March 20, 2003March 20, 2003

9:35 AM9:35 AM

Little 109Little 109

CES 4141CES 4141

Forrest MastersForrest Masters

A Recap of Stiffness by Definition and the Direct Stiffness Method

Farther Down the Yellow Brick Farther Down the Yellow Brick Road..Road..

Vitrual W ork

Force M ethod

S lope Deflection

M om ent-Area

Classical M ethods

Stiffness by Definition

Trusses

Beams

Direct Stiffness

M atrix Methods

Structural Analysis

Our Emphasis This Week: Trusses..Our Emphasis This Week: Trusses..

Composed of slender, Composed of slender, lightweight members lightweight members

All loading occurs on All loading occurs on jointsjoints

No moments or No moments or rotations in the jointsrotations in the joints

Axial Force MembersAxial Force Members Tension (+) Tension (+) Compression (-)Compression (-)

StiffnessStiffness KKijij = the amount of force required = the amount of force required

at i to cause a unit displacement at i to cause a unit displacement at j, with displacements at all at j, with displacements at all other DOF = zeroother DOF = zero

A function of:A function of:– System geometrySystem geometry– Material properties (E, I)Material properties (E, I)– Boundary conditions (Pinned, Roller Boundary conditions (Pinned, Roller

or Free for a truss)or Free for a truss) NOT a function of external loadsNOT a function of external loads

K = AE/LK = AE/L

From Strength of Materials..From Strength of Materials..

Combine two equations to get a Combine two equations to get a stiffness stiffness elementelement

F LA E

F = k * F = k * kF

F

A EL

SpringSpring

Axial Axial DeformationDeformation

kA E

L

Units ofUnits ofForce per Force per

LengthLength

Go to the Board..Go to the Board..

Let’s take a Let’s take a look at last look at last week’s week’s homework to homework to shed some shed some light on the light on the Stiffness by Stiffness by DefinitionDefinition ProcedureProcedure

DOFDOF

From Stiffness by From Stiffness by DefinitionDefinition

We can create aWe can create a stiffness matrixstiffness matrix that that accounts for the material and accounts for the material and geometric properties of the structuregeometric properties of the structure

A square, symmetric matrix KA square, symmetric matrix Kijij = K = Kjiji Diagonal terms always positiveDiagonal terms always positive The stiffness matrix is independent of The stiffness matrix is independent of

the loads acting on the structure. the loads acting on the structure. ManyMany loading casesloading cases can be tested can be tested without recalculating the stiffness without recalculating the stiffness matrixmatrix

Stiffness by Definition only uses a small part of the Stiffness by Definition only uses a small part of the information available to tackle the probleminformation available to tackle the problem

However ..

Stiffness by Definition Stiffness by Definition OnlyOnly Considers..Considers..

Stiffnesses from Stiffnesses from Imposed Imposed DisplacementsDisplacements

Unknown Unknown Displacements Displacements

Known LoadingsKnown Loadings

For each released DOF, we get one equation For each released DOF, we get one equation that adds to the stiffness, displacement and that adds to the stiffness, displacement and loading matricesloading matrices

But what aboutBut what about ReactionsReactions andand Known Known Displacements?Displacements?

K * r = RStiffness Matrix

Unknown Displacements

Known External Forces

A Better Method: A Better Method: Direct StiffnessDirect Stiffness

Consider all DOFs Stiffness ByConsider all DOFs Stiffness By Direct Direct Definition Stiffness Definition Stiffness

..now we have ..now we have more equationsmore equations to work with to work with

PINPIN

ROLLERROLLER

00

11

22

22

A Simple ComparisonA Simple Comparison66

55

4433

2211

Stiffness by DefinitionStiffness by Definition 2 Degrees of 2 Degrees of

FreedomFreedom Direct StiffnessDirect Stiffness 6 Degrees of 6 Degrees of

FreedomFreedom DOFs 3,4,5,6 = 0DOFs 3,4,5,6 = 0 Unknown Reactions Unknown Reactions

(to be solved) (to be solved) included in Loading included in Loading MatrixMatrix

Remember.. Remember.. More DOFs = More EquationsMore DOFs = More Equations

Node Naming ConventionNode Naming Convention Unknown or Unknown or

“Unfrozen” Degrees “Unfrozen” Degrees of Freedom are of Freedom are numbered first…numbered first…

r1, r2r1, r2 Unknown or Unknown or

“Unfrozen” Degrees “Unfrozen” Degrees of Freedom followof Freedom follow

r3, r4, r5, r6r3, r4, r5, r6

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If Possible.. If Possible.. X-direction before Y-directionX-direction before Y-direction

