P4 Stress and Strain Dr. A.B. Zavatsky HT08 Lecture 5 Plane Stress Transformation Equations

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1 P4 Stre and Strain Dr. A.B. Zavatk HT08 Lecture 5 Plane Stre Tranformation Equation Stre element and lane tre. Stree on inclined ection. Tranformation equation. Princial tree, angle, and lane. Maimum hear tre. 1

2 Normal and hear tree on inclined ection To obtain a comlete icture of the tree in a bar, we mut conider the tree acting on an inclined (a ooed to a normal ) ection through the bar. Inclined ection Normal ection P P Becaue the tree are the ame throughout the entire bar, the tree on the ection are uniforml ditributed. P Inclined ection P Normal ection

3 N P V P The force P can be reolved into comonent: Normal force N erendicular to the inclined lane, N P co Shear force V tangential to the inclined lane V P in If we know the area on which the force act, we can calculate the aociated tree. area A area A area ( A / co ) area ( A / co ) 3

4 Force N Area Area co P co A/ co ( 1 co ) ma occur when 0 P A co Force V P in P in co Area Area A/ co A in co ( in ) ma ± / occur when -/ 45 4

5 Introduction to tre element Stre element are a ueful wa to rereent tree acting at ome oint on a bod. Iolate a mall element and how tree acting on all face. Dimenion are infiniteimal, but are drawn to a large cale. P P / A Area A z PA / PA / 5

6 Maimum tree on a bar in tenion P a P ma P / A a Maimum normal tre, Zero hear tre 6

7 Maimum tree on a bar in tenion P ab P / 45 ma / / b Maimum hear tre, Non-zero normal tre 7

8 Stre Element and Plane Stre When working with tre element, kee in mind that onl one intrinic tate of tre eit at a oint in a treed bod, regardle of the orientation of the element ued to ortra the tate of tre. We are reall jut rotating ae to rereent tree in a new coordinate tem

9 Normal tree,, z (tenion i oitive) z z z z z z Shear tree, z z, z z Sign convention for ab Subcrit a indicate the face on which the tre act (oitive face i erendicular to the oitive direction) Subcrit b indicate the direction in which the tre act Strictl,, z zz 9

10 When an element i in lane tre in the lane, onl the and face are ubjected to tree ( z 0 and z z z z 0). Such an element could be located on the free urface of a bod (no tree acting on the free urface). Plane tre element in D, z 0 10

11 Stree on Inclined Section The tre tem i known in term of coordinate tem. We want to find the tree in term of the rotated coordinate tem 1 1. Wh? A material ma ield or fail at the maimum value of or. Thi value ma occur at ome angle other than 0. (Remember that for uniaial tenion the maimum hear tre occurred when 45 degree. ) 11

12 Tranformation Equation Stree Force A A 11 A ec 1 A ec A tan 1 A tan Force can be found from tree if the area on which the tree act i known. Force comonent can then be ummed. Left face ha area A. Bottom face ha area A tan. Inclined face ha area A ec. 1

13 1 A A 11 A ec 1 A ec A tan 1 Sum force in the 1 Aec Sum force in the 11 Uing direction : A tan ( A) co ( A) in ( A tan ) in ( A tan ) co 0 1 direction : ( A) in ( A) co ( Atan ) co ( Atan ) in 0 Aec co in and imlifing give : ( ) in co ( co in ) in co 13

14 Uing the following trigonometric identitie co 1 co 1 co in in co in give the tranformation equation for lane tre : 1 11 ( ) in co co in HLT, age 108 For tree on the 1 1 face, ubtitute 90 co for : in Summing the ereion for and 1 give : Can be ued to find 1, intead of eqn above. 14

15 Eamle: The tate of lane tre at a oint i rereented b the tre element below. Determine the tree acting on an element oriented 30 clockwie with reect to the original element. 50 MPa 80 MPa 80 MPa 5 MPa 50 MPa Define the tree in term of the etablihed ign convention: -80 MPa 50 MPa -5 MPa We need to find 1, 1, and 11 when -30. Subtitute numerical value into the tranformation equation: 1 1 co co in ( 30 ) ( 5) in ( 30 ) 5.9 MPa 15

16 co in co 30 5 ( ) in co ( 80 50) in 1 ( ) ( ) in ( 30 ) 4.15 MPa ( 30 ) ( 5) co ( 30 ) 68.8 MPa 5.8 MPa 4.15 MPa o 5.8 MPa Note that 1 could alo be obtained (a) b ubtituting 60 into the equation for 1 or (b) b uing the equation MPa 68.8 MPa 1 (from Hibbeler, E. 15.) 16

