19.3
Eixo Circular - Tensão em Faixa Linear
Considere um cenário onde um eixo circular está sujeito a um torque que permanece dentro dos limites da Lei de Hooke, evitando qualquer deformação per…
Considere um caso em que o torque aplicado ao eixo circular está dentro do limite da lei de Hooke, portanto, não há deformação permanente.
Agora, lembre-se da expressão para tensão de cisalhamento. Multiplicando-o pelo módulo de rigidez e usando a Lei de Hooke para tensão de cisalhamento e deformação, uma expressão para tensão de cisalhamento em um eixo pode ser determinada.
Lembre-se de que a soma dos momentos das forças elementares exercidas em qualquer seção transversal do eixo deve ser igual à magnitude do torque exercido no eixo.
Substituindo a tensão de cisalhamento e reorganizando os termos, obtém-se uma expressão com um termo integral, que representa o momento polar de inércia da seção transversal em relação ao seu centro.
Outros rearranjos e substituições para a tensão máxima de cisalhamento fornecem a fórmula de torção elástica para a tensão de cisalhamento em um eixo circular uniforme rígido.
No entanto, para um eixo oco com r1 e r2 como raios interno e externo, o momento polar de inércia é expresso como uma diferença na quarta potência de dois raios.
View the full transcript and gain access to JoVE Core videos
Q1: What is the elastic torsion formula for shearing stress in a circular shaft?
The elastic torsion formula determines shearing stress in a rigid uniform circular shaft by combining Hooke's Law with the polar moment of inertia. It derives from multiplying shearing strain by the modulus of rigidity, then ensuring the sum of moments on any cross-section equals the applied torque. This formula applies when torque remains within Hooke's law limit, preventing permanent deformation.
Q2: How does the polar moment of inertia differ between solid and hollow circular shafts?
For a solid circular shaft, the polar moment of inertia is calculated from the fourth power of the radius. For a hollow shaft with inner radius r1 and outer radius r2, the polar moment of inertia is expressed as the difference in the fourth power of both radii. This difference accounts for the removed material in the hollow center.
Q3: Why is the modulus of rigidity important in deriving shaft stress equations?
The modulus of rigidity relates shearing stress to shearing strain through Hooke's Law. When multiplied by the shearing strain expression, it enables derivation of the shearing stress equation for the shaft. This material property is essential for connecting elastic deformation behavior to the applied torque.
Q4: What role does the sum of moments play in the elastic torsion formula?
The sum of moments of elementary forces exerted on any cross-section of the shaft must equal the magnitude of the applied torque. This equilibrium condition is fundamental to deriving the elastic torsion formula. Substituting this relationship into the stress equation produces the integral term representing polar moment of inertia.
Q5: When can the elastic torsion formula be applied to a circular shaft?
The elastic torsion formula applies when torque remains within Hooke's law limit, ensuring no permanent deformation occurs. Under these conditions, the shaft exhibits linear elastic behavior, and shearing stress is directly proportional to shearing strain. This linear range assumption is critical for formula validity.
Q6: How does maximum shearing stress relate to the polar moment of inertia in torsion?
Maximum shearing stress is inversely proportional to the polar moment of inertia. After substituting for maximum shearing stress in the equilibrium equation, the polar moment of inertia emerges as a key geometric property. Larger polar moments of inertia reduce maximum stress for the same applied torque, which is why hollow shafts are often preferred in design of transmission shafts.
Q7: What assumptions must be satisfied for the elastic torsion formula to hold?
The shaft must be rigid and uniform with constant cross-section. Torque must remain within Hooke's law limit to prevent plastic deformation. The formula assumes linear elastic material behavior where shearing stress is proportional to shearing strain. These conditions ensure the derived relationships between torque, stress, and geometric properties remain valid.