analysis of statically determinate trusses session...

Analysis of Statically Determinate TrussesSession 07 - 10

Subject : S0695 / STATICS

Year : 2008

Bina Nusantara

What are Trusses ?

are structures consisting of two or more straight, slender membersconnected to each other at their endpoints.

Bina Nusantara

What for are Trusses used ?

Trusses are often used to support roofs,

bridges, power-line towers, and appear

in many other applications.

Bina Nusantara

3.1. Type of Trusses

3.1.1. Roof Trusses

Are often used as a part of building frame. Trusses used to support roofs are selected on the basis of the span, the slope an the roof material. Roof trusses are supported either by columns of wood, masonry, concrete, or steel.

Bina Nusantara

3.1. Type of Trusses

3.1.1. Roof Trusses

Purlins

A

B

C

D

E

Roof Truss

Span

Bay

Roof

Bina Nusantara

3.1. Type of Trusses

3.1.1. Roof Trusses

There are common types of Roof trusses as follow :

Bina Nusantara

3.1. Type of Trusses

3.1.2. Bridge Trusses

Are often used as an infrastructure facility. Trusses used to support load on deck , then floor beams and finally to the joints of the supporting

trusses. Trusses serves resist to the lateral forces caused by wind load and the

sideways caused by vehicle moving.

Bina Nusantara

3.1. Type of Trusses

3.1.2. Bridges Trusses

Typical Bolted Truss Joint

Gusset Plate

Channel Section

Bina Nusantara

3.1. Type of Trusses

3.1.2. Bridge TrussesThere are common types of Bridge trusses as follow :

Bina Nusantara

3.2. Trusses Classification

3.2.1. Coplanar Trusses

3.2.1.1. Simple Truss

Simple truss constructed by starting with a basic tringular element. The simpliet framework that is

rigid and stable is a triangle

A B

C

Bina Nusantara

3.2. Trusses Classification

3.2.1. Coplanar Trusses 3.2.1.2. Compound Truss

Is formed by connecting two or more simple trusses

AC

BD

E F

Bina Nusantara

3.2. Trusses Classification

3.2.1. Coplanar Trusses

3.2.1.3. Complex Truss

That can not classified as being simple or compund trusses

A C B

D E

F

Bina Nusantara

3.2. Trusses Classification

3.2.2. Space Trusses

AB

CD

Bina Nusantara

3.4. Stability

Determinacy

m + r = 2j Statically determinatem + r > 2j Statically indeterminate

Degree of determinacy

( m + r ) -2jm = # membersj = # jointr = # exteral reaction

Bina Nusantara

3.4. Stability

A truss can be unstable if it is statically determinate or indeterminate. Stability will have to be determinated either by inspection or by forces analysis.

If m + r < 2j a truss will be unstable, it will be collapse.

Bina Nusantara

3.4. Stability

• External Stability

Truss is externally unstable if all its reactions

are concurrent or parralel

Bina Nusantara

3.4. Stability

• External Stability unstable concurrent reactions

A

C

B

D E

VA

HA

HB

Bina Nusantara

3.4. Stability

• External Stability unstable parallel reactions

A

D E

VA VB VC

CB

Bina Nusantara

3.4. Stability

• Internal Stability

If the the structure doesn’t hold on its join

in a fixed position , it will be unstable

Bina Nusantara

3.4. Stability

• Internal Stability

AC

BD

E F HG

Bina Nusantara

3.4. Stability

• Internal Stability

A

C

B

D E

F

O

Bina Nusantara

3.4. Stability

Truss is in unstable condition …..

• If m + r < 2j a truss will be unstable, it will be collapse.

• If m + r > 2j a truss will be unstable, it becomes if truss support reaction are concurrent or parallel , or if some components of the truss form collapsible mechanism.

Bina Nusantara

3.4. Trusses Analysis

Main concepts

• Forces only act at the pin joints

• Member forces are always in the direction of the member.

• There are no moments in a member (there can be moments on the full truss or a section of the truss if it has more than one member).

