The document discusses:
1) Transformation of stresses and strains in 2D planes including equations of transformation, principal stresses, and Mohr's circles.
2) Columns including column theory, buckling theory, and Euler's formula.
3) Finding stresses on inclined planes for uniaxial, biaxial, and general 2D stress elements including principal stresses and maximum shear stress.
Module1 flexibility-2-problems- rajesh sirSHAMJITH KM
This document discusses the flexibility method for analyzing structures. It provides the definitions of flexibility and stiffness influence coefficients and describes how to develop flexibility matrices for truss, beam, and frame elements using the physical and energy approaches. It then shows how to assemble the total flexibility matrix of a structure and use it to analyze simple structures like plane trusses, continuous beams, and plane frames. The document includes an example problem of a two-member structure to illustrate the flexibility method steps, such as determining static indeterminacy, developing member and system flexibility matrices, evaluating joint displacements and member end actions.
This document discusses approximate analysis methods for multi-storey frames under vertical and lateral loads. It introduces the substitute frame method, portal method, and cantilever method for analyzing frames. An example problem demonstrates using the substitute frame method to analyze a frame for vertical loads, distributing fixed end moments using distribution factors. Homework is assigned to analyze another frame using the cantilever method under given loading conditions.
This document provides an overview of mechanics of solids (or strength of materials) including definitions of key terms like stress, strain, elasticity and their relationships. It discusses stress analysis for axially loaded members and introduces various stress conditions in 2D and 3D spaces. Stress is defined as internal resistance against deformation while strain is a measure of deformation. Different material types like isotropic, orthotropic and their elastic relationships are also covered.
This document discusses the direct stiffness method for structural analysis. It begins by introducing the direct stiffness method and its key aspects, including using member stiffness matrices to express actions and displacements at both ends of each member. It then provides examples of applying the direct stiffness method to analyze a plane truss member and plane frame member. This involves deriving the member stiffness matrices in local coordinates, and transforming displacement, load, and stiffness matrices between local and global coordinate systems using rotation matrices.
This document discusses the flexibility method for structural analysis. The flexibility method involves determining flexibility coefficients by applying unit loads corresponding to redundant forces and calculating the resulting displacements. These flexibility coefficients are then used to calculate the redundant forces needed to satisfy compatibility conditions. The flexibility matrices for different structural elements are developed. Joint displacements, member end actions, and support reactions can be determined by incorporating the flexibility coefficients into the basic computations. Examples are provided to illustrate the flexibility method for a continuous beam with one redundant and for determining various outputs like redundants, joint displacements, and reactions.
Module1 1 introduction-tomatrixms - rajesh sirSHAMJITH KM
This document provides an introduction to matrix methods for structural analysis. It discusses key concepts such as flexibility and stiffness matrices, and their application to trusses, beams, and frames. It also covers types of framed structures, static indeterminacy, actions and displacements, equilibrium, compatibility conditions, and the relationships between flexibility, stiffness, actions and displacements. Matrix methods allow the analysis of statically indeterminate structures by transforming them into a set of simultaneous equations that can be solved using computer programs.
This document discusses the stiffness method for structural analysis. It begins by introducing the stiffness method and its key concepts, such as using joint displacements as the primary unknowns and establishing a relationship between member forces and displacements through stiffness matrices. It then provides examples of calculating stiffness matrices for different structural elements, including beams, trusses, frames, grids, and space frames. The document also explains how to develop the total stiffness matrix of a structure by assembling the member stiffness matrices using a displacement transformation matrix. Finally, it formalizes the stiffness method through the governing equations relating member forces, displacements, and loads.
This document provides an overview of structural dynamics and free vibration analysis of single degree of freedom (SDOF) systems. It defines key terms like natural frequency, damping, and logarithmic decrement. Methods are presented for analyzing the free vibration of undamped and damped SDOF systems under initial displacement conditions. Examples are provided to demonstrate calculating the natural frequency, time period, amplitude, and displacement over time of vibrating SDOF structures.
Module1 flexibility-2-problems- rajesh sirSHAMJITH KM
This document discusses the flexibility method for analyzing structures. It provides the definitions of flexibility and stiffness influence coefficients and describes how to develop flexibility matrices for truss, beam, and frame elements using the physical and energy approaches. It then shows how to assemble the total flexibility matrix of a structure and use it to analyze simple structures like plane trusses, continuous beams, and plane frames. The document includes an example problem of a two-member structure to illustrate the flexibility method steps, such as determining static indeterminacy, developing member and system flexibility matrices, evaluating joint displacements and member end actions.
This document discusses approximate analysis methods for multi-storey frames under vertical and lateral loads. It introduces the substitute frame method, portal method, and cantilever method for analyzing frames. An example problem demonstrates using the substitute frame method to analyze a frame for vertical loads, distributing fixed end moments using distribution factors. Homework is assigned to analyze another frame using the cantilever method under given loading conditions.
This document provides an overview of mechanics of solids (or strength of materials) including definitions of key terms like stress, strain, elasticity and their relationships. It discusses stress analysis for axially loaded members and introduces various stress conditions in 2D and 3D spaces. Stress is defined as internal resistance against deformation while strain is a measure of deformation. Different material types like isotropic, orthotropic and their elastic relationships are also covered.
This document discusses the direct stiffness method for structural analysis. It begins by introducing the direct stiffness method and its key aspects, including using member stiffness matrices to express actions and displacements at both ends of each member. It then provides examples of applying the direct stiffness method to analyze a plane truss member and plane frame member. This involves deriving the member stiffness matrices in local coordinates, and transforming displacement, load, and stiffness matrices between local and global coordinate systems using rotation matrices.
This document discusses the flexibility method for structural analysis. The flexibility method involves determining flexibility coefficients by applying unit loads corresponding to redundant forces and calculating the resulting displacements. These flexibility coefficients are then used to calculate the redundant forces needed to satisfy compatibility conditions. The flexibility matrices for different structural elements are developed. Joint displacements, member end actions, and support reactions can be determined by incorporating the flexibility coefficients into the basic computations. Examples are provided to illustrate the flexibility method for a continuous beam with one redundant and for determining various outputs like redundants, joint displacements, and reactions.
Module1 1 introduction-tomatrixms - rajesh sirSHAMJITH KM
This document provides an introduction to matrix methods for structural analysis. It discusses key concepts such as flexibility and stiffness matrices, and their application to trusses, beams, and frames. It also covers types of framed structures, static indeterminacy, actions and displacements, equilibrium, compatibility conditions, and the relationships between flexibility, stiffness, actions and displacements. Matrix methods allow the analysis of statically indeterminate structures by transforming them into a set of simultaneous equations that can be solved using computer programs.
This document discusses the stiffness method for structural analysis. It begins by introducing the stiffness method and its key concepts, such as using joint displacements as the primary unknowns and establishing a relationship between member forces and displacements through stiffness matrices. It then provides examples of calculating stiffness matrices for different structural elements, including beams, trusses, frames, grids, and space frames. The document also explains how to develop the total stiffness matrix of a structure by assembling the member stiffness matrices using a displacement transformation matrix. Finally, it formalizes the stiffness method through the governing equations relating member forces, displacements, and loads.
This document provides an overview of structural dynamics and free vibration analysis of single degree of freedom (SDOF) systems. It defines key terms like natural frequency, damping, and logarithmic decrement. Methods are presented for analyzing the free vibration of undamped and damped SDOF systems under initial displacement conditions. Examples are provided to demonstrate calculating the natural frequency, time period, amplitude, and displacement over time of vibrating SDOF structures.
This document discusses mechanics of solids, specifically torsion and bending moment. It begins with an overview and objectives of Module II which covers torsion of circular elastic and inelastic bars, as well as axial force, shear force, and bending moment diagrams. It then provides explanations, equations, and examples relating to torsion, including assumptions in torsion theory, the torsion equation, torsional rigidity, power transmission in torsion, and example problems calculating torque and shaft diameters. It also discusses statically indeterminate shafts and torsion of inelastic circular bars.
This document analyzes an indeterminate truss using the flexibility method. It first calculates the static degree of indeterminacy as 2. It then selects members OA and OD as redundant members. The truss is analyzed when each redundant member is given a force of 1 kN. A flexibility matrix equation is developed and solved to determine the redundant forces, which are then used to calculate the final forces in each member. The final forces show member OA in tension and member OD in compression.
This document provides an overview of plastic analysis for structural elements. It discusses key concepts like plastic hinges, plastic section modulus, shape factors, and load factors. Plastic analysis is used to determine the ultimate or collapse load of a structure by considering the redistribution of moments that occurs after sections yield. Common failure mechanisms for determinate and indeterminate beams involve the formation of one or more plastic hinges. Methods for plastic analysis include the static/equilibrium method and kinematic/mechanism method. Examples are given for calculating the collapse load of simple structural configurations using these methods.
This document discusses analysis of statically indeterminate structures using the force method. It begins by introducing degree of static indeterminacy and examples of structures with different degrees, such as beams, trusses, and frames. It then covers kinematic indeterminacy and examples. Key analysis methods discussed include the method of consistent deformations, Clapeyron's theorem of three moments, and minimum potential energy theorems. Structural elements and their typical deformations are also summarized.
This document discusses the basics of plastic design of steel structures. It introduces plastic analysis as a way to calculate the actual failure load of a structure, which can be greater than the elastic load capacity. Idealized stress-strain curves are used to model the behavior of elastic-plastic materials. The behavior of cross-sections under bending is examined, including the formation of plastic hinges that allow rotation beyond the yield point. Assumptions and methods for plastic analysis of cross-section properties like yield moment and plastic moment are also presented.
