Stress and Strain ( , 3.14) MAE 316 Strength of Mechanical Components NC State University Department of Mechanical & Aerospace Engineering

Transcription

1 ( , 3.14) MAE 316 Strength of Mechanical Components NC State Universit Department of Mechanical & Aerospace Engineering 1

2 Introduction MAE 316 is a continuation of MAE 314 (solid mechanics) Review topics Beam theor Columns Pressure vessels Principle stresses New topics Contact Stress Press and shrink fits Fracture mechanics Fatigue

3 Normal Stress (3.9) Normal Stress (aial loading) F A Sign Convention > 0 Tensile (member is in tension) < 0 Compressive (member is in compression) 3

4 Shear Stress (3.9) Shear stress (transverse loading) Single shear average shear stress P ave A F A Double shear ave P A F A F A 4

5 Review (3.3) Bearing stress P A b b P td Single shear case Bearing stress is a normal stress 5

6 Strain (3.8) Normal strain (aial loading) δ ε L Hooke s Law Eε Where E Modulus of Elasticit (Young s modulus) 6

7 Review (3.1) Torsion (circular shaft) Shear strain γ ρφ L T Shear stress Tρ J Angle of twist φ TL GJ Where G Shear modulus (Modulus of rigidit) and J Polar moment of inertia of shaft cross-section 7

8 Thin-Walled Pressure Vessels (3.14) Thin-walled pressure vessels Clindrical Spherical t pr t i 1 l Circumferential hoop stress pri t Longitudinal stress 1 pr i t 8

9 Beams in Bending (3.10) Beams in pure bending Strain ε ρ z ν ε ε Where ν Poisson s Ratio and ρ ρ radius of curvature Stress M I Where I nd moment of inertia of the cross-section 9

10 Beam Shear and Bending ( ) Beams (non-uniform bending) Shear and bending moment dv d dm w V d Shear stress Design of beams for bending avg M I c VQ It ma ma M S ma Where Q 1 st moment of the cross-section Factor of Safet allowable applied 10

11 Eample Draw the shear and bending-moment diagrams for the beam and loading shown. 11

12 Eample Problem An etruded aluminum beam has the cross section shown. Knowing that the vertical shear in the beam is 150 kn, determine the shearing stress a (a) point a, and (b) point b. 1 Shear Stress in Beams

13 Combined Stress in Beams In MAE 314, we calculated stress and strain for each tpe of load separatel (aial, centric, transverse, etc.). When more than one tpe of load acts on a beam, the combined stress can be found b the superposition of several stress states. 13

14 Combined Stress in Beams Determine the normal and shearing stress at point K. The radius of the bar is 0 mm. 14

15 D and 3D Stress ( ) MAE 316 Strength of Mechanical Components NC State Universit Department of Mechanical & Aerospace Engineering 15

16 Plane (D) Stress (3.6) Consider a state of plane stress: z z z 0. φ φ φ φ φ φ φ Slice cube at an angle φ to the -ais (new coordinates, ) φ φ 16

17 Plane (D) Stress (3.6) Sum forces in direction and direction and use trig identities to formulate equations for transformed stress. + ' + cos( ϕ) + sin( ϕ) + ' cos( ϕ) sin( ϕ) φ φ φ φ ' ' sin( ϕ) + cos( ϕ) φ 17

18 Plane (D) Stress (3.6) Plotting a Mohr s Circle, we can also develop equations for principle stress, maimum shearing stress, and the orientations at which the occur. ma,min ma,min + ± ± + + ma min ma min R ave R ave + R R tan ( ϕ ) P tan ( ϕ ) S 18

19 Plane (D) Strain Mathematicall, the transformation of strain is the same as stress transformation with the following substitutions. ε and γ ε + ε ε ε γ ε ' + cos( ϕ) + sin( ϕ) ε+ ε ε ε γ ε ' cos( ϕ) sin( ϕ) ( ) ( ) ( ) γ ' ' ε ε sin ϕ + γ cos ϕ ε ma,min tan ( ϕ ) ε P + ε γ ε ε ± ε ε γ + 19

20 3D Stress (3.7) Now, there are three possible principal stresses. Also, recall the stress tensor can be epressed in matri form. z z z z z 0

21 3D Stress (3.7) 1 We can solve for the principle stresses ( 1,, 3 ) using a stress cubic equation. Where i 1,,3 and the three constant I 1, I, and I 3 are epressed as follows I I I i i i 3 1 z z z z z z z z z z z I I I

22 3D Stress Let n, n and n z be the direction cosines of the normal vector to surface ABC with respect to,, and z directions respectivel. 0 ) ( ) ( ) ( ) ( 0 A n A n A n A n F z z 0 ) ( ) ( ) ( ) ( 0 A n A n A n A n F z z 0 ) ( ) ( ) ( ) ( 0 A n A n A n A n F z z z z z z

23 3D Stress (3.7) How do we find the maimum shearing stress? The most visual method is to observe a 3D Mohr's Circle. Rank principle stresses largest to smallest: 1 > > 3 ma

24 3D Stress (5.5) A plane that makes equal angles with the principal planes is called an octahedral plane. oct 1 ( ) 3 1 oct ( ) + ( ) + ( )

25 3D Stress (3.7) For the stress state shown below, find the principle stresses and maimum shear stress. Stress tensor

26 3D Stress (3.7) Draw Mohr s Circle for the stress state shown below Stress tensor

27 Curved Beams (3.18) MAE 316 Strength of Mechanical Components NC State Universit Department of Mechanical & Aerospace Engineering 7

28 Curved Beams (3.18) Thus far, we have onl analzed stress in straight beams. However, there man situations where curved beams are used. Hooks Chain links Curved structural beams 8

29 Curved Beams (3.18) Assumptions Pure bending (no shear and aial forces present will add these later) Bending occurs in a single plane The cross-section has at least one ais of smmetr What does this mean? -M/I no longer applies Neutral ais and ais of smmetr (centroid) are no longer the same Stress distribution is not linear 9

30 Curved Beams (3.18) Fleure formula for tangential stress: M Ae( r ) n Where M bending moment about centroidal ais (positive M puts inner surface in tension) distance from neutral ais to point of interest A cross-section area e distance from centroidal ais to neutral ais r n radius of neutral ais 30

31 Curved Beams (3.18) If there is also an aial force present, the fleure formula can be written as follows. P M + A Ae( r ) Table 3-4 in the tetbook shows r n formulas for several common cross-section shapes. n 31

32 Eample Plot the distribution of stresses across section A-A of the crane hook shown below. The cross section is rectangular, with b0.75 in and h 4 in, and the load is F 5000 lbf. 3

33 Curved Beams (3.18) Calculate the tangential stress at A and B on the curved hook shown below if the load P 90 kn. 33