The objectives of this experiment are to perform a torsion test of a steel bar, examine the strain and stress distribution due to torsion in a bar with a circular cross-section, experimentally determine the shearing modulus of steel, and observe the failure mode of a ductile material in torsion.

B. Theory

### 1.0 Equilibrium

Shearing stresses develop in a material when torque is applied to a bar. Our goal in this section is to derive an equation that relates applied toque to the shearing stresses that develop in our bar. The torques and the shear stresses are related through equilibrium.

Using a polar coordinate system in terms of (?, ?) centered at the center of the circular cross-section, a small, or infinitesimal, amount of torque from the stress at a point on the cross-section is given by*dT *??*dF *???*dA*.

The total torque is found by integrating (essentially adding up) the infinitesimal torques over the cross-sectional area:*T *?? *dT *??*A*??*dA*.

The preceding statement of equilibrium is valid whether or not the material remains linear elastic. However, when the material is linear elastic, and shear stress and strain are related through Hooke’s law for shear:

??*G*?,

which leads to*T *??*A*?*G*?*dA*.

Unfortunately, the preceding equation is not very useful until we can find an expression for the shear strain, ?.

The shearing strain in a twisted bar that has a circular cross-section can be determined based on some simple observations and assumptions of elastic deformation:

-the bar does not elongate,

-plane cross-sections remain plane, and

-radial lines on the cross-sections remain radial lines after deformation.

To illustrate these observations, consider a bar that is clamped on one end. Before the torsional load is applied, points A and C lie on the outside radius at the fixed end of the bar. Point B lies on the free end of the bar, and, on a side view, the angle BAC creates a right angle.

T

After a torque is applied to the end of the bar, point B moves to B*. The reduction in angle from the right angle is the shearing strain, ?.

The objectives of this experiment are to perform a torsion test of a steel bar, examine the strain and stress distribution due to torsion in a bar with a circular cross-section, experimentally determine the shearing modulus of steel, and observe the failure mode of a ductile material in torsion.

B. Theory

### 1.0 Equilibrium

Shearing stresses develop in a material when torque is applied to a bar. Our goal in this section is to derive an equation that relates applied toque to the shearing stresses that develop in our bar. The torques and the shear stresses are related through equilibrium.

Using a polar coordinate system in terms of (?, ?) centered at the center of the circular cross-section, a small, or infinitesimal, amount of torque from the stress at a point on the cross-section is given by*dT *??*dF *???*dA*.

The total torque is found by integrating (essentially adding up) the infinitesimal torques over the cross-sectional area:*T *?? *dT *??*A*??*dA*.

The preceding statement of equilibrium is valid whether or not the material remains linear elastic. However, when the material is linear elastic, and shear stress and strain are related through Hooke’s law for shear:

??*G*?,

which leads to*T *??*A*?*G*?*dA*.

Unfortunately, the preceding equation is not very useful until we can find an expression for the shear strain, ?.

The shearing strain in a twisted bar that has a circular cross-section can be determined based on some simple observations and assumptions of elastic deformation:

-the bar does not elongate,

-plane cross-sections remain plane, and

-radial lines on the cross-sections remain radial lines after deformation.

To illustrate these observations, consider a bar that is clamped on one end. Before the torsional load is applied, points A and C lie on the outside radius at the fixed end of the bar. Point B lies on the free end of the bar, and, on a side view, the angle BAC creates a right angle.

T

After a torque is applied to the end of the bar, point B moves to B*. The reduction in angle from the right angle is the shearing strain, ?.

Attachments:

lab-6–251.docx