This document discusses analysis of statically indeterminate structures using the force method. It begins by introducing statically and kinematically indeterminate structures. It then discusses the degree of static indeterminacy for different types of structures like beams, trusses, frames, and grids. It also discusses the different types of deformations that can occur in these structures. The document then covers the concepts of equilibrium, compatibility, and the force method of analysis using the method of consistent deformation. Several examples are provided to illustrate the calculation of degree of static indeterminacy for beams, trusses and frames. It also discusses kinematic indeterminacy and provides examples of its calculation for different structures.
The document discusses the design of two-way slabs and staircases. It provides guidance on initial proportioning of slab thickness, analysis of bending moments using code provided coefficients, design of flexural reinforcement, checking for deflection and shear limits, and detailing of reinforcement. Specific examples are presented to demonstrate the design of simply supported and torsionally restrained slab panels with reinforcement calculated and laid out. Staircases are also briefly mentioned including different types like waist slab and folded plate.
The document describes the flexibility method for analyzing statically indeterminate beams. It discusses:
- James Clerk Maxwell published the first treatment of the flexibility method in 1864, which was later extended by Otto Mohr.
- The method introduces compatibility equations involving displacements at redundant forces to provide additional equations for solving statically indeterminate structures.
- For a two-span beam example, the redundant reaction at the middle support is chosen, compatibility equations are written, and the flexibility matrix method is demonstrated to solve for redundant forces.
This document discusses the flexibility matrix method for analyzing statically indeterminate structures. It begins by introducing the flexibility matrix method and its formulation. The flexibility matrix relates displacements in a structure to applied forces. Examples are provided to demonstrate applying the flexibility matrix method to analyze pin-jointed plane trusses, continuous beams, and rigid jointed portal frames involving 3 or fewer unknowns. The steps of the method are outlined and illustrated through worked examples.
The document discusses the slope deflection method of structural analysis. It begins by deriving the fundamental slope deflection equations that relate end moments, slopes, and deflections of a beam. It then presents an example problem demonstrating the full procedure of applying the slope deflection method, which involves writing slope deflection equations for each member, establishing joint equilibrium equations, solving for unknown displacements, and substituting these into the slope deflection equations to determine end moments. The method provides a general approach for the analysis of continuous beams and frames.
The document discusses the direct stiffness method for analyzing truss structures. This method treats each individual truss element as a structure and develops the element stiffness matrix. Transformation matrices are used to relate element deformations to structure deformations. The total structure stiffness matrix is obtained by assembling the individual element stiffness matrices based on how the elements are connected at joints in the structure. This direct stiffness method forms the basis for computer programs to analyze truss structures.
The document discusses Karmarkar's projective scaling method for solving linear programming problems. It provides an overview of the method, including expressing LP problems in a standard form suitable for the method. The method involves iterative steps of computing new values for variables to move toward the optimal solution interior to the feasible region. An example applying the method to a sample LP problem is shown over multiple iterations.
The document discusses the displacement or stiffness method for analyzing structures. It begins by introducing the stiffness method, which allows the same approach to analyze both statically determinate and indeterminate structures. The key concepts are that nodal displacements are the unknowns, and equilibrium equations relating the unknown displacements to known stiffness coefficients are written and solved.
It then defines stiffness, stiffness coefficients, and stiffness matrices. The stiffness of a member is the force required to produce a unit displacement. Stiffness coefficients relate the forces and displacements at nodes. Element stiffness matrices are developed and combined to form the overall structure stiffness matrix, which relates the total nodal forces and displacements.
This document discusses stresses in beams, including:
1) Bending stresses in beams, shear flow, and shearing stress formulae for beams.
2) Inelastic bending of beams and deflection of beams using various calculation methods.
3) Elementary treatment of statically indeterminate beams like fixed and continuous beams.
It provides theories, formulae, and examples for calculating stresses in beams undergoing bending loads.
This document provides an overview of structural dynamics and free vibration analysis of single degree of freedom systems. It defines key concepts like natural frequency, damping, and logarithmic decrement. Methods for analyzing undamped and damped free vibration are presented. Examples show how to calculate the natural frequency, time period, amplitude, and displacement as a function of time for undamped systems subjected to initial displacement or velocity conditions. Analysis of damped systems models the response as a decaying exponential function.
1) The document defines trigonometric functions using right triangles and the unit circle. It provides formulas for trig functions, inverse trig functions, and laws of sines, cosines, and tangents.
2) Tables give values of trig functions for angles on the unit circle, along with properties like domain, range, and periodicity.
3) The cheat sheet is a reference for definitions, formulas, and properties of trigonometric functions.
This document provides a summary of common mathematical and calculus formulas:
1) It lists many basic mathematical formulas such as logarithmic, exponential, trigonometric, and algebraic formulas.
2) It also presents various differentiation formulas including the chain rule, product rule, quotient rule, and formulas for deriving trigonometric, exponential, and logarithmic functions.
3) Integration formulas and theorems are covered including integration by parts, substitutions, and the Fundamental Theorem of Calculus.
Second-order Cosmological Perturbations Engendered by Point-like MassesMaxim Eingorn
R. Brilenkov and M. Eingorn, Second-order cosmological perturbations engendered by point-like masses, ApJ 845 (2017) 153: http://iopscience.iop.org/article/10.3847/1538-4357/aa81cd
In the ΛCDM framework, presenting nonrelativistic matter inhomogeneities as discrete massive particles, we develop the second‐order cosmological perturbation theory. Our approach relies on the weak gravitational field limit. The derived equations for the second‐order scalar, vector, and tensor metric corrections are suitable at arbitrary distances, including regions with nonlinear contrasts of the matter density. We thoroughly verify fulfillment of all Einstein equations, as well as self‐consistency of order assignments. In addition, we achieve logical positive results in the Minkowski background limit. Feasible investigations of the cosmological backreaction manifestations by means of relativistic simulations are also outlined.
The document presents mathematical analysis of the Birch and Swinnerton-Dyer conjecture using power series. It derives formulas for the coefficients of the power series involving the sums and derivatives of the coefficients. It then applies these formulas and the Legendre polynomial to obtain an equation relating the coefficients to the value of the parameter λ.
This document discusses methods for calculating stress in soil, including Boussinesq's and Westergaard's equations for point loads, and formulas for uniformly distributed loads, line loads, and loads on circular areas. It also introduces concepts like pressure bulbs and isobars, which are spatial surfaces representing equal vertical pressure beneath a loaded area. Methods like Newmark charts can be used to determine stress distributions and the significant depth of a pressure bulb corresponding to a given percentage of the foundation contact pressure.
The document defines trigonometric functions using right triangles and the unit circle. It lists properties of the trig functions including domain, range, period, formulas, and identities. It also covers inverse trig functions, the laws of sines, cosines, and tangents, and the unit circle.
This document discusses mechanics of solids, specifically torsion and bending moment. It begins with an overview and objectives of Module II which covers torsion of circular elastic and inelastic bars, as well as axial force, shear force, and bending moment diagrams. It then provides explanations, equations, and examples relating to torsion, including assumptions in torsion theory, the torsion equation, torsional rigidity, power transmission in torsion, and example problems calculating torque and shaft diameters. It also discusses statically indeterminate shafts and torsion of inelastic circular bars.
This document analyzes an indeterminate truss using the flexibility method. It first calculates the static degree of indeterminacy as 2. It then selects members OA and OD as redundant members. The truss is analyzed when each redundant member is given a force of 1 kN. A flexibility matrix equation is developed and solved to determine the redundant forces, which are then used to calculate the final forces in each member. The final forces show member OA in tension and member OD in compression.
This document provides an overview of plastic analysis for structural elements. It discusses key concepts like plastic hinges, plastic section modulus, shape factors, and load factors. Plastic analysis is used to determine the ultimate or collapse load of a structure by considering the redistribution of moments that occurs after sections yield. Common failure mechanisms for determinate and indeterminate beams involve the formation of one or more plastic hinges. Methods for plastic analysis include the static/equilibrium method and kinematic/mechanism method. Examples are given for calculating the collapse load of simple structural configurations using these methods.
This document discusses analysis of statically indeterminate structures using the force method. It begins by introducing degree of static indeterminacy and examples of structures with different degrees, such as beams, trusses, and frames. It then covers kinematic indeterminacy and examples. Key analysis methods discussed include the method of consistent deformations, Clapeyron's theorem of three moments, and minimum potential energy theorems. Structural elements and their typical deformations are also summarized.
This document discusses the basics of plastic design of steel structures. It introduces plastic analysis as a way to calculate the actual failure load of a structure, which can be greater than the elastic load capacity. Idealized stress-strain curves are used to model the behavior of elastic-plastic materials. The behavior of cross-sections under bending is examined, including the formation of plastic hinges that allow rotation beyond the yield point. Assumptions and methods for plastic analysis of cross-section properties like yield moment and plastic moment are also presented.
This document discusses analysis of statically indeterminate structures using the force method. It begins by introducing statically and kinematically indeterminate structures. It then discusses the degree of static indeterminacy for different types of structures like beams, trusses, frames, and grids. It also discusses the different types of deformations that can occur in these structures. The document then covers the concepts of equilibrium, compatibility, and the force method of analysis using the method of consistent deformation. Several examples are provided to illustrate the calculation of degree of static indeterminacy for beams, trusses and frames. It also discusses kinematic indeterminacy and provides examples of its calculation for different structures.
The document discusses the design of two-way slabs and staircases. It provides guidance on initial proportioning of slab thickness, analysis of bending moments using code provided coefficients, design of flexural reinforcement, checking for deflection and shear limits, and detailing of reinforcement. Specific examples are presented to demonstrate the design of simply supported and torsionally restrained slab panels with reinforcement calculated and laid out. Staircases are also briefly mentioned including different types like waist slab and folded plate.
