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(Last) (First) PROBLEM # 1 (25 points) A tree in the wind: Assume the wind generates a net force = (10 − 50 + 20) N at the point A on the tree, and the tree deforms elastically. The point A is located at (3 + 1) m. Two points, M and N, are located at the cross section aa which is 1 m away from the ground. The cross section aa is a circular shape and has a radius of 20 cm. Ignore the weight of the tree.
a) Determine the stress state at M due to the wind force. Sketch the stress components on the stress element.
b) Determine the stress state at N due to the wind force. Sketch the stress components on the stress element.
Summarize your results of the stress components at M and N in the given answer table.
a a
Tx g Tx
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polarmomentof inertia Ip T T zsy m4
second area moment Iy Iz i26xw m
Moment aetingonthecross section
look 402 102 502 k t K
502 30J look Loading streesatan stress a N
Fx low A FIT_80Pa r F Sopa
Fg 50N Try O I y 4Ffy 533Pa
Fz 20N Text 447T_213Pa 1 2 0
Mx 50Nm Txz MILIT 3984Pa Try M pI 3984Pa
My 30NM Tx _O Mt 4762Pa
haze 100mm T M2 15873Pa T O
Name (Print) ______________________________________________
Stress Components M N
Txz 213 3984 3771Pa
Try 533 3984 4517Pa
O O O O O 4517Pa 3771Pa O O O
ME 323 – Mechanics of Materials Final Examination December 8th, 2020
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PROBLEM # 3 (25 points) Consider a state of plane stress in a structural component represented by the stress element provided below where = 60 ksi and is unknown at this point.
a) Suppose that the structural component is made of a ductile material with a yield strength of = 100 ksi.
a. Using the maximum shear stress theory, determine the maximum value of for which the ductile material in the structural component does not fail.
b. For the maximum value of found above, does the maximum distortional energy theory predict failure of the material?
b) Suppose now that the structural component is instead made of a brittle material with equal ultimate strengths in tension and compression of = = 100 ksi. Using Mohr’s failure criteria, determine the maximum value of for which the brittle material in the structural component does not fail.
Tawe TE 30 Ksi R VREY IE V302 tIxy Gave
A A Since R True Jp Tpz have opposile
Signs Imax abs Imax in plane
R V 307 From this we have at failure MSD
II Imax abs
1 302 max
40 KsiV 302
b Since MSS is more conservative
than MDE MDE will predict e
b Since Jp 8 Tp have opposite signs the S.O.S Occurs in the
4th quadrant
ZR Tu
1ma 5302 40 Ksi
ME 323 – Mechanics of Materials Final Examination December 8th, 2020
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PROBLEM # 4 (25 points) Part A – 9 points A rectangular cross-section bar is subjected to an axial force P, and a thin-wall pressure vessel is subjected to internal pressure.
Referring to the ten Mohr’s circles shown above, circle the number of the correct in-plane Mohr’s circle for the state of stress at
• Point A: #1 #2 #3 #4 #5 #6 #7 #8 #9 #10
• Point B: #1 #2 #3 #4 #5 #6 #7 #8 #9 #10
• Point C: #1 #2 #3 #4 #5 #6 #7 #8 #9 #10
A B C
o O
ME 323 – Mechanics of Materials Final Examination December 8th, 2020
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PROBLEM # 4 (cont.) The schematics shown below correspond to beams with a span L and loading with either a couple at one of the two ends or a sinusoidally distributed load.
Part B – 3 points Which schematic corresponds to a beam whose bending moment is equal to
( ) = − 0 2
2 sin ( ) −
3
Circle the correct answer (a) (b) (c) (d) (e) (f) (g) (h) (i) (j)
HINT: Identify which supports and loads are compatible with the given bending moment. Part C – 3 points If the deflection at = /2 is to be determined using Castigliano’s second theorem, which schematic(s) correspond to a beam which will require the use of a redundant load
Circle the correct answers (a) (b) (c) (d) (e) (f) (g) (h) (i) (j) Part D – 3 points If the slope at = is to be determined using Castigliano’s second theorem, which schematic(s) correspond to a beam which will require the use of a dummy load
Circle the correct answers (a) (b) (c) (d) (e) (f) (g) (h) (i) (j)
(f)
(g)
(c)
(d)
(h)
(i)
(a)
(b)
(e) (j)
O Van I PExEcost Tana tpo Esinf Ncosto Vcu 10 Tko 0 MG to fixedat o andX L
O O O
00 OO O
ME 323 – Mechanics of Materials Final Examination December 8th, 2020
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PROBLEM # 4 (cont.) Part E – 7 points
The rod shown on the left figure has a variable cross section. A finite element model comprised of three rod-elements and four nodes is produced and shown on the right figure. The finite element model has the following stiffness matrix, where only three elements of the matrix are given:
[] =
______ 8 ______ ______
______ ______ 9 ______
______ ______ ______ 2 ]
(a) Determine the remaining 13 matrix elements and fill the blank spaces. The two ends of rod are attached to fixed walls, and one concentrated force of value P is applied to node 3, as shown on the right figure. (b) Determine the displacement of node 3. (c) Determine the reaction force on the rod due to the wall at node 4.
1 2 P
l 7 O
O 7 2
O O Z
7 97 I top 82k 793 0 az FU3 7Uzt9Us P 4Igt9az P
Urga GI23 R4 293 16123
ME 323 – Mechanics of Materials Final Examination December 8th, 2020
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