**6. Strength and durability analysis**

Measurement of the changes in stresses Δσ when electric current - from 0 to 15 kA - was introduced into the superconducting systems showed that in the support cylinder the stress reached 110 MPa, and 40 MPa when the discharge chamber was heated. During the experimental testing and operation, the elements of the discharge chambers of tokamaks are exposed to mechanical *Qm*(τ), temperature *Qt*(τ) and electromechanical *Qem*(τ) loads, some of which cause the presence of repeated elastoplastic deformations in the areas of stress concentration. As a result, it was necessary to study and substantiate the static and cyclic strength, for which a series of experimental studies was carried out involving the single-frequency and dual-frequency loading of austenitic chromium-nickel stainless steel at a wide temperature range *t* from −196°C up to 400°С. The resulting characteristics of the material's resistance to static and cyclic deformation and destruction are an integral part of the general design justification in relation to the strength and durability of the *N* elements of the discharge chamber.

The calculations and tests were carried out in relation to the tokamak installation in the presence of a strong magnetic field, the purpose of which was to conduct a physical experiment to study the behavior of plasma under conditions close to those of thermonuclear ignition at minimal technical and economic costs. The use of strong magnetic fields to confine plasma makes it possible to significantly reduce the size of the electromagnetic system of the tokamak and the amount of energy stored in it, and the use of combined adiabatic compression - significantly increases the potential of the experiment. However, in order to increase the magnetic field, a number of complex engineering problems need to be solved.

Structurally, the SFT consists of a discharge chamber in the form of a closed torus of noncircular cross-section, along which 32 sections of the toroidal field coil (TFC) are located. The poloidal field coils (PFC) are located outside the TFC. A structural diagram of the electromagnetic system (EMS) is shown in **Figure 15**.

The interaction of TFC currents with toroidal and poloidal fields leads (by analogy with **Figure 12**) to the occurrence of significant ponderomotive forces acting in the TFC plane and overturning moments tending to rotate the TFC section planes around their horizontal axes. The total vertical force, *Q*, disrupting the TFC, is equal to 128 MN, the resulting centripetal force is 108 MN, and the magnitude of torque relative to the vertical axis of the installation is 30 MN·m. In addition to the forces caused by the interaction of magnetic fields and currents, significant forces arise in the TFC, which are a consequence of heating of the conductor.

Ensuring strength of the TFC under the influence of these forces is one of the most difficult tasks involved in developing an EMF. Each section of the TFC is made in the form of a single-turn conductor (bus) made of zirconium bronze, placed in a band made of high-strength non-magnetic steel. The TFC sections are interconnected in such a way that, as a result, a closed thick-walled toroidal shell is formed. This structure is capable of handling both azimuthal and centripetal loads.

*Theoretical and Experimental Analysis of Structural Properties of Load-Bearing Components… DOI: http://dx.doi.org/10.5772/intechopen.94531*

**Figure 15.** *Structural diagram of the SFT electromagnetic system. 1 - bandage; 2 - bus; 3 - poloidal field coil.*

It has been established by analysis that it is not possible to solve the problem of ensuring EMS strength by means of traditional safety factors and acceptable stresses. The development of the structure was therefore carried on the basis of the maximum strength characteristics of the material. This approach is acceptable, since the unit is designed for a limited number of impulses.

The selection of the structure of the section, bandage and other load-bearing elements was made by considering the stress-strain state of a number of design options, taking into account the elastoplastic behavior of materials under the action of ponderomotive forces and thermal stresses. The problem of studying the stressstrain state of a structure while a conductor is undergoing elastoplastic deformation can be solved using the finite element method, by applying the theory of plastic flow. Analysis of the stress state has shown that the effect of overturning moments on maximum stresses and strains is insignificant. **Figure 16** shows the distribution of maximum stresses in the TFC unit resulting from the action of ponderomotive forces on the coil in the unit plane, taking into account the heating of the conductor. An assessment of the service life of the EMS has shown that it is able to withstand a given number (1000) of full-scale operating cycles.

The determination of pulse loads on the inner surface of the magnet coil based on the measurements of stresses arising from such loads in individual zones of the structure is an inverse problem of experimental mechanics.

