LECTURE Third Editio FAILURE CRITERIA: MOHR S CIRCLE AND PRINCIPAL STRESSES A. J. Clark School of Egieerig Departmet of Civil ad Evirometal Egieerig Chapter 7.4 b Dr. Ibrahim A. Assakkaf SPRING 3 ENES Mechaics of Materials Departmet of Civil ad Evirometal Egieerig Uiversit of Marlad, College Park LECTURE. FAILURE CRITERIA: MOHR S CIRCLE AND PRINCIPAL STRESSES (7.4) Slide No. 1 The Stress Trasformatio Equatios for Plae Stress Eample ENES Assakkaf The stresses show i Figure 1a act at a poit o the free surface of a stressed bod. Determie the ormal stresses ad t ad the shearig stress t at this poit if the act o the rotated stress elemet show i Figure 1b.
LECTURE. FAILURE CRITERIA: MOHR S CIRCLE AND PRINCIPAL STRESSES (7.4) Slide No. The Stress Trasformatio Equatios for Plae Stress ENES Assakkaf Eample (cot d) 7MPa 4MPa t t Figure 1 t 1MPa 15 (a) (b) LECTURE. FAILURE CRITERIA: MOHR S CIRCLE AND PRINCIPAL STRESSES (7.4) Slide No. 3 The Stress Trasformatio Equatios for Plae Stress ENES Assakkaf Eample (cot d) The give values are as follows: = 1 MPa, = 7 MPa, θ = 15, θ = 15 + 9 t = 15 t = + 4 MPa t t t 9 15 15
LECTURE. FAILURE CRITERIA: MOHR S CIRCLE AND PRINCIPAL STRESSES (7.4) Slide No. 4 The Stress Trasformatio Equatios for Plae Stress Eample (cot d) Applig Eq. 1 for the give values = cos θ + si θ + = 1cos siθ cosθ ( 15) 7si ( 15) + (4) si( 15) cos( 15) ENES Assakkaf = 5.981 MPa = 5.981 MPa (Tesio) = cos θ + si θ + t = 1cos siθ cosθ ( 15) 7si ( 15) + (4) si( 15) cos( 15) = 85.98 MPa = 86 MPa (comprssio) t LECTURE. FAILURE CRITERIA: MOHR S CIRCLE AND PRINCIPAL STRESSES (7.4) Slide No. 5 The Stress Trasformatio Equatios for Plae Stress Eample (cot d) t = ( ) siθ cosθ + ( cos θ si θ ) = ( 1 ( 7) ) si(15) cos(15) + 4( cos ( 15) si ( 15) ) = 19.64 MPa t 7MPa 4MPa t t =86MPa t =19.64MPa ENES Assakkaf =5.98MPa 1MPa 15
LECTURE. FAILURE CRITERIA: MOHR S CIRCLE AND PRINCIPAL STRESSES (7.4) Slide No. 6 Pricipal Stresses ad Maimum Pricipal Stresses ENES Assakkaf The trasformatio equatios (Eq. 1 or 13) provides a meas for determiig the ormal stress ad the shearig stress t o differet plaes through a poit O i stressed bod. Cosider, for eample, the state of stress at a poit O of the free surface of a structure or machie compoet (Fig. 13). LECTURE. FAILURE CRITERIA: MOHR S CIRCLE AND PRINCIPAL STRESSES (7.4) Slide No. 7 Pricipal Stresses ad Maimum Pricipal Stresses ENES Assakkaf Surfaces perpedicular to z-ais are stress-free. z 7 ksi 1ksi o 5ksi (a) Figure 13 (b)
LECTURE. FAILURE CRITERIA: MOHR S CIRCLE AND PRINCIPAL STRESSES (7.4) Slide No. 8 Pricipal Stresses ad Maimum Pricipal Stresses ENES Assakkaf As the elemet is rotated through a agle θ about a ais perpedicular to the stressfree surface, the ormal stress ad the shearig stress t o differet plaes var cotiuousl as show i Figure 14. For desig purposes, critical stress at the poit are usuall the maimum tesile (or compressive) ad shearig stresses. LECTURE. FAILURE CRITERIA: MOHR S CIRCLE AND PRINCIPAL STRESSES (7.4) Slide No. 9 Pricipal Stresses ad Maimum Pricipal Stresses ENES Assakkaf The pricipal stresses are the maimum ormal stress ma ad miimum ormal stress mi. I geeral, these maimum ad miimum or pricipal stresses ca be determied b plottig curves similar to those of Fig. 14. But this process is time-cosumig, ad therefore, geeral methods are eeded.
