Statically Indeterminate Problems Problems Involving Two .
Statically Indeterminate ProblemsandProblems Involving Two Materials(Strength of Materials)Dave Morgan dave.morgan@sait.ca Statically Indeterminate Problems and Problems Involving Two Materials – p. 1/30
Statically Indeterminate ProblemsWe have used the laws of statics to analyseproblems such as the one illustrated:A10 kNBC30 kNStatically Indeterminate Problems and Problems Involving Two Materials – p. 2/30
Statically Indeterminate ProblemsWe have used the laws of statics to analyseproblems such as the one illustrated:20 kNAΣFy 0, so there is a reaction force of20 kN at A10 kNBC30 kNStatically Indeterminate Problems and Problems Involving Two Materials – p. 2/30
Statically Indeterminate ProblemsWe have used the laws of statics to analyseproblems such as the one illustrated:20 kNA10 kNBΣFy 0, so there is a reaction force of20 kN at ATo find the internal forces in the segmentBC:C30 kNStatically Indeterminate Problems and Problems Involving Two Materials – p. 2/30
Statically Indeterminate ProblemsWe have used the laws of statics to analyseproblems such as the one illustrated:20 kNAΣFy 0, so there is a reaction force of20 kN at ABTo find the internal forces in the segmentBC:Insert a section M-M throughsegment BC10 kNMMC30 kNStatically Indeterminate Problems and Problems Involving Two Materials – p. 2/30
Statically Indeterminate ProblemsWe have used the laws of statics to analyseproblems such as the one illustrated:20 kNΣFy 0, so there is a reaction force of20 kN at AA10 kNBMMTo find the internal forces in the segmentBC:Insert a section M-M throughsegment BCConsider only the segment from C toM-MC30 kNStatically Indeterminate Problems and Problems Involving Two Materials – p. 2/30
Statically Indeterminate ProblemsWe have used the laws of statics to analyseproblems such as the one illustrated:20 kNΣFy 0, so there is a reaction force of20 kN at AA10 kNB30 kNMMC30 kNTo find the internal forces in the segmentBC:Insert a section M-M throughsegment BCConsider only the segment from C toM-MΣFy 0, so there is an internal forceof 30 kN at MStatically Indeterminate Problems and Problems Involving Two Materials – p. 2/30
Statically Indeterminate ProblemsWe have used the laws of statics to analyseproblems such as the one illustrated:20 kNΣFy 0, so there is a reaction force of20 kN at AA10 kNB30 kNMMC30 kNTo find the internal forces in the segmentBC:Insert a section M-M throughsegment BCConsider only the segment from C toM-MΣFy 0, so there is an internal forceof 30 kN at MTBC 30 kN (tension)Statically Indeterminate Problems and Problems Involving Two Materials – p. 2/30
Statically Indeterminate Problems20 kNASimilarly, we can find the internal forcewithin segment AB:10 kNBC30 kNStatically Indeterminate Problems and Problems Involving Two Materials – p. 3/30
Statically Indeterminate ProblemsSimilarly, we can find the internal forcewithin segment AB:20 kNANN10 kNInsert a section N-N through segmentABBC30 kNStatically Indeterminate Problems and Problems Involving Two Materials – p. 3/30
Statically Indeterminate ProblemsSimilarly, we can find the internal forcewithin segment AB:20 kNANN10 kNBInsert a section N-N through segmentABConsider only the segment from A toN-NC30 kNStatically Indeterminate Problems and Problems Involving Two Materials – p. 3/30
Statically Indeterminate ProblemsSimilarly, we can find the internal forcewithin segment AB:20 kNANN20 kN10 kNBCInsert a section N-N through segmentABConsider only the segment from A toN-NΣFy 0, so there is an internal forceof 20 kN at N-N30 kNStatically Indeterminate Problems and Problems Involving Two Materials – p. 3/30
Statically Indeterminate ProblemsSimilarly, we can find the internal forcewithin segment AB:20 kNANN20 kN10 kNBC30 kNInsert a section N-N through segmentABConsider only the segment from A toN-NΣFy 0, so there is an internal forceof 20 kN at N-NTAB 20 kN (tension)Statically Indeterminate Problems and Problems Involving Two Materials – p. 3/30
