T/j = τ/r

Definition

The equation t/j = τ/r expresses the relationship between the shear stress ($$\tau$$) in a material subjected to torsion and its geometric properties, specifically its polar moment of inertia ($$J$$$). In this equation, $$t$$ represents the torque applied, $$j$$ represents the polar moment of inertia of the cross-section, $$r$$ is the radius at which the shear stress is being evaluated, and $$\tau$$ is the shear stress itself. This relationship is crucial for understanding how non-circular members behave under torsional loads and helps engineers design components that can withstand these stresses without failing.

5 Must Know Facts For Your Next Test

1. In non-circular members, the shear stress distribution varies across the cross-section, which can lead to complex failure modes.
2. The polar moment of inertia, $$J$$, is crucial in calculating how much torsion a non-circular member can safely handle before yielding.
3. As the radius ($$r$$) increases, for a constant torque ($$t$$), the shear stress ($$\tau$$) decreases, demonstrating why larger sections are often used to improve strength.
4. For solid shafts versus hollow shafts, the polar moment of inertia influences how effectively a hollow shaft can transmit torque while maintaining lower weight.
5. Understanding this equation helps in optimizing design for various applications like beams, shafts, and other structural elements subjected to torsional forces.

Review Questions

• How does changing the radius ($$r$$) impact the shear stress ($$\tau$$) according to the equation t/j = τ/r?
  • According to the equation t/j = τ/r, increasing the radius ($$r$$) leads to a decrease in shear stress ($$\tau$$) for a given torque ($$t$$). This means that if you have a larger radius in a torsional member, it can handle more torque without experiencing as much shear stress. This concept is particularly important when designing components where reducing weight while maintaining strength is crucial.
• Compare the effects of torsion on circular versus non-circular members in terms of shear stress distribution.
  • In circular members, the shear stress distribution is uniform across any cross-section; however, in non-circular members, this distribution varies significantly. For example, in a rectangular or I-beam section, maximum shear stress occurs at different points rather than uniformly. Understanding this difference is essential for predicting failure points accurately and applying proper safety factors in design.
• Evaluate how the relationship defined by t/j = τ/r can be applied to improve engineering designs for non-circular members under torsional loads.
  • By evaluating the relationship defined by t/j = τ/r, engineers can optimize designs for non-circular members by selecting appropriate materials and shapes that maximize polar moment of inertia while minimizing weight. This analysis allows designers to predict how different configurations will respond under load, ensuring that components remain within safe limits of shear stress. Consequently, this leads to safer and more efficient structures that perform better under torsional stresses.

Related terms

Shear Stress: A measure of the internal forces within a material, defined as the force per unit area acting parallel to the surface.

Polar Moment of Inertia: A geometric property of a cross-section that reflects how its area is distributed with respect to an axis; it plays a critical role in determining an object's resistance to torsion.

Torsional Rigidity: The resistance of a structural member to twisting; it is influenced by both the material properties and the geometry of the member.