1.2Stresses in the Members of a Structure19SampleProblem1.2The steel tie bar shown is to be designed to carry a tension force ofmagnitude P=120kNwhen bolted between doublebrackets atAandB.Thebarwillbefabricatedfrom20-mm-thickplatestock.Forthegrade of steel to be used, the maximum allowable stresses areg=175MPa,T=100MPa,andh=350MPa.Designthetiebarbydetermining the required values of (a)the diameterd of thebolt, (b)thedimension b at each end of the bar, and (c) the dimension h of the bar.STRATEGY:Use free-body diagrams to determine the forces neededtoobtainthestresses intermsofthedesigntensionforce.Settingthesestresses equal to theallowable stresses providesforthedeterminationoftherequireddimensions.MODELINGandANALYSIS:a.Diameter of the Bolt.Since the boltis in double shear (Fig.1)F=PF,=P=60kN.Fig. 1 Sectioned bolt.Fr-60kN60kNt=20mm100MPad=27.6mmindAldUsed=28mmAtthis point,check thebearing stress betweenthe20-mm-thickplate(Fig.2)andthe28-mm-diameterboltP120kNOK=214MPa<350MPaObtd(0.020m)(0.028m)Fig.2Tie bar geometry.b.Dimension bat Each Endof the Bar.We consider one of theend portionsof thebar inFig.3.Recalling thatthethickness of thesteel plate is t=20 mm and that the average tensile stress must notexceed175MPa,writeP60kN=120KN=175MPa=a=17.14mmota(0.02m)a1b=d+2a=28mm+2(17.14mm)b=62.3mmC.Dimensionhofthe Bar.Weconsider a section inthe centralFig. 3End section of tie bar.portion of the bar (Fig.4).Recalling that the thickness of the steel platet=20mmist=20mm,wehaveP120kN175MPah=34.3mmth(0.020 m)hUseh=35mm=120KNREFLECT and THINK:We sized d based on boltshear,andthenchecked bearing on the tie bar.Had the maximum allowable bearingFig.4Mid-body section oftiebar.stressbeen exceeded,wewould havehad torecalculate d based onthe bearing criterion
1.2 Stresses in the Members of a Structure 19 REFLECT and THINK: We sized d based on bolt shear, and then checked bearing on the tie bar. Had the maximum allowable bearing stress been exceeded, we would have had to recalculate d based on the bearing criterion. Sample Problem 1.2 The steel tie bar shown is to be designed to carry a tension force of magnitude P 5 120 kN when bolted between double brackets at A and B. The bar will be fabricated from 20-mm-thick plate stock. For the grade of steel to be used, the maximum allowable stresses are s 5 175 MPa, t 5 100 MPa, and sb 5 350 MPa. Design the tie bar by determining the required values of (a) the diameter d of the bolt, (b) the dimension b at each end of the bar, and (c) the dimension h of the bar. STRATEGY: Use free-body diagrams to determine the forces needed to obtain the stresses in terms of the design tension force. Setting these stresses equal to the allowable stresses provides for the determination of the required dimensions. MODELING and ANALYSIS: a. Diameter of the Bolt. Since the bolt is in double shear (Fig. 1), F1 5 1 2 P 5 60 kN. t 5 F1 A 5 60 kN 1 4p d2 100 MPa 5 60 kN 1 4p d2 d 5 27.6 mm Use d 5 28 mm ◀ At this point, check the bearing stress between the 20-mm-thick plate (Fig. 2) and the 28-mm-diameter bolt. sb 5 P td 5 120 kN 10.020 m210.028 m2 5 214 MPa , 350 MPa OK b. Dimension b at Each End of the Bar. We consider one of the end portions of the bar in Fig. 3. Recalling that the thickness of the steel plate is t 5 20 mm and that the average tensile stress must not exceed 175 MPa, write s 5 1 2P ta 175 MPa 5 60 kN 10.02 m2a a 5 17.14 mm b 5 d 1 2a 5 28 mm 1 2(17.14 mm) b 5 62.3 mm ◀ c. Dimension h of the Bar. We consider a section in the central portion of the bar (Fig. 4). Recalling that the thickness of the steel plate is t 5 20 mm, we have s 5 P th 175 MPa 5 120 kN 10.020 m2h h 5 34.3 mm Use h 5 35 mm ◀ A B d F1 P P F1 F1 1 2 b h t 5 20 mm d P P' 120 kN a t a d b 1 2 P1 2 P 5 120 kN t 5 20 mm h Fig. 1 Sectioned bolt. Fig. 2 Tie bar geometry. Fig. 3 End section of tie bar. Fig. 4 Mid-body section of tie bar. bee98233_ch01_002-053.indd 19 11/15/13 9:42 AM
