10.5. Shell and flange sizing

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10.5. Shell and flange sizing

In the case of a steel tubular tower, the main output of the design exercise is the sizing of the shell segments and of the welded and bolted connections. In the following two sections, an overview of the limit state verifications and of the structural criteria to be satisfied for these components is provided. The physical principles are easily extended to other tower configurations, with the addition of specific verifications. For lattice towers, the size of the individual members and the qualification of the joints are the principal design outcome. They can be sized by performing member and joint checks as specified by appropriate standards (eg, Refs [19,31]). In the case of concrete towers, the size and arrangement of the reinforcement, of the prestressing tendons, and the characteristics of the concrete (eg, density and compression strength) are additional quantities that need to be determined following specific standard guidance (eg, Refs [22,34,36]).

Figure 10.22 Examples of code-to-code verification efforts for the development of FAST v8: (a) comparison of time-series of jacket OTM at the mudline among various codes; (b) the data in terms of PSD.

10.5.1. Load resistance factored design: FLS, ULS, and SLS verifications

From what has been stated thus far, it should not be surprising that the tower must satisfy a large number of structural and modal performance criteria; Ref. [1], for example, prescribes both operational, parked, and fault-loading situations under a number of turbulent and steady wind models and sea state conditions. These DLCs should be translated into stress and strain fields that must be verified against ULS, FLS, and SLS limit states within the LRFD approach. In general, different segments of the tower may be driven by different limit states: for example, the upper segments may be mostly driven by FLS requirements, whereas the bottom portion may be mostly ULS-driven. These factors may change depending on the site conditions, water depths, SbS configuration, and turbine sizes. Accounting for additional effects, such as corrosion and secondary steel (see Section 10.6.1), makes the verification phase all the more complicated.

10.5.1.1. Safety factors

The generic ULS limit state verification may be expressed as follows (cf. Ref. [23]):

$γ_{n} S (F_{d}) \leq R (f_{d})$

[10.14a]

$or explicitly$

$γ_{f} F_{k} \leq \frac{f_{k}}{γ_{n} γ_{m}}$

[10.14b]

where S(F_d) is the probability distribution of the generic, design (factored) load within the LRFD approach; R(f_d) is the analogous function for the material factored resistance; F_d (F_k) is the factored (unfactored) characteristic load; f_d (f_k) is the material factored (unfactored) resistance; γ_n is the consequence of failure PSF (or “importance” factor); γ_f is the generic load PSF; and γ_m is the material PSF.

Because many DLCs involve response to turbulent inflow under a range of mean wind speeds, and wave spectrum forcing, the exceedance probability for the characteristic load must be computed based on the expected wind and wave distributions. Guidance on the determination of the characteristic loads through the extrapolation of the results of limited-duration AHSE simulations is given in the standards (cf. Refs [15,23]).

The standards also provide recommended values for γ_f (see also Table 10.4) and γ_n [1,11,15,21,23]. The load factors represent the uncertainty in the load stochastic distribution and in the load assessment. In general, design standards allow for the adoption of lower than prescribed load PSFs in those cases where the load magnitudes are established by measurements, yielding a high degree of confidence. The important factors are based on the redundancy and fail-safe characteristics of the various components. Following the classification in Ref. [23], towers are considered components of class 2 (see Table 10.5), and γ_n = 1 may be assumed.

Material PSFs can be either taken from specific, recognized design codes (eg, Refs [43,82–84]), or minimum values may be taken per the main design standards such as Ref. [23] (see Table 10.6).

For SLS verifications, which include critical deflection analyses such as blade-to-tower clearance, ULS load factors may be used in combination with PSFs for consequence of failure and material resistance as indicated by the appropriate standards, or as shown in Tables 10.5 and 10.6.

Table 10.4

Examples of ULS, SLS, and FLS load PSFs

Limit state	Unfavorable loads			Favorable loads
Limit state	Normal^a abnormal transport and installation			All DLCs
ULS/SLS	1.35	1.1	1.5	0.9
FLS	1.0	1.0	1.0	1.0

From IEC 61400–1. Wind Turbines – Part 1: Design Requirements, 2005.

