Laminar flow instability in multiple channels

Temperature-viscosity-induced laminar flow instability (LFI) in two gaseous heated parallel channels with interchannel heat exchange is purely excursive, rather than oscillatory. Constant total flow always leads to a stable system, while constant pressure drop could have an instability, depending on the sign of (∂ΔP/∂W)_Q. The system was studied numerically with negative heat perturbations yielding bounded excursions, and positive heat perturbations giving unbounded excursions asymptotically approaching zero. A nine-channel system was probed, giving excursive behavior with the ultimate growth rate the same for single and multichannel systems.     

Characteristics of Cocurrent Two-Phase Downflow in Tubes

An experimental study was made in a cocurrent downflow for air-water system in tube on flow pattern, void fraction and pressure drop. In addition to wetted wall flow which is a distinguished feature in downflow, the same kinds of flow patterns as in up-flow were observed. They were represented on the flow map of downflow with the same variables as those of upflow. The flow maps showed that gas phase is relatively hard to exist in the form of bubble in the cocurrent downflow.

General correlations of the void fraction and the pressure drop in a cocurrent down-flow were obtained by applying the following equations which have been established in upflow, i.e. α/(1-α)(1-Kα)=β/(1-β) for void fraction and φl=(1.α)- ^z for pressure drop. The determined values of K and Z by using the experimental results in the present study and other experimental works in cocurrent downflow wereK=2.0–0.4/β for β≤0.2, K=-0.25+1.250 for β≥0.2 and Z=0.90.

Comparisons between the predicted values by the presented correlations and experimental data showed satisfactory agreement.     

Study on the characteristic points of boiling curve by using wavelet analysis and genetic neural network

Local singularity of a signal includes a lot of important information. Wavelet transform can overcome the shortages of Fourier analysis, i.e., the weak localization in the local time- and frequency-domains. It has the capacity to detect the characteristic points of boiling curves. Based on the wavelet analysis theory of signal singularity detection, Critical Heat Flux (CHF) and Minimum Film Boiling Starting Point (q_min) of boiling curves can be detected by using the wavelet modulus maxima detection. Moreover, a genetic neural network (GNN) model for predicting CHF is set up in this paper. The database used in the analysis is from the 1960s, including 2365 data points which cover a range of pressure (P), from 100 to 1000 kPa, mass flow rate (G) from 40 to 500 kg m⁻² s⁻¹, inlet sub-cooling (ΔT_sub) from 0 to 35 K, wall superheat (ΔT_sat) from 10 to 500 K and heat flux (Q) from 20 to 8000 kW m⁻². GNN mode has some advantages of its global optimal searching, quick convergence speed and solving non-linear problem. The methods of establishing the model and training of GNN are discussed particularly. The characteristic point predictions of boiling curve are investigated in detail by GNN. The results predicted by GNN have a good agreement with experimental data. At last, the main parametric trends of the CHF are analyzed by applying GNN. Simulation and analysis results show that the network model can effectively predict CHF.     

Comparison of experimental and finite element analytical results for the strength and the deformation of pipes with local wall thinning subjected to bending moment

Fracture behaviors of pipes with local wall thinning are very important for the integrity of power plant piping system. In this study, monotonic bending tests without internal pressure are conducted on 48.6 mm diameter Schedule 80 (thickness, 5.1 mm) STS370 full-scale carbon steel pipe specimens. Fracture strengths of locally wall-thinned pipes were calculated by elasto-plastic analysis using finite element method. The elasto-plastic analysis was performed by FE code ANSYS. We simulated various types of local wall thinning that can be occurred at pipe surface due to coolant flow. Locally wall thinned shapes were machined to be different in size along the circumferential or axial direction of straight pipes. We investigated fracture strengths and failure modes of locally wall thinned pipes by four point bending test. And, the allowable limit of pipes with local wall thinning was investigated. In addition, we compared the simulated results by finite element analysis with experimental data. The failure mode, fracture strength and fracture behavior obtained from FE analyses showed well agreement with experimental results. From the test results, we identified three types of failure modes into ovalization, local buckling and crack initiation. These failure modes could be classified according to thinned depth, thinned length and thinned angle of a pipe. For locally wall-thinned specimens, maximum moments (M_max) were estimated by using the net-section stress criterion. Pipes with local wall thinning can be estimated using σ_u instead of σ_f because of 1.19σ_f  σ_u. Also, the axial strain affects failure modes occurred on local wall thinning. the allowable limit of local wall thinning for carbon steel pipe used can be given as follows; in the case of M_max ≥ M_y, if 10 ≤ l < 25 mm, d/t can be allowed to about 55%, and if 25 ≤ l < 100 mm, d/t can be allowed to about 50%. Also, if 100 ≤ l ≤ 120 mm, d/t can be allowed to about 29%.     

