Summary of Topics & Formulae

ANALYSIS & DESIGN OF MACHINE ELEMENTS – I MET 301

SUMMARY OF TOPICS & FORMULAE

Prepared by Prof. A. K. Sengupta Department of Engineering Technology NJIT January 18, 2008

TABLE OF CONTENTS

1. Resolving a force in two orthogonal directions ...... 1 2. Moment of a force about a given point ...... 1 3. Static equilibrium in two dimensions ...... 1 4. Axial tension and compression: Stress & strain & Hook’s law...... 2 5. Transverse loading: internal shear force & bending moment ...... 3 6. Shear force & bending moment diagrams ...... 3 7. Bending stress (s)...... 4 8. Neutral axis (NA) ...... 5 9. Moment of inertia (I) ...... 5 10. Transverse shear stress (t) in beams ...... 6 11. Torsion of circular section & shear stress...... 7 12. Effect of combined stress in 2D: Mohr circle ...... 8 13. Stress manipulation in 3D ...... 9 14. Strain in two dimensions ...... 9 15. Theories of failure ...... 10 16. Stress concentration ...... 11 17. Design for cyclic loading...... 11 18. Design for finite life...... 12 19. Shear stress in a shaft for combined bending & torsion...... 13 20. Design of shaft with cyclic load...... 13 21. Shaft with bending loads in two planes...... 13 22. Design of keys, keyways & couplings ...... 14 23. Design of slender compression members or columns ...... 15 24. Deflection by double integration...... 16 25. Deflection by strain energy method...... 16 26. Shaft on three supports R1, R2 & R3 ...... 17 27. Critical speed of a rotating shaft ...... 17

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1. RESOLVING A FORCE IN TWO ORTHOGONAL DIRECTIONS

Y If we know the magnitude of the force F, and its angle (q) with x-axis, then the component of the force in X and Y F directions are:

Fx= F.cosq, FY=F.sin(q) ` q

x

FX=F.cos(q)

2. MOMENT OF A FORCE ABOUT A GIVEN POINT

Sign convention: Clockwise (CW) = positive Counter clockwise (CCW) = negative

3. STATIC EQUILIBRIUM IN TWO DIMENSIONS

When several forces (and moments) are acting on a body, the body will maintain static equilibrium if the following three equalities are satisfied simultaneously: (i) Sum of all forces acting on the body in X direction = 0; i.e., SFx = 0 (ii) Sum of all forces acting on the body in Y direction = 0; i.e., SFY = 0 (iii)Sum of moments about any point on the body due to all forces acting on the body = 0; i.e., SM = 0

These equations are frequently used to find unknown reaction forces and moments due to externally applied known forces and moments.

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P s = 4. AXIAL TENSION AND COMPRESSION: A STRESS, STRAIN & HOOK’S LAW

When a member is axially loaded with a force P, the P force acting on any cross-section across the length s = of the component = P. A

The normal stress at any cross section can be P obtained by s = , where A = cross-sectional area. A

Sign conventions: Tensile stress is positive, compressive stress is negative.

Statically indeterminate problems: Some times, for axially loaded members, the load shared by each member cannot be determined from static analysis of forces. In those cases, additional deformation relationships can be used to find the load shared.

P Normal stress:s = , P = Axial load, A = cross sectional area and, A d Normal strain: e = , d = elon`gation, L = original length. l Hooks law: Stress (s) is proportional to strain (e)

s =E.e, E = Elastic Modulus or Young’s Modulus

P d Thus, = E . A L PL Rearranging: d = AE

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5. TRANSVERSE LOADING: INTERNAL SHEAR FORCE & BENDING MOMENT

Transversely loaded members are often called beams. Beams can be supported as a cantilever beam or as a simple supported beam, as shown below.

Cantilever type support generates P P w w a vertical reaction force R, and a MR reaction moment MR to support the beam. Simply support generates M M Free-body only vertical reaction forces, RA & Cantilever R RB beam support diagram

The magnitudes and locations of applied loads and moments are P P known. w w

Types of loads: M M P = Concentrated applied load Simple beam Free-body w= Distributed applied load RA RB RB support diagram M = Applied moment For these two types of supports, the unknown reaction force and reaction mome nt at the support can be readily determined by using the static equilibrium conditions.