Table 10.5

Examples of minimum γ_n as a function of component class

Component class	γ_n			Comment
Component class	ULS	FLS	SLS	Comment
1	0.9	1.0	1.0	“Fail-safe” structural components whose failure does not result in the failure of a major part of a wind turbine (eg, replaceable bearings)
2	1.0	1.15	1.0	“Non fail-safe” structural components whose failures may lead to the failure of a major part of a wind turbine
3	1.3	1.3	1.3	“Non fail-safe” mechanical components that link actuators and brakes to main structural components for the purpose of implementing non-redundant turbine protection functions

From IEC 61400–1. Wind Turbines – Part 1: Design Requirements, 2005.

Table 10.6

Examples of minimum γ_m as a function of failure mode

Failure mode	γ_m
Failure mode	ULS	FLS	SLS
Yielding of ductile materials Global buckling of curved shells Rupture from exceeding tensile or compression strength	1.1 1.2 1.3	1.1 (welded and structural steel) to 1.7 (composites)	1.0 (if elastic properties proven by full-scale testing); 1.0 (otherwise)

From IEC 61400–1. Wind Turbines – Part 1: Design Requirements, 2005.

The generic FLS verification may be expressed as follows (cf. Ref. [23]):

$D_{fat} = \sum_{i} \frac{n_{i}}{{\hat{N}}_{i}} \leq 1$

[10.15]

with

${\hat{N}}_{i} = N_{i} (σ_{a, i} γ_{f} γ_{n} γ_{m})$

[10.16]

where D_fat is the fatigue damage; n_i is the number of cycles at the i-th load range; and

{\hat{N}}_{i}

is the “factored” N_i, ie, the number of cycles at failure corresponding to the PSF-augmented i-th load range, (γ_f γ_nγ_mσ_a,_i). N_i values can be calculated from σ−N curves for the material and structural element under consideration as described in Section 10.5.2. The FLS PSFs are given by the same design and certification standards, as for example shown in Tables 10.4–10.6

Finally, per Ref. [18], one should design the OWT and its components with the overall exposure category L2 (see Section 10.3.1), which also requires that a 500-year robustness check be carried out with unity PSFs.

10.5.2. Approximate derivation of structural loads and shell design

Analytically, one could account for the various contributions to the tower loads as in Eqs. [10.17]–[10.20]. The normal (axial, along the z axis) force can be written as:

$N_{d} (z) = γ_{f} F_{z RNA} - γ_{fg} m_{RNA} g - γ_{fg} \int_{z}^{L} ρ A g d z$

[10.17]

where N_d is the design (factored) normal load at the tower station of interest; F_zRNA is the aerodynamic force from the RNA along the z axis; γ_f is the generic load PSF; m_RNA is the RNA mass; g is the gravity acceleration; γ_fg is the gravitational load PSF; ρ is the material density; L is the tower length, or SSt length; A is the cross-sectional area; and z is the coordinate along the tower span.

The shear components along x and y (reference system is that of Fig. 10.14) can be taken as:

$T_{x} (z) = γ_{f} F_{x RNA} + γ_{fa} \int_{z}^{L} f_{a} \cdot \hat{i} d ζ$

[10.18a]

$T_{y} (z) = γ_{f} F_{y RNA} + γ_{fa} \int_{z}^{L} f_{a} \cdot \hat{j} d ζ$

[10.18b]

where F_xRNA is the force from the RNA along the x axis; F_yRNA is the force from the RNA along the y-axis; f_a is given in Eq. [10.10]; γ_fa is the aerodynamic load PSF;

\hat{i}

is the unit vector along the x axis;

\hat{j}

is the unit vector along the y axis. The bending moment components can be written as:

$M_{x} (z) = M_{x RNA} - F_{y RNA} (z_{RNA} - z) - \int_{z}^{L} [f_{a} \cdot \hat{j} ζ + ρ A g (y (ζ) - y (z))] d ζ$

[10.19a]

$M_{y} (z) = M_{y RNA} + F_{x RNA} (z_{RNA} - z) + \int_{z}^{L} [f_{a} \cdot \hat{i} ζ + ρ A g (x (ζ) - x (z))] d ζ$

[10.19b]

where M_x is the component of the bending moment along the x axis at the station of interest; M_y is the component of the bending moment load along the y axis at the station of interest; M_xRNA is the RNA aerodynamic moment along the x axis; M_yRNA is the RNA aerodynamic moment along the y-axis; ζ is the dummy coordinate along the z axis. Finally the torque about the z axis is given by:

$M_{z} (z) = M_{z RNA} - F_{x RNA} (y (z_{RNA}) - y (z)) + F_{y RNA} (x (z_{RNA}) - x (z))$

[10.20]

where M_z is the torsion moment load along the z axis at the station of interest.

