Machine Design Lecture 4

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mec 391-mechanical design

BY

ENGR. FAZLAR RAHMAN Assistant Professor

ME Faculty, IUBAT

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mec -391 (mechanical design)

LOAD AND STRESS ANALYSIS

Plane Stress:

 _{Plane stress is defined to be a state of stress in which the} normal and shear stresses are zero in the direction of perpendicular to a plane.

 Stresses in the Z-direction will be zero for a XY-plane. But strain in Z-direction is not zero.

strain in Z-direction is not zero. Plane Strain:

 _{It is defined to be a state of strain in which the strain in} the direction normal to a plane is assumed to be zero.

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STRESS AT A POINT IN A BODY

 The normal stress (perpendicular to the surface) is

Fazlar Rahman; ME Faculty; MEC 391; LEC_4;

 The normal stress (perpendicular to the surface) is denoted by ‘σ ’ and shear stress (parallel to the surface) is denoted by ‘τ ’ .

 _{‘σ ‘ outward to the surface is considered tensile stress} and positive. ‘σ‘ into the surface is compressive & negative.

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STRESS AT A POINT IN A BODY (Continue)

 To determine the state of stress at a point in the body, it would be necessary to examine all surfaces by making different planar slices through the point.

 _{The state of stress at point can be examined by only three} mutually perpendicular surfaces.

 Double subscripts are used to denote the shear stresses.  Double subscripts are used to denote the shear stresses.  _{First subscript indicates direction of surface normal and}

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STRESS AT A POINT IN A BODY (Continue)

 _{There total nine stress components: σ}_x _{, σ}_y _{, σ}_z_,

τ

_xy _,

τ

_xz _,

τ

_yx ,

τ

_yz,

τ

_zx and

τ

_zy .

 For equilibrium cross-shear are equal to each other.

τ

_xy =

τ

_yx ;

τ

_xz =

τ

_zx and

τ

_zy =

τ

_yz .

 _{Reduces stress component from nine to six components:}

 _{Reduces stress component from nine to six components:} σ_x , σ_y , σ_z,

τ

_xy ,

τ

_yz and

τ

_zx .

 Very common state of stress occurs when stresses on one surface are zero. When this occurs the state of stress is called plane stress.

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PLANE STRESS TRANSFORMATION EQUATIONS

 Consider an element ‘dx dy dz’ is cut by an oblique plane.  _{Cutting plane stress element at an arbitrary angle ‘φ’ and}

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PLANE STRESS TRANSFORMATION EQUATIONS (Continue)

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 Normal stress σ is maximum at,

 _{The two values of 2}

f

_{are the}

f

_p _{are the} principal directions.

 _{Substitute value of Cos2φ}_p _{and Sin2φ}_p _{in both equations.}  _{The stresses in the principal directions are the principal}

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mec -391 (mechanical design)

 Shear stress τ is maximum at,  Shear stress τ is maximum at,

f

_s _{are the directions max/min shear} stress.

 _{Substitute value of Cos2φ}_s _{and Sin2φ}_s _{in both equations.}  _{The max shear stress and principal stress in max shear}

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Problem-1:

A 12.0 in long beam support load of 15,000 lbf at 3.0 in from the left end. A designer has selected a 3.0 in C-channel, which is made of 7075-T6 aluminium extrusion. Find margin of safety of the beam. Assume beam is simply supported at both ends. Assume F = 66 ksi and F = 72 ksi

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PLANE STRESS TRANSFORMATION EQUATIONS (Continue) Problem-2:

A 15.0 in long beam support a load of 19,500 lbf at 5.0 in from the left end of the beam. A designer has selected an I-Beam, which is made of 7075-T6 aluminium extrusion. Cross-section of the beam is shown below. The points of interest are labelled (a, b, c and d) at distances y from the

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PLANE STRESS TRANSFORMATION EQUATIONS (Continue) Problem-2 (continue):

At the critical axial location along the beam find the following information,

(a). Determine transverse shear stress at each interested point and find profile of shear distribution.