Stiffness by Definition vs Stiffness by Definition vs Direct StiffnessDirect Stiffness

KK1111 KK1212 KK1313 KK1414 KK1515 KK1616

KK2121 KK2222 KK2323 KK2424 KK2525 KK2626

KK3131 KK3232 KK3333 KK3434 KK3535 KK3636

KK4141 KK4242 KK4343 KK4444 KK4545 KK4646

KK5151 KK5252 KK5353 KK5454 KK5555 KK5656

KK6161 KK6262 KK6363 KK6464 KK6565 KK6666

rr11

rr22

rr33

rr44

rr55

rr66

RR11

RR22

RR33

RR44

RR55

RR66

==

66 55

44 3322 11

Stiffness by Definition Solution in Stiffness by Definition Solution in REDREDDirect Stiffness Solution in Direct Stiffness Solution in REDRED/YELLOW/YELLOW

The Fundamental The Fundamental ProcedureProcedure Calculate the Calculate the Stiffness MatrixStiffness Matrix

Determine Local Stiffness Matrix, KeDetermine Local Stiffness Matrix, Ke Transform it into Global Coordinates, KGTransform it into Global Coordinates, KG Assemble all matricesAssemble all matrices

Solve for the Solve for the Unknown Displacements Unknown Displacements Use unknown displacements to solve for the Use unknown displacements to solve for the

Unknown ReactionsUnknown Reactions Calculate the Calculate the Internal ForcesInternal Forces

To continue..To continue..

You need your You need your Direct Stiffness – Direct Stiffness – Truss Application HandoutTruss Application Handout to follow to follow the remaining lecture. If you forgot the remaining lecture. If you forgot it, look on your neighbor’s, pleaseit, look on your neighbor’s, please

I have your I have your new homework new homework (if you (if you don’t have it already)don’t have it already)

Go to Go to http://www.ce.ufl.edu/~kgurl for the handout for the handoutFOR MORE INFO ..

OverviewOverviewFirst, we will First, we will

decompose the entire decompose the entire structure into a set of structure into a set of finite elementsfinite elements

Next, we will build a Next, we will build a stiffness matrixstiffness matrix for for each element (6 each element (6 Here)Here)

Later, we will combine Later, we will combine all of the local all of the local stiffness matrices into stiffness matrices into ONE ONE globalglobal stiffness stiffness matrixmatrix

Node 1Node 1

Node 2Node 2

11

4422

33 55

Element Stiffness Matrix in Local Element Stiffness Matrix in Local CoordinatesCoordinates

Remember KRemember Kijij = the amount of force required at i = the amount of force required at i to cause a unit displacement at j, with to cause a unit displacement at j, with displacements at all other DOF = zerodisplacements at all other DOF = zero

For a truss element (which has 2 DOF)..For a truss element (which has 2 DOF)..

K11*v1 + K12*v2 = S1K11*v1 + K12*v2 = S1K21*v1 + K22*v2 = S2K21*v1 + K22*v2 = S2

S1S1

S2S2

v2v2v1v1

K1K111

K1K122

K2K211

K2K222

vv11vv22

== SS11SS22Gurley refers to the Gurley refers to the axial displacementaxial displacement as “ as “vv” ”

and the and the internal forceinternal force as “ as “SS” in the local ” in the local coordinate systemcoordinate system

Element Stiffness Matrix in Local Element Stiffness Matrix in Local CoordinatesCoordinates

Use Stiffness by Definition to finding Ks of Local Use Stiffness by Definition to finding Ks of Local SystemSystem

Node 1Node 1

Node 2Node 2

AEAE LL

KK2222

KK1212

AEAE LL

KK1111

KK2121

KK1212 = - AE / L = - AE / LKK2222 = AE / L = AE / L

KK1111 = AE / L = AE / LKK2121 = - AE / L = - AE / L

Element Stiffness Matrix in Local Element Stiffness Matrix in Local Coordinates Cont..Coordinates Cont..

Put the local stiffness elements in Put the local stiffness elements in matrix matrix formform

Simplified..Simplified..

