X.D.1.6 Interaction criteria

One often finds in the literature semi-empirical failure criteria corresponding to the combination of elementary failure modes with different load components. For example, in [otNCE21], one finds two criteria for the verification of ultimate failure of bolt under combined tensile, shear and bending loads:

The derivation of a reserve factor from expressions (X.D.1) or (X.D.2) is not straightforward. Statring from the definition of reserve factor (value by which loads can be multiplied to reach failure), one verifies that for (X.D.1) it corresponds to the value $x$ such that

Similarly, for interaction expression (X.D.2), RF is the solution in $x$ of

An analytic expression of the solution $x$ of equations (X.D.3) or (X.D.4) as a function of the different parameters is generally not available. Then, one must try other ways to calcula $x$ numerically. For the three methods “Interaction_2_SR”, “Interaction_3_SR” and “Interaction_N_SR”, we propose a dichotomic solver. The two first methods are specializations of the general case solved by predefined criterion “Interaction_N_SR”. This corresponds to the resolution in $x$ of equation:

The $2N$ arguments of the “Interaction_N_SR” predefined criterion are parameters in the following order: ${\alpha }_1$, ${\beta }_1$, ${\alpha }_2$... and ${\beta }_N$:

• The ${\alpha }_i$ parameters are real scalar Result objects. The keys of the different Result objects must match.
• The ${\beta }_i$ parameters are real values.

(“Interaction_2_SR”, “Interaction_3_SR” criteria need 4 and 6 arguments respectively.) “Interaction_N_SR” predefined criterion returns an Array of two values:

• First element is a real value corresponding to $S{R}_{max}=1∕R{F}_{min}=1∕x_{min}$.
• Second element is a Result object correspond to the different Strength Ratios calculated for the keys of Result objects.

A modified version of the “Interaction_abg_N_SR” predefined criterion corresponds to the resolution in $x$ of the following equation:

The $3N$ arguments of the “Interaction_abg_N_SR” predefined criterion are parameters in the following order: ${\alpha }_1$, ${\beta }_1$, ${\gamma }_1$, ${\alpha }_2$... and ${\gamma }_N$:

• The ${\alpha }_i$ parameters are real scalar Result objects. The keys of the different Result objects must match.
• The ${\beta }_i$ parameters are real values.
• The ${\gamma }_i$ parameters are real values.