Statically Indeterminate ProblemsStructures where forces can be determined using thestatic equilibrium equations alone (ΣFx 0,ΣFy 0 and ΣMA 0) are called staticallydeterminate structures. The previous example is astatically determinate structure.Statically Indeterminate Problems and Problems Involving Two Materials – p. 4/30
Statically Indeterminate ProblemsStructures where forces can be determined using thestatic equilibrium equations alone (ΣFx 0,ΣFy 0 and ΣMA 0) are called staticallydeterminate structures. The previous example is astatically determinate structure.Structures where the forces cannot be determined inthis way are called statically indeterminatestructures.Statically Indeterminate Problems and Problems Involving Two Materials – p. 4/30
Statically Indeterminate ProblemsStructures where forces can be determined using thestatic equilibrium equations alone (ΣFx 0,ΣFy 0 and ΣMA 0) are called staticallydeterminate structures. The previous example is astatically determinate structure.Structures where the forces cannot be determined inthis way are called statically indeterminatestructures.Statically indeterminate structures are often analysedusing the conditions of axial deformation given byPLδ AEStatically Indeterminate Problems and Problems Involving Two Materials – p. 4/30
Statically Indeterminate ProblemsExample: Consider a bar AB supported at bothends by fixed supports, with an axial force of 12 kNapplied at C as illustrated. Find the reactions at thewalls500 mmA12 kN400 mmCBStatically Indeterminate Problems and Problems Involving Two Materials – p. 5/30
Statically Indeterminate ProblemsExample: Consider a bar AB supported at bothends by fixed supports, with an axial force of 12 kNapplied at C as illustrated. Find the reactions at thewalls500 mmASolution:12 kN400 mmCBFirst, draw a free body diagram:Statically Indeterminate Problems and Problems Involving Two Materials – p. 5/30
Statically Indeterminate ProblemsExample: Consider a bar AB supported at bothends by fixed supports, with an axial force of 12 kNapplied at C as illustrated. Find the reactions at thewalls500 mmASolution:12 kN400 mmFirst, draw a free body diagram:500 mmRABCA12 kN400 mmCBRBStatically Indeterminate Problems and Problems Involving Two Materials – p. 5/30
Statically Indeterminate Problems500 mmRAA12 kN400 mmCBRBStatically Indeterminate Problems and Problems Involving Two Materials – p. 6/30
Statically Indeterminate Problems500 mmRAAΣFx12 kN400 mmCBRB RA RB 12 0Statically Indeterminate Problems and Problems Involving Two Materials – p. 6/30
Statically Indeterminate Problems500 mmRAAΣFx 12 kN400 mmCBRB RA RB 12 0RA RB 12Statically Indeterminate Problems and Problems Involving Two Materials – p. 6/30
Statically Indeterminate Problems500 mmRAA12 kN400 mmCBRB RA RB 12 0 RA RB 12Two unknowns and a single equation; the problem isstatically indeterminate.ΣFxStatically Indeterminate Problems and Problems Involving Two Materials – p. 6/30
Statically Indeterminate Problems500 mmRAA12 kN400 mmCBRB RA RB 12 0 RA RB 12Two unknowns and a single equation; the problem isstatically indeterminate.The supports at A and B are fixed so δAB 0.ΣFxStatically Indeterminate Problems and Problems Involving Two Materials – p. 6/30
Statically Indeterminate Problems500 mmRAA12 kN400 mmCBRB RA RB 12 0 RA RB 12Two unknowns and a single equation; the problem isstatically indeterminate.The supports at A and B are fixed so δAB 0. δAC δCB 0ΣFxStatically Indeterminate Problems and Problems Involving Two Materials – p. 6/30
Statically Indeterminate Problems500 mmRAA12 kN400 mmCBRB RA RB 12 0 RA RB 12Two unknowns and a single equation; the problem isstatically indeterminate.The supports at A and B are fixed so δAB 0. δAC δCB 0 400A 500 RBAE 0 RAEΣFxStatically Indeterminate Problems and Problems Involving Two Materials – p. 6/30
