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8/11/2019 Chapter 3- Self Assesment Truss Structure
1/35
Mechatronics Engineering DepartmentFaculty of Engineering
National University of Sciences and Technology
EM 415
SPECIAL TOPICS IN MECHATRONICS
CHAPTER 3 EXAMPLE
Mr. Anas Bin Aqeel
Department of Mechatronics Engineering
National University of Sciences and Technology
Pakistan
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Instructions
This file is based on Chapter 3 Example 2 given in your notes. It is taken from a previous exam paper.
Open the file in Powerpoint and launch slideshow (from menu: Slideshow - from Beginning or click at the bottom right of
the window)
The example is broken down into a series of steps to illustrate how to tackle a question like this. At various points, you will be
given multiple choice questions to work through which will show you how to develop the stiffness matrix for each element,
construct the global stiffness matrix and solve to determine the displacement and stresses of the elements.
Click on the Start button to progress
Start
8/11/2019 Chapter 3- Self Assesment Truss Structure
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8/11/2019 Chapter 3- Self Assesment Truss Structure
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Here the node numbers are shown in circles at the nodes and the element numbers in square boxes in
the middle of the element.
Note we have also added the X and Y axis defining the global co-ordinate system.
Stage 1
1 2
34
Continue
1
2 3
A pin jointed frame has been represented using three 1-D
truss elements, Figure 3.9. Use the Finite Element Method
and the elemental stiffness matrix to calculate the
displacements and elemental stresses. The structure is
made from Aluminium with E = 70000 MPa, with the
following cross sectional areas for the rod elements: A12=
300 mm2
, A23= 150 mm2
, A42= 150 mm2
.
Figure 3.9 X
Y
8/11/2019 Chapter 3- Self Assesment Truss Structure
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Before starting, decide on a set of units, and to work consistently
with these all the way through. We could convert everything to SI
units (kg, m, s, N, Pa). In this case, the dimensions are given in
mm, so it makes more sense to use milimetres for length, but
what units should we use for force and stress?
Click on the button A, B or C or D, click clue to get a hint or jumpstraight to the answer
Stage 3: Decide what units to use
Clue
Go straight to answer
A pin jointed frame has been represented using three 1-D
truss elements, Figure 3.9. Use the Finite Element Method
and the elemental stiffness matrix to calculate the
displacements and elemental stresses. The structure is
made from Aluminium with E = 70000 MPa, with the
following cross sectional areas for the rod elements: A12=
300 mm2
, A23= 150 mm2
, A42= 150 mm2
.
Figure 3.9
A
B
C
D
N, GPa
kN, MPa
N, Pa
N, MPa
Units
Force = ?
Length = mm
Stress/E = ?
8/11/2019 Chapter 3- Self Assesment Truss Structure
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Decide on the base units of mass, length and time and then derive the
units for the others. In this case, we have decided to use mm for length,
try the suggestion above for mass and time
Click on the back button to return to the question
Stage 3: Clue
Back
A pin jointed frame has been represented using three 1-D
truss elements, Figure 3.9. Use the Finite Element Method
and the elemental stiffness matrix to calculate the
displacements and elemental stresses. The structure is
made from Aluminium with E = 70000 MPa, with the
following cross sectional areas for the rod elements: A12=
300 mm2
, A23= 150 mm2
, A42= 150 mm2
.
Figure 3.9
Units
Mass = g
Length = mm
Time = ms
Force =
Stress/E =
8/11/2019 Chapter 3- Self Assesment Truss Structure
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Press continue to go proceed
Answer: CORRECT
Continue
The base units are kg, mm and ms, giving the units shown above
A pin jointed frame has been represented using three 1-D
truss elements, Figure 3.9. Use the Finite Element Method
and the elemental stiffness matrix to calculate the
displacements and elemental stresses. The structure is
made from Aluminium with E = 70000 MPa, with the
following cross sectional areas for the rod elements: A12=
300 mm2
, A23= 150 mm2
, A42= 150 mm2
.
