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MMU-307 DESIGN OF MACHINE ELEMENTS

Review of Statics and Mechanics of Materials

Asst. Prof. Özgür ÜNVER October 22nd, 2019

Stress-Strain Curve

𝜖 = 𝑆𝑡𝑟𝑎𝑖𝑛𝛿 = Deflection𝑙 = Length𝜎 = Stress𝐸 = Young’s Modulus𝜈 = Poisson’s Ratio𝜏 = 𝑆ℎ𝑒𝑎𝑟𝐺 = 𝑆ℎ𝑒𝑎𝑟 𝑀𝑜𝑑𝑢𝑙𝑢𝑠 𝑜𝑓 𝐸𝑙𝑎𝑠𝑡𝑖𝑐𝑖𝑡𝑦

http://www.youtube.com/watch?v=67fSwIjYJ-Ehttp://www.youtube.com/watch?v=E5-hwTspJK0&feature=related

Free Body Diagrams

Simplify the analysis and your thinking

Isolate each element

Establishes the directions of reference axes

Provides magnitudes and directions of the known forces

Helps in assuming the directions of unknown forces

Provides a place to store one thought while proceeding to the next.

Free Body DiagramsExample: An automobile Scissors Jack Consists of six links pivoted and/or geared together and seventh link in the form a lead screw

Reaction ForcesIf the system is motionless or, at most, has constant velocity, then the system

has zero acceleration. Under this condition the system is said to be in

equilibrium. In this case the forces and moments acting on the system

balance such that;

Reaction forces are equal and opposing to applied forces to maintain static

equilibrium

Reaction Moments

Reaction moments are equal and opposing to applied moments to maintain

static equilibrium

Question

How can you calculate the forces on the tent and the spile?

Can you calculate the wind speed causing your tent fly away?

Strain Gauge

Question: How do we measure stresses in real life?

Answer: Strain Gauges!

Mohr CircleHere we have 3 stresses, 6 shear

variables, however; τyx = τxy τzy = τyz τxz

= τzx equal therefore we have six

unknowns, σx , σy , σz, τxy , τyz, and τzx .

If σz = τzx = τzy = 0 then it is called plane

stress!

Tensile Test

Mohr Circle for Plane StressIf 3D element is cut by an oblique plane with a normal n at an

arbitrary angle φ counterclockwise from the x axis, we obtain the

element given below;

Rearranging the equations;

Mohr Circle Daigram for Plane Stress

The transformation equations are based on a positive φ being

counterclockwise.

Shear stresses tending to rotate the element clockwise (cw) are

plotted above the σ axis.

Mohr Circle Daigram for Plane StressTwo particular values for the angle 2φp, one of which defines the maximum normal stress σ1 and the other, the minimum normal stress σ2. These two stresses are called the principal stresses, and their corresponding directions, the principal directions.

Example: Mohr Circle Diagram for Plane Stress

General 3D Stress

Uniformly Distributed Stresses

Elongation is linear with axial force

Small (elastic) displacements

For a body of uniform cross sectional area, A:

Beam acts like linear spring with spring constant kx

Bending Moments

Bending moments are twisting forces (moments) applied along two parallel axes

Unit of moment is N-m

No standard sign convention,

Therefore; you must define the

direction!

What is the difference between Moment & Torque?

Deflection of Beams

Double Integration Method

Moment Area Method

Superposition Method

Energy Methods

Beam Bending (Deflection of Beams)

Displacement = y

Angle = θ = dy/dx

Curvature = κ = d2y/dx2

Moment = M = E I d2y/dx2

Shear force = V = dM/dx

Distrib. Load = q = - dV/dx

Signs depend on physical orientation of forces and moments

Double Integration Method

Boundary Conditions

Cantilever Beam Solution

Boundary Conditions of Supports

http://www.youtube.com/watch?v=TUE7DKNBIrUOver hard/soft surface?

Example

How would you decide on the

shape of the cantilever?

Beams in Bending

The second moment of area (area moment of inertia, moment of inertia of plane area, second moment of inertia) is a property of a cross section that can be used to predict the resistance of beams to bending and deflection, around an axis that lies in the cross-sectional plane.

Torsional Spring ConstantRotation is linear with axial torque

Small (elastic) displacements

For a body of uniform cross sectional area:

where G is shear modulus of elasticity, and J is the torsion constant