Mechanics of solids and structures, second edition a revision of a popular textbook, this volume emphasizes the. He primarily works in the field of solid mechanics. For the shear diagram shown, what is the maximum bending moment. The variational derivation of the cbmf produced the existing sets of elasticity equations along with the new boundary compatibility conditions. This compatibility condition is an equation which must be satisfied by the strains at all. This shortcoming prevented the formulation of a direct method to calculate stress and strain, which are currently obtained indirectly by differentiating the displacement. Solid mechanics is a vast subject because of the wide range of solid materials available, such as steel, wood, concrete, biological materials, textiles, geological materials, and plastics. Find materials for this course in the pages linked along the left. We have researched and understood the compatibility condition for linear. Significance of strain in formulation in theory of solid mechanics nasatm2003212584 december 2003. While in the mechanics of materials course, one was introduced. In these instances, a coupled set of equilibrium, constitutive, and compatibility equations must be solved. Mechanics of solids mechanics of solids basic principles.
What would you calculate if you were asked to model the component for a design application. For each of the following applications, outline briefly. This suggests that the three displacements may be removed from the system of equations without loss of information. All problems in pdf format applied mechanics of solids. In the general 3d case, the strain compatibility equations are the set. Take the case of a cantilever propped at its free end.
The compatibility condition cc is a significant ingredient in the theory of solid mechanics. The new student friendly examples simply replaced example problems that the students had a harder time relating to. Please show all of your work, including which equations you are using, and circle your final answer. The strain formulation in elasticity and the compatibility condition in structural mechanics have neither been understood nor have they been utilized. Polymer mechanics, and solid mechanics in general, are topics concerned with the evolution of stresses and strains when a component is exposed to external loads in the form of forces, displacements, and temperatures. Significance of strain in formulation in theory of solid.
However, if three strains are given, these expression provide three. Solid mechanics extensively uses tensors to describe stresses, strains, and the relationship between them. Using these equations, we can find the state of strain at any. It is shown by operational method that the boundary value problem of the theory of elasticity related to stresses, which can be reduced to three strains compatibility equations and to three. An exception to this was the varying stress field in the loaded beam, but there a simplified set of elasticity equations was used. The compatibility conditions in linear elasticity are obtained by observing that there are six straindisplacement relations that are functions of only three unknown displacements. This is primarily aimed at the part iiiv engineering undergraduate student. Solid mechanics is one of the important branches of physical science concerned with the deformation and motion of continuous solid media under applied external loadings such as forces, displacements, and accelerations that result in inertial force in the bodies, thermal changes, chemical interactions, electromagnetic forces, and so on. Patnaik ohio aerospace institute brook park, ohio 44142 shantaram s.
It begins by establishing the fundamentals of continuum mechanics, explaining key areas such as the equations used, stresses and strains, and two and three dimensional problems. Compatibility condition civil engineering ce notes edurev. Solid mechanics is the branch of mechanics, physics, and mathematics that concerns itself with how solid matter under external actions works such as external forces, temperature changes, applied displacements, etc. Concepts of stress, strain and elasticity, beams, columns, plates, shells, elasticity, general theory, waves, stress concentrations and fracture, linear and angular momentum principles, geometry of deformation, stressstrain relations, equations of linear elasticity, mechanical theory, some elementary twodimensional. For the love of physics walter lewin may 16, 2011 duration. What does a compatibility equation mean in structural. The number of independent compatibility equations in terms of stresses, involved in formulating the basic problem in the mechanics of deformable solids in.
Compatibility is the study of the conditions under which such a displacement field can be guaranteed. Lectures notes on mechanics of solids course code bme203 prepared by prof. The first two sets of equations are universal independent of the. Patnaik ohio aerospace institute 22800 cedar point rd. Introduction to finite element analysis in solid mechanics pdf version 2. Es240 solid mechanics fall 2009 92611 linear elasticity5 0,0. Further, because solid mechanics poses challenging mathematical and. It is part of a larger study known as continuum mechanics. Even though a method corresponding to equation 1 could not be developed. Compatibility conditions of structural mechanics surya n. Because the plate is thin, the stress distribution may be very closely approximated by assuming that the foregoing is likewise true throughout the plate.
Lecturenoteson intermediatefluidmechanics joseph m. Lecture notes solid mechanics civil and environmental. Compatibility equations are those additional equations which can be made considering equilibrium of the structure, to solve statically indeterminate structures. In the case of a finite element system in which the equilibrium and compatibility equations are satisfied only at node points along the interface, the fundamental. We can think of an elastic solid as a highly redundant frameworkeach particle is builtin to its neighbours.
This document is highly rated by civil engineering ce students and has been viewed 171 times. Pdf compatibility condition in theory of solid mechanics. Mar 20, 2017 compatibility equations are those additional equations which can be made considering equilibrium of the structure, to solve statically indeterminate structures. Introduction when forces are applied to a body solid, liquid or gas, internal forces are set up in the body and it deforms andor moves. Compatibility if the displacements are given, these three equations can be used to calculate three strains. Sufficiency is achieved through the compatibility conditions cc. Compatibility conditions are particular cases of integrability conditions and were.
