s k = β’ ββ β’ 2.00 β₯ m 2 characterstic snow load on the ground per meter sqauer β’ 1.60 β₯ area of the roof. β’ β₯ 1.20 β’ β₯ β£ 0.80 β¦ The characterstic Snow Load in the selected regions : Using the Design situations Eq.6.10b β‘ 34 β€ β’ 28 β₯ β’ β₯ β’ 21 β₯ Aroof β’ 18 β₯ kN S β Sn ⋅ ββ = β π ⋅ D β’ 15 β₯ m β’ 12 β₯ β’ β₯ β’ 9β₯ β£ 6β¦ Assumption :Snow as accompanying variabel action The chracterstics snow load on the tank shell is the product of the characterstic snow load on the dom roof by the ratio of the area of the roofing to the the perimeter of the shell. Hydrostatic pressure : We assume almost empty tank, negligble hydrostatic pressure for buckling analysis of the tank shell. 4.1.3. Wind Load An assumption has made that the height of the structure is equal to the height of the shell, see SS-EN 1991-1-4 figure 6.1 and 7.2.9(8) Appendix Page 8 of 55 Appendix A. Mathcad calculation document β‘ 21 β€ β’ 19.5 β₯ β’ β₯ β’ 18 β₯ β’ 16 β₯ β’ 14 β₯ z β β’ 12 β₯ m β’ β₯ β’ 10 β₯ β’ 8 β₯ β’ 6 β₯ β’ 4 β₯ β’ β₯ β£ 2 β¦ z11 21m: Top shell course z1 =2m: Bottom shell course External pressure buckling coefficent; Cb , (EN 1993-4-1 section 5.3.2.5 Table 5.2) The shell is adequatly stiffened by the roof strucuture emin β 6.00 mm t β emin D r β β= 13 m 2 Tank shell radius: r Wind pressure distribution coefficent, Cw EN 1993-4-1 section 5.3.2.5 clause (8) Cθ β 1.25 l ωββ ⋅ r β β 2.2 Cw β max β1 , ββββββββ β = 1.09 β β ββ ‾‾‾‾‾‾‾‾ ‾‾ β β r r β ββ β ⋅ β β β1 + β0.1 ⋅ Cb ⋅ β β l t ββ ββ β β β β ....eq(5) External pressure buckling factors for medium cylinders-length from EN1993-1-6 , Table D.3 Assumption: Case 3, tank with out anchors, BC1-BC2 ‾‾ r = 20.69 β t The length of shell segment should be characterized interms of the dimensionless parameter, ω . From EN 1991-1-6 D.1.3.1 Equation D.19 β ‾‾‾‾‾‾ β Cθ r β β kw β 0.46 β1 + 0.1 ⋅ ββ⋅ β β β = 0.99 ββ β ω t β ββ co ββzββ β 1 D β 26 m l β Hp1 = 5.78 m Max. spacing of secondary stiffnners Min. shell thickness Cb β 1 co β 1 Maximum wind pressure coefficient, kw From EN1993-1-6 The orographic factor neglected, determine it through annex A.3 of SS-EN 1991-1-4 The basic wind velocities, detrmine for selected regions , From figure C-4 of EKS 11 (BFS 2015:6) m m m m m m vb1 β 21 ββvb2 β 22 ββvb3 β 23 ββvb4 β 24 ββvb5 β 25 ββvb6 β 26 ββ sec sec sec sec sec sec Appendix Page 9 of 55 Appendix A. Mathcad calculation document β‘ 21 β€ β’ 22 β₯ β’ β₯ 23 m vb β β’ β₯ ββ β’ 24 β₯ sec β’ 25 β₯ β’β£ 26 β₯β¦ kg ρ β 1.25 ββ m3 1 ⋅ ρ ⋅ vb 2 qb β β 2 The air density(SS-EN 1991-1-4 4.5(1) note2) β‘ 0.276 β€ β’ 0.303 β₯ β’ β₯ 0.331 β₯ kN =β’ ββ β’ 0.360 β₯ m 2 β’ 0.391 β₯ β’β£ 0.423 β₯β¦ The basic velocity pressures at the selected locations (SS-EN 1991-1-4 equation 4.10) The peak velocity pressure (EKS 11 chapter 11 part 4.5(1) replaces equation (4.8) of SS-EN 1991-1-4 with this we calculate for the selected geograhpical locations) Wind Load: Terrain category type -II Terrain category, type II: Area with low vegitation such as grass and isolated obstacles (trees,buildings)with separations of at least 20 obstacle heights. (SS-EN 1991-1-4 Terrain category II, Table 4.1) zo β 0.05 m The roughness length, zo , For terrain category, type II (SS-EN 1991-1-4 Table 4.1 ) zmin β 2 m β‘ 0.166 β€ β’ 0.168 β₯ β’ β₯ β’ 0.170 β₯ β’ 0.173 β₯ β’ 0.177 β₯ 1 IV β ββββ = β’ 0.182 β₯ β₯ β β z ββ β’ βco ⋅ ln ββββ β’ 0.189 β₯ β β zo β β β’ 0.197 β₯ β’ 0.209 β₯ β’ 0.228 β₯ β’ β₯ β£ 0.271 β¦ zo_II β 0.05 m The turbulance intensity at a height z ,(EKS 11 4.5 (1) Article 7) (SS-EN 1991-1-4 Terrain category II, Table 4.1) β zo β 0.07 kr β 0.19 βββ β = 0.19 β zo_II β .........eq() The terrain factor( SS-EN 1991-1-4 equation 4.5) Appendix Page 10 of 55 Appendix A. Mathcad calculation document For the shell : The Peak factor for the peak velocity pressure ( qp ββzββ ), resonance determination is not neccessary. 2β β β‘ βzβ β€ β β ......................................eq(6) qp ββzββ β β‘β£ 1 + 2 ⋅ kp ⋅ IV ββzββ β€β¦ ⋅ β’ kr ⋅ ln βββ ⋅ co ββzββ β₯ ⋅ qb ββ β£ β¦ ββ β zo β kp β 3.0 1 IV ββzββ β βββββ β β z ββ βco ββzββ ⋅ ln ββββ β β zo β β .......................................................................eq(7) 2 β β β βzβ β β 2 β z βββ β qp β βkr ⋅ ln βββ ⋅ coβ + 2 ⋅ kp ⋅ βkr ⋅ co ⋅ ln ββββ ⋅ qb T βββ β zo β β β β zo β β ββ vb1 β‘ 0.72 β’ 0.71 β’ β’ 0.70 β’ 0.68 β’ 0.65 qp = β’ 0.63 β’ β’ 0.60 β’ 0.56 β’ 0.51 β’ 0.45 β’ β£ 0.36 vb2 0.79 0.78 0.76 0.74 0.72 0.69 0.65 0.61 0.56 0.50 0.39 vb3 0.87 0.85 0.84 0.81 0.78 0.75 0.71 0.67 0.62 0.54 0.43 vb4 vb5 0.95 1.03 0.93 1.01 0.91 0.99 0.88 0.96 0.85 0.92 0.82 0.89 0.78 0.84 0.73 0.79 0.67 0.73 0.59 0.64 0.46 0.50 .....................................eq(8) vb6 1.11 β€ z=21m β₯ 1.09 z=19.5m β₯ 1.07 β₯ z=18m 1.04 β₯ z=16m 1.00 β₯ z=14m kN β₯ 0.96 ββz=12m β₯ 2 z=10m 0.91 β₯ m z=8m 0.86 β₯ z=6m 0.79 β₯ β₯ z=4m 0.69 β₯ z=2m 0.55 β¦ The peak velocity pressure on the tank shell from bottom to top, for all six basic wind velocity β¨ β© β¨ β© pressures , for example qpβ¨0β© for vb1 =21m/sec and qpβ¨1β© for vb2 =22m/sec β‘ 0.72 β€ β’ 0.71 β₯ β’ β₯ β’ 0.70 β₯ β’ 0.68 β₯ β’ 0.65 β₯ kN β¨β¨0β©β© qp = β’ 0.63 β₯ ββ β’ β₯ m2 β’ 0.60 β₯ β’ 0.56 β₯ β’ 0.51 β₯ β’ 0.45 β₯ β’ β₯ β£ 0.36 β¦ β‘ 0.79 β€ β’ 0.78 β₯ β’ β₯ β’ 0.76 β₯ β’ 0.74 β₯ β’ 0.72 β₯ kN β¨β¨1β©β© qp = β’ 0.69 β₯ ββ β’ β₯ m2 β’ 0.65 β₯ β’ 0.61 β₯ β’ 0.56 β₯ β’ 0.50 β₯ β’ β₯ β£ 0.39 β¦ For the purpose of tank shell buckling design we use the equivalent uniform external pressure on the tank shell, qeq_Ed Selecteing the max. value of the peak wind velocity at the top of each shell courses for the regions under study. Appendix Page 11 of 55 Appendix A. Mathcad calculation document qw_max_Ed β qp qeq_Ed β kw ⋅ qw_max_Ed ...................................................................eq(9) vb1 β‘ 0.71 β’ 0.70 β’ β’ 0.69 β’ 0.67 β’ 0.64 qeq_Ed β kw ⋅ qw_max_Ed = β’ 0.62 β’ β’ 0.59 β’ 0.55 β’ 0.51 β’ 0.45 β’ β£ 0.35 vb2 0.78 0.77 0.75 0.73 0.71 0.68 0.64 0.61 0.56 0.49 0.38 vb3 0.86 0.84 0.82 0.80 0.77 0.74 0.70 0.66 0.61 0.54 0.42 vb4 vb5 vb6 0.93 1.01 1.09 β€ 0.92 0.99 1.07 β₯ β₯ 0.90 0.97 1.05 β₯ 0.87 0.94 1.02 β₯ 0.84 0.91 0.99 β₯ kN 0.81 0.88 0.95 β₯ ββ β₯ m2 0.77 0.83 0.90 β₯ 0.72 0.78 0.85 β₯ 0.66 0.72 0.78 β₯ 0.58 0.63 0.68 β₯ β₯ 0.46 0.50 0.54 β¦ Internal suction pressure on the tank Shell due to internal vacuum pressure, qs_Ed according to EN-1993-1-6 β¨ β© β 0β qs_Ed β 0.4 ⋅ kw ⋅ βqp β ....................................................................................eq(10) β© Constant internal negative pressure is considered 40 % of the peak wind velocity pressure for the six wind velocity pressures 0 β© qp : is the value of the Peak wind velocity pressures at the top of the tank shell for the six wind velocity regions . β¨ 0 β¨ qs_Ed β 0.4 ⋅ qeq_Ed β¨ β© vb1 β‘1β€ β‘ 0.29 β’1β₯ β’ 0.29 β’ β₯ β’ β’1β₯ β’ 0.29 β’1β₯ β’ 0.29 β’1β₯ β’ 0.29 0 β’ β₯ qs_Ed β 1 ⋅ 0.4 ⋅ kw ⋅ qp = β’ 0.29 β’ β₯ β’ β’1β₯ β’ 0.29 β’1β₯ β’ 0.29 β’1β₯ β’ 0.29 β’1β₯ β’ 0.29 β’ β₯ β’ β£1β¦ β£ 0.29 vb2 0.31 0.31 0.31 0.31 0.31 0.31 0.31 0.31 0.31 0.31 0.31 vb3 0.34 0.34 0.34 0.34 0.34 0.34 0.34 0.34 0.34 0.34 0.34 vb4 vb5 0.37 0.40 0.37 0.40 0.37 0.40 0.37 0.40 0.37 0.40 0.37 0.40 0.37 0.40 0.37 0.40 0.37 0.40 0.37 0.40 0.37 0.40 vb6 0.44 β€ 0.44 β₯ β₯ 0.44 β₯ 0.44 β₯ 0.44 β₯ kN 0.44 β₯ ββ β₯ m2 0.44 β₯ 0.44 β₯ 0.44 β₯ 0.44 β₯ β₯ 0.44 β¦ Characteristic Equivalent wind load on the tank shell along its height on each shell courses, for different wind velocity regions is the sum of wind velocity pressure and the internal negative pressure acting on the tank shell Appendix Page 12 of 55 Appendix A. Mathcad calculation document vb1 β‘ 1.00 β’ 0.99 β’ β’ 0.97 β’ 0.95 β’ 0.93 pEd β qeq_Ed + qs_Ed = β’ 0.90 β’ β’ 0.87 β’ 0.84 β’ 0.79 β’ 0.73 β’ β£ 0.64 vb2 1.10 1.08 1.07 1.04 1.02 0.99 0.96 0.92 0.87 0.80 0.70 vb3 1.20 1.18 1.17 1.14 1.11 1.08 1.05 1.00 0.95 0.88 0.76 vb4 vb5 1.31 1.42 1.29 1.40 1.27 1.38 1.24 1.35 1.21 1.32 1.18 1.28 1.14 1.24 1.09 1.19 1.03 1.12 0.96 1.04 0.83 0.90 vb6 1.53 β€ 1.51 β₯ β₯ 1.49 β₯ 1.46 β₯ 1.42 β₯ kN 1.38 β₯ ββ β₯ m2 1.34 β₯ 1.28 β₯ 1.21 β₯ 1.12 β₯ β₯ 0.98 β¦ 6. Combination