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1 009 The McGraw-Hill Companies, Inc. ll rights reserved. Fifth SI Edition CHTER MECHNICS OF MTERIS Ferdinand. Beer E. Russell Johnston, Jr. John T. DeWolf David F. Mazurek Stress and Strain xial oading ecture Notes: J. Walt Oler Texas Tech University 응력과변형률 - 축 하중

2 Contents ( 목차 ) 응력과변형률, 축하중 (Stress & Strain: xial oading) 수직응력 (Normal Strain) 응력 - 변형률시험 (Stress-Strain Test) 응력 - 변형률선도, 연성재료 (Stress- Strain Diagram: Ductile Materials) 응력변형률선도, 취성재료 (Stress-Strain Diagram: Brittle Materials ) 훅법칙, 탄성계수 (Hooke s aw: Modulus of Elasticity) 탄성및소성거동 (Elastic vs. lastic Behavior) 피로 (Fatigue) 축하중에의한변형 (Deformations Under xial oading) 예제.(Example.01) 견본문제.1(Sample roblem.1) 부정정 (Static Indeterminacy) 예제.04(Example.04) 열응력 (Thermal Stresses) 푸와송비 (oisson s Ratio) 일반화된훅법칙 (Generalized Hooke s aw) 체적탄성계수 ( Dilatation: Bulk Modulus) 전단변형률 ( Shearing Strain) 예제. (Example.) E, ν, G 의관계 (Relation mong E, ν, and G) 견본문제.5(Sample roblem.5) 복합재료 (Composite Materials) 생브낭의원리 (Saint-Venant s rinciple) 응력집중 : 구멍 (Stress Concentration: Hole) Copyright 011 by Tae Hyun Baek, Kunsan National University. ll rights reserved. 응력집중 : 필릿 (Stress Concentration: Fillet) 예제.1(Example.1) 탄소성재료 ( Elastoplastic Materials lastic Deformations) 잔류응력 (Residual Stresses) 예제.14,.15,.16(Example.14,.15,.16) 5-

3 Stress & Strain: xial oading ( 응력및변형률 : 축하중 ) 구조물이나기계는하중에의한응력과더불어 구조물의변형을고려하여설계 변형이일어나는구조물은부정정 (statically indeterminate) - 반력과부재의힘을고려 부재내의응력분포계산시부재의변형을고려해야함 장전반부 : 축하중을받는구조물부재의변형 장후반부 : 비틀림및순수굽힘하중 Copyright 011 by Tae Hyun Baek, Kunsan National University. ll rights reserved. 5-

4 Normal Strain ( 수직변형률 ) Fig..1 σ stress δ ε normal strain Fig.. σ δ ε Copyright 011 by Tae Hyun Baek, Kunsan National University. ll rights reserved. Fig..4 σ δ δ ε 5-4

5 응력 - 변형률시험 (Stress-Strain Test) Fig.7 인장시험기 Fig.8 인장시편 (Test specimen with tensile load) Copyright 011 by Tae Hyun Baek, Kunsan National University. ll rights reserved. 5-5

6 응력 - 변형률선도 : 연성재료 (Stress-Strain Diagram: Ductile Materials) Copyright 011 by Tae Hyun Baek, Kunsan National University. ll rights reserved. 5-6

7 응력 - 변형률선도 : 취성재료 (Stress-Strain Diagram: Brittle Materials) Fig.1 Stress-strain diagram for a typical brittle material. Copyright 011 by Tae Hyun Baek, Kunsan National University. ll rights reserved. 5-7

8 탄성및소성거동 (Elastic vs. lastic Behavior) Fig..18 응력제거시변형률이없어지는경우, 재료는탄성적으로거동 (behave elastically ) 재료가탄성적으로거동하는응력의최대값 : 탄성한계 (elastic limit). 응력이제거된후변형률이영 (zero) 으로돌아가지않을경우, 재료는소성거동 (behave plastically) Copyright 011 by Tae Hyun Baek, Kunsan National University. ll rights reserved. 5-8

