Post on 18-Mar-2018
Problem 1 (28 points): The member shown below is subject to a distributed load. It is known that the member has a uniform cross section area π΄π΄, second area moment πΌπΌ, and a shear shape factor πππ π . The member is composed of a material whose Youngβs modulus is πΈπΈ, and shear modulus πΊπΊ.
Using Castiglianoβs theorem, determine the reaction forces in the beam. Please include the shear, normal and bending shear strain energies in your solution. Leave your answer in terms of π€π€, πΏπΏ,π»π»,πΈπΈ,πΊπΊ,π΄π΄, πΌπΌ, and πππ π .
Solution: FBD:
Ξ£πΉπΉππ = 0 = ππ + πΉπΉπ΄π΄ β π€π€π»π» (1)
Ξ£πΉπΉππ = 0 = ππ = 0 (2)
Ξ£ππππ = 0 β ππ + πΉπΉπ΄π΄π»π» β π€π€π»π» Γπ»π»2
= 0 (3)
There are two equations (1), and (3), and three unknowns ππ,πΉπΉπ΄π΄, and ππ. The system is indeterminate of order one. It is best to start energy equations where there are least number of unknowns, namely πΉπΉπ΄π΄.
0 β€ π¦π¦ β€ π»π»
Ξ£πΉπΉππ = 0 β ππ
Ξ£πΉπΉππ = 0 = ππ β πΉπΉπ΄π΄ + π€π€π¦π¦ β ππ
Ξ£ππππ = 0
β ππ = βπΉπΉπ΄π΄π¦π¦ +π€π€2π¦π¦2
πππππππΉπΉπ΄π΄
= 0; πππππππΉπΉπ΄π΄
= 1; πππππππΉπΉπ΄π΄
= βπ¦π¦
0 β€ π₯π₯ β€ πΏπΏ
Ξ£πΉπΉππ = 0 = ππ + πΉπΉπ΄π΄ β π€π€π»π» β ππ = π€π€π»π» β πΉπΉπ΄π΄ (7)
Ξ£πΉπΉππ = 0 = ππ (8)
Ξ£ππππ = 0 = ππ + πΉπΉπ΄π΄π»π» βπ€π€π»π» Γπ»π»2
(9)
β ππ =π€π€π»π»2
2β πΉπΉπ΄π΄π»π»
πππππππΉπΉπ΄π΄
= β1; πππππππΉπΉπ΄π΄
= 0; πππππππΉπΉπ΄π΄
= βπ»π»
From Castiglianoβs theorem,
πΏπΏπΉπΉπ΄π΄ = οΏ½πππΈπΈπ΄π΄
πππππππΉπΉπ΄π΄
π»π»
0πππ¦π¦ + οΏ½
πππΈπΈπΌπΌ
πππππππΉπΉπ΄π΄
π»π»
0πππ¦π¦ + οΏ½
πππ π πππΊπΊπ΄π΄
πππππππΉπΉπ΄π΄
π»π»
0πππ¦π¦ + οΏ½
πππΈπΈπ΄π΄
πππππππΉπΉπ΄π΄
πΏπΏ
0πππ₯π₯ + οΏ½
πππΈπΈπΌπΌ
πππππππΉπΉπ΄π΄
πΏπΏ
0πππ₯π₯
+ οΏ½πππ π πππΊπΊπ΄π΄
πππππππΉπΉπ΄π΄
πΏπΏ
0πππ₯π₯ = 0 (10)
Non zero terms:
οΏ½πππΈπΈπΌπΌ
πππππππΉπΉπ΄π΄
π»π»
0πππ¦π¦ =
1πΈπΈπΌπΌοΏ½ οΏ½βπΉπΉπ΄π΄π¦π¦ +
π€π€2π¦π¦2οΏ½ (βπ¦π¦)
π»π»
0πππ¦π¦ =
πΉπΉπ΄π΄π»π»3
3πΈπΈπΌπΌβπ€π€π»π»4
8πΈπΈπΌπΌ#(11)
οΏ½πππ π πππΊπΊπ΄π΄
πππππππΉπΉπ΄π΄
π»π»
0πππ¦π¦ =
πππ π πΊπΊπ΄π΄
οΏ½ {πΉπΉπ΄π΄ β π€π€π¦π¦}(1)π»π»
0πππ¦π¦ =
πππ π πΉπΉπ΄π΄π»π»πΊπΊπ΄π΄
βπππ π π€π€π»π»2
2πΊπΊπ΄π΄#(12)
οΏ½πππΈπΈπ΄π΄
πππππππΉπΉπ΄π΄
πΏπΏ
0πππ₯π₯ =
