INTRODUCTIONChapter 1

OutlineMechanics of MaterialsMethods of AnalysisEngineering DesignReview of Static EquilibriumInternal Force ResultantsProblem Formulation and Solution Application to Simple Structures

Practical Examples

Mechanics of materialsA course on statics treats the external behavior of bodies, whereas mechanics of materials course deals with internal behavior of variously loaded solid bodies such as shafts, bars, beams, plates, shells, columns, structures and machines that are assemblies of these components. Stress and deflection analyses, and the mechanical properties of materials are the main aspects of solid mechanics course.

Mechanics of materialsThis course is based upon an understanding of the equilibrium of rigid bodies under the action of forces. This course concerned with the relationships between external loads (forces and moments) and internal forces and deformations or displacements induced in the body. Stress and strain are fundamental quantities connected this subject.

Historical developmentLeonardo da Vinci (1452 1519) and Galileo Galilei (15641642) investigated the behavior of bars under loads.Robert Hooke (16151703) was the first to show that a body is deformed if a force acts upon it.Sir Isaac Newton (16421727) developed the concepts of Newtonian mechanics.Leonard Euler (17071783) presented the mathematical theory of columns in 1744.Thomas Young (17731829) established a coefficient of elasticity called, Youngs modulus.

Historical developmentCoulomb, Poisson, Navier, St.Venant, Cauchy, and many other scientists and engineers were responsible for advances in mechanics of materials during the nineteenth century.Over the years, Stephan P. Timoshenko (18781972) made many original contributions to the field of applied mechanics, and wrote many textbooks in this area.

METHODS OF ANALYSISTwo methods or approaches are popular:Mechanics of materials theory (also known as technical theory, or solid mechanics approach), andThe theory of elasticity approach.

METHODS OF ANALYSISThe solid mechanics approach is simpler and makes assumptions that are based upon experimental evidence and the lessons of engineering practice.Reasonably quick solution of the basic problem is possible, for example, determination of strain.

METHODS OF ANALYSISThe theory of elasticity approach establishes every step rigorously from the mathematical point of view and hence seeks to verify the validity of the assumptions introduced to determine the quantities, for example, strains.This technique can provide exact results where configurations of loading and shape are simple. However, this approach yields solutions with considerable difficulty.

Basic Principles of AnalysisEquilibrium Conditions The equations of static equilibrium of forces must be satisfied throughout the member.Material BehaviorThe stressstrain or forcedeformation relations (Hookes law) must apply to the behavior of the material of which the member is made of.Geometry of DeformationThe conditions of compatibility of deformations must be satisfied: that is, each deformed portion of the member must fit together with adjacent portions.Boundary conditions are used in the method of analysis.

Basic Principles of AnalysisInstead of the equilibrium method, energy methods (based on strain energy theory) and numerical methods (for example, finite element analysis) can be used as alternative methods of analysis.

Major Steps in Engineering Design ProcedureEvaluate the mode of possible failure of the member.Determine a relationship between the applied load and the resulting effect such as stress or deformation.Determine the maximum usable value of a significant quantity such as stress or deformation that could conceivably cause failure.Employ this value in connection with the equation found in step 2 or, if required, in any of the formulas associated with the various theories of failure.Determine the safety factor or verify if the design is safe.

REVIEW OF STATIC EQUILIBRIUMTypes of loads: External loads and Internal loads

External loads are due to surface forces and body forcesSurfaces forces can be for example, a concentrated load acting at a point or a distributed load both acting on the surface of a bodyBody forces act on a volumetric portion of the body, forexample, magnetic force or gravitational forceReaction forces caused by the supportsInternal loads can be considered as forces of interaction between the constituent material particles of the body

CONDITIONS OF EQUILIBRIUMWhen a system of forces acting upon a body has zero resultant, the body is said to be in force equilibrium. The equations of static equlibrium require:

Fx = 0; Fy = 0, and Fz = 0

Mx = 0; My = 0, and Mz = 0

In other words, for a body to be in static equilibrium, the sum of all forces acting upon a body in any direction is zero and also the sum of all moments taken about any axis is also zero.

Planar Equations of EquilibriumFor a planar body to be in equilibrium, any one of the following sets of 3 equations may be used to solve for the unknown variables.

Fx = 0, Fy = 0, and MA = 0, where the resultant moment is with respect to any axis z or any point A in the xy-plane, or

Fx = 0, MA = 0, and MB = 0, provided that the line connecting the points A and B is not perpendicular to the x axis, or

MA = 0, MB = 0, and MC = 0, where points A, B, and C are not collinear

Support reactions and applications of equilibrium of planar bodies

Freebody diagramsSelect the free body to be used.Detach this body from its supports and separate it from any other bodies. (If internal force resultants are to be found, use the method of sections).Show on the sketch all of the external forces acting on the body. Location, magnitude, and direction of each force should be marked on the sketch.Label significant points and include dimensions. Any other detail, however, should be omitted.

INTERNAL FORCE RESULTANTSIsolate the bodies. Sketch the isolated body and show all external forces acting on it: draw a free-body diagram. Apply the equations of equilibrium to the diagram to determine the unknown external forces.Cut the body at a section of interest by an imaginary plane, isolate one of the segments, and repeat step 2 for that segment. If the entire body is in equilibrium, any part of it must be in equilibrium. That is, there must be internal forces transmitted across the cut sections.

Summary:

External loads or forces are balanced by internal loads or forces

Components of internal forces

Units

Units

Applications to simple structures