Learn · Mechanics of Materials
Lecture 103 of 104Week 12 : Buckling
Secant formula
Prof. Saravanan.U · IIT Madras
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All the lectures
- Introduction to the course - Why this course?
- Introduction to the course - Concepts and equations in this course
- Introduction to the course - Objectives and prerequisite
- Mathematical Preliminaries - Linear Algebra
- Mathematical Preliminaries - Vector Algebra
- Representation of Vector
- Tutorial 1
- Concept of Force
- Definition of a body
- Motion and Displacement field
- Traction
- Properties of traction
- Defnition of stress tensor and linear function
- Tensor Algebra
- Meaning of components of the stress tensor
- Tutorial 2
- Transformation of stress components
- Mohr's Circle derivation
- Example 1: Construction of Mohr's circle
- Example 2: Extremum normal and shear stress
- Example 3: Transformation of stress components
- Uniaxial stress
- Hydrostatic, pure shear and deviatoric stress
- Biaxial and Plane state of stress
- Extreme stress for 3D stresses
- Extremum shear stress
- Stresses in the Octahedral plane
- Tutorial 3
- Stretch ratio and strain
- Curves and arc Length
- Gradient
- Deformation and displacement Gradient
- Right Cauchy Green Deformation tensor
- Homogeneous deformation
- Engineering strain
- Change in Angle
- Transformation of strain components/ Strain Rosette
- 2D Equilibrium equations
- 3D Equilibrium equations
- Compatibility condition
- Constitutive relation
- Young's Modulus and Poisson's Ratio
- Shear Modulus
- Bulk Modulus
- Restriction on material parameters
- Strain energy, load potential and total potential
- Thermal strain
- Stepped shaft subjected to axial force
- Inhomogeneous bar subjected to axial force
- Stepped shaft subjected to raise in temperature
- Traction in member subjected to bending
- Governing equilibrium equations
- Displacement field
- Bending equation
- Radius of curvature
- Shear force and bending moment diagram
- Variation of axial stress
- Deflected shape and rotation of cross section
- Expression to find shear stress
- Finding centroid of a cross section
- Parallel axis theorem and its application
- Vertical shear stress in I section
- Horizontal shear stress in I section
- Connection design
- Definition of shear center
- Shear center of Channel section
- Expression to find shear center
- Shear force and bending moment diagram
- Finding allowable load
- Modified bending equation
- Bending of a composite beam
- Connection design
- Moment of Intertia about arbitrarily oriented axis
- Example: Angle section
- Bending equation for bending about principal axis
- Bending equation about arbitrary axis
- Neutral axis
- Load not about principal axis
- Load about principal axis
- Displacement field
- Torsion equation
- Example problems: Closed sections
- Twisting of open sections
- Example problems: Open sections
- Twisting of thin walled closed sections
- Example problems: Thin walled sections
- Cylindrical polar coordinate system
- Displacement field
- Governing differential equation and solution
- Example problems : Thick walled cylindrical vessel
- Thin walled pressure vessels
- General concepts in failure criteria
- Different failure modes
- Tresca Condition
- vonMises condition
- Maximum normal stress or rankine condition
- Mohr - Columb condition
- Drucker-Prager Condition
- Numerical Example: Pressure vessel and failure theory
- General concepts in buckling
- Euler critical load for simply supported column
- Euler critical load for column with any boundary condition
- Secant formula
- Determination of maximum load carrying capacity of a simple truss