1πΈπΈπ΄π΄
οΏ½ (π€π€π»π» β πΉπΉπ΄π΄)(β1)πΏπΏ
0πππ₯π₯ =
πΉπΉπ΄π΄πΏπΏπΈπΈπ΄π΄
βπ€π€π»π»πΏπΏπΈπΈπ΄π΄
#(13)
οΏ½πππΈπΈπΌπΌ
πππππππΉπΉπ΄π΄
πΏπΏ
0πππ₯π₯ =
1πΈπΈπΌπΌοΏ½ οΏ½
π€π€π»π»2
2β πΉπΉπ΄π΄π»π»οΏ½ (βπ»π»)
πΏπΏ
0πππ₯π₯ =
πΉπΉπ΄π΄π»π»2πΏπΏπΈπΈπΌπΌ
βπ€π€π»π»3πΏπΏ
2πΈπΈπΌπΌ#(14)
Adding (11) -(14), and plugging into (10),
πΉπΉπ΄π΄π»π»3
3πΈπΈπΌπΌβπ€π€π»π»4
8πΈπΈπΌπΌ+πππ π πΉπΉπ΄π΄π»π»πΊπΊπ΄π΄
βπππ π π€π€π»π»2
2πΊπΊπ΄π΄+πΉπΉπ΄π΄πΏπΏπΈπΈπ΄π΄
βπ€π€π»π»πΏπΏπΈπΈπ΄π΄
+πΉπΉπ΄π΄π»π»2πΏπΏπΈπΈπΌπΌ
βπ€π€π»π»3πΏπΏ
2πΈπΈπΌπΌ= 0
β πΉπΉπ΄π΄ οΏ½π»π»3
3πΈπΈπΌπΌ+πππ π π»π»πΊπΊπ΄π΄
+πΏπΏπΈπΈπ΄π΄
+π»π»2πΏπΏπΈπΈπΌπΌ
οΏ½ =π€π€π»π»4
8πΈπΈπΌπΌ+πππ π π€π€π»π»2
2πΊπΊπ΄π΄+π€π€π»π»πΏπΏπΈπΈπ΄π΄
+π€π€π»π»3πΏπΏ
2πΈπΈπΌπΌ
β πΉπΉπ΄π΄ =οΏ½π€π€π»π»
4
8πΈπΈπΈπΈ+ πππ π π€π€π»π»2
2πΊπΊπ΄π΄+ π€π€π»π»πΏπΏ
πΈπΈπ΄π΄+ π€π€π»π»3πΏπΏ
2πΈπΈπΈπΈοΏ½
οΏ½π»π»3
3πΈπΈπΈπΈ+ πππ π π»π»
πΊπΊπ΄π΄+ πΏπΏ
πΈπΈπ΄π΄+ π»π»2πΏπΏ
πΈπΈπΈπΈοΏ½
From (1),
ππ = βπΉπΉπ΄π΄ + π€π€π»π» = π€π€π»π» βοΏ½π€π€π»π»
4
8πΈπΈπΈπΈ+ πππ π π€π€π»π»2
2πΊπΊπ΄π΄+ π€π€π»π»πΏπΏ
πΈπΈπ΄π΄+ π€π€π»π»3πΏπΏ
2πΈπΈπΈπΈοΏ½
οΏ½π»π»3
3πΈπΈπΈπΈ+ πππ π π»π»
πΊπΊπ΄π΄+ πΏπΏ
πΈπΈπ΄π΄+ π»π»2πΏπΏ
πΈπΈπΈπΈοΏ½
From (3),
ππ = βπΉπΉπ΄π΄π»π» +π€π€π»π»2
2=π€π€π»π»2
2βπ»π» οΏ½π€π€π»π»
4
8πΈπΈπΈπΈ+ πππ π π€π€π»π»2
2πΊπΊπ΄π΄+ π€π€π»π»πΏπΏ
πΈπΈπ΄π΄+ π€π€π»π»3πΏπΏ
2πΈπΈπΈπΈοΏ½
οΏ½π»π»3
3πΈπΈπΈπΈ+ πππ π π»π»
πΊπΊπ΄π΄+ πΏπΏ
πΈπΈπ΄π΄+ π»π»2πΏπΏ
πΈπΈπΈπΈοΏ½
Problem 2 (25 points) SOLUTION
The closed, thin-walled tank shown above has an inner radius of r and wall thickness t and contains a gas of pressure of p. The state of stress at point βaβ on the tank is represented by stress components shown above for an unknown stress element rotation angle of ΞΈ . Note that
Ο β²x β²y is in the direction shown; however, its magnitude is also
unknown. For this problem, you are asked for the following (in no particular order):
β’ Determine the principal components of stress, Ο1 and Ο 2 .
β’ Determine the magnitude of Ο β²x β²y .
β’ Draw the Mohrβs circle for this state of stress. Show the location of the x 'β axis in your Mohrβs circle. From this, determine the rotation angle ΞΈ .