of actions The loads combinations which normally are design deriving at the ultimate limit state according to equation 6.10b for the three safty classes β¨ β© β n β¨ β©β Ed οΌ γd_Rc ⋅ βξ ⋅ γG_sup ⋅ G + γQ ⋅ ψo ⋅ S + γQ ⋅ pEdβ¨mβ©β .....................................................eq(11) n- The nth row of the charactrestic snow load vector m- The nth column of the characterstic wind load matrix _Rc- the reliability classes ,1,2 and 3 ξ : reduction factor ξ β 0.89 γG_sup : Partial cofficient unfavorable permanent load γG_sup β 1.35 γQ : Partial cofficient unfavorable variable load γQ β 1.5 ψo : combination factor for variable load: in this case for snow load is accompanying variable action based on the selected snow zone ψo β 0.8 ψ0_wind : Combination factor for vriable load: when wind is accompanying variable action 6.1 Reliabilty class-1, γd_1 β 0.83 ψ0_1 β ψo ψ0_wind β 0.3 For empty tank: EKS 11 1. Design self weight of the Roofing kN GD_1 β γd_1 ⋅ ξ ⋅ γG_sup ⋅ G = 6.91 β m .................................................eq(12) 2. Design Snow Load The design snow load for the the sleceted snow zones : SD_1 β γd_1 ⋅ γQ ⋅ ψ0_i ⋅ S ......................................................................................eq(13) Appendix Page 13 of 55 Appendix A. Mathcad calculation document SD_1 β γd_1 ⋅ γQ ⋅ ψ0_i ⋅ S ......................................................................................eq(13) β‘ 33.5 β€ β’ 27.41 β₯ β’ β₯ β’ 21.32 β₯ 18.27 β₯ kN SD_1 β γd_1 ⋅ γQ ⋅ ψ0_i ⋅ S = β’ β β’ 13.32 β₯ m β’ 10.66 β₯ β’ β₯ β’ 6.85 β₯ β£ 4.57 β¦ When Snow load is an accompanying variable action Where the Snow load is assumed to be a leading variable action at location where the basic wind m velocity vb β 22 ββ and snow load zone with sec characterstic snow load on the ground , kN sk β 4.5 ββ m2 β¨ β© 1 kN SD_1leading β γd_1 ⋅ γQ ⋅ S = β‘β£ 34.26 β€β¦ β m 3. Design Wind Load β‘ 1.24 β’ 1.23 β’ β’ 1.21 β’ 1.18 β’ 1.16 pEd_1 β γd_1 ⋅ γQ ⋅ pEd = β’ 1.12 β’ β’ 1.09 β’ 1.04 β’ 0.99 β’ 0.91 β’ β£ 0.79 1.37 1.35 1.33 1.30 1.27 1.23 1.19 1.14 1.08 1.00 0.87 1.49 1.47 1.45 1.42 1.39 1.35 1.30 1.25 1.18 1.09 0.95 1.62 1.60 1.58 1.55 1.51 1.47 1.42 1.36 1.29 1.19 1.03 1.76 1.74 1.72 1.68 1.64 1.59 1.54 1.48 1.40 1.29 1.12 1.91 β€ 1.88 β₯ β₯ 1.86 β₯ 1.82 β₯ 1.77 β₯ kN 1.72 β₯ ββ β₯ m2 1.67 β₯ 1.60 β₯ 1.51 β₯ 1.40 β₯ β₯ 1.21 β¦ β‘ 0.41 β€ β’ 0.40 β₯ β’ β₯ β’ 0.40 β₯ β’ 0.39 β₯ β’ 0.38 β₯ Wind load : accompanying variable kN β¨β¨1β©β© pEd_1accompany β γd_1 ⋅ γQ ⋅ ψ0_wind ⋅ pEd = β’ 0.37 β₯ ββaction at locations where basic wind β’ β₯ m2 m β’ 0.36 β₯ velocity, vb β 22 ββ sec β’ 0.34 β₯ β’ 0.32 β₯ β’ 0.30 β₯ β’ β₯ β£ 0.26 β¦ 6.2 Reliabilty class-2, For γd_2 β 0.91 For empty tank: EKS 11 1. Design self weight of the Roofing Appendix Page 14 of 55 Appendix A. Mathcad calculation document kN GD_1 β γd_2 ⋅ ξ ⋅ γG_sup ⋅ G = 7.57 β m 2. Design Snow Load The design snow load for the sleceted snow zones are β‘ 36.73 β€ β’ 30.05 β₯ β’ β₯ β’ 23.37 β₯ 20.03 β₯ kN SD_2 β γd_2 ⋅ γQ ⋅ ψ0_i ⋅ S = β’ β β’ 14.61 β₯ m β’ 11.69 β₯ β’ β₯ β’ 7.51 β₯ β£ 5.01 β¦ β¨ β© 1 kN SD_2leading β γd_2 ⋅ γQ ⋅ S = β‘β£ 37.56 β€β¦ β m When Snow load is an accompanying variable action Where the Snow load is assumed to be a leading variable action at location where the basic wind m velocity vb β 22 ββ and snow load zone with sec characterstic snow load on the ground , kN sk β 4.5 ββ m2 3. Design Wind Load The wind is assumed to be the leading variable action for lower snow zones β‘ 1.36 β’ 1.35 β’ β’ 1.33 β’ 1.30 β’ 1.27 pEd_2 β γd_2 ⋅ γQ ⋅ pEd = β’ 1.23 β’ β’ 1.19 β’ 1.14 β’ 1.08 β’ 1.00 β’ β£ 0.87 1.50 1.48 1.46 1.43 1.39 1.35 1.31 1.25 1.19 1.10 0.95 1.64 1.61 1.59 1.56 1.52 1.48 1.43 1.37 1.30 1.20 1.04 1.78 1.76 1.73 1.70 1.66 1.61 1.56 1.49 1.41 1.30 1.13 1.93 1.91 1.88 1.84 1.80 1.75 1.69 1.62 1.53 1.42 1.23 2.09 β€ 2.06 β₯ β₯ 2.03 β₯ 1.99 β₯ 1.94 β₯ kN 1.89 β₯ ββ β₯ 2 1.83 β₯ m 1.75 β₯ 1.66 β₯ 1.53 β₯ β₯ 1.33 β¦ β‘ 0.45 β€ β’ 0.44 β₯ β’ β₯ β’ 0.44 β₯ β’ 0.43 β₯ β’ 0.42 β₯ kN β¨β¨1β©β© pEd_2accompany β γd_2 ⋅ γQ ⋅ ψ0_wind ⋅ pEd = β’ 0.41 β₯ ββ β’ β₯ m2 β’ 0.39 β₯ β’ 0.38 β₯ β’ 0.36 β₯ β’ 0.33 β₯ β’ β₯ β£ 0.29 β¦ 6.3 Reliabilty class-3, γd_3 β 1 For empty tank: EKS 11 Appendix Page 15 of 55 Appendix A. Mathcad calculation document 6.3 Reliabilty class-3, γd_3 β 1 For empty tank: EKS 11 1. Design self weight of the Roofing kN GD_1 β γd_3 ⋅ ξ ⋅ γG_sup ⋅ G = 8.32 β m 2. Design Snow Load The design snow load for the the sleceted snow zones are β‘ 40.36 β€ β’ 33.02 β₯ β’ β₯ β’ 25.68 β₯ 22.01 β₯ kN SD_3 β γd_3 ⋅ γQ ⋅ ψ0_i ⋅ S = β’ β β’ 16.05 β₯ m β’ 12.84 β₯ β’ β₯ β’ 8.26 β₯ β£ 5.5 β¦ β¨ β© 1 kN SD_3leading β γd_3 ⋅ γQ ⋅ S = β‘β£ 41.28 β€β¦ β m Where the Snow load is assumed to be a leading variable action at location where the basic wind m velocity vb β 22 ββ and snow load zone with sec characterstic snow load on the ground , kN sk β 4.5 ββ m2 3. Design Wind Load β‘ 1.50 β’ 1.48 β’ β’ 1.46 β’ 1.43 β’ 1.39 pEd_3 β γd_3 ⋅ γQ ⋅ pEd = β’ 1.35 β’ β’ 1.31 β’ 1.26 β’ 1.19 β’ 1.10 β’ β£ 0.95 1.64 1.62 1.60 1.57 1.53 1.49 1.44 1.38 1.30 1.20 1.05 1.80 1.77 1.75 1.71 1.67 1.62 1.57 1.51 1.43 1.32 1.14 1.96 1.93 1.90 1.86 1.82 1.77 1.71 1.64 1.55 1.43 1.25 2.12 2.10 2.07 2.02 1.97 1.92 1.86 1.78 1.68 1.56 1.35 2.30 β€ 2.27 β₯ β₯ 2.24 β₯ 2.19 β₯ 2.14 β₯ kN 2.08 β₯ ββ β₯ 2 2.01 β₯ m 1.92 β₯ 1.82 β₯ 1.68 β₯ β₯ 1.46 β¦ β‘ 0.49 β€ β’ 0.49 β₯ β’ β₯ β’ 0.48 β₯ β’ 0.47 β₯ β’ 0.46 β₯ kN β¨β¨1β©β© pEd_3accompany β γd_3 ⋅ γQ ⋅ ψ0_wind ⋅ pEd = β’ 0.45 β₯ ββ β’ β₯ m2 β’ 0.43 β₯ β’ 0.41 β₯ β’ 0.39 β₯ β’ 0.36 β₯ β’ β₯ β£ 0.31 β¦ Load combnations Appendix Page 16 of 55 Appendix A. Mathcad calculation document Load combnations 6. Shell design Loads and Combinations The loads combinations which normally are design deriving at the ultimate limit state according to equation 6.10b for the three safety classes β¨ β© β 0 β¨ β©β Ed οΌ γd ⋅ βξ ⋅ γG_sup ⋅ G + γQ ⋅ ψo ⋅ S + γQ ⋅ pEdβ¨0β©β ...................................................eq(14) ξ : reduction factor ξ β 0.89 γG_sup : Partial cofficient unfavorable permanent load γG_sup β 1.35 γQ : Partial cofficient unfavorable variable load γQ β 1.5 ψo : combination factor for variable load: in this case for snow load is accompanying variable action based on the selected snow zone ψ0_wind : Combination factor for vriable load: when wind is accompanying variable action 6.1 Reliabilty class-1, γd_1 β 0.83 For empty tank: EKS 11 1. Design self weight of the Roofing kN GD_1 β γd_1 ⋅ ξ ⋅ γG_sup ⋅ G = 6.91 β m 2. Design Snow Load The design snow load for the the sleceted snow zones : Appendix Page 17 of 55 ψo β 0.8 ψ0_1 β ψo ψ0_wind β 0.3 Appendix A. Mathcad calculation document β‘ 33.5 β€ β’ 27.41 β₯ β’ β₯ β’ 21.32 β₯ 18.27 β₯ kN SD_1 β γd_1 ⋅ γQ ⋅ ψ0_i ⋅ S = β’ β β’ 13.32 β₯ m β’ 10.66 β₯ β’ β₯ β’ 6.85 β₯ β£ 4.57 β¦ When Snow load is an accompanying variable action 3. Design Wind Load β‘ 1.24 β’ 1.23 β’ β’ 1.21 β’ 1.18 β’ 1.16 pEd_1 β γd_1 ⋅ γQ ⋅ pEd = β’ 1.12 β’ β’ 1.09 β’ 1.04 β’ 0.99 β’ 0.91 β’ β£ 0.79 6.2 Reliabilty class-2, For γd_2 β 0.91 1.37 1.35 1.33 1.30 1.27 1.23 1.19 1.14 1.08 1.00 0.87 1.49 1.47 1.45 1.42 1.39 1.35 1.30 1.25 1.18 1.09 0.95 1.62 1.60 1.58 1.55 1.51 1.47 1.42 1.36 1.29 1.19 1.03 1.76 1.74 1.72 1.68 1.64 1.59 1.54 1.48 1.40 1.29 1.12 1.91 β€ 1.88 β₯ β₯ 1.86 β₯ 1.82 β₯ 1.77 β₯ kN 1.72 β₯ ββ β₯ m2 1.67 β₯ 1.60 β₯ 1.51 β₯ 1.40 β₯ β₯ 1.21 β¦ For empty tank: EKS 11 1. Design self weight :Roof kN GD_2 β γd_2 ⋅ ξ ⋅ γG_sup ⋅ G = 7.57 β m 2. Design Snow Load The design snow load for the the sleceted snow zones are β‘ 36.73 β€ β’ 30.05 β₯ β’ β₯ β’ 23.37 β₯ 20.03 β₯ kN SD_2 β γd_2 ⋅ γQ ⋅ ψ0_i ⋅ S = β’ β β’ 14.61 β₯ m β’ 11.69 β₯ β’ β₯ β’ 7.51 β₯ β£ 5.01 β¦ When Snow load is an accompanying variable action 3. Design Wind Load The wind is assumed to be the leading variable action for lower snow load zones β‘ 1.36 β’ 1.35 β’ β’ 1.33 β’ 1.30 β’ 1.27 1.50 1.48 1.46 1.43 1.39 1.64 1.78 1.93 2.09 β€ 1.61 1.76 1.91 2.06 β₯ Appendix Page 18β₯ of 55 1.59 1.73 1.88 2.03 β₯ 1.56 1.70 1.84 1.99 β₯ 1.52 1.66 1.80 1.94 β₯ Appendix A. Mathcad calculation document β‘ 1.36 β’ 1.35 β’ β’ 1.33 β’ 1.30 