东南大学：《建筑力学 Architectural Mechanics》课程教学课件（英文讲稿）A17 Strength Theory

Strength Theorymi@se.er.cn

Strength Theory mi@seu.edu.cn

Contents·Strength Condition for Simple Stress States（简单应力状态的强度理论回顾·FractureCriteriaforBrittleMaterials（脆性材料强度理论)·YieldCriteriaforDuctileMaterials（塑性材料强度理论)·Summary ofFour StrengthTheories（四大强度理论）·Remarks on StrengthTheory（强度理论补充说明）·Measurement of Strain &StrainRosettes（应变计与应变花）2

• Strength Condition for Simple Stress States（简单应 力状态的强度理论回顾） • Fracture Criteria for Brittle Materials（脆性材料强度 理论） • Yield Criteria for Ductile Materials（塑性材料强度理 论） • Summary of Four Strength Theories（四大强度理论） • Remarks on Strength Theory（强度理论补充说明） • Measurement of Strain & Strain Rosettes（应变计与 应变花） Contents 2

Strength Condition for Simple Stress States.Tensile/compressivestress:. Torsional shearing stressFNT≤[]a≤[]代maxAmaxWmaxmax. Bending shearing stress: Bending normal stressM,FsS2≤[]≤[α]TOmaxmax1.bWmaxmax: Strength condition at locations under complicated stress state:→FdiCStrength theory: Preliminary: find the three principal stress components a, O, and o.3

• Tensile/compressive stress: • Bending normal stress: • Torsional shearing stress: • Bending shearing stress: • Strength condition at locations under complicated stress state: Strength theory x y z dx dz dy F F • Preliminary: find the three principal stress components σa , σb and σc . 3 Strength Condition for Simple Stress States max   max FN A           max   P max T W             * S max max Z z F S I b           max   max Z z M W          

Fracture Criteria for Brittle Materials: Brittle materials fail suddenly in tensile tests. The failure condition ischaracterized by the ultimate strength ou1. Maximum tensile stress criteria2.Maximum tensile strain criteriaFailure criteria: , = SuFailure criteria: O, =OuStrength condition:Strength condition: Oi ≤Ou0102 +0)≤8u61EA· Accurate for materials=9, -v(α2 +0,)≤0uprimarily under tensile stressstate.: Only applicable to linearly: No influences from 02, 0, areelastic brittle materials (untiltaken into considerationfailure)· O, must be tensile stress: Only accurate for few brittlematerials4

1. Maximum tensile stress criteria • Brittle materials fail suddenly in tensile tests. The failure condition is characterized by the ultimate strength σU . Failure criteria:   1  U Strength condition:   1  U • Accurate for materials primarily under tensile stress state. • No influences from σ2 , σ3 are taken into consideration. • σ1 must be tensile stress. 2. Maximum tensile strain criteria 1 U Failure criteria:    Strength condition:     1 1 2 3 1 2 3 U U E E                    • Only applicable to linearly elastic brittle materials (until failure) • Only accurate for few brittle materials 4 Fracture Criteria for Brittle Materials

Yield Criteria for Ductile Materials3. Maximum shearing stress (Tresca) criteriaOb1D at yielding : Tmax = Ty = Oy /2+y2D(o, ±0,0, ±0,0。=0)For o, and o, with the same signda+y-yOao; max(lo.l/o,)≤0yTmax=max2922For o, and o, with opposite signs,? Conservative criteria and0.-0b; a-0,<0Twidely adoptedmax22No influences from , is3D(α。+0,0, + 0,0±0)taken into consideration0i-03; 0,-0,≤0yTmax225

3. Maximum shearing stress (Tresca) criteria max 1D at yielding : 2 Y Y      For σa and σb with the same sign, max max , ; max ,   2 2 2 a b Y a b Y                 For σa and σb with opposite signs, max ; 2 2 a b Y a b Y             • Conservative criteria and widely adopted. • No influences from σ2 is taken into consideration.   1 3 max 1 3 ; 3 0, 0, 0 2 2 : Y a c Y D  b                  5 Yield Criteria for Ductile Materials 2 0, 0, 0 : D   a b c    

Yield Criteria for Ductile Materials4. Maximum distortion energy (von Mises)OUcriteria:+YA(1+vC(g。-0,)+(c,-0)+(o。-0a)a6E-OYOGa+OY1D at yielding(, =0y,0, = 0, = 0) (1+v)Da=uyQYB3E2D(o。±0,0, ±0,0。=0)Economic criteria(1+v-0,0,+o?=ud3E· All three principal0-0,0,+0,=0,stress components areua=uytaken into3D(g。± 0,, ±0,0。+0))二Ua=uyconsiderationF[(0。-0,) +(0, -0.) +(0。-0.)]<0y6