The document describes the flexibility method for analyzing statically indeterminate beams. It discusses:
- James Clerk Maxwell published the first treatment of the flexibility method in 1864, which was later extended by Otto Mohr.
- The method introduces compatibility equations involving displacements at redundant forces to provide additional equations for solving statically indeterminate structures.
- For a two-span beam example, the redundant reaction at the middle support is chosen, compatibility equations are written, and the flexibility matrix method is demonstrated to solve for redundant forces.
This document discusses the flexibility matrix method for analyzing statically indeterminate structures. It begins by introducing the flexibility matrix method and its formulation. The flexibility matrix relates displacements in a structure to applied forces. Examples are provided to demonstrate applying the flexibility matrix method to analyze pin-jointed plane trusses, continuous beams, and rigid jointed portal frames involving 3 or fewer unknowns. The steps of the method are outlined and illustrated through worked examples.
The document discusses the slope deflection method of structural analysis. It begins by deriving the fundamental slope deflection equations that relate end moments, slopes, and deflections of a beam. It then presents an example problem demonstrating the full procedure of applying the slope deflection method, which involves writing slope deflection equations for each member, establishing joint equilibrium equations, solving for unknown displacements, and substituting these into the slope deflection equations to determine end moments. The method provides a general approach for the analysis of continuous beams and frames.
The document discusses the direct stiffness method for analyzing truss structures. This method treats each individual truss element as a structure and develops the element stiffness matrix. Transformation matrices are used to relate element deformations to structure deformations. The total structure stiffness matrix is obtained by assembling the individual element stiffness matrices based on how the elements are connected at joints in the structure. This direct stiffness method forms the basis for computer programs to analyze truss structures.
The document discusses Karmarkar's projective scaling method for solving linear programming problems. It provides an overview of the method, including expressing LP problems in a standard form suitable for the method. The method involves iterative steps of computing new values for variables to move toward the optimal solution interior to the feasible region. An example applying the method to a sample LP problem is shown over multiple iterations.
The document discusses the displacement or stiffness method for analyzing structures. It begins by introducing the stiffness method, which allows the same approach to analyze both statically determinate and indeterminate structures. The key concepts are that nodal displacements are the unknowns, and equilibrium equations relating the unknown displacements to known stiffness coefficients are written and solved.
It then defines stiffness, stiffness coefficients, and stiffness matrices. The stiffness of a member is the force required to produce a unit displacement. Stiffness coefficients relate the forces and displacements at nodes. Element stiffness matrices are developed and combined to form the overall structure stiffness matrix, which relates the total nodal forces and displacements.
This document discusses stresses in beams, including:
1) Bending stresses in beams, shear flow, and shearing stress formulae for beams.
2) Inelastic bending of beams and deflection of beams using various calculation methods.
3) Elementary treatment of statically indeterminate beams like fixed and continuous beams.
It provides theories, formulae, and examples for calculating stresses in beams undergoing bending loads.
This document provides an overview of structural dynamics and free vibration analysis of single degree of freedom systems. It defines key concepts like natural frequency, damping, and logarithmic decrement. Methods for analyzing undamped and damped free vibration are presented. Examples show how to calculate the natural frequency, time period, amplitude, and displacement as a function of time for undamped systems subjected to initial displacement or velocity conditions. Analysis of damped systems models the response as a decaying exponential function.
1) The document defines trigonometric functions using right triangles and the unit circle. It provides formulas for trig functions, inverse trig functions, and laws of sines, cosines, and tangents.
2) Tables give values of trig functions for angles on the unit circle, along with properties like domain, range, and periodicity.
3) The cheat sheet is a reference for definitions, formulas, and properties of trigonometric functions.
This document provides a summary of common mathematical and calculus formulas:
1) It lists many basic mathematical formulas such as logarithmic, exponential, trigonometric, and algebraic formulas.
2) It also presents various differentiation formulas including the chain rule, product rule, quotient rule, and formulas for deriving trigonometric, exponential, and logarithmic functions.
3) Integration formulas and theorems are covered including integration by parts, substitutions, and the Fundamental Theorem of Calculus.
Second-order Cosmological Perturbations Engendered by Point-like MassesMaxim Eingorn
R. Brilenkov and M. Eingorn, Second-order cosmological perturbations engendered by point-like masses, ApJ 845 (2017) 153: http://iopscience.iop.org/article/10.3847/1538-4357/aa81cd
In the ΛCDM framework, presenting nonrelativistic matter inhomogeneities as discrete massive particles, we develop the second‐order cosmological perturbation theory. Our approach relies on the weak gravitational field limit. The derived equations for the second‐order scalar, vector, and tensor metric corrections are suitable at arbitrary distances, including regions with nonlinear contrasts of the matter density. We thoroughly verify fulfillment of all Einstein equations, as well as self‐consistency of order assignments. In addition, we achieve logical positive results in the Minkowski background limit. Feasible investigations of the cosmological backreaction manifestations by means of relativistic simulations are also outlined.
The document presents mathematical analysis of the Birch and Swinnerton-Dyer conjecture using power series. It derives formulas for the coefficients of the power series involving the sums and derivatives of the coefficients. It then applies these formulas and the Legendre polynomial to obtain an equation relating the coefficients to the value of the parameter λ.
This document discusses methods for calculating stress in soil, including Boussinesq's and Westergaard's equations for point loads, and formulas for uniformly distributed loads, line loads, and loads on circular areas. It also introduces concepts like pressure bulbs and isobars, which are spatial surfaces representing equal vertical pressure beneath a loaded area. Methods like Newmark charts can be used to determine stress distributions and the significant depth of a pressure bulb corresponding to a given percentage of the foundation contact pressure.
The document defines trigonometric functions using right triangles and the unit circle. It lists properties of the trig functions including domain, range, period, formulas, and identities. It also covers inverse trig functions, the laws of sines, cosines, and tangents, and the unit circle.
The document defines trigonometric functions using right triangles and the unit circle. It provides formulas for domain, range, period, identities, inverses, and laws involving trig functions like the Law of Sines, Cosines, and Tangents. Key formulas include definitions of sine, cosine, tangent and their inverses, as well as the Pythagorean, double angle, and sum and difference identities.
This document contains important derivatives formulas for bachelor level mathematics. It includes the derivatives of common functions like sin, cos, tan, coth, and expressions involving constants, logarithms, and exponential functions. It also includes the derivatives of composite functions, Leibniz's theorem for computing derivatives of products, and taking higher order derivatives. The document was created by Atiq ur Rehman and is available online at http://www.mathcity.org.
This document provides examples and solutions to problems involving indeterminate forms and improper integrals. It covers various limits that are of the form 0/0 or ∞/∞ and how to evaluate them using techniques like L'Hopital's rule. Some example limits are evaluated directly, while others require applying L'Hopital's rule multiple times. The document also contains problems evaluating definite integrals that result in indeterminate forms, along with their solutions.
Identification of the Mathematical Models of Complex Relaxation Processes in ...Vladimir Bakhrushin
The approach to solving the problem of complex relaxation spectra is presented.
Presentation for the XI International Conference on Defect interaction and anelastic phenomena in solids. Tula, 2007.
Formul me-3074683 Erdi Karaçal Mechanical Engineer University of GaziantepErdi Karaçal
1. The document discusses various topics related to stress analysis and design including moment of inertias, stresses, deflection analysis, design for static strength, fatigue design, tolerances and fits, power screws, and bolted joints.
2. Formulas are provided for calculating stresses and strains under different loading conditions as well as determining critical loads, deflections, endurance limits, and stresses in various mechanical elements.
3. Design considerations for different materials, loading types, and failure theories are outlined for static and fatigue strength analysis. Guidelines for screw thread stresses, efficiency, and joint stiffness are also summarized.
This document provides definitions and formulas for trigonometric functions. It defines the trig functions using both right triangles and the unit circle. It lists important properties like domain, range, period, and formulas for sums, differences, inverses, and the laws of sines, cosines, and tangents.
This document provides definitions and formulas for trigonometric functions. It defines the trig functions using right triangles and the unit circle. It lists important properties like domain, range, period, and formulas for sums, differences, inverses, and the laws of sines, cosines, and tangents.
1. The document discusses principal stresses and planes, describing how to determine the maximum and minimum normal stresses (principal stresses) and their corresponding planes from a state of plane stress.
2. It introduces Mohr's circle as a graphical method to determine principal stresses and maximum shear stresses from the stresses on any plane.
3. Equations are derived relating the principal stresses and maximum shear stress to the normal and shear stresses on any plane using trigonometric functions of the angle between the plane and principal planes.
This document discusses the analysis of laminated composite structures. It outlines the basic assumptions made in the analysis including linear strain-displacement and stress-strain relationships. It defines the strain-displacement relations and stress-strain relations for each layer of a laminate. Stress resultants and force-displacement relations are defined through laminate stiffness and compliance equations. Special classes of laminates are identified and the engineering properties of laminates are discussed. The analysis of laminated composite structures is then introduced.
This document provides examples and solutions for differential equations. It begins with concepts like general solutions involving complex conjugate roots. It then provides worked problems finding general and particular solutions of differential equations with various roots. It discusses repeated roots, finding constants from initial conditions, and complex conjugate roots. It also covers nonhomogeneous and nonlinear differential equations, and using an exponential substitution to solve a differential equation.
Construction Materials and Engineering - Module IV - Lecture NotesSHAMJITH KM
The document discusses various basic components of building construction including substructure, superstructure, foundation, plinth, beams, columns, walls, arches, roofs, slabs, lintels, parapets, staircases, doors, windows and other elements. It provides descriptions of each component, their functions and materials typically used. Foundations discussed include isolated spread footing, wall/strip footing, combined footing, cantilever/strap footing and mat/raft footing for shallow foundations and pile, well/caisson and pier foundations for deep foundations. Flooring materials and requirements are also summarized along with technical terms for doors and windows.
