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# BIOMECHANICS BMED 4540

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This 88 page Class Notes was uploaded by Joshuah Labadie on Monday October 19, 2015. The Class Notes belongs to BMED 4540 at Rensselaer Polytechnic Institute taught by Shiva Kotha in Fall. Since its upload, it has received 36 views. For similar materials see /class/224823/bmed-4540-rensselaer-polytechnic-institute in Biomedical Engineering at Rensselaer Polytechnic Institute.

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Date Created: 10/19/15

Biomechanics Homewerk 2 Due 2132012 Problem 1 Randy s shoulder is labeled S and two points on the cable are labeled P and B The coordinates of the three points are given n the table below The tension in the cable is known to be 150 N Find the moment about Randy s shoulder of the force exerted by the cable on Randy s left hand Point X m y m z m S 08 13 15 P 0 23 0 B 13 11 16 Problem 2 Determine the magnitudes of the forces T1 and T2 for each of the cablepulley systems shown if W 50 lb Be sure to draw the correct FBDs Wl100N W980N Wo1500N a5cm b10cm c15cm Theta30 deg FindFmandFj Problem 3 FM F Problem 4 0 Given the 2 adjacent F in figures Theta is an we unknown constant Calculate me and W me FAILy 5 quot n Fina A B C a r 39 r Rd 4 1 Problem 1 v H A s f LE I lt mwz E 39 mm m2 E a 26g A A I force F 1 Problem 2 b A Mmmllmmhi 139 M L 1 3L 2 g 3 underforccP E j g ifrx Y 713 I Problem3 P E s6 X gmxo k v n wo Assume mmcpamdr L M4 325mm Ima mam 7 1C 39 Y 39f39 N 410 n u Problem 4 up F 7 f I 1 12mm 4 4 2535gt49 f f 27 3 w m L 7 X force uniform Of4X 10quotN Assuming Ex 100 GP and V 030 nd 6 Swan 8 and the deformed dimensions assuming homogcnous i g b Lgtlts b 1 G quotI W f 39 Problems lmy 7L o7 vi 0amp1 I Arccmngukn bar aluminium we 15 x 21 cm in cross scction and a circular m1 rod 1 cm in x 03 4 0 39 quot quot quot39V 39 lnmlnn nyh39 4 A Amt lt1 HEX unloaded con gumion nd The strcSS in cach b The extensional main in each and We ps am fM c The amount of lengthening in each W J 00 LaE7oGPafom1uminiummdEocpafmma M 15 20 M 0 0 H I W 46W U 7074 f Xv 5g TQ w 2 zWQ C M 7 74 7 1 A I X b vm km MW KM J F 1 gi 0 mm ZAa 39 3 a I 143 Ag i Kto gt7LgtX7b 2th V quot M 776 6a 1 7 7 95 A A m V 5 pyj mny A 39 703WJ 17 5Q 0 50 XI 0 5Y3Hgt in V a 2 023m v T M AHMIM Name RCS ID Biomechanics Quiz 6 5th April 2012 Problem 1 The Plastic block is boned to a rigid support wall and to a vertical rigid plate to which a 200 KN load P is applied Knowing that the de ection of plate is 1mm Calculate The shear modulus of the plastic block Dimensions in mm J E F W t A Vlgx L quot k jmw 00735quot 247 640 7047 Mquot 38 J raw0 R t 1H Problem 2 v 18 W Skin tissue is subjected to a biaxial loadin as known The 0 18 E dimension of the skin is 5cm by 5 cm by 2 mi thickness The qt 1 01277 X A F applied forces arealkg in x direction and1amp4 kg in z direction J Wm and the elongation of the tissue is 05 cm and 1 cm in x and z 7gt 77 lt 3 directions respectively Calculate 1 Young s modulus and Poisson s ratio 2 Lame constants shear modulus and bulk modulus of the CU J tissue r XX E 3 Is the skin at plane stress or plane strain status s 6 2gt 4 u I amp6U agxxl y E mesU WW NE igxw ggll gt 39 O lE 39 Iii33L 18 021285 7g O F 91 032 17 217ZH7 YE 3 1C Q oi V M is i f fx k v 3036 3 fzl tfx plng W9 X if Biomechanics Homework 11 15 killsR Problem 1 Jiml For the beam and loading shown design the cross section A r i of the beam knowing that the grade of timber used