Note that the second order P−Δ effect was accounted for in Eqs. [10.19a]–[10.20], and that potentially different PSFs for the various loading components were employed following standards such as Ref. [3]. The RNA forces must be calculated following what was stated in Section 10.4.1.1. The shear and moment components can then be combined to arrive at characteristic design values:

$T_{d} (z) = \sqrt{T_{x} {(z)}^{2} + T_{y} {(z)}^{2}}$

[10.21a]

$M_{d} (z) = \sqrt{M_{x} {(z)}^{2} + M_{y} {(z)}^{2}}$

[10.21b]

Finally, the normal stresses (σ_z,Ed along the meridional, and σ_θ,Ed circumferential directions), the shear stress (τ_zθ,Ed), and the Von-Mises equivalent stress (σ_vm) for the generic cross-section of a tubular tower can conservatively be written as:

$σ_{z, Ed} = \frac{N_{d}}{A} + \frac{M_{d} D_{sh}}{2 J_{x x}}$

[10.22a]

$σ_{θ, Ed} = γ_{f} (k_{w} - 1) q_{\max} \frac{D_{sh} - t_{s}}{2 t_{s}}$

[10.22b]

$τ_{z θ, Ed} = 2 T_{d} / A + \frac{M_{z}}{2 A_{mid} t_{s}}$

[10.22c]

$σ_{vm} = \sqrt{σ_{z, Ed}^{2} + σ_{θ, Ed}^{2} - σ_{z, Ed} σ_{θ, Ed} + 3 τ_{z θ, Ed}^{2}}$

[10.22d]

where the argument z was dropped without losing generality; k_w is the dynamic pressure factor to calculate hoop stresses—it is a function of cylinder dimensions and external pressure buckling factor per Ref. [2]; t_s is the shell thickness; A_mid is the area inscribed by the mid-thickness line; and all the other symbols were above introduced except for q_max, which is the q_max expressed as in Eq. [10.23]:

$q_{\max} = 0.5 * ρ_{a} {| U |}^{2}$

[10.23]

with ρ_a being the air density, and U the wind velocity, which may also include structural motion components (normally negligible).

While the above treatment is sufficient for conceptual and preliminary assessments, only rigorous loads and FEA analyses can fully support a more detailed design. In particular, fully coupled loads' analyses via AHSE simulations are the only way to capture important interactions and vibrational dynamics and to perform FLS and ULS verifications for the certification of OWT SSts.

The FLS verification must determine the accumulated damage over the SSt design lifetime (usually of 20 years), accounting for the appropriate operational and non-operational DLCs. From what was stated earlier, the focus should be on wind–wave misalignment and idling DLCs where the vibrational damping is negligible. Based on the Palmgren–Miner rule [85,86], the total damage requirement can be expressed as:

$D_{fat} = \sum_{i} \frac{n_{i}}{N_{i}} \leq 1$

[10.24]

where D_fat is the fatigue damage; n_i is the number of cycles at the i-th load range; and N_i is the number of cycles at failure for the i-th load range level. N_i values can be calculated from σ–N curves for the material and element under consideration (eg, weld butt, steel sheet, flange, bolts; see also Fig. 10.23). The typical σ–N curve is derived from material coupon testing and it can, at first approximation, be expressed by:

$σ_{a, i} = C N_{i}^{- 1 / m}$

[10.25]

where σ_a,i is the i-th bin stress range; C is the constant in the σ–N curve, approximately equal to the material ultimate strength; and m is the inverse exponent in the σ–N curve. Material standard specification provide values for m; for steels m ranges between 3 and 5.

