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**Unformatted text preview: **Test2 (100 pts) AME3143 Monday, September 13,2017 NAME: ”8:3 [ L4 7%.? a? ﬁg
Problem 1 (20 points) Graphical Problem Mohr’s Circle Axial Case. In an axial loaded structure, an element of material is subjected to normal stresses 6;, cry, and to shear stresses on the x y-face 1:1y (see ﬁgure). 2D stress element for axial case
with stress element lined up with axial load has the following quantities (see ﬁgure): 0; = 8 ksi, 0,, = 0 ksi, 1:“ = 0 ksi . (a) (16 pts) Construct Mohr Circles using the ra hical method as shown in mm for
using Mohr circle equations in your work.) (12 pts). Show all details (x—face; y-face, center C of circle, Radius R of
circle, max normal stress, min normal stress, max shear stress, min shear stress) on the Mohr circles that you construct. Show all work for full credit and deﬁne all terms that you use Il‘ii I‘ll-lum;( Iii-III- MIIIIIIIIIL ' [WEI-IIIIH
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(319/ a 2 All“ (1 pt) Radius R of circle = I? q ’55}: (b) (4 pts) Mohr Circle‘s Maximums and Minimums ‘ Maximum normal stress: 95% 5 e (1 pt); Minimum normal stress: (:3 "/33; (1 pt) ‘ ,. r' _,-v- f‘ P
Maximum shear stress : [lf%é e, ( 1 pt); Minimum shear stress: yﬂéer t, (1 pt) Problem 2 (20 points) Stresses on Inclined Sections (Mohr circle equations of Chapter 2). Consider the bar AB subjected to axial
loads P as shown in Fig. 1. Two “micron” cubes are cut out to display stress elements as shown in Figs 2 and 3. The faces of the “micron” cubes have area A0. The stress element in Fig 3 has the angle 9withwres nect 0 that of F-'-. 2.
.. l' V r l . -:" .- \ '5'. .. . . _______ * Fig. 2. Stress Element at
angle zero. Let the faces of ﬁns element have area A"’ Fig. 3. Stress Element at angle 6. Let the faces have the same area An. Fig. 4. Cut at angie 6. (a) 2 pts Determine the stresses Ox, ﬂy, and Txy for the stress element in Fig. 2 and indicate these in Fig. 2:
6x = (0.5 pt) ; cry : (0.5 pt) ; Txy = (0.5 pt) . 0.5 points for drawing these on Fig. 2 (b) 6 pts We perform a cut at angle (9 on the stress element of Fig. 2 as shown in Fig. 4 to obtain the cut section in Figs 5a,b,c. The
uncut face has area Ao. Indicate the stresses, areas, and forces on the faces of the “cut” section in Fig. 5a,b,c. [Draw arrowsfor
stresses and arses. ,,., X ' . 5c Forces in X1 and (c) 6 pts Write equilibrium equations for the forces acting in X1 and y1 directions (see Fig. 5c) and solve for the unknown stresses on
and Txlyl. Use trig relations ccs2(8) =0.5+0.5 cos(28) and sin(6)cos(0)=0.5 sin(26) to get on and Txlyl as functions of cos(23) and
sin(28). Give force balance equations 2Fx1=0 (2 pt) and 21‘}; :0 (2 pt). (d) 4 pts on =? and Txlyl = ? (2 pts) O'xl = and (2 pts) 155in : (e) 2 pts Show how to ﬁnd oyi: Give equation for ﬁnding it (1 pt) and Gyl =? Gyl = (1 90' ' ' |"‘
Quiz 2 (20 points) Wednesday September 13, 2017 ANIE 3143 NANIE: f5? Mfg/{Vt
Stresses on Inclined Sections (Mohr circle equations of Chapter 2). Consider the bar AB subjected to axial loads 1’ as shown in
Fig. 1. Two “micron” cubes are cut out to display stress elements as shown in Figs 2 and 3. The faces of the “micron" cubes have area As. The stress element in Fig 3 has the angle Qwith respect to that of Fig. 2. Figs 2. tress Element at «*9
angle ze 0. Let the faces of ”‘4 6‘0
this ole nt have area An. cut Fig. 3. Stress Element at angle 9. Let the faces have Fig?! I?" the same area An. Fig. 4. Cut at angle 6. ._..1 (a) 2 pts D termine the stresses 6,, Gy, and 11,1 for the stress element in Fig. 2 and indicate these in Fig. 2:
ex = i3 (0.5 pt) ; (1,1 = O (0.5 pt) ; Txy = g 2 (0.5 pt) . 0.5 points for drawing these on Fig. 2 (b) 6 pts We perform a cut at angle 0 on the stress eiemeat of Fig 2 as shown 111 Fig. 4 to obtain the cut section in Figs 5a,b c
The uncut face has a1 ea A0. Indicate the s asses,” areas, and force on the faces of the “eu ” section 111 Fig. Sa,b ,c. [Draw
arrows or stresses and orces. 1' Fig. Sa Stresses (2 pt! Fig. 5b Areas (2. pt) FigL‘Sc Forces in 111 and 31 directions (2. pt) (c) 6 pts Write equilibrium equations for the forces acting in X1 and y; diiections (see Fig 5c) and solve for the unlcno'am
stresses 1m and 111,11 Use tiig leiations coszt’e) =0 5+0. 5 eos(28) and si11(0)cos(0)=0 5 9.11109) to get" ex] and 11131 as functions W of cos(20) and 513198). Give force balance equations EFx1=ﬂ (2 pt) and BF}; =0 (2 pt) (W [7} (27% ((139
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Wm-ﬂ._ﬁﬁmgumam ... e313: Problem 3 (28 points) Consider the statically indeterminate structure ACE with loads P as show in Fig. 1. Determine equations
for the stresses in the bars 1 and 2 where barl is bar AC and bar2 is bar CB. Draw free body diagrams (FED) that are needed
in solving for the stresses. Make “cuts” and indicate what you are doing. Show equilibrium equation as needed to solve the
probiem. You must use fundamental axial mode] in all cases. Let unknown force in bar AC he P1 and unknown force in bar
CB be P2. [You must know from axial mode] whichdirection these are pointing] Describe all terms that you use. Do M