Figure 3.9
Units
Mass [M] = g
Length [L]= mm
Time [T] = ms
N
s
mkg
10s
10m10kg223
3-3
force
MPa10N/m
10m
N
area
force 6223
stress
8/11/2019 Chapter 3- Self Assesment Truss Structure
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Answer: INCORRECT
A pin jointed frame has been represented using three 1-D
truss elements, Figure 3.9. Use the Finite Element Method
and the elemental stiffness matrix to calculate the
displacements and elemental stresses. The structure is
made from Aluminium with E = 70000 MPa, with the
following cross sectional areas for the rod elements: A12=
300 mm2
, A23= 150 mm2
, A42= 150 mm2
.
Figure 3.9
Start with the base units of mass, length and time given above
and work out the corresponding units of force and stress
Go back to question
Units
Mass = g
Length = mm
Time = ms
Force =
Stress/E =
8/11/2019 Chapter 3- Self Assesment Truss Structure
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A pin jointed frame has been represented using three 1-D
truss elements, Figure 3.9. Use the Finite Element Method
and the elemental stiffness matrix to calculate the
displacements and elemental stresses. The structure is
made from Aluminium with E = 70000 MPa, with the
following cross sectional areas for the rod elements: A12=
300 mm
2
, A23= 150 mm
2
, A42= 150 mm
2
.
Figure 3.9
Answer: INCORRECT
Start with the base units of mass, length and time given above
and work out the corresponding units of force and stress.
Go back to question
Mass = g
Length = mm
Time = ms
Force =
Stress/E =
Units
8/11/2019 Chapter 3- Self Assesment Truss Structure
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Press continue to go proceed
Answer
Continue
A pin jointed frame has been represented using three 1-D
truss elements, Figure 3.9. Use the Finite Element Method
and the elemental stiffness matrix to calculate the
displacements and elemental stresses. The structure is
made from Aluminium with E = 70000 MPa, with the
following cross sectional areas for the rod elements: A12=
300 mm
2
, A23= 150 mm
2
, A42= 150 mm
2
.
Figure 3.9
The base units are kg, mm and ms, giving the units shown above
Units
Mass [M] = g
Length [L]= mm
Time [T] = ms
N
s
mkg
10s
10m10kg223
3-3
force
MPa10N/m
10m
N
area
force 6223
stress
8/11/2019 Chapter 3- Self Assesment Truss Structure
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Next we will consider each element in turn, starting with Element 1.
First sketch the element as shown and then calculate the angle.
What is the angle in this case?
Click on the button A, B or C or D, click clue to get a hint or jumpstraight to the answer
Stage 3: Consider each element in turn
Clue
Go straight to answer
1 1 2
A pin jointed frame has been represented using three 1-D
truss elements, Figure 3.9. Use the Finite Element Method
and the elemental stiffness matrix to calculate the
displacements and elemental stresses. The structure is
made from Aluminium with E = 70000 MPa, with the
following cross sectional areas for the rod elements: A12=
300 mm
2
, A23= 150 mm
2
, A42= 150 mm
2
.
Figure 3.9
X
Y
A
B
C
D
0
45
90
180
Element 1
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The angle of the element is defined as the angle measured at the node
with the lowest number ANTICLOCKWISE from the global X axis to the
element.
Click on the back button to return to the question
Stage 3: Clue
Back
A pin jointed frame has been represented using three 1-D
truss elements, Figure 3.9. Use the Finite Element Method
and the elemental stiffness matrix to calculate the
displacements and elemental stresses. The structure is
made from Aluminium with E = 70000 MPa, with the
following cross sectional areas for the rod elements: A12=
300 mm
2
, A23= 150 mm
2
, A42= 150 mm
2
.
Figure 3.9
Node with lower node number
1 1 2 X
Y
Element 1
8/11/2019 Chapter 3- Self Assesment Truss Structure
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Press continue to go proceed
Answer: CORRECT
Continue
The element is aligned with the global X axis, so the angle between the X axis and the element is zero.
A pin jointed frame has been represented using three 1-D
truss elements, Figure 3.9. Use the Finite Element Method
and the elemental stiffness matrix to calculate the
displacements and elemental stresses. The structure is
made from Aluminium with E = 70000 MPa, with the
following cross sectional areas for the rod elements: A12=
300 mm
2
, A23= 150 mm
2
, A42= 150 mm
2
.