It is common practice to factor σ_a,i in Eq. [10.25] for the load, material resistance, and consequence of failure PSFs to arrive at a “factored” N_i value (

{\hat{N}}_{i}

) as shown in Eq. [10.16]. Hence, Eq. [10.24] can be replaced by Eq. [10.15] and Eq. [10.25] rewrites as:

$σ_{a, i} = \frac{C {\hat{N}}_{i}^{- 1 / m}}{γ_{f} γ_{n} γ_{m}}$

[10.25a]

Note that in Eq. [10.25a], a load factor γ_f potentially different from 1 is included. This means that the calculated fatigue loads that will complement the analysis are considered unfactored.

To account for the presence of non-zero mean loads (hence stresses), Eq. [10.26] (known as linear Goodman's correction) can be used:

$σ_{eq, i} = σ_{a, i} \frac{σ_{u}}{σ_{u} - σ_{m, i}}$

[10.26]

where σ_eq,_i is the i-th bin equivalent stress range after Goodman's correction, and that can replace σ_a,i in Eq. [10.25a]; σ_u is the ultimate strength; and σ_m,_i is the i-th bin stress-range mean.

Figure 10.23 σ–N curves for various detail categories. From European Committee for Standardisation, Eurocode 3: Design of Steel Structures—Part 1–9: Fatigue, 2005.

In order to compute each n_i, the loads output by the AHSE simulations for the various DLCs are to be binned and combined together via their probability of occurrence, which is based on the expected site joint-probability of wind and wave distributions and best estimates of system availability. In the case of an MP tower, additional damage may be due to pile-driving actions and should be added to the fatigue budget reckoning. Through load binning and a rainflow cycle-counting method [23,87], a representation of load-ranges (S_a,i) versus number of cycles (n_i) can be achieved. However, the direct use of Eq. [10.24] (or better Eq. [10.15]) would require the transformation of each load range into stress ranges (σ_a,i). Material non-linearities may require multiple FEA runs to achieve this transformation. It may be more efficient to employ damage equivalent loads (DELs). For a generic load component, the DEL represents the zero-mean load range that, if applied for a select number of cycles, would yield the same damage as that produced by the application of the actual load cycles. In other words, the DEL is a measure of the damage accumulated in a specific structural part due to the oscillating loads that can be rainflow-counted from the FLS DLC outputs and combined as described above for the lifetime of the OWT. Therefore, by equating the actual and the equivalent damages as expressed in Eq. [10.24], and making use of Eq. [10.25] where the stress symbols are used to indicate loads, it can be shown that:

$DEL = {(\frac{n_{DEL}}{\sum_{i} \frac{n_{i}}{s_{a, i}^{- m}}})}^{\frac{- 1}{m}}$

[10.27]

where n_DEL is the reference number of cycles for the DEL load range; and s_a,i is the i-th bin load range.

The DELs can finally be transformed into damage equivalent stresses (DESs) via detailed FEAs, which render a 3D description of the stress field, including stress concentrations and hot-spots. The obtained DES (or, better yet, the σ–N curve that it will refer to) should be modified by the same knockdown factors on the load, material resistance due to quality of construction, importance of the detail, and material characterization as shown in Eq. [10.25a]. Additional PSFs may be included based on expected fatigue corrosion rates (see Section 10.6.1). As discussed in Section 10.5.1.1, load, material resistance, and consequence of failure PSFs are recommended by the primary OWT standards (eg, Refs [23,1]), while further knockdown factors for specific elements (such as welds and flanges) are directly incorporated in the σ–N curves (eg, detail category curves as those in Fig. 10.23) given by the appropriate codes of reference (eg, Refs [30,39,44]; see also Section 10.3).

The DES, together with Eq. [10.25a], yields the number of cycles at failure for the DES stress range (N_DEL), and D_fat can be calculated as:

$D_{fat} = \sum_{i} \frac{n_{i}}{{\hat{N}}_{i}} = \frac{n_{DEL}}{N_{DEL}}$

[10.28]

Eq. [10.28] may be used to readily calculate the FLS material utilization at various stations along the tower and in the welds, in the door reinforcements, and in the bolted connections.

For the ULS verification, data obtained from AHSE simulations are further post-processed to arrive at ultimate loads. For operational DLCs, this process may require extrapolation of stochastically based data as directed by the standards [1,23]. The range of DLCs to investigate includes fault transients, extreme gusts, wind directional changes, as well as emergency stops, and possible shocks from ship impacts and sea ice as discussed in Section 10.4.1.5.