skip any steps? Use coordinate s stem with axial axis-x bein to the ri ht 'ust as we did in class). Fig. 1. Structure ACB held between two walls has lead (force) P
acting at point C as shown. The barl (bar AC) has properties
Young’s modulus El, cross-sectional area A], length L1, and
Poisson’s ratio v1. The bar2 {bar CB) has properties Young’s
modulus E2, cross-sectional area A2, length L2, and Poisson’s ratio
v2. Let L=Ll+L2. A. (2 pt) What is the equation of compatibility needed in solving this statically indeterminate structure? (Deﬁne all terms that you use in this equation.) B. (8 pts) Solving this problem requires solving two equations with two unknowns. Determine those two equations with the two unknowns. Method: 6 pts (method must include FBDs just like we did in class using fundamental axial model in all cases). If you use an equation
other than the three fundamental equations (just like we did in class), show how you got it from the three fundamental equations (it is worth 1 pt). Solutions: What are the two equations with two unknowns?:
(1 pt)
(1 pt) C. (6 pts) Find the force P1 acting on bar] in terms of P and ﬁnd the force P2 acting on bar2 in terms of P
Method: 4 pts Solutions: (1 pt) Pl? ; (1 pt) P2: D. (4 pts) Find the normal stresses acting in barl in terms of P and ﬁnd the normal stresses acting in bar2 in terms of P
Method: (2 pl) Solutions: (1 pt) 61: ; (1 pt) 62:
Normal stresses in barl (bar AC) Normal stresses in bar2 (bar CB) . f“
Q3 (20 points) Friday September 15, 2017 AME 3143 NAME: I (9 t 1/ ‘5‘ M2; Consider the statically indeterminate structure ACB with loads P as show in Fig. 1. Determine equations for the stresses in the
bars I and 2 where barl is bar AC and bar2 is bar CB. Draw free body diagrams (FBD) that are needed in solving for the
stresses. Make “cuts” and indicate what you are doing. Show equilibrium equation as needed to solve the problem. You must
use fundamental axial model in all cases. Let unknown force in bar AC be P1 and unknown force in bar CB be P2. [You must know from axial model which direction these are pointing] Describe all terms that you use. Do NOT skip any steps?
Use coordinate s stem with axial axis-x bein to the ri ht 'ust as we did in class .
Fig. 1. Structure ACB held between two walls has lead (force) P
acting at point C as shown. The barl (bar AC) has properties
Young’s modulus E1, cross—sectional area A1, length L1, and
Poisson’s ratio v1. The bar2 (bar CB) has properties Young’s
modulus E2, cross-sectional area A2, length L2, and Poisson’s ratio
v2. Let L=Li+L2. A. (2 pt) What' Is the equation of compatibility needed In solving this statically indeterminate structure? gilt ﬂ 9 a.“ {i if j rid! 6“ :3 '3‘ [isgé‘yxﬂ tﬂbaf/ gait." (Deﬁne all terms that you use in this equation.) B. (8 pts) Solving this problem requires solving two equations with two unknowns. Determine those two equations with the two unknowns. M__ethod: 6 pts (method must include FBDs just like we did' In class using fundamental axial model in all cases). If you use an equation
other than the three fundamental equations (just like weA did' In class), show—how’you got it from the three fundamen nations (it is worth 1‘ 5’ emu /3{ fM «W we a (1 130$”
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C. (6 pts) Find {he tore: P1 actliig on bar] 111. terms of P and ﬁnd the force P2 acting on beu'Zé':> In terms of P :1 Fen- 61$;
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Solutions: (1 p051: F M if a” Normal stresses in barl (bar AC) “We Normal stresses in bar-2 (bar CB) Problem 4 (20 points)
ts Draw the axial model bar with loads on it must be as written out in class lecture and (b) 4 pts (i) 3 pts. Draw stress-strain diagram with 3 fundamental equations written on it must be as written out in class
lecture and iven in email to class . ii 1 t Then add the additional fundamental e nation as iven on ower— oint slide c 6 ts. Deﬁnition of normal strain (must be as written out in class lecture and iven in email to class; see ts. Deﬁnition of normal stress as written out in class lecture and _iven in email to class; see a t slide ts. Hooke’s Law for axial case as written out in class lecture and _iven in email to class; see u t slide ts. Poisson’s Ratio as written out in class lecture and iven in email to class; see as written out in class lecture and 'ven in email to class; see t slide Quiz 1 (20 points) Monday September 11, 2017 AME 3143 NAME: 3 /d 725
- ts Draw the axial model ar with loads on it must be as written out inclass lecture and _iven in email to class (b) 4 pts Draw stress-strain diagram with 4 fundamental equations written on it must be as written out in class lecture and st C22;fl(5/V€J!faﬂjr 3222251252222”
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(Bf/3‘“! ff fin—«2 inf/L22“ gaifM-qupj @Wﬁﬂﬁ Cred, Car/2,722,222; 5;; 24%;“ -- Problem 5 (26 points). Temp Effects Problem: A rigid bar of weight W hangs from three equally spaced wires, two of
steel and one of aluminum (see ﬁgure). The area of each wire is A. Before they were loaded and before temperature
increase, all three wires had the same length L. The original temperature before temperature increase is To. The weight
W is loaded at temperature Ta and then the temperature is increased from To to 'I}. The ﬁnal temperature is TI: Tad-AT.