6. SHEAR FORCE & BENDING MOMENT DIAGRAMS

Due to transverse loading, shear forces and bending moments are generated internally within beam.

The internal shear force (V) at any section of the beam: V = sum of all external forces (including the reaction force at support) either to the left or right from the section. (Both will have same magnitude but opposite direction as the beam is in static equilibrium) Sign convention of V = positive, if the force is upward to the left of the section. To find the maximum V, often Vs along the length of the beam are determined and a diagram of variation of V is drawn, which is known as shear force diagram.

The internal Bending Moment (M) at any section of the beam: M = sum of moments about that section of all external forces (including the reaction force at support) either to the left or to the right from the section. (Both will have same magnitude but opposite direction as the beam is in static equilibrium) Sign convention of M = positive, if the moment of a force causes compression in the upper layer of the beam. To find the maximum M, often Ms along the length of the beam are determined and a diagram of variation of M is drawn, which is known as bending moment diagram.

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7. BENDING STRESS (s)

s

s

Bending stress due to bending moment M will be zero at the neutral axis (NA). Bending stress is tensile in one side of the NA, and compressive on the other side of NA.

At any point in the loaded beam, bending stress (s) can be calculated from the following formula: Where, M = Internal bending moment at the point stress is being calculated, M I = Moment of Inertia of the beam cross section about the neutral s = v axis (NA), I v = distance of the point from NA where the stress is determined.

Max bending stress will occur at the outermost layer of the beam (v=maximum), furthest away from the NA. M s max = c Where, I c = distance of the furthest outer layer of the beam from the M neutral axis (NA) Also, s = Z = I/c = known as section modulus max Z

If the radius of curvature of the beam r, due to bending is known, bending stress (s) can be found from the following alternative formula:

Where, E E = Elastic Modulus s = v r r = Radius of curvature due to bending at that section v = distance of the point from the neutral axis (NA) where the stress is determined

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8. NEUTRAL AXIS (NA)

NA passes through the Center of gravity (CG) of the beam cross section. For rectangular or circular cross-section of the beam, CG is at the geometric center of the section.

For a composite section, the location of the CG can be determined by the following formula,

A y + A y + A y +... y = 1 1 2 2 3 3 A1 + A2 + A3 +...

9. MOMENT OF INERTIA (I) I, for rectangular and circular sections about their NA can be found using following formulae:

I for I-sections, Box sections and channel sections can be found using following formulae:

Transfer of axis for Moment of Inertia This formula is used to find MI of a T or other sections, whose NA or CG is not located at the geometric symmetrically. 2 I1 = I 0 + Ay

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10. TRANSVERSE SHEAR STRESS (t) IN BEAMS

(i) The transverse shear stress t at a section BB’ is given by the following formula: V VQ t = vA = t Ib a Ib Where, t V= Internal shear force, I = MI about NA of the beam section b= Width of the section v = Distance of CG of the section BB’CC’ from NA Aa= Area of the section BB’CC’

Q = vAa = Area moment of the section BB’CC’ (ii) Max. transverse shear stress, tmax always occurs at NA (because Q is max at NA) (iii) When finding transverse shear stress (t) for a composite section, the following formula can V V t = vA = (v A + v A ...) be used: Ib å a Ib 1 a1 2 a2

(iv) Max Transverse shear stresses (tmax): Solid rectangular 3 V t = A = area of the cross-section = b.d cross-section max 2 A

Solid circular 4 V p 2 t = A = area of the cross-section = d cross-section max 3 A 4 Circular cross V A = area of the cross-section section with thin t = 2 p 2 2 * max = (d - d ) wall A 4 o i V A = t.d I cross-section t= thickness of the web, and t max = A d= total depth of the I-beam

* A more exact analysis gives the values of 1.38 V/A and 1.23 V/A for the transverse shear stress at the center and ends, respectively, of the neutral axis.