Construction Materials and Engineering - Module III - Lecture NotesSHAMJITH KM
The document discusses various construction materials and methods. It covers topics like masonry, bricks, stone masonry, types of bonds, hollow block masonry, partition walls, modern construction methods, and damp proof courses. Masonry involves arranging masonry units like stone or bricks with mortar. There are different types of bonds used in brick masonry like stretcher bond, header bond, English bond and Flemish bond. Modern methods include framed construction, prefabricated construction and earthquake resistant construction. Damp proof courses are provided to prevent entry of moisture into buildings.
Construction Materials and Engineering - Module II - Lecture NotesSHAMJITH KM
This document provides information on various construction materials including paints, plastics, rubber, and aluminum. It discusses the ingredients, properties, types, and applications of paints. It also outlines the classification, characteristics, uses, advantages, and limitations of plastics. Details are provided on types of rubber like natural and synthetic rubber. Applications of aluminum in construction are also mentioned.
Construction Materials and Engineering - Module I - Lecture NotesSHAMJITH KM
This document provides information on various construction materials used in building, including their classification and properties. It discusses stones, classified as igneous, sedimentary and metamorphic based on their geological formation. Bricks and tiles are described as clay products manufactured through processes of preparation, moulding, drying and burning. The characteristics of good building stones and various stone varieties are also summarized.
Computing fundamentals lab record - PolytechnicsSHAMJITH KM
The document is a lab record for a computing fundamentals course. It contains instructions for students on proper lab conduct and procedures. It also outlines 25 experiments to be completed, covering topics like computer hardware, operating systems, word processing, spreadsheets, programming, and calculations. General instructions are provided for safety and proper use of equipment in the computing lab.
Cement is a binding agent that undergoes hydration when mixed with water. There are various types of cement including ordinary Portland cement (OPC), rapid hardening cement, and sulphate resisting cement. Cement provides early strength through C3S and later strength through C2S. Heat is generated during cement hydration through an exothermic reaction. Proper storing, grading of aggregates, minimizing segregation, and adding admixtures can improve the properties of concrete.
നബി(സ)യുടെ നമസ്കാരം - രൂപവും പ്രാര്ത്ഥനകളുംSHAMJITH KM
- \_n(k) regularly led prayers and provided guidance during prayer gatherings.
- He taught to pray with humility and focus, avoiding idle thoughts or actions that distract from prayer.
- The summary provides guidance on proper prayer etiquette like standing, bowing, and order of movements based on hadith sources.
Design of simple beam using staad pro - doc fileSHAMJITH KM
The document describes designing a simple beam using STAAD.Pro software. It involves generating the beam geometry, applying loads and supports, analyzing the beam, and reviewing the results, which include the loading diagram, shear force diagram, bending moment diagram, deflection pattern, input file, concrete takeoff, and concrete design details. The key steps are 1) creating the beam model in STAAD.Pro, 2) applying the loading and support conditions, 3) analyzing the beam, and 4) reviewing the output results.
The document describes designing a simple beam using STAAD.Pro software. It involves generating the beam geometry, applying loads and supports, analyzing the beam, and designing the beam for concrete. Key steps include assigning the beam properties, applying a fixed support at one end and distributed and point loads, obtaining the loading diagram, shear force and bending moment diagrams, and running the concrete design. The output includes structural drawings, input files, concrete takeoff, and beam design details.
Python programs - PPT file (Polytechnics)SHAMJITH KM
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Mechanics of structures module4
1. Mechanics of structuresMechanics of structures
Stresses in 2D plane,
ColumnsColumns
Dr. Rajesh K. N.
Assistant Professor in Civil EngineeringAssistant Professor in Civil Engineering
Govt. College of Engineering, Kannur
Dept. of CE, GCE Kannur Dr.RajeshKN
1
2. Module IV
Transformation of stresses and strains (two dimensional case only)
Module IV
Transformation of stresses and strains (two-dimensional case only) -
equations of transformation - principal stresses - mohr's circles of
stress and strain - strain rosettes - compound stresses - superposition
d it li it tiand its limitations –
Eccentrically loaded members - columns - theory of columns -
buckling theory - Euler's formula - effect of end conditions -
eccentric loads and secant formula
Dept. of CE, GCE Kannur Dr.RajeshKN
2
3. Analysis of plane stress and plane strainy p p
cosnA A θ=
P
L
AA
θ
P
A
σ = cos
n
n
A
A
θ
=
AAn
A
2
2cos cos
cosn
P P
A A
θ θ
σ σ θ= = =
cosP θ
θ
nA A
sin sin cos sin2P Pθ θ θ σ θ
τ
P
θ
sinP θAn
θ
Dept. of CE, GCE Kannur Dr.RajeshKN
3
2
n
nA A
τ = = =
4. 2 2
σ σ τ+
( ) ( )
2 22
cos sin cos
R n nσ σ τ
σ θ σ θ θ
= +
= +
Resultant stress on inclined plane
2 2
cos cos sin
cos
σ θ θ θ
σ θ
= +
=
0 2 cos sin 0nd
d
σ
σ θ θ
θ
= ⇒ =For maximum normal stress,
0
sin2 0 0σ θ θ= ⇒ =
0
When 0
P
θ σ σ= = =
0 2 0ndτ
θFor maximum shear stress
,maxWhen 0 , n
A
θ σ σ= = =
0 0
0 cos2 0
2 90 45
n
d
σ θ
θ
θ θ
= ⇒ =
⇒ = ⇒ =
For maximum shear stress,
Dept. of CE, GCE Kannur Dr.RajeshKN
4
0
,maxWhen 45 ,
2 2
n
P
A
σ
θ τ= = =
6. Element under pure shear
τ
B
θ σ
nτ
τ
θ
τ
θ nσ
ττ θ
cos sinBC AC ABσ τ θ τ θ= − −
A C
ττ
. .cos . .sin
sin cos cos sin
sin2
n
n
BC AC ABσ τ θ τ θ
σ τ θ θ τ θ θ
σ τ θ
= − −
= − −
. . .sin . .cosn BC AC AB
AC AB
τ τ θ τ θ= − +
sin2nσ τ θ= −
2 2
. .sin . .cos
cos sin
n
n
AC AB
BC BC
τ τ θ τ θ
τ τ θ τ θ
= − +
= −
Dept. of CE, GCE Kannur Dr.RajeshKN
6
cos2
n
nτ τ θ=
7. Element under biaxial normal stress
B
nτ
yσ
θ nσ
n
σ
xσθxσ
A C
xσ
yσ
sin cosBC AB ACτ σ θ σ θ= −cos sinBC AB ACσ σ θ σ θ= +
yσ
. . .sin . .cos
. .sin . .cos
n x y
n x y
BC AB AC
AB AC
BC BC
τ σ θ σ θ
τ σ θ σ θ
=
= −
. .cos . .sin
. .cos . .sin
n x y
n x y
BC AB AC
AB AC
BC BC
σ σ θ σ θ
σ σ θ σ θ
= +
= +
( )
cos sin sin cos
sin2
n x y
BC BC
τ σ θ θ σ θ θ
θ
= −2 2
cos sin
y
n x y
BC BC
σ σ θ σ θ= +
Dept. of CE, GCE Kannur Dr.RajeshKN
7
( ) 2
n x yτ σ σ= −
8. Element under a general two-dimensional stress
xyτ
xyτ
B
nτ
τ
σ
θ θ nσ
xσ
xyτ
xσ
xyτ
xyτ
A C
x
xyτ
. .cos . .sin . .cos . .sinn x y xy xyBC AB AC AC ABσ σ θ σ θ τ θ τ θ= + − −
yσ
xy
yσ
. .cos . .sin . .cos . .sinn x y xy xy
AB AC AC AB
BC BC BC BC
σ σ θ σ θ τ θ τ θ= + − −
2 2
cos sin sin2n x y xyσ σ θ σ θ τ θ= + −
+
Dept. of CE, GCE Kannur Dr.RajeshKN
8
cos2 sin2
2 2
x y x y
n xy
σ σ σ σ
σ θ τ θ
+ −
= + −
9. i iBC AB AC AB ACθ θ θ θ. . .sin . .cos . .cos . .sin
sin cos cos sin
n x y xy xyBC AB AC AB AC
AB AC AB AC
τ σ θ σ θ τ θ τ θ
τ σ θ σ θ τ θ τ θ
= − + −
= − + −
2 2
. .sin . .cos . .cos . .sin
cos sin sin cos cos sin
n x y xy xy
n x y xy xy
BC BC BC BC
τ σ θ σ θ τ θ τ θ
τ σ θ θ σ θ θ τ θ τ θ
= − + −
= − + −n x y xy xy
sin2 cos2
2
x y
n xy
σ σ
τ θ τ θ
−
= +
2
n xy
Note: In the above derivations,
Sign convention for θ: Anticlockwise angle is +ve.g g
Sign convention for shear stress: With respect to a point inside
the element, clockwise shear stress is +ve.
Dept. of CE, GCE Kannur Dr.RajeshKN
9
the element, clockwise shear stress is +ve.
10. σ σ σ σ+ −
cos2 sin2
2 2
x y x y
n xy
σ σ σ σ
σ θ τ θ
+
= + −
σ σ−
Th b ti i th l d t ti l ( h ) t
sin2 cos2
2
x y
n xy
σ σ
τ θ τ θ= +
The above equations give the normal and tangential (shear) stresses on
any plane inclined at θ with the vertical.