Applications of Artificial Neural Network for the Prediction of Flow Boiling Curves

An artificial neural network (ANN) was applied successfully to predict flow boiling curves. The databases used in the analysis are from the 1960's, including 1,305 data points which cover these parameter ranges: pressure P=100–1,000 kPa, mass flow rate G=40–500 kg/m²-s, inlet subcooling ΔT_sub =0–35°C, wall superheat ΔT_w = 10–300°C and heat flux Q=20–8,000kW/m². The proposed methodology allows us to achieve accurate results, thus it is suitable for the processing of the boiling curve data. The effects of the main parameters on flow boiling curves were analyzed using the ANN. The heat flux increases with increasing inlet subcooling for all heat transfer modes. Mass flow rate has no significant effects on nucleate boiling curves. The transition boiling and film boiling heat fluxes will increase with an increase in the mass flow rate. Pressure plays a predominant role and improves heat transfer in all boiling regions except the film boiling region. There are slight differences between the steady and the transient boiling curves in all boiling regions except the nucleate region. The transient boiling curve lies below the corresponding steady boiling curve.     

CFD investigation of vertical rod bundles of supercritical water-cooled nuclear reactor

The commercial CFD code STAR-CD v4.02 is used as the numerical simulation tool for the supercritical water-cooled nuclear reactor (SCWR). The numerical simulation is based on the real full 3D rod bundles’ geometry of the nuclear reactors. For satisfying the near-wall resolution of y⁺ ≤ 1, the structure mesh with the stretched fine mesh near wall is employed. The validation of the numerical simulation for mesh generation strategy and the turbulence model for the heat transfer of supercritical water is carried out to compare with 3D tube experiments. After the validation, the same mesh generation strategy and the turbulence model are employed to study three types of the geometry frame of the real rod bundles. Through the numerical investigations, it is found that the different arrangement of the rod bundles will induce the different temperature distribution at the rods’ walls. The wall temperature distributions are non-uniform along the wall and the values depend on the geometry frame. At the same flow conditions, downward flow gets higher wall temperature than upward flow. The hexagon geometry frame has the smallest wall temperature difference comparing with the others. The heat transfer is controlled by P/D ratio of the bundles.     

Monte-Carlo calculation of the calibration factors for the interfacial area concentration and the velocity of the bubbles for double sensor conductivity probe

This paper presents a study of the estimation of the correction factors for the interfacial area concentration and the bubble velocity in two phase flow measurements using the double sensor conductivity probe. Monte-Carlo calculations of these correction factors have been performed for different values of the relative distance (ΔS/D) between the tips of the conductivity probe and different values of the relative bubble velocity fluctuation parameter. Also this paper presents the Monte-Carlo calculation of the expected value of the calibration factors for bubbly flow assuming a log-normal distribution of the bubble sizes. We have computed the variation of the expected values of the calibration factors with the relative distance (ΔS/D) between the tips and the velocity fluctuation parameter. Finally, we have performed a sensitivity study of the variation of the average values of the calibration factors for bubbly flow with the geometrical standard deviation of the log-normal distribution of bubble sizes. The results of these calculations show that the total interfacial area correction factor is very close to 2, and depends very weakly on the velocity fluctuation, and the relative distance between tips. For the velocity calibration factor, the Monte-Carlo results show that for moderate values of the relative bubble velocity fluctuation parameter (H_max ≤ 0.3) and values of the relative distance between tips not too small (ΔS/D ≥ 0.2), the correction velocity factor for the bubble sensor conductivity probe is close to unity, ranging from 0.96 to 1.     