Bina Nusantara

3.4. Trusses Analysis

The three assumptions (or maybe better called idealizations) are :

1. Each joint consists of a single pin to which the respective members are connected individually.

2. No member extends beyond a joint.

3. Support forces and external loads are only applied at joints.

Bina Nusantara

3.4. Trusses Analysis

Assume for member analysis

COMPRESSIVE FORCE -A B

SA-BSA-B

Join

SA-B

A BTENSILE FORCE +SA-B

Bina Nusantara

3.4. Trusses Analysis

If the members forces become higher…..

SA-B

A B

SA-B Separation

A B

SA-BSA-B

A B

SA-BSA-B Buckling

Crumbling

Bina Nusantara

3.4. Trusses Analysis

3.4.1. Graphics Method

By drawing equilibrium diagrams to find force

members, it called Cremona Methods

Bina Nusantara

3.4. Trusses Analysis

3.4.1. Graphics MethodDrawing Step :1. Define the scale for diagram that you will draw2.Draw all the external reaction3.Start from join that has max. 2 unknown force members4. Make a simple complete polygon for that join.5. Pay attention about the assume for tensile or compressive force ( + / - )6. Repeat these steps for another join.7. Measure the all segmen, and fit with its scale

Bina Nusantara

3.4. Trusses Analysis

3.4.1. Graphics Method

Example :

Scale 1 P ~ 4 cm

Joint A VA – HA – S3 – S1 ; HA = 0

VA

AB

C

D

S2

S3

S5

S1S4

P8 m

3 m

VA = ½ PVB = ½ P

HA = 0

VB

P

S3

S1

P

( fit with scale )

Tension/Compressive

S5

S4

S3

S2

S1

Members ForceNumber of

Members

….cm ~ ….. P

….cm ~ ….. P

Tension / +

Compressive / -

Joint C S1 – S4 – S2 – P

S4

S2

4 cm ~ 1 PTension / +

….cm ~ ….. PTension / +

Joint D S4 – S3 – S5

S5

….cm ~ ….. PCompressive / -

HA

Check :

Joint B S2 – S5 – VB OK !!!

ΣV = 0 VA + VB = P

Bina Nusantara

3.4. Trusses Analysis

3.4.2. Analytics Method3.4.2.1. Method of Joint

The method of joints examines each joint as an independent staticstructure. The summation of all forces acting on the joint must equate to

zero. Both member forces and external forces are applied to the joint and then the force equilibrium equations are applied. For two dimensions, the equations are

ΣFx = ΣFy = 0

For three dimensions each joint must also be in equilibrium. For 3D problems, however, the vector form is generally easier. The equation becomes

ΣF = 0 (three dimensional vector form )

Bina Nusantara

3.4. Trusses Analysis

3.4.2. Analytics Method3.4.2.1. Method of Joint

AB

C

D

S2

S3

S5

S1S4

P8 m

3 m

VA = ½ PVB = ½ P

HA = 0

VA = ½P

A S3

HA = 0S1

DS5S4S3

CS2S1

S4

P

B

VB = ½P

S2

S5

Bina Nusantara

3.4. Trusses Analysis

3.4.2. Analytics Method3.4.2.1. Method of Joint

VA = ½P

A S3HA = 0

S1

Y

X

Y

X

VA = ½ P

HA = 0

S3

S1

Σ KY = 0

VA + S3Y = 0

VA + S3. sin α = 0

½ P + S3 . 3/5 = 0

α

S3Y

S3X

S3 = - 5/6 P

Σ KX = 0

HA + S3X + S1 = 0

HA + S3.cos α + S1 = 0

0 + -5/6 . 4/5 + S1 = 0

S1 = + 2/3 P

JOINT “A”

Bina Nusantara

3.4. Trusses Analysis

3.4.2. Analytics Method3.4.2.1. Method of Joint

Y

X

P

S1

Σ KY = 0

-P + S4 = 0

S4 = + 1 P

Σ KX = 0

- S1 + S2 = 0

-2/3 P + S2 = 0

S2 = + 2/3 P

JOINT “C”