To find maximum/minimum value of normal stress
σ∂
( )0nσ
θ
∂
=
∂
( )sin2 2 cos2 0x y xyσ σ θ τ θ⇒ − − − =
2
tan2 xyτ
θ
−
⇒
( )
tan2 y
x y
θ
σ σ
⇒ =
−
Dept. of CE, GCE Kannur Dr.RajeshKN
10
11. 2
σ σ−⎛ ⎞ sin2 xyτ
θ
−
=
xyτ−
( )
2
2
x y
xy
σ σ
τ
⎛ ⎞
+⎜ ⎟
⎝ ⎠
2θ
2
2
sin2
2
y
x y
xy
θ
σ σ
τ
=
−⎛ ⎞
+⎜ ⎟
⎝ ⎠
( )
2
x yσ σ−
2
cos2 x yσ σ
θ
−
=
2
2
2
2
x y
xy
σ σ
τ
−⎛ ⎞
+⎜ ⎟
⎝ ⎠
cos2 sin2
2 2
x y x y
n xy
σ σ σ σ
σ θ τ θ
+ −
= + −
2
2
max
2 2
x y x y
xy
σ σ σ σ
σ τ
+ −⎛ ⎞
⇒ = + +⎜ ⎟
⎝ ⎠⎝ ⎠
2
2
i
x y x yσ σ σ σ
σ τ
+ −⎛ ⎞
= − +⎜ ⎟
Dept. of CE, GCE Kannur Dr.RajeshKN
11
min
2 2
xyσ τ+⎜ ⎟
⎝ ⎠
12. 222
2
max,min 1,3
2 2
x y x y
xy
σ σ σ σ
σ σ τ
+ −⎛ ⎞
= = ± +⎜ ⎟
⎝ ⎠
( )
2
tan2 xy
x y
τ
θ
σ σ
−
=
−
2
2
max,min 1,3
2 2
x y x y
xy
σ σ σ σ
σ σ τ
+ −⎛ ⎞
= = ± +⎜ ⎟
⎝ ⎠
( )
2
tan2 xy
x y
τ
θ
σ σ
−
=
−
Principal stresses Principal planes
σ σ−
Maximum shear stress
We know, sin2 cos2
2
x y
n xy
σ σ
τ θ τ θ= +
F i h t 0nτ∂
For maximum shear stress, 0n
θ
=
∂
( )cos2 2 sin2 0x y xyσ σ θ τ θ⇒ − − =( )x y xy
( )tan2
x yσ σ
θ
−
⇒ =
Dept. of CE, GCE Kannur Dr.RajeshKN
12
tan2
2 xy
θ
τ
⇒
13. 2
( )
2
For max normal stress, tan2 xy
n
x y
τ
θ
σ σ
−
=
−
( )For max shear stress, tan2
2
x y
s
σ σ
θ
τ
−
=
1
tan2θ
−
=
2 xyτ tan2 nθ
( )0
cot 2 tan 2 90n nθ θ= − = ±( )n n
0
2 2 90s nθ θ= ±s n
0
45s nθ θ= ±
Hence, planes of maximum shear stress are at 450 to the principal
planes
Dept. of CE, GCE Kannur Dr.RajeshKN
13
p
14. To get maximum shear stress
( )
2
2
2
x y
xy
σ σ
τ
−⎛ ⎞
+⎜ ⎟
⎝ ⎠
2
2
cos2 xy
s
x y
τ
θ
σ σ
τ
=
−⎛ ⎞
+⎜ ⎟
g
τ
( )
2
x yσ σ− 2
y⎜ ⎟
⎝ ⎠
2 sθ
2
y
xyτ+⎜ ⎟
⎝ ⎠
sin2 x yσ σ
θ
−
=xyτ 2
2
sin2
2
2
s
x y
xy
θ
σ σ
τ
=
−⎛ ⎞
+⎜ ⎟
⎝ ⎠
,max 2 2
2 2
2
2
x y x y xy
n xy
x y x y
σ σ σ σ τ
τ τ
σ σ σ σ
τ τ
− −
= +
− −⎛ ⎞ ⎛ ⎞
+ +⎜ ⎟ ⎜ ⎟
2
σ σ−⎛ ⎞
2
2 2
xy xyτ τ+ +⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠
2
σ σ⎛ ⎞2
,max
2
x y
n xy
σ σ
τ τ
⎛ ⎞
= +⎜ ⎟
⎝ ⎠
2
,max,min
2
x y
n xy
σ σ
τ τ
−⎛ ⎞
= ± +⎜ ⎟
⎝ ⎠
Dept. of CE, GCE Kannur Dr.RajeshKN
14
15. 2
⎛ ⎞ 2
1,3
2
We kno
2
w, x y x y
xy
σ σ σ σ
σ τ
+ −⎛ ⎞
= ± +⎜ ⎟
⎝ ⎠
2
2
1 3 ,max2 2
2
x y
xy n
σ σ
σ σ τ τ
−⎛ ⎞
− = + =⎜ ⎟
⎝ ⎠
1 3
,max
2
n
σ σ
τ
−
=
2
To get normal stress on planes of maximum shear stress
2 2
2 2
2 2
x y x y xy x y
n xy
x y x y
σ σ σ σ τ σ σ
σ τ
σ σ σ σ
+ − −
= + −
− −⎛ ⎞ ⎛ ⎞2 2
2
2 2
x y x y
xy xy
σ σ σ σ
τ τ
⎛ ⎞ ⎛ ⎞
+ +⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠
x yσ σ+
th l f h t
Dept. of CE, GCE Kannur Dr.RajeshKN
15
2
x y
nσ = , on the planes of max shear stressaverageσ=
16. Problem: Find the principal stresses (including principal planes) and
maximum shear stress (including its plane)
2
60 N
2
80 N mm
Principal stresses
2
120 N mm
2
60 N mm
2
120 N mm
2
2
max,min 1,3
2 2
x y x y
xy
σ σ σ σ
σ σ τ
+ −⎛ ⎞
= = ± +⎜ ⎟
⎝ ⎠
Principal stresses
2
80 N mm
2
60 N mm
⎝ ⎠
2
2
max min 1 3
120 80 120 80
60σ σ
− − − +⎛ ⎞
= = ± +⎜ ⎟
⎝ ⎠
max,min 1,3 60
2 2
σ σ ± +⎜ ⎟
⎝ ⎠
100 63 24σ σ= = − ±max,min 1,3 100 63.24σ σ= = ±
2
max 1 163.24 N mmσ σ∴ = = −max 1
2
min 3and 36.75 N mmσ σ= = −
Dept. of CE, GCE Kannur Dr.RajeshKN
16
19. cos2 sin2x y x yσ σ σ σ
σ θ τ θ
+ −
= +
Mohr’s circle
cos2 sin2
2 2
n xyσ θ τ θ= + −
cos2 sin2x y x yσ σ σ σ
σ θ τ θ
+ −
= i
sin2 cos2
2
x y
n xy
σ σ
τ θ τ θ
−
= +
cos2 sin2
2 2
n xyσ θ τ θ− = − i
ii2
n xy
Squaring and adding the above equations,
( ) ( )
2 2
22
2 2
x y x y
n n xy
σ σ σ σ
σ τ τ
+ −⎛ ⎞ ⎛ ⎞
− + = +⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠2 2⎝ ⎠ ⎝ ⎠
( ) ( )
2 2 2
0n av n Rσ σ τ− + − =
This is equation of a circle with centre and radius ( )
2
2
2
x y
xy
σ σ
τ
−⎛ ⎞
+⎜ ⎟
⎝ ⎠
( ),0avσ
Dept. of CE, GCE Kannur Dr.RajeshKN
19
Let us draw this circle!
20. Mohr’s circle
xyτ
( ),y xyσ τ
xσ
xyτ
xy
xyτ
2
2
2
y x
xy
σ σ
τ
−⎛ ⎞
+⎜ ⎟
⎝ ⎠
yσ
xyτ
σy
τxyτ
2
y xσ σ+
y
σx σσ3 σ1
α
2
y xσ σ−
τxy
( ),x xyσ τ− 1 2
tan 2xy
y x
τ
α θ
σ σ
− −
= =
−
Dept. of CE, GCE Kannur Dr.RajeshKN
20: Principal stressesσ1, σ3
measured clockwiseα
21. Mohr’s circle
xyτ
xσ
xyτ
y
xyτ
( ),x xyσ τ
yσ
xyτ
τ
2
2
2
y x
xy
σ σ
τ
−⎛ ⎞
+⎜ ⎟
⎝ ⎠
σyσx
τxyτ
σ3
αx
σ
3
σ1
2
y xσ σ+
2
y xσ σ−
τxy
2
1 2
tan 2xy
y x
τ
α θ
σ σ
−
= =
−( ),y xyσ τ−
Dept. of CE, GCE Kannur Dr.RajeshKN
21
y
measured anticlockwiseα
22. •Mohr’s circle is a graphical representation of the state of stress in an•Mohr s circle is a graphical representation of the state of stress in an
element.
E i h i l h l d h•Every point on the circle represents the normal and shear stress on a
plane.
•While x-coordinate of a point on the circle represents the normal
stress on a plane, y-coordinate represents the shear stress on that plane.
•Procedure for construction of Mohr’s circle
Dept. of CE, GCE Kannur Dr.RajeshKN
22
23. Maximum shear stress from Mohr’s circle
xyτ
( ),y xyσ τ
xσ
xyτ
xy
xyτ
Max shear
2
2
2
y x
xy
σ σ
τ
−⎛ ⎞
+⎜ ⎟
⎝ ⎠ yσ
xyτ
Max shear
stress
1 3
max
2
n
σ σ
τ
−
=
σy
τxyτ
,max
2
n
y
σx σσ3 σ1
2θ
τxy
( ),x xyσ τ−
Dept. of CE, GCE Kannur Dr.RajeshKN
23
25. 2
60 N
2
80 N mmProblem: Find principal stresses, principal
2
120 N mm
2
60 N mm
2
120 N mm
planes and max shear stress analytically. Draw
Mohr’s circle and verify graphically.