Figure 3.9
= 0
1 1 2 X
Y
Element 1
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8/11/2019 Chapter 3- Self Assesment Truss Structure
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A pin jointed frame has been represented using three 1-D
truss elements, Figure 3.9. Use the Finite Element Method
and the elemental stiffness matrix to calculate the
displacements and elemental stresses. The structure is
made from Aluminium with E = 70000 MPa, with the
following cross sectional areas for the rod elements: A12=
300 mm2, A23
= 150 mm2, A42
= 150 mm2.
Figure 3.9
1 1 2 X
Y
Element 1
Node with lower node number
Answer: INCORRECT
Remember the angle of the element is defined as the angle
measured at the node with the lowest number ANTICLOCKWISE
from the global X axis to the element.
Go back to question
8/11/2019 Chapter 3- Self Assesment Truss Structure
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Press continue to go proceed
Answer
Continue
The element is aligned with the global X axis, so the angle between the X axis and the element is zero.
A pin jointed frame has been represented using three 1-D
truss elements, Figure 3.9. Use the Finite Element Method
and the elemental stiffness matrix to calculate the
displacements and elemental stresses. The structure is
made from Aluminium with E = 70000 MPa, with the
following cross sectional areas for the rod elements: A12=
300 mm2, A23
= 150 mm2, A42
= 150 mm2.
Figure 3.9
= 0
1 1 2 X
Y
Element 1
8/11/2019 Chapter 3- Self Assesment Truss Structure
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We know all of the quantities to work out the stiffness matrix for
element 1. In your exam, you will be given a formula sheet with
the standard formula for a truss finite element given above
Next
A pin jointed frame has been represented using three 1-D
truss elements, Figure 3.9. Use the Finite Element Method
and the elemental stiffness matrix to calculate the
displacements and elemental stresses. The structure is
made from Aluminium with E = 70000 MPa, with the
following cross sectional areas for the rod elements: A12=
300 mm2, A23
= 150 mm2, A42
= 150 mm2.
Figure 3.9
Stage 3: Element 1
cosc
sins
scss-cs-
csccs-c-
s-cs-scs
cs-c-csc
L
EA
22
22
22
22
j
j
i
i
V
U
V
U
UK
Truss Finite Element: For a truss finite element
aligned at angle to global X axis with a linear shape
function. Relative to the global co-ordinate system:
8/11/2019 Chapter 3- Self Assesment Truss Structure
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Clue
Multi-choice
Go straight to answer
A pin jointed frame has been represented using three 1-D
truss elements, Figure 3.9. Use the Finite Element Method
and the elemental stiffness matrix to calculate the
displacements and elemental stresses. The structure is
made from Aluminium with E = 70000 MPa, with the
following cross sectional areas for the rod elements: A12=
300 mm2, A23
= 150 mm2, A42
= 150 mm2.
Figure 3.9
Calculate the stiffness matrix for element 1
Stage 3: Element 1
cosc
sins
scss-cs-
csccs-c-
s-cs-scs
cs-c-csc
L
EA
22
22
22
22
j
j
i
i
V
U
V
U
UK
Truss Finite Element: For a truss finite element
aligned at angle to global X axis with a linear shape
function. Relative to the global co-ordinate system:
= 0
1 1 2 X
YElement 1
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8/11/2019 Chapter 3- Self Assesment Truss Structure
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Stage 3: Element 1
A CB
The stiffness matrix is:
0000
042000042000-
0000
042000-042000
1 =][k
0000
042042-
0000
024-042
1 =][k
42000042000-0
0000
420000420000
0000
1 =][k
A pin jointed frame has been represented using three 1-D
truss elements, Figure 3.9. Use the Finite Element Method
and the elemental stiffness matrix to calculate the
displacements and elemental stresses. The structure is
made from Aluminium with E = 70000 MPa, with the
following cross sectional areas for the rod elements: A12=
300 mm2, A23
= 150 mm2, A42
= 150 mm2.
