Strength Theory.

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Irma Nichols
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1 Strength Theory

2 Contents Strength Condition for Simple Stress Sttes( 简单应力状态的强度理论回顾 ) Frcture Criteri for Brittle Mterils( 脆性材料强度理论 ) Yield Criteri for Ductile Mterils( 塑性材料强度理论 ) Summry of Four Strength Theories( 四大强度理论 ) Remrks on Strength Theory( 强度理论补充说明 ) Mesurement of Strin & Strin Rosettes( 应变计与应变花 )

3 Strength Condition for Simple Stress Sttes Tensile/compressive stress: F N A Bending norml stress: Torsionl shering stress: T WP M Wz Z Bending shering stress: * FS S Z Ib z Strength condition t loctions under complicted stress stte: F F Strength theory dy z y dz d Preliminry: find the three principl stress components σ, σ b nd σ c. 3

4 Frcture Criteri for Brittle Mterils Brittle mterils fil suddenly in tensile tests. The filure condition is chrcterized by the ultimte strength σ U. 1. Mimum tensile stress criteri Filure criteri: 1 U Strength condition: 1 U Accurte for mterils primrily under tensile stress stte. No influences from σ, σ 3 re tken into considertion. σ 1 must be tensile stress.. Mimum tensile strin criteri Filure criteri: Strength condition: U U E E Only pplicble to linerly elstic brittle mterils (until filure) Only ccurte for few brittle mterils 4 U

5 Yield Criteri for Ductile Mterils Conservtive criteri nd widely dopted. No influences from σ is 3. Mimum shering stress (Tresc) criteri 1D t yielding : Y Y For σ nd σ b with the sme sign, b Y, ;, b Y For σ nd σ b with opposite signs, b Y ; b Y tken into considertion. 3D 0, b 0, c 0 D 0, 0, 0 : b c 13 Y ; 1 3 Y : 5

6 Yield Criteri for Ductile Mterils Economic criteri. All three principl stress components re tken into considertion. 4. Mimum distortion energy (von Mises) criteri: 1 u d b b c 1D t yielding, 0 : u Y 1 3E 1 ud b b 3E u u Y D 0, 0, 0 : d Y b b Y 1 6E b b c Y b 3D 0, 0, c 0 : b b c c c u d c u Y Y 6

7 Summry of Four Strength Theories All of the four types of strength theory cn be written in universl form, in terms of n effective stress σ r : 1. Mimum tensile stress criteri: r1 1. Mimum tensile strin criteri: ( ) r Mimum shering stress: 4. Mimum distortion energy criteri: r r4 b b c c Note: the limit stress hs been replced by the llowble stress. 7

8 Remrks on Strength Theory Filure mechnism depends on not only mechnicl behvior but lso the stress sttes. Under most stress sttes, select the imum tensile stress / strin criteri for brittle mterils nd the imum shering stress / distortion energy criteri for ductile mterils. Eperiments demonstrte: ll mterils fil by rpture under triil tensile stress stte nd yielding under tri-il compressive stress stte. Hence, brittle or ductile filure criteri should be selected ccordingly. For dngerous points under uni-il stress stte, imum tensile stress criteri is typiclly selected. For dngerous points under pure shering stress stte, imum shering stress criteri is typiclly selected. 8

9 Smple Problem For the plne stress stte shown, nlyze the strength condition bsed on the four types of strength theory. Solution y y b 1 4 b 1 1 4, 0, Mimum tensile stress criteri: 1 r y Y y X 9

10 . Mimum tensile strin criteri: ( ) 1 1 r 1 3 r 4 3. Mimum shering stress: r3 1 3 r Mimum distortion energy criteri: 1 r 4 b b c c r r

11 Eercise Bsed on the imum shering stress criteri, which one of the following stress sttes is most dngerous? σ σ σ () (b) σ σ σ (c) (d) σ 11

12 Smple Problem For the bem shown below: () Sketch the stress stte t points A, B nd C; (b) Sketch the Mohr s circle for points A, B nd C; (c) Sketch the stress stte long principle directions for points A, B nd C; (d) By stisfying the imum tensile stress criteri, determine the orienttion of the frcture slot t points A, B nd C. F F A B C A B C A 3 1 B 1 C 3 1 C A B A B C 1

13 Smple Problem For the thin-wlled pressure vessel shown below, D = 1000 mm, t = 10 mm, p = 3.6 MP nd [σ] = 170 MP. denote the inner dimeter, wll thickness nd pressure respectively. Check the strength condition ccording to the imum distortion energy criteri. t l p pd t pd 4 t pd circ 180 MP t pd b il 90 MP 4t p 3.6 MP 0 c rdil 1 r 4 b b c c D 156 MP 170 MP

14 Smple Problem For the bem shown, q = 40 kn/m, F = 480 kn, nd [] = 160 MP. Check the strength condition long longitudinl lines 1, nd 3 ccording to the imum distortion energy criteri. Solution A F S (kn) F 1 m 6 m 1 m q D F B 1 mm 1. Digrm of internl forces. 40 mm 3 1 y 840 mm z 800 mm. Stress sttes long Lines 1,, nd M (knm)

15 3. Strength condition t Line 1: 1 Mz y, M MD 800 knm, y 40 mm I z I z (40 1) r MP [ ] Strength condition t Line : S z FS s z, Fs FsA Fs B 640 kn, b 1mm Ib z [( ) ] [( ) 0010 ] MP MP, 0, 73.5MP [( ) ( ) ( ) 17MP 160 MP [ ] r m m

16 5. Strength condition t Line 3: Criticl cross-section is identified by both lrge shering stress nd bending moment ( = 1 m)! 3 3 My MP [ ] I.1610 S F S (kn) 640 M (knm) 3 z * z ( ) m * 3 3 FS s z MP 3 3 bi z r 4 MP MP [ ]

17 Mesurement of Strin & Strin Rosettes Strin gges indicte norml strin through chnges in resistnce. y y cos y sin Norml nd shering strins my be obtined from norml strins in ny three directions, y y y 1 cos 1 sin 1 y y y cos sin y y y 3 cos 3 sin 3 With 45 o rosette, ε nd ε y re mesured directly. γ y is obtined indirectly with, y OB y 17

18 Contents Strength Condition for Simple Stress Sttes( 简单应力状态的强度理论回顾 ) Frcture Criteri for Brittle Mterils( 脆性材料强度理论 ) Yield Criteri for Ductile Mterils( 塑性材料强度理论 ) Summry of Four Strength Theories( 四大强度理论 ) Remrks on Strength Theory( 强度理论补充说明 ) Mesurement of Strin & Strin Rosettes( 应变计与应变花 ) 18

Case (a): Ans Ans. Case (b): ; s 1 = 65(4) Ans. s 1 = pr t. = 1.04 ksi. Ans. s 2 = pr 2t ; s 2 = 65(4) = 520 psi

Case (a): Ans Ans. Case (b): ; s 1 = 65(4) Ans. s 1 = pr t. = 1.04 ksi. Ans. s 2 = pr 2t ; s 2 = 65(4) = 520 psi 8 3. The thin-wlled cylinder cn be supported in one of two wys s shown. Determine the stte of stress in the wll of the cylinder for both cses if the piston P cuses the internl pressure to be 65 psi. The