1. Reference problem

1.1. Material properties

The material properties are as follows:

1. Young’s modulus: \(E\mathrm{=}2.E+05\mathit{MPa}\);

2. Poisson’s ratio: \(\nu =0.3\);

3. Thermal expansion coefficient: \(\alpha =1.E–05\mathrm{m.}°{C}^{-1}\).

The characteristics specific to the calculation RCC -M are:

1. material constants for calculating \(\mathrm{Ke}\): \(n=0.2\), \(m=\mathrm{2 }\);

2. Young’s modulus of reference: \({E}_{\mathit{REFE}}\mathrm{=}2.E+05\mathit{MPa}\);

3. allowable stress: \(\mathrm{Sm}=200\mathrm{MPa}\).

The Wöhler curve is defined analytically: \({N}_{\mathrm{adm}}=\frac{{5.10}^{5}}{{S}_{\mathrm{alt}}}\)

Note:

To validate the consideration of the elastoplastic concentration factor \(\mathrm{Ke}\) , some calculations are carried out with a lower or higher allowable stress: \(\mathit{Sm}=50\mathit{MPa}\) (C modeling) and \(\mathit{Sm}=2000\mathit{MPa}\) (D and E modeling) .

1.2. Evolution of constraints

The constraints on the analysis segment are not calculated but read directly from a table. The only non-zero component of the stress tensor is \({\sigma }_{\mathrm{yy}}\). Two transients are considered:

Instant	Thermal constraints			Stress due to pressure			Mechanical stress (moments and efforts)			Constraints total es
	Abscissor			Abscissor			Abscissor			Abscissor
	0	1	2	0	1	2	0	1	2	0	1	2
1	50	100	150	40	0	40	0	0	-80	90	100	110
2	0	50	-100	0	50	0	0	0	10	0	100	-90
3	0	0	50	10	-10	-200	90	-40	50	100	-50	-100
4	0	0	0	0	0	0	0	0	0	0	0	0

Table 1.2-1 : Definition of constraints \({\mathrm{\sigma }}_{\mathit{yy}}\) (in \(\mathit{MPa}\)) for the moments of situation 1 as a function of the curvilinear abscissa

Instant	Thermal constraints			Stress due to pressure			Mechanical stress (moments and efforts)			Constraints total es
	Abscissor			Abscissor			Abscissor			Abscissor
	0	1	2	0	1	2	0	1	2	0	1	2
1, 5	50	100	150	40	0	2**0**	0	0	-80	90	100	9**0**
2, 5	0	50	-100	0	50	0	0	0	10	0	100	-90
3, 5	0	0	50	10	-10	-200	90	-40	50	100	-50	-100

Table 1.2-2: Definition of the constraints \({\sigma }_{\mathit{yy}}\) (in \(\mathit{MPa}\)) for the moments of situation 2 as a function of the curvilinear abscissa

These transients do not aim to represent a specific real transient, but to cover all possible constraints (constant, linear or non-linear evolution of the stress in thickness).

Two other situations are taken into account in this test case. They are provided in the form of two torsors and a unit stress tensor.

Situation 3 is composed of a state A that corresponds to instant 3 of situation 1 and of a state B that corresponds to instant 2 of situation 1. It also consists of a thermal transient which is that of situation 1.

Situation 4 is composed of a state A that corresponds to the instant 1.5 of the situation 2 and of a state B that corresponds to the instant 3.5 of the situation 2. It also consists of a thermal transient which is that of situation 2.

State	Stress due to pressure			Mechanical stress (moments and efforts)			Restraints total mechanical constraints**
	Abscissor			Abscissor			Abscissor
	0	1	2	0	1	2	0	1	2
A	10	-10	-200	90	-40	50	100	-50	-100
B	0	50	0	0	0	10	0	100	-90

Table 1.2-3: Definition of the constraints \({\mathrm{\sigma }}_{\mathit{yy}}\) (in \(\mathit{MPa}\)) of situation 3 as a function of the curvilinear abscissa

State	Stress due to pressure			Mechanical stress (moments and efforts)			Restraints total mechanical constraints**
	Abscissor			Abscissor			Abscissor
	0	1	2	0	1	2	0	1	2
A	40	0	2**0**	0	0	-80	90	100	9**0**
B	10	-10	-200	90	-40	50	100	-50	-100

Table 1.2-4: Definition of constraints \({\sigma }_{\mathit{yy}}\) (in \(\mathit{MPa}\)) of situation 4 as a function of the curvilinear abscissa