For a truss elementFor a truss element

Displacement Transformation Displacement Transformation MatrixMatrix

Structures are composed of many members in Structures are composed of many members in many orientationsmany orientations

We must move the stiffness matrix from a We must move the stiffness matrix from a locallocal to a to a globalglobal coordinate system coordinate system

S1S1

S2S2

v2v2v1v1

r1r1r2r2

r4r4r3r3

xxyy

LOCALLOCAL

GLOBALGLOBAL

How do we do that?How do we do that? Meaning if I give you a point (x,y) in Meaning if I give you a point (x,y) in

Coordinate System Z, how do I find the Coordinate System Z, how do I find the coordinates (x’,y’) in Coordinate System Z’coordinates (x’,y’) in Coordinate System Z’

xxyy

x’x’

y’y’Use aUse a

Displacement Displacement Transformation Transformation

MatrixMatrix

To change the coordinates of a truss..To change the coordinates of a truss..

Each node has one Each node has one displacement in the local displacement in the local system concurrent to the system concurrent to the element (v1 and v2)element (v1 and v2)

In the global system, every In the global system, every node has two displacements node has two displacements in the x and y directionin the x and y direction

r1r1r2r2

r4r4r3r3

xxyy

v1v1

v2v2

v1 will be expressed by r1 and r2v1 will be expressed by r1 and r2v2 will be expressed by r3 and r4v2 will be expressed by r3 and r4

Displacement Transformation Matrix Displacement Transformation Matrix Cont..Cont..

The relationship between v and The relationship between v and r is the vector sum:r is the vector sum:

v1 = r1*cos v1 = r1*cos xx + r2*cos + r2*cos YY

v2 = r3*cos v2 = r3*cos xx + r4*cos + r4*cos YYx

Y

v1

r1

r2

We can simplify the cosine terms:We can simplify the cosine terms:Lx = cos Lx = cos xx Ly = cos Ly = cos yy v1 = r1*Lx + v1 = r1*Lx +

r2*Ly r2*Ly v2 = r3*Lx + v2 = r3*Lx +

r4*Lyr4*LyPut in matrix formPut in matrix form

Displacement Transformation Matrix Displacement Transformation Matrix Cont..Cont..

v1 = r1*Lx + r2*Ly v1 = r1*Lx + r2*Ly v2 = r3*Lx + r4*Lyv2 = r3*Lx + r4*Ly

v1

v2

Lx

0

Ly

0

0

Lx

0

Ly

r1

r2

r3

r4

aLx

0

Ly

0

0

Lx

0

Ly

Transformation matrix, aTransformation matrix, a gives us the gives us therelationship we soughtrelationship we sought

So.. So.. v = a*rv = a*r

Force Transformation MatrixForce Transformation Matrix

Similarly, we can perform a Similarly, we can perform a transformation on the internal forcestransformation on the internal forces

R1

R2

R3

R4

Lx

Ly

0

0

0

0

Lx

Ly

S1

S2

S1S1

S2S2

R1R1R2R2

R3R3R4R4

Element Stiffness Matrix in Global Element Stiffness Matrix in Global CoordinatesCoordinates

Let’s put it all together.. We know that the Let’s put it all together.. We know that the

Internal force = stiffness * local displacementInternal force = stiffness * local displacement (S = k * v)(S = k * v)Units: Force = (Force/Length) * LengthUnits: Force = (Force/Length) * Lengthlocal disp = transform matrix * global displocal disp = transform matrix * global disp (v = a * r)(v = a * r)Substitute local displacementSubstitute local displacementInternal force = stiffness * transform matrix * global dispInternal force = stiffness * transform matrix * global disp

(S = k * a * r)(S = k * a * r)Premultiply by the transpose of “a”Premultiply by the transpose of “a”aaTT * S= a * S= aTT * k * a * r * k * a * r

and substitute R = aand substitute R = aTT * S to get * S to get R = aR = aTT * k * a * r * k * a * r

Element Stiffness Matrix in Global Element Stiffness Matrix in Global Coordinates Cont..Coordinates Cont..

is an important relationship is an important relationship between the loading, stiffness between the loading, stiffness and displacements of the and displacements of the

structurestructurein terms of the global systemin terms of the global system

RR = = aaTT * k * a * k * a * * rrStiffnessStiffness

termterm

We have a stiffness term, We have a stiffness term, KeKe, for each element , for each element in the structure in the structure

We use them to build the global stiffness We use them to build the global stiffness matrix, matrix, KGKG

KeKe = = aaTT * k * a * k * a

Element Stiffness Matrix in Global Element Stiffness Matrix in Global Coordinates Cont..Coordinates Cont..

