Tr u ss Di D i spl plac ace eme men n t
DISPLACEMENT of STATICALLY DETERMINATE TRUSSES (Part 1)
Ahmad Ahma d Shufni Shufni bin Othm Othman an Mei 2014
Tr u ss Di D i spl plac ace eme men n t
Learning Objectives Upon completion of this course, students should be able to;
Determine value and types of displacement that occur in frame / trusses using the Method of Unit Load ; •
Horizontal Horizontal displacement
•
Vertical displacement
•
Combination of horizontal horizontal and vertical displacement
Influences of Temperature Change and Fabrication Errors (inaccurate (inaccurate length of members)
Tr u ss Di D i spl plac ace eme men n t
Learning Objectives Upon completion of this course, students should be able to;
Determine value and types of displacement that occur in frame / trusses using the Method of Unit Load ; •
Horizontal Horizontal displacement
•
Vertical displacement
•
Combination of horizontal horizontal and vertical displacement
Influences of Temperature Change and Fabrication Errors (inaccurate (inaccurate length of members)
Tr u ss Di D i spl plac ace eme men n t
Vertical
Diagonal
Top Chord Web Members
Depth or Rise
(verticals & diagonals)
Panel
l
Joint, Panel point or Node
Bottom Chord
Span Flat Truss Truss or Parallel Paral lel Chord Chor d T Truss russ
l
Tr u ss Di D i spl plac ace eme men n t
Introduction •
•
•
•
•
•
•
When a frame structure or truss carries some external loads, all of its members 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 displace ment 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, depending upon the direction of the load system and rigidity of the truss members. There are two very important type of displacement ; i. Vertical displacement ii Hori ontal di displacement
Tr uss Displacement
Vertical and Horizontal Displacement It is a total displacement, suffer by a joint from its original position, in the vertical or horizontal direction due to the action of external forces. 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.
Tr uss Displacement
Method of Determining The Displacement There are many methods used for determining the displacement of the joints in a frame or truss; 1. 2. 3. 4.
Virtual Work Method Unit Load Method Strain Energy Method Castigliano Theorem Method
Tr uss Displacement
Method of Determining The Displacement The Principle of Virtual Work Based upon the Principle of Minimum Total Potential Energy , we can see that any small variation about equilibrium must do no work. Thus, the Principle of Virtual Work states that: A body is in equil ibri um if , and onl y if , the vir tual wor k of al l f orces acting on the body is zer o.
In this context, the word ‘virtual’ means ‘having the effect of, but not the actual form of, what is specified’. Thus we can imagine ways in which to impose virtual work, without worrying about how it might be achieved in the physical world.
Tr uss Displacement
Method of Determining The Displacement Virtual Work There are two ways to define virtual work, as follows. 1. Principle of Virtual Displacements: Vir tual work is the work done by the actual f orces acti ng on the body moving thr ough a vir tual displacement.
This means we solve an equilibrium problem through geometry. 2. Principle of Virtual Forces: Vir tual work is the work done by a vir tual f orce acti ng on the body moving thr ough the actual displacements.
This means we solve a geometry problem through equilibrium.
Tr uss Displacement
Method of Determining The Displacement Vir tual D isplacements
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 apply a virtual displacement. If the virtual displacement at the location of and in the direction of F is δy , then the force F does virtual work δW = F δ y . In summary, virtual displacements are not real, they can be physically 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.
Tr uss Displacement
Method of Determining The Displacement Principle of Vi r tual Displacements:
We can prove the Principle of Virtual Work quite simply, as follows. Consider a particle P under the influence of a number of forces F 1 F n which have a resultant force, F R . Apply a virtual displacement of δy to P , moving it to P’ , as shown: ,…. ,
The virtual work done by each of the forces is:
Tr uss Displacement
Method of Determining The Displacement Principle of Vi r tual Displacements:
Where δy is the virtual displacement along the line of action of F 1 and so on. Now if the particle P is in equilibrium, then the forces F 1 F n have no resultant. That is, there is no net force. Hence we have: ,…. ,
Proving that when a particle is in equilibrium the virtual work of all the forces acting on it sum to zero. Conversely, a particle is only in equilibrium if the virtual work done during a virtual displacement is zero.
Tr uss Displacement
Method of Determining The Displacement Vir tual F orces (Uni t L oad M ethod)
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 of δF we then have the virtual work δW = y δF . 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.
Tr uss Displacement
Method of Determining The 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 (σ = P0/A), where A are the cross-sectional area of the members. (+ve for tension and –ve for compression)
Tr uss Displacement
Method of Determining The 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.
Tr uss Displacement
Unit Load Method (cont.) 5. Now, the displacement (vertical or horizontal) of the joint can be determine by the equation ; P 0 P 1 L P 0 P 1 L P 0 P 1 L ............. AE 1 AE 2 AE 3
Where :
P 0 P 1 L A E
P 0 = forces in all the truss member, due to the external loading. P 1 = forces in the member of the truss due to this unit load. L = lengths of the members of the truss. A = Cross-sectional area of the member. E = Young’s Modulus of elasticity of the truss material.
