Final Answer:
To determine analytically the safety factors guarding against static failure, one must calculate safety factors (Option a), apply the Coulomb-Mohr theory (Option c), and utilize the modified Mohr theory (Option d). These steps involve mathematical calculations based on the principles of material strength, stress analysis, and critical failure conditions. By calculating safety factors and applying these theories, engineers can assess the stability of structures and materials, ensuring they can withstand various loads and environmental conditions while minimizing the risk of static failure.
Step-by-step explanation:
a) Calculating safety factors involves assessing the ratio of the material's strength to the applied stress. This provides a numerical value indicating the margin of safety against potential static failure.
c) The Coulomb-Mohr theory evaluates the influence of shear stress and normal stress on material failure. It establishes criteria for failure based on a combination of these stresses.
d) The modified Mohr theory expands on the Coulomb-Mohr theory by considering additional factors such as intermediate principal stress. It provides a more refined analysis of material stability.