The resolving equation connecting the stresses determined from measurements in a certain zone (**Figure 17a**) with the required load vector on a part of the surface is expressed in the form of a system of Fredholm integral equations of the first kind. It is pointless to attempt to resolve this system - this would be an incorrectly posed problem, −as small perturbations of the initial data can result in arbitrarily large perturbations of the solution. A regularizing algorithm is therefore chosen. The

#### **Figure 16.**

*Isolines of equivalent stresses acting in the bandage (1) and the bus (2). The asterisks mark the points of local maxima.*

#### **Figure 17.**

*The principle stresses in a narrow section of the TFC bandage (a) and the pressure distribution on the wall of the toroidal chamber (b)* S *- measurement area;* L *- load area.*

restoration of the magnetic pressure in the model was carried out using measurements of strains which were made in a narrow section of the bandage. The measurements of the strains were used to determine the axial stresses in the connector of the toroidal chamber subject to the force of a magnetic field. **Figure 17a** illustrates the axial stresses in the section of the bandage: «these define the error margin (curves 1–3) and are constructed on the basis of experimental data. The corresponding distribution of the magnetic field pressure on the inner contour of the bus is shown in **Figure 17b**, curves 1–3. These results are characterized by a satisfactory level of reconstruction and are consistent with the a priori ideas on the distribution of magnetic pressure.

It is proposed that zirconium bronze be used for the current-carrying elements that are subject to cyclic heating during the operation of the installation. In order to *Theoretical and Experimental Analysis of Structural Properties of Load-Bearing Components… DOI: http://dx.doi.org/10.5772/intechopen.94531*

#### **Figure 18.**

*Calculated low-cycle fatigue curves and experimental data for zirconium bronze at temperatures of 200 and 245°С.*

determine its performance, its resistance to low-cycle deformation and fracture at elevated temperatures was studied.

The calculated low-cycle fatigue curves for zirconium bronze under rigid isothermal cyclic loading are shown in **Figure 18**, in which curve 3 represents the normative equation and characterizes the lower, conservative limit for low-cycle durability, with safety factors for strains *n*e = 2 and for durability *n*N = 10 [9, 10].

As part of the commissioning work on the creation of SFT, significant emphasis is given to experimental studies of the stress-strain state of the elements of the installation. One specific feature affecting the measurement of strain in the SFT unit is the presence of pulsed magnetic fields of up to 20 T, together with a change in the temperature of the current-carrying elements of up to 250°C and a high level of measured strains (up to 1.2%).

In connection with the impulsive growth of strains and the possibility of the transition of the elements of the wedge part of the bandage and the bus to a plastic state (see **Figures 15** and **16**), it seems possible to use the brittle strainsensitive coatings method for the purpose of studying the stress fields. Estimates of mechanical stresses in the steel bandage of the model were carried out using the brittle strain-sensitive coatings method at magnetic fields reaching 14 T, with a yield point - of about 12 T.

The analysis and modeling of deformation processes of the elements of a thermonuclear installation in operating mode showed that fretting fatigue arises on the contact surface between the bandage and the bus as a result of the difference in displacements and the presence of contact interactions. This requires special study, since according to [7, 8] it has a significant impact on the installation's integral strength and service life parameters.

As noted above, the characteristics of the mechanical properties of bronze were determined by experiment, using specimens from supplied semi-circular forgings with an average radius of 270 mm and a cross-section of 110 × 140 mm. In order to carry out cyclic tests under conditions that simulate the operation of the busses in contact with the bandage at room temperature, a special loading device was developed that creates a transverse load on the specimen.

When quantifying parameters through the characteristics of the mechanical properties of the studied zirconium bronzes which are included in the Eqs. (1)–(3), this equation takes the form [2–4, 9].

$$
\varepsilon\_a = 0, 25e\_c \cdot N^{-m\_p} + 0, 435 \frac{\mathcal{S}\_c}{E} N^{-m\_c}, \tag{8}
$$

where *е*к is the fracture strain under a monotonic loading, *m*p = 0.65, *m*e = 0.06 are the characteristics of the material.

The value is equal to

$$e\_c = \ln \frac{1}{1 - \nu\_c},\tag{9}$$

where *ψ*c is the relative narrowing of the specimen in the neck, *ψ*c = 0.7÷0.75. Tear fracture resistance at the specimen neck

$$S\_{\varepsilon} = \sigma\_u \left( \mathbf{1} + \mathbf{1}, 4\psi\_{\varepsilon} \right), \tag{10}$$

where σu is the ultimate strength (σu = 310–330 MPa), *Е* is the elasticity modulus (*Е* = (1.3 ÷ 1.5) ⋅ 105 MPa.

From the data obtained from service life assessments, taking into account the fretting effect, the amplitude of the fracture strains *е*а in the Eq. (8) needs to be reduced by the reduction factor *K*k. This factor reaches values of 2–2.25.

$$
\varepsilon\_{ak} = \varepsilon\_a \nmid K\_\kappa \tag{11}
$$

which is comparable with the safety factor for durability *n*e = 2, applied in the norms for nuclear reactor calculations [9, 10].