β’ 1.27 pEd_2 β γd_2 ⋅ γQ ⋅ pEd = β’ 1.23 β’ β’ 1.19 β’ 1.14 β’ 1.08 β’ 1.00 β’ β£ 0.87 6.3 Reliabilty class-3, γd_3 β 1 1.50 1.48 1.46 1.43 1.39 1.35 1.31 1.25 1.19 1.10 0.95 1.64 1.61 1.59 1.56 1.52 1.48 1.43 1.37 1.30 1.20 1.04 1.78 1.76 1.73 1.70 1.66 1.61 1.56 1.49 1.41 1.30 1.13 1.93 1.91 1.88 1.84 1.80 1.75 1.69 1.62 1.53 1.42 1.23 2.09 β€ 2.06 β₯ β₯ 2.03 β₯ 1.99 β₯ 1.94 β₯ kN 1.89 β₯ ββ β₯ m2 1.83 β₯ 1.75 β₯ 1.66 β₯ 1.53 β₯ β₯ 1.33 β¦ For empty tank: EKS 11 1. Design self weight of the Roofing kN GD_3 β γd_3 ⋅ ξ ⋅ γG_sup ⋅ G = 8.32 β m 2. Design Snow Load The design snow load for the the sleceted snow zones are β‘ 40.36 β€ β’ 33.02 β₯ β’ β₯ β’ 25.68 β₯ 22.01 β₯ kN SD_3 β γd_3 ⋅ γQ ⋅ ψ0_i ⋅ S = β’ β β’ 16.05 β₯ m β’ 12.84 β₯ β’ β₯ β’ 8.26 β₯ β£ 5.5 β¦ 3. Design Wind Load β‘ 1.50 β’ 1.48 β’ β’ 1.46 β’ 1.43 β’ 1.39 pEd_3 β γd_3 ⋅ γQ ⋅ pEd = β’ 1.35 β’ β’ 1.31 β’ 1.26 β’ 1.19 β’ 1.10 β’ β£ 0.95 1.64 1.62 1.60 1.57 1.53 1.49 1.44 1.38 1.30 1.20 1.05 1.80 1.77 1.75 1.71 1.67 1.62 1.57 1.51 1.43 1.32 1.14 1.96 1.93 1.90 1.86 1.82 1.77 1.71 1.64 1.55 1.43 1.25 2.12 2.10 2.07 2.02 1.97 1.92 1.86 1.78 1.68 1.56 1.35 2.30 β€ 2.27 β₯ β₯ 2.24 β₯ 2.19 β₯ 2.14 β₯ kN 2.08 β₯ ββ β₯ m2 2.01 β₯ 1.92 β₯ 1.82 β₯ 1.68 β₯ β₯ 1.46 β¦ β© β© β© β¨ β¨ β¨ 7. Design Load combnations 6 6 6 m kN 1. vb β 26 ββ and vs Snow zone-2, sk β 1.5 ββ : SD_1 , SD_2 , SD_3 2 sec m Appendix Page 19 of 55 Appendixm β© β© β¨ β¨ β© β¨ 6 6 6 kN 1. vb β 26 ββ and vs Snow zone-2, sk β 1.5 ββ : SD_1 , SD_2 , SD_3 2 seccalculation document m A. Mathcad β¨ β© β‘ β€ 6 β¨ β© Comb_1 β β£ pEd_1β¨5β© SD_1 GD_1 β¦ β© β‘ β€ 6 β¨ β© Comb_2 β β£ pEd_2β¨5β© SD_2 GD_2 β¦ β¨ Design Load combinations β¨ β© β‘ β€ 6 β¨ β© Comb_3 β β£ pEd_3β¨5β© SD_3 GD_3 β¦ 6 kN SD_1 = β‘β£ 6.85 β€β¦ β m kN GD_1 = 6.91 β m Comb_2 β‘ 2.09 β€ β’ 2.06 β₯ β’ β₯ β’ 2.03 β₯ β’ 1.99 β₯ β’ 1.94 β₯ kN β¨β¨5β©β© pEd_2 = β’ 1.89 β₯ ββ β’ β₯ m2 β’ 1.83 β₯ β’ 1.75 β₯ β’ 1.66 β₯ β’ 1.53 β₯ β’ β₯ β£ 1.33 β¦ 6 kN SD_2 = β‘β£ 7.51 β€β¦ β m kN GD_2 = 7.57 β m Comb_3 β‘ 2.30 β€ β’ 2.27 β₯ β’ β₯ β’ 2.24 β₯ β’ 2.19 β₯ β’ 2.14 β₯ kN β¨β¨5β©β© pEd_3 = β’ 2.08 β₯ ββ β’ β₯ m2 β’ 2.01 β₯ β’ 1.92 β₯ β’ 1.82 β₯ β’ 1.68 β₯ β’ β₯ β£ 1.46 β¦ 6 kN SD_3 = β‘β£ 8.26 β€β¦ β m kN GD_3 = 8.32 β m β¨ β© β¨ β© β¨ β© Comb_1 β‘ 1.91 β€ β’ 1.88 β₯ β’ β₯ β’ 1.86 β₯ β’ 1.82 β₯ β’ 1.77 β₯ kN β¨β¨5β©β© pEd_1 = β’ 1.72 β₯ ββ β’ β₯ m2 β’ 1.67 β₯ β’ 1.60 β₯ β’ 1.51 β₯ β’ 1.40 β₯ β’ β₯ β£ 1.21 β¦ β© β© β© β¨ β¨ β¨ 6 6 6 m kN 2. vb β 25 ββ and vs Snow zone-2, sk β 1.5 ββ : SD_1 , SD_2 , SD_3 2 sec m β¨ β© β‘ β€ 6 β¨ β© Comb_4 β β£ pEd_1β¨4β© SD_1 GD_1 β¦ β© β‘ β€ 6 β¨ β© Comb_5 β β£ pEd_2β¨4β© SD_2 GD_2 β¦ β¨ Design Load combinations Appendix Page 20 of 55 Appendix β© β‘ β€ 6 β¨ β© Comb_5 β β£ pEd_2β¨4β© SD_2 GD_2 β¦ β¨ A. Mathcad calculation document β¨ β© β‘ β€ 6 β¨ β© Comb_6 β β£ pEd_3β¨4β© SD_3 GD_3 β¦ Comb_6 β‘ 2.12 β€ β’ 2.10 β₯ β’ β₯ β’ 2.07 β₯ β’ 2.02 β₯ β’ 1.97 β₯ kN β¨β¨4β©β© pEd_3 = β’ 1.92 β₯ ββ β’ β₯ m2 β’ 1.86 β₯ β’ 1.78 β₯ β’ 1.68 β₯ β’ 1.56 β₯ β’ β₯ β£ 1.35 β¦ β¨ β© kN GD_1 = 6.91 β m kN GD_2 = 7.57 β m β¨ β© 6 kN SD_2 = β‘β£ 7.51 β€β¦ β m 6 kN SD_3 = β‘β£ 8.26 β€β¦ β m kN GD_3 = 8.32 β m β© Comb_5 β‘ 1.93 β€ β’ 1.91 β₯ β’ β₯ β’ 1.88 β₯ β’ 1.84 β₯ β’ 1.80 β₯ kN β¨β¨4β©β© pEd_2 = β’ 1.75 β₯ ββ β’ β₯ m2 β’ 1.69 β₯ β’ 1.62 β₯ β’ 1.53 β₯ β’ 1.42 β₯ β’ β₯ β£ 1.23 β¦ 6 kN SD_1 = β‘β£ 6.85 β€β¦ β m β¨ Comb_4 β‘ 1.76 β€ β’ 1.74 β₯ β’ β₯ β’ 1.72 β₯ β’ 1.68 β₯ β’ 1.64 β₯ kN β¨β¨4β©β© pEd_1 = β’ 1.59 β₯ ββ β’ β₯ m2 β’ 1.54 β₯ β’ 1.48 β₯ β’ 1.40 β₯ β’ 1.29 β₯ β’ β₯ β£ 1.12 β¦ β© β© β¨ β¨ β¨ β© 3 3 3 m kN 3. vb β 24 ββ and vs Snow zone-2, sk β 3.0 ββ : SD_1 , SD_2 , SD_3 2 sec m β¨ β© β‘ β€ 3 β¨ β© Comb_7 β β£ pEd_1β¨3β© SD_1 GD_1 β¦ β© β‘ β€ 3 β¨ β© Comb_8 β β£ pEd_2β¨3β© SD_2 GD_2 β¦ β¨ Design Load combinations Appendix Page 21 of 55 Appendix β¨ β© β‘ β€ 3 β¨ β© Comb_8 β β£ pEd_2β¨3β© SD_2 GD_2 β¦ β‘ β€ 3 β¨ β© Comb_9 β β£ pEd_3β¨3β© SD_3 GD_3 β¦ β¨ β© A. Mathcad calculation document β¨ β© kN GD_2 = 7.57 β m kN GD_3 = 8.32 β m β© 3 kN SD_3 = β‘β£ 22.01 β€β¦ β m β© β© β¨ β¨ β© 2 2 2 kN Snow zone-3, sk β 3.5 ββ : SD_1 , SD_2 , SD_3 m2 β¨ m 4. vb β 23 ββ and sec 3 kN SD_2 = β‘β£ 20.03 β€β¦ β m β¨ Comb_9 β‘ 1.96 β€ β’ 1.93 β₯ β’ β₯ β’ 1.90 β₯ β’ 1.86 β₯ β’ 1.82 β₯ kN β¨β¨3β©β© pEd_3 = β’ 1.77 β₯ ββ β’ β₯ m2 β’ 1.71 β₯ β’ 1.64 β₯ β’ 1.55 β₯ β’ 1.43 β₯ β’ β₯ β£ 1.25 β¦ kN GD_1 = 6.91 β m β© Comb_8 β‘ 1.78 β€ β’ 1.76 β₯ β’ β₯ β’ 1.73 β₯ β’ 1.70 β₯ β’ 1.66 β₯ kN β¨β¨3β©β© pEd_2 = β’ 1.61 β₯ ββ β’ β₯ m2 β’ 1.56 β₯ β’ 1.49 β₯ β’ 1.41 β₯ β’ 1.30 β₯ β’ β₯ β£ 1.13 β¦ 3 kN SD_1 = β‘β£ 18.27 β€β¦ β m β¨ Comb_7 β‘ 1.62 β€ β’ 1.60 β₯ β’ β₯ β’ 1.58 β₯ β’ 1.55 β₯ β’ 1.51 β₯ kN β¨β¨3β©β© pEd_1 = β’ 1.47 β₯ ββ β’ β₯ m2 β’ 1.42 β₯ β’ 1.36 β₯ β’ 1.29 β₯ β’ 1.19 β₯ β’ β₯ β£ 1.03 β¦ β¨ β© β‘ β€ 2 β¨ β© Comb_10 β β£ pEd_1β¨2β© SD_1 GD_1 β¦ β© β‘ β€ 2 β¨ β© Comb_11 β β£ pEd_2β¨2β© SD_2 GD_2 β¦ β¨ Design Load combinations β¨ β© β‘ β€ 2 β¨ β© Comb_12 β β£ pEd_3β¨2β© SD_3 GD_3 β¦ β‘ 1.49 β€ β’ 1.47 β₯ β’ β₯ β’ 1.45 β₯ β’ 1.42 β₯ β’ 1.39 β₯ Appendix Page 22 of 55 Appendix A. Mathcad calculation document Comb_11 β‘ 1.64 β€ β’ 1.61 β₯ β’ β₯ β’ 1.59 β₯ β’ 1.56 β₯ β’ 1.52 β₯ kN β¨β¨2β©β© pEd_2 = β’ 1.48 β₯ ββ β’ β₯ m2 β’ 1.43 β₯ β’ 1.37 β₯ β’ 1.30 β₯ β’ 1.20 β₯ β’ β₯ β£ 1.04 β¦ kN GD_1 = 6.91 β m β¨ β© Comb_10 β‘ 1.49 β€ β’ 1.47 β₯ β’ β₯ β’ 1.45 β₯ β’ 1.42 β₯ 2 kN β’ 1.39 β₯ SD_1 = β‘β£ 21.32 β€β¦ β kN β¨ β© m pEd_1β¨2β© = β’ 1.35 β₯ ββ β’ β₯ m2 β’ 1.30 β₯ β’ 1.25 β₯ β’ 1.18 β₯ β’ 1.09 β₯ β’ β₯ β£ 0.95 β¦ kN GD_2 = 7.57 β m β¨ β© 2 kN SD_2 = β‘β£ 23.37 β€β¦ β m β© β© β¨ β¨ β© β¨ β¨ β© β‘ 1.80 β€ β’ 1.77 β₯ β’ β₯ β’ 1.75 β₯ β’ 1.71 β₯ β’ 1.67 β₯ 2 kN kN kN β¨β¨2β©β© Comb_12 pEd_3 = β’ 1.62 β₯ ββ SD_3 = β‘β£ 25.68 β€β¦ β GD_3 = 8.32 β β’ β₯ m2 m m β’ 1.57 β₯ β’ 1.51 β₯ β’ 1.43 β₯ β’ 1.32 β₯ β’ β₯ β£ 1.14 β¦ 1 1 1 m kN 5. vb β 22 ββ and Snow zone-6, sk β 4.5 ββ : SD_1 , SD_2 , SD_3 2 sec m Snow load is accompanyingvariable action and wind is leading avariable action. β¨ β© β‘ β€ 1 β¨ β© Comb_13 β β£ pEd_1β¨1β© SD_1 GD_1 β¦ β¨ β© β‘ β€ 1 β¨ β© Comb_14 β β£ pEd_2β¨1β© SD_2 GD_2 β¦ β© β‘ β€ 1 β¨ β© Comb_15 β β£ pEd_3β¨1β© SD_3 GD_3 β¦ β¨ Design Load combinations Appendix Page 23 of 55 Appendix A. Mathcad calculation document Comb_14 β‘ 1.50 β€ β’ 1.48 β₯ β’ β₯ β’ 1.46 β₯ β’ 1.43 β₯ β’ 1.39 β₯ 1 kN kN β¨β¨1β©β© pEd_2 = β’ 1.35 β₯ ββ SD_2 = β‘β£ 30.05 β€β¦ β β’ β₯ m2 m β’ 1.31 β₯ β’ 1.25 β₯ β’ 1.19 β₯ β’ 1.10 β₯ β’ β₯ β£ 0.95 β¦ Comb_15 β‘ 1.64 β€ β’ 1.62 β₯ β’ β₯ β’ 1.60 β₯ β’ 1.57 β₯ β’ 1.53 β₯ 1 kN kN β¨β¨1β©β© pEd_3 = β’ 1.49 β₯ ββ SD_3 = β‘β£ 33.02 β€β¦ β β’ β₯ m2 m β’ 1.44 β₯ β’ 1.38 β₯ β’ 1.30 β₯ β’ 1.20 β₯ β’ β₯ β£ 1.05 β¦ kN GD_1 = 6.91 β m β¨ β© Comb_13 β‘ 1.37 β€ β’ 1.35 β₯ β’ β₯ β’ 1.33 β₯ β’ 1.30 β₯ β’ 1.27 β₯ 1 kN kN β¨β¨1β©β© pEd_1 = β’ 1.23 β₯ ββ SD_1 = β‘β£ 27.41 β€β¦ β 2 β’ β₯ m m β’ 1.19 β₯ β’ 1.14 β₯ β’ 1.08 β₯ β’ 1.00 β₯ β’ β₯ β£ 0.87 β¦ β¨ β© kN GD_2 = 7.57 β m β© β© β¨ β¨ 3 3 3 kN Snow zone-4, sk β 3.0 ββ : SD_1 , SD_2 , SD_3 2 m β© m vb β 21 ββ and sec β¨ 6. β¨ β© kN GD_3 = 8.32 β m β¨ β© β‘ β€ 3 β¨ β© Comb_16 β β£ pEd_1β¨0β© SD_1 GD_1 β¦ β¨ β© β‘ β€ 3 β¨ β© Comb_17 β β£ pEd_2β¨0β© SD_2 GD_2 β¦ β© β‘ β€ 3 β¨ β© Comb_18 β β£ pEd_3β¨0β© SD_3 GD_3 β¦ β¨ Design Load combinations β‘ 1.24 β€ β’ 1.23 β₯ β’ β₯Appendix Page 24 of 55 β’ 1.21 β₯ β’ 1.18 β₯ β’ 1.16 β₯ Appendix A. Mathcad calculation document 3 kN SD_1 = β‘β£ 18.27 β€β¦ β m kN GD_1 = 6.91 β m Comb_17 β‘ 1.36 β€ β’ 1.35 β₯ β’ β₯ β’ 1.33 β₯ β’ 1.30 β₯ β’ 1.27 β₯ kN β¨β¨0β©β© pEd_2 = β’ 1.23 β₯ ββ β’ β₯ m2 β’ 1.19 β₯ β’ 1.14 β₯ β’ 1.08 β₯ β’ 1.00 β₯ β’ β₯ β£ 0.87 β¦ 3 kN SD_2 = β‘β£ 20.03 β€β¦ β m kN GD_2 = 7.57 β m Comb_18 β‘ 1.50 β€ β’ 1.48 β₯ β’ β₯ β’ 1.46 β₯ β’ 1.43 β₯ β’ 1.39 β₯ kN β¨β¨0β©β© pEd_3 = β’ 1.35 β₯ ββ β’ β₯ m2 β’ 1.31 β₯ β’ 1.26 β₯ β’ 1.19 β₯ β’ 1.10 β₯ β’ β₯ β£ 0.95 β¦ β¨ β© β¨ β© 3 kN SD_3 = β‘β£ 22.01 β€β¦ β m β¨ Comb_16 β© β‘ 1.24 β€ β’ 1.23 β₯ β’ β₯ β’ 1.21 β₯ β’ 1.18 β₯ β’ 1.16 β₯ kN β¨β¨0β©β© pEd_1 = β’ 1.12 β₯ ββ β’ β₯ m2 β’ 1.09 β₯ β’ 1.04 β₯ β’ 0.99 β₯ β’ 0.91 β₯ β’ β₯ β£ 0.79 β¦ kN GD_3 = 8.32 β m 4.1 .3 Wind Load: Terrain category, type -I An assumption has made that the height of the structure is equal to the