LECTURE. FAILURE CRITERIA: MOHR S CIRCLE AND PRINCIPAL STRESSES (7.4) Slide No. 1 Pricipal Stresses ad Maimum 4 3 ENES Assakkaf Variatio of Stresses as Fuctios of θ Stress s Stress tt t 7 ksi Stress (Ksi) 1 6 1 18 4 3 36 t θ 1 ksi 5 ksi -1 - Agle θ (Degrees) Figure 14 LECTURE. FAILURE CRITERIA: MOHR S CIRCLE AND PRINCIPAL STRESSES (7.4) Slide No. 11 Pricipal Stresses ad Maimum Pricipal Stresses Pricipal Stresses for Special Loadig Coditios: Bar uder aial load P = ad A Shaft uder Pure Bedig P A ENES Assakkaf ma ma = (14) T ma c ma = ma = (15) J
LECTURE. FAILURE CRITERIA: MOHR S CIRCLE AND PRINCIPAL STRESSES (7.4) Slide No. 1 Pricipal Stresses ad Maimum Pricipal Stresses for Aiall Loaded Bar P ENES Assakkaf Figure 15 Origial Area, A θ Iclied Area, A P F LECTURE. FAILURE CRITERIA: MOHR S CIRCLE AND PRINCIPAL STRESSES (7.4) Slide No. 13 Pricipal Stresses ad Maimum Pricipal Stresses for Aiall Loaded A P Bar N = P cosθ V = Psiθ = A cosθ Figure 16a θ θ θ V N ENES Assakkaf N P cosθ P P = = = cos θ = A A A A cosθ ( 1+ cos θ )
LECTURE. FAILURE CRITERIA: MOHR S CIRCLE AND PRINCIPAL STRESSES (7.4) Slide No. 14 Pricipal Stresses ad Maimum A Pricipal Stresses for Aiall Loaded Bar Figure 16b N P N = P cosθ V = Psiθ = A cosθ θ θ θ V ENES Assakkaf V P siθ P P = = = siθ cosθ = si θ A A A A cosθ LECTURE. FAILURE CRITERIA: MOHR S CIRCLE AND PRINCIPAL STRESSES (7.4) Slide No. 15 Pricipal Stresses ad Maimum Pricipal Stresses for Aiall Loaded Bar is maimum whe θ = or 18 is maimum whe θ = 45 or 135 Also ma ma = (16) Therefore P = ad A ma ma = P A ENES Assakkaf (17)
LECTURE. FAILURE CRITERIA: MOHR S CIRCLE AND PRINCIPAL STRESSES (7.4) Slide No. 16 Pricipal Stresses ad Maimum Fig.17 t da cos α ENES Assakkaf Pricipal Stresses for Shaft uder Pure Torsio α t da (c) α da si α da α A (b) (a) LECTURE. FAILURE CRITERIA: MOHR S CIRCLE AND PRINCIPAL STRESSES (7.4) Slide No. 17 Pricipal Stresses ad Maimum ENES Assakkaf Pricipal Stresses for Shaft uder Pure Torsio = siα cosα si α = ( cos α si α ) cos α = t = (18) (19) t da cos α α t da α da T ma c ma = ma = () J da si α
LECTURE. FAILURE CRITERIA: MOHR S CIRCLE AND PRINCIPAL STRESSES (7.4) Slide No. 18 Pricipal Stresses ad Maimum Developmet of Pricipal Stresses Equatios Recall Eq 13 t = si θ + cos θ ENES Assakkaf + = + cos θ + si θ (13a) (13b) LECTURE. FAILURE CRITERIA: MOHR S CIRCLE AND PRINCIPAL STRESSES (7.4) Slide No. 19 Pricipal Stresses ad Maimum Developmet of Pricipal Stresses Equatios or ENES Assakkaf Differetiatig the first equatio with respect to θ ad equate the result to zero, gives d + d = + cos θ + si θ dθ dθ set = ( ) si θ + cos θ = 1 ta θ = or = ta p θ p (1)