Statically Indeterminate Problems500 mmRAA12 kN400 mmCBRB RA RB 12 0 RA RB 12Two unknowns and a single equation; the problem isstatically indeterminate.The supports at A and B are fixed so δAB 0. δAC δCB 0 400A 500 RBAE 0 RAE 400RB 500RAΣFxStatically Indeterminate Problems and Problems Involving Two Materials – p. 6/30
Statically Indeterminate Problems500 mmRAA12 kN400 mmCBRB RA RB 12 0 RA RB 12Two unknowns and a single equation; the problem isstatically indeterminate.The supports at A and B are fixed so δAB 0. δAC δCB 0 400A 500 RBAE 0 RAE 400RB 500RANow we have two equations and two unknowns; we cansolve for RA and RBΣFxStatically Indeterminate Problems and Problems Involving Two Materials – p. 6/30
Statically Indeterminate ProblemsRA RB 12400RB 500RAStatically Indeterminate Problems and Problems Involving Two Materials – p. 7/30
Statically Indeterminate ProblemsRA RB 12400RB 500RA RA 12 RBStatically Indeterminate Problems and Problems Involving Two Materials – p. 7/30
Statically Indeterminate ProblemsRA RB 12400RB 500RA RA 12 RB400RB 500 (12 RB )Statically Indeterminate Problems and Problems Involving Two Materials – p. 7/30
Statically Indeterminate ProblemsRA RB 12400RB 500RA RA 12 RB 400RB 500 (12 RB ) 900RB 6000Statically Indeterminate Problems and Problems Involving Two Materials – p. 7/30
Statically Indeterminate ProblemsRA RB 12400RB 500RA RA 12 RB 400RB 500 (12 RB ) 900RB 6000 RB 6.667 kNStatically Indeterminate Problems and Problems Involving Two Materials – p. 7/30
Statically Indeterminate ProblemsRA RB 12400RB 500RA RA 12 RB 400RB 500 (12 RB ) 900RB 6000 RB 6.667 kN RA 5.333 kNStatically Indeterminate Problems and Problems Involving Two Materials – p. 7/30
Statically Indeterminate Problems500 mm400 mm6.667 kN5.333 kNA12 kNCBStatically Indeterminate Problems and Problems Involving Two Materials – p. 8/30
Statically Indeterminate ProblemsExercise: Find RA and RB for the problem illustrated:baABCPStatically Indeterminate Problems and Problems Involving Two Materials – p. 9/30
Statically Indeterminate ProblemsExercise: Find RA and RB for the problem illustrated:baABCPSolution: Draw free body diagramStatically Indeterminate Problems and Problems Involving Two Materials – p. 9/30
Statically Indeterminate ProblemsExercise: Find RA and RB for the problem illustrated:baABCPSolution: Draw free body diagramaARAbBCPRBStatically Indeterminate Problems and Problems Involving Two Materials – p. 9/30
Statically Indeterminate ProblemsaARAbBCPRBSolution:Statically Indeterminate Problems and Problems Involving Two Materials – p. 10/30
Statically Indeterminate ProblemsaAbBCRASolution:PRA RB P RB0Statically Indeterminate Problems and Problems Involving Two Materials – p. 10/30
Statically Indeterminate ProblemsabABCRAPSolution: RA RB P 0RA P RBRBStatically Indeterminate Problems and Problems Involving Two Materials – p. 10/30
Statically Indeterminate ProblemsabABCRAPSolution: RA RB P 0RA P RBδAC δCB 0RBStatically Indeterminate Problems and Problems Involving Two Materials – p. 10/30
Statically Indeterminate ProblemsabABCRAPSolution:RA RB P 0RA P RBδAC δCB 0 0 RA aAE RB bAERBStatically Indeterminate Problems and Problems Involving Two Materials – p. 10/30
Statically Indeterminate ProblemsabABCRAPSolution:RA RB P 0RA P RBδAC δCB 0 RB bAE 0aRA bRB RA aAE RBStatically Indeterminate Problems and Problems Involving Two Materials – p. 10/30
Statically Indeterminate ProblemsabABCRAPRA RB P 0RA P RBδAC δCB 0 RB bAE 0 aRA bRB a (P RB ) bRBSolution: RA aAE RBStatically Indeterminate Problems and Problems Involving Two Materials – p. 10/30
Statically Indeterminate ProblemsabABCRAPRA RB P 0RA P RBδAC δCB 0 RB bAE 0 aRA bRB a (P RB ) bRB aP (a b) RBSolution: RA aAE RBStatically Indeterminate Problems and Problems Involving Two Materials – p. 10/30
Statically Indeterminate ProblemsabABCRAPRA RB P 0RA P RBδAC δCB 0 RB bAE 0 aRA bRB a (P RB ) bRB aP RB (a b) RB aa b PSolution: RA aAE RBStatically Indeterminate Problems and Problems Involving Two Materials – p. 10/30