height of the shell, seeComb_35 SS-EN 1991-1-4 figure 6.1 and 7.2.9(8) Terrain category, type I: Lakes or flat and horizontal area with negligible vegetation and without obstacles zo β 0.01 m zmin β 1 m co ββzββ β 1 The roughness length, zo , For terrain category, type I (SSEN 1991-1-4 Table 4.1 ) The orographic factor neglected, determine it through co β 1 annex A.3 of SS-EN 1991-1-4 Appendix Page 25 of 55 Appendix A. Mathcad calculation document β‘ 0.131 β€ β’ 0.132 β₯ β’ β₯ β’ 0.133 β₯ β’ 0.136 β₯ β’ 0.138 β₯ 1 IV β ββββ = β’ 0.141 β₯ β₯ β β z ββ β’ βco ⋅ ln ββββ β’ 0.145 β₯ β β zo β β β’ 0.150 β₯ β’ 0.156 β₯ β’ 0.167 β₯ β’ β₯ β£ 0.189 β¦ The turbulance intensity at a height z ,(EKS 11 4.5(1) Article 7) zo β 0.01 m (SS-EN 1991-1-4 Terrain category I, Table 4.1) zo_II β 0.05 m (SS-EN 1991-1-4 Terrain category II, Table 4.1) β zo β 0.07 kr β 0.19 βββ β = 0.17 β zo_II β The terrain factor( SS-EN 1991-1-4 equation 4.5) The basic wind velocities, detrmine for selected regions , From figure C-4 of EKS 11 (BFS 2015:6) The peak velocity pressure (EKS 11 chapter 11 part 4.5(1) replaces equation (4.8) of SS-EN 1991-1-4 with this we calculate for the selected geograhpical locations) For the shell The Peak factor for the peak velocity pressure ( qp ββzββ ), resonance determination is not neccessary 2 β β β βzβ β β 2 β z βββ β qp β βkr ⋅ ln βββ ⋅ coβ + 2 ⋅ kp ⋅ βkr ⋅ co ⋅ ln ββββ ⋅ qb T βββ β zo β β β β zo β β ββ kp β 3.0 vb1 β‘ 0.83 β’ 0.82 β’ β’ 0.80 β’ 0.78 β’ 0.76 qp = β’ 0.74 β’ β’ 0.71 β’ 0.67 β’ 0.63 β’ 0.57 β’ β£ 0.48 vb2 0.91 0.90 0.88 0.86 0.84 0.81 0.78 0.74 0.69 0.63 0.52 vb3 0.99 0.98 0.96 0.94 0.91 0.88 0.85 0.81 0.76 0.68 0.57 vb4 vb5 1.08 1.18 1.07 1.16 1.05 1.14 1.02 1.11 1.00 1.08 0.96 1.04 0.92 1.00 0.88 0.95 0.82 0.89 0.75 0.81 0.62 0.67 vb6 1.27 β€ z=21m 1.25 β₯ z=19.5m β₯ 1.23 β₯ z=18m 1.20 β₯ z=16m β₯ 1.17 z=14m kN 1.13 β₯ ββz=12m β₯ 2 z=10m 1.09 β₯ m z=8m 1.03 β₯ z=6m 0.97 β₯ β₯ z=4m 0.87 β₯ z=2m 0.73 β¦ The peak velocity pressure on the tank shell from bottom to top, z=2-4-6-8....19.5m-21m. for β¨ β© β¨ β© Region-5 with wind velocity, qpβ¨4β© for vb5 =25m/sec and qpβ¨5β© vb6 =26m/sec Appendix Page 26 of 55 Appendix A. Mathcad calculation document β‘ 1.18 β€ β’ 1.16 β₯ β’ β₯ β’ 1.14 β₯ β’ 1.11 β₯ β’ 1.08 β₯ kN β¨β¨4β©β© qp = β’ 1.04 β₯ ββ β’ β₯ m2 β’ 1.00 β₯ β’ 0.95 β₯ β’ 0.89 β₯ β’ 0.81 β₯ β’ β₯ β£ 0.67 β¦ β‘ 1.27 β€ β’ 1.25 β₯ β’ β₯ β’ 1.23 β₯ β’ 1.20 β₯ β’ 1.17 β₯ kN β¨β¨5β©β© qp = β’ 1.13 β₯ ββ β’ β₯ m2 β’ 1.09 β₯ β’ 1.03 β₯ β’ 0.97 β₯ β’ 0.87 β₯ β’ β₯ β£ 0.73 β¦ For the purpose of tank shell buckling design we use the equivalent uniform external pressure on the tank shell, qeq_Ed Selecting the max. value of the peak wind velocity at the top of each shell courses for the regions under study. qw_max_Ed β qp vb1 vb2 vb3 vb4 vb5 vb6 β‘ 0.82 0.90 0.98 1.07 1.16 1.25 β€ β’ 0.81 0.88 0.97 1.05 1.14 1.23 β₯ β’ β₯ β’ 0.79 0.87 0.95 1.04 1.12 1.21 β₯ β’ 0.77 0.85 0.93 1.01 1.10 1.19 β₯ β’ 0.75 0.82 0.90 0.98 1.07 1.15 β₯ kN qeq_Ed β kw ⋅ qw_max_Ed = β’ 0.73 0.80 0.87 0.95 1.03 1.11 β₯ ββ β’ β₯ m2 β’ 0.70 0.77 0.84 0.91 0.99 1.07 β₯ β’ 0.66 0.73 0.80 0.87 0.94 1.02 β₯ β’ 0.62 0.68 0.75 0.81 0.88 0.95 β₯ β’ 0.56 0.62 0.68 0.74 0.80 0.86 β₯ β’ β₯ β£ 0.47 0.51 0.56 0.61 0.66 0.72 β¦ Internal suction pressure on the tank shell due to internal vacuum pressure, qs_Ed according to EN-1993-1-6 constant internal negative pressure is considered 40 % of the peak wind velocity pressure at the top of the tank shell. 0 β¨ qs_Ed β 0.4 ⋅ qeq_Ed 0 qp : is the value of the peak wind velocity pressures at the top of the tank shell for the six wind velocity regions . β¨ β© β© β¨ β© β 0β qs_Ed β 0.4 ⋅ kw ⋅ βqp β β¨ β© 0 kN qp = β‘β£ 0.83 0.91 0.99 1.08 1.18 1.27 β€β¦ ββ m2 Appendix Page 27 of 55 Appendix A. Mathcad calculation document β¨ β© vb1 β‘1β€ β‘ 0.33 β’1β₯ β’ 0.33 β’ β₯ β’ β’1β₯ β’ 0.33 1 β’ β₯ β’ 0.33 β’1β₯ β’ 0.33 0 β’ β₯ qs_Ed β 1 ⋅ 0.4 ⋅ kw ⋅ qp = β’ 0.33 β’ β₯ β’ β’1β₯ β’ 0.33 β’1β₯ β’ 0.33 β’1β₯ β’ 0.33 β’1β₯ β’ 0.33 β’ β₯ β’ β£1β¦ β£ 0.33 vb2 0.36 0.36 0.36 0.36 0.36 0.36 0.36 0.36 0.36 0.36 0.36 vb3 0.39 0.39 0.39 0.39 0.39 0.39 0.39 0.39 0.39 0.39 0.39 vb4 vb5 0.43 0.46 0.43 0.46 0.43 0.46 0.43 0.46 0.43 0.46 0.43 0.46 0.43 0.46 0.43 0.46 0.43 0.46 0.43 0.46 0.43 0.46 vb6 0.50 β€ 0.50 β₯ β₯ 0.50 β₯ 0.50 β₯ 0.50 β₯ kN 0.50 β₯ ββ β₯ m2 0.50 β₯ 0.50 β₯ 0.50 β₯ 0.50 β₯ β₯ 0.50 β¦ Characteristic Equivalent wind load on the tank shell along its height on each shell courses, for the six wind velocity regions is the sum of wind velocity pressure and the internal negative pressure acting on the tank shell Wind is the leading variable,Design situation 6.10b. vb1 β‘ 1.15 β’ 1.13 β’ β’ 1.12 β’ 1.10 β’ 1.08 pEd β qeq_Ed + qs_Ed = β’ 1.05 β’ β’ 1.03 β’ 0.99 β’ 0.95 β’ 0.89 β’ β£ 0.80 vb2 1.26 1.24 1.23 1.21 1.18 1.16 1.13 1.09 1.04 0.98 0.87 vb3 1.37 1.36 1.34 1.32 1.29 1.26 1.23 1.19 1.14 1.07 0.96 vb4 vb5 1.50 1.62 1.48 1.61 1.46 1.59 1.44 1.56 1.41 1.53 1.38 1.49 1.34 1.45 1.29 1.41 1.24 1.34 1.16 1.26 1.04 1.13 vb6 1.76 β€ 1.74 β₯ β₯ 1.72 β₯ 1.69 β₯ 1.65 β₯ kN 1.62 β₯ ββ β₯ m2 1.57 β₯ 1.52 β₯ 1.45 β₯ 1.36 β₯ β₯ 1.22 β¦ 6. Shell design Loads and Combinations The loads combinations which normally are design deriving at the ultimate limit state according to equation 6.10b for the three safety classes β¨ β© β 0 β¨ β©β Ed οΌ γd ⋅ βξ ⋅ γG_sup ⋅ G + γQ ⋅ ψo ⋅ S + γQ ⋅ pEdβ¨0β©β ξ : reduction factor ξ β 0.89 γG_sup : Partial cofficient unfavorable permanent load γG_sup β 1.35 γQ : Partial cofficient unfavorable variable load γQ β 1.5 ψo : combination factor for variable load: in this case for snow load is accompanying variable action based on the selected snow zone Appendix Page 28 of 55 ψo β 0.8 ψ0_1 β ψo Appendix A. Mathcad calculation document ψ0_wind : Combination factor for variable load: when wind is accompanying variable action 6.1 Reliabilty class-1, γd_1 β 0.83 ψ0_wind β 0.3 For empty tank: EKS 11 1. Design self weight of the Roofing kN GD_1 β γd_1 ⋅ ξ ⋅ γG_sup ⋅ G = 6.91 β m 2. Design Snow Load The design snow load for the the sleceted snow zones : β‘ 33.5 β€ β’ 27.41 β₯ β’ β₯ β’ 21.32 β₯ 18.27 β₯ kN SD_1 β γd_1 ⋅ γQ ⋅ ψ0_i ⋅ S = β’ β β’ 13.32 β₯ m β’ 10.66 β₯ β’ β₯ β’ 6.85 β₯ β£ 4.57 β¦ β¨ β© 1 kN SD_1leading β γd_1 ⋅ γQ ⋅ S = β‘β£ 34.26 β€β¦ β m When Snow load is an accompanying variable action Where the Snow load is assumed to be a leading variable action at location where the basic wind m velocity vb β 22 ββ and snow load zone with sec characterstic snow load on the ground , kN sk β 4.5 ββ m2 3. Design Wind Load β‘ 1.43 β’ 1.41 β’ β’ 1.39 β’ 1.37 β’ 1.34 pEd_1 β γd_1 ⋅ γQ ⋅ pEd = β’ 1.31 β’ β’ 1.28 β’ 1.23 β’ 1.18 β’ 1.11 β’ β£ 0.99 1.56 1.55 1.53 1.50 1.47 1.44 1.40 1.35 1.30 1.22 1.09 1.71 1.69 1.67 1.64 1.61 1.57 1.53 1.48 1.42 1.33 1.19 1.86 1.84 1.82 1.79 1.75 1.71 1.67 1.61 1.54 1.45 1.29 2.02 2.00 1.98 1.94 1.90 1.86 1.81 1.75 1.67 1.57 1.40 2.19 β€ 2.16 β₯ β₯ 2.14 β₯ 2.10 β₯ 2.06 β₯ kN 2.01 β₯ ββ β₯ m2 1.96 β₯ 1.89 β₯ 1.81 β₯ 1.70 β₯ β₯ 1.52 β¦ β‘ 0.47 β€ β’ 0.46 β₯ Page 29 of 55 β’ Appendix β₯ β’ 0.46 β₯ β’ 0.45 β₯ β’ 0.44 β₯ Appendix A. Mathcad calculation document β‘ 0.47 β€ β’ 0.46 β₯ β’ β₯ β’ 0.46 β₯ β’ 0.45 β₯ β’ 0.44 β₯ kN β¨β¨1β©β© pEd_1accompany β γd_1 ⋅ γQ ⋅ ψ0_wind ⋅ pEd = β’ 0.43 β₯ ββ β’ β₯ m2 β’ 0.42 β₯ β’ 0.41 β₯ β’ 0.39 β₯ β’ 0.36 β₯ β’ β₯ β£ 0.33 β¦ 6.2 Reliabilty class-2, For γd_2 β 0.91 Wind load : accompanying variable action at locations where basic wind m velocity, vb β 22 ββ sec For empty tank: EKS 11 1. Design self weight of the Roofing kN GD_2 β γd_2 ⋅ ξ ⋅ γG_sup ⋅ G = 7.57 β m 2. Design Snow Load The design snow load for the the sleceted snow zones are β‘ 36.73 β€ β’ 30.05 β₯ β’ β₯ β’ 23.37 β₯ 20.03 β₯ kN SD_2 β γd_2 ⋅ γQ ⋅ ψ0_i ⋅ S = β’ β β’ 14.61 β₯ m β’ 11.69 β₯ β’ β₯ β’ 7.51 β₯ β£ 5.01 β¦ β¨ β© 1 kN SD_2leading β γd_2 ⋅ γQ ⋅ S = β‘β£ 37.56 β€β¦ β m When Snow load is an accompanying variable action Where the Snow load is assumed to be a leading variable action at location where the basic wind m velocity vb β 22 ββ and snow load zone with sec characterstic snow load on the ground , kN sk β 4.5 ββ m2 3. Design Wind Load The