LECTURE. FAILURE CRITERIA: MOHR S CIRCLE AND PRINCIPAL STRESSES (7.4) Slide No. Pricipal Stresses ad Maimum Developmet of Pricipal Stresses Equatios ENES Assakkaf Substitutig the epressio for θ p ito Eq. 13a, ields + p1, p ± + = () Eq. 1 gives the two pricipal stresses i the -plae, ad the third stress p3 = z =. LECTURE. FAILURE CRITERIA: MOHR S CIRCLE AND PRINCIPAL STRESSES (7.4) Slide No. 1 Pricipal Stresses ad Maimum Pricipal Stresses Pricipal stresses ma ad mi ca be computed from + p1, p ± + = (a) where subscript p refers to the plaes of maimum ad miimum values of. ENES Assakkaf
LECTURE. FAILURE CRITERIA: MOHR S CIRCLE AND PRINCIPAL STRESSES (7.4) Slide No. Pricipal Stresses ad Maimum ENES Assakkaf Locatio of the Plae of Pricipal Stresses θ p = 1 1 ta (b) LECTURE. FAILURE CRITERIA: MOHR S CIRCLE AND PRINCIPAL STRESSES (7.4) Slide No. 3 Pricipal Stresses ad Maimum Notes o Pricipal Stresses Equatio 1. Eq. gives the agle θ p ad θ p + 9 betwee -plae (or -plae) ad the mutuall perpedicular plaes o which the pricipal stresses act.. Whe ta θ p is positive, θ p is positive, ad the rotatio is couterclockwise from the - ad-plaes to the plaes o which the two pricipal stresses act. ENES Assakkaf
LECTURE. FAILURE CRITERIA: MOHR S CIRCLE AND PRINCIPAL STRESSES (7.4) Slide No. 4 Pricipal Stresses ad Maimum Notes o Pricipal Stresses Equatio ENES Assakkaf 3. Whe ta θ p is egative, θ p is egative, ad the rotatio is clockwise. 4. The shearig stress is zero o plaes eperiecig maimum ad miimum values of ormal stresses. 5. If oe or both of the pricipal stresses from Eq. is egative, the algebraic maimum stress ca have a smaller absolute value tha the miimum stress. LECTURE. FAILURE CRITERIA: MOHR S CIRCLE AND PRINCIPAL STRESSES (7.4) Slide No. 5 Pricipal Stresses ad Maimum ENES Assakkaf Developmet of Maimum Shearig Stress Equatio = si θ cos θ or Recall Eq 13b: t + d t d = si θ + cos θ dθ dθ = ta θ = set ( ) cos θ si θ = or θ = ta 1 (3)
LECTURE. FAILURE CRITERIA: MOHR S CIRCLE AND PRINCIPAL STRESSES (7.4) Slide No. 6 Pricipal Stresses ad Maimum Developmet of Pricipal Shearig Stress Equatio ENES Assakkaf Substitutig the epressio for θ ito Eq. 13b, ields p ± + Eq. 4 gives the maimum i-plae shearig stress. = (4) LECTURE. FAILURE CRITERIA: MOHR S CIRCLE AND PRINCIPAL STRESSES (7.4) Slide No. 7 Pricipal Stresses ad Maimum ENES Assakkaf Maimum I-Plae Maimum i-plae shearig stress ca be computed from p = ± + (4a) where the subscript p refers to the plae of maimum i-plae shearig stress p.