Statically Indeterminate ProblemsabABCRAPRA RB P 0RA P RBδAC δCB 0 RB bAE 0 aRA bRB a (P RB ) bRB aP RB RA (a b) RB aa b P ba b PSolution: RA aAE RBStatically Indeterminate Problems and Problems Involving Two Materials – p. 10/30
Problems Involving Two MaterialsSteel-reinforced concrete is used in the construction ofmany structures:BridgesBasementsHigh-Rise BuildingsStadia, such as the SaddleDome or theSpeed-Skating OvalStatically Indeterminate Problems and Problems Involving Two Materials – p. 11/30
Problems Involving Two MaterialsConcrete has a high load-bearing capacity incompression but is not very strong under a tensileload.Statically Indeterminate Problems and Problems Involving Two Materials – p. 12/30
Problems Involving Two MaterialsConcrete has a high load-bearing capacity incompression but is not very strong under a tensileload.Steel rod has high load-bearing capacity in tensionbut buckles easily under compression.Statically Indeterminate Problems and Problems Involving Two Materials – p. 12/30
Problems Involving Two MaterialsConcrete has a high load-bearing capacity incompression but is not very strong under a tensileload.Steel rod has high load-bearing capacity in tensionbut buckles easily under compression.Combining steel rod and concrete gives a buildingmaterial with both good tensile and compressiveload-bearing qualities.Statically Indeterminate Problems and Problems Involving Two Materials – p. 12/30
Problems Involving Two MaterialsSteel in a concrete column also helps the concrete’scompressive strength:Statically Indeterminate Problems and Problems Involving Two Materials – p. 13/30
Problems Involving Two MaterialsSteel in a concrete column also helps the concrete’scompressive strength: PLWhen a column is loaded, it deforms δ AEStatically Indeterminate Problems and Problems Involving Two Materials – p. 13/30
Problems Involving Two MaterialsSteel in a concrete column also helps the concrete’scompressive strength: PLWhen a column is loaded, it deforms δ AEUnder compression, δ is negative and there isnegative axial strainStatically Indeterminate Problems and Problems Involving Two Materials – p. 13/30
Problems Involving Two MaterialsSteel in a concrete column also helps the concrete’scompressive strength: PLWhen a column is loaded, it deforms δ AEUnder compression, δ is negative and there isnegative axial strainConsequently, there is a positive transverse strain(ǫt µǫa )Statically Indeterminate Problems and Problems Involving Two Materials – p. 13/30
Problems Involving Two MaterialsSteel in a concrete column also helps the concrete’scompressive strength: PLWhen a column is loaded, it deforms δ AEUnder compression, δ is negative and there isnegative axial strainConsequently, there is a positive transverse strain(ǫt µǫa )The concrete is under tension laterallyStatically Indeterminate Problems and Problems Involving Two Materials – p. 13/30
Problems Involving Two MaterialsSteel in a concrete column also helps the concrete’scompressive strength: PLWhen a column is loaded, it deforms δ AEUnder compression, δ is negative and there isnegative axial strainConsequently, there is a positive transverse strain(ǫt µǫa )The concrete is under tension laterallyHorizontal steel-reinforcing increases the lateraltensile strength of the columnStatically Indeterminate Problems and Problems Involving Two Materials – p. 13/30
Problems Involving Two MaterialsA concrete footing ispoured:It contains steel rebarthroughoutSteel extrudes fromthe top of the footingThis will be attachedto the steel for thecolumn.Statically Indeterminate Problems and Problems Involving Two Materials – p. 14/30
Problems Involving Two MaterialsSteel is tied for the columnStatically Indeterminate Problems and Problems Involving Two Materials – p. 15/30
Problems Involving Two MaterialsA frame is built around thesteel and the concretecolumn is pouredStatically Indeterminate Problems and Problems Involving Two Materials – p. 16/30