wind is assumed to be the leading variable action for lower snow zones Appendix Page 30 of 55 Appendix A. Mathcad calculation document β‘ 1.56 β’ 1.55 β’ β’ 1.53 β’ 1.50 β’ 1.47 pEd_2 β γd_2 ⋅ γQ ⋅ pEd = β’ 1.44 β’ β’ 1.40 β’ 1.35 β’ 1.29 β’ 1.21 β’ β£ 1.09 1.72 1.70 1.68 1.65 1.62 1.58 1.54 1.49 1.42 1.33 1.19 1.88 1.85 1.83 1.80 1.77 1.73 1.68 1.62 1.55 1.46 1.30 2.04 2.02 2.00 1.96 1.92 1.88 1.83 1.77 1.69 1.59 1.42 2.22 2.19 2.17 2.13 2.09 2.04 1.98 1.92 1.83 1.72 1.54 2.40 β€ 2.37 β₯ β₯ 2.34 β₯ 2.30 β₯ 2.26 β₯ kN 2.21 β₯ ββ β₯ m2 2.15 β₯ 2.07 β₯ 1.98 β₯ 1.86 β₯ β₯ 1.67 β¦ β‘ 0.51 β€ β’ 0.51 β₯ β’ β₯ β’ 0.50 β₯ β’ 0.49 β₯ β’ 0.48 β₯ kN β¨β¨1β©β© pEd_2accompany β γd_2 ⋅ γQ ⋅ ψ0_wind ⋅ pEd = β’ 0.47 β₯ ββ β’ β₯ m2 β’ 0.46 β₯ β’ 0.45 β₯ β’ 0.43 β₯ β’ 0.40 β₯ β’ β₯ β£ 0.36 β¦ 6.3 Reliabilty class-3, γd_3 β 1 1. Design self weight of the Roofing kN GD_3 β γd_3 ⋅ ξ ⋅ γG_sup ⋅ G = 8.32 β m 2. Design Snow Load The design snow load for the the sleceted snow zones are β‘ 40.36 β€ β’ 33.02 β₯ β’ β₯ β’ 25.68 β₯ 22.01 β₯ kN SD_3 β γd_3 ⋅ γQ ⋅ ψ0_i ⋅ S = β’ β β’ 16.05 β₯ m β’ 12.84 β₯ β’ β₯ β’ 8.26 β₯ β£ 5.5 β¦ β¨ β© 1 kN SD_3leading β γd_3 ⋅ γQ ⋅ S = β‘β£ 41.28 β€β¦ β m Where the Snow load is assumed to be a leading variable action at location where the basic wind m velocity vb β 22 ββ and snow load zone with sec characterstic snow load on the ground , kN sk β 4.5 ββ m2 Appendix Page 31 of 55 Appendix A. Mathcad calculation document 3. Design Wind Load β‘ 1.72 β’ 1.70 β’ β’ 1.68 β’ 1.65 β’ 1.62 pEd_3 β γd_3 ⋅ γQ ⋅ pEd = β’ 1.58 β’ β’ 1.54 β’ 1.49 β’ 1.42 β’ 1.34 β’ β£ 1.19 1.89 1.86 1.84 1.81 1.78 1.74 1.69 1.63 1.56 1.47 1.31 2.06 2.04 2.01 1.98 1.94 1.90 1.85 1.78 1.71 1.60 1.43 2.24 2.22 2.19 2.16 2.11 2.07 2.01 1.94 1.86 1.74 1.56 2.43 2.41 2.38 2.34 2.29 2.24 2.18 2.11 2.02 1.89 1.69 2.63 β€ 2.60 β₯ β₯ 2.57 β₯ 2.53 β₯ 2.48 β₯ kN 2.42 β₯ ββ β₯ m2 2.36 β₯ 2.28 β₯ 2.18 β₯ 2.05 β₯ β₯ 1.83 β¦ β‘ 0.57 β€ β’ 0.56 β₯ β’ β₯ β’ 0.55 β₯ β’ 0.54 β₯ β’ 0.53 β₯ kN β¨β¨1β©β© pEd_3accompany β γd_3 ⋅ γQ ⋅ ψ0_wind ⋅ pEd = β’ 0.52 β₯ ββ β’ β₯ m2 β’ 0.51 β₯ β’ 0.49 β₯ β’ 0.47 β₯ β’ 0.44 β₯ β’ β₯ β£ 0.39 β¦ 7. Design Load Combinations: Terrain Category-I β© β© β© β¨ β¨ β¨ 6 6 6 m kN 1. vb β 26 ββ and vs Snow zone-1, sk β 1.5 ββ : SD_1 , SD_2 , SD_3 2 sec m β¨ β© β‘ β€ 6 β¨ β© Comb_16 β β£ pEd_1β¨5β© SD_1 GD_1 β¦ β© β‘ β€ 6 β¨ β© Comb_17 β β£ pEd_2β¨5β© SD_2 GD_2 β¦ β¨ Design Load combinations:T1W26Sk1.5RcFb- β¨ β© β‘ β€ 6 β¨ β© Comb_18 β β£ pEd_3β¨5β© SD_3 GD_3 β¦ Appendix Page 32 of 55 Appendix β¨ β© β‘ β€ 6 β¨ β© Comb_18 β β£ pEd_3β¨5β© SD_3 GD_3 β¦ A. Mathcad calculation document 6 kN SD_1 = β‘β£ 6.85 β€β¦ β m kN GD_1 = 6.91 β m Comb_17 β‘ 2.40 β€ β’ 2.37 β₯ β’ β₯ β’ 2.34 β₯ β’ 2.30 β₯ β’ 2.26 β₯ kN β¨β¨5β©β© pEd_2 = β’ 2.21 β₯ ββ β’ β₯ m2 β’ 2.15 β₯ β’ 2.07 β₯ β’ 1.98 β₯ β’ 1.86 β₯ β’ β₯ β£ 1.67 β¦ 6 kN SD_2 = β‘β£ 7.51 β€β¦ β m kN GD_2 = 7.57 β m Comb_18 β‘ 2.63 β€ β’ 2.60 β₯ β’ β₯ β’ 2.57 β₯ β’ 2.53 β₯ β’ 2.48 β₯ kN β¨β¨5β©β© pEd_3 = β’ 2.42 β₯ ββ β’ β₯ m2 β’ 2.36 β₯ β’ 2.28 β₯ β’ 2.18 β₯ β’ 2.05 β₯ β’ β₯ β£ 1.83 β¦ 6 kN SD_3 = β‘β£ 8.26 β€β¦ β m kN GD_3 = 8.32 β m β¨ β© β¨ β© β¨ β© Comb_16 β‘ 2.19 β€ β’ 2.16 β₯ β’ β₯ β’ 2.14 β₯ β’ 2.10 β₯ β’ 2.06 β₯ kN β¨β¨5β©β© pEd_1 = β’ 2.01 β₯ ββ β’ β₯ m2 β’ 1.96 β₯ β’ 1.89 β₯ β’ 1.81 β₯ β’ 1.70 β₯ β’ β₯ β£ 1.52 β¦ β© β© β¨ β¨ β¨ β© 5 5 5 m kN 2. vb β 25 ββ and vs Snow zone-2, sk β 2.0 ββ : SD_1 , SD_2 , SD_3 2 sec m β¨ β© β‘ β€ 5 β¨ β© Comb_19 β β£ pEd_1β¨5β© SD_1 GD_1 β¦ β© β‘ β€ 5 β¨ β© Comb_20 β β£ pEd_2β¨5β© SD_2 GD_2 β¦ β¨ Design Load combinations β¨ β© β‘ β€ 5 β¨ β© Comb_21 β β£ pEd_3β¨5β© SD_3 GD_3 β¦ β‘ 2.02 β€ β’ 2.00 β₯ β’ β₯ β’ 1.98 β₯ β’ 1.94 β₯ β’ 1.90 β₯ Appendix Page 33 of 55 Appendix A. Mathcad calculation document Comb_20 β‘ 2.22 β€ β’ 2.19 β₯ β’ β₯ β’ 2.17 β₯ β’ 2.13 β₯ β’ 2.09 β₯ kN β¨β¨4β©β© pEd_2 = β’ 2.04 β₯ ββ β’ β₯ m2 β’ 1.98 β₯ β’ 1.92 β₯ β’ 1.83 β₯ β’ 1.72 β₯ β’ β₯ β£ 1.54 β¦ kN GD_1 = 6.91 β m β¨ β© Comb_19 β‘ 2.02 β€ β’ 2.00 β₯ β’ β₯ β’ 1.98 β₯ β’ 1.94 β₯ β’ 1.90 β₯ 5 kN kN β¨β¨4β©β© pEd_1 = β’ 1.86 β₯ ββ SD_1 = β‘β£ 10.66 β€β¦ β β’ β₯ m2 m β’ 1.81 β₯ β’ 1.75 β₯ β’ 1.67 β₯ β’ 1.57 β₯ β’ β₯ β£ 1.40 β¦ kN GD_2 = 7.57 β m β¨ β© 5 kN SD_2 = β‘β£ 11.69 β€β¦ β m β© β© β© β¨ β¨ β¨ β¨ β© β‘ 2.43 β€ β’ 2.41 β₯ β’ β₯ β’ 2.38 β₯ β’ 2.34 β₯ 5 kN kN β’ 2.29 β₯ SD_3 = β‘β£ 12.84 β€β¦ β GD_3 = 8.32 β kN β¨ β© m m Comb_21 pEd_3β¨4β© = β’ 2.24 β₯ ββ β’ β₯ m2 β’ 2.18 β₯ β’ 2.11 β₯ β’ 2.02 β₯ β’ 1.89 β₯ β’ β₯ β£ 1.69 β¦ 3 3 3 m kN 3. vb β 24 ββ and vs Snow zone-2, sk β 3.0 ββ : SD_1 , SD_2 , SD_3 2 sec m β¨ β© β‘ β€ 3 β¨ β© Comb_21 β β£ pEd_1β¨4β© SD_1 GD_1 β¦ β© β‘ β€ 3 β¨ β© Comb_22 β β£ pEd_2β¨4β© SD_2 GD_2 β¦ β¨ Design Load combinations β¨ β© β‘ β€ 3 β¨ β© Comb_23 β β£ pEd_3β¨4β© SD_3 GD_3 β¦ β‘ 2.02 β€ β’ 2.00 β₯ β’ β₯ β’ 1.98 β₯ β’ 1.94 β₯ β’ 1.90 β₯ Appendix Page 34 of 55 Appendix A. Mathcad calculation document Comb_21 β‘ 2.02 β€ β’ 2.00 β₯ β’ β₯ β’ 1.98 β₯ β’ 1.94 β₯ β’ 1.90 β₯ kN β¨β¨4β©β© pEd_1 = β’ 1.86 β₯ ββ β’ β₯ m2 β’ 1.81 β₯ β’ 1.75 β₯ β’ 1.67 β₯ β’ 1.57 β₯ β’ β₯ β£ 1.40 β¦ Comb_22 β‘ 2.22 β€ β’ 2.19 β₯ β’ β₯ β’ 2.17 β₯ β’ 2.13 β₯ β’ 2.09 β₯ 3 kN kN β¨β¨4β©β© pEd_2 = β’ 2.04 β₯ ββ SD_2 = β‘β£ 20.03 β€β¦ β 2 β’ β₯ m m β’ 1.98 β₯ β’ 1.92 β₯ β’ 1.83 β₯ β’ 1.72 β₯ β’ β₯ β£ 1.54 β¦ kN GD_2 = 7.57 β m β‘ 2.43 β€ β’ 2.41 β₯ β’ β₯ β’ 2.38 β₯ β’ 2.34 β₯ β’ 2.29 β₯ 3 kN kN β¨β¨4β©β© pEd_3 = β’ 2.24 β₯ ββ SD_3 = β‘β£ 22.01 β€β¦ β β’ β₯ m2 m β’ 2.18 β₯ β’ 2.11 β₯ β’ 2.02 β₯ β’ 1.89 β₯ β’ β₯ β£ 1.69 β¦ kN GD_3 = 8.32 β m kN GD_1 = 6.91 β m β© β© β¨ β¨ 2 2 2 kN Snow zone-3.5, sk β 3.5 ββ : SD_1 , SD_2 , SD_3 m2 β© β‘ β€ 2 β¨ β© Comb_24 β β£ pEd_1β¨2β© SD_1 GD_1 β¦ Design Load combinations β¨ β© m 4. vb β 23 ββ and sec β¨ Comb_23 β¨ β© β¨ β© β¨ β© 3 kN SD_1 = β‘β£ 18.27 β€β¦ β m β¨ β© β‘ β€ 2 β¨ β© Comb_25 β β£ pEd_2β¨2β© SD_2 GD_2 β¦ β¨ β© β‘ β€ 2 β¨ β© Comb_26 β β£ pEd_3β¨2β© SD_3 GD_3 β¦ Appendix Page 35 of 55 Appendix A. Mathcad calculation document β‘ 1.71 β€ β’ 1.69 β₯ β’ β₯ β’ 1.67 β₯ β’ 1.64 β₯ 2 kN β’ 1.61 β₯ SD_1 = β‘β£ 21.32 β€β¦ β kN β¨β¨2β©β© m pEd_1 = β’ 1.57 β₯ ββ β’ β₯ m2 β’ 1.53 β₯ β’ 1.48 β₯ β’ 1.42 β₯ β’ 1.33 β₯ β’ β₯ β£ 1.19 β¦ β¨ β© kN GD_1 = 6.91 β m Comb_24 Comb_26 β‘ 2.06 β€ β’ 2.04 β₯ β’ β₯ β’ 2.01 β₯ β’ 1.98 β₯ 2 kN β’ 1.94 β₯ SD_3 = β‘β£ 25.68 β€β¦ β kN β¨β¨2β©β© m pEd_3 = β’ 1.90 β₯ ββ β’ β₯ m2 β’ 1.85 β₯ β’ 1.78 β₯ β’ 1.71 β₯ β’ 1.60 β₯ β’ β₯ β£ 1.43 β¦ kN GD_2 = 7.57 β m β¨ β© Comb_25 β‘ 1.88 β€ β’ 1.85 β₯ β’ β₯ β’ 1.83 β₯ β’ 1.80 β₯ β’ 1.77 β₯ 2 kN kN β¨β¨2β©β© pEd_2 = β’ 1.73 β₯ ββ SD_2 = β‘β£ 23.37 β€β¦ β β’ β₯ m2 m β’ 1.68 β₯ β’ 1.62 β₯ β’ 1.55 β₯ β’ 1.46 β₯ β’ β₯ β£ 1.30 β¦ β¨ β© kN GD_3 = 8.32 β m β© β© β¨ β¨ β© 0 0 2 m kN vb β 22 ββ and Snow zone-4, sk β 4.5 ββ : SD_1leading , SD_2leading , SD_3leading sec m2 Snow load is the leading variable action and wind is an accompanying action. β¨ 5. β¨ β© β‘ β€ 0 β¨ β© Comb_27 β β£ pEd_1accompanyβ¨0β© SD_1leading GD_1 β¦ β© β‘ β€ 0 β¨ β© Comb_28 β β£ pEd_2accompanyβ¨0β© SD_2leading GD_2 β¦ β¨ Design Load combinations β¨ β© β‘ β€ 0 β¨ β© Comb_29 β β£ pEd_3accompanyβ¨0β© SD_3leading GD_3 β¦ β‘ 0.47 β€ β’ 0.46 β₯ β’ β₯ β’ 0.46 β₯ β’ 0.45 β₯ β’ 0.44 β₯ Appendix Page 36 of 55 Appendix A. Mathcad calculation document β© β¨ 0 kN SD_3leading = β‘β£ 41.28 β€β¦ β m kN GD_3 = 8.32 β m β© kN GD_2 = 7.57 β m β¨ Comb_29 β‘ 0.57 β€ β’ 0.56 β₯ β’ β₯ β’ 0.55 β₯ β’ 0.54 β₯ β’ 0.53 β₯ kN β¨β¨0β©β© pEd_3accompany = β’ 0.52 β₯ ββ β’ β₯ m2 β’ 0.51 β₯ β’ 0.49 β₯ β’ 0.47 β₯ β’ 0.44 β₯ β’ β₯ β£ 0.39 β¦ kN GD_1 = 6.91 β m 0 kN SD_2leading = β‘β£ 37.56 β€β¦ β m β© Comb_28 β‘ 0.51 β€ β’ 0.51 β₯ β’ β₯ β’ 0.50 β₯ β’ 0.49 β₯ β’ 0.48 β₯ kN β¨β¨0β©β© pEd_2accompany = β’ 0.47 β₯ ββ β’ β₯ m2 β’ 0.46 β₯ β’ 0.45 β₯ β’ 0.43 β₯ β’ 0.40 β₯ β’ β₯ β£ 0.36 β¦ 0 kN SD_1leading = β‘β£ 34.26 β€β¦ β m β¨ Comb_27 β‘ 0.47 β€ β’ 0.46 β₯ β’ β₯ β’ 0.46 β₯ β’ 0.45 β₯ β’ 0.44 β₯ kN β¨β¨0β©β© pEd_1accompany = β’ 0.43 β₯ ββ β’ β₯ m2 β’ 0.42 β₯ β’ 0.41 β₯ β’ 0.39 β₯ β’ 0.36 β₯ β’ β₯ β£ 0.33 β¦ β© β© β¨ β¨ β¨ β© 1 1 1 m kN 5. vb β 22 ββ and Snow zone-4, sk β 4.5 ββ : SD_1 , SD_2 , SD_3 sec m2 Snow load is accompanyingvariable action and wind is leading avariable action. β¨ β© β‘ β€ 1 β¨ β© Comb_30 β β£ pEd_1β¨1β© SD_1 GD_1 β¦ β© β‘ β€ 1 β¨ β© Comb_31 β β£ pEd_2β¨1β© SD_2 GD_2 β¦ β¨ Design Load combinations β¨ β© β‘ β€ 1 β¨ β© Comb_32 β β£ pEd_3β¨1β© SD_3 GD_3 β¦ β‘ 1.56 β€ β’ 1.55 β₯ β’ β₯ β’ 1.53 β₯ β’ 1.50 β₯ β’ 1.47 β₯ Appendix Page 37 of 55 Appendix A. Mathcad calculation document Comb_30 β‘ 1.56 β€ β’ 1.55 β₯ β’ β₯ β’ 1.53 β₯ β’ 1.50 β₯ β’ 1.47 β₯ kN β¨β¨1β©β© pEd_1 = β’ 1.44 β₯ ββ β’ β₯ m2 β’ 1.40 β₯ β’ 1.35 β₯ β’ 1.30 β₯ β’ 1.22 β₯ β’ β₯ β£ 1.09 β¦ Comb_31 β‘ 1.72 β€ β’ 1.70 β₯ β’ β₯ β’ 1.68 β₯ β’ 1.65 β₯ β’ 1.62 β₯ 1 kN kN β¨β¨1β©β© pEd_2 = β’ 1.58 β₯ ββ SD_2 = β‘β£ 30.05 β€β¦ β 2 β’ β₯ m m β’ 1.54 β₯ β’ 1.49 β₯ β’ 1.42 β₯ β’ 1.33 β₯ β’ β₯ β£ 1.19 β¦ kN GD_2 = 7.57 β m Comb_32 β‘ 1.89 β€ β’ 1.86 β₯ β’ β₯ β’ 1.84 β₯ β’ 1.81 β₯ β’ 1.78 β₯ 1 kN kN β¨β¨1β©β© pEd_3 = β’ 1.74 β₯ ββ SD_3 = β‘β£ 33.02 β€β¦ β β’ β₯ m2 m β’ 1.69 β₯ β’ 1.63 β₯ β’ 1.56 β₯ β’ 1.47 β₯ β’ β₯ β£ 1.31 β¦ kN GD_3 = 8.32 β m kN GD_1 = 6.91 β m β© β© β¨ β¨ 3 3 3 kN Snow zone-4, sk β 3.0 ββ : SD_1 , SD_2 , SD_3 m2 β© m vb β 21 ββ and sec β¨ 6. β¨ β© β¨ β© β¨ β© 1 kN SD_1 = β‘β£ 27.41 β€β¦ β m β¨ β© β‘ β€ 3 β¨ β© Comb_33 β β£ pEd_1β¨0β© SD_1 GD_1 β¦ β© β‘ β€ 3 β¨ β© Comb_34 β β£ pEd_2β¨0β© SD_2 GD_2 β¦ β¨ Design Load combinations β¨ β© β‘ β€ 3 β¨ β© Comb_35 β β£ pEd_3β¨0β© SD_3 GD_3 β¦ β‘ 1.43 β€ β’ 1.41 β₯ β’ β₯ β’ 1.39 β₯ β’ 1.37 β₯ β’ 1.34 β₯ Appendix Page 38 of 55 Appendix A. Mathcad calculation document 3 kN SD_1 = β‘β£ 18.27 β€β¦ β m kN GD_1 = 6.91 β m Comb_34 β‘ 1.56 β€ β’ 1.55 β₯ β’ β₯ β’ 1.53 β₯ β’ 1.50 β₯ β’ 1.47 β₯ kN β¨β¨0β©β© pEd_2 = β’ 1.44 β₯ ββ β’ β₯ m2 β’ 1.40 β₯ β’ 1.35 β₯ β’ 1.29 β₯ β’ 1.21 β₯ β’ β₯ β£ 1.09 β¦ 3 kN SD_2 = β‘β£ 20.03 β€β¦ β m kN GD_2 = 7.57 β m Comb_35 β‘ 1.72 β€ β’ 1.70 β₯ β’ β₯ β’ 1.68 β₯ β’ 1.65 β₯ β’ 1.62 β₯ kN β¨β¨0β©β© pEd_3 = β’ 1.58 β₯ ββ β’ β₯ m2 β’ 1.54 β₯ β’ 1.49 β₯ β’ 1.42 β₯ β’ 1.34 β₯ β’ β₯ β£ 1.19 β¦ 3 kN SD_3 = β‘β£ 22.01 β€β¦ β m kN GD_3 = 8.32 β m β¨ β© β¨ β© β¨ Comb_33 β© β‘ 1.43 β€ β’ 1.41 β₯ β’ β₯ β’ 1.39 β₯ β’ 1.37 β₯ β’ 1.34 β₯ kN β¨β¨0β©β© pEd_1 = β’ 1.31 β₯ ββ β’ β₯ m2 β’ 1.28 β₯ β’ 1.23 β₯ β’ 1.18 β₯ β’ 1.11 β₯ β’ β₯ β£ 0.99 β¦ 4.1 .3 Wind Load:Terrain category type-0 An assumption has made that the height of the structure is equal to the height of the shell, See SS-EN 1991-1-4 figure 6.1 and 7.2.9(8) Terrain category, type 0: Sea or coastal area exposed to the open sea zo β 0.003 m The roughness length, zo , For terrain category, type 0 (SSEN 1991-1-4 Table 4.1 ) zmin β 1 m β‘ 0.113 β€ β’ 0.114 β₯ β’ β₯ β’ 0.115 β₯ β’ 0.117 β₯ β’ 0.118 β₯ Appendix Page 39 of 55 Appendix A. Mathcad calculation document β‘ 0.113 β€ β’ 0.114 β₯ β’ β₯ β’ 0.115 β₯ β’ 0.117 β₯ β’ 0.118 β₯ 1 IV β ββββ = β’ 0.121 β₯ β₯ β β z ββ β’ βco ⋅ ln ββββ β’ 0.123 β₯ β β zo β β β’ 0.127 β₯ β’ 0.132 β₯ β’ 0.139 β₯ β’ β₯ β£ 0.154 β¦ zo_II β 0.05 m The turbulance intensity at a height z ,(EKS 11 4.5(1) Article 7) (SS-EN 1991-1-4 Terrain category II, Table 4.1) The terrain factor( SS-EN 1991-1-4 equation 4.5) β zo β 0.07 kr β 0.19 βββ β = 0.156 β zo_II β .......................................................................................(20) The basic wind velocities, detrmine for selected regions , From figure C-4 of EKS 11 (BFS 2015:6) The peak velocity pressure (EKS 11 chapter 11 part 4.5(1) replaces equation (4.8) of SS-EN 1991-1-4, with this the peak velocity pressure acting at the top of each courses of the tank shell can be calculated for the selected geograhpical locations) kp β 3.0 The Peak factor for the peak velocity pressure ( qp ββzββ ), resonance determination is not neccessary 2β β β‘ βzβ β€ β β β β β β β‘ β β β€ qp βzβ β β£ 1 + 2 ⋅ kp ⋅ IV βzβ β¦ ⋅ β’ kr ⋅ ln βββ ⋅ co βzβ β₯ ⋅ qb ββ β£ β¦ ββ β zo β 1 IV ββzββ β βββββ β β z ββ βco ββzββ ⋅ ln ββββ β β zo β β Substituting and rearranging the above two equations , eq.1and eq.2 , The peak velocity pressure acting on the shell becomes: 2 β β β βzβ β β 2 β z βββ β qp β βkr ⋅ ln βββ ⋅ coβ + 2 ⋅ kp ⋅ βkr ⋅ co ⋅ ln ββββ ⋅ qb T βββ β zo β β β β zo β β ββ Appendix Page 40 of 55 Appendix A. Mathcad calculation document vb1 β‘ 0.88 β’ 0.87 β’ β’ 0.86 β’ 0.84 β’ 0.82 qp = β’ 0.80 β’ β’ 0.77 β’ 0.74 β’ 0.69 β’ 0.64 β’ β£ 0.55 vb2 0.97 0.96 0.94 0.92 0.90 0.87 0.84 0.81 0.76 0.70 0.60 vb3 1.06 1.04 1.03 1.01 0.98 0.95 0.92 0.88 0.83 0.76 0.65 vb4 vb5 1.15 1.25 1.14 1.23 1.12 1.22 1.10 1.19 1.07 1.16 1.04 1.13 1.00 1.09 0.96 1.04 0.91 0.98 0.83 0.90 0.71 0.77 vb6 1.35 β€ z=21m 1.33 β₯ z=19.5m β₯ 1.32 β₯ z=18m 1.29 β₯ z=16m 1.26 β₯ z=14m kN β₯ 1.22 ββz=12m β₯ 2 z=10m 1.18 β₯ m z=8m 1.13 β₯ z=6m 1.06 β₯ β₯ z=4m 0.98 β₯ z=2m 0.84 β¦ Note: The hieght of the tank on this mathcad document is along z-axis is similar to the y-axis on the FE-Modelling β‘ 0.97 β€ β‘ 1.06 β€ β’ 0.96 β₯ β’ 1.04 β₯ β’ β₯ β’ β₯ β’ 0.94 β₯ β’ 1.03 β₯ The peak velocity pressure on each shell course β’ 0.92 β₯ β’ 1.01 β₯ β’ 0.90 β₯ β’ 0.98 β₯ of the tank shell from bottom to top, kN kN β¨β¨1β©β© β¨β¨2β©β© β’ β₯ z=2-4-6-8....19.5m-21m. for Region-3 with qp = 0.87 ββ qp = β’ 0.95 β₯ ββ 2 β’ β₯ β’ β₯ wind velocity vb2 =22m/sec and vb3 =23m/sec m m2 β’ 0.84 β₯ β’ 0.92 β₯ β’ 0.81 β₯ β’ 0.88 β₯ β’ 0.76 β₯ β’ 0.83 β₯ β’ 0.70 β₯ β’ 0.76 β₯ β’ β₯ β’ β₯ β£ 0.60 β¦ β£ 0.65 β¦ For the purpose of tank shell buckling design we use the equivalent uniform external pressure on the tank shell, qeq_Ed Selecteing the max. value of the peak wind velocity at the top of each shell courses for the regions under study. qw_max_Ed β qp vb1 vb2 vb3 vb4 vb5 vb6 β‘ 0.87 0.96 1.04 1.14 1.23 1.33 β€ β’ 0.86 0.94 1.03 1.12 1.22 1.32 β₯ β’ β₯ β’ 0.85 0.93 1.02 1.11 1.20 1.30 β₯ β’ 0.83 0.91 0.99 1.08 1.17 1.27 β₯ β’ 0.81 0.89 0.97 1.06 1.14 1.24 β₯ kN qeq_Ed β kw ⋅ qw_max_Ed = β’ 0.78 0.86 0.94 1.02 1.11 1.20 β₯ ββ β’ β₯ m2 β’ 0.76 0.83 0.91 0.99 1.07 1.16 β₯ β’ 0.73 0.80 0.87 0.95 1.03 1.11 β₯ β’ 0.68 0.75 0.82 0.89 0.97 1.05 β₯ β’ 0.63 0.69 0.75 0.82 0.89 0.96 β₯ β’ β₯ β£ 0.54 0.59 0.65 0.70 0.76 0.82 β¦ Internal suction pressure on the tank shell due to internal vacuum pressure, qs_Ed according to EN-1993-1-6 Appendix Page 41 of 55 Appendix A. Mathcad calculation document β© β 0β qs_Ed β 0.4 ⋅ kw ⋅ βqp β β© β¨ Constant interal negative pressure is assumed to be 40 % of the peak wind velocity pressure for the six basic wind velocity pressures 0 β© qp : is the value of the peak wind velocity pressures at the top of the tank shell for the six basic wind velocity regions . β¨ β¨ qs_Ed β 0.4 ⋅ qeq_Ed 0 β¨ β© vb1 β‘1β€ β‘ 0.35 β’1β₯ β’ 0.35 β’ β₯ β’ β’1β₯ β’ 0.35 β’1β₯ β’ 0.35 β’1β₯ β’ 0.35 0 β’ β₯ qs_Ed β 1 ⋅ 0.4 ⋅ kw ⋅ qp = β’ 0.35 β’ β₯ β’ β’1β₯ β’ 0.35 1 β’ β₯ β’ 0.35 β’1β₯ β’ 0.35 β’1β₯ β’ 0.35 β’ β₯ β’ β£1β¦ β£ 0.35 vb2 0.38 0.38 0.38 0.38 0.38 0.38 0.38 0.38 0.38 0.38 0.38 vb3 0.42 0.42 0.42 0.42 0.42 0.42 0.42 0.42 0.42 0.42 0.42 vb4 vb5 0.45 0.49 0.45 0.49 0.45 0.49 0.45 0.49 0.45 0.49 0.45 0.49 0.45 0.49 0.45 0.49 0.45 0.49 0.45 0.49 0.45 0.49 vb6 0.53 β€ 0.53 β₯ β₯ 0.53 β₯ 0.53 β₯ 0.53 β₯ kN 0.53 β₯ ββ β₯ m2 0.53 β₯ 0.53 β₯ 0.53 β₯ 0.53 β₯ β₯ 0.53 β¦ Characterstic Equivalent wind load on the tank shell along its height at the top of each shell courses, for different wind velocity regions is the sum of wind velocity pressure and the internal negative pressure acting on the tank shell. vb1 β‘ 1.22 β’ 1.21 β’ β’ 1.19 β’ 1.18 β’ 1.16 pEd β qeq_Ed + qs_Ed = β’ 1.13 β’ β’ 1.11 β’ 1.07 β’ 1.03 β’ 0.98 β’ β£ 0.89 vb2 1.34 1.32 1.31 1.29 1.27 1.24 1.21 1.18 1.13 1.07 0.97 vb3 1.46 1.45 1.43 1.41 1.39 1.36 1.33 1.29 1.24 1.17 1.06 vb4 vb5 1.59 1.73 1.58 1.71 1.56 1.69 1.54 1.67 1.51 1.64 1.48 1.61 1.44 1.57 1.40 1.52 1.35 1.46 1.28 1.38 1.16 1.26 vb6 1.87 β€ 1.85 β₯ β₯ 1.83 β₯ 1.80 β₯ 1.77 β₯ kN 1.74 β₯ ββ β₯ 2 1.70 β₯ m 1.65 β₯ 1.58 β₯ 1.50 β₯ β₯ 1.36 β¦ 6. Shell design Loads and Combinations The design loads combinations which normally are design deriving at the ultimate limit state according to equation 6.10b for the three safty classes β¨ β© β 0 β¨ β©β Ed οΌ γd ⋅ βξ ⋅ γG_sup ⋅ G + γQ ⋅ ψo ⋅ S + γQ ⋅ pEdβ¨0β©β ξ : self-weight reduction factor ξ β 0.89 γG_sup : Partial cofficient unfavorable permanent load γG_sup β 1.35 γQ : Partial cofficient unfavorable variable load γQ β 1.5 