has an allowable normal stress of 1750 psi i3ft 6ft Problem 2 P P 05m Determine the largest permissible value of P l 1 for the beam and loading shown knowing A f i i D 1 I 2m that the allowable normal stress is 6 ksi in m l U 5 in tension and 18 ksi in compression 4 in Problem 3 q 39 p u Given the cantilever beam has an uniformly distributed load of 50 kNm and the beam is 05 m long nd the de ection of the beam at point P The exuml stiffness is 60 MNm2 E122 L H Problem 4 qa Given the cantilever beam is loaded as shown in the gure q is 500 kN and the beam is 075 m long nd the de ection of the beam at the end of the beam The exural stiffness is 90 MNm2 Elzz Biomechanics Homework 10 3in 3in 3in IS laps l5 lips Problem 1 Two vertical forces are applied to a beam of the cross section shown Determine the maximum tensile and compressive stresses in the portion BC of the beam Problem 2 Two equal and opposite couples of magnitude M 25kNm are applied to a channel shaped beam AB cross section of beam shown in gure aloongside Observing that the couples cause the beam to bend in a horizontal plane determine the stress at point CDE Problem 3 39 Given that the load applied P is 10 kN the distance between point D d if E A and C d is 50mm The cross section of the beam is a square of mm length 25mm 7 Determine the stresses generated at point F and G 3im3ilh3in r 2 co o A 312 M7 5 a T 8 5 QO 9 m 398 39 398 E 35 10 y 103 Y 3T 3quot Neuf m Lzl39s u 3 in z 2 Lav 44w base 2 i egt I b A d r 523c3ltls22 12c w 11 516 A Jf 37001 cmm 72 h I 1911 26 72 204 3 5m ym 3 in 10mm 10mm W l39 A m1 5m Aq Ma Q Goo 3o 12 o3 50mm A Q 00 50 Sr lo5 1 10mm C3 300 5 Lsxm Lama f Isco 379w 375m1 Ya W 25mm I NewIL ms In 25m mm 1 Base 27mm 1 x amm qwam msuo nm IL I 175quot I3 739 30300334 Scone 1225y10 mm I l I1 1 5252105 m 5125 0 3 X 35Mquot 0035 JIM 45 0025 h quotIquot 2 wwer v I39 I r na quot w5563 Wm 2 c lt90 lt71 1 96 2 WWW WWWquoty I Tlt O39tfxi a 010 93 l quot Z g A 3 Wquot Clix C V 2 733 4 quotEx may 7 V x be W 500 3 N a 3 gtF 1 J E m at 9 W 0 h 1 7amp6 E 21 903 T 000 TL T L W 5 1Kng 967 G 30Cch 1 TL 7 6 L Jr 1 3 CIA Ci W a 93 TL 4 w J L GE VG L A L EWCC q QLGD 6T1 EU z quot QR 1 Biomechanics Homework 2 Sa w mg Problem 1 WW 5 my 2 M a Mg wm 110 Wu ZngpxrmW Problem 2 5 F50 Uf fV oufrj5 l f 2 WW7 7 F130 hum Palt 7 7 2 957 77 a L def0 5274 0 Jim b r760 c vr7 uk 7quot fl 2745CJ7 7 7 Jr 2 5 0 a 27quot fwd 2 7 2539 f D Midi4 palow 739 f7 2 057739r n 9 21 f z 269 d Problem 3 WW Um We 327 it Q 070quot uW39MQlg J HC FM3999D 9 p 1 F13 WW WWW auxAm Wq f M 39 x dadm Lw WT EYK S39IMQ M m We 43M 13 V V Df mw Mfl j ftwb Problem 4 V t I M 194w W 0 accwz MW Llm VT I w c u 0 A 39 TAMAA 4A9 Wo W w 39aL 1 w y rajMg Fuy z tubMIL FM W W 9 i T quot2 0 F r I T l 0 quot9 VHW E F Vy W 71hr q 9 O u I ich Biomechanics Homework 4 Due 3512 Problem 1 Given the 2D state of stress shown below If the shear stress is 0 and normal stresses equal to ZOMPa Find the value of normal and shear stress if we rotate the frame of reference by 60 degrees Note that the stresses can be converted into hydrostatic compressive stress and deviatoric components shear stress 0w W SIGN xx CONVENTION L OW O39yy 39 quot O39xx Problem 2 A state of pure shear is one in which the normal stresses are zero Consider a case where the shear stress is 5 MPa Find the values of principal stresses Problem 3 Given normal stress in x direction 3 MPa and in y direction 1 MPa shear stress 2 MPa Find the values of the maximum shear stress Problem 4 Due to geometric constraints it is necessary