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11. TORSION OF CIRCULAR SECTION & SHEAR STRESS

When a torque T is applied to a circular section, the elastic deformation produces an angular twist f, and shear stress t, within the member. At any radial distance r1, the shear stress t, can be obtained from: T t = r t J 1 Where, J = polar moment of inertia of the circular section t = 2I

The shear stress t = 0 at the axis, as r1=0, and the shear stress t = tmax, at r1=r, the outermost layer. T Thus, t = r max J p d 4 p r 4 For, solid circular section: J = = 32 2 For, hollow circular section: p (d 4 - d 4 ) p (r 4 - r 4) J = o i = o i t 32 2 Putting the values of J, in the above equation: t For solid circular section: T T d 16T t t max = r = = J (p d 4 /32) 2 p d 3 For hollow circular section: t

T T do 16T 16T t max = r = 4 4 = 4 4 do = 3 4 J (p (d0 - di ) / 32) 2 p (d0 - di ) p d0 (1- l ) d where, l = i = ratio of inner to outer diameter do The shear stress t and the angle of twist f is related by

fGr1 t = , where G = shear modulus of elasticity, l = length of the shaft and r1= radius. l Relationship between power (kW/HP), torque (T) and rpm (n)

63,025HP T = in - lb n 9,550,000 kW T = N - mm n

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12. EFFECT OF COMBINED Y STRESS IN 2D: MOHR CIRCLE sy txy The two dimensional general state of X sx sx stress at a point is shown, where, sx= txy normal stress in X direction, s = normal y sy stress in Y direction, and txy= shear t stress. (savg ,tmax) tmax The corresponding Mohr circle can be drawn by plotting the X(sx,txy), and Y x, xy X axis X (s t ) (s ,t ) points in the s-t plane. The txy y xy 2f sy 2q circle drawn, using the XY line as -s sx s diameter, is the Mohr circle. o s2 avg s1 txy s s +s Y axis The center of the circle s = x y avg 2 Y( sy,txy) Radius of the circle 2 (savg ,-tmax) æs -s ö -t R = ç x y ÷ +t 2 ç ÷ xy è 2 ø æ 2t ö The angle, 2q = tan -1ç xy ÷ , and the angle, 2f = 90-2q ç ÷ ès x -s y ø

All Mohr circle angles are double the actual angles

Principal normal stresses will occur along the diameter s1s2 at an angle q from the original X direction. 2 s x +s y æs x -s y ö 2 s1= savg + R = + ç ÷ +t ç ÷ xy s2 2 è 2 ø s1 X 2 q s +s æs -s ö s1 s = s – R = x y - ç x y ÷ +t 2 s1 2 avg ç ÷ xy 2 è 2 ø s2 Maximum shear stress will occur along the Y vertical diameter at an angle f from the original X savg direction. tmin savg f 2 x æs -s ö tmax t = R = ç x y ÷ +t 2 savg max ç ÷ xy è 2 ø savg

Stresses in any arbitrary direction u-v which is at an angle f with respect to the x-y axes system, can be readily found if savg, R & q are known:

su= savg+Rcos[2(q+f)], sv = savg - Rcos[2(q+f)], and tuv = Rsin[2(q+f)]

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Stress relationship for a rotation of axis system by 45o

(a) When the XY coordinate system is rotated by 45o counter-clockwise to X’Y’ system s -s s +s s +s Then, t = x y , s =s -t and s =s +t , where s = x y = x' y' x'y' 2 x' avg xy y' avg xy avg 2 2 (b) When the X’Y’ coordinate system is rotated by 45o clockwise to XY system s -s s +s s +s Then, t = y' x' , s =s +t and s =s -t , where s = x' y' = x y xy 2 x avg x'y' y avg x'y' avg 2 2

X'(sx',tx'y')

(s x-s y)/2 X(sx,txy)

y' x' (s y'-s x')/2 (s -s )/2 (s x-s y)/2 Y(sy,-txy)

Y'(sy',-tx'y' )

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13. STRESS MANIPULATION IN 3-D

For a general state of stress in 3D, shown in (a), normal and shear stresses may be present in 3 orthogonal directions. It can be shown that at a certain orientation, three principal normal stresses, orthogonal to each other, are equivalent to the stress condition at (a). S1, S2 and S3 are the three principal normal stresses, which are three roots of the following cubic equation:

S3 – a.S2 + b.S – c = 0, where, a = sx+ sy+ sz 2 2 2 b = sxsy+ sysz+ szsx–txy –tyz –tzx 2 2 2 c = sxsysz + 2txytyztzx– sx tyz –sy tzx –sz txy

æ S - S S - S S - S ö The maximum shear stress t = Max of ç 1 2 2 3 3 1 ÷ max ç , , ÷ è 2 2 2 ø

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14. STRAIN IN TWO DIMENSIONS

If the sx & sy are normal stresses, then normal strains ex & ey can be determined from:

1 e x = (s x - ms y ), and E 1 e = (s - ms ) y E y x

Where m = Poisson’s ratio, which is a material property

Solving the above two equations, we can find the normal stresses in two directions.

