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Normal stress is a fundamental concept in material mechanics, representing the internal forces acting on an object per unit area. It results when forces are applied perpendicular (normal) to the surface of a material. Normal stress has a significant role in determining a material's behavior under load.

Mathematically, normal stress \( \sigma \) is given by:

• \( \sigma = \frac{F}{A} \)

where \( F \) is the force acting normal to the surface, and \( A \) is the cross-sectional area.

In practical terms, understanding normal stress helps in evaluating whether a material or component can withstand the applied loads without failure. The evaluation of normal stress is crucial, especially in designing structures and machines to ensure safety and performance under operational conditions.

Ultimate tensile stress (UTS), also known as tensile strength, is the maximum amount of tensile stress that a material can support before experiencing failure by breaking. When studying machine components or structural elements, knowing the UTS tells us how much stress they can endure before being pulled apart.

This property is essential in determining how durable and strong an engineering material is. UTS is commonly measured in kilopounds per square inch (ksi) or megapascals (MPa).

In the exercise, the ultimate tensile stress of cast iron is given as \( \sigma_{UT} = 8 \text{ ksi} \). This value means that, before breaking, the material can withstand normal stresses up to 8 ksi. It is important for engineers to consider UTS when designing to prevent tensile failure, especially for materials like cast iron, which are brittle and prone to cracking.

Ultimate compressive stress (UCS) refers to the maximum compressive force that a material can take before it fails. Unlike tensile stress, compressive stress pushes together the particles of a material.

Understanding UCS helps engineers and designers ensure that a material like cast iron can withstand loading without a risk of compressive failure. Compressive stress is also measured in units like ksi or MPa.

In your exercise related to the Mohr's Criterion, the ultimate compressive stress is \( \sigma_{UC} = 20 \text{ ksi} \). This indicates that the cast iron component can withstand a stress level of up to 20 ksi in compression before failure. For this reason, this property is especially important when the material is going to face significant compressive forces during its service life.