to find out if the material chosen can sustain required mechanical loading Assume that the material will be under planner stress with Fxx 30 N Fyy 60N and ny25 N The yield stresses of the material are ZOMPa in shear 30MPa in tension and 50MPa in compression With the increase of factor of safety allowed maximal tensile compressive and shear stress decrease proportionally Please find the largest factor of safety at which the material cannot be used and determine whether the material fails due to tension compression or shear Assume that the material will be under planar stress i 39l Fry II D X X II 4 3mm I l l I l l I I 4 3mm gt 3 3 235 Fm gwwww kcmw a mwmgww u g ws m At t mm a w Q mmmmmgmm3wV ma mw may M111 i 1llll 1 Mxmw wu w M ida mvm mw b gav a n5 gv ww tgw v5 311mm wUmmm mmm sing W w st w 1533 5 e f W quot1 39 at 27 7 i 9 PM 0 2 9 J 4999 if 7 7931 H lt 7 32 quotif 3 Sax Aw Em r 93 4quot 539 1 94 ms 2 M H 5w 5 191 i 24539E rB0 655 quot 3 2M 2 gw quot grim lt 2m gigagl l 399 ayg w 0 2 k 3 5 ax W 29 4 m ic S gmww k 5quot quotquotquot C 363 KBiggg m 33km 1 WELMOQQ N A V M A W 7 AW 77 7 7 Mh 7 6034 77 v v Q m a M v Eyw39 Em m 39 245m ifwvmsgx vrr Em mama Pi 3 31 ziz w sifii ng ig ljr 9353f wgfqasw gamma gt m umgva rig gagm V az zgtff a ram m 13 imWQ w 57Wquot rr 2amp5 A 599 O 5 gnaw COQ 39 g 1 Q Sm w max RM 342J k5 527 0 08 7 if 7 Wii 7777 7 ja im gig gter W 77 7 Egg gait 7777 7 7 U A i k i M f i 7quot 05133i M if 53 1 EW 39 aim if j m wquot ig gm A 7 jg i1 4 W i Biomechanics Homework 8 a Problem 1 A1 The elastic modulus for the bar of length l 2cm is 70 GPa Find the elongation of 2 A2 the bar under tensile force 2kN The bar has a cross section of 4mm2 and 2mm i P Problem 2 10 E1 The elastic modulus for the bar of length l 3cm is 70 GPa E1 and 200 GPa E Find the elongation of the bar under tensile force 2kN The bar has a cross section of 8mm2 E2 Problem 3 Assume Youngs Modulus E 16 GPa and Poisson39s raio V 0325 for bone computethe aXial load necessary to casue a strain of 400 111 1w ie microstrain 1 X 10 396 in the normal adult diaphysial region of the femurAssume the diaphysial region to be a cylinder of radius of 12mm Problem 4 A rectangular bar 2 X 2 X 20 cm in dimension is subjected to an aXial force uniform of 4 X 106N Assuming E 100 GPa and v 030 nd Gxx exx eyy all and the deformed dimensions assuming homogenous strains Problem 5 A rectangular bar aluminium bar 15 X 21 cm in cross section and a circular steel rod 1 cm in radius are each subjected to an aXial force of ZOkN Assuming that both are 30cm long in their unloaded configuration find a The stress in each b The extensional strain in each and c The amount of lengthening in each Let E 70 GPa for aluminium and 200 GPa for steel W 1 2m d ZAQMM 6039 L a all Cb V lomm 0 mm era 116 2 3x 03x 039 E jaupq Wquot 033 P GAOkN 640K10311 0122 ouz39 437d0q 392 3Xoquot3 2 9 l3xm393x392 QAZGmm 39 Ehr 8 033 x13gtlto 5 2 O39AXVOB39 OAxm 324o OOotbmm 0394x10 3m 2203 08 lPo Stho 5844 kFCL o47 STA 4r 048x2 0 aPo 03 z E 30 3898 We 82 V1 l 413quot 43 E 9 80 19 O2 A We 02wa 635g ZPQ QR VCH 2L izx o e QU WL 393 E l 3U2v3 Expo qu 1AM 3 c 41844 56023 7 7C 7 7 T 3 Us 4 W s i E 339616XWB W 1556 1590 4 SX0339TL 67 x HZ MPCL out emnaq On 0 2 4 Mm Biomechanics Homework 3 Due 2202012 Problem 1 A long bone is sectioned cut perpendicular to its long axis left figureexposing the forces and moments on the cut plane as shown on the right Identify moments on the cut plane that tend to twist the long bone Problem 2 There is a bone fracture fixation plate with