Figure 3.9
cosc
sins
scss-cs-
csccs-c-
s-cs-scs
cs-c-csc
L
EA
22
22
22
22
j
j
i
i
V
U
V
U
UK
Truss Finite Element: For a truss finite element
aligned at angle to global X axis with a linear shape
function. Relative to the global co-ordinate system:
= 0
1 1 2 X
YElement 1
8/11/2019 Chapter 3- Self Assesment Truss Structure
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8/11/2019 Chapter 3- Self Assesment Truss Structure
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Answer: INCORRECT
Back
A pin jointed frame has been represented using three 1-D
truss elements, Figure 3.9. Use the Finite Element Method
and the elemental stiffness matrix to calculate the
displacements and elemental stresses. The structure is
made from Aluminium with E = 70000 MPa, with the
following cross sectional areas for the rod elements: A12=
300 mm2, A23
= 150 mm2, A42
= 150 mm2.
Figure 3.9
cosc
sins
scss-cs-
csccs-c-
s-cs-scs
cs-c-csc
L
EA
22
22
22
22
j
j
i
i
V
U
V
U
UK
Truss Finite Element: For a truss finite element
aligned at angle to global X axis with a linear shape
function. Relative to the global co-ordinate system:
= 0
1 1 2 X
YElement 1
Be careful with the sine and cosine terms, remember alpha = o, so s = 0 and c = 1
8/11/2019 Chapter 3- Self Assesment Truss Structure
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Answer: INCORRECT
Back
A pin jointed frame has been represented using three 1-D
truss elements, Figure 3.9. Use the Finite Element Method
and the elemental stiffness matrix to calculate the
displacements and elemental stresses. The structure is
made from Aluminium with E = 70000 MPa, with the
following cross sectional areas for the rod elements: A12=
300 mm2, A23
= 150 mm2, A42
= 150 mm2.
Figure 3.9
cosc
sins
scss-cs-
csccs-c-
s-cs-scs
cs-c-csc
L
EA
22
22
22
22
j
j
i
i
V
U
V
U
UK
Truss Finite Element: For a truss finite element
aligned at angle to global X axis with a linear shape
function. Relative to the global co-ordinate system:
= 0
1 1 2 X
YElement 1
Be careful with the units, remember we are working in mm and MPa, so you
can use the quantities exactly as they appear in the question.
8/11/2019 Chapter 3- Self Assesment Truss Structure
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Press continue to go to next question
Answer:
Continue
A pin jointed frame has been represented using three 1-D
truss elements, Figure 3.9. Use the Finite Element Method
and the elemental stiffness matrix to calculate the
displacements and elemental stresses. The structure is
made from Aluminium with E = 70000 MPa, with the
following cross sectional areas for the rod elements: A12=
300 mm2, A23
= 150 mm2, A42
= 150 mm2.
Figure 3.9
= 0
1 1 2 X
YElement 1
2
2
1
1
1
0000
042000042000-
0000
042000-042000
V
U
V
U
Uk =][
From the question, we have E = 70000 MPa, A12= 300 mm2, and L12= 500mm.
We know the angle alpha = 0, so sin() = 0 and cos() = 1
We can plug all these numbers into the formula for [K] to give the result shown
8/11/2019 Chapter 3- Self Assesment Truss Structure
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We can now repeat the process for element 2.
Next
A pin jointed frame has been represented using three 1-D
truss elements, Figure 3.9. Use the Finite Element Method
and the elemental stiffness matrix to calculate the
displacements and elemental stresses. The structure is
made from Aluminium with E = 70000 MPa, with the
following cross sectional areas for the rod elements: A12=
300 mm2, A23
= 150 mm2, A42
= 150 mm2.
Figure 3.9
Stage 3: Element 2
cosc
sins
scss-cs-
csccs-c-
s-cs-scs
cs-c-csc
L
EA
22
22
22
22
j
j
i
i
V
U
V
U
UK
Truss Finite Element: For a truss finite element
aligned at angle to global X axis with a linear shape
function. Relative to the global co-ordinate system:
2
3
2
X
YElement 2
8/11/2019 Chapter 3- Self Assesment Truss Structure
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Clue
Multi-choice
Go straight to answer
A pin jointed frame has been represented using three 1-D
truss elements, Figure 3.9. Use the Finite Element Method
and the elemental stiffness matrix to calculate the
displacements and elemental stresses. The structure is
made from Aluminium with E = 70000 MPa, with the
following cross sectional areas for the rod elements: A12=
300 mm2, A23
= 150 mm2, A42
= 150 mm2.