Let’s expand all of terms to get Let’s expand all of terms to get a Ke that we can use.a Ke that we can use.KeKe = = aaTT * k * a * k * a

KeA E

L

Lx

Ly

0

0

0

0

Lx

Ly

1

1

1

1

Lx

0

Ly

0

0

Lx

0

Ly

KeA E

L

Lx2

LxLy

Lx2

Lx Ly

LxLy

Ly2

Lx Ly

Ly2

Lx2

Lx Ly

Lx2

LxLy

Lx Ly

Ly2

LxLy

Ly2

(14) From notes(14) From notesGreat formula to Great formula to plug into your plug into your calculatorcalculator

Element Stiffness Matrix in Global Element Stiffness Matrix in Global Coordinates Cont..Coordinates Cont..

Let’s use a Let’s use a problem to illustrate problem to illustrate the rest of the the rest of the procedureprocedure

We will start by We will start by calculating KE’s for calculating KE’s for the two elementsthe two elements

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3 ft3 ft

4 ft4 ft

Node Node 11

Node Node 33

Node Node 22Element Element 22

Element Element 11

Assembly of the Global Stiffness Matrix Assembly of the Global Stiffness Matrix (KG)(KG)

r1r1

r2r2

r3r3

r4r43 ft3 ft

Element 1Element 1LL = 3= 3Lx Lx = = x / L = (3-0) / 3 = 1x / L = (3-0) / 3 = 1Ly Ly = = y / L = (0-0) / 3 = 0y / L = (0-0) / 3 = 0

NearNear FarFar

r1r1 r2r2 r3r3 r4r4r1r1r2r2r3r3r4r4

Ke1 A E

0.333

0

0.333

0

0

0

0

0

0.333

0

0.333

0

0

0

0

0

Pick a Pick a NearNear and a and a FarFar

Plug Lx, Ly and L Plug Lx, Ly and L into equation 14 to into equation 14 to getget

Assembly of the Global Stiffness Matrix Assembly of the Global Stiffness Matrix (KG)(KG)

r1r1r2r2

r5r5r6r6

3 ft3 ft

4 ft4 ft5 ft5 ft

Element 2Element 2LL = 5= 5Lx Lx = = x / L = (3-0) / 5 = x / L = (3-0) / 5 = 0.60.6Ly Ly = = y / L = (4-0) / 5 = y / L = (4-0) / 5 = 0.80.8

NearNear

FarFar

Ke2 A E

0.072

0.096

0.072

0.096

0.096

0.128

0.096

0.128

0.072

0.096

0.072

0.096

0.096

0.128

0.096

0.128

r1r1 r2r2 r5r5 r6r6r1r1r2r2r5r5r6r6

The Entire Local Stiffness The Entire Local Stiffness Matrix in Global TermsMatrix in Global Terms

Ke2 A E

0.072

0.096

0.072

0.096

0.096

0.128

0.096

0.128

0.072

0.096

0.072

0.096

0.096

0.128

0.096

0.128

r1r1 r2r2 r5r5 r6r6r1r1r2r2r5r5r6r6

0.072

0.096

0

0

0.072

0.096

0.096

0.128

0

0

0.096

0.128

0

0

0

0

0

0

0

0

0

0

0

0

0.072

0.096

0

0

0.072

0.096

0.096

0.128

0

0

0.096

0.128

r1r1r2r2r3r3r4r4r5r5r6r6

r1 r2 r3 r4 r5 r1 r2 r3 r4 r5 r6r6

Notice that there Notice that there aren’t any terms aren’t any terms in the local in the local matrix for matrix for r3r3 and and r4r4

ShorthandShorthandReal Real MatrixMatrix

Assembly of the Global Assembly of the Global Stiffness Matrix (KG)Stiffness Matrix (KG)

Summing Ke1 and Ke2Summing Ke1 and Ke2

r1r1

KG A E

0.405

0.096

0.333

0.000

0.072

0.096

0.096

0.128

0.000

0.000

0.096

0.128

0.333

0.000

0.333

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.072

0.096

0.000

0.000

0.072

0.096

0.096

0.128

0.000

0.000

0.096

0.128

r2r2 r3r3 r4r4 r5r5 r6r6r1r1r2r2r3r3r4r4r5r5r6r6

==KK rr RR

How does this relate to Stiffness by Definition?How does this relate to Stiffness by Definition?

