2.1 Displacement (External Load)

7/26/2019 2.1 Displacement (External Load)

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Truss Displacement

Ahmad Shufni bin Othman

Mei 2014

DISPLACEMENT of

STATICALLY

DETERMINATE

TRUSSES(Part 1)


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Truss Displacement

Depth

or Rise

Panel

Spanl l

Flat Truss or Parallel Chord Truss

Vertical Diagonal

Web

Members

(verticals & diagonals)

Top Chord

Bottom Chord

Joint,

Panel point

or Node


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Truss Displacement

When a frame structure or truss carries some external loads, all of itsmembers are subjected to either compressive or tensile axial forces.

We also know that whenever a body is subjected to some external

forces (load), it undergoes some deformation.

Therefore, when a frame or trusses are subjected to external forces,

all the members will undergo a change in their lengths. This change in lengths, will cause some displacement of all its joints,

except those which are rigidly fixed.

The net displacement of any joint may be found out by studying the

combine effect of changes of all members of the truss.

The truss joint may suffer displacement in any direction, dependingupon the direction of the load system and rigidity of the truss

members.

There are two very important type of displacement ;

i. Vertical displacement

ii. Horizontal displacement

Introduction


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Truss Displacement

It is a total displacement, suffer by a joint from its originalposition, in the vertical or horizontal direction due to the

action of external forces.

Vertical and Horizontal Displacement

The vertical or horizontal displacement of a joint may be

found out first by finding out the individual vertical or

horizontal displacements of all the joint, due to the changes

in the various members caused by external forces, and then by

combining up the result.


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Truss Displacement

There are many methods used for determining the displacement

of the joints in a frame or truss;

1. Virtual Work Method2. Unit Load Method

3. Strain Energy Method

4. Castigliano Theorem Method

Method of Determining The Displacement


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Truss Displacement

Vir tual Displacements

A virtual displacement is a displacement that is only imagined

to occur.

So given any real force, F, acting on a body to which we applya virtual displacement. If the virtual displacement at the

location of and in the direction ofFis y , then the force F

does virtual work W= F y .

In summary, virtual displacements are not real, they can bephysically impossible but they must be compatible with the

geometry of the original structure and they must be small

enough so that the original geometry is not significantly altered.



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Truss Displacement

Principle of Virtual Displacements:

We can prove the Principle of Virtual

Work quite simply, as follows.

Consider a particle Punder the

influence of a number of forcesF1,. , Fn which have a resultant

force, FR. Apply a virtual

displacement of yto P, moving it to

P, as shown:

The virtual work done by each of the forces is:



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Truss Displacement

Vir tual Forces (Uni t Load Method)

A virtual force is a force imagined to be applied and is then

moved through the actual deformations of the body, thus

causing virtual work.

So if at a particular location of a structure, we have a deflection,

y, and impose a virtual force at the same location and in the

same direction ofFwe then have the virtual work W = yF .

Virtual forces must form an equilibrium set of their own. For

example, if a virtual force is applied to the end of a spring there

will be virtual stresses in the spring as well as a virtual reaction.



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Truss Displacement

Unit Load Method

1. First, determine all the forces (P0 )

in all the truss member, due to the

external loading. (using the

method of joint or method of

section)

2. Then, find out the stress in the

various members ( = P0/A),

where A are the cross-sectional

area of the members. (+ve for

tension andve for compression)



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Truss Displacement

Unit Load Method (cont.)

3. Remove all the external load on

the truss and apply a unit load

(vertical or horizontal) to act at a

joint whose displacement is

required to be found.

4. Now, find out all the forces (P1

) in

the various member of the truss

due to this unit load.



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Truss Displacement

Unit Load Method (cont.)

5. Now, the displacement (vertical or horizontal) of the joint can

be determine by the equation ;

Where :

P1= forces in the member of the truss due to this unit load.

EA

LPP

AE

LPP

AE

LPP

AE

LPP

10

3

10

2

10

1

10 .............

P0= forces in all the truss member, due to the external loading.

L = lengths of the members of the truss.

A = Cross-sectional area of the member.

E= Youngs Modulus of elasticity of the truss material.


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Truss Displacement

Example 1 : Solution for VERTICAL DISPLACEMENT

1. First, determine all the forces (P0 )in all the truss member, due to the

external loading (4 kN).

