Analysis of Aerospace Structures_ azargoshasb.pdf
HamedAzarkeshb
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Jul 23, 2024
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About This Presentation
Analysis of Aerospace Structures
Slide Content
Analysis of Aerospace Structures
Prov: H. Azargoshasb
Sharif University of Technology _ Aerospace Engineering Faculty
Prov: H. Azargoshasb
Sharif University of Technology _ Aerospace Engineering Faculty
2Objective
The Purpose of the Course is to teach the Principles
of Solid and Structural Mechanics that can be used
to Design and Analyze Aerospace Structures, in
particular Aircraft Structures.
The Purpose of the Course is to teach the Principles
of Solid and Structural Mechanics that can be used
to Design and Analyze Aerospace Structures, in
particular Aircraft Structures.
3TABLE OFCONTENTS
•Introduction
•Multi Disciplines Design Process, System Engineering, Phases of Aircraft Design Process
•Design Requirements, RFP Proposal, Design Project Planning _ Gantt Chart, Flow Chart
•Structure Design Steps, Structure Architecture, Standards and, Structure Design Cycle Process
•Chapter 0. Characteristics of Aircraft Structures and Materials
•Introduction
•Types of Aircraft Structures
•Basic Structural Elements in Aircraft Structure
•Definitions
•Aircraft Materials
•Chapter I. Pure Torsion
•Elastic, Inelastic, Homogeneous, and Nonhomogeneous Bars,
•Circular and Noncircular Cross Sections,
•Shear Flow, Shear Stress, Residual Stress, Power Transmission, and Angle of Twist
•Introduction
•Multi Disciplines Design Process, System Engineering, Phases of Aircraft Design Process
•Design Requirements, RFP Proposal, Design Project Planning _ Gantt Chart, Flow Chart
•Structure Design Steps, Structure Architecture, Standards and, Structure Design Cycle Process
•Chapter 0. Characteristics of Aircraft Structures and Materials
•Introduction
•Types of Aircraft Structures
•Basic Structural Elements in Aircraft Structure
•Definitions
•Aircraft Materials
•Chapter I. Pure Torsion
•Elastic, Inelastic, Homogeneous, and Nonhomogeneous Bars,
•Circular and Noncircular Cross Sections,
•Shear Flow, Shear Stress, Residual Stress, Power Transmission, and Angle of Twist
4TABLE OFCONTENTS
•Chapter II. Pure Bending
•Elastic, Inelastic, Homogeneous, and Nonhomogeneous Beams,
•Symmetric and Unsymmetric Bending, Neutral Axis Location, Deflection,
•Bending Stresses in Straight and Curved Beams
•Chapter III. Transverse Shear Loading of Beams With OPEN Cross Sections
•Elastic, Homogeneous Unstiffened and Stiffened Beams,
•Symmetric and Unsymmetric Cross Sections,
•Shear Center, Shear Flow, Shear and Bending Stresses, and Deflection
•Chapter IV. Transverse Shear Loading of Beams With CLOSED Cross Sections
•Elastic, Homogeneous Unstiffened and Stiffened Beams,
•Uniform and Tapered Beams, Symmetric and Unsymmetric Cross Sections,
•Shear Center, Shear Flow, Shear and Bending Stresses, and Deflection
•Chapter V. Combined Transverse Shear, Bending, and Torsion Loading
•Shear and Normal Stresses in Closed Thin-Walled Beams
•Chapter VI. Internal Pressure
•In-Plane Stresses in Monocoque and Semi-Monocoque Pressure Vessels
•Chapter II. Pure Bending
•Elastic, Inelastic, Homogeneous, and Nonhomogeneous Beams,
•Symmetric and Unsymmetric Bending, Neutral Axis Location, Deflection,
•Bending Stresses in Straight and Curved Beams
•Chapter III. Transverse Shear Loading of Beams With OPEN Cross Sections
•Elastic, Homogeneous Unstiffened and Stiffened Beams,
•Symmetric and Unsymmetric Cross Sections,
•Shear Center, Shear Flow, Shear and Bending Stresses, and Deflection
•Chapter IV. Transverse Shear Loading of Beams With CLOSED Cross Sections
•Elastic, Homogeneous Unstiffened and Stiffened Beams,
•Uniform and Tapered Beams, Symmetric and Unsymmetric Cross Sections,
•Shear Center, Shear Flow, Shear and Bending Stresses, and Deflection
•Chapter V. Combined Transverse Shear, Bending, and Torsion Loading
•Shear and Normal Stresses in Closed Thin-Walled Beams
•Chapter VI. Internal Pressure
•In-Plane Stresses in Monocoque and Semi-Monocoque Pressure Vessels
5Books on Aircraft Structures
•Sun, C.T., "Mechanics of Aircraft Structures," John Wiley & Sons, 1998; ISBN 0-471-17877-2
•Megson, T.H.G., "Aircraft Structures for Engineering Students," Third Edition, Arnold (also John Wiley & Sons),
1999; ISBN 0-470-34937-9
•Curtis, H.T., "Fundamentals of Aircraft Structural Analysis," Irwin, 1997; ISBN 0-256-19260-X
•Lomax, T.L., "Structural Loads Analysis for Commercial Transport Aircraft: Theory and Practice," AIAA
Education Series, AIAA, Inc., 1801 Alexander Bell Drive, Reston, VA 22091, 1996; ISBN 1-56347-114-0.
•Niu, M.C.Y., "Airframe Structural Design," CommilitPress LTD., 22/F, Sing Pao Building, 101 King's Road, North
Point, Hong Kong. U.S. order/inquiry: Technical Book Company, 2056 Westwood Blvd., Los Angeles, CA 90025,
1988; ISBN 962-7128-04-X.
•Allen, H.A. and Haisler, W.E., "Introduction to Aerospace Structural Analysis," John Wiley & Sons, Inc., 1985;
ISBN 0-471-88839-7.
•Peery, D.J. and Azar, J.J., "Aircraft Structures," McGraw-Hill, Inc., 1982; ISBN 0-07-049196-8.
•Bruhn, E.F., "Analysis and Design of Flight Vehicle Structures," S.R. Jacobs & Associates, Inc., 1973.
•Abraham, L.H., "Structural Design of Misslesand Spacecraft," McGaw-Hill, Inc., 1962.
•Sun, C.T., "Mechanics of Aircraft Structures," John Wiley & Sons, 1998; ISBN 0-471-17877-2
•Megson, T.H.G., "Aircraft Structures for Engineering Students," Third Edition, Arnold (also John Wiley & Sons),
1999; ISBN 0-470-34937-9
•Curtis, H.T., "Fundamentals of Aircraft Structural Analysis," Irwin, 1997; ISBN 0-256-19260-X
•Lomax, T.L., "Structural Loads Analysis for Commercial Transport Aircraft: Theory and Practice," AIAA
Education Series, AIAA, Inc., 1801 Alexander Bell Drive, Reston, VA 22091, 1996; ISBN 1-56347-114-0.
•Niu, M.C.Y., "Airframe Structural Design," CommilitPress LTD., 22/F, Sing Pao Building, 101 King's Road, North
Point, Hong Kong. U.S. order/inquiry: Technical Book Company, 2056 Westwood Blvd., Los Angeles, CA 90025,
1988; ISBN 962-7128-04-X.
•Allen, H.A. and Haisler, W.E., "Introduction to Aerospace Structural Analysis," John Wiley & Sons, Inc., 1985;
ISBN 0-471-88839-7.
•Peery, D.J. and Azar, J.J., "Aircraft Structures," McGraw-Hill, Inc., 1982; ISBN 0-07-049196-8.
•Bruhn, E.F., "Analysis and Design of Flight Vehicle Structures," S.R. Jacobs & Associates, Inc., 1973.
•Abraham, L.H., "Structural Design of Misslesand Spacecraft," McGaw-Hill, Inc., 1962.
6Books on Aircraft Structures
•Williams, D., "An Introduction to the Theory of Aircraft Structures," Edward Arnold (Publishers) LTD., London,
England, 1960.
•Kuhn, P., "Stresses in Aircraft and Shell Structures," McGraw-Hill Book Company, Inc., 1956.
•Gerard, G., "Minimum Weight Analysis of Compression Structures," New York University Press, 1956.
•Niles, A.S. and Newell, J.S., "Airplane Structures," Volume I, Fourth Edition, John Wiley & Sons, Inc., 1954.
•Shanley, F.R., "Weight-Strength Analysis of Aircraft Structures," The Rand Corporation, First Edition, 1952.
•Steinbacher, F.R. and Gerard, G., "Aircraft Structural Mechanics," Pitman Aeronautical Publications, 1952.
•Peery, D.J., "Aircraft Structures," McGraw-Hill BookkCompany, Inc., 1950.
•Barton, M.V., "Fundamentals of Aircraft Structures," Prentice-Hall, Inc., 1948.
•Mangurian, G.N. and Johnston, N.M., "Aircraft Structural Analysis," Prentice-Hall, Inc., 1947.
•Niles, A.S. and Newell, J.S., "Airplane Structures," Volume II, Third Edition, John Wiley & Sons, Inc., 1947.
•Sechler, E.E. and Dunn, L.G., "Airplane Structural Analysis and Design," John Wiley & Sons, Inc., 1942
•Williams, D., "An Introduction to the Theory of Aircraft Structures," Edward Arnold (Publishers) LTD., London,
England, 1960.
•Kuhn, P., "Stresses in Aircraft and Shell Structures," McGraw-Hill Book Company, Inc., 1956.
•Gerard, G., "Minimum Weight Analysis of Compression Structures," New York University Press, 1956.
•Niles, A.S. and Newell, J.S., "Airplane Structures," Volume I, Fourth Edition, John Wiley & Sons, Inc., 1954.
•Shanley, F.R., "Weight-Strength Analysis of Aircraft Structures," The Rand Corporation, First Edition, 1952.
•Steinbacher, F.R. and Gerard, G., "Aircraft Structural Mechanics," Pitman Aeronautical Publications, 1952.
•Peery, D.J., "Aircraft Structures," McGraw-Hill BookkCompany, Inc., 1950.
•Barton, M.V., "Fundamentals of Aircraft Structures," Prentice-Hall, Inc., 1948.
•Mangurian, G.N. and Johnston, N.M., "Aircraft Structural Analysis," Prentice-Hall, Inc., 1947.
•Niles, A.S. and Newell, J.S., "Airplane Structures," Volume II, Third Edition, John Wiley & Sons, Inc., 1947.
•Sechler, E.E. and Dunn, L.G., "Airplane Structural Analysis and Design," John Wiley & Sons, Inc., 1942
7Grading
•Final Exam (6points)
•Homework (4points)
•Midterm, Quizizz (6points)
•Project (Report = 2.5and presentation = 1.5)
•Attendance In The Classroom(1+points)
•Class participation(1+points)
•Final Exam (6points)
•Homework (4points)
•Midterm, Quizizz (6points)
•Project (Report = 2.5and presentation = 1.5)
•Attendance In The Classroom(1+points)
•Class participation(1+points)
8Course software
•Abaqus or Another FEM software
•Matlabor Python
•Excel
•etc
•Abaqus or Another FEM software
•Matlabor Python
•Excel
•etc
9Introduction-Objective
✓Air vehicle design?
✓The position of design in the production of air vehicles?
✓Multidisciplinary design?
✓Systems engineering?
✓Different steps of the design process?
✓Design requirements?
✓Request for proposal? &Proposal?
✓Design Project Planning? &Gantt Chart, Flow Chart ?
✓The position of structural design in the design of air vehicles?
✓The position of structural analysis in the design of the air vehicle structure?
✓Architectural processes of an air vehicle?
✓Standards and guidelines?
✓Air vehicle design?
✓The position of design in the production of air vehicles?
✓Multidisciplinary design?
✓Systems engineering?
✓Different steps of the design process?
✓Design requirements?
✓Request for proposal? &Proposal?
✓Design Project Planning? &Gantt Chart, Flow Chart ?
✓The position of structural design in the design of air vehicles?
✓The position of structural analysis in the design of the air vehicle structure?
✓Architectural processes of an air vehicle?
✓Standards and guidelines?
1
0
Introduction-Multi Disciplines DesignProcess
0
Introduction-Multi Disciplines DesignProcess
1
1
Introduction-Multi Disciplines DesignProcess
1
Introduction-Multi Disciplines DesignProcess
* یسدنهم هخاش دنچ تلاماعت* نامزمه یسدنهم* یراک کرتشم مهف و نابز داجیا *مجسنم یمیت شلات داجیا
یدنمزاین لوصحم
کینویوا یسدنهم
کیلوردیه یسدنهم
هزاس یسدنهم
زا رگید یاه هزوح
یسدنهم
یا هخاش دنچ یسدنهم یاهدنیآرف
تاصخشم حطس
(Specification)
یحارط حطس
(Design)
ارجا و یزاس هدایپ حطس
(Implementation)
12Introduction-System Engineering
یدنمزاین لوصحم
کینویوا یسدنهم
کیلوردیه یسدنهم
هزاس یسدنهم
زا رگید یاه هزوح
یسدنهم
یا هخاش دنچ یسدنهم یاهدنیآرف
تاصخشم حطس
(Specification)
یحارط حطس
(Design)
ارجا و یزاس هدایپ حطس
(Implementation)
12Introduction-System Engineering
* هتفای راتخاس دنیآرف داجیا*فلتخم یاه هزوح رد مهافتءوس داجیا زا یریگولج
یدنمزاین لوصحممتسیس یسدنهم
کیلوردیه یسدنهم
هزاس یسدنهم
یسدنهم زا رگید یاه هزوح یسدنهم یاهدنیآرف
یا هخاش دنچ
تاصخشم حطس
(Specification)
یحارط حطس
(Design)
ارجا و یزاس هدایپ حطس
(Implementation)
کینویوا یسدنهم
هلئسم تاحیضوت
هدش دیدناک یاه لح
بولطم لح یبایزرا
هدش هدایپ لح
؟دراد یتاصخشم هچ و تسیچ هلئسم
لح شور دنچ(یحارط )؟درک باختنا ناوتیم هلئسم نیا یارب
یحارط نیرتهب
یحارط نیرتهب ساسارب ارجا
تامازلا اب هدش ارجا یحارط ایآ
؟دراد تقباطم
13Introduction-System Engineering
یدنمزاین لوصحممتسیس یسدنهم
کیلوردیه یسدنهم
هزاس یسدنهم
یسدنهم زا رگید یاه هزوح یسدنهم یاهدنیآرف
یا هخاش دنچ
تاصخشم حطس
(Specification)
یحارط حطس
(Design)
ارجا و یزاس هدایپ حطس
(Implementation)
کینویوا یسدنهم
هلئسم تاحیضوت
هدش دیدناک یاه لح
بولطم لح یبایزرا
هدش هدایپ لح
؟دراد یتاصخشم هچ و تسیچ هلئسم
لح شور دنچ(یحارط )؟درک باختنا ناوتیم هلئسم نیا یارب
یحارط نیرتهب
یحارط نیرتهب ساسارب ارجا
تامازلا اب هدش ارجا یحارط ایآ
؟دراد تقباطم
13Introduction-System Engineering
یدنمزاین لوصحممتسیس یسدنهم
تامازلا تیریدم
یدرکلمع لیلحت
یرامعم فیرعت
یقطنم
یکیزیف یحارط
یگچراپکی
دیئات و یسررب
و تیحلاص
یجنسرابتعا
هلئسم تاحیضوت
هدش دیدناک یاه لح
بولطم لح
یبایزرا
هدش هدایپ لح
لح هار
تاصخشم(Specification) یزاس هدایپ(Implementation)یحارط(Design)
دنیآرفیسدنهم
یا هخاشدنچ
دنیآرفیسدنهممتسیس
* شورRFLP هخرچ یحارط دنیآرف رب ینتبم روحم دنیآرف متسیس یسدنهم یعونVتسا.
1
4
Introduction-System Engineering
تامازلا تیریدم
یدرکلمع لیلحت
یرامعم فیرعت
یقطنم
یکیزیف یحارط
یگچراپکی
دیئات و یسررب
و تیحلاص
یجنسرابتعا
هلئسم تاحیضوت
هدش دیدناک یاه لح
بولطم لح
یبایزرا
هدش هدایپ لح
لح هار
تاصخشم(Specification) یزاس هدایپ(Implementation)یحارط(Design)
دنیآرفیسدنهم
یا هخاشدنچ
دنیآرفیسدنهممتسیس
* شورRFLP هخرچ یحارط دنیآرف رب ینتبم روحم دنیآرف متسیس یسدنهم یعونVتسا.
1
4
Introduction-System Engineering
* غولب یاه هخرچ رتشیب هچ رهرارکتVدنک یم قبطنم یرتشم تامازلا اب ار لوصحم رتشیب ، .یدرکلمع تامازلا فیرعت هب طقف ،غولب لوا یاه هخرچ تسا نکممدسرب یتامدقم.
اه متسیس رتشیب هچ ره هیزجت
لوصحم رتشیب هچ ره ندش کیدزن
یرتشم تامازلا ساسارب
تامازلا تیریدم
یدرکلمع لیلحت
یرامعم فیرعت
یقطنم
یکیزیف یحارط
یگچراپکی
دیئات و یسررب
و تیحلاص
یجنسرابتعا
•Requirement (لوصحم حور)
یفیصوتینتمزاهیلکزلایتاماهک
رارقتساربیحارطودییاتملوصح
لامعادوش."هچ؟میزاسب!"
•Functional (لوصحم حور)
یا هعومجمزایتاصخشمهکهبیگنوگچدرکلمع
یاه متسیسلوصحمساساربتامازلا
دزادرپ یم.ربیحارطودییاتلوصحمعالام
دوش."هنوگچ؟میزاسب!"
•Logical(لوصحم مسج)
راتخاسیقطنمیاه متسیسلوصحم؟تسیچ"ابهچ
یراتخاس؟میزاسب!"(،تازیهجت،تاعطقطباراه...)
همزلاقلخ(یحارط)کیمتسیسهبتروصیف،یکیزداجیا
ییاه طیحممزلاهارمهبتاطابترانیبامساهنآت.