two screws a and a bone fracture fixation plate with 4 screws b Determine the shear stress per screw in each situation For both a and b the weight of the patient is OON The diameter of the screws is 5mm Assume the patient is standing on one leg Q 69 GQ 362 Problem 3 A circular cylindrical rod with radius r 126 cm is tested in a uniagtltial tension test Before applying a tensile force of F 1 N two points A and B that are at a distance to 30 cm gage length are marked on the rod After the force is applied the distance between A and B is measured as it 315 cm Determine the tensile strain and average tensile stress generated in the rod Problem 4 experiment was designed to determine the elastic modulus of the huma bone cortical tissue Three almost Identical bone Clmens were used is shown in Figure 731 which has a square 2 x 2 mm crosssection Two sections A and B are marked on each specimen at a fixed distance apart Each specimen was then subjected to tensile loading of varying magnitudes and the lengths between the marked sections were again measured electronically The following data were obtained Determine the tensile stresses and strains developed in each specimen plot a stress strain diagram for bone and determint the elastic modulus E for the bone Applied Force F N Mea Length sured Ga em 5017 5033 505 ge m Q a mm dd G 2 631 pm we NW a 1 3 24 a A gt a J 392 Rs 7 7 59 6 A 3 gm z r awg i r 7 4 1 ac r g zav x Jr 5 Q mnwmmwmwmmm WWWWmmmmwu M f A xga quot mi 1 532 Clt S 11 DZ P aalm ml o hie I S 1739 M35 2 quot 3 a 391 quotI 4ng UB L73 quot l2 39239 viii Gm 42 a 2 Q gig v 3r V Ei zjfn FW M Wmu 1 mm 6 quot W P E baz EQ g bm 231 p M 13quot Wag9a TU wwxm 15 3 2 i pt w a m w z A ww r Q In lt3 W WWW J I 97 CidL 9 C w W 3ka 34 3 0 4 cmrd i254 mg I A I w a rx mg AL 4 g f3 L 1 J r we w j K 1 quotgr 6 R 63 I A z W f I JO 5 Jr 1 x v X J 5 02quot 1 quotiii L 7 cf MI Uy w acaw a N2 a L w 7 A quot L M I quot quot39 mm 9quot M7 r 2 5 r g a a v fr quot 7 9 W q i39lli ambs l X gr 2 0L M M 7 IVA in 3 391 ng 48 6 f if R K x WW mum 2am am quot 7 Mquot q we 0 L M1Av EMA So xfx bo 0 17 80 3 ix 0 v L a M 1 3 e m ifm u 9 5 1 gt Nb fl 39l l 7 I a C a 12131 W 253 if 9 o a 0 AK 3 L M ML WM y b A T 57 V6 AH 10 k 3951 m7 17x sza quotZglL X 39 7LX 1L v E Fw F WVLZ 3137ltOlt 0 y A l 7 ak H VuFv y3lt L U jg if JrimLvaL mle chBoLba mt V SHE 0 L ya Ll i39w 5quot f D 7 A5quot X X390X Vu7lt Wm 170 Lxs 6me K 97 L j 1 ML MAXV 1 Xg 39O g bl f 39 mEV 1 mm ijgLTVZQDKL XIlyk L J AMA r 391 LAM 3 L w x g L 33 7 it K HIOWVWL 0 1 2L cu Jiltf but h AM 1 0E1 Biomechanics Homework 7 Problem 1 A 2m length of an aluminum pipe of 240mm outer diameter and 10m wall thickness is used as a short column and carries a centric axial load of l 640 kN Knowing that E 73 GPa and V 033 Determine 7 7 i 7 a The change in the length of the pipe b The change in its outer diameter c The change in its wall thickness Problem 2 The gure ABCD is a square piece of skin which is subjected to a biaXial loading as shown It is known that 62 60 and that the change in the length of the plate is the Xdirection must be zero that is 8X 0 Denoting by E the modulus of elasticity and by V Poisson s ration Determine a The required magnitude of 6X 7 b The ratio of 60 82 ForCase l E50kPaV04c70 2kPa Case2 E30kPaV04860 2kPa Problem 3 Assume that the articular cartilage is elastic and isotropic calculate the lame constants 9 u and also the Bulk modulus K The elastic modulus E is given to be 200 KPa and V 02 Problem 4 A uniaxial test is performed on a bone specimen having a