LECTURE. FAILURE CRITERIA: MOHR S CIRCLE AND PRINCIPAL STRESSES (7.4) Slide No. 8 Pricipal Stresses ad Maimum Locatio of the Plae of Maimum ENES Assakkaf θ = 1 1 ta (4b) LECTURE. FAILURE CRITERIA: MOHR S CIRCLE AND PRINCIPAL STRESSES (7.4) Slide No. 9 Pricipal Stresses ad Maimum Notes o Pricipal Stresses ad Maimum I-Plae Equatio 1. The two agles θ p ad θ differ b 9, therefore, θ p ad θ are 45 apart.. This meas that the plaes i which the maimum i-plae shearig stress occur are 45 from the pricipal plaes. ENES Assakkaf
LECTURE. FAILURE CRITERIA: MOHR S CIRCLE AND PRINCIPAL STRESSES (7.4) Slide No. 3 Pricipal Stresses ad Maimum ENES Assakkaf Notes o Pricipal Stresses ad Maimum I-Plae Equatio 3. The directio of the maimum shearig stress ca be determied b drawig a wedge-shaped block with two sides parallel to the plaes havig the maimum ad miimum pricipal stresses, ad with the third side at a agle of 45. The directio of the maimum shearig stress must oppose the larger of the two pricipal stresses. LECTURE. FAILURE CRITERIA: MOHR S CIRCLE AND PRINCIPAL STRESSES (7.4) Slide No. 31 Pricipal Stresses ad Maimum θ p Fig.18 ENES Assakkaf Wedge-shaped Block > p1 P θ p ma p1 45 45 p1 ma p p
LECTURE. FAILURE CRITERIA: MOHR S CIRCLE AND PRINCIPAL STRESSES (7.4) Slide No. 3 Pricipal Stresses ad Maimum Useful Relatioships ENES Assakkaf The maimum value of t is equal to oe half the differece betwee the two i-plae pricipal stresses, that is p1 p p = (5) For plae stress, the sum of the ormal stresses o a two orthogoal plaes through a poit i a bod is a costat or i ivariat. = + p1 + p (6) LECTURE. FAILURE CRITERIA: MOHR S CIRCLE AND PRINCIPAL STRESSES (7.4) Slide No. 33 Pricipal Stresses ad Maimum Useful Relatioships ENES Assakkaf Whe a state of plae eists, oe of the pricipal stresses is zero. If the values of p1 ad p from Eq. 5 have the same sig, the the third pricipal stress p3 equals zero, will be either the maimum or miimum ormal stresses. Three possibilities: ( )/, ma = ( ) /, ma = ( )/ ma = p1 p p1 p
LECTURE. FAILURE CRITERIA: MOHR S CIRCLE AND PRINCIPAL STRESSES (7.4) Slide No. 34 Pricipal Stresses ad Maimum Eample 3 ENES Assakkaf Normal ad shearig stresses o horizotal ad vertical plaes through a poit i a structural member subjected to plae stress are show i Figure 19. Determie ad show o a sketch the pricipal ad maimum shearig stresses. LECTURE. FAILURE CRITERIA: MOHR S CIRCLE AND PRINCIPAL STRESSES (7.4) Slide No. 35 Pricipal Stresses ad Maimum Eample 3 (cot d) ENES Assakkaf Fig.19 4ksi 6ksi 1ksi The give values for use i Eqs. ad 4 are: = +1 ksi = - 4 ksi = - 6 ksi
LECTURE. FAILURE CRITERIA: MOHR S CIRCLE AND PRINCIPAL STRESSES (7.4) Slide No. 36 Pricipal Stresses ad Maimum Eample 3 (cot d) Usig Eq. a for the give values: + p = ± + 1 + ( 4) 1 ( 4) = ± + ( 6) = 4 ± 1 Therefore, = 4 + 1 = + 14 ksi = 14 ksi (T) P1 p p3 = = = 4 1 = 6 ksi = 6 ksi (C) z ENES Assakkaf LECTURE. FAILURE CRITERIA: MOHR S CIRCLE AND PRINCIPAL STRESSES (7.4) Slide No. 37 Pricipal Stresses ad Maimum Eample 3 (cot d) Sice the p1 ad p have opposite sig, the maimum shearig stress is p 1 p 14 ( 6) ma = = = = + 1 ksi The locatio θ p of the pricipal stresses is computed from Eq. b θ p = 1 ta 1 1 ( 6) ( 4) = 18.43 ENES Assakkaf
LECTURE. FAILURE CRITERIA: MOHR S CIRCLE AND PRINCIPAL STRESSES (7.4) Slide No. 38 Pricipal Stresses ad Maimum Eample 3 (cot d) 4ksi 6ksi 6ksi = 1 ksi = p ma Sketch of pricipal ad ma shearig stresses 1ksi 18.43 4 ksi 1 ksi 45 6 ksi + = 1 4 = = 4 14 ksi ENES Assakkaf ksi (T) LECTURE. FAILURE CRITERIA: MOHR S CIRCLE AND PRINCIPAL STRESSES (7.4) Slide No. 39 Pricipal Stresses ad Maimum Eample 4 Normal ad shearig stresses o horizotal ad vertical plaes through a poit i a structural member subjected to plae stress are show i Figure. Determie ad show o a sketch the pricipal ad maimum shearig stresses ENES Assakkaf
