lec 2 mathematical foundation

8/6/2019 Lec 2 Mathematical Foundation

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STRUCTURAL ENGINEERING

Mathematical Foundation


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Tensor, Matrices, Vectors

Tensor

Is a general name for physical measure (quantity) thatis generally described by a whole set of values and

these values may be dependent on space coordinates. Tensors are specified by their rank or order depending

on the no. of components they posses

In 3D- space a tensor of a rank N has 3N components

0th

order : 1 Component (Scalar) 1st order: 3 Components (Vector)

2nd order: 9 Components (Matrix)

3rd order: 27 Components (Tensor)


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Tensors are geometric entities introduced into

mathematics and physics to extend the notion of

scalars, (geometric) vectors, and matrices

Because they express a relationship between

vectors, tensors themselves are independent of a

particular choice of coordinate system


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Types of Notations

1- Matrix Notation

2- Index Notation

1- Matrix Notations

A matrix is a rectangular arrangement of numbers.

For example,


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The horizontal and vertical lines in a matrix are

called rows and columns, respectively

The numbers in the matrix are called its entries or its

elements

To specify a matrix's size, a matrix with m rows and

n columns is called an m-by-n matrix or m n

matrix, while m and n are called its dimensions In Example the Matrix is a 4x3 matrix


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More than one index is used to describe arrays or

number with two or more dimensions, such as the

elements of a matrix. The (i, j)th entry of a matrix A

is most commonly written as ai,j

where the first subscript is the row number and the

second is the column number

Free index & Dummy Index


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Free index

Must Appear ONCE in each term of expression or

equation

Dummy Index

Appears TWICE on a term can be replaced by

alternating index


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Matrix Addition

[A] + [B] = [C]

Ai,j + Bi,j = Ci,j

The sumA

+B

of two m-by-n matricesA

andB

iscalculated entry wise: (A + B)i,j = Ai,j + Bi,j


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Scalar Multiplication

The scalar multiplication cA of a matrix A and a

number c is given by multiplying every entry of A

by c

(cA)i,j = c Ai,j


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Transpose of Matrix

The transpose of an m-by-n matrix A is the n-by-m

matrix ATformed by turning rows into columns and

vice versa

(AT)i,j = Aj,I


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Matrix Multiplication

Multiplication of two matrices is defined only if the

number of columns of the left matrix is the same as

the number of rows of the right matrix

If A is an mxn matrix and B is an nxp matrix, then

their matrix productAB is the mxp matrix whose

entries are given by dot-product of the

corresponding row ofA

and the correspondingcolumn of B


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Matrix multiplication satisfies the rules (AB)C =A(BC) (associativity) and

(A+B)C = AC+BC as well as C(A+B) = CA+CB

matrix multiplication is not commutativeAB BA


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Identity Matrix

The identity matrix In of size n is the nxn matrix in

which all the elements on the main diagonal are

equal to 1 and all other elements are equal to 0

It is called identity matrix because multiplication

with it leaves a matrix unchanged

MIn = ImM = M


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Symmetric Matrix

In linear algebra, a symmetric matrix is a square

matrix that is equal to its transpose

The entries of a symmetric matrix are symmetric

with respect to the main diagonal


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Row & Column Matrix

In linear algebra, a row vector or row matrix is a

1 n matrix, that is, a matrix consisting of a single

row

The transpose of a row vector is a column matrix or

vector


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Scalar Matrix

A diagonal matrix with all its main diagonal entries

equal and rest are zeros is a scalar matrix


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Trace of a Matrix

In linear algebra, the trace of an nxn square matrix

A is defined to be the sum of the elements on the

main diagonal

Let Tbe a linear operator represented by the matrix

Then tr (T) = 2 + 1 1 = 2.


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Determinant of Matrix

In algebra, the determinant is a special number

associated with any square matrix

The determinant of a matrix A, is denoted det(A),

or without parentheses: det A

denotes the determinant of the matrix

has determinant det A = ad bc


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Inverse of Matrix


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Orthogonal Matrix

In linear algebra, an orthogonal matrix is a square

matrix with real entries whose columns (or rows) are

orthogonal unit vectors If [A]-1 = [A]T then [A] is orthogonal

If det [A] = +1 then [A] is Proper orthogonal

If det [A] = -1 then [A] is Improper orthogonal

For orthogonal Matrix [A][A]T = [I]


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Vector

Physical quantities which have both magnitude anddirection, such as force, in contrast to scalarquantities, which have no direction.

Where vx, vy, and vz are the magnitudes of thecomponents of v.

Where v1, v2, , vn 1, vn are the components ofv.


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Dot Product


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Cross Product

The cross product of two vectors a and b is denoted

by a b

a = a1i + a2j + a3k = (a1, a2, a3)And

= b1i + b2j + b3k = (b1, b2, b3).


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Simultaneous Equations

In mathematics, simultaneous equations are a set

of equations containing multiple variables

This set is often referred to as a system of

equations

A solution to a system of equations is a particular

specification of the values of all variables that

simultaneously satisfies all of the equations


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Solution of Equation

Solving this involves subtractingx + y= 6 from

2x + y= 8 (using the elimination method) to remove

the y-variable, then simplifying the resulting

equation to find the value ofx, then substituting the

x-value into either equation to find y.


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Solution of Simultaneous Equations

Methods of Solution of Simultaneous Equations

1- Cramers Rule

2- Gauss Elimination Method

3- Gamss-Seidel Iteration

4- Choleskys Method


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Flexibility Method

In structural engineering, the flexibility method is

the classical consistent deformation method for

computing member forces and displacements in

structural systems

Its modern version formulated in terms of the

members' flexibility matrices also has the name the

matrix force method due to its use of member

forces as the primary unknowns

Flexibility is the inverse of stiffness


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The relationship between actions and displacements play an important role

structural

analysis. A convenient way to see this relationship is through a linear, elastic

spring

The action A will compre

ss (translate

) the

spring an amount D.T

his can be

expressed

through the simple expression:

In this equation F is the flexibility of the spring, and this quantity is defined as the

displacement produced by a unit value of the action A.


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This relationship can also be expressed as

A = KD

Here K is the stiffness of the spring and is defined as the action required toproduce a unit

displacement in the spring. The flexibility and stiffness of the spring are

inverse to one another.

lec 2 mathematical foundation

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