• Solid Mechanics: Solved Examples

Text-book Solutions

SOM, Solid Mechanics, Fracture Mechanics, Statics and Dynamics

This page addressess the analytical solution of text-books in the field of Strength of Materials, Solid Mechanics, Engineering Mechanics, Mechanical Vibrations, Kinematics & Dynamics of Mechanisms and Machinery...

Navier-Stokes Equation for Solid Deformation

Analogy between Solid Deformation and Fluid Flow

At the fundamental level, the governing equations of fluid mechanics ([relatively LARGER] deformation and kinematics of fluids particles) and structural mechanics ([relatively SMALLER] deformation and kinematics of solid particles) are equivalent.

Navier-Stokes Equation for Solid Mechanics

Navier-Stokes Equation for Solid Mechanics

Navier-Stokes Equation for Solid Mechanics

Buckling Calulation - Fixed Beam with Lateral Loads

Buckling Fixed Beam

Buckling Fixed Beam

Buckling Fixed Beam

Rod sliding against frictionless wall and ground

While this problem may look trivial, the solution requires good experience in solving dynamics problem. The application of moment balance about sliding contact point will not yield desired result.Frictionless Sliding Rod

Direction of Reaction Force on Inclined Plane

This type of situation is very frequently encountered while solving statics and dynamics problems. What is the angle made by normal reaction with inclined plane or what is the component of gravity force perpendicular to the inclined plane? Just remember the theorem: "The angle between two lines is equal to angle between perpendicular to those lines".

Inclined Plane

Mechanical Vibrations: Lagrangian Approach

Lagrangian approach is a powerful and straightforward method to derive equation of motion (EOM) for complex systems. The EOM for pring-mass-damper pendulum (also known as "sprung pendulum") is described below.

Inclined Plane

Inelastic Collision and Spring Compression

Collision is based on conservation of momentum and loss of energy defined by coefficient of restitution. This solution describes a case where one of the colliding objects is attached to a spring.

Inelastic Collision

Inelastic Collision

Crack Growth Calculations - Fracture Mechanics

Paris law is widely used in fracture mechanics calculations for number of cycles of variable loading required for crack to grow to critical size. Note that the rate of growth of crack per cycle, da/dN = A * δKm. Very often m is not an integer. The unit of δK is [MPa.m0.5]. Hence, special precautions should be taken to use consistent units for A, m and δK.

Fracture Mechanics

4-Bar Mechanism with Slider

Four bar slider

4-Bar Mechanism with Coupler Point

Four bar coupler point

Slipping of Ladder

A ladder is leaning against a wall at certain angle having different coefficient of friction at wall and at the floor. The method to calculate maximum height a person of known weight can climb without slipping is described below.

Sliding of a ladder

Bending of Circulate Plates

Bending Circular Plate

  • Q = Transverse shear force [N/m]
  • q = transverse distributed load intensity [N/m2]
  • D = Flexural rigidity [N-m]
  • w = out of plane deflection at any radius 'r' [m]
  • R = Radius of the plate [m]
  • t = thickness of the plate [m]
  • E = Modulus of elasticity [Pa]
  • ν = Poisson’s ratio

Bending Profile Equation of Circular Plate

Strength of Materials: Book and Contents

The book by H. Timoshenko is considered as master-piece on this subject. The content of volume 2 are described below.

SOM Timoshenko Cover





Some Solved Examples from the Book

SOM Conical-Ring-1

SOM Conical-Ring-2

Book on Fracuture Mechanics

Fracture Mechanics Cover

Fracture Mechanics TOC 1

Fracture Mechanics TOC 2

Fracture Mechanics TOC 3

Fracture Mechanics TOC 4

Kinematics: Motions in Machines

The absolute motion analysis methods (also known as parametric method) requires specification of position s(P) of point of interest 'P' as a function of angle or Cartesion coordinates. Usually, sine or cosine rules are applied to derive an analytical equation. Chain rule is used while taking derivatives of the position to get instantaneous velocity and acceleration of point 'P'.

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