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