central region intially 6 mm long and 2 mm in diameter Five data points were recorded 7 Calculate the Young s Modulus E and yield stress Axial Force N 94 190 284 376 440 Change inLength mm 900E003 180E002 270E002 500E002 940E003 Biomechanics Homework 3 SQHUHQHS Problem 1 A long bone is sectioned cut perpendicular to its long axis left figureexposing the forces and moments on the cut plane as shown on the right Identify moments on the cut plane that tend to twist the long bone ANS MX Problem 2 There is a bone fracture fixation plate with two screws a and a bone fracture fixation plate with 4 screws b Determine the shear stress per screw in each situation For both a and b the weight of the patient is 7OON The diameter of the screws is 5mm Assume the patient is standing on one leg Q 69 GQ 362 During a single leg stance a person can apply hisher entire weight to the ground via a single fool In such situations the total weight of the person is applied back on the person through the same foot which has a compressive effect on the leg its bones and joints In the case ofa patient with a fractured leg bone this force is transferred from below to above the fracture through the screws of fixation device Answex 4700 7 3 Nl395 5 57 6 to m 355 um am for me syerem consisted Dme 111 and iv a serene me amng force on screw 5 F msread ofFJ Yen an da free body dtagmn In check it some smdems made inmates here Note that this problem is reality is much more complex than h39s t I The screws also have to sustain a moment force atthe interface ofthe bone and the plate The finalsolutions are lal 3565 MPa lbl 1785 MPa Problem 3 A circular cylindrical rod with radius r 126 cm is tested in a uniaxial tension test Before applying a tensile force of F 1000 N two points A and B that are at a distance to 30 cm gage length are marked on the rod After the force is applied the distance between A and B is measured as it 315 cm Determine the tensile strain and average tensile stress generated in the rod Answer Tensile 3mm Average Iensile sn ess 2005 105N m1 Problem 4 An experiment was designed to determine the elastic modulus of the human bone cortical tissue Three almost identical bone specimens were prepared The specimen size and shape used is shown in Figure 731 which has a square 2 x 2 mm crosssection Two sections A and B are marked on each specimen at a fixed distance apart Each specimen was then subjected to tensile loading of varying magnitudes and the lengths between the marked sections were again measured electronically The following data were obtained Determine the average elastic modulus E for the bone Applied Force r N Measured Gage Length emm 0 5 240 5017 480 5033 720 505 Note E stressstrain Answer Fazgtlt106 AW EFa2AIx101 1 60 00034 17647 2 120 00066 18182 3 180 001 18 x108 16 15 The average Young s Modulus is 1 4 52213179431 1010P02179430pa 12 y 3 3 Or from the least square tting we have L37 1 quot DH E18025101 Pa1s02SGPa ms u 4 02 quot 4 E 7 a Sham Biomechanics Homework 9 39z in 3i 3 m Problem 1 95 I Find the centroids and moments of inertia for the structure shown alongside in Problem 2 01mm 111mm quotI r I 1 Find the centroids and moments of inertia for the structure shown alongside Problem 3 A hollow LEHI cylinder has an inner radius a 20mm and outer radius c 25mm and length L l m The applied torque T is 750 N m with an angle of twist A0 z L 50 For a hollow cyliner of outer radius 39rO39 and inner radius 39ri39 J H r04 r14 2 Calculate G39Zelmax G39Zzlmax nd the value of shear modulus G and calculate