ULS structural checks amount to ensuring that the material utilization be below 1, and are based on the verification of steel members under compression-bending. In summary, one should prove that, for the various tower segments, the stresses are kept below the allowable yield strength and that stability at a global and local level is guaranteed. These constraints can be expressed by the following equations [3,2]:

$σ_{vm} \leq \frac{f_{y}}{γ_{m} γ_{n}}$

[10.29]

$\frac{N_{d}}{κ N_{p}} + \frac{β_{m} M_{d}}{M_{p}} + Δ_{n} \leq 1$

[10.30]

$σ_{z, Ed} \leq σ_{z, Rd}$

[10.31a]

$σ_{θ, Ed} \leq σ_{θ, Rd}$

[10.31b]

$τ_{z θ, Ed} \leq τ_{z θ, Rd}$

[10.31c]

${(\frac{σ_{z, Ed}}{σ_{z, Rd}})}^{k_{z}} + {(\frac{σ_{θ, Ed}}{σ_{θ, Rd}})}^{k_{θ}} - (\frac{σ_{z, Ed} σ_{θ, Ed}}{σ_{z, Rd} σ_{θ, Rd}}) k_{i} + {(\frac{τ_{z θ, Ed}}{τ_{z θ, Rd}})}^{k_{τ}} \leq 1$

[10.31d]

Eq. [10.29] states the constraint on the yield resistance of the material (f_y).

Eq. [10.30] is the global (Eulerian) buckling constraint; where N_d and M_d are the design axial and bending moment loads; N_p and M_p are the associated characteristic, buckling critical, resistance values; κ and β_m are reduction factors for flexural buckling and bending moment coefficient, respectively [3]; Δ_n is a function of the slenderness (

\bar{λ}

) of the tower as in Eq. [10.32]:

$Δ_{n} = 0.25 κ {\bar{λ}}^{2}$

[10.32]

Eq. [10.31] is the local (shell) buckling constraint; σ_z,Ed, σ_θ,Ed, τ_zθ,Ed are the design values of the axial, hoop, and shear stress, respectively; σ_z,Rd, σ_θ,Rd, τ_zθ,Rd are the corresponding characteristic buckling strengths; k_z, k_θ, k_τ, and k_i are constants given by the standards [2].

In the calculation of the resistance values in Eqs. [10.30]–[10.31], the standards of reference offer guidance on the selection of manufacturing quality factors.

In the case of MP towers, the stability of the pile at the seabed must be verified. This is done for the largest (ULS) pile-head loads, where head displacement and rotation are checked against allowable values from the standards. Additionally, the embedment length must develop sufficient vertical friction to counteract the maximum normal force. These checks are commonly performed by the foundation engineering group. That group is also responsible for ensuring sufficient stiffness in the SSI superelement to provide the necessary boundary condition to the SSt.

In addition to FLS and ULS checks, an important structural check for both blade and tower design is the verification of potential blade strike on the tower, which may be considered part of SLS. Blade maximum deflection may occur under a transient event as in the case of extreme gusts or fault situations. In order to verify that a safety margin remains in the deflection of the blade before tower strike, the maximum deflection across all ULS DLCs must be determined. Standards such as Refs [23,15] offer guidance on the PSFs to employ in this calculation: Ref. [23] uses the same load PSF as for any other ULS DLC, whereas Ref. [15] states that the blade clearance shall not be less than 30% of the value under unloaded (at rest) conditions.

For other types of towers, such as lattice and concrete towers, additional structural checks are necessary. For lattices, each member and each joint must be assessed under all DLCs (eg, against structural constraints described in Ref. [19]), which is a critical and time-consuming activity. For concrete towers, for all ULS DLCs, one should verify: concrete's compressive strength; reinforcement's bending strength; punching, anchorage, and pull-out strength of the reinforcement; and the shear strength of structural members with and without shear reinforcement. For SLS and FLS, analyses should assess deformation limits, crack width limits, and stresses on the prestressed members. Possible guideline standards are listed in Table 10.1, but it is to be noted that the high load cycle number fatigue for prestressed concrete structures is still a focus of significant research.