•Physical(لوصحم مسج )متسیس زا یلح هار( .یلاعف هب یعقاو کنیل ای یزاجم کنیلیراجت ت)
•Integration(مسج راتخاس رد نزاوت و یگچراپکی یبایزرالوصحم)
راتخاسیقطنمدیابهبیوحنداجیادوشهکردتخاسرایاه متسیس
لوصحمهنوگچیه،لخادتهچردکیتاتساوهچردکلمعدریکیمانید
دشابن.
•Verification(حم درکلمع و راتفر دیئات و یسرربلوص)
ابهدافتسازایزاس هیبشو،تستدرکلمعلوصحمدروم
یسرربرارقهتفرگوردتروص،یتسردیئاتددوش یم.
•Validation and Qualification
(زلا اب متسیس راتفر قابطنا یسرربیرتشم تاما)
ساساربتامازلا،یرتشمیسرربمدوش ی
هکایآدرکلمعحیحصیاه متسیسم،لوصح
یاهزاینیرتشمارهدروآربدنک یمیاریخ.
15Introduction-System Engineering
اه متسیس رتشیب هچ ره هیزجت
لوصحم رتشیب هچ ره ندش کیدزن
یرتشم تامازلا ساسارب
تامازلا تیریدم
یدرکلمع لیلحت
یرامعم فیرعت
یقطنم
یکیزیف یحارط
یگچراپکی
دیئات و یسررب
و تیحلاص
یجنسرابتعا
•Requirement (لوصحم حور)
یفیصوتینتمزاهیلکزلایتاماهک
رارقتساربیحارطودییاتملوصح
لامعادوش."هچ؟میزاسب!"
•Functional (لوصحم حور)
یا هعومجمزایتاصخشمهکهبیگنوگچدرکلمع
یاه متسیسلوصحمساساربتامازلا
دزادرپ یم.ربیحارطودییاتلوصحمعالام
دوش."هنوگچ؟میزاسب!"
•Logical(لوصحم مسج)
راتخاسیقطنمیاه متسیسلوصحم؟تسیچ"ابهچ
یراتخاس؟میزاسب!"(،تازیهجت،تاعطقطباراه...)
همزلاقلخ(یحارط)کیمتسیسهبتروصیف،یکیزداجیا
ییاه طیحممزلاهارمهبتاطابترانیبامساهنآت.
•Physical(لوصحم مسج )متسیس زا یلح هار( .یلاعف هب یعقاو کنیل ای یزاجم کنیلیراجت ت)
•Integration(مسج راتخاس رد نزاوت و یگچراپکی یبایزرالوصحم)
راتخاسیقطنمدیابهبیوحنداجیادوشهکردتخاسرایاه متسیس
لوصحمهنوگچیه،لخادتهچردکیتاتساوهچردکلمعدریکیمانید
دشابن.
•Verification(حم درکلمع و راتفر دیئات و یسرربلوص)
ابهدافتسازایزاس هیبشو،تستدرکلمعلوصحمدروم
یسرربرارقهتفرگوردتروص،یتسردیئاتددوش یم.
•Validation and Qualification
(زلا اب متسیس راتفر قابطنا یسرربیرتشم تاما)
ساساربتامازلا،یرتشمیسرربمدوش ی
هکایآدرکلمعحیحصیاه متسیسم،لوصح
یاهزاینیرتشمارهدروآربدنک یمیاریخ.
15Introduction-System Engineering
Sharif University of Technology
Aerospace Engineering Faculty
16Introduction-Phases of Aircraft Design Process
Aerospace Engineering Faculty
16Introduction-Phases of Aircraft Design Process
17Introduction-Design Requirements
18Introduction-RFP -Proposal
19Introduction-RFP -Proposal
20Introduction-Design Project Planning _ Gantt Chart, Flow Chart
•Geometry Design or Structure Architecture
•Determination of The Design Standard
•Loading
•Determination of Boundary Conditions
•Material Selection
•Structural Analysis
•Damage And Stability Analysis
•Documenting
A Different and Repeatable Process in Different Phases of Design
21Introduction-Structure Design Steps
•Determination of The Design Standard
•Loading
•Determination of Boundary Conditions
•Material Selection
•Structural Analysis
•Damage And Stability Analysis
•Documenting
A Different and Repeatable Process in Different Phases of Design
21Introduction-Structure Design Steps
22Introduction-Structure Architecture
23Introduction-Standards and Guidelines
✓ نامدتتتتتتیچ اتتتتتتفایرد
یاه ناملاهزاس یلصا
✓هزاس هیلوا یسدنه لدم
✓یراذگرابهزاس بلص
✓ باختنالایرتم
✓هزاس هیلوا نامدیچ
✓هزاس ییاهن هسدنه
✓ ییاهن لدمFEM
✓یراذگرابهزاس کیتسلاا
✓تایئزج یحارط اهج هدامآ
24Introduction-Preliminary Structure Design Cycle Process
یاه ناملاهزاس یلصا
✓هزاس هیلوا یسدنه لدم
✓یراذگرابهزاس بلص
✓ باختنالایرتم
✓هزاس هیلوا نامدیچ
✓هزاس ییاهن هسدنه
✓ ییاهن لدمFEM
✓یراذگرابهزاس کیتسلاا
✓تایئزج یحارط اهج هدامآ
24Introduction-Preliminary Structure Design Cycle Process
2
5
Chapter 0.-Objective
✓Types of Air vehicle ?
✓Types of Aircraft Structures?
✓The difference between an air vehicle and a ground vehicle?
✓Airframe?
✓Basic Structural Elements in Aircraft Structure?
✓wing, fuselage, landing gears, tail units (horizontal and vertical stabilizers), canard, nacelle,
and control surfaces such as aileron, rudder, flap, slat, tabs and elevator?
✓Spar, rib, stringer, Bulkhead, skin, former, web, longeron, frame, cap, flange, joggle, ..?
✓truss, semimonocoque, and monocoque?
✓Aircraft Cutaway?
✓Aircraft Materials?
✓aluminum, titanium, and steel alloys, advanced fiber composites?
5
Chapter 0.-Objective
✓Types of Air vehicle ?
✓Types of Aircraft Structures?
✓The difference between an air vehicle and a ground vehicle?
✓Airframe?
✓Basic Structural Elements in Aircraft Structure?
✓wing, fuselage, landing gears, tail units (horizontal and vertical stabilizers), canard, nacelle,
and control surfaces such as aileron, rudder, flap, slat, tabs and elevator?
✓Spar, rib, stringer, Bulkhead, skin, former, web, longeron, frame, cap, flange, joggle, ..?
✓truss, semimonocoque, and monocoque?
✓Aircraft Cutaway?
✓Aircraft Materials?
✓aluminum, titanium, and steel alloys, advanced fiber composites?
26
•An aircraft is a vehicle that is used for flight in the air
•vehiclelikethisistypicallybuiltbyassemblingmanycomponentstructuressuchaswing,fuselage,
landinggears,stabilizers,etc.
•Each component structure is typically built by assembling many substructures.
•Each substructure can be made out of different materials.
•Themaindifferencebetweenaircraftstructuresandmaterialsandcivilengineeringstructures
andmaterialsliesintheirweight.
•Themaindrivingforceinaircraftstructuraldesignandaerospacematerialdevelopmentisto
reduceweight.
•Ingeneral,materialswithhighstiffness,highstrength,andlightweightaremostsuitablefor
aircraftapplications.
•Atypicalaircraftdesigncycleinvolvesthreemajorsteps(i)conceptualdesign,(ii)preliminary
design,and(iii)detaildesign.
Chapter 0.-Introduction
•An aircraft is a vehicle that is used for flight in the air
•vehiclelikethisistypicallybuiltbyassemblingmanycomponentstructuressuchaswing,fuselage,
landinggears,stabilizers,etc.
•Each component structure is typically built by assembling many substructures.
•Each substructure can be made out of different materials.
•Themaindifferencebetweenaircraftstructuresandmaterialsandcivilengineeringstructures
andmaterialsliesintheirweight.
•Themaindrivingforceinaircraftstructuraldesignandaerospacematerialdevelopmentisto
reduceweight.
•Ingeneral,materialswithhighstiffness,highstrength,andlightweightaremostsuitablefor
aircraftapplications.
•Atypicalaircraftdesigncycleinvolvesthreemajorsteps(i)conceptualdesign,(ii)preliminary
design,and(iii)detaildesign.
Chapter 0.-Introduction
27
•In any of these design stages, different factors such as aerodynamics, avionics, propulsion,
and structural integrity are simultaneously taken into account.
•Assuch,aircraftstructuresarenotdesignedforstructuralsafetyandintegrityonly;many
nonstructuralrequirementsimposeadditionalrestrictionsindesigningaircraftstructuralcomponents.
•Forinstance,anairfoilischosenaccordingtoaerodynamicliftanddragcharacteristics.
•Assuch,thesizeandshapeofanaircraftstructuralcomponentareusuallypredetermined.
•Suchrestrictionssignificantlylimitthenumberofsolutionsforstructuralproblemsintermsofglobal
configurations.
•Often,thesolutionsresorttotheuseofspecialmaterialsdevelopedforapplicationsinaerospace
vehicles.
•Thenonstructuralandweight‐savingdesignrequirementsgenerallyleadtotheuseofshell‐like
structures(monocoqueconstructions)andstiffenedshellstructures(semimonocoque
constructions).
Chapter 0.-Introduction
•In any of these design stages, different factors such as aerodynamics, avionics, propulsion,
and structural integrity are simultaneously taken into account.
•Assuch,aircraftstructuresarenotdesignedforstructuralsafetyandintegrityonly;many
nonstructuralrequirementsimposeadditionalrestrictionsindesigningaircraftstructuralcomponents.
•Forinstance,anairfoilischosenaccordingtoaerodynamicliftanddragcharacteristics.
•Assuch,thesizeandshapeofanaircraftstructuralcomponentareusuallypredetermined.
•Suchrestrictionssignificantlylimitthenumberofsolutionsforstructuralproblemsintermsofglobal
configurations.
•Often,thesolutionsresorttotheuseofspecialmaterialsdevelopedforapplicationsinaerospace
vehicles.
•Thenonstructuralandweight‐savingdesignrequirementsgenerallyleadtotheuseofshell‐like
structures(monocoqueconstructions)andstiffenedshellstructures(semimonocoque
constructions).
Chapter 0.-Introduction
28
•The geometrical details of aircraft structures are much more complicated than those of civil
engineering structures. They usually require the assemblage of thousands of parts.
•Technologies for joining the parts are especially important for aircraft construction.
•Because of their highstiffness/weightand strength/weightratios, aluminumand titaniumalloys
have been the dominant aircraft structural materials for many decades.
•However, the recent advent of advanced fiber‐reinforced composites has changed the outlook.
•Composites may now achieve weight savings of 30-40% over aluminum or titanium counterparts.
•As a result, composites have been used increasingly in aircraft structures.
Chapter 0.-Introduction
•The geometrical details of aircraft structures are much more complicated than those of civil
engineering structures. They usually require the assemblage of thousands of parts.
•Technologies for joining the parts are especially important for aircraft construction.
•Because of their highstiffness/weightand strength/weightratios, aluminumand titaniumalloys
have been the dominant aircraft structural materials for many decades.
•However, the recent advent of advanced fiber‐reinforced composites has changed the outlook.
•Composites may now achieve weight savings of 30-40% over aluminum or titanium counterparts.
•As a result, composites have been used increasingly in aircraft structures.
Chapter 0.-Introduction
29
•Most aircraft are built as fixed‐wing vehicles and are commonly known as airplanes.
•Other categories include rotorcrafts, glider, lighter‐than‐airvehicles, etc.
•Presence of air is essential for generating lift on thesevehicles.
•Assuch,structuraldesignofsuchvehiclesdependsonhowairloadistransmittedtothestructural
elements.
•Fixed‐WingAircraft
•Afixed‐wingaircraftisakindofairvehiclethatisheavier‐than‐airbutcanflyintheairby
generatingliftusingthewings.
•Anaircraftwithapoweredengineisgenerallycalledanairplane.
•Theunpoweredversionoffixed‐wingaircraftiscalledgliders.
Chapter 0.-Types of Aircraft Structures
•Most aircraft are built as fixed‐wing vehicles and are commonly known as airplanes.
•Other categories include rotorcrafts, glider, lighter‐than‐airvehicles, etc.
•Presence of air is essential for generating lift on thesevehicles.
•Assuch,structuraldesignofsuchvehiclesdependsonhowairloadistransmittedtothestructural
elements.
•Fixed‐WingAircraft
•Afixed‐wingaircraftisakindofairvehiclethatisheavier‐than‐airbutcanflyintheairby
generatingliftusingthewings.
•Anaircraftwithapoweredengineisgenerallycalledanairplane.
•Theunpoweredversionoffixed‐wingaircraftiscalledgliders.
Chapter 0.-Types of Aircraft Structures
30
•Rotorcraft
•Arotorcraftorrotary‐wingaircraftisaheavier‐than‐airvehiclethatgeneratesliftusingrotarywings
orrotorblades,whichrevolvearoundarotor.
•Dependingonhowrotorbladesfunction,rotorcraftsarecategorizedashelicopters,autogyros
(gyrocopter),orgyrodynes(VTOL).
•Recently,small‐scalemultirotorrotorcraftsarewidelyusedforsurveillanceorvideo‐capturing
purposes.
•Designingbladesfortherotorcraftisfarmorecomplexthandesigningafixed‐wingaircraftbecause
ofthecomplexaerodynamicforces.
Chapter 0.-Types of Aircraft Structures
•Rotorcraft
•Arotorcraftorrotary‐wingaircraftisaheavier‐than‐airvehiclethatgeneratesliftusingrotarywings
orrotorblades,whichrevolvearoundarotor.
•Dependingonhowrotorbladesfunction,rotorcraftsarecategorizedashelicopters,autogyros
(gyrocopter),orgyrodynes(VTOL).
•Recently,small‐scalemultirotorrotorcraftsarewidelyusedforsurveillanceorvideo‐capturing
purposes.
•Designingbladesfortherotorcraftisfarmorecomplexthandesigningafixed‐wingaircraftbecause
ofthecomplexaerodynamicforces.
Chapter 0.-Types of Aircraft Structures
31
•Lighter‐than‐Air Vehicles
•Aircraftsuchasballoons(hot‐airballoon),nonrigidblimps,andairships(alsoknownasdirigibles)
aredesignedtocontainsufficientamountoflighter‐than‐airgases(typicallyhelium)sothatliftcan
begeneratedfromtheliftinggas.
Chapter 0.-Types of Aircraft Structures
•Lighter‐than‐Air Vehicles
•Aircraftsuchasballoons(hot‐airballoon),nonrigidblimps,andairships(alsoknownasdirigibles)
aredesignedtocontainsufficientamountoflighter‐than‐airgases(typicallyhelium)sothatliftcan
begeneratedfromtheliftinggas.
Chapter 0.-Types of Aircraft Structures
32
•Drones
•Dronesaresmall‐scaleairvehiclesthatcanbefixed‐wingtypeorrotary‐wingtype.
•Thesizeofadroneissignificantlysmallerthanatypicalairplaneorrotorcraft.
•Assuch,mostdronesarepoweredbyelectricalsources.
•Otherthantheirsize,theliftingmechanismofadroneissimilartotheconventionalfixed‐wingor
rotary‐wingvehicles.
Chapter 0.-Types of Aircraft Structures
•Drones
•Dronesaresmall‐scaleairvehiclesthatcanbefixed‐wingtypeorrotary‐wingtype.
•Thesizeofadroneissignificantlysmallerthanatypicalairplaneorrotorcraft.
•Assuch,mostdronesarepoweredbyelectricalsources.
•Otherthantheirsize,theliftingmechanismofadroneissimilartotheconventionalfixed‐wingor
rotary‐wingvehicles.
Chapter 0.-Types of Aircraft Structures
33
•BASIC STRUCTURAL ELEMENTS IN AIRCRAFT STRUCTURE
•Anaircrafthasmanyintegratedparts,asshowninFigures.
•Ingeneral,thesepartscanbecategorizedintobasicstructuralelementssuchaswing,fuselage,
landinggears,tailunits(horizontalandverticalstabilizers),andcontrolsurfacessuchas
aileron,rudder,andelevator.
Chapter 0.-Basic Structural Elements in Aircraft Structure
•BASIC STRUCTURAL ELEMENTS IN AIRCRAFT STRUCTURE
•Anaircrafthasmanyintegratedparts,asshowninFigures.
•Ingeneral,thesepartscanbecategorizedintobasicstructuralelementssuchaswing,fuselage,
landinggears,tailunits(horizontalandverticalstabilizers),andcontrolsurfacessuchas
aileron,rudder,andelevator.
Chapter 0.-Basic Structural Elements in Aircraft Structure
34
•Fuselage
•Thefuselageisthemainstructuralelementofafixed‐wingaircraft.
•Itprovidesspaceforcargo,controlsystemandpilots,passengersandcabincrews,andother
accessoriesandequipment.
•Insingle‐engineaircraft,thefuselagealsocarriesthepowerplant.
•AsshowninFigure,afuselagecanbeconstructedinvariousconfigurationssuchastruss,
semimonocoque,andmonocoque.
Chapter 0.-Basic Structural Elements in Aircraft Structure
•Fuselage
•Thefuselageisthemainstructuralelementofafixed‐wingaircraft.
•Itprovidesspaceforcargo,controlsystemandpilots,passengersandcabincrews,andother
accessoriesandequipment.
•Insingle‐engineaircraft,thefuselagealsocarriesthepowerplant.
•AsshowninFigure,afuselagecanbeconstructedinvariousconfigurationssuchastruss,
semimonocoque,andmonocoque.
Chapter 0.-Basic Structural Elements in Aircraft Structure
35
•VariousConfigurations
OfFuselage
Chapter 0.-Basic Structural Elements in Aircraft Structure
•VariousConfigurations
OfFuselage
Chapter 0.-Basic Structural Elements in Aircraft Structure
36Chapter 0.-Basic Structural Elements in Aircraft Structure
37
•Wing
•Themainfunctionofthewingistopickuptheairandpowerplantloadsandtransmitthemtothe
fuselage.