Problems Involving Two MaterialsStatically Indeterminate Problems and Problems Involving Two Materials – p. 17/30
Problems Involving Two MaterialsPC ·LCAC ·ECto calculate the deformationWe can use δC of concrete under a loadStatically Indeterminate Problems and Problems Involving Two Materials – p. 18/30
Problems Involving Two MaterialsPC ·LCAC ·ECto calculate the deformationWe can use δC of concrete under a loadSWe can use δS APSS ·L·ES to calculate the deformationof steel under a loadStatically Indeterminate Problems and Problems Involving Two Materials – p. 18/30
Problems Involving Two MaterialsPC ·LCAC ·ECto calculate the deformationWe can use δC of concrete under a loadSWe can use δS APSS ·L·ES to calculate the deformationof steel under a loadHow can we calculate the deformation of asteel-reinforced concrete column?Statically Indeterminate Problems and Problems Involving Two Materials – p. 18/30
Problems Involving Two MaterialsPC ·LCAC ·ECto calculate the deformationWe can use δC of concrete under a loadSWe can use δS APSS ·L·ES to calculate the deformationof steel under a loadHow can we calculate the deformation of asteel-reinforced concrete column?EC is not the same as ES so we cannot simplyPLapply δ AEfor the whole columnStatically Indeterminate Problems and Problems Involving Two Materials – p. 18/30
Problems Involving Two MaterialsPC ·LCAC ·ECto calculate the deformationWe can use δC of concrete under a loadSWe can use δS APSS ·L·ES to calculate the deformationof steel under a loadHow can we calculate the deformation of asteel-reinforced concrete column?EC is not the same as ES so we cannot simplyPLapply δ AEfor the whole columnWe cannot solve this problem directly using theequations of statics, so this is astatically-indeterminate problemStatically Indeterminate Problems and Problems Involving Two Materials – p. 18/30
Problems Involving Two MaterialsExample:A concrete column has adiameter of 300 mm. The column has 6 steelreinforcing rods.Each rod has a cross-sectional area of200 mm2 . (See plan view)ES 210 GPa and EC 25 GPaThe column is 1.15 m long and has a load of1.37 MN is applied to a rigid steel plate atthe top of the column (the plate distributesthe load evenly over the top of the column).D 300 mmA 200 mm21.37 MN1.15 mStatically Indeterminate Problems and Problems Involving Two Materials – p. 19/30
Problems Involving Two MaterialsExample:A concrete column has adiameter of 300 mm. The column has 6 steelreinforcing rods.Each rod has a cross-sectional area of200 mm2 . (See plan view)ES 210 GPa and EC 25 GPaThe column is 1.15 m long and has a load of1.37 MN is applied to a rigid steel plate atthe top of the column (the plate distributesthe load evenly over the top of the column).D 300 mmA 200 mm21.37 MN1.15 mFind the stress in the steel and in theconcrete, and the deformation under the loadStatically Indeterminate Problems and Problems Involving Two Materials – p. 19/30
Problems Involving Two MaterialsSolution: Let PS be the total reaction force of thesix steel rods and PC the reaction force of theconcrete.1370 kNPS6PS6PCStatically Indeterminate Problems and Problems Involving Two Materials – p. 20/30
Problems Involving Two MaterialsSolution: Let PS be the total reaction force of thesix steel rods and PC the reaction force of theconcrete.ΣFy PS PC 1370 1370 kN0PS6PS6PCStatically Indeterminate Problems and Problems Involving Two Materials – p. 20/30
Problems Involving Two MaterialsSolution: Let PS be the total reaction force of thesix steel rods and PC the reaction force of theconcrete.ΣFy PS PC 1370 0PS PC 1370 kN1370 kNPS6PS6PCStatically Indeterminate Problems and Problems Involving Two Materials – p. 20/30
Problems Involving Two MaterialsSolution: Let PS be the total reaction force of thesix steel rods and PC the reaction force of theconcrete.ΣFy PS PC 1370 0PS PC 1370 kNWe have a single equation with two unknowns, sothe problem is statically indeterminate.1370 kNPS6PS6PCStatically Indeterminate Problems and Problems Involving Two Materials – p. 20/30