S39Zel max and S39ZZ Problem 4 A solid circular member is to be subjected to an applied torque of 500 N m Find the required diameter of the member so as not to exceed the maximum stress Gze of 75 Mpa 4 L2 Problem 5 33 A solid circular member has a initial diameter of 25 mm which is followed by a solid circular member with diameter 10 mm Given that the torque applied T is A1 A2 1000 N m total length L is 5 cm and shear modulus of the material is 30 GPa nd the angle of twist at at end of the circular member Problem 6 L24 A hollow LEHI cylinder has an inner radius a 20mm and outer radius c 5 K K LT 25mm Given that the torque applied T is 1000 N m total length L is 15 cm A G G i 2 and shear modulus of the material is 30 GPa and 60 GPa nd the angle of twist at at end of the circular member Biomechanics Homework 1 Due 8 am 262012 Question 1 Given 2 vectors a3i2j5k andb2ij Find A ab ab ab axb p090 Question 2 Given 2vectors a3i2jk andb2ij2k Find the angle between the 2 vectors using the definition of a scalar product Question 3 A beam is loaded as shown Pick the appropriate Free Body Diagram from the given choices choices are on the next pagei sco a o z TEE A TED VS 0 i 3 061013 N B 013O VLU 5 0 a Tag C Loleio TTBD SI 0 393 CquotSO3 Question 4 A frame is loaded as shown Pick the appropriate Free Body Diagram from the given choices 200 II sou lh Ey Ex Av me u Ax A 1m It m n 500 lb Aquot c Ex M mu m Ax Question 5 In the example shown below write down the equilibrium condition to compute the tension in rope BE for a 3m 4 quot3 066 6 Low quotL 5190 a sTep quot USi era I Liam M L J 8 Question 6 0 Find the resultant of the system ofthe 3 couples shown in the diagram Biomechanics Sa ut mg Problem 1 Given the 2D state of stress shown below If the shear stress is 0 and normal stresses equal to ZOMPa Find the value of normal and shear stress if we rotate the frame of reference by 60 degrees Note that the stresses can be converted into hydrostatic compressive stress and deviatoric components shear stress 2 9 SIGN xx CONVENTION O39xx L W O39yy Given cm 20 MPa SW 20 MPa and SW 0 MPa and alpha is 60 degrees Substitute these values into the given formulae to get 63039y on L2 a o 22Lc052a 0 511120 sinZa 0 603206 a fm7yy qu 039 Y 2 2 1quot cosZoc o xy sinZa M 20 MPa Uw 20 MPa and ny 0 MPa Problem 2 A state of pure shear is one in which the normal stresses are zero Consider a case where the shear stress is 5 MPa Find the values of principal stresses The formula for Principal Stresses is given below r O39xx ny tv5 2 61 a TQMmm Given m M 0 and CW 5 MPa 61 5 MPa and 62 5 MPa The radical sign gives the Problem 3 Given normal stress in x direction 3 MPa and in y direction 1 MPa shear stress 2 MPa Find the values of the maximum shear stress The formula for Maximum Shear Stresses is given below 2 I 62x 5 2 I maxmm t 2 ny Given Uxx 3 MPa Cw 1 MPa and m 2 MPa vamex 2236 MPa Problem 4 Due to geometric constraints it is necessary to find out if the material chosen can sustain required mechanical loading Assume that the material will be under planner stress with Fxx 30 N Fyy 60N and ny25 N The yield stresses of the material are ZOMPa in shear 30MPa in tension and 50MPa in compression With the increase of factor of safety allowed maximal tensile compressive and shear stress decrease proportionally Please find the largest factor of safety at which the material cannot be used and determine whether the material fails due to tension compression or shear Assume that the material will be under planar stress l Fm 1n 4 