Problems1.1Twosolidcylindrical rodsABandBCareweldedtogetheratBandloadedasshown.Knowingthatd,=30mmandd,=50mmfind the average normal stress at the midsection of (a)rod AB,(b) rodBC.125KNB60KN125N0.9m1.2mFig.P1.1and P1.21.2Twosolid cylindricalrodsABandBCarewelded togetheratBandloadedasshown.Knowingthattheaveragenormalstressmustnotexceed 150MPa in either rod,determine the smallest allowablevaluesofthediametersdiandd1.3Twosolidcylindrical rodsABandBCareweldedtogetheratBandloaded as shown.Knowing thatP=1okips,find the averagenor-mal stress at themidsection of (a)rodAB, (b)rodBC.30in-1.25in.B12kips25in0.75inFig.P1.3and P1.41.4Twosolidcylindrical rodsABandBCareweldedtogetheratBand loaded as shown.Determine the magnitude of the force Pforwhichthetensilestresses inrodsABand BCareequal.20
20 1.1 Two solid cylindrical rods AB and BC are welded together at B and loaded as shown. Knowing that d1 5 30 mm and d2 5 50 mm, find the average normal stress at the midsection of (a) rod AB, (b) rod BC. Problems Fig. P1.1 and P1.2 d1 d2 125 kN 125 kN 60 kN C A B 0.9 m 1.2 m 1.3 Two solid cylindrical rods AB and BC are welded together at B and loaded as shown. Knowing that P 5 10 kips, find the average normal stress at the midsection of (a) rod AB, (b) rod BC. Fig. P1.3 and P1.4 0.75 in. 1.25 in. 12 kips P B C 25 in. 30 in. A 1.2 Two solid cylindrical rods AB and BC are welded together at B and loaded as shown. Knowing that the average normal stress must not exceed 150 MPa in either rod, determine the smallest allowable values of the diameters d1 and d2. 1.4 Two solid cylindrical rods AB and BC are welded together at B and loaded as shown. Determine the magnitude of the force P for which the tensile stresses in rods AB and BC are equal. bee98233_ch01_002-053.indd 20 11/15/13 9:42 AM
1.5A strain gage located at Con the surface of boneAB indicates that1200Nthe average normal stress in the bone is 3.8o MPa when the boneis subjected to two1200-Nforces as shown.Assumingthecrosssection of the bone at Cto be annular and knowing that its outerdiameter is25 mm, determine the inner diameter of the bone'scrosssectionatC.1.6TwobrassrodsABandBC,eachofuniformdiameter,will beCbrazed together at B to form a nonuniform rod of total length1o0 m thatwill be suspended from a supportatA as shown.Knowingthat thedensityof brassis8470kg/m,determine(a)thelengthofrodABforwhichthemaximumnormalstressinABCisminimum,(b)thecorrespondingvalueof themaximumnormal stress.51200NFig.P1.5-15mm100m10mmFig.P1.61.7Eachofthefourverticallinkshasan8X36-mmuniformrectan-gularcrosssection,andeachofthefourpinshasal6-mmdiameter.Determinethe maximumvalue oftheaveragenormal stress in thelinksconnecting(a)pointsBandD,(b)pointsCandE0.4m0.25m02120kFig.P1.721
21 1.5 A strain gage located at C on the surface of bone AB indicates that the average normal stress in the bone is 3.80 MPa when the bone is subjected to two 1200-N forces as shown. Assuming the cross section of the bone at C to be annular and knowing that its outer diameter is 25 mm, determine the inner diameter of the bone’s cross section at C. 1.6 Two brass rods AB and BC, each of uniform diameter, will be brazed together at B to form a nonuniform rod of total length 100 m that will be suspended from a support at A as shown. Knowing that the density of brass is 8470 kg/m3 , determine (a) the length of rod AB for which the maximum normal stress in ABC is minimum, (b) the corresponding value of the maximum normal stress. Fig. P1.5 1200 N 1200 N C A B Fig. P1.6 100 m 15 mm 10 mm b a B C A 1.7 Each of the four vertical links has an 8 3 36-mm uniform rectangular cross section, and each of the four pins has a 16-mm diameter. Determine the maximum value of the average normal stress in the links connecting (a) points B and D, (b) points C and E. Fig. P1.7 0.2 m 0.25 m 0.4 m 20 kN C B A D E bee98233_ch01_002-053.indd 21 11/15/13 9:42 AM