Solution ProcedureSolution Procedure

Now, we can examine the full systemNow, we can examine the full system

ReactionsReactions Known displacementsKnown displacements@ reactions ( = 0 )@ reactions ( = 0 )

Unknown DeflectionsUnknown DeflectionsLoads acting on the nodesLoads acting on the nodesR1 0.405 0.096 -0.333 0.000 -0.072 -0.096 r1R2 0.096 0.128 0.000 0.000 -0.096 0.128 r2R3 -0.333 0.000 0.333 0.000 0.000 0.000 r3R4 0.000 0.000 0.000 0.000 0.000 0.000 r4R5 -0.072 -0.096 0.000 0.000 0.072 0.096 r5R6 -0.096 -0.128 0.000 0.000 0.096 0.128 r6

= X

Solution Procedure cont..Solution Procedure cont..

To find the unknowns, we must subtend the matricesTo find the unknowns, we must subtend the matrices

Rk

Ru

AEK11

K21

K12

K22

ru

rk

K11K11

K22K22

K12K12

K21K21==

Two Two Important Important EquationsEquations

Rk = AE ( K11*ru + Rk = AE ( K11*ru + K12*rk )K12*rk )Ru = AE ( K21*ru + Ru = AE ( K21*ru + K22*rk )K22*rk )

(24)(24)(25)(25)

Going to be ZERO. Going to be ZERO. Why?Why?

Solution Procedure cont..Solution Procedure cont..

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3 ft3 ft

4 ft4 ft

10 kips10 kips

We will apply a load at DOF We will apply a load at DOF 22Then use equation (24)Then use equation (24)Rk = AE ( K11*ru + K12*rk )Rk = AE ( K11*ru + K12*rk )

0

10

AE0.405

0.096

0.096

0.128

r1

r2

AE K12

0

0

0

0

0 = AE ( 0.405*r1 + 0.096*r2)0 = AE ( 0.405*r1 + 0.096*r2)-10 = AE ( 0.096*r1 + 0.128*r2)-10 = AE ( 0.096*r1 + 0.128*r2)

r1 = 22.52/AEr1 = 22.52/AEr2 = -95.02/AEr2 = -95.02/AEsolvedsolved

00

Solution Procedure cont..Solution Procedure cont..

With the displacements, we can use equation (25) With the displacements, we can use equation (25) to find the reactions at the pinned endsto find the reactions at the pinned endsRu = AE ( K21*ru + K22*rk )Ru = AE ( K21*ru + K22*rk )

R3

R4

R5

R6

AE

0.333

0

0.072

0.096

0

0

0.096

0.128

22.52AE

95.02AE

AE K22

0

0

0

0

00

R3 = -7.5 kipsR3 = -7.5 kips R4 = 0 kipsR4 = 0 kipsR5 = 7.5 kipsR5 = 7.5 kips R6 = 10 kipsR6 = 10 kips

Internal Member Force RecoveryInternal Member Force Recovery

To find the internal force inside of an To find the internal force inside of an element, we must return to the local element, we must return to the local coordinate systemcoordinate systemRemember the equation Remember the equation S = k * a * rS = k * a * r ? ?

S1

S2

AEL

1

1

1

1

Lx

0

Ly

0

0

Lx

0

Ly

r1

r2

r3

r4

But S1 always But S1 always Equals –S2Equals –S2

soso SAEL

Lx Ly Lx Ly( )

r1

r2

r3

r4

Internal Member Force Recovery Internal Member Force Recovery Cont..Cont..

For Element 1For Element 1

For Element 2For Element 2

S1AE3

1 0 1 0( )

22.52AE

95.02AE

0

0

r1r1r2r2r3r3r4r4

= -7.5 kips= -7.5 kips

r1r1r2r2r5r5r6r6

= 12.5 kips= 12.5 kipsS2AE5

0.6 0.8 0.6 0.8( )

22.52AE

95.02AE

0

0

ConclusionConclusionWe solvedWe solved Element StiffnessesElement Stiffnesses Unknown Unknown

DisplacementsDisplacements ReactionsReactions Internal ForcesInternal Forces

I will cover another example I will cover another example in the laboratoryin the laboratory

Matrices.Matrices...