•Thewingcross‐sectiontakestheshapeofanairfoil,whichisdesignedbasedonaerodynamic
considerations.
•Ingeneral,wingsareconstructedbasedonmonospar,multispar,orboxbeamconfigurations,as
showninFigures.
Chapter 0.-Basic Structural Elements in Aircraft Structure
•Wing
•Themainfunctionofthewingistopickuptheairandpowerplantloadsandtransmitthemtothe
fuselage.
•Thewingcross‐sectiontakestheshapeofanairfoil,whichisdesignedbasedonaerodynamic
considerations.
•Ingeneral,wingsareconstructedbasedonmonospar,multispar,orboxbeamconfigurations,as
showninFigures.
Chapter 0.-Basic Structural Elements in Aircraft Structure
38
•WingConfigurations
•Thesethreedesignconfigurationsareconsideredasthebasicdesigns,andaircraftmanufacturers
mayadoptamodifiedconfiguration.
•Inthemonosparwingconfiguration,onlyonemainspanwisememberispresent.Ribsorbulkheads
areusedtoprovidethenecessaryaerodynamiccontourorshapetotheairfoil.
•Themultisparwingconfigurationhasmorethanonemainlongitudinalmemberinits
construction.Toattainthedesiredaerodynamicshape,ribsorbulkheadsareoftenincluded.
•Theboxbeamwingconfigurationhastwomainlongitudinalmembersandconnectingbulkheads
toattaintherequiredairfoilcontour.
Chapter 0.-Basic Structural Elements in Aircraft Structure
•WingConfigurations
•Thesethreedesignconfigurationsareconsideredasthebasicdesigns,andaircraftmanufacturers
mayadoptamodifiedconfiguration.
•Inthemonosparwingconfiguration,onlyonemainspanwisememberispresent.Ribsorbulkheads
areusedtoprovidethenecessaryaerodynamiccontourorshapetotheairfoil.
•Themultisparwingconfigurationhasmorethanonemainlongitudinalmemberinits
construction.Toattainthedesiredaerodynamicshape,ribsorbulkheadsareoftenincluded.
•Theboxbeamwingconfigurationhastwomainlongitudinalmembersandconnectingbulkheads
toattaintherequiredairfoilcontour.
Chapter 0.-Basic Structural Elements in Aircraft Structure
39
•تامازلا ساسارب لاب عاونا
•هفيظويلصالاباميپاوهنيماتشخبمظعايآربدرومزاينياربزاورپاسا.
•اه لابهتسبهبتامازلاهطوبرمردلاكشانوگانوگهتخاسيمدنوش.
Chapter 0.-Basic Structural Elements in Aircraft Structure
•تامازلا ساسارب لاب عاونا
•هفيظويلصالاباميپاوهنيماتشخبمظعايآربدرومزاينياربزاورپاسا.
•اه لابهتسبهبتامازلاهطوبرمردلاكشانوگانوگهتخاسيمدنوش.
Chapter 0.-Basic Structural Elements in Aircraft Structure
40
• تاصخشميسدنهلاب مهم
• لاب احاسم(:(Sلااب زا امن رد لاب احاسم.
• هناهد(نپسا ) لاب(:(b هلصافميقتسمكون زا اسار لابكونپچ لاب.
• رتو(درُك ) لاب(c:) هلصافميقتسمرظن دروم عطقم رد لاب رارف هبل ات هلمح هبل.درُک
یا هزاس و لاب یاتسار رب دومعدرُکیکیمانیدورئآ نایرج یاتسار رددشاب یم .هک هیواز
نیباه نآ هیواز نامه اقیقح رد ،پیئوسدشاب یم.
• ابسنيرظن
َ
م(:(AspectR لاب هناهد روذجمميسقتنآ احاسم رب .
??????
2
??????
=
??????
ҧ??????
•بیرضیرغلا(:(TaperRابسنلاب هشیر رتو لوط هب لاب کون رتو لوط .
????????????
????????????
• اماخض ابسن:(TicknessR)رظندروم عطقم رد نآ رتو لوط هب لاب اماخض ابسن.
??????
??????
•هيوازپييوس(?????? :) نياهيواز ردياه ايعقوميفلتخم لاب رتو زافيرعتدوش يمهك
نيرت فورعماه نآهيوازپييوس ردكياسا رتو مراهچ.
•هيواز دروخربايهيواز لاب بصن(??????
� :)هيوازنيب رتوهشير لحم رد قفا طخ و لاب
هندب هب لاب لاصتا
Chapter 0.-Basic Structural Elements in Aircraft Structure
• تاصخشميسدنهلاب مهم
• لاب احاسم(:(Sلااب زا امن رد لاب احاسم.
• هناهد(نپسا ) لاب(:(b هلصافميقتسمكون زا اسار لابكونپچ لاب.
• رتو(درُك ) لاب(c:) هلصافميقتسمرظن دروم عطقم رد لاب رارف هبل ات هلمح هبل.درُک
یا هزاس و لاب یاتسار رب دومعدرُکیکیمانیدورئآ نایرج یاتسار رددشاب یم .هک هیواز
نیباه نآ هیواز نامه اقیقح رد ،پیئوسدشاب یم.
• ابسنيرظن
َ
م(:(AspectR لاب هناهد روذجمميسقتنآ احاسم رب .
??????
2
??????
=
??????
ҧ??????
•بیرضیرغلا(:(TaperRابسنلاب هشیر رتو لوط هب لاب کون رتو لوط .
????????????
????????????
• اماخض ابسن:(TicknessR)رظندروم عطقم رد نآ رتو لوط هب لاب اماخض ابسن.
??????
??????
•هيوازپييوس(?????? :) نياهيواز ردياه ايعقوميفلتخم لاب رتو زافيرعتدوش يمهك
نيرت فورعماه نآهيوازپييوس ردكياسا رتو مراهچ.
•هيواز دروخربايهيواز لاب بصن(??????
� :)هيوازنيب رتوهشير لحم رد قفا طخ و لاب
هندب هب لاب لاصتا
Chapter 0.-Basic Structural Elements in Aircraft Structure
41
• تاصخشميسدنهلاب مهم
• هیوازشچيپ(?????? :)هيوازنيب رتو طخكون رتو طخ اب لابهشيرنآ.
•هيوازلاردهياد(??????
�):هيوازنيبلاباميپاوهوطخقفارديامنوربور.مزلاهبركذاساهكنياهيوازيمدناوتابثم
(لابليامتمهبلااب)واييفنم(لابليامتمهبنيياپ)دشاب.رديتلاحهكنياهيوازيفنماساهبنآهيوازلاردهنا
زينيمدنيوگ.
Chapter 0.-Basic Structural Elements in Aircraft Structure
• تاصخشميسدنهلاب مهم
• هیوازشچيپ(?????? :)هيوازنيب رتو طخكون رتو طخ اب لابهشيرنآ.
•هيوازلاردهياد(??????
�):هيوازنيبلاباميپاوهوطخقفارديامنوربور.مزلاهبركذاساهكنياهيوازيمدناوتابثم
(لابليامتمهبلااب)واييفنم(لابليامتمهبنيياپ)دشاب.رديتلاحهكنياهيوازيفنماساهبنآهيوازلاردهنا
زينيمدنيوگ.
Chapter 0.-Basic Structural Elements in Aircraft Structure
42
• لكشيا هزاسهندب هب نآ لاصتا و لاب
• لكش هب لابريت :يزاين زا هدافتسا هبياضعايا هزاسرگيديارب راهماه نآدرادن دوجو.
• لابدسيرب :اه لاب هبيياهنتناكما لمحتياهراب زا دنرادن ار دوخ رب دراواه لباك وياه هليميدلاوفياربيوقتا
اه نآ هدافتسايمدوش.
Chapter 0.-Basic Structural Elements in Aircraft Structure
• لكشيا هزاسهندب هب نآ لاصتا و لاب
• لكش هب لابريت :يزاين زا هدافتسا هبياضعايا هزاسرگيديارب راهماه نآدرادن دوجو.
• لابدسيرب :اه لاب هبيياهنتناكما لمحتياهراب زا دنرادن ار دوخ رب دراواه لباك وياه هليميدلاوفياربيوقتا
اه نآ هدافتسايمدوش.
Chapter 0.-Basic Structural Elements in Aircraft Structure
43
• ايعقوميريگرارقلاب:
•لااب لاب
•طسو لاب
• لابنيياپ
Chapter 0.-Basic Structural Elements in Aircraft Structure
• ايعقوميريگرارقلاب:
•لااب لاب
•طسو لاب
• لابنيياپ
Chapter 0.-Basic Structural Elements in Aircraft Structure
44
•لاب یلصا ءاضعا:
•اهراپسا ،هكياضعايلصا هزاسياه لابيزورما و دنتسهنيرتشيب لمحت ار لاب هب هدراو رابيمدننك.
•اهرگنيرتسا ،هك ردياتسار لاب هناهدهديشكيمدنوش.
•اه بير ورگيدياه هزاس ،هدنهد لكشهك ردياتسار لاب رارف هبل ات هلمح هبل زا رتوهديشكدنا هدش.
• ،لاب هتسوپهك هباه بير واهرگنيرتسا شخب دوخ و اسا هدش لصتميكچوك لمحت ار لاب هب دراو راب زايمدنك.
Chapter 0.-Basic Structural Elements in Aircraft Structure
•لاب یلصا ءاضعا:
•اهراپسا ،هكياضعايلصا هزاسياه لابيزورما و دنتسهنيرتشيب لمحت ار لاب هب هدراو رابيمدننك.
•اهرگنيرتسا ،هك ردياتسار لاب هناهدهديشكيمدنوش.
•اه بير ورگيدياه هزاس ،هدنهد لكشهك ردياتسار لاب رارف هبل ات هلمح هبل زا رتوهديشكدنا هدش.
• ،لاب هتسوپهك هباه بير واهرگنيرتسا شخب دوخ و اسا هدش لصتميكچوك لمحت ار لاب هب دراو راب زايمدنك.
Chapter 0.-Basic Structural Elements in Aircraft Structure
45
• راتخاسيساساياربنامديچياضعا هزاسيالاب:
•راتخاسكتهراپسا،
•ردنياراتخاساهنتكيراپساهبناونعهزاسيلصاهدننك لمحترابردلوطهناهدلابهديشكدوش يم.
Chapter 0.-Basic Structural Elements in Aircraft Structure
• راتخاسيساساياربنامديچياضعا هزاسيالاب:
•راتخاسكتهراپسا،
•ردنياراتخاساهنتكيراپساهبناونعهزاسيلصاهدننك لمحترابردلوطهناهدلابهديشكدوش يم.
Chapter 0.-Basic Structural Elements in Aircraft Structure
46
• راتخاسيساساياربنامديچياضعا هزاسيالاب:.
•راتخاسدنچهراپسا،
•ردنياراتخاسزاودايدنچراپساهبناونعياه هزاسيلصاهدننك لمحتبرارد
لابهدافتسادوش يم.
Chapter 0.-Basic Structural Elements in Aircraft Structure
• راتخاسيساساياربنامديچياضعا هزاسيالاب:.
•راتخاسدنچهراپسا،
•ردنياراتخاسزاودايدنچراپساهبناونعياه هزاسيلصاهدننك لمحتبرارد
لابهدافتسادوش يم.
Chapter 0.-Basic Structural Elements in Aircraft Structure
47
• راتخاسيساساياربنامديچياضعا هزاسيالاب:.
•راتخاسهبعج،ریت
•ردنيا،راتخاس الاومعمزاودراپسايلصاردهزاسلابهدافتساهدشدوش يم.
•ابلاصتاياه بيريلصاهبرگيدكيامواقملابشيازفاهتفاياسا.
•زاياه هقروراد جومنيبياه بيريلصاوزينزاهتسوپفاصينوريبياربلمحترتهبياهرابيششكويراشف
هدافتسادوش یم.
•رديضعبزادراومزاياه هدننك ايوقترت يوقياج هبنياياه هقروراد جومياربلاصتااه بيرمه هبهدافتساگ يمددر.
•ردياهاميپاوهيربرفاسم الاومعمنياعونراتخاسياربهزاسلابراك هبيمدور.
Chapter 0.-Basic Structural Elements in Aircraft Structure
• راتخاسيساساياربنامديچياضعا هزاسيالاب:.
•راتخاسهبعج،ریت
•ردنيا،راتخاس الاومعمزاودراپسايلصاردهزاسلابهدافتساهدشدوش يم.
•ابلاصتاياه بيريلصاهبرگيدكيامواقملابشيازفاهتفاياسا.
•زاياه هقروراد جومنيبياه بيريلصاوزينزاهتسوپفاصينوريبياربلمحترتهبياهرابيششكويراشف
هدافتسادوش یم.
•رديضعبزادراومزاياه هدننك ايوقترت يوقياج هبنياياه هقروراد جومياربلاصتااه بيرمه هبهدافتساگ يمددر.
•ردياهاميپاوهيربرفاسم الاومعمنياعونراتخاسياربهزاسلابراك هبيمدور.
Chapter 0.-Basic Structural Elements in Aircraft Structure
48
•راپسا
•راپسانيرت يلصاوضعهزاسلابوهدننك لمحتشخبمظعارابهدراوهبنآاساهكردلوطهناهدلابهديشكشهدورد
هشيرلابهبهزاسهندبلصتمدوش يم.
•هتسبهبطيارشوياهرايعميحارط،اهراپسا،يزلف،يبوچواييتيزوپماكدنتسه.
•هزورمارتشيبناگدنزاساميپاوهزاياهراپسايموينيمولآابعطاقمهچراپكيوايدنچهكتهدافتسادننك يم.
Chapter 0.-Basic Structural Elements in Aircraft Structure
•راپسا
•راپسانيرت يلصاوضعهزاسلابوهدننك لمحتشخبمظعارابهدراوهبنآاساهكردلوطهناهدلابهديشكشهدورد
هشيرلابهبهزاسهندبلصتمدوش يم.
•هتسبهبطيارشوياهرايعميحارط،اهراپسا،يزلف،يبوچواييتيزوپماكدنتسه.
•هزورمارتشيبناگدنزاساميپاوهزاياهراپسايموينيمولآابعطاقمهچراپكيوايدنچهكتهدافتسادننك يم.
Chapter 0.-Basic Structural Elements in Aircraft Structure
49
•راپسا
Chapter 0.-Basic Structural Elements in Aircraft Structure
نامدیچ زا ییامناه هیلایتیزوپماک راپسا کی عطقم حطس رد
ییاه لکشیزلف راپسا زا
حطسعطقمیاهراپسایبوچ
•راپسا
Chapter 0.-Basic Structural Elements in Aircraft Structure
نامدیچ زا ییامناه هیلایتیزوپماک راپسا کی عطقم حطس رد
ییاه لکشیزلف راپسا زا
حطسعطقمیاهراپسایبوچ
50
•راپسا
Chapter 0.-Basic Structural Elements in Aircraft Structure
ایلپ ناج(بو ) اب راپسایاه هدننک ایوقتیدومع راپساییاپرخ
راپسا کیFail Safe اببوهدش چرپ اب راپسا کیایلپبوراد جوم(یسونیس)
•راپسا
Chapter 0.-Basic Structural Elements in Aircraft Structure
ایلپ ناج(بو ) اب راپسایاه هدننک ایوقتیدومع راپساییاپرخ
راپسا کیFail Safe اببوهدش چرپ اب راپسا کیایلپبوراد جوم(یسونیس)
51
•بیر
•اه بیرياضعايا هزاسدنتسههكردياتساررتولاب
رارقهتفرگوابلاصتاهباهراپساواهرگنيرتسا،سراتخا
يلكلابارليكشتيمدنهد.
•هفيظواه بيرنداد لكشهبهتسوپلابوزيناقتنالرابزا
هتسوپهباهرگنيرتساواهراپسااسا.
•زااه بيررداه مُدوحوطسيلرتنكزينهدافتساش يمدو.
•اه بير الاومعمزابوچوايزلفهتخاسدنوش يم.
•ياهاميپاوهيارادياهراپسايبوچدنناوت يمزا
ياه بيريبوچواييزلفهدافتسادننكامااهاميپاوهي
يارادياهراپسايزلف اامومعزابيرياهيزلفهرهب
دنرب يم.
Chapter 0.-Basic Structural Elements in Aircraft Structure
زا يا هنومنياه بيريبوچيياپرخ و
يا هيلا هيلايزاس كبسهدش
•بیر
•اه بیرياضعايا هزاسدنتسههكردياتساررتولاب
رارقهتفرگوابلاصتاهباهراپساواهرگنيرتسا،سراتخا
يلكلابارليكشتيمدنهد.
•هفيظواه بيرنداد لكشهبهتسوپلابوزيناقتنالرابزا
هتسوپهباهرگنيرتساواهراپسااسا.
•زااه بيررداه مُدوحوطسيلرتنكزينهدافتساش يمدو.
•اه بير الاومعمزابوچوايزلفهتخاسدنوش يم.
•ياهاميپاوهيارادياهراپسايبوچدنناوت يمزا
ياه بيريبوچواييزلفهدافتسادننكامااهاميپاوهي
يارادياهراپسايزلف اامومعزابيرياهيزلفهرهب
دنرب يم.
Chapter 0.-Basic Structural Elements in Aircraft Structure
زا يا هنومنياه بيريبوچيياپرخ و
يا هيلا هيلايزاس كبسهدش
52Chapter 0.-Basic Structural Elements in Aircraft Structure
53Chapter 0.-Basic Structural Elements in Aircraft Structure
•لِگاج ردبیر
•لاومعمردلحملاصتاجنلفبیرابپَک،راپساکییگتفرورفردبیراهجداجیایحطس مه(هبروظنمیگرارقیرهتسوپ
ربیوراهنآ)،داجیادوش یمهکهبنآ ااحلاطصالِگاجهتفگدوش یم.
