Simple Truss Problems Linear Algebraic Systems

Simple Truss Problemsand

Linear Algebraic Systems

ES100

February 22, 1999

T.S. Whitten

Definition of a truss

● A truss is a rigid frame consisting of slendermembers connected at their endpoints.

truss

A

D

C

B

E

Simple Trusses

The simplest configuration for a stable truss is atriangle as shown in red above.

The Triangular Truss

45°

2 m

A

B

C

Ay=500 N

500 N

2 m

Ax=500 N

Cy=500 N

The Triangular Truss

45°

2 m

A

B

C

Ay=500 N

500 N

2 m

Ax=500 N

Cy=500 N

500 N

FBA

FBC

Free Body Diagram (FBD)500 N

2 m

0 N

Static Equilibrium

500 N

FBA

FBC

B

Static Equilibrium

500 N

FBA

FBC

B● The joint B is not moving and is therfore

said to be in static equilibrium.

Static Equilibrium

500 N

FBA

FBC



● Physically speaking, this means that thereare no unbalanced forces so if we add allof the forces acting in the x-direction,their sum should be zero.

Static Equilibrium

500 N

FBA

FBC



● Physically speaking, this means that thereare no unbalanced forces so if we add allof the forces acting in the x-direction,their sum should be zero.

● The same is true for forces acting in the y-direction

Solving For Forces

500 N

FBA

FBC

B

Solving For Forces

500 N

FBA

FBC

B

045sin500;0 =°−=∑ BCx FF

N1.707=BCF

Solving For Forces

500 N

FBA

FBC

B

045sin500;0 =°−=∑ BCx FF

N1.707=BCF

045cos;0 =−°=∑ BABCy FFF

N500=BAF

Unique vs. Singular Systems

● Some systems of equations do not haveunique solutions.

y

x

unique solution

singular system

Statically Indeterminate Example

● Previously, we showed a system of twoequations that had two unknowns. Now,adding member BD and CD:

A C

B



A C

B D

C



A C

B D

C

0500 =+−=∑ BDxBCx FFF

At pin B:

0=−=∑ yy BABCy FFF



A C

B D

C

0500 =+−=∑ BDxBCx FFF

At pin B:

0=−=∑ yy BABCy FFF

Hence, we have onlytwo equations forthree unknowns.

Expressing Sets of Linear AlgebraicEquations in Matrix Form

● Summation of the forces in the x and y directionscan be written as:

0)1(45cos

500)0(45sin

=−°−=+°−

BABC

BABC

FF

FF

● These two equations can be equivalently expressedin matrix form as...

−=

⋅

°°−

0

500

145cos

045sin

32144 344 21variables

BA

BC

tscoefficien

F

F

Multiplication of Two Matrices

Generalized System of LinearEquations

mnmnmm

nn

nn

bxaxaxa

bxaxaxa

bxaxaxa

=+++

=+++=+++

...

...

...

2211

22222121

11212111

MMMM

Generalized MatrixRepresentation of Linear System

=

mnmm

n

n

aaa

aaa

aaa

L

MLMM

L

L

21

22221

11211

A

=

nx

x

x

M2

1

x

=

mb

b

b

M2

1

b

In matrix A, m is the index that identifies the row and n is theindex that identifies the column. Thus, the requirement that thenumber of unknowns must equal the number of equations in orderfor a unique solution to exist, is at the root of matrixmultiplication. i.e. m must be equal to n

Solution Techniques

● One method of solving involves successiveelimination of variables until only oneequation and one unknown variable remains.Gauss Elimination

● Cramer’s Method is based on finding matrixdeterminants for the system

● Another technique particularly suited toMATLAB is based on the matrix inversemethod

Solution of Linear System UsingMATLAB

Start script file

Ask user fornumber of equations

Call function build.mto create matrix A

Call function build.mto create matrix B

Send A and B to functionlinsolve1.m

Display results

end file

Script matalg.m

● Calls the function build.m twice• build.m performs a dedicated task to inout data

● Calls a separate function, linsolve1.m to dothe dedicated task of computing the solution

● Displays the answer

Function build.m

● Function uses a for loop to iterate throughmatrix position.

Function linsolve1.m

● This function introduces use of the MATLAB

backslash( \ ), matrix operator to solvelinear systems of the general form:

bAx =

MATLAB Demo

run matalg.m

Simple Truss Problems Linear Algebraic Systems

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