E s = (e + me ), and x 1- m 2 x y E s = (e + me ) y 1- m 2 y x

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15. THEORIES OF FAILURE

Uniaxial Tensile Stress Testing During uniaxial stress testing of ductile materials, the first mechanical failure occurs by yielding, at the material’s yield strength Syp. The maximum stress that the material can withstand before breakage occurs at the ultimate tensile strength, Su, and Su > Syp. For ductile materials, Syp and Su values are same in tension and compression.

During uniaxial stress testing of brittle materials, the first mechanical failure occurs by fracture, at the material’s ultimate tensile strength Su. For brittle materials, Syp is greater than Su and thus Syp is non-existent. Also, for brittle materials, fracture strength in compression Suc is higher than fracture strength in tension Sut.

General 3D state of stress For these types of stresses, predicting failure is not as straight forward as in case of uniaxial stresses. The following theories of failure are developed to predict failure in such general state of stress. To apply these theories, first the principal normal stresses S1, S2 and S3 are computed, and then the theories are applied, with a factor of safety Nfs. For 2D stresses, one of the principal normal stresses = 0.

A positive value of principal normal stress means the principal stress is tensile, and a negative value means that the principal stress is compressive.

S uc S ut £ S1 £ N fs N fs

Maximum Normal Stress theory S uc S ut £ S2 £ Applicable for brittle materials N fs N fs

S uc S ut £ S3 £ N fs N fs

S yp S1 - S2 £ N fs Maximum Shear Stress theory S yp Applicable for ductile materials S - S £ 2 3 N fs

S yp S3 - S1 £ N fs 2 Maximum Strain Energy: æ S ö S 2 + S 2 + S 2 - 2m(S S + S S + S S ) £ ç yp ÷ Applicable for ductile materials 1 2 3 1 2 2 3 3 1 ç ÷ è N fs ø Maximum Distortion Energy 2 theory: Applicable for ductile æ S ö materials S 2 + S 2 + S 2 - S S - S S - S S £ ç yp ÷ 1 2 3 1 2 2 3 3 1 ç ÷ (Also known as von Mises-Hencky è N fs ø theory)

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16. STRESS CONCENTRATION

If there is a sudden change in geometry at a point, then the actual stress at that point is K times the calculated stress, where K>1. The value of K, the stress concentration factor, for combinations of geometry and loading type, can be obtained from the graphs given in pages 139- 145 in the text book.

When the material is ductile, loads are not cyclic or are not applied suddenly or the application is not working in a low temperature condition, then the effect of stress concentration factors can be ignored, or K = 1, even if there is a sudden change in geometry.

For brittle materials all types of applications, and for ductile materials when, loads are applied suddenly, or for low temperature applications, the actual stress = K times the theoretically calculated stress.

For ductile materials, when the load is cyclic, a stress concentration factor Kf is used which is lesser than K.

K f -1 Kf can be determined from the formula q = . K -1 Here, q is the notch sensitivity factor or notch sensitivity index, which depends on the material and its surface condition. The value of q can be obtained from handbooks and 0

17. DESIGN FOR CYCLIC LOADING

For pure cyclic stress (Sr), that varies cyclically from 0 to Sr to 0 to –Sr,

Se Sr K f £ , where Se = endurance limit of the material, & Nfs = Factor of safety. N fs If the stress changes cyclically between Smax and Smin, then the equivalent steady stress (Savg) and

Smax + S min Smax - S min equivalent cyclic stress (Sr) can be given by S = and S = . In such avg 2 r 2 situations the design equations are as followings:

æ S ö S ç yp ÷ yp Soderberg’s equation: Savg + Sr K f ç ÷ £ è Se ø N fs æ S ö S ç u ÷ u Goodman’s equation: Savg + Sr K f ç ÷ £ è Se ø N fs æ S ö S S ç u ÷ u yp Modified Goodman’s equation: Savg + Sr K f ç ÷ £ & Savg + Sr K f £ è Se ø N fs N fs

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18. DESIGN FOR FINITE LIFE

(i) For a completely reversing cyclic stress Sr, the S-N curve is represented by, B Sr = AN , where N = number of stress reversals.