LECTURE. FAILURE CRITERIA: MOHR S CIRCLE AND PRINCIPAL STRESSES (7.4) Slide No. 4 Pricipal Stresses ad Maimum Eample 4 (cot d) ENES Assakkaf Fig. 36MPa 7 MPa 4 MPa The give values for use i Eqs. ad 4 are: = +7 MPa = +36 MPa = - 4 MPa LECTURE. FAILURE CRITERIA: MOHR S CIRCLE AND PRINCIPAL STRESSES (7.4) Slide No. 41 Pricipal Stresses ad Maimum Eample 4 (cot d) Usig Eq. a for the give values: + p = ± + 7 + ( + 36) 7 ( + 36) = ± + ( 4) = 54 ± 3 Therefore, = 54 + 3 = + 84 ksi = 84 MPa (T) P1 p p3 = = = 54 3 = + 4 ksi = 4 MPa (T) z ENES Assakkaf
LECTURE. FAILURE CRITERIA: MOHR S CIRCLE AND PRINCIPAL STRESSES (7.4) Slide No. 4 Pricipal Stresses ad Maimum Eample 4 (cot d) ENES Assakkaf Sice the p1 ad p have the same sig, the maimum shearig stress is 1 84 84 ma = p = = = + 4 MPa The locatio θ p of the pricipal stresses is computed from Eq. b θ p = 1 ta 1 = 1 ta 1 ( 4) = 6.57 7 36 LECTURE. FAILURE CRITERIA: MOHR S CIRCLE AND PRINCIPAL STRESSES (7.4) Slide No. 43 p Pricipal Stresses ad Maimum Eample 4 (cot d) 36 MPa 6ksi 4 MPa Sketch of pricipal ad ma shearig stresses 7 MPa 6.57 p1 p 84 4 p = = = 3 MPa = 4 MPa ma ma 3 MPa 54 MPa 45 4 MPa 4 MPa 45 p3 = 84 MPa 4 MPa + 7 + 36 = = = 54 ENES Assakkaf 84 MPa MPa (T)
LECTURE. FAILURE CRITERIA: MOHR S CIRCLE AND PRINCIPAL STRESSES (7.4) Slide No. 44 Mohr s Circle for Plae Stress ENES Assakkaf Itroductio Mohr s circle is a pictorial or graphical iterpretatio of the trasformatio equatios for plae stress. The process ivolves the costructio of a circle i such a maer that the coordiates of each poit o the circle represet the ormal ad shearig stresses o oe plae through the stressed LECTURE. FAILURE CRITERIA: MOHR S CIRCLE AND PRINCIPAL STRESSES (7.4) Slide No. 45 Maimum ENES Assakkaf Maimum shearig stress occurs for = ave ma = R = + ta θ s = o Note : defies two agles separated b 9 ad o offset fromθ p b 45 + = ave =
LECTURE. FAILURE CRITERIA: MOHR S CIRCLE AND PRINCIPAL STRESSES (7.4) Slide No. 46 Mohr s Circle for Plae Stress ENES Assakkaf Itroductio Poit, ad the agular positio of the radius to the poit gives the orietatio of the plae. The proof that ormal ad shearig compoets of stress o arbitrar plae through a poit ca be represeted as poits o a circle follows from Eqs. 13a ad 13b. LECTURE. FAILURE CRITERIA: MOHR S CIRCLE AND PRINCIPAL STRESSES (7.4) Slide No. 47 Mohr s Circle for Plae Stress ENES Assakkaf Plae Stresses usig Mohr s Circle Recall Eqs. 13a ad 13b, + = + cos θ + si θ (13a) t = si θ + cos θ (13a) Squarig both equatios, addig, ad simplifig gives + = t + + (7)
LECTURE. FAILURE CRITERIA: MOHR S CIRCLE AND PRINCIPAL STRESSES (7.4) Slide No. 48 Mohr s Circle for Plae Stress ENES Assakkaf Plae Stresses usig Mohr s Circle The previous equatio is ideed a equatio of a circle i terms of the variable ad t. The circle is cetered o the the ais at a distace ( - )/ from the ais, ad the radius of the circle is give b R = + (8) LECTURE. FAILURE CRITERIA: MOHR S CIRCLE AND PRINCIPAL STRESSES (7.4) Slide No. 49 Mohr s Circle for Plae Stress ENES Assakkaf Plae Stresses usig Mohr s Circle Normal stresses are plotted as horizotal coordiates, with tesile stresses (positive) plotted to the right of the origi ad compressive stresses (egative) plotted to the left. Shearig stresses are plotted as vertical coordiates, with those tedig to produce a clockwise rotatio of the stress elemet
