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Unformatted text preview: Chapter 1. Introduction and Mathematical Preliminaries 1. Scope Behavior of structures deals with (a) Micro state of stresses, strains, and displacements at a point (b) Macro global behavior, collapse mechanism, etc Theory of elasticity is concerned with (a) Equilibrium of forces (although equations may be expressed in terms of stress, a state of equilibrium must be established by forces) (b) Kinematics and compatibility examine straindisplacement relationship (c) Constitutive equations stressstrain relationship (d) Boundary conditions domain (e) Uniqueness applicability of solutions 2. Vector Algebra 1 1 2 2 3 3 x x y y z z i j k A A e A e A e Ae A e A e Ai A j A k = + + = + + = + + % % % % % % % % % % where 2 x k A ,A ,A are scalar components of the vector and 2 x e ,e ,k % % % are unit vectors. Unit vectors are mutually orthogonal only in the righthand Cartesian coordinate system. Unless specifically mentioned the coordinate system adopted, the coordinate system is assumed to be the righthand Cartesian coordinate system in all subsequent discussions herein. ( 29 1 2 2 2 2 1 2 3 / A A A A = + + length of a vector 1 1 2 2 3 3 cos AB C A B A B A B A B A B = = + + = % % scalar quantity termed to be dot product or scalar product 1 1 cos x x x y z A x A A e A A A A A = = + + = = % % vector projection on to the xaxis Particularly, if A % is a unit vector, x e % , then the dot product of two unit vectors is equal to the direction cosine of the angle between these two unit vectors. ( 29 cos , i j i j ij e e e e = = % % % % These are procedures frequently used in elementary mechanics introduced in statics and mechanics of materials. 1 2 3 1 2 3 1 2 3 e e e C A B A A A B B B = = % % % vector; referred to as vector product or cross product sin AB C A B = area of a parallelogram 3. Scalar and Vector Fields ( 29 1 2 3 f x ,x ,x temperature, potential, etc. ( 29 1 2 3 A x ,x ,x % vector fields, velocity, etc. 1 2 3 1 2 3 1 2 3 f f f f f f grad f f e e e , , x x x x x x = = + + = % % % % vector grad f f 3 1 2 1 2 3 A A A div A A x x x = = + + % % scalar ( 29 ( 29 ( 29 1 2 3 1 2 3 i j k curl A A x x x A A A = = % % % % % % vector 2 $ Integral Theorem Two integral theorems relating vector fields are particularly useful in structural mechanics for transforming between contour, area, and volume integral. Greens theorem : Consider two functions ( 29 P x,y and ( 29 Q x,y which are continuous and have continuous first partial derivatives ( 1 C continuity) in a domain D , Greens theorem states that ( 29 ( 29 ,x ,y C A Pdx Qdy Q P dxdy + = where A is a closed region of D bounded by C . It should be noted that A should not have any holes in it. This Greens theorem is the basis of an old instrument, planimeter, measuring the...
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This note was uploaded on 09/24/2011 for the course CIVL 7690 taught by Professor Staff during the Summer '10 term at Auburn University.
 Summer '10
 Staff

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