•لِگاج ردبیر
•لاومعمردلحملاصتاجنلفبیرابپَک،راپساکییگتفرورفردبیراهجداجیایحطس مه(هبروظنمیگرارقیرهتسوپ
ربیوراهنآ)،داجیادوش یمهکهبنآ ااحلاطصالِگاجهتفگدوش یم.
54Chapter 0.-Basic Structural Elements in Aircraft Structure
• حرطبیریا هزاس وبیریکیمانیدورئآ لاب ردکَب اپئوس
• حرطبیریا هزاس وبیریکیمانیدورئآ لاب ردکَب اپئوس
55
•لاب هتسوپ
•ردياهاميپاوهنرد
ُ
ميزورما،هفيظوهتسوپلابلمحتشخبيكچوكزارابيلامعاهبنآولاقتناشخبمظعانآهب
ياضعايا هزاسرگيددنناماهرگنيرتساواه بيراسا.
•روط هبلومعمردیياه لابابيحارطرسكيريگردهكياراداه ميسوياه هليمهدننك راهميلخادواييجراخنتسيند،مزلا
اساهتسوپلابيمكايوقتدوشاتيياناوتلمحتياهرابيلامعاارهتشاددشاب.
Chapter 0.-Basic Structural Elements in Aircraft Structure
لابياراد هبعج راتخاسريت اب وناكمايزاس هريخذ
راتخاسيا هزاس لابيزلفياراد هتسوپهدننك لمحت
•لاب هتسوپ
•ردياهاميپاوهنرد
ُ
ميزورما،هفيظوهتسوپلابلمحتشخبيكچوكزارابيلامعاهبنآولاقتناشخبمظعانآهب
ياضعايا هزاسرگيددنناماهرگنيرتساواه بيراسا.
•روط هبلومعمردیياه لابابيحارطرسكيريگردهكياراداه ميسوياه هليمهدننك راهميلخادواييجراخنتسيند،مزلا
اساهتسوپلابيمكايوقتدوشاتيياناوتلمحتياهرابيلامعاارهتشاددشاب.
Chapter 0.-Basic Structural Elements in Aircraft Structure
لابياراد هبعج راتخاسريت اب وناكمايزاس هريخذ
راتخاسيا هزاس لابيزلفياراد هتسوپهدننك لمحت
56
• ژانپمِا /رزیلابَتسِا
•روظنمزاژانپمِاهكرديخربزاعجارمزانآهبناونعمُدایهدننکرادیاپ(رزیلابَتسِا)زيندايدوش يم،هعومجم ُدياه ميقفاو
يدومعرارقهتفرگردياهتناياهاميپاوهلوادتماسا.نیااه مُدیراتخاسهیبشلابدنراد.
• لحميريگرارق دادعت وياه مديقفا واييدومعزيندناوت يمدشاب توافتم.
•ياه شخبيلصاژانپمِا طورخم ،يمد(tail (Rear) Cone) حوطس ،يكيمانيدورئآ حوطس و اباثيكيمانيدورئآكرحتم.
Chapter 0.-Basic Structural Elements in Aircraft Structure
يخرب زاياهراتخاس موسرمياربژانپمِا فلتخم ياه الاحيريگرارقاه مُد
• ژانپمِا /رزیلابَتسِا
•روظنمزاژانپمِاهكرديخربزاعجارمزانآهبناونعمُدایهدننکرادیاپ(رزیلابَتسِا)زيندايدوش يم،هعومجم ُدياه ميقفاو
يدومعرارقهتفرگردياهتناياهاميپاوهلوادتماسا.نیااه مُدیراتخاسهیبشلابدنراد.
• لحميريگرارق دادعت وياه مديقفا واييدومعزيندناوت يمدشاب توافتم.
•ياه شخبيلصاژانپمِا طورخم ،يمد(tail (Rear) Cone) حوطس ،يكيمانيدورئآ حوطس و اباثيكيمانيدورئآكرحتم.
Chapter 0.-Basic Structural Elements in Aircraft Structure
يخرب زاياهراتخاس موسرمياربژانپمِا فلتخم ياه الاحيريگرارقاه مُد
57
•ياه شخب رد ژانپمِا فلتخميياميپاوهلوادتم راتخاس اب
•دنراد لاب هیبش یراتخاس اه مُد.
Chapter 0.-Basic Structural Elements in Aircraft Structure
یدومع هدننک رادیاپ( هیبشمُد ) راتخاس ورِدار
•ياه شخب رد ژانپمِا فلتخميياميپاوهلوادتم راتخاس اب
•دنراد لاب هیبش یراتخاس اه مُد.
Chapter 0.-Basic Structural Elements in Aircraft Structure
یدومع هدننک رادیاپ( هیبشمُد ) راتخاس ورِدار
58
•دراناک
•دراناککیحطس(هچلاب)یقفاویورهندباساهکردیولجلابیلصارارقدراداتتابثولرتنکیلوطارفمهاردنک.
هتسبهبعون،بصننکمماساکیحطسیسدنه،اباثکرحتمایریغتمدشابونکمماساحوطسنکیلرتاررد
دوخیاجدهدبایدهدن.
Chapter 0.-Basic Structural Elements in Aircraft Structure
•دراناک
•دراناککیحطس(هچلاب)یقفاویورهندباساهکردیولجلابیلصارارقدراداتتابثولرتنکیلوطارفمهاردنک.
هتسبهبعون،بصننکمماساکیحطسیسدنه،اباثکرحتمایریغتمدشابونکمماساحوطسنکیلرتاررد
دوخیاجدهدبایدهدن.
Chapter 0.-Basic Structural Elements in Aircraft Structure
59
• حوطسلرتنك
•هفيظوحوطسيلرتنكاميپاوه،ايادهنآردتاهج
بولطمردنيح،اساخربزاورپواياسشناسا.
•نيا،حوطس الاومعمتروصبييلاولهبياه شخب
فلتخملصتميمدنوش.
•نياحوطسهبروطيلكهبودهتسدهدمعيسقتم
دنوش يم:حوطسلرتنكيلصاوحوطسلرتنككيكم.
Chapter 0.-Basic Structural Elements in Aircraft Structure
• حوطسلرتنك
•هفيظوحوطسيلرتنكاميپاوه،ايادهنآردتاهج
بولطمردنيح،اساخربزاورپواياسشناسا.
•نيا،حوطس الاومعمتروصبييلاولهبياه شخب
فلتخملصتميمدنوش.
•نياحوطسهبروطيلكهبودهتسدهدمعيسقتم
دنوش يم:حوطسلرتنكيلصاوحوطسلرتنككيكم.
Chapter 0.-Basic Structural Elements in Aircraft Structure
60
• حوطسلرتنكيلصا
•حوطسيلرتنكيلصادنترابعزانورليااه،اهروتيولاوردار.
•لحمبصناهروتيولاوردارهببيترتردهبلرارفياه مد
يقفاويدومعاسا.
•اه نورلياروط هبلومعمردهبلرارفاه لاببصندنوش يمو
ناكماناروداميپاوهلوحروحميلوطهندبارمهارف
دننك يم.
Chapter 0.-Basic Structural Elements in Aircraft Structure
هنومن دنچ يامننورليا ردياه لابفلتخم
• حوطسلرتنكيلصا
•حوطسيلرتنكيلصادنترابعزانورليااه،اهروتيولاوردار.
•لحمبصناهروتيولاوردارهببيترتردهبلرارفياه مد
يقفاويدومعاسا.
•اه نورلياروط هبلومعمردهبلرارفاه لاببصندنوش يمو
ناكماناروداميپاوهلوحروحميلوطهندبارمهارف
دننك يم.
Chapter 0.-Basic Structural Elements in Aircraft Structure
هنومن دنچ يامننورليا ردياه لابفلتخم
61
• حوطس عاونالرتنكامیپاوه یکمک
Chapter 0.-Basic Structural Elements in Aircraft Structure
• حوطس عاونالرتنكامیپاوه یکمک
Chapter 0.-Basic Structural Elements in Aircraft Structure
62
• حوطسلرتنك يكمك-پلف
Chapter 0.-Basic Structural Elements in Aircraft Structure
هنومنيزاس هنيهب هدشپلفرلواف پلفهلمح هبل
• حوطسلرتنك يكمك-پلف
Chapter 0.-Basic Structural Elements in Aircraft Structure
هنومنيزاس هنيهب هدشپلفرلواف پلفهلمح هبل
63
• حوطسلرتنك يكمك-اَلسِا
Chapter 0.-Basic Structural Elements in Aircraft Structure
• حوطسلرتنك يكمك-اَلسِا
Chapter 0.-Basic Structural Elements in Aircraft Structure
64
• حوطسلرتنك يكمك-رلیوپسا
Chapter 0.-Basic Structural Elements in Aircraft Structure
رليوپسا ؛يورلابرليوپسا ؛يورهندب
• حوطسلرتنك يكمك-رلیوپسا
Chapter 0.-Basic Structural Elements in Aircraft Structure
رليوپسا ؛يورلابرليوپسا ؛يورهندب
65
• حوطسلرتنك يكمكبت
•ييوريندراوهدشردنيحزاورپزابناجنايرجاوههبحوطسيلرتنك،ثعابدوش يمهكيخربعقاومنابلخدناوتنبيتحار ه
نياحوطساراكرحدهد.نينچمهنكمماساهبليلاديفلتخميكيزاحوطسيلرتنكشيبزادحهبنيمارفبلخنا
ساسحدشاب.ياربلحنياتلاكشمزايحوطسهبمانبتهدافتسايمدوش.
Chapter 0.-Basic Structural Elements in Aircraft Structure
• حوطسلرتنك يكمكبت
•ييوريندراوهدشردنيحزاورپزابناجنايرجاوههبحوطسيلرتنك،ثعابدوش يمهكيخربعقاومنابلخدناوتنبيتحار ه
نياحوطساراكرحدهد.نينچمهنكمماساهبليلاديفلتخميكيزاحوطسيلرتنكشيبزادحهبنيمارفبلخنا
ساسحدشاب.ياربلحنياتلاكشمزايحوطسهبمانبتهدافتسايمدوش.
Chapter 0.-Basic Structural Elements in Aircraft Structure
66
• حوطسلرتنك يكمكبت
Chapter 0.-Basic Structural Elements in Aircraft Structure
• حوطسلرتنك يكمكبت
Chapter 0.-Basic Structural Elements in Aircraft Structure
67
• حوطسلرتنكيكمك- هنومنيا بت زاميرت هتفرگ رارقيورردار
•
Chapter 0.-Basic Structural Elements in Aircraft Structure
• حوطسلرتنكيكمك- هنومنيا بت زاميرت هتفرگ رارقيورردار
•
Chapter 0.-Basic Structural Elements in Aircraft Structure
68
• حوطسلرتنكيكمك- اهج سنلااب بتكمك هباكرح حطسيلرتنك
Chapter 0.-Basic Structural Elements in Aircraft Structure
• حوطسلرتنكيكمك- اهج سنلااب بتكمك هباكرح حطسيلرتنك
Chapter 0.-Basic Structural Elements in Aircraft Structure
69
• حوطسلرتنكيكمك- هدافتسا دروم يوورس بتياربكمك هباكرح حطسيلرتنك
Chapter 0.-Basic Structural Elements in Aircraft Structure
• حوطسلرتنكيكمك- هدافتسا دروم يوورس بتياربكمك هباكرح حطسيلرتنك
Chapter 0.-Basic Structural Elements in Aircraft Structure
70
• حوطسلرتنكيكمك- سنلااب دض بتاي دضوورس
Chapter 0.-Basic Structural Elements in Aircraft Structure
• حوطسلرتنكيكمك- سنلااب دض بتاي دضوورس
Chapter 0.-Basic Structural Elements in Aircraft Structure
71
• حوطسلرتنكيكمك-یرنف بت
Chapter 0.-Basic Structural Elements in Aircraft Structure
• حوطسلرتنكيكمك-یرنف بت
Chapter 0.-Basic Structural Elements in Aircraft Structure
72
• دورف هبارا(ریگنیدنَل)
•هفيظوهبارادورفندروآ مهارفناكمااسشنواكرح
نميااميپاوهيورنيمزوايدنابياهوانرباميپاوهاسا.
يخربزااهاميپاوهزينناكمااسشنواساخربزاطسح
بآاردنراد.
•ياه هبارادورفديابهبيردقمكحتسمويوقدنشابكه
دنناوتبياهرابيلامعاهباميپاوهردهظحلدورفارحتلم
هدركونامزمهيارادنزويمكزیندنشاب.زانياوررد
اخاسنآاهزاداوميفلتخمنوچ،دلاوفموينيمولآو
مويزينمهدافتسايمدوش.
•كيتسلاياههبارادورفزينزاداوميارادماكحتسالاابرد
رباربكاكطصاهتخاسيمدنوش.
•هباراياهدورف الاومعمزاياه متسيسهدننك بذجبرضه
ياربندرك يثنخياهرابيلامعازينهدافتسادننك يم.
Chapter 0.-Basic Structural Elements in Aircraft Structure
• دورف هبارا(ریگنیدنَل)
•هفيظوهبارادورفندروآ مهارفناكمااسشنواكرح
نميااميپاوهيورنيمزوايدنابياهوانرباميپاوهاسا.
يخربزااهاميپاوهزينناكمااسشنواساخربزاطسح
بآاردنراد.
•ياه هبارادورفديابهبيردقمكحتسمويوقدنشابكه
دنناوتبياهرابيلامعاهباميپاوهردهظحلدورفارحتلم
هدركونامزمهيارادنزويمكزیندنشاب.زانياوررد
اخاسنآاهزاداوميفلتخمنوچ،دلاوفموينيمولآو
مويزينمهدافتسايمدوش.
•كيتسلاياههبارادورفزينزاداوميارادماكحتسالاابرد
رباربكاكطصاهتخاسيمدنوش.
•هباراياهدورف الاومعمزاياه متسيسهدننك بذجبرضه
ياربندرك يثنخياهرابيلامعازينهدافتسادننك يم.
Chapter 0.-Basic Structural Elements in Aircraft Structure
73
• هزاسهدنراد هگنروتوم/لِسان
•ياه فلاغيا هتسبدنتسههكروتومولعتمتاق
نآرداه نآياجدنريگ يم.
•روط هبلومعمياربشهاكاسپردنيااه هزاس
ياراديراتخاسينحنملكشدنتسه.
•ردبلغاياهاميپاوهكتهروتومنيااه هزاس
ردولجوايبقعهندبرارقهداددنوش يم.
•ردياهاميپاوهدنچهروتومنيااه شخبهب
اه لابوايژانپمِالصتمدنوش يم.
•كيلِسانلماشروتوموتاقلعتم،نآهياپياه
،روتومياضعايا هزاس،فلتخمهراويدمواقمرد
رباربترارحوهتسوپينوريباسا.
•رديخربدراومزااه لِسانياربیریگ رارقهبارا
دورفزينهدافتسادوش يم.
Chapter 0.-Basic Structural Elements in Aircraft Structure
• هزاسهدنراد هگنروتوم/لِسان
•ياه فلاغيا هتسبدنتسههكروتومولعتمتاق
نآرداه نآياجدنريگ يم.
•روط هبلومعمياربشهاكاسپردنيااه هزاس
ياراديراتخاسينحنملكشدنتسه.
•ردبلغاياهاميپاوهكتهروتومنيااه هزاس
ردولجوايبقعهندبرارقهداددنوش يم.
•ردياهاميپاوهدنچهروتومنيااه شخبهب
اه لابوايژانپمِالصتمدنوش يم.
•كيلِسانلماشروتوموتاقلعتم،نآهياپياه
،روتومياضعايا هزاس،فلتخمهراويدمواقمرد
رباربترارحوهتسوپينوريباسا.
•رديخربدراومزااه لِسانياربیریگ رارقهبارا
دورفزينهدافتسادوش يم.
Chapter 0.-Basic Structural Elements in Aircraft Structure
74Chapter 0.-Basic Structural Elements in Aircraft Structure
رد ترارح ربارب رد مواقم هراويدلسان(لاوریاف)
ياه ياپهدنراد هگنروتوم
روتوم لوادتم یاه ششوپ
رد ترارح ربارب رد مواقم هراويدلسان(لاوریاف)
ياه ياپهدنراد هگنروتوم
روتوم لوادتم یاه ششوپ
75Chapter 0.-Basic Structural Elements in Aircraft Structure
76
•Aircraft Cutaway
Chapter 0.-Definitions
•Aircraft Cutaway
Chapter 0.-Definitions
77Chapter 0.-Definitions
78Chapter 0.-Basic Structural Elements in Aircraft Structure
79Chapter 0.-Definitions
•Airframe
•Airframe
80
•General Function of Aircraft Structures
•The structures of most flight vehicles are thin walled structures (shells)
•Resistsappliedloads(Aerodynamicloadsactingonthewingstructure)
•Providestheaerodynamicshape
•Protectsthecontentsfromtheenvironment
Chapter 0.-Definitions
•Primary structure
•Acriticalload-bearingstructureonanaircraft.Ifthisstructureisseverelydamaged,the
aircraftcannotfly.
•Secondary structure
•Structuralelementsmainlytoprovideenhancedaerodynamics.Fairings,forinstance,are
foundwherethewingmeetsthebodyoratvariouslocationsontheleadingortrailing
edgeofthewing.
•General Function of Aircraft Structures
•The structures of most flight vehicles are thin walled structures (shells)
•Resistsappliedloads(Aerodynamicloadsactingonthewingstructure)
•Providestheaerodynamicshape
•Protectsthecontentsfromtheenvironment
Chapter 0.-Definitions
•Primary structure
•Acriticalload-bearingstructureonanaircraft.Ifthisstructureisseverelydamaged,the
aircraftcannotfly.
•Secondary structure
•Structuralelementsmainlytoprovideenhancedaerodynamics.Fairings,forinstance,are
foundwherethewingmeetsthebodyoratvariouslocationsontheleadingortrailing
edgeofthewing.
81Chapter 0.-Definitions
•Monocoque structure
•Unstiffenedshells.mustberelativelythicktoresistbending,
compressive,andtorsionalloads.
