Module 6-10 (Axial and Torsional Loads)

Saint-Venant's Principle

Stress and strain distributions become uniform at a relatively short distance from the point of load application.

Axial Load Diagram

A diagram that shows the internal normal force along the length of a member, with positive indicating tension and negative compression.

Displacement in Compound Axial Members

The relative displacement of one point to another equals the sum of deformations of each segment, considering different materials, areas, and internal forces.

Static Indeterminacy (Axial)

A problem where equilibrium equations alone are insufficient to solve for all internal forces, requiring additional compatibility equations based on deformation geometry.

General Solution Process for Indeterminate Axial Systems

Steps: 1. Equilibrium equations, 2. Geometry of deformation, 3. Force-deformation relationships, 4. Compatibility equations, 5. Solve.

Coaxial or Parallel Axial Members

Two members sharing the same axis or acting in parallel. Their deformations are equal or differ by a specified gap.

Series Axial Members

Members connected end-to-end. The total deformation is the sum of individual deformations.

Thermal Deformation

Change in length due to temperature change. In a statically determinate system, thermal deformation occurs without thermal stress.

Thermal Stress

Stress caused when thermal deformation is prevented, typically in statically indeterminate systems.

General Solution Process for Thermal plus Indeterminate Systems

Similar to axial indeterminate, but the force-deformation relationship includes both mechanical deformation and thermal deformation components.

Second Moment of Area (Area Moment of Inertia)

A geometric property quantifying how area is distributed about an axis. Always positive, depends on the chosen axis, with units of length to the fourth power.

Parallel Axis Theorem

Relates the moment of inertia about any axis to the moment of inertia about a parallel centroidal axis, used for composite shapes.

Polar Moment of Inertia

The sum of moments of inertia about two perpendicular axes intersecting at a point. For circular cross-sections, it is directly used in torsion.

Radius of Gyration

A distance representing how the area is distributed about an axis, calculated as the square root of moment of inertia divided by area.

Torsional Shear Stress Distribution

Shear stress varies linearly from zero at the center to a maximum at the outer surface of a circular shaft.

Complementary Property of Shear Stress

Shear stress never acts alone: a shear stress on a cross-sectional surface is always accompanied by an equal magnitude shear stress on a longitudinal surface.

Torsional Load Diagram

A diagram showing internal torque along a shaft using double-headed arrows. Sign convention: arrow away from the free-body diagram is positive.

Power Transmission by Shafts

Power is the product of torque and angular velocity. For a given rotational frequency, torque can be determined from power.

Angle of Twist

The relative rotation angle between two cross-sections of a shaft under torsion, analogous to axial deformation.

Torsional Indeterminacy

Occurs when equilibrium alone cannot determine internal torques, requiring compatibility of twist angles (e.g., total twist zero or sum of twists equals a known value).

Composite Shafts in Torsion (Parallel Connection)

Two shafts (e.g., a core and a tube) connected rigidly at ends. They have the same angle of twist, and the total torque is shared between them.

Common Misconception: Axial deformation formula applies to any cross-section shape

Reality: The formula assumes uniform stress distribution. Near changes in cross-section or load application, Saint-Venant's principle states that stress becomes uniform only after a certain distance.

Common Misconception: Thermal stress occurs in all temperature changes

Reality: Thermal stress only develops when deformation is constrained. In a free, statically determinate member, temperature change causes strain but no stress.

Common Misconception: Area moment of inertia is the same as mass moment of inertia

Reality: In mechanics of deformable bodies, area moment of inertia is a purely geometric property, not related to mass.

Common Misconception: Torsion formulas apply to any cross-sectional shape

Reality: The linear shear stress distribution and polar moment formulas apply only to circular shafts (solid or hollow). Non-circular sections warp and require different analysis.

Common Misconception: Torque sign convention based on clockwise/counterclockwise

Reality: In torsion, sign is based on double-headed arrow direction relative to the free-body diagram (away = positive, toward = negative), following the right-hand rule.

Exam-Ready Concept: Statically determinate vs indeterminate

Determinate: can solve using only equilibrium. Indeterminate: need compatibility from deformation (e.g., equal elongations, total twist zero). Always check the number of unknown reactions.

Exam-Ready Concept: Deformation compatibility for rigid bodies

When a rigid bar rotates, the deformations of attached axial members are related by similar triangles (for small rotations).

Exam-Ready Concept: Gap problems in axial members

If an initial gap exists, members do not share load until deformation closes the gap. After closure, compatibility includes the gap as an initial offset.

Exam-Ready Concept: Maximum safe load in compound axial members (series)

For members in series, the allowable load is governed by the weakest member (lowest allowable stress or force). For parallel members, load sharing depends on stiffness (product of modulus and area).

Exam-Ready Concept: Relationship between elastic constants

For isotropic materials, the modulus of rigidity can be derived from the modulus of elasticity and Poisson's ratio. This relationship is often tested indirectly.

Exam-Ready Concept: Power transmission unit conversions

Power equals torque times angular velocity. If rotational frequency is given, convert to radians per second. Know that one horsepower equals 746 watts.

Exam-Ready Concept: Sign convention for angle of twist

Positive angle follows the right-hand rule: fingers curl in torque direction, thumb points away from the section. Maintain consistent sign when summing twists.

Exam-Ready Concept: Composite shafts in torsion – series vs parallel

Series: same torque through each segment, total twist equals sum of twists. Parallel (e.g., core and tube): same twist angle, total torque equals sum of torques.