Design of Machine Elements: Volume II by J.B.K. Das
This book is a comprehensive guide to the design and analysis of various machine elements, such as curved beams, springs, gears, shafts, bearings, clutches, brakes, belts, chains and flywheels. The book covers the fundamental concepts, description, terminology, force analysis and methods of analysis and design of these elements using both analytical and numerical approaches. The book also includes numerous solved examples, problems and exercises to reinforce the theoretical concepts and practical applications.
The book is intended for undergraduate and postgraduate students of mechanical engineering, as well as practicing engineers and designers who are involved in the design and development of machine elements. The book is divided into two volumes: Volume I deals with the design of machine elements based on strength, stiffness and failure criteria; while Volume II focuses on the design of machine elements based on kinematics, dynamics and tribology.
The book is written by J.B.K. Das, who is a professor of mechanical engineering at National Institute of Technology Karnataka (NITK), Surathkal. He has over 35 years of teaching and research experience in the field of machine design. He has authored several books and papers on various aspects of machine design and engineering mechanics.One of the topics covered in this book is the design and analysis of curved beams. Curved beams are structural elements that have a curved shape and are subjected to bending moments. Unlike straight beams, curved beams have a non-uniform stress distribution over the cross section and a variable radius of curvature along the length. Therefore, the analysis of curved beams requires some modifications to the basic theory of bending.
The book introduces the Winkler's theory of curved beams, which is based on the following assumptions:
The cross section has an axis of symmetry in a plane along the length of the beam.
Plane cross sections remain plane after bending.
The modulus of elasticity is the same in tension as in compression.
The book derives the expressions for the normal stress, shear stress, and bending moment in a curved beam subjected to pure bending or combined bending and axial force. The book also provides formulas for some common cross sections, such as rectangular, circular, I-shaped, and T-shaped sections. The book illustrates the application of these formulas with several solved examples and problems.Curved beams are widely used in various applications and industries, such as bridges, cranes, aircraft, and machinery. For example, curved beams are often employed in arch bridges to span large distances, tower cranes to lift heavy loads, wings to generate lift, and gears to transmit motion[^1^]. These applications require curved beams to have high performance and efficiency in terms of strength, stability, and durability.
To optimize the design and analysis of curved beams, it is important to consider the effects of bending and torsion on the stress state and deformation of the beam. Bending and torsion are the main loading conditions that curved beams are subjected to in service. Bending causes normal stress that varies along the cross section of the beam, while torsion causes shear stress that also varies along the cross section. The combination of bending and torsion results in a complex stress state that can lead to failure or excessive deflection of the beam.
Therefore, it is necessary to use appropriate methods and tools to calculate the stress and deflection of curved beams under different loading scenarios. Some of the methods and tools that can be used are:
The Winkler-Bach formula: This is a simple formula that relates the normal stress due to bending to the bending moment, the radius of curvature, and the position of the centroid of the cross section[^2^]. This formula is valid for pure bending and symmetric cross sections.
The Prandtl's membrane analogy: This is a graphical method that compares the torsion problem to a membrane problem with the same boundary conditions[^2^]. This method can be used to find the shear stress due to torsion for any cross section shape.
The principle of superposition: This is a general principle that states that the total stress at any point is equal to the sum of the stresses due to bending and torsion separately[^2^]. This principle can be used to find the combined stress state for any loading condition.
The finite element method: This is a numerical method that divides the beam into small elements and solves a system of equations for each element[^3^]. This method can handle complex geometries, loading conditions, and material properties.
By using these methods and tools, engineers and designers can optimize the performance and efficiency of curved beams in various applications and industries. 061ffe29dd
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