Consider A Point O In A Solid Body In A Plane State

Consider A Point O In A Solid Body In The State Of Plane Stress On The

Consider a point O in a solid body in the state of plane stress on the (x, y) plane. Figure 7 shows the stress components acting at O along planes A and B. Using Mohr’s circle:

(a) Determine the angle θ between the two planes.

(b) Find the principal stresses σ₁ and σ₂ and the angles formed between the principal directions and the plane A.

(c) Find the maximum in-plane shear stress and the normal stress on the planes of maximum in-plane shear stress.

Paper For Above instruction

The analysis of stress states within a solid body, especially those involving plane stress conditions, is crucial for understanding material behavior under various loading scenarios. In the case of a point O under plane stress, the state can be characterized fully by normal and shear stresses acting on various planes passing through O. Employing Mohr’s circle offers an effective graphical and analytical approach to determine key parameters such as principal stresses, their orientations, maximum shear stresses, and the associated normal stresses on those planes.

To address the problem systematically, we start with the assumption that at point O, the known stress components on a particular plane (say plane A) are σₓ and τₓy, representing normal and shear stresses, respectively. The primary goal is to use Mohr’s circle for plane stress to find the angles and stresses requested. Since the problem references Figure 7 for the actual stress components, an exact numerical solution would typically depend on those figures. However, the methodology remains consistent regardless of specific values.

(a) Determining the angle θ between the two planes

The angle θ between planes A and B is related to the orientation difference between the two associated planes through the Mohr’s circle method. If the normal stress and shear stress on plane A are known, the stress components on plane B can be found by rotating the state of stress by an angle 2θ in Mohr’s circle. Therefore, the geometric relation in Mohr’s circle allows us to determine θ via:

tan 2θ = (2τₓy) / (σₓ - σ_y)

where σₓ and σ_y are the normal stresses on the planes, and τₓy is the shear stress component. Once θ is known, it indicates the angular difference between the two planes in physical space.

(b) Finding principal stresses σ₁ and σ₂ and their angles relative to plane A

The principal stresses are the maximum and minimum normal stresses at point O, occurring on planes oriented at specific angles. They are obtained from Mohr’s circle by locating the extreme points along the circle's horizontal axis. The principal stresses are given by:

σ₁, σ₂ = (σₓ + σ_y)/2 ± sqrt[((σₓ - σ_y)/2)² + τₓy²]

The angle θ_p between the principal directions and the original plane A is determined by:

tan 2θ_p = (2τₓy) / (σₓ - σ_y)

which is similar to the expression used in part (a). The principal directions are at angles θ_p and θ_p + 90° relative to the original plane, corresponding to the maximum and minimum normal stresses.

(c) Finding the maximum in-plane shear stress and normal stress on the maximum shear planes

The maximum in-plane shear stress, τ_max, is given by the radius of Mohr’s circle:

τ_max = sqrt[((σₓ - σ_y)/2)² + τₓy²]

This maximum shear stress acts on planes oriented at 45° to the principal planes. The normal stress on these planes of maximum shear is:

σ_avg = (σ₁ + σ₂) / 2

and the normal stress component on the plane of maximum shear is:

σ_n = σ_avg ± τ_max

depending on orientation. The insight from Mohr’s circle shows that the planes experiencing maximum shear stresses are oriented at 45° relative to principal planes, and the maximum shear stress magnitude provides critical information about potential failure modes, especially shear failure.

Conclusion

By applying Mohr’s circle to the given state of plane stress at point O, engineers can effectively determine the relative orientations of stress planes, principal stresses, and maximum shear stresses. This comprehensive understanding guides in assessing the safety and durability of materials under complex loading conditions. The graphical method simplifies complex tensor analysis into visual and algebraic solutions, enabling precise failure predictions and optimal design configurations.

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