A & B are determined from:

log( Se ) - log( 0.9Su ) B = , where Se = endurance limit, and Su = ultimate tensile strength. 3 S A = e 10(6B)

Once, A & B are known, for a given completely reversing stress sr , the number of stress 1 æ S ö B reversals before failure can be found by: N = ç r ÷ è A ø

(ii) For a combined reversing (Sr) and steady (Savg) stress situation, the equivalent completely

K f Sr Sult reversing stress SR = . For this type of loading, SR should substitute Sr of the equation Sult - Savg shown in (i)

n1 n2 n3 (iii) Miner’s equation: + + + .... =1, where n1, n2, n3, … are actual number of N1 N2 N3 reversals with SR1, SR2, SR3,….. equivalent completely reversing stress levels, and N1, N2, N3, … are maximum number of reversals before failure with SR1, SR2, SR3,….. equivalent completely reversing stress levels.

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19. SHEAR STRESS IN A SHAFT FOR COMBINED BENDING & TORSION For a shaft carrying bending moment (M) and torque (T), the maximum shear stress (tmax),: 2 æs ö 2 t max = ç ÷ +t è 2 ø 32M 16T 16 For solid shaft:s = and t = thust = M 2 +T 2 pd 3 pd 3 max pd 3 For hollow shaft:

32M 16T 16 2 2 s = and t = thust max = M +T pd 3(1- l4 ) pd 3 (1-l4 ) pd 3 (1- l4 ) o o o d where, l = i do

20. DESIGN OF SHAFT WITH CYCLIC LOAD

Based on maximum distortion energy theory, the design equation is: 2 2 2 æ æ S öö æ æ S öö æ S ö çs +s K ç yp ÷÷ + 3çt +t K ç yp ÷÷ = ç yp ÷ ç av r fb ç ÷÷ ç av r ft ç ÷÷ ç ÷ è è Se øø è è Se øø è N fs ø Where,

s max +s min sav= Steady normal stress = 2

s max -s min sr = Cyclic normal stress = 2 Kfb = Fatigue stress concentration factor in bending

t max +t min tav= Steady shear stress = 2

t max -t min tr = Cyclic shear stress = 2 Kft = Fatigue stress concentration factor in torsion Syp = Yield stress Se = Endurance limit Nfs = Factor of Safety

21. SHAFT WITH BENDING LOADS IN TWO PLANES

(i) Resolve each bending load in vertical and horizontal direction. (ii) Determine the bending moment diagram separately for horizontal and vertical loads 2 2 (iii) The resultant bending moment at any point on the beam = M R = M v + M H

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22. DESIGN OF KEYS, KEYWAYS & COUPLINGS

From the HP or kW rating and rpm n, torque T can be determined using the following formula: 63,025HP 9,550,000 kW T = in -lb or, T = N - mm n n Then, the tangential force F = T/r, where r = radius at which the tangential force is required.

(i) For designing keys and keyways, tangential force transmitted by the key is assumed to be acting at the outer diameter of the shaft, ie., r = do/2, and F = 2T/do

If, L = length of the key or the keyway a = width of the key or the keyway, and b = depth of the key

F F t yp Then, shear stress in the key = t = = £ , As a.L N fs

s yp F s yp It is assumed t yp = , thus , £ 2 a.L 2N fs

F F 2F s yp F s yp Bearing stress in the key = t = = = £ or, £ A æ b ö bL N bL 2N b ç ÷.L fs fs è 2 ø For square Key, a=b, and hence key designed from either shearing or bearing stress, will result in same dimension, and square keys are equally strong from shearing and bearing.

(ii) In rigid couplings, (a) Bolts may fail due to shearing or bearing: 2T Tangential force carried by each bolt = F = , where, n = number of bolts, & dp = nd p pitch circle diameter of the bolts.