Problems Involving Two MaterialsSolution: Let PS be the total reaction force of thesix steel rods and PC the reaction force of theconcrete.ΣFy PS PC 1370 0PS PC 1370 kNWe have a single equation with two unknowns, sothe problem is statically indeterminate.The concrete and the steel rods deform (contract)by the same amount, δ, so.1370 kNPS6PS6PCStatically Indeterminate Problems and Problems Involving Two Materials – p. 20/30
Problems Involving Two MaterialsSolution: Let PS be the total reaction force of thesix steel rods and PC the reaction force of theconcrete.ΣFy PS PC 1370 0PS PC 1370 kNWe have a single equation with two unknowns, sothe problem is statically indeterminate.The concrete and the steel rods deform (contract)by the same amount, δ, so.1370 kNPS6PS6PCPC · LCPS · LS δ AS · ESAC · ECStatically Indeterminate Problems and Problems Involving Two Materials – p. 20/30
Problems Involving Two MaterialsSolution:PS ·LSAS ·ES PC ·LCAC ·EC1370 kNPS6PS6PCStatically Indeterminate Problems and Problems Involving Two Materials – p. 21/30
Problems Involving Two MaterialsSolution: PS ·LSAS ·ES PS 1150(6 200) ES PC ·LCAC ·EC“1370 kNPC 1150 ”2π 300 (6 200) EC4PS6PS6PCStatically Indeterminate Problems and Problems Involving Two Materials – p. 21/30
Problems Involving Two MaterialsSolution:PS ·LSAS ·ES PS 1150(6 200) ES PS 11501200 (200 103 ) PC ·LCAC ·EC“PC 1150 ”2π 300 (6 200) EC4“PC 1150”2π 300 1200 (25 103 )41370 kNPS6PS6PCStatically Indeterminate Problems and Problems Involving Two Materials – p. 21/30
Problems Involving Two MaterialsSolution:PS ·LSAS ·ES PS 1150(6 200) ES PS 11501200 (200 103 ) PS1200 200 PC ·LCAC ·EC“PC 1150 ”2π 300 (6 200) EC4“PC 1150”2π 300 1200 (25 103 )4PC69486 251370 kNPS6PS6PCStatically Indeterminate Problems and Problems Involving Two Materials – p. 21/30
Problems Involving Two MaterialsSolution:PC ·LCAC ·ECPS ·LSAS ·ES PS 1150(6 200) ES PS 11501200 (200 103 ) PS1200 200 PC69486 25PS 1200 20069486 25 “PC 1150 ”2π 300 (6 200) EC4“PC 1150”2π 300 1200 (25 103 )41370 kNPS6PS6PC· PCStatically Indeterminate Problems and Problems Involving Two Materials – p. 21/30
Problems Involving Two MaterialsSolution:PC ·LCAC ·ECPS ·LSAS ·ES PS 1150(6 200) ES PS 11501200 (200 103 ) PS1200 200 PC69486 25 PS 1200 20069486 25 PS 0.13816PC“PC 1150 ”2π 300 (6 200) EC4“PC 1150”2π 300 1200 (25 103 )41370 kNPS6PS6PC· PCStatically Indeterminate Problems and Problems Involving Two Materials – p. 21/30
Problems Involving Two MaterialsSolution:PC ·LCAC ·ECPS ·LSAS ·ES PS 1150(6 200) ES PS 11501200 (200 103 ) PS1200 200 PC69486 25 PS 1200 20069486 25 PS 0.13816PC“PC 1150 ”2π 300 (6 200) EC4“PC 1150”2π 300 1200 (25 103 )41370 kNPS6PS6PC· PCWe now have two equations for the two unknowns,PS and PC .Statically Indeterminate Problems and Problems Involving Two Materials – p. 21/30
Problems Involving Two MaterialsSolution:PS PC 1370PS 0.13816PC1370 kNPS6PS6PCStatically Indeterminate Problems and Problems Involving Two Materials – p. 22/30
Problems Involving Two MaterialsSolution: PS PC 1370PS 0.13816PC0.13816PC PC 13701370 kNPS6PS6PCStatically Indeterminate Problems and Problems Involving Two Materials – p. 22/30
Problems Involving Two MaterialsSolution:PS PC 1370PS 0.13816PC 0.13816PC PC 1370 PC 13701 0.138161370 kNPS6PS6PCStatically Indeterminate Problems and Problems Involving Two Materials – p. 22/30
Problems Involving Two MaterialsSolution:PS PC 1370PS 0.13816PC 0.13816PC PC 1370 PC 13701 0.13816 PC 1203.7 kN1370 kNPS6PS6PCStatically Indeterminate Problems and Problems Involving Two Materials – p. 22/30
Problems Involving Two MaterialsSolution:PS PC 1370PS 0.13816PC 0.13816PC PC 1370 PC 13701 0.13816 PC 1203.7 kN PS 166.3 kN1370 kNPS6PS6PCStatically Indeterminate Problems and Problems Involving Two Materials – p. 22/30