•Semi -Monocoque structure
•Constructionswithstiffeningmembersthatmayalsoberequired
todiffuseconcentratedloadsintothecover.Moreefficienttypeof
constructionthatpermitsmuchthinnercoveringshell.
•Monocoque structure
•Unstiffenedshells.mustberelativelythicktoresistbending,
compressive,andtorsionalloads.
•Semi -Monocoque structure
•Constructionswithstiffeningmembersthatmayalsoberequired
todiffuseconcentratedloadsintothecover.Moreefficienttypeof
constructionthatpermitsmuchthinnercoveringshell.
82
•WING LAYOUT
Chapter 0.-Definitions
•WING LAYOUT
Chapter 0.-Definitions
83
•Function of Skin
•Reactstheappliedtorsionandshearforces
•transmitsaerodynamicforcestothelongitudinalandtransversesupportingmembers
•actswiththelongitudinalmembersinresistingtheappliedbendingandaxialloads
•actswiththetransversemembersinreactingthehoop,orcircumferential,loadwhenthe
structureispressurized.
•Function of Spar
•resistbendingandaxialloadsandformthewingboxforstabletorsionresistance
•Function of Stiffener or Stringers
•resistbendingandaxialloadsalongwiththeskin
•dividetheskinintosmallpanelsandtherebyincreaseitsbucklingandfailingstresses
•actwiththeskininresistingaxialloadscausedbypressurization
Chapter 0.-Definitions
•Function of Skin
•Reactstheappliedtorsionandshearforces
•transmitsaerodynamicforcestothelongitudinalandtransversesupportingmembers
•actswiththelongitudinalmembersinresistingtheappliedbendingandaxialloads
•actswiththetransversemembersinreactingthehoop,orcircumferential,loadwhenthe
structureispressurized.
•Function of Spar
•resistbendingandaxialloadsandformthewingboxforstabletorsionresistance
•Function of Stiffener or Stringers
•resistbendingandaxialloadsalongwiththeskin
•dividetheskinintosmallpanelsandtherebyincreaseitsbucklingandfailingstresses
•actwiththeskininresistingaxialloadscausedbypressurization
Chapter 0.-Definitions
84Chapter 0.-Definitions
85
•Simplifications
•Thebehaviorofthesestructuralelementsisoftenidealizedtosimplifytheanalysisoftheassembled
component.
•Severallongitudinalmaybelumpedintoasingleeffectivelongitudinaltoshortencomputations.
•Thewebs(skinandsparwebs)carryonlyshearingstresses.
•Thelongitudinalelementscarryonlyaxialstress.
•Thetransverseframesandribsarerigidwithintheirownplanes,sothatthecrosssectionis
maintainedunchangedduringloading.
Chapter 0.-Definitions
•Simplifications
•Thebehaviorofthesestructuralelementsisoftenidealizedtosimplifytheanalysisoftheassembled
component.
•Severallongitudinalmaybelumpedintoasingleeffectivelongitudinaltoshortencomputations.
•Thewebs(skinandsparwebs)carryonlyshearingstresses.
•Thelongitudinalelementscarryonlyaxialstress.
•Thetransverseframesandribsarerigidwithintheirownplanes,sothatthecrosssectionis
maintainedunchangedduringloading.
Chapter 0.-Definitions
86
•Traditionalmetallicmaterialsusedinaircraftstructuresarealuminum,titanium,andsteelalloys.
•Inthepastthreedecades,applicationsofadvancedfibercompositeshaverapidlygainedmomentum.
•Todate,somenewcommercialjets,suchastheBoeing787,alreadycontaincompositematerialsupto
50%oftheirstructuralweight.
•Selectionofaircraftmaterialsdependsonmanyconsiderationsthatcan,ingeneral,be
categorizedascostandstructuralperformance.
•Costincludesinitialmaterialcost,manufacturingcost,andmaintenancecost.
•Thekeymaterialpropertiesthatarepertinenttomaintenancecostandstructuralperformance
areasfollows:
•Density(weight),Stiffness(Young'smodulus),Strength(ultimateandyieldstrengths),Durability
(fatigue),Damagetolerance(fracturetoughnessandcrackgrowth),Corrosion.
•Seldomisasinglematerialabletodeliveralldesiredpropertiesinallcomponentsoftheaircraftstructure.A
combinationofvariousmaterialsisoftennecessary.
Chapter 0.-Aircraft Materials
•Traditionalmetallicmaterialsusedinaircraftstructuresarealuminum,titanium,andsteelalloys.
•Inthepastthreedecades,applicationsofadvancedfibercompositeshaverapidlygainedmomentum.
•Todate,somenewcommercialjets,suchastheBoeing787,alreadycontaincompositematerialsupto
50%oftheirstructuralweight.
•Selectionofaircraftmaterialsdependsonmanyconsiderationsthatcan,ingeneral,be
categorizedascostandstructuralperformance.
•Costincludesinitialmaterialcost,manufacturingcost,andmaintenancecost.
•Thekeymaterialpropertiesthatarepertinenttomaintenancecostandstructuralperformance
areasfollows:
•Density(weight),Stiffness(Young'smodulus),Strength(ultimateandyieldstrengths),Durability
(fatigue),Damagetolerance(fracturetoughnessandcrackgrowth),Corrosion.
•Seldomisasinglematerialabletodeliveralldesiredpropertiesinallcomponentsoftheaircraftstructure.A
combinationofvariousmaterialsisoftennecessary.
Chapter 0.-Aircraft Materials
87Chapter 0.-Aircraft Materials
Mechanical properties of metals at room temperature in aircraft structures
Mechanical properties of metals at room temperature in aircraft structures
88
•AluminumAlloys
•Aluminumalloyshaveplayedadominantroleinaircraftstructuresformanydecades.
•Theyoffergoodmechanicalpropertieswithlowweight.
•Amongthealuminumalloys,the2024and7075alloysareperhapsthemostused.
•The2024alloys(2024‐T3,T42)haveexcellentfracturetoughnessandslowcrackgrowthrateaswellas
goodfatiguelife.
•ThecodenumberfollowingTforeachaluminumalloyindicatestheheattreatmentprocess.
•The7075alloys(7075‐T6,T651)havehigherstrengththanthe2024butlowerfracturetoughness.
•The2024‐T3isusedinthefuselageandlowerwingskins,whicharepronetofatigueduetoapplications
ofcyclictensilestresses.
•Fortheupperwingskins,whicharesubjectedtocompressivestresses,fatigueislessofaproblem,and
7075‐T6isused.
•Therecentlydevelopedaluminumlithiumalloysofferimprovedpropertiesoverconventionalaluminum
alloys.Theyareabout10%stifferand10%lighterandhavesuperiorfatigueperformance.
Chapter 0.-Aircraft Materials
•AluminumAlloys
•Aluminumalloyshaveplayedadominantroleinaircraftstructuresformanydecades.
•Theyoffergoodmechanicalpropertieswithlowweight.
•Amongthealuminumalloys,the2024and7075alloysareperhapsthemostused.
•The2024alloys(2024‐T3,T42)haveexcellentfracturetoughnessandslowcrackgrowthrateaswellas
goodfatiguelife.
•ThecodenumberfollowingTforeachaluminumalloyindicatestheheattreatmentprocess.
•The7075alloys(7075‐T6,T651)havehigherstrengththanthe2024butlowerfracturetoughness.
•The2024‐T3isusedinthefuselageandlowerwingskins,whicharepronetofatigueduetoapplications
ofcyclictensilestresses.
•Fortheupperwingskins,whicharesubjectedtocompressivestresses,fatigueislessofaproblem,and
7075‐T6isused.
•Therecentlydevelopedaluminumlithiumalloysofferimprovedpropertiesoverconventionalaluminum
alloys.Theyareabout10%stifferand10%lighterandhavesuperiorfatigueperformance.
Chapter 0.-Aircraft Materials
89
•TitaniumAlloys
•TitaniumsuchasTi-6Al-4V(thenumberindicatestheweightpercentageofthealloying
element)withadensityof4.5g/cm3islighterthansteel(7.8g/cm3)butheavierthan
aluminum(2.7g/cm3).
•Itsultimateandyieldstressesarealmostdoublethoseofaluminum7075‐T6.
•Itscorrosionresistanceingeneralissuperiortobothsteelandaluminumalloys.
•Whilealuminumisusuallynotforapplicationsabove350°F,titanium,ontheotherhand,
canbeusedcontinuouslyupto1000°F.
•Titaniumisdifficulttomachine,andthusthecostofmachiningtitaniumpartsishigh.
•Nearnetshapeformingisaneconomicwaytomanufacturetitaniumparts.
•Despiteitshighcost,titaniumhasfoundincreasinguseinmilitaryaircraft.Forinstance,
theF‐15contained26%(structuralweight)titanium.
Chapter 0.-Aircraft Materials
•TitaniumAlloys
•TitaniumsuchasTi-6Al-4V(thenumberindicatestheweightpercentageofthealloying
element)withadensityof4.5g/cm3islighterthansteel(7.8g/cm3)butheavierthan
aluminum(2.7g/cm3).
•Itsultimateandyieldstressesarealmostdoublethoseofaluminum7075‐T6.
•Itscorrosionresistanceingeneralissuperiortobothsteelandaluminumalloys.
•Whilealuminumisusuallynotforapplicationsabove350°F,titanium,ontheotherhand,
canbeusedcontinuouslyupto1000°F.
•Titaniumisdifficulttomachine,andthusthecostofmachiningtitaniumpartsishigh.
•Nearnetshapeformingisaneconomicwaytomanufacturetitaniumparts.
•Despiteitshighcost,titaniumhasfoundincreasinguseinmilitaryaircraft.Forinstance,
theF‐15contained26%(structuralweight)titanium.
Chapter 0.-Aircraft Materials
90
•Fiber‐ReinforcedComposites
•Materialsmadeintofiberformscanachievesignificantlybettermechanicalpropertiesthantheir
bulkcounterparts.Anotableexampleisglassfiberversusbulkglass.Thetensilestrengthofglass
fibercanbetwoordersofmagnitudehigherthanthatofbulkglass.
•Inthiscentury,fibersciencehasmadegiganticstrides,andmanyhigh‐performancefibershave
beenintroduced.
•Fibersalonearenotsuitableforstructuralapplications.Toutilizethesuperiorpropertiesof
fibers,theyareembeddedinamatrixmaterialthatholdsthefiberstogethertoformasolid
bodycapableofcarryingcomplexloads.
•Matrixmaterialsthatarecurrentlyusedforformingcompositesincludethreemajorcategories:
polymers,metals,andceramics.Theresultingcompositesareusuallyreferredtoaspolymer
matrixcomposites(PMCs),metalmatrixcomposites(MMCs),andceramicmatrixcomposites
(CMCs).
Chapter 0.-Aircraft Materials
•Fiber‐ReinforcedComposites
•Materialsmadeintofiberformscanachievesignificantlybettermechanicalpropertiesthantheir
bulkcounterparts.Anotableexampleisglassfiberversusbulkglass.Thetensilestrengthofglass
fibercanbetwoordersofmagnitudehigherthanthatofbulkglass.
•Inthiscentury,fibersciencehasmadegiganticstrides,andmanyhigh‐performancefibershave
beenintroduced.
•Fibersalonearenotsuitableforstructuralapplications.Toutilizethesuperiorpropertiesof
fibers,theyareembeddedinamatrixmaterialthatholdsthefiberstogethertoformasolid
bodycapableofcarryingcomplexloads.
•Matrixmaterialsthatarecurrentlyusedforformingcompositesincludethreemajorcategories:
polymers,metals,andceramics.Theresultingcompositesareusuallyreferredtoaspolymer
matrixcomposites(PMCs),metalmatrixcomposites(MMCs),andceramicmatrixcomposites
(CMCs).
Chapter 0.-Aircraft Materials
91Chapter 0.-Aircraft Materials
Mechanical properties of fibers
Mechanical properties of fibers
92
•Fiber‐ReinforcedComposites
•Therangeofservicetemperatureofacompositeisoftendeterminedbyitsmatrixmaterial.
•PMCsareusuallyforlowertemperature(lessthan300°F)applications,andCMCsare
intendedforapplicationsinhot(higherthan1500°F)environments,suchasjetengines.
•Fibercompositesarestiff,strong,andlightandarethusmostsuitableforaircraftstructures.
•Theyareoftenusedintheformoflaminatesthatconsistofanumberofunidirectional
laminaewithdifferentfiberorientationstoprovidemultidirectionalloadcapability.
•Compositelaminateshaveexcellentfatiguelife,damagetolerance,andcorrosionresistance.
•Laminateconstructionsofferthepossibilityoftailoringfiberorientationstoachieveoptimal
structuralperformanceofthecompositestructure.
Chapter 0.-Aircraft Materials
•Fiber‐ReinforcedComposites
•Therangeofservicetemperatureofacompositeisoftendeterminedbyitsmatrixmaterial.
•PMCsareusuallyforlowertemperature(lessthan300°F)applications,andCMCsare
intendedforapplicationsinhot(higherthan1500°F)environments,suchasjetengines.
•Fibercompositesarestiff,strong,andlightandarethusmostsuitableforaircraftstructures.
•Theyareoftenusedintheformoflaminatesthatconsistofanumberofunidirectional
laminaewithdifferentfiberorientationstoprovidemultidirectionalloadcapability.
•Compositelaminateshaveexcellentfatiguelife,damagetolerance,andcorrosionresistance.
•Laminateconstructionsofferthepossibilityoftailoringfiberorientationstoachieveoptimal
structuralperformanceofthecompositestructure.
Chapter 0.-Aircraft Materials
93Chapter 0.-Aircraft Materials
Longitudinal mechanical properties of fiber composites.
Longitudinal mechanical properties of fiber composites.
94
✓analysisofstructuralmembersunderpuretorsion
✓reviewofmechanicsofmaterials
✓torsionofmemberswithcircularcrosssections
✓elasticandinelasticstressstatesareconsideredforbothhomogeneousandnon-homogeneousmembers
✓elastictorsionalanalysisofmemberswithsolidorhollownon-circularcrosssections
✓elastictorsionalanalysisofthin-walledmemberswithopencrosssections
✓torsionalanalysisofthin-walledstructureswithclosedcrosssections
✓torsionalanalysisofmulti-cell,thin-walledstiffenedmembers,typicalofwingandfuselagestructures
Chapter 1.-objectives
PURETORSION
✓analysisofstructuralmembersunderpuretorsion
✓reviewofmechanicsofmaterials
✓torsionofmemberswithcircularcrosssections
✓elasticandinelasticstressstatesareconsideredforbothhomogeneousandnon-homogeneousmembers
✓elastictorsionalanalysisofmemberswithsolidorhollownon-circularcrosssections
✓elastictorsionalanalysisofthin-walledmemberswithopencrosssections
✓torsionalanalysisofthin-walledstructureswithclosedcrosssections
✓torsionalanalysisofmulti-cell,thin-walledstiffenedmembers,typicalofwingandfuselagestructures
Chapter 1.-objectives
PURETORSION
•Applications
•RotatingMachinery
•Propellershaft
•Driveshaft
•StructuralSystems
•Landinggearstrut
•Flapdrivemechanism
•Characteristic of Circular Bars
•Whenacircularbaristwisted,itscrosssectionremainsplaneandcircular.Thischaracteristicisdueto
axisymmetricshapeofthecrosssection,andappliestocircularbarsthatareeithersolidorhollow,
homogeneousornon-homogeneous,elasticorinelastic.
•AXISYMMETRICSECTION:Asectionisconsideredaxisymmetricifitsappearance-whenviewedfroma
fixedposition-remainsthesameasitisrotatedaboutitsaxisthroughanarbitraryangle.
Torsional Loads on Circular Shafts
95Chapter 1.-Torsion of Circular Bars
•RotatingMachinery
•Propellershaft
•Driveshaft
•StructuralSystems
•Landinggearstrut
•Flapdrivemechanism
•Characteristic of Circular Bars
•Whenacircularbaristwisted,itscrosssectionremainsplaneandcircular.Thischaracteristicisdueto
axisymmetricshapeofthecrosssection,andappliestocircularbarsthatareeithersolidorhollow,
homogeneousornon-homogeneous,elasticorinelastic.
•AXISYMMETRICSECTION:Asectionisconsideredaxisymmetricifitsappearance-whenviewedfroma
fixedposition-remainsthesameasitisrotatedaboutitsaxisthroughanarbitraryangle.
Torsional Loads on Circular Shafts
95Chapter 1.-Torsion of Circular Bars
96Chapter 1.-Torsion of Circular Bars, Elastic and Homogeneous
•ShaftDeformations
✓Fromobservation,theangleoftwistoftheshaftisproportionalto
theappliedtorqueandtotheshaftlength;
✓Whensubjectedtotorsion,everycross-sectionofacircularshaft
remainsplaneandundistorted.
✓Cross-sectionsforhollowandsolidcircularshaftsremainplainand
undistortedbecauseacircularshaftisaxisymmetric.
✓Cross-sectionsofnoncircular(non-axisymmetric)shaftsaredistorted
whensubjectedtotorsion.
??????∝??????,??????∝�
•ShaftDeformations
✓Fromobservation,theangleoftwistoftheshaftisproportionalto
theappliedtorqueandtotheshaftlength;
✓Whensubjectedtotorsion,everycross-sectionofacircularshaft
remainsplaneandundistorted.
✓Cross-sectionsforhollowandsolidcircularshaftsremainplainand
undistortedbecauseacircularshaftisaxisymmetric.
✓Cross-sectionsofnoncircular(non-axisymmetric)shaftsaredistorted
whensubjectedtotorsion.
??????∝??????,??????∝�
97Chapter 1.-Torsion of Circular Bars
•Rule of Thumb
•Thetorsion-inducedshearstrainisalwaysa
linearfunctionofrwiththemaximumvalueat
theedgeofthecrosssection.
•Thisistrueforallpossibleconditions,whether
thebariselasticorinelastic,homogeneousor
non-homogeneousasshownintheseexamples.
•Shear Strain Calculation
•Basedonthecharacteristicstatedabove,theshear
strainvariationislinear,andisdescribedbythe
equation: ,whereristheradialposition
measuredfromthecenterofthecrosssectionandcis
theradiusofthecrosssection.