2
80 N mm
2
60 N mm
( )( )120,60−
τ60 18.44
2
71 56
θ
80120
60
σ σ3σ1
3σ
1σ
35.78
71.56=
τ
9.22
60 1
,maxnτ
Dept. of CE, GCE Kannur Dr.RajeshKN
25
( )80, 60− −
26. Problems: Find principal stresses, principal planes and max shear stress
analytically. Draw Mohr’s circle and verify graphically.analytically. Draw Mohr s circle and verify graphically.
1510
80
5015
15
15
50
50
10
10
10
50
50
30
15
30
15
80
10
10
15
50
5
20
20
20
50
5
50
20
Dept. of CE, GCE Kannur Dr.RajeshKN
26
27. Transformation of strains
σ σ σ σ+ − sin2 cos2x yσ σ
τ θ τ θ
−
+cos2 sin2
2 2
x y x y
n xy
σ σ σ σ
σ θ τ θ
+
= + −
sin2 cos2
2
y
n xyτ θ τ θ= +
The above equations, which give the normal and tangential (shear) stresses on
any plane inclined at θ with the vertical, are called stress transformation
equations.equations.
Similarly strain transformation equations can be derived as follows:
2 2
cos sin sin cosn x y xyε ε θ ε θ γ θ θ= + −
cos2 sin2
2 2 2
x y x y xy
n
ε ε ε ε γ
ε θ θ
+ −
= + −OR,
Dept. of CE, GCE Kannur Dr.RajeshKN
27
sin2 cos2
2 2 2
x y xyn
ε ε γγ
θ θ
−
= +and,
28. Principal strains:
Planes on which
i i l t i tPrincipal strains: principal strains act:
2 2
max min 1 3
x y x y xyε ε ε ε γ
ε ε
+ −⎛ ⎞ ⎛ ⎞
= = ± +⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠ ( )
tan2 xyγ
β
−
=
max,min 1,3
2 2 2⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠ ( )x yε ε−
Strain Rosettes
M t f l t i i i l• Measurement of normal strains is simple.
• Strain gages are placed as a cluster, along several gage lines through
a pointa point
• This arrangement of strain gages is called a strain rosette
• If three measurements are taken at a rosette (in three directions), the
information is sufficient to get the complete state of plane strain at a
point
Dept. of CE, GCE Kannur Dr.RajeshKN
p
29. 0
3 90θ =
1 2 3, ,θ θ θε ε ε are measured from strain gages
0
2 45θ =
1 2 3, ,θ θ θε ε ε are measured from strain gages
0 45 90ε ε ε45 degree rosette:
1 0θ =
0 45 90, ,ε ε ε45 degree rosette:
0 60 120, ,ε ε ε60 degree rosette:
Rectangular strain rosette
From strain transformation equations,
g
(45 degree rosette)2 2
cos sin sin cosn x y xyε ε θ ε θ γ θ θ= + −
Hence, for a 45 degree rosette, 0 0 0x xε ε ε= + − =
45 0.5 0.5 0.5x y xyε ε ε γ= + −y y
90 0 0y yε ε ε= + − =
Dept. of CE, GCE Kannur Dr.RajeshKN
From the above, we can get , ,x y xyε ε γ
30. Problem: Using a 60 degree rosette, the following strains are obtained at
i t D t i t i t d i i l t ia point. Determine strain components and principal strains.
0 60 12040 , 980 , 330ε μ ε μ ε μ= = =
, ,x y xyε ε γ
We have, 2 2
cos sin sin cosn x y xyε ε θ ε θ γ θ θ= + −
2 2
0 cos 0 sin 0 sin cos0x y xyε ε ε γ θ∴ = + −
i.e., 40 0 0 40x xε ε= + − ⇒ =
0 x y xyγ
60 980 0.25 0.75 0.433x y xyε ε ε γ⇒ = + −
120 330 0.25 0.75 0.433x y xyε ε ε γ⇒ = + +
40 , 860 , 750x y xyε μ ε μ γ μ= = = −
y y
From the above, we can get
Principal strains and their planes can be obtained from:
2 2
ε ε ε ε γ+ −⎛ ⎞ ⎛ ⎞ tan2 xyγ
β
−
=
Dept. of CE, GCE Kannur Dr.RajeshKN
max,min 1,3
2 2 2
x y x y xyε ε ε ε γ
ε ε
+ ⎛ ⎞ ⎛ ⎞
= = ± +⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠
( )
tan2
x y
β
ε ε
=
−
31. Theory of columnsy
Compression member: A structural member loaded in compression
Column: A vertical compression member
Strut: An inclined compression member – as in roof trusses
Stanchion: A compression member made of rolled steel section
Classification of columns based on mode of failure
Short columns: Failure by crushing under axial
compression
L ( l d ) l F il b l l b di
Intermediate (medium length) columns: Failure by
Long (slender) columns: Failure by lateral bending
(buckling)
Dept. of CE, GCE Kannur Dr.RajeshKN
31
Intermediate (medium length) columns: Failure by
combination of buckling and crushing
32. Equilibrium:qu b u :
Stable, neutral, unstable
Dept. of CE, GCE Kannur Dr.RajeshKN
32
33. L d th t b i d b th b b f f il
Critical load
Load that can be carried by the member before failure
Least load that causes elastic instability
depends on
dimensions of the member end conditions
modulus of elasticity
Slenderness ratio: Ratio of length to the least radius of gyration.
Buckling tendency varies with slenderness ratio
modulus of elasticity
Buckling tendency varies with slenderness ratio.
Dept. of CE, GCE Kannur Dr.RajeshKN
33
34. Euler’s theory – Leonhard Euler (1757)y ( )
2
d y
Both ends hinged PEI
M
R
=
2
d y
EI P
2
d y
2
d y
EI M
dx
⇒ = A2
y
EI Py
dx
⇒ = −
2
0
d y
EI Py
dx
+ =
2
2
0
d y P
y+ =2
y
dx EI
P P⎛ ⎞ ⎛ ⎞
Solution for the above differential equation is:
y
XX
l1 2cos sin
P P
y C x C x
EI EI
⎛ ⎞ ⎛ ⎞
= +⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠
When 0, 0x y= = 1 0C⇒ =
B
x
When 0, 0x y 1
When , 0x l y= = 20 sin
P
C l
EI
⎛ ⎞
⇒ = ⎜ ⎟
⎝ ⎠
P
B⎝ ⎠
sin 0 0, ,2 ,3 ,4 ...
P P
l l
EI EI
π π π π
⎛ ⎞
= ⇒ =⎜ ⎟
⎝ ⎠
Dept. of CE, GCE Kannur Dr.RajeshKN
34
P
,where 0,1,2,3,4...
P
l n n
EI
π= =
35. 2 2
2
n EI
P
l
π
=
2 sin
P
y C x
EI
⎛ ⎞
= ⎜ ⎟
⎝ ⎠
2 sin
n x
C
L
π⎛ ⎞
= ⎜ ⎟
⎝ ⎠
The least practical
2
EI
P
π
The least practical
value for P is:
Critical load2cr
EI
P
l
π
=
sin
x
y C
π⎛ ⎞
= ⎜ ⎟The corresponding mode shape is:
Dept. of CE, GCE Kannur Dr.RajeshKN
35
2 siny C
L
= ⎜ ⎟
⎝ ⎠
The corresponding mode shape is:
36. Assumptions in Euler’s theoryAssumptions in Euler s theory
• Material is homogeneous and isotropic
• Axis of column is perfectly straight when unloaded
• Line of thrust coincides exactly with the unstrained axis of the column
• Column fails by buckling alone
• Flexural rigidity EI is uniform
S lf h f l l d• Self weight of column is neglected
• Stresses are within elastic limit
Dept. of CE, GCE Kannur Dr.RajeshKN
36
37. Euler’s theory
P
y
2
d y
EI M
One end fixed and the other end free
( )
2
d y
EI P yδ⇒ = −
P
A
δ
2
y
EI M
dx
= ( )2
EI P y
dx
δ⇒ =
2
d y
EI P Pδ
2
d y P P
y
δ
+ = y2
y
EI Py P
dx
δ+ = 2
y
dx EI EI
+ = y
lSolution for the above differential equation is:
1 2cos sin
P P
y C x C x
EI EI
δ
⎛ ⎞ ⎛ ⎞
= + +⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠
xEI EI⎝ ⎠ ⎝ ⎠
When 0, 0x y= = 1C δ⇒ = −
x
B
Dept. of CE, GCE Kannur Dr.RajeshKN
37
38. dy P P P P⎛ ⎞ ⎛ ⎞
dy P
1 2sin cos
dy P P P P
C x C x
dx EI EI EI EI
⎛ ⎞ ⎛ ⎞
= − +⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠
When 0, 0
dy
x
dx
= = 2 20 0 0
P
C C
EI
⇒ = = ⇒ =
When x l y δ= = cos
P
lδ δ δ
⎛ ⎞
⇒ +⎜ ⎟
3 5
cos 0
P P
l l
π π π⎛ ⎞
⇒⎜ ⎟
When ,x l y δ= = cos l
EI
δ δ δ⇒ = − +⎜ ⎟
⎝ ⎠
cos 0 , , ...