10.5.3. Flanges and main detail components

Flanges and other detail components make up an important portion of the structural performance and functionality of the tower. Bolted connections and weldments are among the most important details as they assure the structural integrity and safety of the structure. Just as important are the details of door and man-hole reinforcements, and of the brackets for supporting the internals and secondary steel. Secondary steel is discussed in Section 10.6.

In general, cracks in welded steel structures almost certainly depart from welds. The reason is that the welding process inevitably leaves behind microscopic defects from which cracks may grow. As a result, one should assume the existence of microcracks, and that most of the fatigue cycles will contribute to the growth of the crack rather than to its nucleation. Additionally, welds are characterized by changes in surface slope, as at the toes of butt welds and at the toes and roots of fillet welds. These areas are characterized by important stress concentration factors (SCFs).

The door reinforcement is also a location of high SCF. In order to restore the structural strength of the tower at the door opening, either a welded reinforcement ring (stiffener), or an increased shell wall thickness, or a combination of the two is commonly utilized. The reinforcement has the double goal of controlling the local stresses, and of supporting the main shell against buckling. Structural codes (eg, Refs [30,39]) describe fatigue curves that can be used for general design and are based on test results of actual characteristic weld shapes (often referred to as detail categories). However, especially for the door opening area, actual stresses and hot-spots near welds need to be assessed via accurate FEA analyses (eg, Fig. 10.24).

Ref. [2] provides guidance on extrapolating “hot-spot” stresses, and discusses the complex geometrically and materially non-linear analyses with imperfection modes (GMNIA) process to assess buckling resistance. Weakening of a structural weld can significantly reduce the overall buckling strength. Therefore fatigue design of the connections between shells, and between flanges and shells, is critical to the lifetime integrity of the OWT, and indirectly even on its ULS capacity.

Additionally, the effect of a corrosive environment can accelerate crack propagation and effectively increase fatigue damage (see Section 10.6.1) around the connection details. Corrosion-fatigue of bolts may contribute to stiffness degradation with the previously mentioned problems of dynamic response in undesirable frequency bands (see Section 10.4.1.2).

Figure 10.24 Examples of FEA of the tower door opening: (a) a mesh of the lower tower segment; (b) results of the analysis verifying shell buckling around the door region. From M. Veljkovic, C. Heistermann, W. Husson, M. Limam, M. Feldmann, J. Naumes, D. Pak, T. Faber, M. Klose, K. Fruhner, L. Krutschinna, C. Baniotopoulos, I. Lavasas, A. Pontes, E. Ribeiro, M. Hadden, R. Sousa, L. da Silva, C. Rebelo, R. Simoes, J. Henriques, R. Matos, J. Nuutinen, H. Kinnunen, High-Strength Tower in Steel for Wind Turbines (Histwin). Final Report EUR 25127 EN. Directorate-General for Research and Innovation – European Commission, Luxembourg, Europe, 2012, Contract No RFSR-CT-2006–00031.

Tubular tower flanges connect tower segments together and the whole tower to the TP, or foundation in the case of onshore installations. They make up a significant contribution to the material and manufacturing costs. Flanges and bolts need to be checked against FLS and ULS limit states. A common way to analyze the bolted flanges follows the well-known Petersen method [89], or “segment-model.” In that model, the flange connection is replaced by an individual circular segment, where a single bolt is considered with the associated collaborating flange and shell regions. The new unit connection is loaded with the equivalent normal force (F_z) deriving from the shell normal stresses as shown in Fig. 10.25 and Eq. [10.33].

$F_{z} = \frac{4 M_{d}}{n_{b} D_{sh,m}} - \frac{N_{d}}{n_{b}}$

[10.33]

where M_d and N_d are the factored bending moment and the factored normal load at the flange elevation; n_b is the number of bolts in the flange connection; and D_sh,m is the shell mid-wall diameter. Because N_d acts in a favorable direction, it is usual practice to assign γ_fg = 0.9 to the gravitational loads that make up N_d. Eq. [10.33] is easily derived considering trigonometry rules, the symmetry in the connection, and a simple moment balance about a horizontal axis.