1.8 Link AC has a uniform rectangular cross sectionin,thick and1BO2im1 in.wide.Determinethe normal stressin the central portion ofthe link.T12in.1.9 Three forces, each of magnitude P = 4kN, are applied to the1201bstructure shown.Determine the cross-sectional area of the uni-form portionofrodBEforwhichthenormalstress inthatportion1201b4in.,Ais+100MPa.30c-10in-8m.-0.100mFig.P1.8a1IBC140.150m0.300m0.250mFig.P1.91.10 Link BD consists ofa single bar I in.wide and in.thick.Knowingthat each pin has a g-in.diameter, determine themaximum valueof the average normal stress in link BD if (a)=0, (b) =90°4kipsC16-6in.FO12in30°DOFig.P1.101.11 ForthePrattbridgetruss and loading shown,determine theaver-age normal stress in member BE,knowingthatthe cross-sectionalareaofthatmemberis5.87in2DH12ftAC-9ft--9ft--9ft--9ft-sokipssokips80kipsFig.P1.7122
22 1.8 Link AC has a uniform rectangular cross section 1 8 in. thick and 1 in. wide. Determine the normal stress in the central portion of the link. 1.9 Three forces, each of magnitude P 5 4 kN, are applied to the structure shown. Determine the cross-sectional area of the uniform portion of rod BE for which the normal stress in that portion is 1100 MPa. 10 in. 8 in. 2 in. 12 in. 4 in. 30 120 lb 120 lb C A B Fig. P1.8 Fig. P1.9 0.100 m 0.150 m 0.300 m 0.250 m P P P E A B C D Fig. P1.10 4 kips 308 u 6 in. 12 in. D C B A 1.10 Link BD consists of a single bar 1 in. wide and 1 2 in. thick. Knowing that each pin has a 3 8-in. diameter, determine the maximum value of the average normal stress in link BD if (a) u 5 0, (b) u 5 908. Fig. P1.11 9 ft 80 kips 80 kips 80 kips 9 ft 9 ft 9 ft 12 ft B D F H C E G A 1.11 For the Pratt bridge truss and loading shown, determine the average normal stress in member BE, knowing that the cross-sectional area of that member is 5.87 in2 . bee98233_ch01_002-053.indd 22 11/15/13 9:42 AM
1.12Theframeshownconsistsoffourwoodenmembers,ABC,DEF,BE, and CF.Knowing that each member has a 2 × 4-in. rectan-gular cross section and that each pin has a 2-in. diameter, deter-mine the maximum value of the average normal stress(a)inmemberBE,(b)inmemberCF.45in.30inAFBCOO4801b4n.4in.40 in.OOIEHDl-15in.-30in.Fig.P1.121.13 An aircraft towbar is positioned bymeans of a single hydrauliccylinderconnectedbya25-mm-diametersteelrodtotwoidentical arm-and-wheel units DEF.Themass of the entiretow bar is20okg,and its center of gravity is located at G.For the positionshown,determinethenormal stressintherodDimensions inmm15610064500F250E850500675825Fig.P1.131.14 Twohydrauliccylindersareusedto controlthepositionoftherobotic arm ABC. Knowing that the control rods attached at AandDeachhavea20-mmdiameterandhappentobeparallel inthe position shown, determine the average normal stress in(a)memberAE,(b)memberDG150mmSOON-300mm-600mmCOBo400mmOD150mm200mmFig.P1.1423
23 1.12 The frame shown consists of four wooden members, ABC, DEF, BE, and CF. Knowing that each member has a 2 3 4-in. rectangular cross section and that each pin has a 1 2-in. diameter, determine the maximum value of the average normal stress (a) in member BE, (b) in member CF. Fig. P1.12 40 in. 45 in. 15 in. 4 in. A B C D E F 4 in. 30 in. 30 in. 480 lb 1.13 An aircraft tow bar is positioned by means of a single hydraulic cylinder connected by a 25-mm-diameter steel rod to two identical arm-and-wheel units DEF. The mass of the entire tow bar is 200 kg, and its center of gravity is located at G. For the position shown, determine the normal stress in the rod. Fig. P1.13 D B E A Dimensions in mm 100 450 250 850 1150 500 675 825 G C F 1.14 Two hydraulic cylinders are used to control the position of the robotic arm ABC. Knowing that the control rods attached at A and D each have a 20-mm diameter and happen to be parallel in the position shown, determine the average normal stress in (a) member AE, (b) member DG. Fig. P1.14 A C B EF G D 150 mm 200 mm 150 mm 300 mm 400 mm 600 mm 800 N bee98233_ch01_002-053.indd 23 11/15/13 9:42 AM