F 4F s yp Shear stress in bolts = t = = 2 £ As p db 2N fs

F F s yp Bearing stress in bolts = s = = £ , Ab tdb N fs where, db= diameter of the bolt, and t = flange thickness of the coupling

(b) The coupling can shear from the hub/flange joint: 2T Tangential force F = , where dh = hub diameter dh

F F s yp Shear stress = t = = £ Ah p d h 2N fs

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23. DESIGN OF SLENDER COMPRESSION MEMBERS OR COLUMNS

For columns with an initial crookedness, the design load P can be determined from the following quadratic equation:

é æ ac ö ù P s yp APcr P2 - s A + 1+ P + = 0 ê yp ç 2 ÷ cr ú 2 ë è i ø û N fs N fs Where, P = the design load syp= yield strength A = cross sectional area a = initial crookedness c = distance from the neutral axis to the edge of the cross section i = I / A = radius of gyration Pc r= Critical load for a centrally loaded column Nfs = factor of safety.

- b ± b 2 - 4ac Roots of quadratic equation ax 2 + bx + c = 0; x = 2a

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24. DEFLECTION & SLOPE BY DOUBLE INTEGRATION Express bending moment (M) as a function of the longitudinal distance, x of the shaft. i.e., M = f(x). If y represents downward deflection, then for a shaft with constant EI, d 2 y EI = -M = - f (x) ...... (1) dx 2 Integrating both sides of (1): dy EI = - f (x) + C ...... (2) dx ò 1 dy Here represents the slope of the shaft, and C1 is the constant of integration. If the dx value of the slope is known for any value of x, then using those values, C1 can be determined from the above equation.

Integrating both sides of (2): EIy = - f (x) + C x + C ...... (3) òò 1 2 Here y represents the downward deflection of the shaft, and C2 is another constant of integration. If the value of the deflection y, is known for any value of x, then using those values, C2 can be determined from the above equation. Usually y at the support = 0. Once C1 and C2 are known, equation (2) and (3) are the slope and deflection equations for any valid value of x.

25. DEFLECTION & SLOPE BY STRAIN ENERGY METHOD This method can take into account when diameter of the shaft is not uniform

1 M p M f dx Deflection y = E ò I

1 M pM mdx and, slope q = E ò I Where Mp= Bending moment due to applied loads Mf = Bending moment due to a fictitious unit load applied at the point where deflection is needed. Mm = Bending moment due to a fictitious unit moment applied at the point where the slope is needed. The above integrations can be calculated graphically. The integration is the volume enclosed by

M p the and Mf diagrams for deflection, or the volume I

M p enclosed by the and Mm diagrams for slope for the I entire length of the shaft. l For a prismoidal solid shown, l Volume = (A + 4A + A ) A 6 1 m 2 l For a pyramid Volume = A 3

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26. SHAFT ON THREE SUPPORTS R1, R2 & R3

(i) Remove R2 and find the downward deflection, y of the shaft at R2, due to the downward applied loads. (ii) Now, remove the applied loads and write an equation relating the upward force (R2) and upward deflection (y1) at the support point in the form R2=K.y1 (iii)Now three types of situations can arise: (a) If all three supports are in the same level, then the magnitude of the reaction force R2 = K.y (b) If R2 is offset by d from R1 & R3, then R2 = K(y-d) (c) If R2 is an elastic support with a spring constant K1 with no offset at no load, then: (y-R2K1)K = R2 or, R2 = yK/(1+K.K1) (iv) When the reaction force R2 is known, then the problem becomes statically determinate, and the R1 & R3 can be found by the application of static equilibrium conditions.

27. CRITICAL SPEED OF A ROTATING SHAFT

g(W1y1 +W2 y2 +W3 y3 + ...... ) f = 2 2 2 cycles / sec W1 y1 +W2 y2 +W3 y3 +...... n = 60 f r.p.m.

Where, W1, W2, W3 are the vertical loads on the shaft, and y1, y2, y3 are deflections at of the loads, W1, W2, W3 due to bending of the shaft. g = acceleration due to gravity, = 32*12 = 386 in/sec2, = 9806 mm/sec2

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