LECTURE. FAILURE CRITERIA: MOHR S CIRCLE AND PRINCIPAL STRESSES (7.4) Slide No. 5 Mohr s Circle for Plae Stress ENES Assakkaf Plae Stresses usig Mohr s Circle plotted above the -ais, ad those tedig to produce couterclockwise rotatio of the stress elemet plotted below the -ais. Sig covetios for iterpretig the ormal ad shearig stresses will be provided, ad illustrated through eamples. LECTURE. FAILURE CRITERIA: MOHR S CIRCLE AND PRINCIPAL STRESSES (7.4) Slide No. 51 Mohr s Circle for Plae Stress ENES Assakkaf Plae Stresses usig Mohr s Circle Mohr s circle for a poit subjected to plae stress ca be draw whe stresses o two mutuall perpedicular plaes through the poit are kow. A θ A t θ
LECTURE. FAILURE CRITERIA: MOHR S CIRCLE AND PRINCIPAL STRESSES (7.4) Slide No. 5 Mohr s Circle for Plae Stress ENES Assakkaf Mohr s Circle Fig.1 p H, ) ( p + C θ θ p t V, ) ( p1 LECTURE. FAILURE CRITERIA: MOHR S CIRCLE AND PRINCIPAL STRESSES (7.4) Slide No. 53 Mohr s Circle for Plae Stress Mohr s Circle ENES Assakkaf Maimum shearig stress occurs for = ave ma = R = ta θ s = Note : defies two agles separated b 9 = offset fromθ b 45 ave p + = + o o ad
LECTURE. FAILURE CRITERIA: MOHR S CIRCLE AND PRINCIPAL STRESSES (7.4) Slide No. 54 Mohr s Circle for Plae Stress ENES Assakkaf Plae Stresses usig Mohr s Circle Drawig Procedure for Mohr s Circle 1. Choose a set of - coordiate aes.. Idetif the stresses, ad = ad list them with proper sig. 3. Draw a set of -coordiate aes with ad positive to the right ad upward, respectivel. 4. Plot the poit (, - ) ad label it poit V (vertical plae). LECTURE. FAILURE CRITERIA: MOHR S CIRCLE AND PRINCIPAL STRESSES (7.4) Slide No. 55 Mohr s Circle for Plae Stress ENES Assakkaf Plae Stresses usig Mohr s Circle Drawig Procedure for Mohr s Circle (cot d) 5. Plot the poit (, ) ad label it poit H (horizotal plae). 6. Draw a lie betwee V ad H. This establishes the ceter ad the radius R of Mohr s circle. 7. Draw the circle. 8. A etesio of the radius betwee C ad V ca be idetified as the -ais or referece lie for the agle measuremets (I.e., θ =).
LECTURE. FAILURE CRITERIA: MOHR S CIRCLE AND PRINCIPAL STRESSES (7.4) Slide No. 56 Mohr s Circle for Plae Stress ENES Assakkaf Sig Covetios I a give face of the stressed elemet, the shearig stresses that teds to rotate the elemet clockwise will be plotted above the -ais i the circle. I a give face of the stressed elemet, the shearig stresses that teds to rotate the elemet couterclockwise will be plotted below the -ais i the circle. LECTURE. FAILURE CRITERIA: MOHR S CIRCLE AND PRINCIPAL STRESSES (7.4) Slide No. 57 Mohr s Circle for Plae Stress ENES Assakkaf Sig Covetios Fig. The followig jigle ma be helpful i rememberig this covetios: I the kitche, the clock is above, ad the couter is below. Beer ad Johsto (199) (a) Clockwise Above (b) Couterclockwise Below
LECTURE. FAILURE CRITERIA: MOHR S CIRCLE AND PRINCIPAL STRESSES (7.4) Slide No. 58 Mohr s Circle for Plae Stress ENES Assakkaf Poits of Iterests o Mohr s Circle 1. Poit D that provides the pricipal stress p1.. Poit E that gives the pricipal stress p. 3. Poit A that provides the maimum iplae shearig stress - p ad the accompaied ormal stress avg that acts o the plae. LECTURE. FAILURE CRITERIA: MOHR S CIRCLE AND PRINCIPAL STRESSES (7.4) Slide No. 59 Mohr s Circle for Plae Stress ENES Assakkaf With the phsical sigificace of Mohr s circle for plae stress established, it ma be applied with simple geometric cosideratios. Critical values are estimated graphicall or calculated. For a kow state of plae stress,, plot the poits X ad Y ad costruct the circle cetered at C. + ave = R = + The pricipal stresses are obtaied at A ad B. = ± R ma,mi ave ta θ p = The directio of rotatio of O to Oa is the same as CX to CA.