Tr uss Displacement Example 1
Determine the vertical and horizontal displacement of joint B for The loaded truss below.
Tr uss 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.
Tr uss Displacement Example 1 : Solution for VERTICAL DISPLACEMENT (cont.)
3. Now, the vertical displacement at joint B can be determine by the equation ; (Members of the truss has the same P 0 P 1 L cross-sectional area and material. AE 1 Therefore A and E are constant) P P L 01 VB
VB
Member
AE
P0
P1
L (m)
AB
+6
+0.75
3
+13.5
BC
+5
0
5
0
CD
-3
0
6
0
BD
-5
-1.25
5
+31.25
+ve for tension –ve for compression
VB
+
44.75
1
P P L AE
VB
VB
0
44.75 AE
1
+ve means that displacement downward (in the direction of the assume unit load)
Tr uss 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 the left) at joint B, whose displacement is required to be determine.
Tr uss Displacement Example 1 : Solution for HORIZONTAL DISPLACEMENT (cont.)
3. Now, the horizontal displacement at joint B can be determine by the equation ; (Members of the truss has the same P 0 P 1 L cross-sectional area and material. AE 1 Therefore A and E are constant) P P L 01 HB
HB
Ahli
P0
AE
P1 H
L
δh/AE
AB
6
-1
3
-18
BC
5
0
5
0
CD
-3
0
6
0
BD
-5
0
3
0
JUMLAH
-18
+ve for tension –ve for compression
1
P P L AE
HB
HB
0
18 AE
1
-ve means that displacement is to the right (opposite direction to the assume unit load)
Tr uss Displacement Example 2
Determine the vertical and horizontal displacement of joint B for the loaded truss below. Each truss member has a cross-sectional area of 1000 mm2 and made from the same material, with the modulus of elasticity 210 Gpa.
Tr uss 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 assume downward) at joint B, whose displacement is required to be determine.
Tr uss Displacement Example 2 : Solution for VERTICAL DISPLACEMENT (cont.)
3. Now, the vertical displacement at joint B can be determine by the equation ; (Members of the truss has the same P 0 P 1 L cross-sectional area and material. AE 1 Therefore A and E are constant) P P L 01 VB
VB
Member P 0 AB 15 BC 15 AD -75 BD 80 CD -25
AE
P1 V 0.375 0.375 -0.625 1 -0.625
+ve f or tension –ve for compression
L
δV
3 3 5 4 5
16.875 16.875 234.375 320.000 78.125 666.251
P 0 P 1 L
VB
VB
VB
0.00317m
AE
666.251 AE
+ve means that displacement downward (in the direction of the assume unit load)
Tr uss Displacement Example 2 : Solution for HORIZONTAL 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 (horizontally and assume to the right) at joint B, whose displacement is required to be determine.
Tr uss Displacement Example 2 : Solution for HORIZONTAL DISPLACEMENT (cont.)
3. Now, the horizontal displacement at joint B can be determine by the equation ; (Members of the truss has the same P 0 P 1 L cross-sectional area and material. AE 1 Therefore A and E are constant) P P L 01 HB
HB
Ahli AB BC AD BD CD
P0 15 15 -75 80 -25
AE
P1 H 1 0 0 0 0
+ve for tension –ve for compression
L
δH
3 3 5 4 5
45 0 0 0 0 45
1
P P L AE
HB
HB
HB
0.000214 m ()
0
1
45 EI
+ve means that displacement is to the right (in the direction of the assume unit load)
Tr uss Displacement Exercise 1:
Determine the vertical and horizontal displacement for the loaded truss shown;
Tr uss Displacement Exercise 1 : (cont.)
Tr uss Displacement Exercise 1 : (cont.)
Tr uss Displacement Exercise 1 : (cont.)
Tr uss Displacement Exercise 1 : (answer)
Tr uss Displacement Exercise 1 : (answer)
Tr uss 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. From Example 1: The Total Displacement at joint B ; 2
B
44.75 18 AE AE
B
AND V H
44.75 AE
18 AE
2
48.23 AE
18 1 AE tan 44 . 75 AE 21.912 (downward to the right)
Tr uss Displacement TOTAL DISPLACEMENT (VERTICAL and HORIZONTAL)
From Example 2: The Total Displacement at joint B ;
B
0.2142 3.17 2
B
0.382mm
AND V H
0.317mm 0.214mm
0.214 3.17 3.862 (downward to the right)
tan1
Tr uss Displacement Exercise 2 :
Determine the TOTAL DISPLACEMENT of the truss from the Previous EXERCISE 1.
Exercise 2 : (answer)
Tr uss 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)
Tr uss Displacement Exercise 3 : (cont.)
Tr uss Displacement Exercise 3 : (cont.)
Tr uss Displacement Exercise 3 : (cont.)
Tr uss Displacement Exercise 3 : (cont.)
