Two forces 𝐏 can be applied separately or at the same time...

Two forces $\mathbf{P}$ can be applied separately or at the same time to a plate that is welded to a solid circular bar of radius $r$. Determine the larg. est compressive stress in the circular bar, (a) when both forces are applied. (b) when only one of the forces is applied.

Key Concepts

Axial Stress

Axial stress arises from forces acting along the member's centroidal axis. It is calculated by dividing the axial force by the cross?sectional area, which for a circular section is determined using the area formula. This stress is uniform across a cross section for members loaded purely in compression or tension.

Bending Stress

Bending stress is produced when a force or couple creates a moment about a section, causing one side of the member to experience tension and the opposite side compression. The maximum bending stress is computed using the formula ? = M·c/I, where M is the bending moment, c is the distance from the neutral axis (often the outer fiber, such as the radius in a circular bar), and I is the moment of inertia.

Section Properties

The geometric properties of a cross section, including area and moment of inertia, are essential for computing both axial and bending stresses. For a solid circular cross section, the area is given by A = ?r² and the moment of inertia by I = (?r?)/4, which are key in quantifying how the section resists axial loads and bending moments.

Combined Loading

When a member is subjected to both an axial force and a bending moment simultaneously, the resulting stress at any point is the sum of the axial stress and the bending stress. This superposition is critical for determining the maximum compressive or tensile stresses present in the section, ensuring that the design remains within safe limits.

Superposition Principle

The principle of superposition allows the individual effects of different load types to be added together linearly provided that the material behavior is linear-elastic. In the context of combined axial and bending loads, this principle is used to sum the stresses due to each load component to find the total stress at a specific point in the member.

Transcript

Please give Ace some feedback