Experimental investigation of the two-phase flow distribution in the outlets of a horizontal multi-branch header

An experimental investigation was conducted to study the two-phase flow distribution in a horizontal header with two inlet turrets and 30 (six banks of five) outlet branches. Tests were performed using air–water mixtures at room temperature and a nominal header pressure of 170.3 kPa. The test matrix included one- and two-turret injection, two inlet water flow rates and four different air flow rates for each water flow rate, giving four different inlet qualities. The outlet flow rates of air and water were measured in all the branches under the condition of equal pressure drop across all branches. The data show that there is significant variation in air and water flow rates among the branches, both in the axial and circumferential directions. The flow distribution among the branches was found to be strongly dependent on the inlet flow rates of air and water, and the type of injection (one or two turrets).     

Droplet entrainment correlation in vertical upward co-current annular two-phase flow

Upward annular two-phase flow in a vertical tube is characterized by the presence of liquid film on the tube wall and entrained droplet laden gas phase flowing through the tube core. Entrainment fraction in annular flow is defined as a fraction of the total liquid flow flowing in the form of droplets through the central gas core. Its prediction is important for the estimation of pressure drop and dryout in annular flow. In the following study, measurements of entrainment fraction have been obtained in vertical upward co-current air–water annular flow covering wide ranges of pressure and flow conditions. Comparison of the experimental data with the existing entrainment fraction prediction correlations revealed their inadequacies in simulating the trends observed under high flow and high pressure conditions. Furthermore, several correlations available in the literature are implicit and require iterative calculations.Analysis of the experimental data showed that the non-dimensional numbers, Weber number (We = ρ_gj_g²D/σ(Δρ/ρ_g)^1/4) and liquid phase Reynolds number (Re_f = ρ_fj_fD/μ_f), successfully collapse the data. In view of this, simple, explicit correlation was developed based on these non-dimensional numbers for the prediction of entrainment fraction. The new correlation successfully predicted the trends under the high flow and high pressure conditions observed in the current experimental data and the data available in open literature. However, in order to use the proposed correlation it is necessary to predict the maximum possible entrainment fraction (or limiting entrainment fraction). In the current analysis, an experimental data based correlation was used for this purpose. However, a better model or correlation is necessary for the maximum possible entrainment fraction. A theoretical discussion on the mechanism and modeling of the maximum possible entrainment fraction condition is presented.     

Turbulent heat mixing of a heavy liquid metal flow in the MEGAPIE target geometry—The heated jet experiment

The MEGAPIE target installed at the Paul–Scherrer Institute is an example of a spallation target using eutectic liquid lead–bismuth (Pb⁴⁵Bi⁵⁵) both as coolant and neutron source. An adequate cooling of the target requires a conditioning of the flow, which is realized by a main flow transported in an annular gap downwards, u-turned at a hemispherical shell into a cylindrical riser tube. In order to avoid a stagnation point close to the lowest part of the shell a jet flow is superimposed to the main flow, which is directed towards to the stagnation point and flows tangentially along the shell.The heated jet experiment conducted in the THEADES loop of the KALLA laboratory is nearly 1:1 representation of the lower part of the MEGAPIE target. It is aimed to study the cooling capability of this specific geometry in dependence on the flow rate ratio (Q_main/Q_jet) of the main flow (Q_main) to the jet flow (Q_jet). Here, a heated jet is injected into a cold main flow at MEGAPIE relevant flow rate ratios. The liquid metal experiment is accompanied by a water experiment in almost the same geometry to study the momentum field as well as a three-dimensional turbulent numerical fluid dynamic simulation (CFD). Besides a detailed study of the envisaged nominal operation of the MEGAPIE target with Q_main/Q_jet = 15 deviations from this mode are investigated in the range from 7.5 ≤ Q_main/Q_jet ≤ 20 in order to give an estimate on the safe operational threshold of the target.The experiment shows that, the flow pattern establishing in this specific design and the turbulence intensity distribution essentially depends on the flow rate ratio (Q_main/Q_jet). All Q_main/Q_jet-ratios investigated exhibit an unstable time dependent behavior. The MEGAPIE design is highly sensitive against changes of this ratio.Mainly three completely different flow patterns were identified. A sufficient cooling of the lower target shell, however, is only ensured if Q_main/Q_jet ≤ 12.5. In this case the jet flow covers the whole lower shell. Although for Q_main/Q_jet ≤ 12.5 the flow is more unstable compared to the other patterns most of the fluctuations close to the centerline are in the high frequency range (>1 Hz), so that they will not lead to severe temperature fluctuations in the lower shell material. In this case the thermal mixing occurs on large scales and is excellent.For flow rate ratios Q_main/Q_jet > 12.5 complex flow patterns consisting of several fluid streaks and vortices were identified. Since in these cases the jet flow does not fully cover the lower shell an adequate cooling of the MEGAPIE target cannot be guaranteed and thus temperatures may appear exceeding material acceptable limits.All conducted experiments show a high sensitivity to asymmetries even far upstream. A comparison of the numerical simulation, which assumed a symmetric flow, with the experimental data was due to the experimentally found asymmetry only partially possible.     