Figure 3.9
Calculate the stiffness matrix for element 2
Stage 3: Element 2
cosc
sins
scss-cs-
csccs-c-
s-cs-scs
cs-c-csc
L
EA
22
22
22
22
j
j
i
i
V
U
V
U
UK
Truss Finite Element: For a truss finite element
aligned at angle to global X axis with a linear shape
function. Relative to the global co-ordinate system:
X
Y
2
3
2Element 2
8/11/2019 Chapter 3- Self Assesment Truss Structure
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Clue
Back
A pin jointed frame has been represented using three 1-D
truss elements, Figure 3.9. Use the Finite Element Method
and the elemental stiffness matrix to calculate the
displacements and elemental stresses. The structure is
made from Aluminium with E = 70000 MPa, with the
following cross sectional areas for the rod elements: A12=
300 mm2, A23
= 150 mm2, A42
= 150 mm2.
Figure 3.9
Start off by drawing the element and working out the angle alpha.
From the question, we have E = 70000 MPa, A23= 150 mm2, and L23= 566mm
(calculated from the Figure).
We can plug all these numbers into the formula shown above for [K]
cosc
sins
scss-cs-
csccs-c-
s-cs-scs
cs-c-csc
L
EA
22
22
22
22
j
j
i
i
V
U
V
U
UK
Truss Finite Element: For a truss finite element
aligned at angle to global X axis with a linear shape
function. Relative to the global co-ordinate system:
X
Y
2
3
2Element 2
= ?
8/11/2019 Chapter 3- Self Assesment Truss Structure
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Stage 3: Element 2
A CB
The stiffness matrix is:
A pin jointed frame has been represented using three 1-D
truss elements, Figure 3.9. Use the Finite Element Method
and the elemental stiffness matrix to calculate the
displacements and elemental stresses. The structure is
made from Aluminium with E = 70000 MPa, with the
following cross sectional areas for the rod elements: A12=
300 mm2, A23
= 150 mm2, A42
= 150 mm2.
Figure 3.9
cosc
sins
scss-cs-
csccs-c-
s-cs-scs
cs-c-csc
L
EA
22
22
22
22
j
j
i
i
V
U
V
U
UK
Truss Finite Element: For a truss finite element
aligned at angle to global X axis with a linear shape
function. Relative to the global co-ordinate system:
X
Y
2
3
2Element 2
13125131251312513125
13125-131251312513125
13125-131251312513125
1312513125-13125-13125
9281928192819281
9281928192819281
9281-928192819281
92819281-92819281
9281928192819281
9281-928192819281
9281-928192819281
92819281-9281-9281
8/11/2019 Chapter 3- Self Assesment Truss Structure
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Press continue to go to next question
Answer: CORRECT
Continue
A pin jointed frame has been represented using three 1-D
truss elements, Figure 3.9. Use the Finite Element Method
and the elemental stiffness matrix to calculate the
displacements and elemental stresses. The structure is
made from Aluminium with E = 70000 MPa, with the
following cross sectional areas for the rod elements: A12=
300 mm2, A23
= 150 mm2, A42
= 150 mm2.
Figure 3.9
X
Y
2
3
2
Element 2
= 135
3
3
2
2
2
9281928192819281
9281-928192819281
9281-928192819281
92819281-9281-9281
V
U
V
U
Uk =][
8/11/2019 Chapter 3- Self Assesment Truss Structure
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Answer: INCORRECT
Back
A pin jointed frame has been represented using three 1-D
truss elements, Figure 3.9. Use the Finite Element Method
and the elemental stiffness matrix to calculate the
displacements and elemental stresses. The structure is
made from Aluminium with E = 70000 MPa, with the
following cross sectional areas for the rod elements: A12=
300 mm2, A23
= 150 mm2, A42
= 150 mm2.