ψo : combination factor for variable load: in this case for snow load is accompanying variable action based on the Appendix Page 42 of 55 selected snow zone ψo β ψ0_1 = 0.8 Appendix A. Mathcad calculation document ψo : combination factor for variable load: in this case for snow load is accompanying variable action based on the selected snow zone ψ0_wind : Combination factor for vriable load: when wind is accompanying variable action 6.1 Design Loads : For Reliabilty class-1, γd_1 β 0.83 ψo β ψ0_1 = 0.8 ψ0_wind β 0.3 EKS 11 For empty tank: 1. Design self weight of the Roofing : for Reliabilty class-1, γd_1 β 0.83 kN GD_1 β γd_1 ⋅ ξ ⋅ γG_sup ⋅ G = 6.91 β m 2. Design Snow Load : for Reliabilty class-2, γd_2 β 0.83 The design snow load for the the sleceted snow zones : β‘ 33.5 β€ β’ 27.41 β₯ β’ β₯ β’ 21.32 β₯ 18.27 β₯ kN SD_1 β γd_1 ⋅ γQ ⋅ ψ0_i ⋅ S = β’ β β’ 13.32 β₯ m β’ 10.66 β₯ β’ β₯ β’ 6.85 β₯ β£ 4.57 β¦ When Snow load is an accompanying variable action Where the Snow load is assumed to be a leading variable action at location where the basic wind m kN velocity vb β 22 ββ and snow load zone with characterstic snow load on the ground , sk β 4.5 ββ sec m2 β¨ β© 1 kN SD_1leading β γd_1 ⋅ γQ ⋅ S = β‘β£ 34.26 β€β¦ β m 3. Design Wind Load: For Reliabilty class-1, γd_1 β 0.83 m Wind load : as a leading variable action at locations where basic wind velocity, vb β 22 ββ sec β‘ 1.52 β’ 1.50 β’ β’ 1.49 β’ 1.46 β’ 1.44 pEd_1 β γd_1 ⋅ γQ ⋅ pEd = β’ 1.41 β’ β’ 1.38 β’ 1.34 β’ 1.29 β’ 1.22 β’ β£ 1.10 1.67 1.65 1.63 1.61 1.58 1.55 1.51 1.47 1.41 1.33 1.21 1.82 1.80 1.78 1.76 1.73 1.69 1.65 1.60 1.54 1.46 1.32 1.98 1.96 1.94 1.91 1.88 1.84 1.80 1.75 1.68 1.59 1.44 2.15 2.13 2.11 2.08 2.04 2.00 1.95 1.89 1.82 1.72 1.56 2.33 β€ 2.30 β₯ β₯ 2.28 β₯ 2.25 β₯ 2.21 β₯ kN 2.16 β₯ ββ β₯ m2 2.11 β₯ 2.05 β₯ 1.97 β₯ 1.86 β₯ β₯ 1.69 β¦ β‘ 0.50 β€ β’ 0.49 β₯ β’ β₯ Page 43 of 55 Appendix β’ 0.49 β₯ β’ 0.48 β₯ β’ 0.47 β₯ Appendix A. Mathcad calculation document β‘ 0.50 β€ β’ 0.49 β₯ β’ β₯ β’ 0.49 β₯ Wind load : accompanying β’ 0.48 β₯ β’ 0.47 β₯ variable action at locations where kN β¨ β© m pEd_1accompany β γd_1 ⋅ γQ ⋅ ψ0_wind ⋅ pEdβ¨1β© = β’ 0.46 β₯ ββ basic wind velocity, v β 22 ββ β’ β₯ m2 b sec β’ 0.45 β₯ β’ 0.44 β₯ β’ 0.42 β₯ β’ 0.40 β₯ β’ β₯ β£ 0.36 β¦ 6.2 Design Loads :For Reliabilty class-2, γd_2 β 0.91 EKS 11 1. Design self weight of the Roofing : for Reliabilty class-2, γd_2 β 0.91 kN GD_2 β γd_2 ⋅ ξ ⋅ γG_sup ⋅ G = 7.57 β m 2. Design Snow Load : for Reliabilty class-2, γd_2 β 0.91 The design snow load for the the sleceted snow zones are β‘ 36.73 β€ β’ 30.05 β₯ β’ β₯ β’ 23.37 β₯ 20.03 β₯ kN SD_2 β γd_2 ⋅ γQ ⋅ ψ0_i ⋅ S = β’ β β’ 14.61 β₯ m β’ 11.69 β₯ β’ β₯ β’ 7.51 β₯ β£ 5.01 β¦ β¨ β© 1 kN SD_2leading β γd_2 ⋅ γQ ⋅ S = β‘β£ 37.56 β€β¦ β m When Snow load is an accompanying variable action Where the Snow load is assumed to be a leading variable action at location where the basic wind m velocity vb β 22 ββ and snow load zone with sec characterstic snow load on the ground , kN sk β 4.5 ββ m2 3. Design Wind Load: for Reliabilty class-2, γd_2 β 0.91 The wind is assumed to be the leading variable action for lower snow zones β‘ 1.66 β’ 1.65 β’ β’ 1.63 β’ 1.61 β’ 1.58 1.83 1.81 1.79 1.76 1.73 2.00 2.17 2.36 2.55 β€ 1.98 2.15 2.33 2.53 β₯ Appendix Page 44β₯ of 55 1.96 2.13 2.31 2.50 β₯ 1.93 2.10 2.28 2.46 β₯ 1.89 2.06 2.24 2.42 β₯ Appendix A. Mathcad calculation document β‘ 1.66 β’ 1.65 β’ β’ 1.63 β’ 1.61 β’ 1.58 pEd_2 β γd_2 ⋅ γQ ⋅ pEd = β’ 1.55 β’ β’ 1.51 β’ 1.47 β’ 1.41 β’ 1.33 β’ β£ 1.21 1.83 1.81 1.79 1.76 1.73 1.70 1.66 1.61 1.55 1.46 1.33 2.00 1.98 1.96 1.93 1.89 1.85 1.81 1.76 1.69 1.60 1.45 2.17 2.15 2.13 2.10 2.06 2.02 1.97 1.91 1.84 1.74 1.58 2.36 2.33 2.31 2.28 2.24 2.19 2.14 2.08 2.00 1.89 1.71 2.55 β€ 2.53 β₯ β₯ 2.50 β₯ 2.46 β₯ 2.42 β₯ kN 2.37 β₯ ββ β₯ m2 2.31 β₯ 2.25 β₯ 2.16 β₯ 2.04 β₯ β₯ 1.85 β¦ β‘ 0.55 β€ β’ 0.54 β₯ β’ β₯ β’ 0.54 β₯ β’ 0.53 β₯ β’ 0.52 β₯ kN β¨β¨1β©β© pEd_2accompany β γd_2 ⋅ γQ ⋅ ψ0_wind ⋅ pEd = β’ 0.51 β₯ ββ β’ β₯ m2 β’ 0.50 β₯ β’ 0.48 β₯ β’ 0.46 β₯ β’ 0.44 β₯ β’ β₯ β£ 0.40 β¦ 6.3 Design Loads :For Reliabilty class-3, γd_3 β 1 Design wind load as a leading variable action Design wind load as an accompanying variable action EKS 11 1. Design self weight of the Roofing: Reliabilty class-3, γd_3 β 1 kN GD_3 β γd_3 ⋅ ξ ⋅ γG_sup ⋅ G = 8.32 β m 2. Design Snow Load : Reliabilty class-3, γd_3 β 1 The design snow load for the sleceted snow zones are β‘ 40.36 β€ β’ 33.02 β₯ β’ β₯ β’ 25.68 β₯ 22.01 β₯ kN SD_3 β γd_3 ⋅ γQ ⋅ ψ0_i ⋅ S = β’ β β’ 16.05 β₯ m β’ 12.84 β₯ β’ β₯ β’ 8.26 β₯ β£ 5.5 β¦ β¨ β© 1 kN SD_3leading β γd_3 ⋅ γQ ⋅ S = β‘β£ 41.28 β€β¦ β m Snow load as an accompanying variable action at location where the basic wind velocity m vb β 22 ββ and snow load zone with characterstic sec kN snow load on the ground , sk β 4.5 ββ m2 Snow load : assumed to be a leading variable action at location where the basic wind velocity m vb β 22 ββ and snow load zone with characterstic sec kN snow load on the ground , sk β 4.5 ββ m2 3. Design Wind Load for: Reliabilty class-3, γd_3 β 1 Appendix Page 45 of 55 Appendix A. Mathcad calculation document 3. Design Wind Load for: Reliabilty class-3, γd_3 β 1 β‘ 1.83 β’ 1.81 β’ β’ 1.79 β’ 1.76 β’ 1.73 pEd_3 β γd_3 ⋅ γQ ⋅ pEd = β’ 1.70 β’ β’ 1.66 β’ 1.61 β’ 1.55 β’ 1.46 β’ β£ 1.33 2.01 1.99 1.97 1.94 1.90 1.86 1.82 1.77 1.70 1.61 1.46 2.19 2.17 2.15 2.12 2.08 2.04 1.99 1.93 1.86 1.76 1.59 2.39 2.36 2.34 2.30 2.26 2.22 2.17 2.10 2.02 1.91 1.74 2.59 2.57 2.54 2.50 2.46 2.41 2.35 2.28 2.19 2.08 1.88 2.80 β€ 2.78 β₯ β₯ 2.75 β₯ 2.70 β₯ 2.66 β₯ kN 2.60 β₯ ββ β₯ m2 2.54 β₯ 2.47 β₯ 2.37 β₯ 2.25 β₯ β₯ 2.04 β¦ β‘ 0.60 β€ β’ 0.60 β₯ β’ β₯ β’ 0.59 β₯ β’ 0.58 β₯ β’ 0.57 β₯ kN β¨β¨1β©β© pEd_3accompany β γd_3 ⋅ γQ ⋅ ψ0_wind ⋅ pEd = β’ 0.56 β₯ ββ β’ β₯ m2 β’ 0.55 β₯ β’ 0.53 β₯ β’ 0.51 β₯ β’ 0.48 β₯ β’ β₯ β£ 0.44 β¦ Design wind load as a leading variable action Design wind load as an accompanying variable action 7. Shell design Load combinations n SD_i : design snow load at reliabilty class i and the corresponding nth row of the characterstic snow load vector,S. β¨ Notation: β© Design load combinations for the tank shell , for terrain category type-0 β¨ β© pEd_iβ¨mβ© : design wind load at reliabilty class-i and the corresponding nth column of the characterstic wind velocity pressure matrix, pED . Appendix Page 46 of 55 Appendix A. Mathcad calculation document β© β© β¨ β¨ β© β¨ 6 6 6 m kN 1. vb β 26 ββ and vs Snow zone-1, sk β 1.5 ββ : SD_1 , SD_2 , SD_3 sec m2 β¨ β© β‘ β€ 6 β¨ β© Comb_36 β β£ pEd_1β¨5β© SD_1 GD_1 β¦ β© β‘ β€ 6 β¨ β© Comb_37 β β£ pEd_2β¨5β© SD_2 GD_2 β¦ β¨ Design Load combinations β¨ β© β‘ β€ 6 β¨ β© Comb_38 β β£ pEd_3β¨5β© SD_3 GD_3 β¦ Comb_36 β‘ 2.33 β€ β’ 2.30 β₯ β’ β₯ β’ 2.28 β₯ β’ 2.25 β₯ β’ 2.21 β₯ kN β¨β¨5β©β© pEd_1 = β’ 2.16 β₯ ββ β’ β₯ m2 β’ 2.11 β₯ β’ 2.05 β₯ β’ 1.97 β₯ β’ 1.86 β₯ β’ β₯ β£ 1.69 β¦ Comb_37 β‘ 2.55 β€ β’ 2.53 β₯ β’ β₯ β’ 2.50 β₯ β’ 2.46 β₯ β’ 2.42 β₯ 6 kN kN β¨β¨5β©β© pEd_2 = β’ 2.37 β₯ ββ SD_2 = β‘β£ 7.51 β€β¦ β β’ β₯ m2 m β’ 2.31 β₯ β’ 2.25 β₯ β’ 2.16 β₯ β’ 2.04 β₯ β’ β₯ β£ 1.85 β¦ kN GD_2 = 7.57 β m Comb_38 β‘ 2.80 β€ β’ 2.78 β₯ β’ β₯ β’ 2.75 β₯ β’ 2.70 β₯ β’ 2.66 β₯ 6 kN kN β¨β¨5β©β© pEd_3 = β’ 2.60 β₯ ββ SD_3 = β‘β£ 8.26 β€β¦ β 2 β’ β₯ m m β’ 2.54 β₯ β’ 2.47 β₯ β’ 2.37 β₯ β’ 2.25 β₯ β’ β₯ β£ 2.04 β¦ kN GD_3 = 8.32 β m kN GD_1 = 6.91 β m β© β© β© β¨ β¨ 5 5 5 m kN vb β 25 ββ and vs Snow zone-3, sk β 1.5 ββ : SD_1 , SD_2 , SD_3 2 sec m β¨ 2. β¨ β© β¨ β© β¨ β© 6 kN SD_1 = β‘β£ 6.85 β€β¦ β m Appendix Page 47 of 55 Appendixm β© β© β¨ β¨ β© β¨ 5 5 5 kN vb β 25 ββ and vs Snow zone-3, sk β 1.5 ββ : SD_1 , SD_2 , SD_3 2 seccalculation document m A. Mathcad β¨ β© β‘ β€ 5 β¨ β© Comb_36 β β£ pEd_1β¨4β© SD_1 GD_1 β¦ β© β‘ β€ 5 β¨ β© Comb_37 β β£ pEd_2β¨4β© SD_2 GD_2 β¦ β¨ Design Load combinations β¨ β© β‘ β€ 5 β¨ β© Comb_38 β β£ pEd_3β¨4β© SD_3 GD_3 β¦ Comb_36 β‘ 2.15 β€ β’ 2.13 β₯ β’ β₯ β’ 2.11 β₯ β’ 2.08 β₯ β’ 2.04 β₯ 5 kN kN β¨β¨4β©β© pEd_1 = β’ 2.00 β₯ ββ SD_1 = β‘β£ 10.66 β€β¦ β β’ β₯ m2 m β’ 1.95 β₯ β’ 1.89 β₯ 6 kN β’ 1.82 β₯ SD_1 = β‘β£ 6.85 β€β¦ β β’ 1.72 β₯ m β’ β₯ β£ 1.56 β¦ Comb_37 β‘ 2.36 β€ β’ 2.33 β₯ β’ β₯ β’ 