Axial static pressure variations in inner and side subchannels of a 61-tube wire-wrap bundle

Measurements of axial distribution of the static pressure in an inner and side subchannel of a 61 wire-wrap tube bundle obtained with water at atmospheric conditions are presented. The wire wrap configuration is different from those used by previous workers and more representative of a bundle for the blanket of a Gas Cooled Fast Reactor. The data display axial static pressure variations which are attributed to the interchannel cross flow induced by the wire-wrap configuration. The static pressure drop over one wire pitch agrees well with the bundle pressure drop based on a bundle average Reynolds number and a friction factor f = 0.436 Re^−0.263 (Re > 2000). The experimental data obtained with water provide a useful benchmark to model and check the accuracy of thermal-hydraulic codes used for the analysis of subchannel flow distribution and pressure drop in wire wrap tube bundle cooled with one-phase fluid.The nodal subchannel code COBRA-IV was modeled by adjusting the forced cross-flow function to match the measured axial static pressure distribution in an inner and side subchannel. Some discrepancy remained in the static pressure profile in the side channel attributed to the flow distortion at the bundle exit.     

Verification of rod safety via confidence intervals for a binomial proportion

The probabilistic safety assessed to a set of N fuel rods assembled in one core of a nuclear power reactor is commonly modelled by ∑_i≤N X_i, where X₁, …, X_N are independent Bernoulli random variables (rv) with individual probability p_i = P (X_i = 1) that the ith rod shows no failure during one cycle. This is the probability of the event that the ith rod will not exceed the failure limit during one cycle. The safety standard presently set by the German Reaktor-Sicherheitskommission (Reactor Safety Commission) requires that the expected number of unfailed rods in the core during one cycle is at least N − 1, i.e., E(∑_i≤N X_i) = ∑_i≤N p_i ≥ N − 1, whereby a confidence level of 0.95 for the verification of this condition is demanded. In this paper, we provide an approach, based on the Clopper–Pearson confidence interval for the proportion p of a binomial B(n, p) distribution, how to verify this condition with a confidence level of at least 0.95. We extend our approach to the case, where the set of N fuel rods is arranged in strata, possibly due to different design in each stratum.     

Simplified method to evaluate upper limit stress intensity factor range of an inner-surface circumferential crack under steady state thermal striping

Simplified method to evaluate the upper limit stress intensity factor (SIF) range of an inner-surface circumferential crack in a thin- to thick-walled cylinder under steady state thermal striping was considered in this paper. The edges of the cylinder were rotation-restrained and the outer surface was adiabatically insulated. The inner surface of the cylinder was heated by a fluid with constant heat transfer coefficient whose temperature fluctuated sinusoidally at constant amplitude ΔT. By combining our analytical temperature solution for the problem and our semi-analytical-numerical SIF evaluation method for the crack, we showed that the desired maximum steady state SIF range can be evaluated with an engineering accuracy after ΔT, the mean radius to wall thickness ratio r_m/W of the cylinder, the thermal expansion coefficient and Poisson's ratio are specified. By applying our method, no transient SIF analysis nor sensitivity analysis of the striping frequency on the SIF range is necessary. Numerical results showed that our method is valid for cylinders in a range of r_m/W = 10–1.     