2. Remove all the external load on the

truss and apply a unit load

(vertically and assume downward)at joint B, whose displacement is

required to be determine.


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Truss Displacement

Example 1 : Solution for HORIZONTAL DISPLACEMENT

1. First, determine all the forces (P0 )in all the truss member, due to the

external loading (4 kN).

2. Remove all the external load on

the truss and apply a unit load

(horizontally and assume to theleft) at joint B, whose

displacement is required to be

determine.


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Truss Displacement

Example 1 : Solution for HORIZONTAL DISPLACEMENT (cont.)

3. Now, the horizontal displacement at joint B can be determineby the equation ;

(Members of the truss has the same

cross-sectional area and material.

Therefore A and E are constant)LPPAE

AE

LPP

HB

HB

10

10

1

18

110

AE

LPPAE

HB

HB

-ve means that

displacement is to the right

(opposite direction to the

assume unit load)

Ahli P0 P1 H L h/AE

AB 6 -1 3 -18

BC 5 0 5 0CD -3 0 6 0

BD -5 0 3 0

JUMLAH -18+ve for tension

ve for compression


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Truss Displacement

Example 2 : Solution for VERTICAL DISPLACEMENT

1. Determine all the forces (P0 )in all the truss member, due

to the external loading.

2. Remove all the external load

on the truss and apply a unit

load (vertically and assumedownward) at joint B,

whose displacement is

required to be determine.


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Truss Displacement

Example 2 : Solution for VERTICAL DISPLACEMENT (cont.)

3. Now, the vertical displacement at joint B can be determineby the equation ;




AE

LPP

VB

VB

10

10

1

Member P0 P1 V L V

AB 15 0.375 3 16.875

BC 15 0.375 3 16.875

AD -75 -0.625 5 234.375

BD 80 1 4 320.000

CD -25 -0.625 5 78.125

666.251

00317.0

251.666

10

m

AE

AE

LPP

VB

VB

VB

+ve means that

displacement downward

(in the direction of the

assume unit load)

+ve for tension

ve for compression


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Truss Displacement

Example 2 : Solution for HORIZONTAL DISPLACEMENT (cont.)

3. Now, the horizontal displacement at joint B can be determineby the equation ;




AE

LPP

HB

HB

10

10

1

Ahli P0 P1 H L H

AB 15 1 3 45

BC 15 0 3 0

AD -75 0 5 0BD 80 0 4 0

CD -25 0 5 0

45

)(000214.0

45

110

m

EI

LPPAE

HB

HB

HB

+ve means that

displacement is to the right

(in the direction of the

assume unit load)

+ve for tension

ve for compression


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Truss Displacement

Exercise 1:

Determine the vertical and horizontal displacement for the

loaded truss shown;


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Truss Displacement

Exercise 1 : (cont.)


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Truss Displacement



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Truss Displacement



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Truss Displacement

Exercise 1 : (answer)


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Truss Displacement



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Truss Displacement

COMBINE DISPLACEMENT(VERTICAL and HORIZONTAL)

Using the theorem pythagoras and vector principle, the combine

or total displacement (vertical and horizontal displacement) can

be determine.

AE

AEAE

B

B

23.48

1875.44 22

From Example 1: The Total Displacement at joint B ;

18

75.44

AE

AE

H

V

right)theto(downward912.21

75.44

18

tan1

AE

AE

AND


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Truss Displacement

mmB

B

382.0

17.3214.022

From Example 2: The Total Displacement at joint B ;

214.0

317.0

mm

mm

H

V

right)theto(downward862.3

17.3214.0tan 1

AND

TOTAL DISPLACEMENT (VERTICAL and HORIZONTAL)


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Truss Displacement

Exercise 2 :

Determine the TOTAL DISPLACEMENT of the truss from thePrevious EXERCISE 1.


T Di l t


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Truss Displacement

Exercise 3 :

Determine the vertical, horizontal and total displacement for the

loaded trusses shown. (All the member has the same

cross-sectional area A and Modulus of Elasticity E)

T Di l t


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Truss Displacement


T Di l t


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Truss Displacement


T Di l t


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Truss Displacement


T Di l t


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Truss Displacement


T Di l t


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Truss Displacement

2.1 Displacement (External Load)

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