17 Princial Stree The maimum and minimum normal tree ( 1 and ) are known a the rincial tree. To find the rincial tree, we mut differentiate the tranformation equation. 1 d 1 d d 1 d tan ( ) ( in ) ( co ) in co in co 0 0 are rincial angle aociated with the rincial tree (HLT, age 108) There are two value of in the range 0-360, with value differing b 180. There are two value of in the range 0-180, with value differing b 90. So, the lane on which the rincial tree act are mutuall erendicular. 17

18 18 in co tan 1 We can now olve for the rincial tree b ubtituting for in the tre tranformation equation for 1. Thi tell u which rincial tre i aociated with which rincial angle. ( ) / R R R R in co R R 1

19 19 Subtituting for R and re-arranging give the larger of the two rincial tree: 1 To find the maller rincial tre, ue 1. 1 Thee equation can be combined to give: 1, ± Princial tree (HLT age 108) To find out which rincial tre goe with which rincial angle, we could ue the equation for in and co or for 1.

20 The lane on which the rincial tree act are called the rincial lane. What hear tree act on the rincial lane? Comare the equation for 11 d d 1 ( ) ( ) in in 11 0 and d co 0 1 co 0 ( ) in co 0 d 0 Solving either equation give the ame ereion for tan Hence, the hear tree are zero on the rincial lane. 0

21 1 1 1 Princial Stree 1, ± tan Princial Angle defining the Princial Plane 1

22 Eamle: The tate of lane tre at a oint i rereented b the tre element below. Determine the rincial tree and draw the correonding tre element. 50 MPa 80 MPa 80 MPa 5 MPa 50 MPa Define the tree in term of the etablihed ign convention: -80 MPa 50 MPa -5 MPa 1, ± 1, MPa ± MPa ( 5) 15 ± 69.6

23 tan tan ( 5) MPa 54.6 MPa o 10.5 o 84.6 MPa 1.0 and , MPa But we mut check which angle goe with which rincial tre. 1 1 co co 10.5 in ( ) ( 5) in ( 10.5 ) 84.6 MPa MPa with MPa with

24 The two rincial tree determined o far are the rincial tree in the lane. But remember that the tre element i 3D, o there are alwa three rincial tree z,, 1,, 3 0 Uuall we take 1 > > 3. Since rincial tree can be comreive a well a tenile, 3 could be a negative (comreive) tre, rather than the zero tre. 4

25 11 d Maimum Shear Stre To find the maimum hear tre, we mut differentiate the tranformation equation for hear. 11 d ( ) ( ) tan in co co in 0 There are two value of in the range 0-360, with value differing b 180. There are two value of in the range 0-180, with value differing b 90. So, the lane on which the maimum hear tree act are mutuall erendicular. Becaue hear tree on erendicular lane have equal magnitude, the maimum oitive and negative hear tree differ onl in ign. 5

26 6 ( ) / R ( ) co in tan 1 1 We can now olve for the maimum hear tre b ubtituting for in the tre tranformation equation for 11. R R R in co ma min ma

27 Ue equation for in and co or 11 to find out which face ha the oitive hear tre and which the negative. What normal tree act on the lane with maimum hear tre? Subtitute for in the equation for 1 and 1 to get 1 1 ma ma ma ma 7

28 Eamle: The tate of lane tre at a oint i rereented b the tre element below. Determine the maimum hear tree and draw the correonding tre element. 50 MPa 80 MPa 80 MPa 5 MPa 50 MPa Define the tree in term of the etablihed ign convention: -80 MPa 50 MPa -5 MPa ma ma ( 5) 69.6 MPa MPa 8

29 80 50 tan ( ) and , MPa 15 MPa But we mut check which angle goe with which hear tre ( ) ( 80 50) in in co ma 69.6 MPa with ma 55.5 min MPa with min MPa 55.5 o o 69.6 MPa ( 34.5) ( 5) co ( 34.5) 69.6 MPa 15 MPa 9

30 30 Finall, we can ak how the rincial tree and maimum hear tree are related and how the rincial angle and maimum hear angle are related. 1 ma ma 1 1 1, ± cot tan 1 tan tan tan

31 tan cot 0 in co 0 co in in in co ( ) ± 90 ± 45 ± 45 co co 0 0 So, the lane of maimum hear tre ( ) occur at 45 to the rincial lane ( ). 31

32 Original Problem 50 MPa Princial Stree 54.6 MPa 80 MPa 80 MPa 84.6 MPa o 10.5 o 84.6 MPa 5 MPa 50 MPa 54.6 MPa -80, 50, , 0, Maimum Shear 15 MPa 15 MPa 15 MPa 55.5 o o 15 MPa ma ma ma MPa ( 84.6) ± ± , MPa ma 69.6, -15 3

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