Problems Involving Two MaterialsSolution:PS PC 1370PS 0.13816PC 0.13816PC PC 1370 PC 13701 0.13816 PC 1203.7 kN PS 166.3 kN1370 kNPS6PS6PCWe can now calculate the stress in the steel and inthe concreteStatically Indeterminate Problems and Problems Involving Two Materials – p. 22/30
Problems Involving Two MaterialsSolution: Find the stress in the concrete:1370 kNPS6PS6PCStatically Indeterminate Problems and Problems Involving Two Materials – p. 23/30
Problems Involving Two MaterialsSolution: Find the stress in the concrete:PC 1098 kN1370 kNPS6PS6PCStatically Indeterminate Problems and Problems Involving Two Materials – p. 23/30
Problems Involving Two MaterialsSolution: Find the stress in the concrete: PC 1098 kNσC PCA1370 kNPS6PS6PCStatically Indeterminate Problems and Problems Involving Two Materials – p. 23/30
Problems Involving Two MaterialsSolution: Find the stress in the concrete: PC 1098 kNσC PCA 1098π 3002 (6 200)4σC1370 kNPS6PS6PCStatically Indeterminate Problems and Problems Involving Two Materials – p. 23/30
Problems Involving Two MaterialsSolution: Find the stress in the concrete: PC 1098 kNσC PCA σC 1098π 3002 (6 200)4 σC 0.05801370 kNkNmm2PS6PS6PCStatically Indeterminate Problems and Problems Involving Two Materials – p. 23/30
Problems Involving Two MaterialsSolution: Find the stress in the concrete: PC 1098 kNσC PCA σC 1098π 3002 (6 200)4 σC 0.0580 σC 58.0 MPa1370 kNkNmm2PS6PS6PCStatically Indeterminate Problems and Problems Involving Two Materials – p. 23/30
Problems Involving Two MaterialsSolution: Find the stress in the concrete: PC 1098 kNσC PCA σC 1098π 3002 (6 200)4 σC 0.0580 σC 58.0 MPaFind the stress in the steel:1370 kNkNmm2PS6PS6PCStatically Indeterminate Problems and Problems Involving Two Materials – p. 23/30
Problems Involving Two MaterialsSolution: Find the stress in the concrete: PC 1098 kNσC PCA σC 1098π 3002 (6 200)4 σC 0.0580 σC 58.0 MPakNmm2Find the stress in the steel:PS 1370 kNPS6152 kNPS6PCStatically Indeterminate Problems and Problems Involving Two Materials – p. 23/30
Problems Involving Two MaterialsSolution: Find the stress in the concrete: PC 1098 kNσC PCA σC 1098π 3002 (6 200)4 σC 0.0580 σC 58.0 MPakNmm2Find the stress in the steel: 1370 kNPS6PS 152 kNσS 152(6 200)PS6PCStatically Indeterminate Problems and Problems Involving Two Materials – p. 23/30
Problems Involving Two MaterialsSolution: Find the stress in the concrete: PC 1098 kNσC PCA σC 1098π 3002 (6 200)4 σC 0.0580 σC 58.0 MPa1370 kNkNmm2Find the stress in the steel:PS6PS 152 kN σS 152(6 200) σS 0.1267PS6PCkNmm2Statically Indeterminate Problems and Problems Involving Two Materials – p. 23/30
Problems Involving Two MaterialsSolution: Find the stress in the concrete: PC 1098 kNσC PCA σC 1098π 3002 (6 200)4 σC 0.0580 σC 58.0 MPa1370 kNkNmm2Find the stress in the steel:PS6PS6PCPS 152 kN σS 152(6 200) σS 0.1267 σS 126.7 MPakNmm2Statically Indeterminate Problems and Problems Involving Two Materials – p. 23/30
Problems Involving Two MaterialsSolution: Find the deformation in the concrete:Statically Indeterminate Problems and Problems Involving Two Materials – p. 24/30
Problems Involving Two MaterialsSolution: Find the deformation in the concrete:δC PC ·LCAC ·ECStatically Indeterminate Problems and Problems Involving Two Materials – p. 24/30
Problems Involving Two MaterialsSolution: Find the deformation in the concrete: δC δC PC ·LCAC ·EC1098 (3.5 103 )2 1200) (25 103 )( π 3004Statically Indeterminate Problems and Problems Involving Two Materials – p. 24/30
Problems Involving Two MaterialsSolution: Find the deformation in the concrete:δC δC δC PC ·LCAC ·EC1098 (3.5 103 )2 1200) (25 103 )( π 30040.00221 mmStatically Indeterminate Problems and Problems Involving Two Materials – p. 24/30
Problems Involving Two MaterialsSolution: Find the deformation in the concrete:δC δC δC PC ·LCAC ·EC1098 (3.5 103 )2 1200) (25 103 )( π 30040.00221 mmFind the deformation in the steel (if we’ve done our calculations correctly,then δS δC ):δS PS ·LSAS ·ESStatically Indeterminate Problems and Problems Involving Two Materials – p. 24/30