•Rule of Thumb
•Thetorsion-inducedshearstrainisalwaysa
linearfunctionofrwiththemaximumvalueat
theedgeofthecrosssection.
•Thisistrueforallpossibleconditions,whether
thebariselasticorinelastic,homogeneousor
non-homogeneousasshownintheseexamples.
•Shear Strain Calculation
•Basedonthecharacteristicstatedabove,theshear
strainvariationislinear,andisdescribedbythe
equation: ,whereristheradialposition
measuredfromthecenterofthecrosssectionandcis
theradiusofthecrosssection.
98Chapter 1.-Torsion of Circular Bars
•Shear Stress and Angle of Twist Calculations
•Fourpossiblescenariosareconsideredforshearstressandangleoftwistcalculations.
•Ineachcasetheequationsusedforthesecalculationsareexplicitlydescribed.
•Itshouldbeobviousthatinallofthesecalculationswearedealingwithmembershavingcircularcross
sections.
•Asyouwillseethedegreeofcomplexityintheanalysisgrowsasweproceedinthefollowingsequence
•ElasticandHomogeneous
•ElasticandNon-homogeneous
•InelasticandHomogeneous
•InelasticandNon-homogeneous
•ResidualStressDistribution
•PowerTransmission
•Restriction:Theappliedtorque(s)mustbeinaplane(s)perpendiculartotheaxisofthebar.
•Shear Stress and Angle of Twist Calculations
•Fourpossiblescenariosareconsideredforshearstressandangleoftwistcalculations.
•Ineachcasetheequationsusedforthesecalculationsareexplicitlydescribed.
•Itshouldbeobviousthatinallofthesecalculationswearedealingwithmembershavingcircularcross
sections.
•Asyouwillseethedegreeofcomplexityintheanalysisgrowsasweproceedinthefollowingsequence
•ElasticandHomogeneous
•ElasticandNon-homogeneous
•InelasticandHomogeneous
•InelasticandNon-homogeneous
•ResidualStressDistribution
•PowerTransmission
•Restriction:Theappliedtorque(s)mustbeinaplane(s)perpendiculartotheaxisofthebar.
99Chapter 1.-Torsion of Circular Bars, Elastic and Homogeneous
✓Torqueappliedtoshaftproducesshearingstressesonthefacesperpendiculartotheaxis.
✓Conditionsofequilibriumrequiretheexistenceofequalstressesonthefacesofthetwoplanes
containingtheaxisoftheshaft
✓Theexistenceoftheaxialshearcomponentsisdemonstratedbyconsideringashaftmadeup
ofaxialslats.
✓Theslatsslidewithrespecttoeachotherwhenequalandoppositetorquesareappliedtothe
endsoftheshaft.
✓Torqueappliedtoshaftproducesshearingstressesonthefacesperpendiculartotheaxis.
✓Conditionsofequilibriumrequiretheexistenceofequalstressesonthefacesofthetwoplanes
containingtheaxisoftheshaft
✓Theexistenceoftheaxialshearcomponentsisdemonstratedbyconsideringashaftmadeup
ofaxialslats.
✓Theslatsslidewithrespecttoeachotherwhenequalandoppositetorquesareappliedtothe
endsoftheshaft.
100Chapter 1.-Torsion of Circular Bars
•ELASTIC,INELASTIC
•Ifthestraininducedinamaterialbytheapplicationofagivenloaddisappearswhenthe
loadisremoved,thematerialissaidtobehaveelastically.Thelargestvalueofthestress
forwhichthematerialbehaveselasticallyiscalledtheelasticlimit.Ifthematerialis
stressedbeyondthislimit,thenitbehavesinelastically(i.e.,uponremovaloftheloadthe
strainwillnotgotozero).
•ISOTROPICMATERIAL
•Amaterialisconsideredisotropicifitsphysicalandmechanicalpropertiesatagiven
pointareindependentofdirection.Mostmetalsareconsideredtobeisotropic.Fibrous
compositematerialstypicallyhavepropertieswhicharedirectiondependent,assuch
theyaregenerallyconsideredtobeanisotropic.
•ELASTIC,INELASTIC
•Ifthestraininducedinamaterialbytheapplicationofagivenloaddisappearswhenthe
loadisremoved,thematerialissaidtobehaveelastically.Thelargestvalueofthestress
forwhichthematerialbehaveselasticallyiscalledtheelasticlimit.Ifthematerialis
stressedbeyondthislimit,thenitbehavesinelastically(i.e.,uponremovaloftheloadthe
strainwillnotgotozero).
•ISOTROPICMATERIAL
•Amaterialisconsideredisotropicifitsphysicalandmechanicalpropertiesatagiven
pointareindependentofdirection.Mostmetalsareconsideredtobeisotropic.Fibrous
compositematerialstypicallyhavepropertieswhicharedirectiondependent,assuch
theyaregenerallyconsideredtobeanisotropic.
101Chapter 1.-Torsion of Circular Bars
•Homogeneous and Non-Homogeneous Bars
•Inthecaseofcircularbars,homogeneousornon-homogeneousreferstoradialmaterial
compositionofthematerial.Ifthematerialvariesalongtheradius,thebarissaidtobe
non-homogeneous.Ifitdoesn'tvarythenitishomogeneous.
•Below are two examples of a homogeneous and non-homogeneous circular shaft.
Homogeneous
Non-homogeneous
•Homogeneous and Non-Homogeneous Bars
•Inthecaseofcircularbars,homogeneousornon-homogeneousreferstoradialmaterial
compositionofthematerial.Ifthematerialvariesalongtheradius,thebarissaidtobe
non-homogeneous.Ifitdoesn'tvarythenitishomogeneous.
•Below are two examples of a homogeneous and non-homogeneous circular shaft.
Homogeneous
Non-homogeneous
102Chapter 1.-Torsion of Circular Bars
•PROPORTIONAL LIMIT
•Proportionallimitisthepointonthestress-straindiagramwherethecurvebecomesnonlinear.The
proportionallimitstressisthevalueofstresscorrespondingtotheelasticlimitofthematerial.Forstrain
levelsbelowtheelasticlimitstrain,Hooke'slawmaybeusedtorelatestresstostrain.Theproportionallimit
iscommonlyassumedtocoincidewiththeyieldpointunlessotherwisestatedintheproblemstatement.
This is a typical shear
stress-strain diagram
Thisisanidealstress-straindiagramwheretheproportionallimit
andtheyieldpointcoincideandstressremainsconstantbeyond
theelasticlimit.Thistypeofmaterialisknownaselasto-plastic.
•PROPORTIONAL LIMIT
•Proportionallimitisthepointonthestress-straindiagramwherethecurvebecomesnonlinear.The
proportionallimitstressisthevalueofstresscorrespondingtotheelasticlimitofthematerial.Forstrain
levelsbelowtheelasticlimitstrain,Hooke'slawmaybeusedtorelatestresstostrain.Theproportionallimit
iscommonlyassumedtocoincidewiththeyieldpointunlessotherwisestatedintheproblemstatement.
This is a typical shear
stress-strain diagram
Thisisanidealstress-straindiagramwheretheproportionallimit
andtheyieldpointcoincideandstressremainsconstantbeyond
theelasticlimit.Thistypeofmaterialisknownaselasto-plastic.
103Chapter 1.-Torsion of Circular Bars
•ELASTOPLASTIC MATERIAL
•Amaterialisconsideredelastoplastic(orelastic-perfectlyplastic)whentheinelasticregionofthestress-
straindiagramisidealizedasastraightline.Ifthematerialbehaveslinearlyintheelasticrange,thenthe
stress-straindiagramconsistsoftwostraightlinesintheelasticandinelasticregionswithdifferentslopes.
•Aslongasthestressislessthantheyieldvalue,Hooke'slaw
maybeusedwhenthematerialbehaveslinearlyelastic.
Whenthestressreachestheyieldstress,thematerialstarts
toyieldandkeepsdeformingplasticallyataconstantstress
level.Iftheloadisremovedatanypointalongthestress-
straindiagram,thentheunloadingpathwillbealonga
straightlinesegmentCDparalleltotheinitialportionAYof
theloadingcurve.Thefactthattheunloadingcurveis
linearisbecausethematerialislinearlyelastic.
•ELASTOPLASTIC MATERIAL
•Amaterialisconsideredelastoplastic(orelastic-perfectlyplastic)whentheinelasticregionofthestress-
straindiagramisidealizedasastraightline.Ifthematerialbehaveslinearlyintheelasticrange,thenthe
stress-straindiagramconsistsoftwostraightlinesintheelasticandinelasticregionswithdifferentslopes.
•Aslongasthestressislessthantheyieldvalue,Hooke'slaw
maybeusedwhenthematerialbehaveslinearlyelastic.
Whenthestressreachestheyieldstress,thematerialstarts
toyieldandkeepsdeformingplasticallyataconstantstress
level.Iftheloadisremovedatanypointalongthestress-
straindiagram,thentheunloadingpathwillbealonga
straightlinesegmentCDparalleltotheinitialportionAYof
theloadingcurve.Thefactthattheunloadingcurveis
linearisbecausethematerialislinearlyelastic.
104Chapter 1.-Torsion of Circular Bars
•Elastic and Homogeneous
•Thetorsion-inducedshearstressvariationinanelastic,homogeneous,andisotropicbarisdetermined
by:
•whereTistheinternaltorqueatthesectiontheshearstressisbeingcalculated,ristheradial
positionofthepointonthecrosssectiontheshearstressissolvedfor,andJisthepolarmomentof
inertiaoftheentirecrosssection.Forsolidbars0<r<candforhollowbarsci<r<co.
•Theaboveshearstressequationisknownastheelastictorsionformula,andwecallitEl-Hocir.This
acronymhelpsustoeasilyrememberunderwhatconditionswecanusetheelastictorsionformula,viz,
elastic,homogeneous,andcircular.
•Theshearstressequationshowsthatforanelasticbar(i.e.,whenthemaximumshearstressislessthan
theproportionallimitshearstressofthematerial),thestressvarieslinearlywithradialposition.Thus,
themaximumshearstressinthiscasewouldbeattheedgeofthecrosssection(i.e.,atthefarthest
distancefromthecenter).
•Elastic and Homogeneous
•Thetorsion-inducedshearstressvariationinanelastic,homogeneous,andisotropicbarisdetermined
by:
•whereTistheinternaltorqueatthesectiontheshearstressisbeingcalculated,ristheradial
positionofthepointonthecrosssectiontheshearstressissolvedfor,andJisthepolarmomentof
inertiaoftheentirecrosssection.Forsolidbars0<r<candforhollowbarsci<r<co.
•Theaboveshearstressequationisknownastheelastictorsionformula,andwecallitEl-Hocir.This
acronymhelpsustoeasilyrememberunderwhatconditionswecanusetheelastictorsionformula,viz,
elastic,homogeneous,andcircular.
•Theshearstressequationshowsthatforanelasticbar(i.e.,whenthemaximumshearstressislessthan
theproportionallimitshearstressofthematerial),thestressvarieslinearlywithradialposition.Thus,
themaximumshearstressinthiscasewouldbeattheedgeofthecrosssection(i.e.,atthefarthest
distancefromthecenter).
105Chapter 1.-Torsion of Circular Bars, Elastic and Homogeneous
•Elastic and Homogeneous
•Herearesomeexamplesofelasticandinelasticstress
variationsforhomogeneousandnon-homogeneous
circularbarsmadeofelastoplasticmaterials.
•Elastic and Homogeneous
•Herearesomeexamplesofelasticandinelasticstress
variationsforhomogeneousandnon-homogeneous
circularbarsmadeofelastoplasticmaterials.
106Chapter 1.-Torsion of Circular Bars, Elastic and Homogeneous
•NetTorqueDuetoInternalStresses
✓Netoftheinternalshearingstressesisaninternaltorque,equal
andoppositetotheappliedtorque,
✓Althoughthenettorqueduetotheshearingstressesisknown,the
distributionofthestressesisnot
✓Distributionofshearingstressesisstaticallyindeterminatemust
considershaftdeformations
✓Unlikethenormalstressduetoaxialloads,thedistributionof
shearingstressesduetotorsionalloadscannotbeassumed
uniform.
??????=න���=න�(??????��)
•NetTorqueDuetoInternalStresses
✓Netoftheinternalshearingstressesisaninternaltorque,equal
andoppositetotheappliedtorque,
✓Althoughthenettorqueduetotheshearingstressesisknown,the
distributionofthestressesisnot
✓Distributionofshearingstressesisstaticallyindeterminatemust
considershaftdeformations
✓Unlikethenormalstressduetoaxialloads,thedistributionof
shearingstressesduetotorsionalloadscannotbeassumed
uniform.
??????=න���=න�(??????��)
107Chapter 1.-Torsion of Circular Bars, Elastic and Homogeneous
•ShearingStrain
✓Consideraninteriorsectionoftheshaft.Asatorsionalload
isapplied,anelementontheinteriorcylinderdeformsinto
arhombus.
✓Sincetheendsoftheelementremainplanar,theshear
strainisequaltoangleoftwist.
✓Itfollowsthat:
✓Shearstrainisproportionaltotwistandradius
??????
�??????�=
�??????
�
and??????=
�
�
??????
�??????�
�??????=�??????or??????=
�??????
�
or??????=
�??????
�
•ShearingStrain
✓Consideraninteriorsectionoftheshaft.Asatorsionalload
isapplied,anelementontheinteriorcylinderdeformsinto
arhombus.
✓Sincetheendsoftheelementremainplanar,theshear
strainisequaltoangleoftwist.
✓Itfollowsthat:
✓Shearstrainisproportionaltotwistandradius
??????
�??????�=
�??????
�
and??????=
�
�
??????
�??????�
�??????=�??????or??????=
�??????
�
or??????=
�??????
�
108Chapter 1.-Torsion of Circular Bars, Elastic and Homogeneous4
2
1
cJ= ( )
4
1
4
2
2
1
ccJ −=
•StressesinElasticRange
✓Multiplyingthepreviousequationbytheshearmodulus,
✓FromHookesLaw, ,so:
✓Theshearingstressvarieslinearlywiththeradialpositioninthesection.
✓Recallthatthesumofthemomentsfromtheinternalstressdistributionis
equaltothetorqueontheshaftatthesection,
✓Theresultsareknownastheelastictorsionformulas,
�??????=
�
�
�??????
�??????�
??????=�????????????=
�
�
??????
�??????�
??????=�??????��=
??????
�??????�
�
�
�
��=
??????
�??????�
�
??????
??????
�??????�=
??????�
??????
and??????=
??????�
??????
2
1
cJ= ( )
4
1
4
2
2
1
ccJ −=
•StressesinElasticRange
✓Multiplyingthepreviousequationbytheshearmodulus,
✓FromHookesLaw, ,so:
✓Theshearingstressvarieslinearlywiththeradialpositioninthesection.
✓Recallthatthesumofthemomentsfromtheinternalstressdistributionis
equaltothetorqueontheshaftatthesection,
✓Theresultsareknownastheelastictorsionformulas,
�??????=
�
�
�??????
�??????�
??????=�????????????=
�
�
??????
�??????�
??????=�??????��=
??????
�??????�
�
�
�
��=
??????
�??????�
�
??????
??????
�??????�=
??????�
??????
and??????=
??????�
??????
109Chapter 1.-Torsion of Circular Bars, Elastic and Homogeneous
•SampleProblem2-1
ShaftBCishollowwithinnerandouter
diametersof90mmand120mm,respectively.
ShaftsABandCDaresolidofdiameterd.For
theloadingshown,determine:
✓(a)theminimumandmaximum
shearingstressinshaftBC,
✓(b)therequireddiameterdofshaftsAB
andCDiftheallowableshearingstress
intheseshaftsis65MPa.
•SampleProblem2-1
ShaftBCishollowwithinnerandouter
diametersof90mmand120mm,respectively.
ShaftsABandCDaresolidofdiameterd.For
theloadingshown,determine:
✓(a)theminimumandmaximum
shearingstressinshaftBC,
✓(b)therequireddiameterdofshaftsAB
andCDiftheallowableshearingstress
intheseshaftsis65MPa.
110Chapter 1.-Torsion of Circular Bars, Elastic and Homogeneous
•SolutionofSampleProblem2-1
I.CutsectionsthroughshaftsABandBCandperformstaticequilibriumanalysistofind
torqueloadings
�
�=�=�kN⋅�−??????
��
??????
��=�kN⋅�=??????
��
�
�=�=�kN⋅�+��kN⋅�−??????
��
??????
��=��kN⋅�
•SolutionofSampleProblem2-1
I.CutsectionsthroughshaftsABandBCandperformstaticequilibriumanalysistofind
torqueloadings
�
�=�=�kN⋅�−??????
��
??????
��=�kN⋅�=??????
��
�
�=�=�kN⋅�+��kN⋅�−??????
��
??????
��=��kN⋅�
111Chapter 1.-Torsion of Circular Bars, Elastic and Homogeneous
•SolutionofSampleProblem2-1
II.ApplyelastictorsionformulastofindminimumandmaximumstressonshaftBC
??????=
�
�
�
�
�
−�
�
�
=
�
�
�.���
�
−�.���
�
=��.����
−�
�
�
??????
�??????�=??????
�=
??????
��∗�
�
??????
=
��kN⋅??????�.���??????
��.����
−�
??????
4
=��.�MPaMPa7.64
MPa2.86
min
max
=
=
??????�??????�
??????�??????�
=
��
��
⇒
??????�??????�
��.�MPa
=
����
����
??????
�??????�=��.�MPa
•SolutionofSampleProblem2-1
II.ApplyelastictorsionformulastofindminimumandmaximumstressonshaftBC
??????=
�
�
�
�
�
−�
�
�
=
�
�
�.���
�
−�.���
�
=��.����
−�
�
�
??????
�??????�=??????
�=
??????
��∗�
�
??????
=
��kN⋅??????�.���??????
��.����
−�
??????