2 2 2
l l
EI EI
= ⇒ =⎜ ⎟
⎝ ⎠
P
l
π
Th l t ti l l i
2
P
l
EI
π
=
2
EI
P
π 2
EI
P
π
The least practical value is:
2el l= Effective length
2
4
EI
P
l
π
= 2
e
EI
P
l
π
=
P⎛ ⎞ ⎛ ⎞
e Effective length
Dept. of CE, GCE Kannur Dr.RajeshKN
38
cos 1 cos
2 e
P x
y x
EI l
π
δ δ δ
⎛ ⎞ ⎛ ⎞
= − + = −⎜ ⎟ ⎜ ⎟
⎝ ⎠⎝ ⎠
39. Euler’s theory Py
( )
2
d y
EI M P H l
One end fixed and the other hinged
H
A
( )2
y
EI M Py H l x
dx
= = − + −
( )
2
d y
EI P H l y( )2
y
EI Py H l x
dx
+ = −
( )cos sin
P P H
C C l
⎛ ⎞ ⎛ ⎞
+ +⎜ ⎟ ⎜ ⎟
y
l
( )1 2cos siny C x C x l x
EI EI P
= + + −⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠
x
⎛ ⎞ ⎛ ⎞
x
1 2sin cos
dy P P P P H
C x C x
dx EI EI EI EI P
⎛ ⎞ ⎛ ⎞
= − + −⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠
When 0 0x y= = 1 10
H H
C l C l⇒ = + ⇒ = −
B
M
Dept. of CE, GCE Kannur Dr.RajeshKN
39
When 0, 0x y 1 1
P P
P
M
40. When 0, 0
dy
x
dx
= = 2 20
P H H EI
C C
EI P P P
⇒ = − ⇒ =
When 0x l y= = 0 cos sin
H P H EI P
l l l
⎛ ⎞ ⎛ ⎞
= − +⎜ ⎟ ⎜ ⎟
tan
P P
l l
⎛ ⎞ ⎛ ⎞
⎜ ⎟ ⎜ ⎟
When , 0x l y= = 0 cos sinl l l
P EI P P EI
+⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠
tan l l
EI EI
=⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠
P
2
20.25 2EI EI
P
π
4.5 radians
P
l
EI
= 2 2
P
l l
= ≈
2
EI
P
π l
l2
e
EI
P
l
π
=
2
el =
Dept. of CE, GCE Kannur Dr.RajeshKN
40
41. Euler’s theory
P
y
2
d y
EI M M P
Both ends fixed A M0
02
y
EI M M Py
dx
= = −
2
d y
EI P M
y2
0d y P M
02
y
EI Py M
dx
+ =
0P P M⎛ ⎞ ⎛ ⎞
y
l
0
2
y
y
dx EI EI
+ =
0
1 2cos sin
P P M
y C x C x
EI EI P
⎛ ⎞ ⎛ ⎞
= + +⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠
⎛ ⎞ ⎛ ⎞
x
1 2sin cos
dy P P P P
C x C x
dx EI EI EI EI
⎛ ⎞ ⎛ ⎞
= − +⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠
x
When 0 0x y= =
0 0
1 10
M M
C C⇒ = + ⇒ = −
B
M
Dept. of CE, GCE Kannur Dr.RajeshKN
41
When 0, 0x y 1 1
P P
P
M0
42. When 0, 0
dy
x
dx
= = 2 20 0
P
C C
EI
⇒ = ⇒ =
When 0x l y= =
0 0
0 cos
M P M
l
P EI P
⎛ ⎞
= − +⎜ ⎟ 0
1 cos 0
M P
l
⎡ ⎤⎛ ⎞
⇒ − =⎢ ⎥⎜ ⎟
cos 1
P
l
⎛ ⎞
⎜ ⎟
When , 0x l y= = P EI P
⎜ ⎟
⎝ ⎠ P EI
⎢ ⎥⎜ ⎟
⎢ ⎥⎝ ⎠⎣ ⎦
0 2 4 6
P
l π π π⇒ =cos 1l
EI
=⎜ ⎟
⎝ ⎠
2
4 EI
P
π
0,2 ,4 ,6 ...l
EI
π π π⇒ =
2
P
l π
2
P
l
=
2
EI l
2l
EI
π=
2
2
e
EI
P
l
π
=
2
e
l
l =
Dept. of CE, GCE Kannur Dr.RajeshKN
42
43. Effective lengthEnd conditions
Both ends hinged
g
l
One end fixed and the
other end free 2l
B th d fi d
One end fixed and the
other hinged 2l
Both ends fixed 2l
Dept. of CE, GCE Kannur Dr.RajeshKN
43
44. Limitations of Euler’s theoryLimitations of Euler s theory
• Applicable to ideal cases only. There may be imperfections in the
l th l d t tl th h th t id f th
2
column, the load may not pass exactly through the centroid of the
column section
• Direct stress is not taken into account
2
2E
e
EI
P
l
π
=• Strength of the material is not taken into account
2
2
E
E
e
P EI
A Al
π
σ = =
2 2 2
EAr Eπ π
22E
e e
EAr E
Al l
r
π π
σ⇒ = =
⎛ ⎞
⎜ ⎟
⎝ ⎠
Dept. of CE, GCE Kannur Dr.RajeshKN
44
45. σEσE
l /
Validity limits of Euler’s formula
le /r
Dept. of CE, GCE Kannur Dr.RajeshKN
45
Critical stress for mild steel with E=2x105 MPa
46. 2
l Eπe
E
l E
OD
r
π
σ
= =
5 2
2 10 NEL t
2
250 N mmPLσ =Stress at limit of proportionality
5 2
2 10 N mmE = ×Let
( )2 5
2 10
89
250
el
r
π ×
∴ = =
250r
i.e., Euler’s theory is applicable for 89el
> for mild steel
r
Dept. of CE, GCE Kannur Dr.RajeshKN
46
47. Rankine’s theoryy
2
2Euler
EI
P
π
=
c cP Aσ=For short compression members,
For long columns, 2Euler
el
g ,
Rankine proposed a general empirical formula:
1 1 1
Rankine c EulerP P P
= + For a short compression member, PE is very large.
P P∴ ≈Rankine cP P∴ ≈
For long columns, 1/PEuler is very large.
2
2
1 el
P EIπ
=
Rankine EulerP P≈
EulerP EIπ
2
1 1 1
R ki
EIP A πσ
= + 2
I Ar=
Dept. of CE, GCE Kannur Dr.RajeshKN
47
2
Rankine c
e
EIP A
l
πσ
48. 1 1 1
2
2 r
Eπ σ
⎛ ⎞
+⎜ ⎟
2
2
1 1 1
Rankine cP A r
EA
l
σ
π
= +
⎛ ⎞
⎜ ⎟
⎝ ⎠
2
2
c
e
Rankine
c
E
lA
P r
E
l
π σ
σ π
+⎜ ⎟
⎝ ⎠=
⎛ ⎞
⎜ ⎟
⎝ ⎠
2 2
c c
Rankine
A A
P
l l
σ σ
σ
= =
⎛ ⎞ ⎛ ⎞
c
a
σ
=
el⎝ ⎠ c
el⎜ ⎟
⎝ ⎠
2
1 1c e el l
a
E r r
σ
π
⎛ ⎞ ⎛ ⎞
+ +⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠
2
a
Eπ
2 2
Crushing loadc
Rankine
A
P
l l
σ
= =
⎛ ⎞ ⎛ ⎞
2
1 el
a
r
⎛ ⎞
+ ⎜ ⎟
⎝ ⎠1 1e el l
a a
r r
⎛ ⎞ ⎛ ⎞
+ +⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠
r⎝ ⎠
Factor that accounts
for bucklingg
Dept. of CE, GCE Kannur Dr.RajeshKN
48
49. Find the length of the column for which Rankine’s and Euler’s formulae
give the same buckling load:
Rankine EulerP P=
2
A EIσ π 1/2
2 2
⎛ ⎞2 2
1
c
ee
A EI
ll
a
r
σ π
=
⎛ ⎞
+ ⎜ ⎟
⎝ ⎠
1/2
2 2
2e
c
Er
l
Ea
π
σ π
⎛ ⎞
= ⎜ ⎟
−⎝ ⎠
Dept. of CE, GCE Kannur Dr.RajeshKN
49
50. Problem: Compare the buckling (crippling) loads given by Rankine’sProblem: Compare the buckling (crippling) loads given by Rankine s
and Euler’s formulae for a tubular strut hinged at both ends, 6 m long
having outer diameter 15 cm and thickness 2 cm. Given,
4 2 2 1
8 10 N mm , 567 N mm ,
1600
cE aσ= × = =
For what length of the column does the Euler’s formula cease to apply?For what length of the column does the Euler s formula cease to apply?
2
2Euler
EI
P
l
π
=
( )4 4 4
150 110 17663604.69 mm
64
I
π
= − =
el
6 m 6000 mmel l= = =( )64
e
2
2
387406.2 NEuler
EI
P
l
π
= = 387.4 kN=
2Euler
el
( )2 2 2π
Dept. of CE, GCE Kannur Dr.RajeshKN
50
( )2 2 2
150 110 8168.141 mm
4
A
π
= − =
51. c A
P
σ
2
1
c
Rankine
e
P
l
a
r
σ
=
⎛ ⎞
+ ⎜ ⎟
⎝ ⎠
2 4
17663604.69 mmI Ar= = 17663604.69
46.503 mm
8168.141
I
r
A
= = =
567 8168.141
406097 78 NP
×
406 098 kN=2
406097.78 N
1 6000
1
1600 46.503
RankineP = =
⎛ ⎞
+ ⎜ ⎟
⎝ ⎠
406.098 kN=
2
2E
Eπ
σ = ( )2 4
8 10
37 317el π ×
∴ = =
2
EP EIπ
σ = =
To find the length of the column above which Euler’s formula is applicable
2E
el
r
σ
⎛ ⎞
⎜ ⎟
⎝ ⎠
37.317
567r
∴ = =2E
eA Al
σ = =
46 503 37 317l 1735 34 mm 1 735 m
Dept. of CE, GCE Kannur Dr.RajeshKN
51
46.503 37.317el∴ = × 1735.34 mm 1.735 m= =
52. Long column under eccentric loading P
σ =For short columns
P
A
σ =
M P e=
For short columns,
e
.M P e
Z Z
σ = =
.M P e=
.P P e
y
A I
σ = ±
Z Z A I
. .