a x b y c z dStart with a basic equation

In order to solve x,y,z .. You must have three equations

a 1

a 2

a 3

b 1

b 2

b 3

c 1

b 2

b 3

x

y

z

a1 x b1 y c1 z d1

a2 x b2 y b2 z d2

a3 x b3 y b3 z d3

But you must put these equations in matrix form

d 1

d 2

d 3

=

41

12

3

B

CA

10 kips5 kips

A Sample Problem solved with Stiffness by Definition and Direct Stiffness

42

For Stiffness by Definition, we are only concerned with the three DOF’s that are free to move:

r1

r2

r3

43

For Column 1, we set r1 = 1 and r2 = r3 = 0

A

B

C

B’Element Change in LengthElement Change in Length

1 6/10 Long2 8/10 Short3 0

Unit DisplacementUnit Displacement

44

For Column 2, we set r2 = 1 and r1 = r3 = 0

A

B

C

B’

Element Change in LengthElement Change in Length

1 8/10 Short2 6/10 Short3 0

Unit DisplacementUnit Displacement

45

For Column 3, we set r3 = 1 and r1 = r2 = 0

A

B

C

Element Change in LengthElement Change in Length

1 02 4/5 Long3 1 Long

C’

Unit DisplacementUnit Displacement

46

K

750

150

225

150

91600

350

225

350

950

r1 r2 r3

r1

r2

r3

The final stiffness matrix is as follows..

r1 r2 r30.14 -0.02 -0.08 r1-0.02 0.152 -0.06 r2-0.08 -0.06 0.18 r3

47

For Direct Stiffness, we are concerned with all six DOF’s in the structural system:

r1

r2

r3

r4

r5

r6

48

In the Direct Stiffness Method, we will use this equation for each elements 1, 2 and 3:

KeA E

L

Lx2

LxLy

Lx2

Lx Ly

LxLy

Ly2

Lx Ly

Ly2

Lx2

Lx Ly

Lx2

LxLy

Lx Ly

Ly2

LxLy

Ly2

Near X Near Y Far X Far Y

Near X

Near Y

Far X

Far Y

DOFLocation

49

Element 1

r5 r6 r1 r2

r5

r6

r1

r2

L = 6Lx = 0.6Ly = -0.8

Ke 1 AE

350

225

350

225

225

875

225

875

350

225

350

225

225

875

225

875

50

Element 1 – Another View

r1 r2 r3 r4 r5 r6

r1

Ke1 AE

350

225

0

0

350

225

225

875

0

0

225

875

0

0

0

0

0

0

0

0

0

0

0

0

350

225

0

0

350

225

225

875

0

0

225

875

r2

r3r4

r5

r6

51

Element 2

r1 r2 r3 r4

r1

r2

r3

r4

L = 8Lx = 0.8Ly = 0.6

Ke 2 AE

225

350

225

350

250

9200

350

9200

225

350

225

350

350

9200

350

9200

52

Element 3

r5 r6 r3 r4

r5

r6

r3

r4

L = 10Lx = 1Ly = 0

Ke 3 AE

110

0

110

0

0

0

0

0

110

0

110

0

0

0

0

0

53

Summing Elements 1 through 3

Ke 1 AE

350

225

350

225

225

875

225

875

350

225

350

225

225

875

225

875

Ke 2 AE

225

350

225

350

250

9200

350

9200

225

350

225

350

350

9200

350

9200

Ke 3 AE

110

0

110

0

0

0

0

0

110

0

110

0

0

0

0

0

+ +

Remember: We must take care to add the correct elements from the local stiffness matrix to the global stiffness matrix.

54

Summing Elements 1 through 3

KG AE

350

225

225

350

225

350

350

225

225

350

875

9200

350

9200

225

875

225

350

225

110

350

0

110

0

350

9200

350

0

9200

0

0

0

350

225

110

0

350

110

225

0

225

875

0

0

225

0

875

0

r1 r2 r3 r4 r5 r6

r1

r2

r3

r4

r5

r6

55

Summing Elements 1 through 3

r1 r2 r3 r4 r5 r60.14 -0.02 -0.08 -0.06 -0.06 0.08 r1-0.02 0.15 -0.06 -0.05 0.08 -0.11 r2-0.08 -0.06 0.18 0.06 -0.10 0.00 r3-0.06 -0.05 0.06 0.05 0.00 0.00 r4-0.06 0.08 -0.10 0.00 0.16 -0.08 r50.08 -0.11 0.00 0.00 -0.08 0.11 r6

Look Familiar? We found the yellow portion in the Stiffness by Definition Method

X

Stiffness by Definition vs Direct Stiffness by Definition vs Direct StiffnessStiffness

KK

K K completedcompleted

rrunknownunknown

RRunknownunknownrrknownknown

RRknownknown=

=X

ReactionsReactionsZero Unless Zero Unless Settlement Settlement

OccursOccurs