4
=��.�MPaMPa7.64
MPa2.86
min
max
=
=
??????�??????�
??????�??????�
=
��
��
⇒
??????�??????�
��.�MPa
=
����
����
??????
�??????�=��.�MPa
112Chapter 1.-Torsion of Circular Bars, Elastic and Homogeneous
•SolutionofSampleProblem2-1
III.Givenallowableshearingstressandappliedtorque,inverttheelastictorsionformulato
findtherequireddiameter
??????
�??????�=
????????????
??????
=
????????????
??????
�
??????
�
⇒ ��MPa=
�kN⋅�
??????
�
??????
�
�=��=��.�"��"
�=��.���
−�
�
•SolutionofSampleProblem2-1
III.Givenallowableshearingstressandappliedtorque,inverttheelastictorsionformulato
findtherequireddiameter
??????
�??????�=
????????????
??????
=
????????????
??????
�
??????
�
⇒ ��MPa=
�kN⋅�
??????
�
??????
�
�=��=��.�"��"
�=��.���
−�
�
113Chapter 1.-Torsion of Circular Bars, Elastic and Homogeneous
•Angle of Twist in Elastic Range
•Recallthattheangleoftwistandmaximumshearingstrainare
related,
•Intheelasticrange,theshearingstrainandsheararerelatedby
HookesLaw,
•Equatingtheexpressionsforshearingstrainandsolvingfortheangle
oftwist,
•Ifthetorsionalloadingorshaftcross-sectionchangesalongthe
length,theangleofrotationisfoundasthesumofsegment
rotations,
??????
�??????�=
�??????
�
??????
�??????�=
??????�??????�
�
=
??????�
??????�
??????=
??????�
??????�
??????=
??????
??????
??????�
??????
??????
??????�
??????
•Angle of Twist in Elastic Range
•Recallthattheangleoftwistandmaximumshearingstrainare
related,
•Intheelasticrange,theshearingstrainandsheararerelatedby
HookesLaw,
•Equatingtheexpressionsforshearingstrainandsolvingfortheangle
oftwist,
•Ifthetorsionalloadingorshaftcross-sectionchangesalongthe
length,theangleofrotationisfoundasthesumofsegment
rotations,
??????
�??????�=
�??????
�
??????
�??????�=
??????�??????�
�
=
??????�
??????�
??????=
??????�
??????�
??????=
??????
??????
??????�
??????
??????
??????�
??????
114Chapter 1.-Torsion of Circular Bars, Elastic and Homogeneous
•the relative twist angle
•The angle of twist at one section relative to another can be found in two ways:
or
•Inthefirstequationweareusinggeometricrelationshipbetweenangleof
twistandshearstrain.Thisisaverypowerfulequationasittellsusthatwe
don'tneedtoknowthetorqueortheresultingshearstressinorderto
calculatetherelativetwistangle.Ifweknowthemaximumshearstrainat
sectionBandknowthedistanceLbetweensectionsAandB,thenwecan
calculatetherelativetwistangleusingthisformula.Noticethatthis
equationcanalsobewrittenintermsofshearstrainatanyradialposition.
Whatisneededisthevalueofranditscorrespondingshearstrain.
•Thesecondequationisusefulwhenwedon'tknowtheshearstrain,butwe
doknowtheinternaltorqueatsectionB.Thenusingthisequationwecan
determinetherelativeangleoftwist.
•Total angle of twist at section B is determined by:
??????∝??????
??????∝�
•the relative twist angle
•The angle of twist at one section relative to another can be found in two ways:
or
•Inthefirstequationweareusinggeometricrelationshipbetweenangleof
twistandshearstrain.Thisisaverypowerfulequationasittellsusthatwe
don'tneedtoknowthetorqueortheresultingshearstressinorderto
calculatetherelativetwistangle.Ifweknowthemaximumshearstrainat
sectionBandknowthedistanceLbetweensectionsAandB,thenwecan
calculatetherelativetwistangleusingthisformula.Noticethatthis
equationcanalsobewrittenintermsofshearstrainatanyradialposition.
Whatisneededisthevalueofranditscorrespondingshearstrain.
•Thesecondequationisusefulwhenwedon'tknowtheshearstrain,butwe
doknowtheinternaltorqueatsectionB.Thenusingthisequationwecan
determinetherelativeangleoftwist.
•Total angle of twist at section B is determined by:
??????∝??????
??????∝�
115Chapter 1.-Torsion of Circular Bars, Elastic and Homogeneous
•Solution of Sample Problem 2-2: Statically Indeterminate Shafts
Giventheshaftdimensionsandtheappliedtorque,wewouldliketofind
thetorquereactionsatAandB.
•From a free-body analysis of the shaft,
✓which is not sufficient to find the end torques. The problem is
statically indeterminate.
•Dividetheshaftintotwocomponentswhichmusthavecompatible
deformations,
•Substituteintotheoriginalequilibriumequation,
??????
�+??????
�=��lb⋅ft
??????=??????
�+??????
�=
??????
��
�
??????
��
−
??????
��
�
??????
��
=�⇒??????
�=
�
�??????
�
�
�??????
�
??????
�
??????
�+
�
�??????
�
�
�??????
�
??????
�=��lb⋅ft
•Solution of Sample Problem 2-2: Statically Indeterminate Shafts
Giventheshaftdimensionsandtheappliedtorque,wewouldliketofind
thetorquereactionsatAandB.
•From a free-body analysis of the shaft,
✓which is not sufficient to find the end torques. The problem is
statically indeterminate.
•Dividetheshaftintotwocomponentswhichmusthavecompatible
deformations,
•Substituteintotheoriginalequilibriumequation,
??????
�+??????
�=��lb⋅ft
??????=??????
�+??????
�=
??????
��
�
??????
��
−
??????
��
�
??????
��
=�⇒??????
�=
�
�??????
�
�
�??????
�
??????
�
??????
�+
�
�??????
�
�
�??????
�
??????
�=��lb⋅ft
116Chapter 1.-Torsion of Circular Bars, Elastic and Homogeneous
•SampleProblem2-3
Twosolidsteelshaftsareconnectedbygears.
Knowingthatforeachshaft??????=11.2∗10
6
psi
andthattheallowableshearingstressis8ksi,
determine:
✓(a)thelargesttorque??????
0
thatmaybe
appliedtotheendofshaftAB,
✓(b)thecorrespondinganglethrough
whichendAofshaftABrotates.
•SampleProblem2-3
Twosolidsteelshaftsareconnectedbygears.
Knowingthatforeachshaft??????=11.2∗10
6
psi
andthattheallowableshearingstressis8ksi,
determine:
✓(a)thelargesttorque??????
0
thatmaybe
appliedtotheendofshaftAB,
✓(b)thecorrespondinganglethrough
whichendAofshaftABrotates.
117Chapter 1.-Torsion of Circular Bars, Elastic and Homogeneous
•SolutionofSampleProblem2-3
I.Applyastaticequilibriumanalysisonthetwoshaftstofindarelationshipbetween
??????
CD
and??????
0
�
�=�=��.���in.−??????
�
�
�=�=��.��in.−??????
��
??????
��=�.�??????
�
•SolutionofSampleProblem2-3
I.Applyastaticequilibriumanalysisonthetwoshaftstofindarelationshipbetween
??????
CD
and??????
0
�
�=�=��.���in.−??????
�
�
�=�=��.��in.−??????
��
??????
��=�.�??????
�
118Chapter 1.-Torsion of Circular Bars, Elastic and Homogeneous
•SolutionofSampleProblem2-3
II.Applyakinematicanalysistorelatetheangularrotationsofthegears;
�
�??????
�=�
�??????
�
??????
�=
�
�
�
�
??????
�=
�.��in.
�.���in.
??????
�
??????
�=�.�??????
�
•SolutionofSampleProblem2-3
II.Applyakinematicanalysistorelatetheangularrotationsofthegears;
�
�??????
�=�
�??????
�
??????
�=
�
�
�
�
??????
�=
�.��in.
�.���in.
??????
�
??????
�=�.�??????
�
119Chapter 1.-Torsion of Circular Bars, Elastic and Homogeneous
•SolutionofSampleProblem2-3
III.Findthe??????
0
forthemaximumallowabletorqueoneachshaftchoosethesmallest( )
( )
( )
( )
in.lb561
in.5.0
in.5.08.2
8000
in.lb663
in.375.0
in.375.0
8000
0
4
2
0
max
0
4
2
0
max
=
==
=
==
T
T
psi
J
cT
T
T
psi
J
cT
CD
CD
AB
AB
•SolutionofSampleProblem2-3
III.Findthe??????
0
forthemaximumallowabletorqueoneachshaftchoosethesmallest( )
( )
( )
( )
in.lb561
in.5.0
in.5.08.2
8000
in.lb663
in.375.0
in.375.0
8000
0
4
2
0
max
0
4
2
0
max
=
==
=
==
T
T
psi
J
cT
T
T
psi
J
cT
CD
CD
AB
AB
120Chapter 1.-Torsion of Circular Bars, Elastic and Homogeneous
•SolutionofSampleProblem2-3
IV.FindthecorrespondingangleoftwistforeachshaftandthenetangularrotationofendA
??????
�/�=
??????
��??????
??????
��??????
=
���lb⋅in.��??????�.
??????
�
�.���in.
�
��.���
�
psi
=�.���rad=2.22
�
??????
�/�=
??????
��??????
??????
��??????
=
�.����lb⋅in.��??????�.
??????
�
�.�in.
�
��.���
�
psi
=�.���rad=�.��
�
??????
�=�.�??????
�=�.��.��
�
=�.��
�
??????
�=??????
�+??????
�/�=�.��
�
+2.22
�
??????
�=��.��
�
•SolutionofSampleProblem2-3
IV.FindthecorrespondingangleoftwistforeachshaftandthenetangularrotationofendA
??????
�/�=
??????
��??????
??????
��??????
=
���lb⋅in.��??????�.
??????
�
�.���in.
�
��.���
�
psi
=�.���rad=2.22
�
??????
�/�=
??????
��??????
??????
��??????
=
�.����lb⋅in.��??????�.
??????
�
�.�in.
�
��.���
�
psi
=�.���rad=�.��
�
??????
�=�.�??????
�=�.��.��
�
=�.��
�
??????
�=??????
�+??????
�/�=�.��
�
+2.22
�
??????
�=��.��
�
121Chapter 1.-Torsion of Circular Bars, Elastic and Non-Homogeneous
•Elastic and Non-Homogeneous
•Thelinearstrainvariationisvalidasstatedbefore;
•however,sincethebarisnon-homogeneous,i.e.,madeoftwoormorematerials,
•wecannotuseEl-Hocir.Thisproblemisstaticallyindeterminateinthesensethatwedon'tknowhow
muchofthetorqueisbeingcarriedbyeachmaterial.
•Thegeneraltorqueformulainitsintegralformmustbebrokenupintotwoormorepartsdependingon
thematerialmakeupofthebar.
•Also,sincethebaristotallylinearlyelastic(i.e.,stressesarebelowtheproportionallimit),Hooke'slaw
maybeusedtorelatestresstostrainineachrespectiveregioninthecrosssection.
•Hereiswhatthetorqueequationlookslikeforabarmadeoftwomaterials:
•Where0and1limitsinthefirstintegraldenotetheradialboundariesofthecorematerial,while1and2denote
thosefortheshellmaterial.Thesubscripts'C'and'S'correspondtothecoreandshellmaterials,respectively.
,
•Elastic and Non-Homogeneous
•Thelinearstrainvariationisvalidasstatedbefore;
•however,sincethebarisnon-homogeneous,i.e.,madeoftwoormorematerials,
•wecannotuseEl-Hocir.Thisproblemisstaticallyindeterminateinthesensethatwedon'tknowhow
muchofthetorqueisbeingcarriedbyeachmaterial.
•Thegeneraltorqueformulainitsintegralformmustbebrokenupintotwoormorepartsdependingon
thematerialmakeupofthebar.
•Also,sincethebaristotallylinearlyelastic(i.e.,stressesarebelowtheproportionallimit),Hooke'slaw
maybeusedtorelatestresstostrainineachrespectiveregioninthecrosssection.
•Hereiswhatthetorqueequationlookslikeforabarmadeoftwomaterials:
•Where0and1limitsinthefirstintegraldenotetheradialboundariesofthecorematerial,while1and2denote
thosefortheshellmaterial.Thesubscripts'C'and'S'correspondtothecoreandshellmaterials,respectively.
,
122
•Elastic and Non-Homogeneous
•Knowingthemaximumshearstrain,wecanusetheaboveequationtosolveforthetorqueata
givensection,orviceversa.
•Noticethatweusedthesamelinearequationforstress-strainrelationsinbothintegralswith
shearmodulusbeingtheonlydifference.
•Examiningthisequationmorecarefully,werealizethatwhilemaximumshearstrainalways
occursatthefarthestdistancefromthecenterofthebar,itisnotnecessaryformaximumshear
stresstobeatthesamelocation.
•Thedifferencebetweentheshearmoduliofthecoreandtheshellcouldbesuchthattheshear
stressinthecoreattheinterfaceoftheshellandcoremightbehigherthanthatattheedgeof
thebarnearitsoutsidesurface.
,
Chapter 1.-Torsion of Circular Bars, Elastic and Non-Homogeneous
•Elastic and Non-Homogeneous
•Knowingthemaximumshearstrain,wecanusetheaboveequationtosolveforthetorqueata
givensection,orviceversa.
•Noticethatweusedthesamelinearequationforstress-strainrelationsinbothintegralswith
shearmodulusbeingtheonlydifference.
•Examiningthisequationmorecarefully,werealizethatwhilemaximumshearstrainalways
occursatthefarthestdistancefromthecenterofthebar,itisnotnecessaryformaximumshear
stresstobeatthesamelocation.
•Thedifferencebetweentheshearmoduliofthecoreandtheshellcouldbesuchthattheshear
stressinthecoreattheinterfaceoftheshellandcoremightbehigherthanthatattheedgeof
thebarnearitsoutsidesurface.
,
Chapter 1.-Torsion of Circular Bars, Elastic and Non-Homogeneous
123
•Rule of Thumb:
•Maximum shear strain occurs atr = cregardless of
whether the shell material has higher or lower
shear rigidity.
•Maximum shear stress occurs in the material with
the highest shear rigidity regardless of its location.
•Here are some examples:
•Since the core and the shell are assumed to be
perfectly attached at the interface, the angle of
twist is the same for both materials; therefore;
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Chapter 1.-Torsion of Circular Bars, Elastic and Non-Homogeneous
•Rule of Thumb:
•Maximum shear strain occurs atr = cregardless of
whether the shell material has higher or lower
shear rigidity.
•Maximum shear stress occurs in the material with
the highest shear rigidity regardless of its location.
•Here are some examples:
•Since the core and the shell are assumed to be
perfectly attached at the interface, the angle of
twist is the same for both materials; therefore;
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Chapter 1.-Torsion of Circular Bars, Elastic and Non-Homogeneous
124
•SampleProblem2-4
SteppedbarABCismadeofAluminumalloywithsectionABfilledwithsteel.Usethepropertiesgiventodothefollowing:
(a)determinethemaximumelastictorquethatcanbeappliedatthefreeend
(b)determinethecorrespondingangleoftwistatthefreeend
(c)determineandsketchtheshearstressdistributionontheshaftcrosssectionforAB
(d)determineandsketchtheshearstressdistributionontheshaftcrosssectionforBC
Chapter 1.-Torsion of Circular Bars, Elastic and Non-Homogeneous
•SampleProblem2-4
SteppedbarABCismadeofAluminumalloywithsectionABfilledwithsteel.Usethepropertiesgiventodothefollowing:
(a)determinethemaximumelastictorquethatcanbeappliedatthefreeend
(b)determinethecorrespondingangleoftwistatthefreeend
(c)determineandsketchtheshearstressdistributionontheshaftcrosssectionforAB
(d)determineandsketchtheshearstressdistributionontheshaftcrosssectionforBC
Chapter 1.-Torsion of Circular Bars, Elastic and Non-Homogeneous
125
•SolutionofSampleProblem2-4_a
✓Thisproblemisanexampleofcircularshaftswithmaterialand/orgeometric
discontinuitiesalongtheaxis.
✓Suchaxialdiscontinutiesrequirethattheshaftbebrokenupintoseveralpartssuchthat
ineachpartwehaveuniformaxialpropertiesandconstantloading.
✓Inthisexamplewehavematerialandgeometricdiscontinutiesbutnotloading
discontinuityastorqueisequaltoTeverywherealongtheshaft.
✓NoticealsowhileportionBCishomogeneous(madeofaluminumonly),portionABisnot.
Thus,toanalyzethisproblemweuseelastictorsionformulaforportionBCandthe
generaltorsionformula(integralform)forportionAB.
Chapter 1.-Torsion of Circular Bars, Elastic and Non-Homogeneous
•SolutionofSampleProblem2-4_a
✓Thisproblemisanexampleofcircularshaftswithmaterialand/orgeometric
discontinuitiesalongtheaxis.
✓Suchaxialdiscontinutiesrequirethattheshaftbebrokenupintoseveralpartssuchthat
ineachpartwehaveuniformaxialpropertiesandconstantloading.
✓Inthisexamplewehavematerialandgeometricdiscontinutiesbutnotloading
discontinuityastorqueisequaltoTeverywherealongtheshaft.
✓NoticealsowhileportionBCishomogeneous(madeofaluminumonly),portionABisnot.
Thus,toanalyzethisproblemweuseelastictorsionformulaforportionBCandthe
generaltorsionformula(integralform)forportionAB.
Chapter 1.-Torsion of Circular Bars, Elastic and Non-Homogeneous
126
•SolutionofSampleProblem2-4_a
✓First,wefindthemaximumelastictorquethat
portionBCcancarry.
✓Itwillcorrespondtothemaximumelasticshear
stressandmaximumshearstraininthissection-
bothatthesurfaceinthiscase.
✓Fromthestress-straincurveinthefigure,weknow
themaximumelasticshearingstressforaluminum
is8000psi.UsingHooke'slaw,themaximum
elasticshearstrain(thestrainatradius'c')canbe
found.