Problems Involving Two MaterialsSolution: Find the deformation in the concrete:δC δC δC PC ·LCAC ·EC1098 (3.5 103 )2 1200) (25 103 )( π 30040.00221 mmFind the deformation in the steel (if we’ve done our calculations correctly,then δS δC ):δS δS PS ·LSAS ·ES 152 (3.5 103 )1200 (200 103 )Statically Indeterminate Problems and Problems Involving Two Materials – p. 24/30
Problems Involving Two MaterialsSolution: Find the deformation in the concrete:δC δC δC PC ·LCAC ·EC1098 (3.5 103 )2 1200) (25 103 )( π 30040.00221 mmFind the deformation in the steel (if we’ve done our calculations correctly,then δS δC ):δS PS ·LSAS ·ES δS 152 (3.5 103 )1200 (200 103 ) δS 0.00222 mmStatically Indeterminate Problems and Problems Involving Two Materials – p. 24/30
Problems Involving Two MaterialsSolution: Find the deformation in the concrete:δC δC δC PC ·LCAC ·EC1098 (3.5 103 )2 1200) (25 103 )( π 30040.00221 mmFind the deformation in the steel (if we’ve done our calculations correctly,then δS δC ):δS PS ·LSAS ·ES δS 152 (3.5 103 )1200 (200 103 ) δS 0.00222 mmThe small difference in deformation is due to rounding errorsStatically Indeterminate Problems and Problems Involving Two Materials – p. 24/30
Problems Involving Two MaterialsExercise:115 mmA hollow square ste
Statically Indeterminate Problems Example: Consider a bar AB supported at both ends by fixed supports, with an axial force of 12 kN applied at C as illustrated. Find the reactions at the walls A C B 500 mm 400 mm 12 kN Solution: First, draw a free body diagram: Statically Indeterminate Problems and Problems Involving Two Materials – p. 5/30
Influence Lines for Statically Indeterminate Beams Reaction at A. 1 Scale Factor 1 E DE EE EE Vf f f §· ¹ Influence Lines for Statically Indeterminate Beams Shear at E. Influence Lines for Statically Indeterminate Beams Moment at E 1 Scale Factor 1
alone are known as statically indeterminate structures. These, then, are structures that have more than 3 unknowns to be solved for. Therefore, in order to solve statically indeterminate stru
The spring at A has stiffness of 250 kN/m and the spring at B has a stiffness of 150 kN/m. Determine the displacement under the load. Units: kN, mm. 3.7 220 A B 900 C A B 900 C . Statically Indeterminate Problems 2-17 STATICALLY INDETERMINATE PROBLEMS a b L P P A B C . 2-18
Statics and Mechanics of Materials Statically Indeterminate Problems Chapter 4-3 . . Example Problem 3 The stress needed to resist a change in length 0f 5.95 mm is The Internal force on the cross section of the rail will be . Department of Mechanical Engineering
Steps in Solving an Indeterminate Structure using the Force Method Determine degree of Indeterminacy Let n degree of indeterminacy (i.e. the structure is indeterminate to the nth degree) Define Primary Structure and the nRedundants Define the Primary Problem Solve for the n Relevant Deflections in Primary Problem Define the n Redundant Problems
2. Analyse indeterminate trusses by approximate methods. 3. Analyse industrial frames and portals by approximate methods. 35.1 Introduction In module 2, force method of analysis is applied to solve indeterminate beams, trusses and frames. In modules 3 and 4, displacement based methods are discussed for the analysis of indeterminate structures.
C. Application of linear superposition to analyze indeterminate axially loaded members. Statically indeterminate structural systems are those where the number of unknown support reactions exceeds the number of global equilibrium equations. In such systems it is not possible to solve for the support
3006 AGMA Toilet Additive 1338 (3006) 19.0% 2914 CERAVON BLUE V10 DC (2914) 0.05% 2922 FORMALDEHYDE REODORANT ALTERNATIVE (2922) 0.6% 3 Water (3) 80.05% Constituent Chemicals 1 Water (3) 80.05% CAS number: 7732-18-5 EC number: 231-791-2 Product number: — EU index number: — Physical hazards Not Classified Health hazards Not Classified Environmental hazards Not Classified 2 Bronopol (INN .