LECTURE. FAILURE CRITERIA: MOHR S CIRCLE AND PRINCIPAL STRESSES (7.4) Slide No. 6 Mohr s Circle for Plae Stress ENES Assakkaf With Mohr s circle uiquel defied, the state of stress at other aes orietatios ma be depicted. For the state of stress at a agle θ with respect to the aes, costruct a ew diameter X Y at a agle θ with respect to XY. Normal ad shear stresses are obtaied from the coordiates X Y. LECTURE. FAILURE CRITERIA: MOHR S CIRCLE AND PRINCIPAL STRESSES (7.4) Slide No. 61 Mohr s Circle for Plae Stress Mohr s circle for cetric aial loadig: ENES Assakkaf P =, = = A P = = = A Mohr s circle for torsioal loadig: Tc Tc = = = = = = J J
LECTURE. FAILURE CRITERIA: MOHR S CIRCLE AND PRINCIPAL STRESSES (7.4) Slide No. 6 Mohr s Circle for Plae Stress Eample 5 The stresses show i Figure 3 act at a poit o the free surface of a stressed bod. Use Mohr s circle to determie the ormal ad shearig stresses at this poit o the iclied plae AB show i the figure. 14 ksi ENES Assakkaf Fig.3 A 5 1 B ksi LECTURE. FAILURE CRITERIA: MOHR S CIRCLE AND PRINCIPAL STRESSES (7.4) Slide No. 63 Mohr s Circle for Plae Stress ENES Assakkaf Eample 5 (cot d) The give values for use i drawig Mohr s circle are: C R = 3 = p1 = ksi = p = 14 ksi 45.4 1 1 z = p3 = θ = ta = 134.76 5 R = 3 134.76 + 14 C = = 17 ksi 14 R = radius = = 3 t = Rsi 45.4 = 3si 45.4 = = C R cos ( 45.4) = 17 3(.7) = 14.89 ksi t ( ) ( ).13 ksi
LECTURE. FAILURE CRITERIA: MOHR S CIRCLE AND PRINCIPAL STRESSES (7.4) Slide No. 64 Mohr s Circle for Plae Stress ENES Assakkaf Eample 6 For the state of plae stress show, (a) costruct Mohr s circle, determie (b) the pricipal plaes, (c) the pricipal stresses, (d) the maimum shearig stress ad the correspodig ormal stress. LECTURE. FAILURE CRITERIA: MOHR S CIRCLE AND PRINCIPAL STRESSES (7.4) Slide No. 65 Mohr s Circle for Plae Stress ENES Assakkaf Eample 6 (cot d) SOLUTION: Costructio of Mohr s circle + ( 5) + ( 1) ave = = = MPa CF = 5 = 3MPa FX = 4MPa R = CX = ( 3) + ( 4) = 5MPa
LECTURE. FAILURE CRITERIA: MOHR S CIRCLE AND PRINCIPAL STRESSES (7.4) Slide No. 66 Eample 6 (cot d) Pricipal plaes ad stresses ma = OA = OC + CA = + 5 ma = 7MPa ma = OB = OC BC = 5 ma = 3MPa ENES Assakkaf FX = CP θ p = 53.1 ta θ p = θ p = 6. 6 4 3 LECTURE. FAILURE CRITERIA: MOHR S CIRCLE AND PRINCIPAL STRESSES (7.4) Slide No. 67 Eample 6 (cot d) ENES Assakkaf Maimum shear stress θ s = θ p + 45 θ s = 71. 6 ma = R ma = 5 MPa = ave = MPa