1
P P e y P ey⎛ ⎞
± +⎜ ⎟2 2
1
y y
A Ar A r
σ ⎛ ⎞
= ± = +⎜ ⎟
⎝ ⎠
Aσ
2
1
A
P
ey
r
σ
=
⎛ ⎞
+⎜ ⎟
⎝ ⎠
2
A
P
al ey
σ
=
⎛ ⎞⎛ ⎞
For long columns,
Dept. of CE, GCE Kannur Dr.RajeshKN
52
2 2
1 1eal ey
r r
⎛ ⎞⎛ ⎞
+ +⎜ ⎟⎜ ⎟
⎝ ⎠⎝ ⎠
53. Secant formula P
A
e
2
2
0
d y
EI Py⇒ + =
2
2
d y
EI Py= −
Both ends hinged
2
y
dx
2
2
0
d y P
y
d EI
+ =
2
y
dx
y
l
2
dx EI
P P⎛ ⎞ ⎛ ⎞
Solution for the above differential equation is:
y
x
1 2cos sin
P P
y C x C x
EI EI
⎛ ⎞ ⎛ ⎞
= +⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠
When 0,x y e= = 1C e⇒ =
B
When 0,x y e 1C e⇒
2sin cos
dy P P P P
e x C x
dx EI EI EI EI
⎛ ⎞ ⎛ ⎞
= − +⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠dx EI EI EI EI⎝ ⎠ ⎝ ⎠
When 0
l dy
x = =
sin
2
l P
EI
C e
⎛ ⎞
⎜ ⎟
⎝ ⎠=
Dept. of CE, GCE Kannur Dr.RajeshKN
53
When , 0
2
x
dx
= = 2
cos
2
C e
l P
EI
=
⎛ ⎞
⎜ ⎟
⎝ ⎠
54. l P⎡ ⎤⎛ ⎞
sin
2
cos sin
cos
l P
EIP P
y e x x
EI EIl P
⎡ ⎤⎛ ⎞
⎢ ⎥⎜ ⎟
⎛ ⎞ ⎛ ⎞⎝ ⎠⎢ ⎥= +⎜ ⎟ ⎜ ⎟⎢ ⎥⎛ ⎞⎝ ⎠ ⎝ ⎠⎢ ⎥⎜ ⎟
2
sin
2
l P
EI
⎡ ⎤⎛ ⎞
⎢ ⎥⎜ ⎟
⎛ ⎞
cos
2 EI
⎢ ⎥⎜ ⎟
⎢ ⎥⎝ ⎠⎣ ⎦
l P⎛ ⎞
max
2
When , cos
2 2
cos
2
EIl l P
x y y e
EI l P
EI
⎢ ⎥⎜ ⎟
⎛ ⎞ ⎝ ⎠⎢ ⎥= = = +⎜ ⎟⎢ ⎥⎛ ⎞⎝ ⎠⎢ ⎥⎜ ⎟
⎢ ⎥⎝ ⎠⎣ ⎦
max .sec
2
l P
y e
EI
⎛ ⎞
⇒ = ⎜ ⎟
⎝ ⎠
max max . .sec
2
l P
M Py P e
EI
⎛ ⎞
= = ⎜ ⎟
⎝ ⎠2 EI⎝ ⎠
P My P Pey l P P ey l P⎡ ⎤⎛ ⎞ ⎛ ⎞
max 2
sec 1 sec
2 2
c c cP My P Pey l P P ey l P
A I A I EI A r EI
σ
⎡ ⎤⎛ ⎞ ⎛ ⎞
= + = + = +⎢ ⎥⎜ ⎟ ⎜ ⎟
⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦
Dept. of CE, GCE Kannur Dr.RajeshKN
54
55. Secant formula P
δ
One end fixed and the other end free
A
δ e
( )
2
2
d y
EI P e yδ= + −
( )
2
2
d y P P
y e
d EI EI
δ+ = +
y
( )2
y
dx
( )2
dx EI EI
P P⎛ ⎞ ⎛ ⎞
Solution for the above differential equation is:
l
( )1 2cos sin
P P
y C x C x e
EI EI
δ
⎛ ⎞ ⎛ ⎞
= + + +⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠
When 0, 0x y= = ( )1C eδ⇒ = − +
x
When 0, 0x y ( )1
( ) 2sin cos
dy P P P P
e x C x
dx EI EI EI EI
δ
⎛ ⎞ ⎛ ⎞
= − + +⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠
B
dx EI EI EI EI⎝ ⎠ ⎝ ⎠
When 0 0
dy
x = = 0C⇒ =
Dept. of CE, GCE Kannur Dr.RajeshKN
55
When 0, 0x
dx
= = 2 0C⇒ =
56. When x l y δ= = ( ) ( )cos
P
e l eδ δ δ
⎛ ⎞
⇒ = + + +⎜ ⎟When ,x l y δ= = ( ) ( )cose l e
EI
δ δ δ⇒ = − + + +⎜ ⎟
⎝ ⎠
( ) 1
P
lδ δ
⎡ ⎤⎛ ⎞
⇒ + ⎢ ⎥⎜ ⎟( ) 1 cose l
EI
δ δ⇒ = + −⎢ ⎥⎜ ⎟
⎢ ⎥⎝ ⎠⎣ ⎦
( )cos
P
e l eδ
⎛ ⎞
⇒ + ⎜ ⎟( )cose l e
EI
δ⇒ + =⎜ ⎟
⎝ ⎠
( ) sec
P
e e lδ
⎛ ⎞
⇒ + = ⎜ ⎟
( )
P
M P P lδ
⎛ ⎞
+ ⎜ ⎟
( ) .sece e l
EI
δ⇒ + = ⎜ ⎟
⎝ ⎠
( )max . .secM P e P e l
EI
δ= + = ⎜ ⎟
⎝ ⎠
⎡ ⎤⎛ ⎞ ⎛ ⎞
max 2
sec 1 secc c cP My P Pey P P ey P
l l
A I A I EI A r EI
σ
⎡ ⎤⎛ ⎞ ⎛ ⎞
= + = + = +⎢ ⎥⎜ ⎟ ⎜ ⎟
⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦
Dept. of CE, GCE Kannur Dr.RajeshKN
56
57. max . .sec
2
el P
M P e
EI
⎛ ⎞
= ⎜ ⎟
⎝ ⎠
In general,
2 EI⎝ ⎠
max 2
1 sec
2
c eP ey l P
A r EI
σ
⎡ ⎤⎛ ⎞
= +⎢ ⎥⎜ ⎟
⎢ ⎥⎝ ⎠⎣ ⎦2A r EI⎢ ⎥⎝ ⎠⎣ ⎦
For short compression members (no buckling), max .M P e=
For long columns (with buckling) el P
M P
⎛ ⎞
⎜ ⎟For long columns (with buckling),
max . .sec
2
e
M P e
EI
= ⎜ ⎟
⎝ ⎠
Dept. of CE, GCE Kannur Dr.RajeshKN
57
58. Problem: A hollow mild steel column with internal diameter 80 mm
and external diameter 100 mm is 2.4 m long, hinged at both ends,
carries a load of 60 kN at an eccentricity of 16mm from the geometrical
axis. Calculate the maximum and minimum stresses in the column.
Also find the maximum eccentricity so that no tension is induced in the
section. 5 2
2 10 N mmE = ×
( )4 4 4
100 80 2898119 mmI
π
= − = ( )2 2 2
100 80 2827 4 mmA
π
= − =( )100 80 2898119 mm
64
I
2400 mml l
( )100 80 2827.4 mm
4
A
2898119
32 015 mm
I
r 2400 mmel l= =32.015 mm
2827.4
r
A
= = =
Dept. of CE, GCE Kannur Dr.RajeshKN
58
59. sec el P
M P e
⎛ ⎞
= ⎜ ⎟
3
3 2400 60 10
60 10 16 secM
⎛ ⎞×
× × × ⎜ ⎟ 0 96 kN
max . .sec
2
e
M P e
EI
= ⎜ ⎟
⎝ ⎠
max 5 6
60 10 16 sec
2 2 10 2.898 10
M = × × × ⎜ ⎟
⎜ ⎟× × ×⎝ ⎠
0.96 kNm=
3 6
⎧max
max
min
cP M y
A I
σ = ±
3 6
max
min
37.78 MPa60 10 0.96 10 50
4.69 MPa2827.4 2898119
σ
⎧× × ×
= ± = ⎨
⎩
To find the maximum eccentricity so that no tension is induced
in the section
max
0cP M y
A I
− =
.
.sec 0
2
e
c
P P l
y
I
e P
A EI
⎛ ⎞
− =⎜ ⎟
⎝ ⎠
20.5 mme =
Dept. of CE, GCE Kannur Dr.RajeshKN
59
60. SummarySummary
Transformation of stresses and strains (two dimensional case only)Transformation of stresses and strains (two-dimensional case only) -
equations of transformation - principal stresses - mohr's circles of
stress and strain - strain rosettes - compound stresses - superposition
d it li it tiand its limitations –
Eccentrically loaded members - columns - theory of columns -
buckling theory - Euler's formula - effect of end conditions -
eccentric loads and secant formula
Dept. of CE, GCE Kannur Dr.RajeshKN
60
61. "A teacher is one who makes himself
progressively unnecessary "progressively unnecessary."
Thomas Carruthers
Dept. of CE, GCE Kannur Dr.RajeshKN
61