An experimental investigation on thermal striping: Mixing phenomena of a vertical non-buoyant jet with two adjacent buoyant jets as measured by ultrasound Doppler velocimetry

An experimental investigation on the thermal mixing phenomena of three quasi-planar vertical jets, with the central jet at a lower relative temperature than the two adjacent jets, was conducted. The central jet was unheated (‘cold’), while the two adjacent jets were heated (‘hot’). The temperature difference and velocity ratio between the heated (h) and unheated (c) jets were, ΔT_hc=5°C, 10°C and r=V_cold,exit/V_hot,exit=1.0 (isovelocity), 0.7, 0.5 (non-isovelocity) respectively. The typical Reynolds number was Re_D=1.8×10⁴, where D is the hydraulic diameter of the exit nozzle. Velocity measurement of a reference single-jet and triple-jet arrangement were taken by ultrasound Doppler velocimetry (UDV) while temperature data were taken by a vertically traversed thermocouple array. Our UDV data revealed that, beyond the exit region, our single-jet data behaved in the classic manner. In contrast, the triple-jet exhibited, for example, up to 20 times the root-mean-square velocity values of the single-jet, especially in the regions in-between the cold and hot jets. In particular, for the isovelocity case (V_exit=0.5 m/s) with ΔT_hc=5°C, we found that the convective mixing predominantly takes place at axial distances, z/D=2.0–4.5, over a spanwise width, x/D|2.25|, centered about the cold jet. An estimate of the turbulent heat flux distribution semi-quantitatively substantiated our results. As for the non-isovelocity case, temperature data showed a localized asymmetry that subsequently delayed the onset of mixing. Convective mixing however, did occur and yielded higher post-mixing temperatures in comparison to the isovelocity case.     

Film boiling heat transfer from a vertical cylinder in forced flow of liquids under saturated and subcooled conditions at pressures

Forced convection film boiling heat transfer on a vertical 3-mm diameter and 180-mm length platinum test cylinder located in the center of the 40-mm inner diameter test channel was measured. Saturated water, and saturated and subcooled R113 were used as the test liquids that flowed upward along the cylinder in the test channel. Flow velocities ranged from 0 to 3 m s⁻¹, pressures from 102 to 490 kPa, and liquid subcoolings for R113 from 0 to 60 K. The heat transfer coefficients for a certain pressure and liquid subcooling are almost independent of flow velocity and of a vertical position on the cylinder for the flow velocities lower than ≈1 m s⁻¹ (the first range), and they become higher for the velocities higher than ≈1 m s⁻¹ (the second range). Slight dependence on a vertical position being nearly proportional to z^−1/4, where z is the height from the leading edge of the test cylinder, exists for the flow velocities in the second range. The heat transfer coefficients at each velocity in the first and second ranges are higher for higher pressure and liquid subcooling. Correlation for the forced convection film boiling heat transfer with radiation contribution on a vertical cylinder was derived by modifying an approximate analytical solution for a two-phase laminar boundary layer model to agree better with the experimental data. It was confirmed that the experimental data of film boiling heat transfer coefficients in water and R113 were described by the correlation within ±20% difference.     

An utilization of liquid sublayer dryout mechanism in predicting critical heat flux under low pressure and low velocity conditions in round tubes

From a theoretical assessment of extensive critical heat flux (CHF) data under low pressure and low velocity (LPLV) conditions, it was found out that lots of CHF data would not be well predicted by a normal annular film dryout (AFD) mechanism, although their flow patterns were identified as annular–mist flow. To predict these CHF data, a liquid sublayer dryout (LSD) mechanism has been newly utilized in developing the mechanistic CHF model based on each identified CHF mechanism. This mechanism postulates that the CHF occurrence is caused by dryout of the thin liquid sublayer resulting from the annular film separation or breaking down due to nucleate boiling in annular film or hydrodynamic fluctuation. In principle, this mechanism well supports the experimental evidence of residual film flow rate at the CHF location, which can not be explained by the AFD mechanism. For a comparative assessment of each mechanism, the CHF model based on the LSD mechanism is developed together with that based on the AFD mechanism. The validation of these models is performed on the 1406 CHF data points ranging over P=0.1–2 MPa, G=4–499 kg m⁻² s⁻¹, L/D=4–402. This model validation shows that 1055 and 231 CHF data are predicted within ±30 error bound by the LSD mechanism and the AFD mechanism, respectively. However, some CHF data whose critical qualities are <0.4 or whose tube length-to-diameter ratios are <70 are considerably overestimated by the CHF model based on the LSD mechanism. These overestimations seem to be caused by an inadequate CHF mechanism classification and an insufficient consideration of the flow instability effect on CHF. Further studies for a new classification criterion screening the CHF data affected by flow instabilities as well as a new bubble detachment model for LPLV conditions, are needed to improve the model accuracy.     

