Post on 09-Jul-2020
transcript
Virtual Work Truss ExampleTemperature and Fabrication Errors
StevenVukazichSanJoseStateUniversity
A
BC
3 m 3 m
4 m
D E
Example Using the Principle of Virtual Work
Consider the idealized truss structure from the previous example. Truss members AD and DE increase in temperature 40℃. Member CE decreases in temperature 30℃. In addition, member BE is fabricated 0.5 cm too short.
Find the vertical displacement of point B using the Principle of Virtual Work
For all truss members use:A = 25 cm2
E = 210 GPa𝛼& = 12×10+,/℃
𝛿/0
Virtual System to Measure 𝛿/0
1. Remove all loads (if any) from the structure;
2. Apply a unit, dimensionless virtual load in-line with the real displacement, 𝛿/0, that we want to find;
3. Perform a truss analysis to find all truss member virtual axial forces, FQi
1
A B C
3 m 3 m
4 m
D E
A
BC
3 m 3 m
4 m
D E
Ax
Ay Dy
1𝑀3
�
�
= 0+
Find Support Reactions
Dy = -0.5
1
A
BC
3 m 3 m
4 m
D E
Ax
Ay Dy
Find Support Reactions
1
1𝐹7
�
�
= 0+
Ay = -0.51𝐹8
�
�
= 0+
A
BC
3 m 3 m
4 m
D E
0.5
Virtual System Support Reactions
0.5 1
A
0.5
FBD of Joint A
FQAB
3
45
𝜃
FQAD
A
3
45
𝜃
35𝐹;3<
1𝐹7
�
�
= 0+
A
0.5
FQAB
45𝐹;3<
FQAD = 0.625
FBD of Joint A
A
3
45
𝜃
35𝐹;3<
A
0.5
FQAB
45𝐹;3<
FQAB = – 0.375
FBD of Joint A
1𝐹8
�
�
= 0+
A
BC
3 m 3 m
4 m
D E
Virtual System Results on a FBD of the Entire Truss
Tension is Positive(0
.5)
(–0.
5)
(0)(– 0.375)
(0.375)
B
0.50.5 1
Virtual truss member forces, FQi
1 ⋅ 𝛿/0 =1𝐹;&𝛼&∆𝑇&𝐿& + 1 𝐹;&∆𝐿&BCDE
FGHIJ
&KL
FM
&KL
Step 2 – Use the Principle of VirtualWork to Find 𝛿/0
From Step 1–virtual analysis
A
BC
3 m 3 m
4 m
D E
For all truss members use:A = 25 cm2
E = 210 GPa𝛼& = 12×10+,/℃
𝛿/0
Use a Table to Organize VirtualWork Calculations
Member 𝜶×𝟏𝟎𝟔 /℃ ∆𝑻 ℃ ∆fabr (cm) L (m) FQ UQ (cm)
AD 12 40 0 5 0.625 0.15AB 12 0 0 3 – 0.375 0BD 12 0 0 4 – 0.5 0DE 12 40 0 3 0.375 0.054BE 12 0 –0.5 5 – 0.625 0.3125BC 12 0 0 3 0 0CE 12 –30 0 4 0.5 – 0.072Total 0.4445
𝐹;3< ∝3< ∆𝑇3<𝐿3< = 0.625 12×10+,/℃ 40℃ 5m100cmm
= 0.15cm
Sample Calculations
𝐹;/X∆𝐿BCDE3< = −0.625 −0.5 = 0.3125cm
R𝐞𝐬𝐮𝐥𝐭𝐬𝐟𝐨𝐫𝜹𝑩𝒗
Positive sign indicates that deflection is in the same direction of the virtual force
𝛿/0 = 0.4445cmupward
AB
C
3 m 3 m
4 m
D E
0.4445cm