9 피로 (Fatigue) 피로특성 : S-N 선도 (S-N diagrams) 로나타냄.. 많은하중사이클을받을경우부재는극한강도 (ultimate strength ) 보다훨씬낮은응력레벨에피로 (fatigue) 에의해파손 내구한도 (endurance limit) 보다낮은응력레벨에서는무한대하중사이클에서도파손이일어나지않음. Copyright 011 by Tae Hyun Baek, Kunsan National University. ll rights reserved. 5-9

10 훅범칙 : 탄성계수 (Hooke s aw: Modulus of Elasticity) Fig.16 Stress-strain diagrams for iron and different grades of steel. 항복응력이하 (below the yield stress) σ Eε E Youngs Modulus or Modulus of Elasticity 강도는함금의종류, 열처리, 제조공정에따라달리변화하나강성또는탄성계수는그렇지않음 (Strength is affected by alloying, heat treating, and manufacturing process but stiffness (Modulus of Elasticity) is not. Copyright 011 by Tae Hyun Baek, Kunsan National University. ll rights reserved. 5-

11 Deformations Under xial oading Fig.. 훅법칙으로부터 σ σ E ε ε E 변형률의정의로부터 δ ε E 위의관계식으로부터변형 (deformation) 에대해풀면, δ E 하중, 단면적또는재료의성질이다를경우 i δ E i i i i Copyright 011 by Tae Hyun Baek, Kunsan National University. ll rights reserved. 5-11

12 예제 (Example.01) 주어진하중에서강봉의변형량을계산하라. 풀이 : 봉을하중작용점을기준으로나눈다. 내력을결정하기위해각부분의자유물체도를작성 전체변형량계산 Copyright 011 by Tae Hyun Baek, Kunsan National University. ll rights reserved. 5-1

풀이 : 봉을 부분으로나눈다 1 1 0.m 581-6 m 0.4m 194-6 m 각부분의자유물체도로부터내력을결정 1 40 N 60 N N 전체변형량계산 1 i i 1 1 δ + + i i Ei E 1 1 9 + 6 00 581 581 1.

13 풀이 : 봉을 부분으로나눈다 m m 0.4m m 각부분의자유물체도로부터내력을결정 1 40 N 60 N N 전체변형량계산 1 i i 1 1 δ + + i i Ei E m ( 40 ) 0. ( 60 ) 0. ( ) 6 δ 1.7 mm Copyright 011 by Tae Hyun Baek, Kunsan National University. ll rights reserved. 5-1

14 견본문제 (Sample roblem.1) 강체봉 BDE 는두개의링크 B 와 CD 에의해지지. 링크 B: 알루미늄 (E 70 Ga), 단면적 (500 mm ) 링크 CD: 강재 (E 00 Ga), 단면적 (600 mm ). 0-kN 하중작용시 (a) B, (b) D, (c) E 지점에서변형량계산 풀이 : BDE의자유물체도로부터링크 B 와 DC에의한힘을계산 링크 B 와 DC 의변형량 (deformation) 또는 B 와 D 에서변위 (displacement) 를계산 B 와 D 의변위로부터 E 에서변위를기하학적으로계산 Copyright 011 by Tae Hyun Baek, Kunsan National University. ll rights reserved. 5-14

15 견본문제 (Sample roblem.1) 풀이 : B의변위 : 자유물체도 : 봉 BDE M F M F B 0 CD D 0 B 0 ( 0kN 0.6m) 0 ( 0kN 0.4m) + F + 90kN tension F CD B 60kN compression 0.m 0.m D 의변위 : δ B δ D E ( 60 N)( 0.m) -6 9 ( 500 m )( 70 a) 514 E 6 m δ B mm ( 90 N)( 0.4m) -6 9 ( 600 m )( 00 a) 00 6 m δ D 0.00 mm Copyright 011 by Tae Hyun Baek, Kunsan National University. ll rights reserved. 5-15