β’ Determine the absolute maximum shear stress for this state of stress.
xa
y
x
y
12 ksi
stressstateatpointa
β²x
β²y
ΞΈ = ?
18 ksi
Ο β²x β²y = ?
r
t
tankcrosssec-on
Problem 2 (25 points):
The closed, thin-walled tank shown above has an inner radius of r and wall thickness t and contains a gas of pressure of p. The state of stress at point βaβ on the tank is represented by stress components shown above for an unknown stress element rotation angle of ΞΈ . Note that
Ο β²x β²y is in the direction shown; however, its magnitude is also
unknown. For this problem, you are asked for the following (in no particular order):
β’ Determine the principal components of stress, Ο1 and Ο 2 .
β’ Determine the magnitude of Ο β²x β²y .
β’ Draw the Mohrβs circle for this state of stress. Show the location of the x 'β axis in your Mohrβs circle. From this, determine the rotation angle ΞΈ .
β’ Determine the absolute maximum shear stress for this state of stress.
xa
y
x
y
12 ksi
stressstateatpointa
β²x
β²y
ΞΈ = ?
18 ksi
Ο β²x β²y = ?
r
t
tankcrosssec-on
2ΞΈ = 180Β°β tanβ1 4
15β12βββ
ββ β= 126.9Β° β ΞΈ = 63.4Β°
AlsofromMohrβscircle:
Ο
max,abs=Οmax βΟmin
2= 20
2= 10 ksi
Ο1 = 20 =ΟmaxΟ ave = 1512Ο 2 = 10Οmin = 0
2ΞΈ
x 'β axis
4
Ο
Ο
Problem 4 (10 points):
Five points (i.e., A, B, C, D, and E) are marked on the structures shown above. The state of stress of each of these points can be represented by a Mohrβs circle and the Mohrβs circle can be used to determine principal stresses Ο1 > Ο2 > Ο3.
Circle the state(s) of stress represented by the Mohrβs circle depicted below.
Page 11 of 11
Problem 4.5 (4 Points): The cantilever beam is loaded as shown below. Which one of the Mohr's circle shown here represents the stress element at point A and B?
Stress element at A
a) b) c) d)
Stress element at B
a) b) c) d)
Problem 4.6 (2 Points): The beam is loaded as shown below. Which one of the Mohr's circle shown here represents the stress element at point A?
a) b) c) d)
ME 323 Examination # 3 Name ___________________________________ (Print) (Last) (First) _____ ___, 2016 Instructor _______________________________
PART C - 8 points:
Beam (i) β Steel Beam (ii) β Aluminum
Beams (i) and (ii) shown above are identical, except that beam (i) is made up of steel and beam (ii) is made up of aluminum. Note that πΈπΈπ π π π π π π π π π β₯ πΈπΈπππ π ππππππππππππ.
Let |ππ|ππππππ,(ππ) and |ππ|ππππππ,(ππππ) represent the maximum magnitude of flexural stress in beams (i) and (ii), respectively. Circle the correct relationship between these two stresses:
a) |ππ|ππππππ,(ππ) > |ππ|ππππππ,(ππππ)
b) |ππ|ππππππ,(ππ) = |ππ|ππππππ,(ππππ)
c) |ππ|ππππππ,(ππ) < |ππ|ππππππ,(ππππ)
Let |ππ|ππππππ,(ππ) and |ππ|ππππππ,(ππππ) represent the maximum magnitude of the xy-component of shear stress in beams (i) and (ii), respectively. Circle the correct relationship between these two stresses:
a) |ππ|ππππππ,(ππ) > |ππ|ππππππ,(ππππ)
b) |ππ|ππππππ,(ππ) = |ππ|ππππππ,(ππππ)
c) |ππ|ππππππ,(ππ) < |ππ|ππππππ,(ππππ)
Let |πΏπΏ|ππππππ,(ππ) and |πΏπΏ|ππππππ,(ππππ) represent the maximum magnitude of deflection in beams (i) and (ii), respectively. Circle the correct relationship between these two stresses:
a) |πΏπΏ|ππππππ,(ππ) > |πΏπΏ|ππππππ,(ππππ)
b) |πΏπΏ|ππππππ,(ππ) = |πΏπΏ|ππππππ,(ππππ)
c) |πΏπΏ|ππππππ,(ππ) < |πΏπΏ|ππππππ,(ππππ)
Let |π΅π΅π¦π¦|(ππ) and |π΅π΅π¦π¦|(ππππ) represent the vertical reaction at B in beams (i) and (ii), respectively. Circle the correct relationship between these two stresses:
a) |π΅π΅π¦π¦|(ππ) > |π΅π΅π¦π¦|(ππππ)
b) |π΅π΅π¦π¦|(ππ) = |π΅π΅π¦π¦|(ππππ)
c) |π΅π΅π¦π¦|(ππ) < |π΅π΅π¦π¦|(ππππ)