(b). Determine bending stress at each interested point. (b). Determine bending stress at each interested point.

(c). Determine the maximum shear stresses at the point of interest and compare them.

(d). Find margin of safety.

Assume beam is simply supported at both ends. Assume F_ty = 72 ksi and F_cy = 66 ksi , F_su = 42 ksi and E = 10.4 x106 _psi

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MOHR’S CIRCLE DIAGRAM

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MOHR’S CIRCLE DIAGRAM (Continue) Example-1:

 Two value of principal stress and two value of φ but how  Two value of principal stress and two value of φ_p but how do we know which value of φ_p corresponds to which value of principal stress. To find it we need to substitute value of φ_p to the corresponding equation.

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MOHR’S CIRCLE DIAGRAM (Continue) Convention of Mohr’s Circle:

 _{A graphical method for visualizing the stress state at a} point by plotting normal stress along X-axis and shear stress along Y-axis.

 _{It represents relation between σ}_x_{, σ}_y _{stresses and}

It represents relation between σ_x, σ_y stresses and principal stresses σ₁, σ₂ .

 Parametric relationship between



and



(with 2

f

as parameter).

 Relationship is a circle with center at ‘C’ and radius ‘R’

C = (



,



) = [(



_x +



_y)/2, 0 ] and

2

x y

xy

R  _  _ 

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MOHR’S CIRCLE DIAGRAM (Continue) Sign Convention of Mohr’s Circle:

 _{Normal tensile stress (σ}_x_{, σ}_y _{positive) is plotted along} positive X-axis and normal compressive stress (σ_x, σ_y negative) is plotted along negative X-axis.

 _{Shear stress that tending to rotate the element in}_{Shear stress that tending to rotate the element in} clockwise (cw) is plotted along positive Y-axis;

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MOHR’S CIRCLE DIAGRAM (Continue)

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MOHR’S CIRCLE DIAGRAM (Continue) Problem-3:

A stress element has

σ

_x= 80 Mpa and

τ

_xy = -50 Mpa as shown in the figure below.

(a). Find principal & shear stresses and their direction by using Mohr’s circle diagram; also show alignment of each element respect to xy coordinates.

element respect to xy coordinates.

(b). Find part (a) by algebraic approach.

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MOHR’S CIRCLE DIAGRAM (Continue) Problem-4:

A plane stress element is subject to stress as shown in the figure below.

(a). Find the principal & maximum shear stresses and their direction by using Mohr’s circle diagram; also show alignment of each element respect to xy coordinates.

(b). Find part (a) by algebraic approach.

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MOHR’S CIRCLE DIAGRAM (Continue) Assignmet-1:

A plane stress element is subject to stress as shown in the figure below.

(a). Find the principal & maximum shear stresses and their direction by using Mohr’s circle diagram; also show alignment of each element respect to xy coordinates.

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THREE DIMENSIONAL STRESSES

 If there is six stress components (σ_x _,

σ

_y_,

σ

_z _,

τ

_xy _,

τ

_yz _and

τ

_zx , in an element, then there will be three principal stresses (σ₁ ,

σ

₂ ,

σ

₃) and three shear stresses (

τ

_1/2,

τ

_2/3,

τ

_1/3 ).

 _{Finding roots of the cubic equation,}

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THREE DIMENSIONAL STRESSES

 In plotting Mohr’s circles for three-dimensional stress, the principal normal stresses are ordered so that

 There are always three extreme-value shear stresses,

 _The maximum shear stress _{is always the greatest of these} three.

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BENDING IN CURVED BEAM

 Crane hooks, U-shaped Frame and Frame of clamp etc are example of curved beam (beam initially curved) .  _{Neutral axis and centroidal axis are not coincident.}

 _{Bending stress does not vary linearly with distance from} the neutral axis.