Figure 3.9
Be careful with the angle, remember this is measured at the node with the lower
number, anticlockwise from the global X axis
cosc
sins
scss-cs-
csccs-c-
s-cs-scs
cs-c-csc
L
EA
22
22
22
22
j
j
i
i
V
U
V
U
UK
Truss Finite Element: For a truss finite element
aligned at angle to global X axis with a linear shape
function. Relative to the global co-ordinate system:
X
Y
2
3
2Element 2
= ?
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Answer: INCORRECT
Back
A pin jointed frame has been represented using three 1-D
truss elements, Figure 3.9. Use the Finite Element Method
and the elemental stiffness matrix to calculate the
displacements and elemental stresses. The structure is
made from Aluminium with E = 70000 MPa, with the
following cross sectional areas for the rod elements: A12=
300 mm2, A23= 150 mm2, A42= 150 mm
2.
Figure 3.9
Be careful with the term L, this is the length of the element, so in this case,
you will need to work this out using trigonometry from the information given
in the figure.
cosc
sins
scss-cs-
csccs-c-
s-cs-scs
cs-c-csc
L
EA
22
22
22
22
j
j
i
i
V
U
V
U
UK
Truss Finite Element: For a truss finite element
aligned at angle to global X axis with a linear shape
function. Relative to the global co-ordinate system:
X
Y
2
3
2Element 2
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Press continue to go to next question
Answer:
Continue
A pin jointed frame has been represented using three 1-D
truss elements, Figure 3.9. Use the Finite Element Method
and the elemental stiffness matrix to calculate the
displacements and elemental stresses. The structure is
made from Aluminium with E = 70000 MPa, with the
following cross sectional areas for the rod elements: A12=
300 mm2, A23= 150 mm2, A42= 150 mm
2.
Figure 3.9
From the question, we have E = 70000 MPa, A12= 300 mm2, and L12= 500mm.
We know the angle alpha = 0, so sin() = 0 and cos() = 1
We can plug all these numbers into the formula for [K] to give the result shown
X
Y
2
3
2
Element 2
= 135
3
3
2
2
2
9281928192819281
9281-928192819281
9281-928192819281
92819281-9281-9281
V
U
V
U
Uk =][
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Press continue to show the stiffness matrix for element 3
Stage 3: Element 3
Continue
A pin jointed frame has been represented using three 1-D
truss elements, Figure 3.9. Use the Finite Element Method
and the elemental stiffness matrix to calculate the
displacements and elemental stresses. The structure is
made from Aluminium with E = 70000 MPa, with the
following cross sectional areas for the rod elements: A12=
300 mm2, A23= 150 mm2, A42= 150 mm
2.
Figure 3.9
Finally, we will repeat the procedure for element 3. You should be able to work
this out using the formula given. Once you have written it down, check it against
the answer given next...
cosc
sins
scss-cs-
csccs-c-
s-cs-scs
cs-c-csc
L
EA
22
22
22
22
j
j
i
i
V
U
V
U
UK
Truss Finite Element: For a truss finite element
aligned at angle to global X axis with a linear shape
function. Relative to the global co-ordinate system:
2
4
3
X
Y
Element 3
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Press continue
Answer:
Continue
A pin jointed frame has been represented using three 1-D
truss elements, Figure 3.9. Use the Finite Element Method
and the elemental stiffness matrix to calculate the
displacements and elemental stresses. The structure is
made from Aluminium with E = 70000 MPa, with the
following cross sectional areas for the rod elements: A12=
300 mm2, A23= 150 mm2, A42= 150 mm
2.
Figure 3.9
Check you have correctly calculated the matrix for element 3.
Note that next to each element, we have written the corresponding
displacement vector {u}this will help when we are constructing the global
stiffness matrix
Element 3
2
4
3
X
Y
4
4
2
2
3
928192819281-9281-
928192819281-9281-
9281-9281-92819281
9281-9281-92819281
V
U
V
U
Uk =][
= 45
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You have finished the first stage of this question.
We will cover the rest in the lectures in Week
3.