Static Failure Chapter 5 PDF

Static Failure Chapter 5  Maximum Shear Stress Theory (MSS) 1 Theory: Yielding begins when the maximum shear stress in a stress element exceeds the maximum shear stress in a tension test specimen of the same material when that specimen begins to yield. For a tension test spec...

Static Failure Chapter 5  Maximum Shear Stress Theory (MSS) 1 Theory: Yielding begins when the maximum shear stress in a stress element exceeds the maximum shear stress in a tension test specimen of the same material when that specimen begins to yield. For a tension test specimen, the maximum shear stress is s1 /2. At yielding, when s1 = Sy, the maximum shear stress is Sy /2. Could restate the theory as follows: Theory: Yielding begins when the maximum shear stress in a stress element exceeds Sy/2. © McGraw Hill 5  Maximum Shear Stress Theory (MSS) 2 For any stress element, use Mohr’s circle to find the maximum shear stress. Compare the maximum shear stress to Sy/2. Ordering the principal stresses such that σ1 ≥ σ2 ≥ σ3, s1 - s 3 Sy t max = ³ or s 1 - s 3 ³ S y (5 - 1) 2 2 Incorporating a factor of safety n. Sy Sy t max = or s 1 - s 3 = (5 - 3) 2n n Or solving for factor of safety. Sy 2 n= t max © McGraw Hill 6  Maximum Shear Stress Theory (MSS) 3 To compare to experimental data, express τmax in terms of principal stresses and plot. To simplify, consider a plane stress state. Let σA and σB represent the two non-zero principal stresses, then order them with the zero principal stress such that σ1 ≥ σ2 ≥ σ3. Assuming σA ≥ σB there are three cases to consider Case 1: σA ≥ σB ≥ 0 Case 2: σA ≥ 0 ≥ σB Case 3: 0 ≥ σA ≥ σB © McGraw Hill 7  Maximum Shear Stress Theory (MSS) 5 Plot three cases on principal stress axes. Case 1: σA ≥ σB ≥ 0 σ A ≥ Sy Case 2: σA ≥ 0 ≥ σB σ A − σ B ≥ Sy Case 3: 0 ≥ σA ≥ σB σB ≤ −Sy Other lines are symmetric cases Inside envelope is predicted safe zone. Fig. 5–7 © McGraw Hill 9  Maximum Shear Stress Theory (MSS) Comparison to experimental data. Conservative in all quadrants. Commonly used for design situations. © McGraw Hill 10  Distortion Energy (Von Mises) Failure Theory Originated from observation that ductile materials stressed hydrostatically (equal principal stresses) exhibited yield strengths greatly in excess of expected values. Theorizes that if strain energy is divided into hydrostatic volume changing energy and angular distortion energy, the yielding is primarily affected by the distortion energy. Fig. 5–8 © McGraw Hill 11  Distortion Energy (DE) Failure Theory 3 Theory: Yielding occurs when the distortion strain energy per unit volume reaches the distortion strain energy per unit volume for yield in simple tension or compression of the same material. Fig. 5–8 © McGraw Hill 12 Distortion Energy (Von Mises) Failure Theory  Coulomb-Mohr Theory 2 From the geometry, derive the failure criteria. B2C2 - B1C1 B3C3 - B1C1 = OC2 - OC1 OC3 - OC1 B2C2 - B1C1 B3C3 - B1C1 = C1C2 C1C3 B1C1 = St 2, B2C2 = (s 1 - s 3 ) 2, and B3C3 = Sc 2 s1 - s 3 St S c St - - 2 2 = 2 2 St s 1 + s 3 St S c Fig. 5–13 - + 2 2 2 2 s1 s3 - =1 (5 - 22) St Sc © McGraw Hill 19  Coulomb-Mohr Theory s1 s 3 - =1 (5 - 22) St Sc To plot on principal stress axes, consider three cases Case 1: σA ≥ σB ≥ 0 For this case, σ1 = σA and σ3 = 0 Eq. (5−22) reduces to s A ³ St (5 - 23) Case 2: σA ≥ 0 ≥ σB For this case, σ1 = σA and σ3 = σB Eq. (5-22) reduces to sA sB - ³1 (5 - 24) St S c Case 3: 0 ≥ σA ≥ σB For this case, σ1 = 0 and σ3 = σB Eq. (5−22) reduces to s B £ - Sc (5 - 25) © McGraw Hill 20  Coulomb-Mohr Theory Plot three cases on principal stress axes. Similar to MSS theory, except with different strengths for compression and tension. Fig. 5–14 Access the text alternative for slide images. © McGraw Hill 21 Example: Coulomb-Mohr Theory Maximum Normal Stress Theory Theory: Failure occurs when the maximum principal stress in a stress element exceeds the strength. Predicts failure when s 1 ³ Sut or s 3 £ - Suc (5 - 28) For plane stress, s A ³ Sut or s B £ - Suc (5 - 29) Incorporating design factor, S Suc s A = ut or sB = - (5 - 30) n n © McGraw Hill 24 Maximum Normal Stress Theory 2 Plot on principal stress axes. Unsafe in part of fourth quadrant. Not recommended for use. Included for historical comparison. Fig. 5–18 © McGraw Hill 25 Modified Mohr criteria Coulomb-Mohr is conservative in 4th quadrant. Modified Mohr criteria adjusts to better fit the data in the 4th quadrant. Fig. 5–19 © McGraw Hill 27  Modified-Mohr Theory Quadrant condition Failure criteria σA ≥ σB ≥ 0 Sut sA = (5 - 32a ) sB n sA ³ 0 ³ sB and £1 Sut sA sA = (5 - 32a ) n > 1 ( uc sB S - Sut ) s A s B 1 sA ³ 0 ³ sB and - = (5 - 32b ) sA Suc Sut Suc n S s B = - uc (5 - 32c ) 0 ≥ σA ≥ σB n © McGraw Hill 28