(a) Consider a hollow cylinder of length l, outer radius r+Δr inner radius r, fixed at one end a

step 1 Hi friends, consider a hollow cylinder of length L and outer radius R plus DR and inner radius f

(a) Consider a hollow cylinder of length l, outer radius...

(a) Consider a hollow cylinder of length $l$, outer radius $r+\Delta r$ inner radius $r$, fixed at one end and twisted at the other by means of a couple of moment $N$. The angular displacement $\varphi$, at a distance $l$ from the fixed end, is proportional to both $l$ and $N$. Consider an element of length $d x$ at the twisted end. It is moved by an angle $\varphi$ as shown. A vertical section is also shown and the twisting of the parallelopipe of length $l$ and area $\Delta r d x$ under the action of the twisting couple can be discussed by elementary means. If $f$ is the tangential force generated then shearing stress is $f / \Delta r d x$ and this must equal Hence, $$ \begin{gathered} G \theta=G \frac{r \varphi}{l}, \text { since } \theta=\frac{r \varphi}{l} \\ f=G \Delta r d x \frac{r \varphi}{l} \end{gathered} $$ The force $f$ has moment $f r$ about the axis and so the total moment is $$ N=G \Delta r \frac{\varphi}{l} r^{2} \int d x=\frac{2 \pi r^{3} \Delta r \varphi}{l} G $$

(a) Consider a hollow cylinder of length $l$, outer radius $r+\Delta r$ inner radius $r$, fixed at one end and twisted at the other by means of a couple of moment $N$. The angular displacement
$\varphi$, at a distance $l$ from the fixed end, is proportional to both $l$ and $N$. Consider an element of length $d x$ at the twisted end. It is moved by an angle $\varphi$ as shown. A vertical section is also shown and the twisting of the parallelopipe of length $l$ and area $\Delta r d x$ under the action of the twisting couple can be discussed by elementary means. If $f$ is the tangential force generated then shearing stress is $f / \Delta r d x$ and this must equal
Hence,
$$
\begin{gathered}
G \theta=G \frac{r \varphi}{l}, \text { since } \theta=\frac{r \varphi}{l} \\
f=G \Delta r d x \frac{r \varphi}{l}
\end{gathered}
$$
The force $f$ has moment $f r$ about the axis and so the total moment is
$$
N=G \Delta r \frac{\varphi}{l} r^{2} \int d x=\frac{2 \pi r^{3} \Delta r \varphi}{l} G
$$

Key Concepts

Differential Element Analysis and Integration

Differentiating the structure into small elements allows for an elementary analysis of the resulting stresses and forces. By considering the effects on a differential element and then integrating these contributions over the entire length or area of the structure, one calculates the overall response, such as the total moment generated by the distributed shear stresses. This method is a powerful tool in structural mechanics and engineering analysis.

Modulus of Rigidity (Shear Modulus)

The modulus of rigidity, commonly known as the shear modulus, is a material property that quantifies its resistance to shear deformation. It is defined as the ratio of shear stress to shear strain within the elastic limit of the material. This constant plays a critical role in determining how much a material will twist under a given torque.

Shear Stress and Strain

Shear stress is defined as the force per unit area acting tangentially to a surface, while shear strain is the deformation or angular displacement resulting from the applied shear force. In a twisting scenario, the shear stress experienced by a small element is proportional to the torque applied and inversely proportional to its cross-sectional area, leading to an angular displacement that is related to the geometry of the member.

Torsion in Cylindrical Shafts

Torsion in cylindrical shafts refers to the phenomenon when a cylindrical object is twisted by applying a moment (or torque) at one end while the other end is restrained. This leads to a distribution of shear stresses across the cross section and a corresponding angular deformation (twist) along the length of the shaft. Understanding the torsional behavior of cylinders is fundamental in mechanics and structural analysis.

Please give Ace some feedback