Y

X

CS2S1

S4

P

S2

S4

Bina Nusantara

3.4. Trusses Analysis

3.4.2. Analytics Method3.4.2.1. Method of Joint

Y

X

Y

S4 = +1 P

Σ KY = 0

- S4 - S3Y - S5Y = 0

- 1 P- (-5/6 P).sin α - S5 sin α = 0

- 1 P- (-5/6 P).3/5 - S5. 3/5 = 0

- ½ P - S5. 3/5 = 0

S5 = - 5/6 P

Σ KX = 0

- S3X + S5X = 0

- (+5/6 P). cos α + S5 cos α = 0

- (+5/6 P). 4/5 + S5.4/5 = 0

- 2/3 P + S5.4/5 = 0

S5 = -5/6 P

JOINT “D”

DS5S4S3

X

S3 = - 5/6 PS5

α α

S3Y

S3X

S5Y

S5X

OK !!!

Bina Nusantara

3.4. Trusses Analysis

3.4.2. Analytics Method3.4.2.1. Method of Joint

Y

S2 = +2/3 P

Σ KY = 0

VB + S5Y = 0

½ P +S5 sin α = 0

½ P +S5. 3/5 = 0

½ P + (- 5/6 P). 3/5 = 0

½ P – ½ P = 0

0 = 0

Σ KX = 0

- S2 - S5X = 0

- (+2/3 P ) - S5 cos α = 0

- (+2/3 P) - (- 5/6 P).4/5 = 0

- 2/3 P + 2/3 P = 0

0 = 0

JOINT “B”Check

X

S5 = - 5/6 P

α

OK !!!

Y

XB

VB = ½P

S2

S5

VB = ½ P

S5Y

S5X

Bina Nusantara

3.4. Trusses Analysis

3.4.2. Analytics Method

3.4.2.2. Cross Section Method

This method uses free-body-diagrams of sections of the truss to

obtain unknown forces

Bina Nusantara

3.4. Trusses Analysis

3.4.2. Analytics Method3.4.2.2. Cross Section Method

AB

C

D

S2

S3

S5

S1S4

P8 m

3 m

VA = ½ PVB = ½ P

HA = 0

I

I

II

II

Cutpart of structure at

max. 3members

Bina Nusantara

3.4. Trusses Analysis

3.4.2. Analytics Method3.4.2.2. Cross Section Method

A

CS2

S3

S1S4

PVA = ½ P

HA = 0

I

I

D

S5

VB = ½ P

I

I

S2

S4

S3

B

Bina Nusantara

3.4. Trusses Analysis3.4.2. Analytics Method3.4.2.2. Cross Section Method

A

CS2

S3

S1S4

PVA = ½ P

HA = 0

I

I

D

S5

VB = ½ P

I

I

S2

S4

S3

B

CROSS SECTION I-I

D ΣMD=0

ΣMD=0-S2.3m + VA.4m – HA.3m = 0 -S2.3m + ½ P.4m – 0.3m = 0-S2.3m + 2 P.m = 0

S2 = + 2/3 P

ΣMC=0

ΣMC=0S3Y.4m+ VA.4m = 0 S3.sin α. 4m + ½ P.4m = 0S3.3/5.4m + 2 P.m = 0

S3 = - 5/6 P

ΣV=0S3Y + VA + P + S4 = 0 (-5/6).(3/5) + ½ P + P+ S4 = 0-3/6 + ½ P + P + S4 = 0

S4 = + 1 P

CΣMC=0

ΣMD=0

ΣMD=0S2.3m + VB.4m = 0S2.3m + ½ P.4m = 0-S2.3m + 2 P.m = 0

S2 = + 2/3 PΣMC=0- S3X.3m+ VB.4m = 0 - S3.cos α. 3m + ½ P.4m = 0- S3.4/5.3m + 2 P.m = 0

S3 = - 5/6 P

ΣV=0- S3Y + VB + S4 = 0 - (-5/6).(3/5) + ½ P + S4 = 03/6 + ½ P + S4 = 0

S4 = + 1 P

OK !!!

OK !!!

OK !!!

