ES 13 WFV Date given: July 20, 2012 Due: July 27, 2012 (Friday) at class hour
PROB SET 2
Instruction: Answer the following questions on letter-sized (8.5”x11”) paper or A4 (8.27”x11.69”), preferably recycled (i.e. one side already used). Only one side of every sheet of paper shall be used u sed to write your solutions to the problems. I.
Axially Loaded Members pts) A. Statically Determinate (choose 1 of 2 problems , 5 pts) 1. The truss is made up of three steel (E = 200 GPa) members, AB, BC, and AC. The 2 2 cross sectional area of AB and BC is 400 mm while AC is 600 mm . P θ
B
800 mm A C
800 mm
600 mm
a. If P = 10 kN and θ = 53.13°, determine the following: a.1. Displacement of C in mm. Note: You can only have displacement/s at axis/axes with no support. a.2. Deformation of each bar. b. If C displaces by 0.15mm due to the applied load P as shown, determine the magnitude of P. 2. Bar ABCD is is subjected to to the following loads as shown. Determine the maximum maximum value of force P that can be applied to the bar if σallow used for design is based on ultimate strength and δallow (of bar ABCD) = ±1.2 mm. Note: Member AB at inner diameter is hollow.
Aluminum
Steel Copper
Properties E (GPa) σyp (MPa) σult (MPa) FS
Aluminum 70 180 310 2
Copper 120 200 400 4
Steel 200 300 580 3
B. Statically Indeterminate (choose 3 of 4 problems , 5 pts each) 1. Three bars AB, AC, and AD, are pinned together to support a load of 20 kN. Horizontal movement is prevented at joint A by a horizontal strut AE. Note: All supports are pin-connected .
Bars AB and AD: E = 70 GPa 2 A = 400 mm Bar AC: E = 200 GPa 2 A = 200 mm
Determine a. σavg in each bar (AB, AC, and AD) b. Force in the strut AE 2. In the figure shown below, each of the two vertical steel posts ( E = 200 GPa, α = -6 12×10 /°C) has a height of 300 mm and a diameter of 50 mm. If the springs each has a stiffness of 0.5 kN/m, determine the resulting stress in the steel posts if the temperature is raised by 70°C. Assume that springs are initially unstretched. Neglect the masses of the rigid members.
3. The respective ends of three wires are joined together as shown. Each wire has a cross2 sectional area of 3 mm and E=200 GPa. The lengths of the wires are 4 m, 4.002 m, and 4.003 m. The three wires carry a total load of 1000 N.
• P Determine: a. The deformation of the shortest wire b. The tension in the most stressed wire c. The tension in the least stressed wire 4. An aluminum cylinder and a bronze cylinder are centered and secured between two rigid slabs by tightening two steel bolts as shown below. Hint: Though slabs are rigid, please take note that they are free to move (or be displaced).
At 10°C no axial loads exist in the assembly. The weight of all components in the assembly can be assumed negligible. The geometric and material properties of the different bars are given in the table below: Property Aluminum Bronze Steel 2 Cross sectional area per bar (mm ) 1800 1200 500 Length (mm) 75 100 215 Modulus of Elasticity (GPa) 70 83 200 -6 -6 -6 Coefficient of Thermal Expansion (1/°C) 23.0×10 19.0×10 11.7×10 If the temperature of the entire assembly is increased to 90°C, and the internal force and deformation in the two steel bars are the same due to symmetry, determine the following:
i. Given that the force in each of the steel bars is the same, what is the degree of static indeterminacy of the assembly? Explain why. a. 0 b. 1 c. 2 d. 3 ii. If the force in each of the steel bars is P S , the force in the aluminum bar is P A , and the force in the bronze bar is P B , which of the following statements is false? Explain your answer. a. 2 PS + P A
=0
b. 2 PS + P B = 0 c. P = P B d. P A + PB − 2 P S = 0 iii. If δ A , δ B ,and δ S are the change in length in the aluminum, brass and steel bar due to
P P S respectively, and internal forces P A , B and
δ TA| , δ TB ,and δ TS are
the change in
length in the aluminum, brass and steel bar respectively due to the increase in temperature from 10°C to 90°C, which of the following statements false? Explain each choice and why it is true/false. a. δ TA| , δ TB , and δ TS will always be positive, since all bars will ex pand due to an increase in temperature. b. δTA = α A LAΔT where α is the coefficient of thermal expansion of aluminum, L A is equal to the initial length of the aluminum bar before the temperature was increased, and ΔT is the increase in temperature which in this case is equal to 80 C°. P L LS is equal to the c. δ s = s s , where P s is the internal force in each of the steel bars, A s E s initial length of the steel bar before the temperature was increased, and A s is the cross sectional area of the steel bar, and E s is the modulus of elasticity of the steel bar. d. 2 (δTS + δ S ) = ( δ TA + δ TB ) − ( δ A + δ B ) iv. Which force given below is closest to the internal force in the steel bar after the temperature is increased from 10°C to 90°C? Show your solution to support your answer. a. 8.3 kN c. 33.2 kN b. 16.6 kN d. 66.4 kN v. Which stress is closest to the stress in the bronze bar after the temperature is increased from 10°C to 90°C? Show your solution to support your answer.
a. 16.6 MPa compressive b. 27.66 MPa compressive
c. 33.2 MPa compressive d. 66.4 MPa compressive
II. Torsional Loading (choose 2 of 3 problems. 5 pts each) 1. A compound shaft consisting of steel (G = 83 GPa) segment and an aluminum (G = 28 GPa) segment is acted upon by two torques as shown.
a. Assuming τall of steel governs, what is the value of T. b. Assuming τall of aluminum governs, what is the value of T. c. Assuming θmax (1.5 m length) governs, what is the value of T. 2.
A hollow rod (G = 80 GPa) is bent as shown in figure below to support a homogeneous plate weighing 300 N. If the rod has an outer diameter of 50 mm, an inner diameter of 40 mm and a linear mass density of 9 kg/m, determine: a. The maximum shear stress due to torsion at a section through a. b. The maximum angle-of-twist for segment AB of the rod.
Fixed support
3.
th
Hibbeler, RC. Mechanics of Materials. International Edition, 4 Ed. Problem 5-56. Use G = 85 GPa. *Please take the problem from the edition specified since numbering may vary for different editions.
Bonus: 1. Determine the moment of inertia I x and I y at the centroidal axes of the section. (2 pts)
y
x
2. Draw the shear and bending moment diagram of the figure below. (3 pts)
A
25 kN·m ∙ B
1m
1m
10 kN/m C
D
3m
E
1m
3. Determine whether the figure below is determinate, indeterminate, or unstable. Show “solutions” (i.e. indicate # of bars, #of reactions, and # of joints). If indeterminate, indicate degree of indeterminacy. (2 pts) [Observe the figure carefully]
a.
b.
Max pts: 37/30 Sir J.U. Tan Tian "What else does this craving, and this helplessness, proclaim but that there was once in man a true happiness, of which all that now remains is the empty print and trace? This he tries in vain to fill with everything around him, seeking in things that are not there the help he cannot find in those that are, though none can help, since this infinite abyss can be filled only with an infinite and immutable object; in other words by God himself." - Blaise Pascal (1623-1662) *French Scientist and Theologian
