2.1 displacement (external load)
TRANSCRIPT
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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.
Method of Determining The Displacement
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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:
Method of Determining The Displacement
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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.
Method of Determining The Displacement
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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)
Method of Determining The Displacement
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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.
Method of Determining The Displacement
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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 ;
(Members of the truss has the same
cross-sectional area and material.
Therefore A and E are constant)LPPAE
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 ;
(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
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
Exercise 1 : (cont.)
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Truss Displacement
Exercise 1 : (cont.)
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Truss Displacement
Exercise 1 : (answer)
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Truss Displacement
Exercise 1 : (answer)
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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.
Exercise 2 : (answer)
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
Exercise 3 : (cont.)
T Di l t
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Truss Displacement
Exercise 3 : (cont.)
T Di l t
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Truss Displacement
Exercise 3 : (cont.)
T Di l t
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Truss Displacement
Exercise 3 : (cont.)
T Di l t
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Truss Displacement