Chapter 1.-Torsion of Circular Bars, Elastic and Non-Homogeneous
•SolutionofSampleProblem2-4_a
✓First,wefindthemaximumelastictorquethat
portionBCcancarry.
✓Itwillcorrespondtothemaximumelasticshear
stressandmaximumshearstraininthissection-
bothatthesurfaceinthiscase.
✓Fromthestress-straincurveinthefigure,weknow
themaximumelasticshearingstressforaluminum
is8000psi.UsingHooke'slaw,themaximum
elasticshearstrain(thestrainatradius'c')canbe
found.
Chapter 1.-Torsion of Circular Bars, Elastic and Non-Homogeneous
127
•SolutionofSampleProblem2-4_a
✓Nowusingtheelastictorsionformula,themaximumelastictorqueforsectionBCis
Chapter 1.-Torsion of Circular Bars, Elastic and Non-Homogeneous
•SolutionofSampleProblem2-4_a
✓Nowusingtheelastictorsionformula,themaximumelastictorqueforsectionBCis
Chapter 1.-Torsion of Circular Bars, Elastic and Non-Homogeneous
128
•SolutionofSampleProblem2-4_a
✓Noticethatthisisnotthefinal
answeryetbecausethistorquemay
begreaterthatthemaximum
elastictorqueforportionAB.
✓Nowfindmaximum allowable
torqueforportionAB.The
maximumstrainwillbeatthe
surfaceasbefore.Sincetheshellis
aluminum,themaximumstrain
willbethesameasabove
Chapter 1.-Torsion of Circular Bars, Elastic and Non-Homogeneous
•SolutionofSampleProblem2-4_a
✓Noticethatthisisnotthefinal
answeryetbecausethistorquemay
begreaterthatthemaximum
elastictorqueforportionAB.
✓Nowfindmaximum allowable
torqueforportionAB.The
maximumstrainwillbeatthe
surfaceasbefore.Sincetheshellis
aluminum,themaximumstrain
willbethesameasabove
Chapter 1.-Torsion of Circular Bars, Elastic and Non-Homogeneous
129
•SolutionofSampleProblem2-4_a
✓Usingthetorqueequationabove,themaximumallowabletorqueforportionABis:
✓Therefore,themaximumelastictorquefortheshaftsothatbothportionsABand
BCremainelasticwillbe:
✓Pleasenotethattheunitfortorqueshouldbein-lbinsteadofpsi.
Chapter 1.-Torsion of Circular Bars, Elastic and Non-Homogeneous
•SolutionofSampleProblem2-4_a
✓Usingthetorqueequationabove,themaximumallowabletorqueforportionABis:
✓Therefore,themaximumelastictorquefortheshaftsothatbothportionsABand
BCremainelasticwillbe:
✓Pleasenotethattheunitfortorqueshouldbein-lbinsteadofpsi.
Chapter 1.-Torsion of Circular Bars, Elastic and Non-Homogeneous
130
•SolutionofSampleProblem2-4_b
✓Theangleoftwistatthefreeendisfoundbyusing:
✓ThetorqueTforeachportionisfoundtobeTmax.
✓Theaboveequationforangleoftwistisvalidforhomogeneousshafts.
✓TobeabletousethisequationinportionAB,weneedtoknowhowmuchofthetorqueiscarriedbysteel
coreandhowmuchiscarriedbythealuminumshell.
✓This,however,hasbeendeterminedpreviouslybyexaminingtheintegralequationfortorqueinportion
AB;thatis,thefirstintegralcorrespondstothetorqueinthecorewhilethesecondintegralcorresponds
tothetorqueintheshell.
✓Thesetorquescanalsobeobtainedbasedonthefactthattheangleoftwistforthesteelandaluminumis
thesameinportionAB.
Chapter 1.-Torsion of Circular Bars, Elastic and Non-Homogeneous
•SolutionofSampleProblem2-4_b
✓Theangleoftwistatthefreeendisfoundbyusing:
✓ThetorqueTforeachportionisfoundtobeTmax.
✓Theaboveequationforangleoftwistisvalidforhomogeneousshafts.
✓TobeabletousethisequationinportionAB,weneedtoknowhowmuchofthetorqueiscarriedbysteel
coreandhowmuchiscarriedbythealuminumshell.
✓This,however,hasbeendeterminedpreviouslybyexaminingtheintegralequationfortorqueinportion
AB;thatis,thefirstintegralcorrespondstothetorqueinthecorewhilethesecondintegralcorresponds
tothetorqueintheshell.
✓Thesetorquescanalsobeobtainedbasedonthefactthattheangleoftwistforthesteelandaluminumis
thesameinportionAB.
Chapter 1.-Torsion of Circular Bars, Elastic and Non-Homogeneous
131
•Thesetorques:
•Theangleoftwistatthefreeendis:
Chapter 1.-Torsion of Circular Bars, Elastic and Non-Homogeneous
•Thesetorques:
•Theangleoftwistatthefreeendis:
Chapter 1.-Torsion of Circular Bars, Elastic and Non-Homogeneous
132
•SolutionofSampleProblem2-4_c-d
✓Tosketchtheshearstressdistributionineachsection,themaximumshearstressmustbefoundforeach
section.Theyare:
Chapter 1.-Torsion of Circular Bars, Elastic and Non-Homogeneous
•SolutionofSampleProblem2-4_c-d
✓Tosketchtheshearstressdistributionineachsection,themaximumshearstressmustbefoundforeach
section.Theyare:
Chapter 1.-Torsion of Circular Bars, Elastic and Non-Homogeneous
133Chapter 1.-Torsion of Circular Bars, Inelastic and Homogeneous
•Plastic Deformations
•Withtheassumptionofalinearlyelasticmaterial,
•If the yield strength is exceeded or the material has a nonlinear
shearing-stress-strain curve, this expression does not hold.
•Shearing strain varies linearly regardless of material properties.
Application of shearing-stress-strain curve allows determination of
stress distribution.
•The integral of the moments from the internal stress distribution is
equal to the torque on the shaft at the section,
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•Plastic Deformations
•Withtheassumptionofalinearlyelasticmaterial,
•If the yield strength is exceeded or the material has a nonlinear
shearing-stress-strain curve, this expression does not hold.
•Shearing strain varies linearly regardless of material properties.
Application of shearing-stress-strain curve allows determination of
stress distribution.
•The integral of the moments from the internal stress distribution is
equal to the torque on the shaft at the section,
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134
•Elastoplastic Materials
•Atthemaximumelastictorque,
•As the torque is increased, a plastic region;
( ) develops around an elastic core ( )
•As , the torque approaches a limiting value,
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Chapter 1.-Torsion of Circular Bars, Inelastic and Homogeneous
•Elastoplastic Materials
•Atthemaximumelastictorque,
•As the torque is increased, a plastic region;
( ) develops around an elastic core ( )
•As , the torque approaches a limiting value,
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Chapter 1.-Torsion of Circular Bars, Inelastic and Homogeneous
135Chapter 1.-Torsion of Circular Bars, Inelastic and Homogeneous
•InelasticandHomogeneous
oAsmentionedearlier,theshearstrainpatternis
lineareventhoughthememberhasbeenstressed
beyondtheelasticlimitandintotheinelastic
range.
oForalinearlyelasticmaterial,shearstress
variationislinearuptotheradialdistance
correspondingtotheelasticrimshowninthe
figuresbelow.
oBeyondthisdistance,Hooke'slawcanNOTbe
usedtorelatestresstostrainvariation.Once
againthematerialisassumedtobeelasto-plastic.
•InelasticandHomogeneous
oAsmentionedearlier,theshearstrainpatternis
lineareventhoughthememberhasbeenstressed
beyondtheelasticlimitandintotheinelastic
range.
oForalinearlyelasticmaterial,shearstress
variationislinearuptotheradialdistance
correspondingtotheelasticrimshowninthe
figuresbelow.
oBeyondthisdistance,Hooke'slawcanNOTbe
usedtorelatestresstostrainvariation.Once
againthematerialisassumedtobeelasto-plastic.
136Chapter 1.-Torsion of Circular Bars, Inelastic and Homogeneous
oLet'sseehowtorsionalanalysisisperformedforacircularbarastheinternaltorqueisincreased
fromzerotothatcorrespondingtothefully-plasticbar.
oNoticethatweareexaminingonlyonesection(slice)ofthebaralongitslength.Therefore,
dependingonwhichsectionweconsidertheinternaltorquemaybedifferent.
▪Ifatthesectionofinterestthemaximumshearstrainisbelowtheelastic-limitshearstrainof
thematerial,thenthebarisfullyelasticandthecorrespondingtorquecanbeobtainedas
shownin(1).
▪Ifthemaximumshearstrainreachestheelasticshearstrain,thenthebarhasreachedthe
elasticlimitstate,andthetorquecalculatedaccordingto(2)isreferredtoasthe"elastic-limit
torque".
oLet'sseehowtorsionalanalysisisperformedforacircularbarastheinternaltorqueisincreased
fromzerotothatcorrespondingtothefully-plasticbar.
oNoticethatweareexaminingonlyonesection(slice)ofthebaralongitslength.Therefore,
dependingonwhichsectionweconsidertheinternaltorquemaybedifferent.
▪Ifatthesectionofinterestthemaximumshearstrainisbelowtheelastic-limitshearstrainof
thematerial,thenthebarisfullyelasticandthecorrespondingtorquecanbeobtainedas
shownin(1).
▪Ifthemaximumshearstrainreachestheelasticshearstrain,thenthebarhasreachedthe
elasticlimitstate,andthetorquecalculatedaccordingto(2)isreferredtoasthe"elastic-limit
torque".
137Chapter 1.-Torsion of Circular Bars, Inelastic and Homogeneous
▪Asthetorqueisincreasedbeyondtheelastic-limittorque,theinternalshearstressmovesintothe
inelasticrange.Inthatcasewemustfirstlocatetheelasticrimbycalculatingre.Oncethatdistanceis
known,thenwecancalculatetheinternaltorquecorrespondingtothespecifiedmax.shearstrain
accordingto(3).
▪Notethatinthiscase,thematerialisassumedtobeelasto-plastic,whichimpliesthatbeyondthe
elasticlimit,stressremainsconstantattheelasticlimitvalue.inelasticrange,thenthemaximum
inelasticorfully-plastictorqueisfoundaccordingto(4).
oTheangleoftwistatagivensectioninaninelasticcircularbarisfoundby
▪whichcorrespondstotheangleoftwistattheonsetofyield(whentheelasticradius=radiusofbar,c).
Or where
▪Asthetorqueisincreasedbeyondtheelastic-limittorque,theinternalshearstressmovesintothe
inelasticrange.Inthatcasewemustfirstlocatetheelasticrimbycalculatingre.Oncethatdistanceis
known,thenwecancalculatetheinternaltorquecorrespondingtothespecifiedmax.shearstrain
accordingto(3).
▪Notethatinthiscase,thematerialisassumedtobeelasto-plastic,whichimpliesthatbeyondthe
elasticlimit,stressremainsconstantattheelasticlimitvalue.inelasticrange,thenthemaximum
inelasticorfully-plastictorqueisfoundaccordingto(4).
oTheangleoftwistatagivensectioninaninelasticcircularbarisfoundby
▪whichcorrespondstotheangleoftwistattheonsetofyield(whentheelasticradius=radiusofbar,c).
Or where
138Chapter 1.-Torsion of Circular Bars, Inelastic and Non-Homogeneous
•InelasticandNon-Homogeneous
oHere,wearedealingwithacircularbarthatismadeoftwoormultiple
materialsdistributedintheradialdirection.
oThebaristwistedbeyondtheelasticlimitofatleastoneofthematerials.
oItisnotnecessaryforallmaterialstobestressedbeyondtheirrespectiveelastic
limits.
oInthiscase,theshearstrainvariationisstilllinearandthetorqueequationis
brokenupatthematerialinterfaceandatthepointwherethetransitionfrom
elastictoinelasticregiontakesplace.
•InelasticandNon-Homogeneous
oHere,wearedealingwithacircularbarthatismadeoftwoormultiple
materialsdistributedintheradialdirection.
oThebaristwistedbeyondtheelasticlimitofatleastoneofthematerials.
oItisnotnecessaryforallmaterialstobestressedbeyondtheirrespectiveelastic
limits.
oInthiscase,theshearstrainvariationisstilllinearandthetorqueequationis
brokenupatthematerialinterfaceandatthepointwherethetransitionfrom
elastictoinelasticregiontakesplace.
139Chapter 1.-Torsion of Circular Bars, Inelastic and Non-Homogeneous
•InelasticandNon-Homogeneous
oLet'sconsiderthecasewherethebariscomposed
oftwodifferentmaterials.Thereareseveral
possibilitiesthatneedtobeinvestigated:
▪(1)Theinnermaterial(core)maybecomeinelastic
whiletheoutershellisstillelastic;
▪(2)theoutershellmaybecomefullyplasticwhilethe
coreremainsfarbelowitsproportionallimit;etc.
oHereisanexampleofanon-homogeneouscircular
barinastateofinelasticstressandstrain.
•InelasticandNon-Homogeneous
oLet'sconsiderthecasewherethebariscomposed
oftwodifferentmaterials.Thereareseveral
possibilitiesthatneedtobeinvestigated:
▪(1)Theinnermaterial(core)maybecomeinelastic
whiletheoutershellisstillelastic;
▪(2)theoutershellmaybecomefullyplasticwhilethe
coreremainsfarbelowitsproportionallimit;etc.
oHereisanexampleofanon-homogeneouscircular
barinastateofinelasticstressandstrain.
140Chapter 1.-Torsion of Circular Bars, Inelastic and Non-Homogeneous
oItisobviousthattheanalysisofthistypeofproblemrequiresmorerigorthanthosedealingwithelastic
torsion.
oAnalysisProcedure:
oAnalysisofcircularnon-homogeneousbarsinwhichmax.shearstrainisspecified.Noticethatthisisa
staticallyindeterminateproblemaswedon'tknowwhatpercentageoftheresultanttorqueisbeing
carriedbyeachmaterial.
1.Knowingthatthemax.shearstrainoccursatthefarthestdistancefromthecenterofcircularcross
section,wecalculatetheactualshearstrainatthematerialinterfaceusingthelinearequation.
Nowweknowthemax.shearstrainineachmaterial.
2.Next,wecalculatetheelastic-limitshearstrainofeachmaterialfromthestrain-stressdiagramor
materialpropertiesprovidedintheproblem,andcompareitwiththecorrespondingactualshear
strain.Forinelastictorsionproblem,themax.shearstraininoneofthematerialsmustexceedits
correspondingelasticlimitvalue.
oItisobviousthattheanalysisofthistypeofproblemrequiresmorerigorthanthosedealingwithelastic
torsion.
oAnalysisProcedure:
oAnalysisofcircularnon-homogeneousbarsinwhichmax.shearstrainisspecified.Noticethatthisisa
staticallyindeterminateproblemaswedon'tknowwhatpercentageoftheresultanttorqueisbeing
carriedbyeachmaterial.
1.Knowingthatthemax.shearstrainoccursatthefarthestdistancefromthecenterofcircularcross
section,wecalculatetheactualshearstrainatthematerialinterfaceusingthelinearequation.
Nowweknowthemax.shearstrainineachmaterial.
2.Next,wecalculatetheelastic-limitshearstrainofeachmaterialfromthestrain-stressdiagramor
materialpropertiesprovidedintheproblem,andcompareitwiththecorrespondingactualshear
strain.Forinelastictorsionproblem,themax.shearstraininoneofthematerialsmustexceedits
correspondingelasticlimitvalue.
141Chapter 1.-Torsion of Circular Bars, Inelastic and Non-Homogeneous
3.Withthisinformation,wecanproceedwithplottingthestressvariationacrossthecrosssectionofthebar.This
givesusamapthatwecanusetosetuptheintegralformofthetorqueequation.Theintegralwouldbesplit
intoseveralsegmentsdependingonthevariationsofstressesandmaterials.
4.Firstweconsiderthecore.Basedonthestressvariationinthecore,wemayneeduptotwoseparateintegrals.
Then,weexaminetheshell.Again,dependingonthestressvariation,wemayneeduptotwoseparate
integrals-onefortheelasticpartandanotherfortheinelasticpart.
5.Onceweidentifythelimitsoneachintegral,andcorrectlywritetheintegrandineachcase,wecanproceed
withtheintegrationtofindthetorquethatcorrespondstothespecifiedmax.shearstrainintheproblem
statement.
6.Noticethattheintegralscorrespondtotheportionoftheresultanttorquecarriedbythecoreandtheshell.So
oncewehavecalculatedthesevalues,wecandeterminethepercentageoftorquecarriedbyeachconstituent.
7.Theangleoftwistiscalculatedinthesamemannerspecifiedintheprevioussection.Themainassumptionis
thatthetwomaterialsareperfectlybondedattheinterface
3.Withthisinformation,wecanproceedwithplottingthestressvariationacrossthecrosssectionofthebar.This
givesusamapthatwecanusetosetuptheintegralformofthetorqueequation.Theintegralwouldbesplit
intoseveralsegmentsdependingonthevariationsofstressesandmaterials.
4.Firstweconsiderthecore.Basedonthestressvariationinthecore,wemayneeduptotwoseparateintegrals.
Then,weexaminetheshell.Again,dependingonthestressvariation,wemayneeduptotwoseparate
integrals-onefortheelasticpartandanotherfortheinelasticpart.
5.Onceweidentifythelimitsoneachintegral,andcorrectlywritetheintegrandineachcase,wecanproceed
withtheintegrationtofindthetorquethatcorrespondstothespecifiedmax.shearstrainintheproblem
statement.
6.Noticethattheintegralscorrespondtotheportionoftheresultanttorquecarriedbythecoreandtheshell.So
oncewehavecalculatedthesevalues,wecandeterminethepercentageoftorquecarriedbyeachconstituent.
7.Theangleoftwistiscalculatedinthesamemannerspecifiedintheprevioussection.Themainassumptionis
thatthetwomaterialsareperfectlybondedattheinterface
142Chapter 1.-Torsion of Circular Bars, Residual Stress Distribution
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