Bina Nusantara

3.4. Trusses Analysis

3.4.2. Analytics Method3.4.2.2. Cross Section Method

A

D

S3

S5

S1S4

VA = ½ P

HA = 0

II

II

CROSS SECTION II-II

BCS2

S5

S1S4

PVB = ½ P

II

II

Determine the force in members for Cross Section II-II !!

Bina Nusantara

3.4. Trusses Analysis

3.4.2. Analytics Method

3.4.2.3. Henneberg Method

We used this method for complex trusses,

where there are no join has max. 2 unknown forces.

Bina Nusantara

3.4. Trusses Analysis

3.4.2. Analytics Method

3.4.2.3. Henneberg Method

m + r = 2j11 + 3 = 2. 7

14 = 14

OK !!! Stable ..

Statically Determinate Truss

A C B

D

E F

G

S1 S2

S3S4

S5

S6

S7

S9S11 S10

S8

P

VA VB

HA

But… which joint has max.

2Unknown forces ??

Bina Nusantara

3.4. Trusses Analysis

3.4.2. Analytics Method

3.4.2.3. Henneberg Method

Step 1st :

A C B

D

E F

G

S1 S2

S3S4

S5

S6

S7

S9

Y

S10S8

P

VA VB

HA

Change the members position & make a new member “Y”

Change Position of member 11 & new member is “Y”

Bina Nusantara

3.4. Trusses Analysis

3.4.2. Analytics Method

3.4.2.3. Henneberg Method

Step 2nd :

A C B

D

E F

G

S1 S2

S3S4

S5

S6

S7

S9

Y

S10S8

P

VA VB

HA

Determine all the members forces with external force(by using any methods )

So

Bina Nusantara

3.4. Trusses Analysis

3.4.2. Analytics Method

3.4.2.3. Henneberg Method

Step 3rd :

A C B

D

E F

G

S1 S2

S3S4

S5

S6

S7

S9

Y

S10S8

P

VA VB

HA

Put 1 unit forceson the member position that has changed its position before. Assume that 1 unit force is a Tension force

1 unit

1 unit

Bina Nusantara

3.4. Trusses Analysis

3.4.2. Analytics Method

3.4.2.3. Henneberg Method

Step 4th :

A C B

D

E F

G

S1 S2

S3S4

S5

S6

S7

S9

Y

S10S8

VA VB

HA

1 unit

1 unitDetermine all the members forces

without external force (by using any methods )

S’

Bina Nusantara

3.4. Trusses Analysis3.4.2. Analytics Method

3.4.2.3. Henneberg Method Step 5th :

Check with member “Y” that used before the structure never has member “Y”

So…S’Y (x) + SoY = 0 x = - SoY

S’Y

Member forces are :So + S’(x) = 0

Determine x value for that structure

Bina Nusantara

3.4. Trusses Analysis3.4.2. Analytics Method

3.4.2.3. Henneberg Method

Step 6th :Put all your results in Step 2nd and step 4th on the table below….

i….

So + S’(x) S’. xS’SoNo of Members

Input So from step 2ndInput S’ from step 4th

x from step 5th

Final member forces

Bina Nusantara

3.5. Space Truss Analysis

A space truss can be unstable if it is statically determinate or indeterminate. Stability will have to be determinated either by inspection or by forces analysis.

If m + r < 3j a truss will be unstable, it will be collapse.

Bina Nusantara

3.5. Space Truss Analysis

Like a plannar trusses , Stable condition for :

m + r = 2j Statically determinate

m + r > 2j Statically indeterminate For

External stablity Check Reactions

Internal Stability Check Members arrangement

Bina Nusantara

3.5. Space Truss Analysis

That 3 dimensions there are

3 equations of equilibrium for each joint

Σ Fx = 0Σ Fy = 0Σ Fz = 0

Z

Y

X

Bina Nusantara

3.5. Space Truss Analysis

AB

CD

Truss ConstructionBracing

Bina Nusantara

3.5. Space Truss Analysis

AB

CD

E

Bina Nusantara

3.5. Space Truss Analysis

Joint E Z

Y

X

Σ Fx = 0Σ Fy = 0Σ Fz = 0