What temperature increase AT in all three wires will result in the entire load being carried by the steel wires? In solving
this problem ﬁnd the following: Find how much axial load P,l is on aluminum wire at ﬁnal Temperature Tf. Find how
much axial load P3 is on each steel wire at ﬁnal Temperature 1}. Find equation for aluminnm’s unstressed length at the final
Temperature Tf. Find equation for aluminum’s stressed length at the ﬁnal Temperature T f, Find equation for steel’s unstressed
length '. the ﬁnal Temperature T f. Find equation for steel’s stressed length at the ﬁnal Temperature 1}. Find equation (with (b) (2 pts) Find Eq. for axial load Pa is on
aluminum wire at ﬁnal LTemperature 1}. Method (1 pt):/ ﬁle!” / Lari? ti 9Z2 riﬂe Etg/ an 5- «1’9er W“ 5"
EquationP= Q l (1 pt)
l/ (c) (2 pts) Find Eq. for axial load P3 is on
' r ., each steel Wire at ﬁnal Temperature T f
f 01517 Method (1.. pt): 2 lg» 5.: ' 5"! law“ -' . , _ -- . Euation PS:
(d) (2 pts) Equ :5 '2'“ ' _ 1isnnstté’Ssed lengtm tthe ﬁnal Temperature 1} Method (1 pt): 1 - - Equation for aluminum 5 unstressed length at the ﬁnal Temperature T f = I "L A / (1 pt) (e) (2 pts) Find Equation for aluminum’ 5 stressedjlength at the ﬁnal Temperature f. Method (1 pt): m)
Maul/Hrs“- 4..., ’1”) (“4” [L41 swan 2:: Lamrr Arenas? Equation for aluminum” s stressed length at the ﬁnal Temperature '1} = L (I (I 7“ Q24} ’61 7‘) (1 pt) (t) (2 pts) Find Equation for steel’s unstressed length at the ﬁnal Temperature 1}. Method (1 pt): L;wg¢l«1ﬁd-L7ﬁL¥ﬂiTLCbﬁ%/ﬂ Equation for steel’s unstressed length at the ﬁnal Tern erature T = L C ( 7L A 7‘“) l t
P f _7§ P
(g) (2 pts) Find Equation for steel’s stressed length at the ﬁnal Temperature T f. Method (1 pt): (7 [J m 4 (12¢ ﬁg; [:5 SfMJﬁ-a-J ”’3 [:r “wwwrfimﬂ 7L 41 did/{ewe 1"” 7% x4;
LI Stir-errand 2: 4,21 enema? :2 QuatJ/Wakﬂ” m ALVA" 4‘57) Equation for Steel’s stressed length at the ﬁnal Temperature 1}“=L ( I 71 Wﬁf: ‘6 7:) (1 pt) (11) (5 pts) Find eq. for the Temperature increase that result 1n the entire load being carried by the steel wires. Method (4 pt): Method beforea roximation (2 pt): . 2: 7i 0/ Af/§j
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(Kmmation of Method (2 pt): Describe ppro oxrgatio‘tﬁl pt);w]1 on can make it (1 pt). To be convincing of why the approximation, you can use the data. W=200 lbs., A=1 111.2,lli‘s=30><10+6 psi; as=5><10 61’°F; 1E'a=10><10+6 psi; ora=15><10. WW WW” "Q” “i ””"W W
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Wé’ﬂ (A? ﬂngfé 230% A55 £23 /A£r.s?— 2940 (Solution) Final Equation (with approxi atio‘il ) for Temperatur in rease (with parameters, not data} AT= 6 g (a; 7;“ ‘93.)33 4g (1 pt) ...

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