Figure 10.25 Reduction of the flange bolted connection to an equivalent, unit–bolt connection (flange segment model). Modified from P. Schaumann, M. Seidel, Failure analysis of bolted steel flanges, in: X. Zhao, R. Grzebieta (Eds.), Proceedings of the 7th International Symposium on-Structural Failure and Plasticity (IMPLAST 2000), Melbourne, Australia, 2000.

The connection capacity is calculated through the plastic hinge theory considering the three failure methods as shown in Fig. 10.26 (ie, bolt rupture, bolt rupture and plastic hinge in shell, plastic hinges in shell and flange), and as described by Eqs. [10.34]–[10.36]:

$F_{u,A} = F_{t,RD}$

[10.34a]

Figure 10.26 Approximation of main failure modes for flange connections. Adapted from P. Schaumann, M. Seidel, Failure analysis of bolted steel flanges, in: X. Zhao, R. Grzebieta (Eds.), Proceedings of the 7th International Symposium on- Structural Failure and Plasticity (IMPLAST 2000), Melbourne, Australia, 2000.

$F_{u,B} = \frac{F_{t,RD} a + M_{Pl, 3,MN}}{a + b}$

[10.34b]

$F_{u,C} = \frac{M_{Pl, 2} + M_{Pl, 3,MN}}{b}$

[10.34c]

with

$F_{t,RD} = \min (\frac{f_{y,b} A_{b}}{γ_{Mb,y}}, \frac{0.9 f_{u,b} A_{b}}{γ_{Mb,u}})$

[10.35]

where a and b are distances from the bolt centerline to the flange edge and the shell mid-wall; F_t,RD is the bolt strength load; M_Pl,3,MN and M_Pl,2 are the shell and flange plastic bending resistances, respectively; F_u,A, F_u,B, and F_u,C are the ultimate loads for the three different failure modes. Note that M_Pl,3,MN accounts for tension bending interaction.

The M_Pl,2 and M_Pl,3,MN plastic hinge moments are given by:

$M_{Pl, 2} = \frac{c_{f} t_{f}^{2}}{4} \frac{f_{y,f}}{γ_{Mf,y}}$

[10.36a]

$M_{Pl, 3} = \frac{c_{s} t_{s}^{2}}{4} \frac{f_{y,s}}{γ_{Ms,y}}$

[10.36b]

$M_{Pl, 3,MN} = [1 - {(\frac{F_{ult}}{N_{Pl, 3}})}^{2}] M_{Pl, 3}$

[10.36c]

where M_Pl,3 is the simple plastic bending resistance, without tension bending interaction; t_f is the flange thickness; f_y,f is the flange characteristic yield strength; γ_Mf,y is the material PSF for the flange characteristic yield strength; t_s is the shell thickness; f_y,s is the shell characteristic yield strength; γ_Ms,y is the material PSF for the shell characteristic yield strength; and N_Pl,3, the plastic resistance of the shell, is given by:

$N_{Pl, 3} = \frac{c_{s} t_{s} f_{y,s}}{γ_{Ms,y}}$

[10.37]

In Eqs. [10.36a]–[10.37], c_s and c_f are defined by:

$c_{s} = \frac{π D_{sh,m}}{n_{b}}$

[10.38a]

$c_{f} = \frac{π D_{bc}}{n_{b}} - d_{b}$

[10.38b]

with D_bc the bolt circle diameter in the flange connection; and d_b the bolt hole diameter.

From Eq. [10.36c], it can be seen how M_Pl,3,MN introduces a non-linearity that requires some iterations to get to the final connection resistance F_ult, which is the minimum of the expressions obtained through Eq. [10.34]. The connection is thus verified if:

$F_{z} \leq F_{ult} = \min (F_{u,A}, F_{u,B}, F_{u,C})$

[10.39]

Bolt stresses vary non-linearly as a function of shell loads due to the preload. Petersen's method assumes a simplified bolt-load behavior as shown in Fig. 10.27(a), and as described by the following equations:

$F_{t 1} = F_{p} + K_{α} λ_{f} F_{z} for F_{z} < F_{z cr}$

[10.40a]

$F_{t 2} = λ_{f} F_{z} for F_{z} \geq F_{z cr}$

[10.40b]

where

$λ_{f} = \frac{a + b}{a}$

[10.40c]

$F_{z cr} = \frac{F_{p}}{λ_{f} K_{β}}$

[10.40d]

where F_p is the bolt preload, and K_α and K_β are the fractions of the total stiffness for bolt and flanges, respectively:

Figure 10.27 Bolt load (F_t) as a function of shell load (F_z): (a) experiments (dashed curve) and approximated trend according to Petersen [89]; (b) experiments (dashed curve) and approximated trend according to Schmidt/Neuper's method [93] with the three linear regions.