What a mess what do these really tell us the strains must be compatible, they cannot be prescribed in. To complete the formulation of the solid mechanics problem, appropriate boundary conditions must be applied. In structures the ad hoc compatibility conditions through virtual cuts and closing gaps are not parallel to the strain formulation in. Mechanics of elastic solids lesson solid mechanics homework answers 1 solid mechanics homework answers. Experimental mechanics of solids is a comprehensive introduction to the topics, technologies and methods of experimental mechanics of solids. Schematic of generic problem in linear elasticity or alternatively the equations of strain compatibility 6 equations, 6 unknowns, see. Compatability equation example solid mechanics youtube. One of the most common practical applications of solid mechanics is the eulerbernoulli beam equation. The cc makes solid mechanics a research discipline that is practiced in the academia and in major research institutions.
Show that the strain field satisfies the equations of compatibility. Pdf compatibility equations in the theory of elasticity. Compatibility conditions for infinitesimal strains. Fundamentals of solid mechanics krzysztof wilmanski.
They do not have any information on the rheology of the body, so these equations apply to viscous fluids, plastics, and elastic solids. As part of this work, cauchy also introduced the equations which express the six components of strain. Compatibility conditions of structural mechanics ntrs nasa. Compatibility equations of a potential vector field. However, if three strains are given, these expression provide three equation for two unknowns, u and v respectively. Apr 04, 2016 for the love of physics walter lewin may 16, 2011 duration. If we begin with the case when the weight of the body is the only body force, the equations to be satisfied are to these equations the boundary conditions 20 should be added. Sol mech course text feb10 solid mechanics at harvard. Feb 18, 2018 solid state workshop recommended for you. The yield stress of steel is 250 mpa 250,000,000 pa. Hopkins national aeronautics and space administration glenn research center cleveland, ohio 445 summary.
The compact expression corresponding to the 6 equations in 3. In addressing any problem in continuum or solid mechanics, three factors must be considered. Ee equilibrium equations ifm integrated force method ifmd dual integrated force method summary the theory of elasticity has camouflaged a deficiency in the compatibility formulation since 1860. Rice, solid mechanics pdf 84p this note covers the following topics.
The deficit is made up by compatibility conditionsstatements that the deformed components must fit together. Morozov, in advanced mechanics of composite materials third edition, 20. Includes color graphics and illustrations throughout for better visual. Brief introduction to the objectives and methods of solid mechanics. One example of this is the demonstra tion of the use of compatibility to solve axially loaded statically indeterminate pr oblems. For such a framework, we expect to get some equations from considerations of equilibrium, but not as many as there are unknowns. Threedimensional problems in elasticity are often very complex. Equilibrium equation compatibility condition stress mechanical load initial deformation 1 for stress analysis the equilibrium equation ee represents the necessary condition. Mech 5312 solid mechanics ii mechanical engineering. Compatibility condition in theory of solid mechanics elasticity, structures, and design optimization december 2007 archives of computational methods in engineering 144. Powers department of aerospace and mechanical engineering university of notre dame notre dame, indiana 465565637.
In 1822 he formalized the stress concept in the context of a general threedimensional theory, showed its properties as consisting of a 3 by 3 symmetric array of numbers that transform as a tensor, derived the equations. The compatibility concept makes solid mechanics a research discipline that is. Aug 22, 2007 the strain formulation in elasticity and the compatibility condition in structural mechanics have neither been understood nor have they been utilized. This creates a problem, where u andor v can be found to have multiple solutions xx u x h w w yy v y h w xy w. Mar 15, 2020 compatibility condition civil engineering ce notes edurev is made by best teachers of civil engineering ce. Mechanics of deformable bodies sridhar krishnaswamy 1. Haftka eml5526 finite element analysis university of florida 2 outline most structural analyypsis fe codes are displacement based in this chapter we discuss interpolation methods and. Formulas in solid mechanics tore dahlberg solid mechanics ikp, linkoping university linkoping, sweden this collection of formulas is intended for use by foreign students in the course tmhl61, damage mechanics and life analysis, as a complement to the textbook dahlberg and. A 8 kn m b 16 kn m c 18 kn m d 26 kn m starting from the left end of the beam, areas begin to cancel after 2 m. Solid mechanics simple english wikipedia, the free. The equilibrium equations and the compatibility conditions required for the analysis of stress in an elastic con. Formulas in solid mechanics tore dahlberg solid mechanicsikp, linkoping university linkoping, sweden this collection of formulas is intended for use by foreign students in the course tmhl61, damage mechanics and life analysis, as a complement to the textbook dahlberg and. The equilibrium equations and the compatibility conditions required for the analysis of.
The bending moment at the ends is zero, and there are no concentrated couples. The problem of solid mechanics is reduced, as follows from the foregoing derivation, to a set of 15 equations, i. Module 4 boundary value problems in linear elasticity. We shall, as a condition of the problem, take the body force and and. Theory of continuum and nano rods, finite deformation elastoplasticity, computational. In the twodimensional case, there are three straindisplacement. Usually, the displacements are known on some parts of the boundaries of the body. Natural phenomena involving solid mechanics are studied in geology, seismology and tectonophysics, in materials science and the physics of condensed matter, and in parts of biology and physiology. Compatibility condition in theory of solid mechanics. The deficit is made up by compatibility conditionsstatements that the deformed components must fit. Compatibility condition in theory of solid mechanics elasticity, structures, and design optimization surya n. In continuum mechanics, a compatible deformation or strain tensor field in a body is that unique tensor field that is obtained when the body is subjected to a continuous, singlevalued, displacement field.
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