2.31 β₯ β’ 2.28 β₯ β’ 2.24 β₯ 6 kN kN β¨β¨4β©β© pEd_2 = β’ 2.19 β₯ ββ SD_2 = β‘β£ 7.51 β€β¦ β β’ β₯ m2 m β’ 2.14 β₯ β’ 2.08 β₯ β’ 2.00 β₯ β’ 1.89 β₯ β’ β₯ β£ 1.71 β¦ kN GD_2 = 7.57 β m Comb_38 β‘ 2.59 β€ β’ 2.57 β₯ β’ β₯ β’ 2.54 β₯ β’ 2.50 β₯ β’ 2.46 β₯ 5 kN kN β¨β¨4β©β© S = β‘β£ 12.84 β€β¦ β pEd_3 = β’ 2.41 β₯ ββ β’ β₯ m 2 D_3 m β’ 2.35 β₯ β’ 2.28 β₯ β’ 2.19 β₯ β’ 2.08 β₯ β’ β₯ β£ 1.88 β¦ kN GD_3 = 8.32 β m β¨ β© β¨ β© β¨ β© β¨ β© kN GD_1 = 6.91 β m β© β© β¨ β¨ β© 3 3 3 m kN vb β 24 ββ and vs Snow zone-2, sk β 3.0 ββ : SD_1 , SD_2 , SD_3 2 sec m β¨ β¨ β© β‘ β€ 3 β¨ β© Comb_39 β β£ pEd_1β¨3β© SD_1 GD_1 β¦ β‘ β€ 3 β¨ β© Comb_40 β β£ pEd_2β¨3β© SD_2 GD_2 β¦ β© Design Load combinations β¨ 3. Appendix Page 48 of 55 Appendix β© β‘ β€ 3 β¨ β© Comb_40 β β£ pEd_2β¨3β© SD_2 GD_2 β¦ β¨ A. Mathcad calculation document β¨ β© β‘ β€ 3 β¨ β© Comb_41 β β£ pEd_3β¨3β© SD_3 GD_3 β¦ kN GD_1 = 6.91 β m β‘ 2.17 β€ β’ 2.15 β₯ β’ β₯ β’ 2.13 β₯ β’ 2.1 β₯ β’ 2.06 β₯ 3 kN kN β¨β¨3β©β© pEd_2 = β’ 2.02 β₯ ββ SD_2 = β‘β£ 20.03 β€β¦ β β’ β₯ m2 m β’ 1.97 β₯ β’ 1.91 β₯ β’ 1.84 β₯ β’ 1.74 β₯ β’ β₯ β£ 1.58 β¦ kN GD_2 = 7.57 β m β© β‘ 1.98 β€ β’ 1.96 β₯ β’ β₯ β’ 1.94 β₯ β’ 1.91 β₯ β’ 1.88 β₯ 3 kN kN β¨β¨3β©β© pEd_1 = β’ 1.84 β₯ ββ SD_1 = β‘β£ 18.27 β€β¦ β β’ β₯ m2 m β’ 1.8 β₯ β’ 1.75 β₯ β’ 1.68 β₯ β’ 1.59 β₯ β’ β₯ β£ 1.44 β¦ Comb_40 β¨ β© β¨ Comb_39 β© β© β¨ β¨ β© β¨ β¨ β© β‘ 2.39 β€ β’ 2.36 β₯ β’ β₯ β’ 2.34 β₯ β’ 2.3 β₯ β’ 2.26 β₯ 3 kN kN kN β¨β¨3β©β© Comb_41 SD_3 = β‘β£ 22.01 β€β¦ β GD_3 = 8.32 β pEd_3 = β’ 2.22 β₯ ββ β’ β₯ m2 m m β’ 2.17 β₯ β’ 2.1 β₯ β’ 2.02 β₯ β’ 1.91 β₯ β’ β₯ β£ 1.74 β¦ 2 2 2 m kN vb β 23 ββ and Snow zone-3, sk β 3.5 ββ : SD_1 , SD_2 , SD_3 2 4. sec m β¨ β© β‘ β€ 2 β¨ β© Comb_42 β β£ pEd_1β¨2β© SD_1 GD_1 β¦ β© β‘ β€ 2 β¨ β© Comb_43 β β£ pEd_2β¨2β© SD_2 GD_2 β¦ β¨ Design Load combinations β¨ β© β‘ β€ 2 β¨ β© Comb_44 β β£ pEd_3β¨2β© SD_3 GD_3 β¦ β‘ 1.82 β€ β’ 1.80 β₯ β’ β₯ β’ 1.78 β₯ β’ 1.76 β₯ β’ 1.73 β₯ Appendix Page 49 of 55 Appendix A. Mathcad calculation document 2 kN SD_1 = β‘β£ 21.32 β€β¦ β m kN GD_1 = 6.91 β m Comb_43 β‘ 2.00 β€ β’ 1.98 β₯ β’ β₯ β’ 1.96 β₯ β’ 1.93 β₯ β’ 1.89 β₯ kN β¨β¨2β©β© pEd_2 = β’ 1.85 β₯ ββ β’ β₯ m2 β’ 1.81 β₯ β’ 1.76 β₯ β’ 1.69 β₯ β’ 1.60 β₯ β’ β₯ β£ 1.45 β¦ 2 kN SD_2 = β‘β£ 23.37 β€β¦ β m kN GD_2 = 7.57 β m β¨ β© β¨ β© Comb_42 β‘ 1.82 β€ β’ 1.80 β₯ β’ β₯ β’ 1.78 β₯ β’ 1.76 β₯ β’ 1.73 β₯ kN β¨β¨2β©β© pEd_1 = β’ 1.69 β₯ ββ β’ β₯ m2 β’ 1.65 β₯ β’ 1.60 β₯ β’ 1.54 β₯ β’ 1.46 β₯ β’ β₯ β£ 1.32 β¦ β© β© β¨ β¨ β¨ 5. β© β¨ β© β‘ 2.19 β€ β’ 2.17 β₯ β’ β₯ β’ 2.15 β₯ β’ 2.12 β₯ β’ 2.08 β₯ 2 kN kN kN β¨β¨2β©β© GD_3 = 8.32 β pEd_3 = β’ 2.04 β₯ ββ SD_3 = β‘β£ 25.68 β€β¦ β Comb_44 β’ β₯ m2 m m β’ 1.99 β₯ β’ 1.93 β₯ β’ 1.86 β₯ β’ 1.76 β₯ β’ β₯ β£ 1.59 β¦ 0 0 0 m kN vb β 22 ββ and Snow zone-4, sk β 4.5 ββ : SD_1leading , SD_2leading , SD_3leading 2 sec m Snow load is the leading variable action and wind is an accompanying action. β¨ β© β‘ β€ 0 β¨ β© Comb_45 β β£ pEd_1accompanyβ¨0β© SD_1leading GD_1 β¦ β¨ β© β‘ β€ 0 β¨ β© Comb_46 β β£ pEd_2accompanyβ¨0β© SD_2leading GD_2 β¦ β© β‘ β€ 0 β¨ β© Comb_47 β β£ pEd_3accompanyβ¨0β© SD_3leading GD_3 β¦ β¨ Design Load combinations β‘ 0.50 β€ β’ 0.49 β₯ β’ β₯ Appendix Page 50 of 55 β’ 0.49 β₯ β’ 0.48 β₯ β’ 0.47 β₯ Appendix A. Mathcad calculation document Comb_45 β‘ 0.50 β€ β’ 0.49 β₯ β’ β₯ β’ 0.49 β₯ β’ 0.48 β₯ β’ 0.47 β₯ kN β¨β¨0β©β© pEd_1accompany = β’ 0.46 β₯ ββ β’ β₯ m2 β’ 0.45 β₯ β’ 0.44 β₯ β’ 0.42 β₯ β’ 0.40 β₯ β’ β₯ β£ 0.36 β¦ Comb_46 β‘ 0.55 β€ β’ 0.54 β₯ β’ β₯ β’ 0.54 β₯ β’ 0.53 β₯ β’ 0.52 β₯ 0 kN kN β¨β¨0β©β© pEd_2accompany = β’ 0.51 β₯ ββSD_2leading = β‘β£ 37.56 β€β¦ β 2 β’ β₯ m m β’ 0.50 β₯ β’ 0.48 β₯ β’ 0.46 β₯ β’ 0.44 β₯ β’ β₯ β£ 0.40 β¦ kN GD_2 = 7.57 β m Comb_47 β‘ 0.60 β€ β’ 0.60 β₯ β’ β₯ β’ 0.59 β₯ β’ 0.58 β₯ β’ 0.57 β₯ 0 kN kN β¨β¨0β©β© pEd_3accompany = β’ 0.56 β₯ ββSD_3leading = β‘β£ 41.28 β€β¦ β β’ β₯ m2 m β’ 0.55 β₯ β’ 0.53 β₯ β’ 0.51 β₯ β’ 0.48 β₯ β’ β₯ β£ 0.44 β¦ kN GD_3 = 8.32 β m kN GD_1 = 6.91 β m β© β© β¨ β¨ β© 1 1 1 m kN vb β 22 ββ and Snow zone-4, sk β 4.5 ββ : SD_1 , SD_2 , SD_3 sec m2 Snow load is accompanyingvariable action and wind is leading avariable action. β¨ β¨ β© β‘ β€ 1 β¨ β© Comb_48 β β£ pEd_1β¨1β© SD_1 GD_1 β¦ β© β‘ β€ 1 β¨ β© Comb_49 β β£ pEd_2β¨1β© SD_2 GD_2 β¦ β¨ Design Load combinations β© β‘ β€ 1 β¨ β© Comb_50 β β£ pEd_3β¨1β© SD_3 GD_3 β¦ β¨ 6. β¨ β© β¨ β© β¨ β© 0 kN SD_1leading = β‘β£ 34.26 β€β¦ β m β‘ 1.67 β€ β’ 1.65 β₯ β’ β₯ β’ 1.63 β₯ β’ 1.61 β₯ β’ 1.58 β₯ Appendix Page 51 of 55 Appendix A. Mathcad calculation document kN GD_1 = 6.91 β m Comb_49 β‘ 1.83 β€ β’ 1.81 β₯ β’ β₯ β’ 1.79 β₯ β’ 1.76 β₯ β’ 1.73 β₯ 1 kN kN β¨β¨1β©β© pEd_2 = β’ 1.7 β₯ ββ SD_2 = β‘β£ 30.05 β€β¦ β 2 β’ β₯ m m β’ 1.66 β₯ β’ 1.61 β₯ β’ 1.55 β₯ β’ 1.46 β₯ β’ β₯ β£ 1.33 β¦ kN GD_2 = 7.57 β m β‘ 2.01 β€ β’ 1.99 β₯ β’ β₯ β’ 1.97 β₯ β’ 1.94 β₯ β’ 1.90 β₯ 1 kN kN β¨β¨1β©β© pEd_3 = β’ 1.86 β₯ ββ SD_3 = β‘β£ 33.02 β€β¦ β β’ β₯ m2 m β’ 1.82 β₯ β’ 1.77 β₯ β’ 1.70 β₯ β’ 1.61 β₯ β’ β₯ β£ 1.46 β¦ kN GD_3 = 8.32 β m β© β© β¨ 3 3 3 kN Snow zone-4, sk β 3.0 ββ : SD_1 , SD_2 , SD_3 m2 β¨ m vb β 21 ββ and sec β© β¨ β© β‘ β€ 3 β¨ β© Comb_51 β β£ pEd_1β¨0β© SD_1 GD_1 β¦ β¨ β© β‘ β€ 3 β¨ β© Comb_52 β β£ pEd_2β¨0β© SD_2 GD_2 β¦ β‘ β€ 3 β¨ β© Comb_53 β β£ pEd_3β¨0β© SD_3 GD_3 β¦ β© Design Load combinations β¨ 7. β¨ Comb_50 β¨ β© β¨ β© β¨ β© Comb_48 β‘ 1.67 β€ β’ 1.65 β₯ β’ β₯ β’ 1.63 β₯ β’ 1.61 β₯ β’ 1.58 β₯ 1 kN kN β¨β¨1β©β© pEd_1 = β’ 1.55 β₯ ββ SD_1 = β‘β£ 27.41 β€β¦ β β’ β₯ m2 m β’ 1.51 β₯ β’ 1.47 β₯ β’ 1.41 β₯ β’ 1.33 β₯ β’ β₯ β£ 1.21 β¦ Appendix Page 52 of 55 Appendix A. Mathcad calculation document Comb_52 β‘ 1.66 β€ β’ 1.65 β₯ β’ β₯ β’ 1.63 β₯ β’ 1.61 β₯ β’ 1.58 β₯ kN β¨β¨0β©β© pEd_2 = β’ 1.55 β₯ ββ β’ β₯ m2 β’ 1.51 β₯ β’ 1.47 β₯ β’ 1.41 β₯ β’ 1.33 β₯ β’ β₯ β£ 1.21 β¦ 3 kN SD_2 = β‘β£ 20.03 β€β¦ β m β‘ 1.83 β€ β’ 1.81 β₯ β’ β₯ β’ 1.79 β₯ β’ 1.76 β₯ β’ 1.73 β₯ kN β¨β¨0β©β© pEd_3 = β’ 1.70 β₯ ββ β’ β₯ m2 β’ 1.66 β₯ β’ 1.61 β₯ β’ 1.55 β₯ β’ 1.46 β₯ β’ β₯ β£ 1.33 β¦ 3 kN SD_3 = β‘β£ 22.01 β€β¦ β m β© β¨ β© β¨ Comb_53 β¨ β© Comb_51 β‘ 1.52 β€ β’ 1.50 β₯ β’ β₯ β’ 1.49 β₯ β’ 1.46 β₯ β’ 1.44 β₯ 3 kN kN β¨β¨0β©β© pEd_1 = β’ 1.41 β₯ ββ SD_1 = β‘β£ 18.27 β€β¦ β 2 β’ β₯ m m β’ 1.38 β₯ β’ 1.34 β₯ β’ 1.29 β₯ β’ 1.22 β₯ β’ β₯ β£ 1.10 β¦ Appendix Page 53 of 55 kN GD_1 = 6.91 β m kN GD_2 = 7.57 β m kN GD_3 = 8.32 β m Appendix A. Mathcad calculation document 5. Fabrication tolerances and imperfections Tolerance on the shell geometry The max. difference between the design and as built profile for minimum plate thickness becomes For emin ≤ 12.5 mm ediff β 16 mm is EN14015 Table 25 emin ≤ 12.5 mm Dimple tolerances Depth of intial dimples, Δwo for the three fabrication tolerance classes lgx β 4 ⋅ ‾‾‾ r ⋅ t = 1.12 m lgθ β 2.3 ⋅ ββl 2 ⋅ r ⋅ tββ 0.25 = 2.92 m lgw β 25 ⋅ t = 150 mm or Gage length in both meridional and circumferencial directions due to meridional compressive stresses but lgw β 25 ⋅ t Uox ≤ Uo_max but with Uox β Uo_max Δwox Uox β ββ lgx For Uoθ ≤ Uo_max Uow ≤ Uo_max Δwox β Uox ⋅ lgx = 6.7 mm Uow β Uo_max Δwow β Uow ⋅ lgw = 0.9 mm b. For fabrication tolerance quality class B Uox ≤ Uo_max Uo_max β 0.006 Δwoθ β Uoθ ⋅ lgθ = 17.53 mm Δwow Uox β ββ lgw For lgw ≤ 500 mm Uoθ β Uo_max Δwoθ Uoθ β ββ lgθ For l :the meridional length of the shell segment The gauge length due to additional across welds, in both the circumferential and merdional directions a. For fabrication tolerance quality class A For lgθ ≤ r Uo_max β 0.010 Uox β Uo_max Appendix Page 54 of 55 Appendix A. Mathcad calculation document Δwox_B β Uox ⋅ lgx = 11.17 mm Δwox_B Uox β ββ lgx For Uoθ ≤ Uo_max Uoθ β Uo_max Δwoθ_B Uoθ β βββ lgθ For Uow ≤ Uo_max Uow β Uo_max Δwow_B Uox β βββ lgw c. For fabrication tolerance quality class - C For Uox ≤ Uo_max Uox β Uo_max Δwox_A Uox β βββ lgx For Uoθ ≤ Uo_max Uoθ β Uo_max Δwoθ_B Uoθ β βββ lgθ For Uow ≤ Uo_max Uow β Uo_max Δwow_C Uox β βββ lgw Δwoθ_B β Uoθ ⋅ lgθ = 29.22 mm Δwow_B β Uow ⋅ lgw = 1.5 mm Uo_max β 0.016 Δwox_A β Uox ⋅ lgx = 17.87 mm Δwoθ_B β Uoθ ⋅ lgθ = 46.75 mm Δwow_C β Uow ⋅ lgw = 2.4 mm Appendix Page 55 of 55 Fakulteten för teknik 391 82 Kalmar | 351 95 Växjö Tel 0772-28 80 00 teknik@lnu.se Lnu.se/fakulteten-for-teknik