3mm 1 a 723 X H I 4 3mm D Given the forces as shown on the cube we can calculate the stresses generated on each cube face Hence we have Gm 309 MPa 6W 609 MPa and M 259 MPa The maxmin principal stresses and maximum shear stress for the given system is 2 I 6x 031 an 039 2 0391 a Oahumm 2 i 2 Cry I dxy maxmiu t Thus the we have m 739 MPa and 62 405 MPa 6mm 572MPa Usingthese values against the yield faiure stress values given in the problem we have safety factor for shearto be 35 which is the lowest Hence that is the largest safety factor one can use forthe material block If a safety factor of gt 35 is used the block will fail in shear loading Biomechanics Homework 6 Problem 1 Given uX A 1X and uy 0 please compute the approximate linear a strain for A 11 If the system has an imaginary cut making a 60 angle with the horizontal What would be the resultant strains sxx syy sxy Problem 2 u Let uX 03X 01Y uy 009Y Compute the values of the components of the 2D Green nite strain EXX Eyy and Exy and the linearized strain sxx eyy sxy Problem 3 Consider the motion X X X1 ke1 Let Xm dSl 5 3e1 4 e2 and dX2 dSz 5 4e1 3e2 be two unit perpendicular material elements in the undeformed con guration a Find the deformed elements dx1 and dxz b Evaluate the stretches of these elements ie ds1 dS1 and ds2 dSz and the angle change between dx1 and dxz f I Problem 4 For a given 30 7 60 7 120 degree y xquot strain rosette the 3 extensional strain f g measurements sxx sxx sxx are P x measured to be 009 0075 and 004 1 Flnd the S Syx and Sxy f strain gauge placements 2 Calculate principal strains and their directions maXimum shear strain and its direction Biomechanics Homework 1 SQHWEEQH Question 1 Given 2 vectors a3i2j5k andb2ij Find A ab5i3j5k ab i j5k ab 8 axb 5i 10jk p090 Question 2 Given 2vectors a3i2jk andb2ij2k Find the angle between the 2 vectors using the definition of a scalar product ab axbcostheta costheta ab a x b theta 27 degrees A beam is loaded as shown Pick the appropriate Free Body Diagram from the given choices choices are on the next pagei Question 3 2 1 ll 081919 Tee B Question 4 5quot H Aframe is loaded as shown Pick the appropriate Free Body Diagram 20 from the given choices E 500 lb y 100 ll Question 5 In the example shown below write down the equilibrium condition to compute the tension in rope BE for a 3m 4 quot3 066 6 Low quotL 5190 a sTep quot USi era I Liam M L J 8 Question 5 Solution F 621 7 J 65 3 62N F455 15 3 3 T BD BDKTJ 1 2 2N r 15 33 T TEE TBEK 45 J 1 22N ZMA Z CAX12 EBAXFEBD gBAXBE 0 H i k i k T i 65 0 6 0 0 3 0 i 0 3 0 0 3 6 2 1 2 2 1 2 2 Using your favorite method 780 2TBD 2TBE 1170 TED TEEk 0 2 MAX 2 0 780 2TBD 2TBE 0 a Z MAZ 0 1170 TED TEE 0 b Multiplying b by 2 and adding to a 4TBD 3120 gt TED 780N 260i 520 j 520kN TEE 390N 1301 260 j 260kN Question 6 Solution Co crr mka of we Pdmb A u t 3 so The inuamppqL ng Jul Drs Use Me M39 Fjs FL0 3 51 471 F 33 43 5 0 E Lolqls D r 5 AA km en39th roams new lt b kuonnn l 39SOrce varies a 0390 3 CPo m o cw Po mt F L 009 L0 65L Aamp39 S H F quot 3053 3905 M20 o39kx u 39 quot Sui4 7250an 5 Mn quot Q 3 Tag J7 H2 T410 Fhs rrqw l e w Lse Mcn39 M 683 J34 M15 7 b k 3 350 g u m 3 729 V H H 4 HF J Hw HLIWfw V w m VJ MWUWTW TV w lw mw PVLVP 4gtwmmmn x FGVEW FQQU L 3V 0 mgw a8 4 cm 3 Lun nmv ZEAES V313 4 693 imaged 6 6x 6 u 9 97 wmmpww b M 3m 5 i Rm g g 2 0 E i g i J a 4 f W km 301 rm I T T TA I 1quotwilmpmb M i J 4 mm of 3 a n3 Li Y M gag 353k 4 3f 253quot 5 m A a Vie Ni 14mph bf J r 4111 W o

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