16 견본문제 (Sample roblem.1) D 의변위 : BB DD BH HD mm 0.00 mm x 7.7 mm EE DD δ E 0.00 mm δ E HE HD 1.98 mm ( 00 mm) x δ E x ( ) mm 7.7 mm 1.98 mm Copyright 011 by Tae Hyun Baek, Kunsan National University. ll rights reserved. 5-16

17 부정정문제 (Static Indeterminacy) 정역학적관계식으로내력과반력을결정할수없는구조물 부정정 (statically indeterminate) 평형을유지하기위해필요로하는지지보다더많은구조물 부정정 δ δ + δ R 0 잉여반력 (redundant reaction) 을상응하는변형을발생시킬수있는미지하중 (unknown load) 으로대체 잉여반력과실제하중에의한변형 (deformation) 을별도로구한후, 서로더하거나중첩 (superpose) Copyright 011 by Tae Hyun Baek, Kunsan National University. ll rights reserved. 5-17

18 예제.04 (Example.04) 하중을가하기전꼭맞게고정된강재봉이하중을받는상태에서 와 B 에발생되는반력을결정 풀이 : B 에서반력을잉여로간주, 봉을지지로부터해제. 가해진하중에의한 B 지점의변위를계산 B 지점에서잉여반력으로인한변위를계산 하중에의한변위와잉여반력으로인한변위는서로같아야하며, 합은영 (zero) 이되어야함. B 지점의반력과작용된하중에의해 지점에서반력을계산 Copyright 011 by Tae Hyun Baek, Kunsan National University. ll rights reserved. 5-18

예제.04 (Example.04) 풀이 : 잉여반력이없다고가정하고작용된하중에의한 B지점의변위를계산 δ 1 1 1 0 1.

ll rights reserved. i i i i E i 400 600 4 6 m 9 N 0.

19 예제.04 (Example.04) 풀이 : 잉여반력이없다고가정하고작용된하중에의한 B지점의변위를계산 δ E Copyright 011 by Tae Hyun Baek, Kunsan National University. ll rights reserved. i i i i E i m 9 N m 잉여반력으로인한 B 지점의변위 δ R 400 i R ii E m i i B m ( 1.95 ) E 50 R B 50 6 m N 6 m 5-19

Example.04 하중과잉여반력에의한변위는서로같아야하며이들을서로더하여영이되도록한다. δ δ + δ 1.15 δ E R B R 577 0 9 ( 1.

20 Example.04 하중과잉여반력에의한변위는서로같아야하며이들을서로더하여영이되도록한다. δ δ + δ 1.15 δ E R B R ( 1.95 ) E N 577 kn R B 0 B 지점의반력과하중에의해 지점의반력계산 F 0 R 00 kn 600kN kn R y kn R R B kn 577 kn Copyright 011 by Tae Hyun Baek, Kunsan National University. ll rights reserved. 5-0

National University. ll rights reserved.

21 열응력 (Thermal Stresses) 온도가변화되면길이가달라지거나열변형률 (thermal strain) 이발생. 팽창 (elongation) 이구속되지않을경우열변형률은발생하나응력은영 (zero) α ( T ) Copyright 011 by Tae Hyun Baek, Kunsan National University. ll rights reserved. δ T 부가적인지지 (additional support) 를잉여로고려, 중첩법적용 α thermal expansion coef. δ δ T + δ 0 δ E 열변형과잉여지지 (redundant support) 에의한변형은서로같아야하며중첩시키면영이되어야함. α( T ) + 0 E Eα σ ( T ) Eα ( T ) 5-1

Microsoft PowerPoint - 11_에너지 방법.ppt

Microsoft PowerPoint - 11_에너지 방법.ppt Fifth SI Edition CHATER MECHANICS OF MATERIAS Ferdinand. Beer E. Russell Johnston, Jr. John T. DeWolf David F. Mazurek ecture Notes: J. Walt Oler Teas Tech niversity 에너지방법 Energy Methods 9 The McGraw-Hill