4. Maximum distortion energy (von Mises) criteria:                         2 2 2 2 2 2 2 2 2 2 2 2 1D at yieldi 1 3 1 3 1 ng , 0 : 2 0, 0, 0 : 3 0, 0, 2 : 1 6 0 Y Y d a a Y b d a b b c c a a b b d Y a b c a c a b c a b b Y d Y a b b c c a u E D D u E u E u u u u                                                                                   Y    • Economic criteria. • All three principal stress components are taken into consideration. 6 Yield Criteria for Ductile Materials

Summary of Four Strength TheoriesAll of the four types of strength theory can be written in a universalform, in terms of an effective stress o,:1. Maximum tensile stress criteria: Ori = O, ≤[α]2. Maximum tensile strain criteria: ,2 = , -v(α2 +0,) ≤[α]3. Maximum shearing stress: Or3 = , -O, ≤[α]4. Maximum distortion energy criteria:[(0。-0,) +(0, -0.) +(0。-.) ]≤[0]ONote: the limit stress has been replaced by the allowable stress7

1. Maximum tensile stress criteria:  r1 1     2. Maximum tensile strain criteria:       r2 1 2 3     ( )   3. Maximum shearing stress：     r3 1 3      4. Maximum distortion energy criteria:         2 2 2 4 1 2         r a b b c c a            All of the four types of strength theory can be written in a universal form, in terms of an effective stress σr : 7 Summary of Four Strength Theories Note: the limit stress has been replaced by the allowable stress

Remarks on Strength Theory: Failure mechanism depends on not only mechanical behavior butalso the stress states.. Under most stress states, select the maximum tensile stress / straincriteria for brittle materials and the maximum shearing stress /distortion energy criteria for ductile materials.: Experiments demonstrate: all materials fail by rapture under tri-axial tensile stress state and yielding under tri-axial compressivestress state. Hence, brittle or ductile failure criteria should beselected accordingly: For dangerous points under uni-axial stress state, maximum tensilestress criteria is typically selected.: For dangerous points under pure shearing stress state, maximumshearing stress criteria is typically selected8

• Failure mechanism depends on not only mechanical behavior but also the stress states. 8 Remarks on Strength Theory • Under most stress states, select the maximum tensile stress / strain criteria for brittle materials and the maximum shearing stress / distortion energy criteria for ductile materials. • Experiments demonstrate: all materials fail by rapture under tri￾axial tensile stress state and yielding under tri-axial compressive stress state. Hence, brittle or ductile failure criteria should be selected accordingly. • For dangerous points under uni-axial stress state, maximum tensile stress criteria is typically selected. • For dangerous points under pure shearing stress state, maximum shearing stress criteria is typically selected

Sample Problem: For the plane stress state shown, analyze thestrength condition based on the four types ofstrength theory.Solutionaa.+0ax1X20Ob0a9X22Obaa+4t2，02a22221. Maximum tensile stress criteriaO2 +42 ≤[]dr1 =0≤[g]=0, =229

• For the plane stress state shown, analyze the strength condition based on the four types of strength theory. 2 2 2 2 2 2 1 4 2 2 a x y x y xy b a b                            2 2 2 2 1 2 3 1 1 4 , 0, 4 2 2 2 2                 • Solution 1. Maximum tensile stress criteria:     2 2 1 1 1 1 4 2 2 r                9 Sample Problem X   Y xy   x

2. Maximum tensile strain criteria0r2 = 01 -v(2 +0,)≤[α](1-v)。. (1+v)2 +4t2 ≤[o]2→r2223. Maximum shearing stress :→0,r3 = V? +4t? ≤[α]Or3 =, -0, ≤[α]4. Maximum distortion energy criteria:a-0,) +(0, -0.) +(o。-0a) ≤[0]O=0r4 = ~α? +3t? ≤[0]10

2. Maximum tensile strain criteria:         2 1 2 3 2 2 2 ( ) 1 1 4 2 2 r r                         3. Maximum shearing stress：     2 2         r r 3 1 3 3        4 4. Maximum distortion energy criteria:             2 2 2 4 2 2 2 2 2 2 2 2 2 4 2 2 4 1 2 1 1 1 4 4 4 2 2 2 2 2 3 r a b b c c a r r                                                                  10