The fracture and aerosol release of impacted HLW-glasses due to accidental canister drops in a disposal site

For the disposal of HLW-canisters in a salt dome, two different accident scenarios have to be considered, canister drops in the reloading hall or in a borehole with drop heights of 10 m and 600 m, and reference drop velocities of 14 m/s and 80 m/s.The experimental program had two parts:

&#x02022; - Laboratory scale drop tests with bare and canistered waste glass probes (scale: 1:10) to obtain basic data.

&#x02022; - Full scale drop tests with inactive HLW-canisters, specified as planned for the German salt repository (H = 1.335 m, Ø = 0.43 m, weight: 550 kg, canister: SST 1.4833, wall: 5 mm).

The size distributions of the broken fines were measured by sieving and those of the filtered airborne particles by particle size analysis. The dominating parameter is the impact velocity (i.e. impact energy), further test parameters show no measurable influence, especially the canister influence on the fracture or aerosol release is negligible.Source terms, evaluated for the respirable fraction (particles with d < 10 μ m are between 2 × 10⁻⁴% for a 10 m drop and 0.1% for a 600 m borehole drop.     

A bulk tungsten divertor row for the outer strike point in JET

In the frame of the ITER-like wall project, a new row of divertor tiles has been developed which consists of 96 bulk tungsten load-bearing septum replacement plates (LB-SRP). Exposed to the outer strike point for most ITER-relevant, high triangularity configurations, they shall be subject to high power loads (locally 10 MW/m² and above). These conditions are demanding, particularly for an inertially cooled design as prescribed. The expected erosion rates are high as well as the risk of melting, especially with transients and repetitive ELM loads. The development is also a real challenge with respect to the inevitable excursions of the tungsten material through the so-called DBTT, ductile-to-brittle transition temperature.A lamella design has been selected to fulfil the requirements with respect to the thermo-mechanical and electromagnetic loads during disruptions (∂T/∂z ≤ 5 × 10⁴ K/m vertically, induction rate of change ∂B/∂t ≤ 100 T/s, and I_halo ≤ 18 kA/module). Care is taken to act on refractory metals solely with compressive forces to a large extent. The dedicated clamping concept is described. Results of a test exposure to an electron beam around 70 MJ/m² substantiate the resort to ‘high temperature’ materials like – among others – high-grade Nimonic^® alloys, molybdenum or ceramic coatings.     

Reactivity feedback coefficients of a material test research reactor fueled with high-density U3Si2 dispersion fuels

The reactivity feedback coefficients of a material test research reactor fueled with high-density U₃Si₂ dispersion fuels were calculated. For this purpose, the low-density LEU fuel of an MTR was replaced with high-density U₃Si₂ LEU fuels currently being developed under the RERTR program. Calculations were carried out to find the fuel temperature reactivity coefficient, moderator temperature reactivity coefficient and moderator density reactivity coefficient. Nuclear reactor analysis codes including WIMS-D4 and CITATION were employed to carry out these calculations. It is observed that the average values of fuel temperature reactivity feedback coefficient, moderator temperature reactivity coefficient and moderator density reactivity coefficient from 20 °C to 100 °C, at the beginning of life, followed the relationships (in units of Δk/k × 10⁻⁵ K⁻¹) −2.116 − 0.118 ρ_U, 0.713 − 37.309/ρ_U and −12.765 − 34.309/ρ_U, respectively for 4.0 ≤ ρ_U (g/cm³) ≤ 6.0.     

Development of stress indices for nonradial branch connections

This paper presents a brief summary of the technical basis for the recommended stress indices for 45 degree lateral connections under internal pressure and in-plane moment loadings.Starting with the historical background of the Pressure Vessel Research Committee's (PVRC) long range program on lateral connections, this paper highlights the various aspects intrinsic to this program such as model selection, analysis technique, finite element discretization, loading conditions, material properties and boundary conditions.A discussion of the stress index method and its application to the pressure vessel and piping design is offered as a prelude to the recommended code indices. Proposed changes to Par. NB-3338.2(C) (1) of Subsection NB of the ASME Code pertaining to lateral nozzles in cylindrical vessels and Par. NB-3650 relative to branch connections in piping systems are presented and discussed in detail.Besides, this paper discusses the need to develop additional data in the range of geometric parameters, 0.5 ≤ d/D ≤ 1.0 and 10 ≤ D/T ≤ 50, and under other remaining loading conditions to broaden the application of the stress index method.