$K_{α} = \frac{K_{b}}{K_{b} + K_{f}}$

[10.41a]

$K_{β} = \frac{K_{f}}{K_{b} + K_{f}}$

[10.41b]

The bolt axial stiffness (K_b) can be written as:

$K_{b} = {(\int_{0}^{L_{b}} \frac{1}{E_{b} A_{b}} d z)}^{- 1} ≃ \frac{E_{b} A_{b}}{2 t_{f}}$

[10.42]

where L_b, ie, the bolt effective length, can be approximated by 2t_f. The equivalent stiffness of the compressed flange pair (K_f) can be written as:

$K_{f} = {(2 \int_{0}^{t_{f}} \frac{1}{E_{f} A_{f}} d z)}^{- 1}$

[10.43]

where A_f is the effective cross-sectional area for the compressed flanges. K_f is sometimes taken as three to six times the K_b, or approximated as [91]:

$K_{f} = \frac{E_{f} A_{f}}{2 t_{f}} ≃ \frac{E_{f} (d_{b}^{2} + 1.36 d_{b} t_{f} + 0.26 t_{f}^{2})}{2 t_{f}}$

[10.44]

Other standards and literature, such as Refs [40,92], can also be consulted to determine the A_f and thus K_f.

Petersen's approach is considered adequate for ULS assessments. Compared to experiments (see Fig. 10.27(a)), the method is conservative below F_zcr, but, because of the assumed ideal-elastic and pure edge-bearing behavior, could become unsafe in the case of ring-flanges with imperfections at and above F_zcr. For more accurate FLS verifications, Schmidt/Neuper's method [93] may be used. That method uses a three-region, piecewise-linear function to approximate the bolt load as shown in Fig. 10.27(b) and as described by Eq. [10.45].

$F_{t 1} = F_{p} + K_{α} F_{z} for F_{z} \leq F_{z 1}$

[10.45a]

$F_{t 2} = F_{p} + K_{α} F_{z 1} + (λ_{f}^{*} F_{z 2} - (F_{p} + K_{α} F_{z 1})) \frac{F_{z} - F_{z 1}}{F_{z 2} - F_{z 1}} for F_{z 1} < F_{z} \leq F_{z 2}$

[10.45b]

$F_{t 3} = λ_{j}^{*} F_{z} for F_{z} \geq F_{z 2}$

[10.45c]

where

$F_{z 1} = F_{p} \frac{a - 0.5 b}{a + b}$

[10.45d]

$F_{z 2} = F_{p} \frac{1}{λ_{f}^{*} K_{β}}$

[10.45e]

$λ_{f}^{*} = \frac{0.7 a + b}{0.7 a}$

[10.45f]

As seen in Fig. 10.27, Schmidt/Neuper's method is less conservative than Petersen's for low shell forces F_z. The third part of the function, which considers the final edge-bearing behavior of the flange connection (F_z ≥ F_z2), is however devised to account for the effects of small imperfections. The range of applicability of Schmidt/Neuper's method is given by:

$\frac{a + b}{t_{f}} \leq 3$

[10.46]

Neither of these methods considers bending stresses in the bolt; therefore caution is advised in the selection of the fatigue detail curve and a final FEA is recommended.

For a state of the art in flange design consult Ref. [94] and later work by the same author.

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Table of Contents for 10.5. Shell and flange sizing

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10.5. Shell and flange sizing

10.5.1. Load resistance factored design: FLS, ULS, and SLS verifications

10.5.1.1. Safety factors

10.5.2. Approximate derivation of structural loads and shell design

10.5.3. Flanges and main detail components

Table of Contents for
10.5. Shell and flange sizing