 Assume, r = radius of outer fiber; r = radius of inner

 Assume, r_o = radius of outer fiber; r_i= radius of inner

fiber ; r_n = radius of neutral axis ; r_c = radius of centroidal axis ; h = depth of section; always r_n < r_c and e = r_c – r_n.  _{And c}_o_{= distance from neutral axis to outer fiber; c}_i ₌

distance from neutral axis to inner fiber ; e = distance from centroidal axis to neutral axis ; M = bending

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BENDING IN CURVED BEAM (Continue)

 Location of the neural axis with respect to the centre of curvature is given by,

 Stress distribution due to resisting internal moment,

 _{Stress distribution is hyperbolic. Critical stresses at the} inner and outer surfaces, where y = c_i and y = c_o respectively,

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BENDING IN CURVED BEAM (Continue)

 In the usual or more general case, such as crane hook, U frame of press, frame of clamp, bending moment is calculated about centroidal axis but not the neutral axis.  Additional axial tensile or compressive stress must be

added to the bending stresses, added to the bending stresses,

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BENDING IN CURVED BEAM (Continue) Problem-5: (SEE HAND ANLAYSIS)

Plot stress distribution across section A-A of the crane hook shown in the figure below. The cross section is rectangular with b = 0.75 in and h = 4.0 in. Load on the hook is 5000 lbf.

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BENDING IN CURVED BEAM (Continue) Problem-6: (SEE HAND ANLAYSIS)

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PRESSURE VESSELS



There are two kind of pressure vessels: o Thin-walled pressure vessel : o Thick-walled pressure vessel.



Example of pressure vessel: Cylindrical Pressure vessels, Spherical pressure vessel, hydraulic cylinders, gun barrels and pipe carrying fluids at high pressure etc.

and pipe carrying fluids at high pressure etc.



There are three types of stresses developed in the pressure vessels depending on wall thickness, shape (cylindrical or spherical).

o Tangential or hoop stress,

σ

_t o Radial stress,

σ

_r

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PRESSURE VESSELS



If inner radius

r

_i ,



Other types are Cylindrical Pressure vessels, Spherical



Other types are Cylindrical Pressure vessels, Spherical

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THICK-WALLED PRESSURE VESSELS

 Cylinder with inside radius r_i, outside radius r_o, internal pressure p_i, and external pressure p_o

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THICK-WALLED PRESSURE VESSELS

 If outside pressure or external pressure p_o= 0 then

 If ends are closed, then

σ

longitudinal stress σ_l also exist ,

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mec -391 (mechanical design)

THIN-WALLED PRESSURE VESSELS

 Cylindrical pressure vessel with wall thickness 1/10 or less of the radius.

 _{Radial stress is quite small compared to tangential stress.}

 Average and maximum tangential stresses are,

 _{Longitudinal stress (if ends are closed),}

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mec -391 (mechanical design)

PRESSURE VESSELS PROBLEM Problem-7:

A pressure vessel has an outside diameter of 24 cm and wall thickness of 10 mm. If the internal pressure is 2400 Kpa, what is the maximum shear stress in the vessel wall?

Problem-8: Problem-8:

A cylinder with 0.30 m internal and 0.40 m external diameters is fabricated of a material whose elastic limit is 250 Mpa.

Determine the following:

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PRESS AND SHRINK FITS

 A contact pressure ‘p’ exists between the members at the

nominal transition radius ‘R’, causing radial stresses

σ

_r = -p in each member at the contact surface.

 Pressure for cylinder,

 _{For the inner member,}p_o = p _andp_i _{= 0}

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PRESS AND SHRINK FITS (Continue)

 The difference in dimensions in radial direction called

radial interference.

 δ_i and δ_o is the change in radii of outer and inner member.

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PRESS AND SHRINK FITS (Continue)

 Radial stress of outer cylinder, (σ_r)_o = - p then

 Similarly,

 Total deformation,

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mec -391 (mechanical design)

PRESS AND SHRINK FITS (Continue) Problem-9:

A compound cylinder with a = 150 mm, b = 200 mm and c = 250 mm as shown in the figure below. E = 200 Gpa, and δ = 0.1 mm is subjected to an internal pressure of 140 Mpa. Determine the distribution of tangential stress throughout the cylinder.