• Deflection or stiffness, rather than stress, is often
the controlling factor in the design of a part.
• Eg Machine tool frames must be extremely rigid
to maintain manufacturing accuracy.
• Parts may require great stiffness in order to
eliminate vibration problems.
5.1 Elastic strain, deflection, stiffness, and Stability (Pg 194)
• Elastic strains is a directly measurable quantity
• Used for measuring stresses.
(Stress is not, in general, a directly measurable Quantity)
• Elastic strains – Big role in Experimental techniques!
• When the elastic constants of a material are known,
experimentally determined strain values can be transposed into
corresponding stress
Fig 5.5 Grid configurations of typical metal foil electrical resistance
Strain gages (Pg 196)
5.6 Deflection and Spring Rate—Simple Cases (Pg 204)
Deflection Load P
Deflection Length L
Deflection 1 / Geometric Rigidity
Property (A or I )
Deflection 1 / Material Property E
Spring rate k = Linear Deflections
Spring rate K = Angular Deflections
Spring Rate :
A person lying on a spring mattress
How much weight will sink the mattress by 1 inch?
Spring Rate = The amount of weight required to deflect a
spring by one inch
Spring rate k = Linear Deflections
Spring rate K = Angular Deflections
Beams are structural members, subjected to
transverse loads.
Examples: Machinery shafts, leaf springs,
automobile frame members etc.
A beam often requires a larger
cross section to limit deflection than
it does to limit stress.
Many steel beams
are made of low-cost alloys because
these have the same modulus of elasticity
(thus, the same resistance to elastic deflection)
as stronger, high-cost steels.
Stepped Shafts
For various reasons, many beams do not have a uniform cross
section.
For example, machinery rotating shafts are usually “stepped”, in
order to accommodate the bearings and other parts assembled on
the shaft.
(a–d) Elastically stable loaded members.
(e) Potentially
elastically unstable
loaded members.
Slender column
Small disturbance of the equilibrium will be corrected by
elastic restoring forces, moments, or both.
For slender column shown in (Fig 5.1e),
- Slender column
- Low elastic modulus E
- Large load
-Compression member will be elastically unstable
-Slightest disturbance will cause buckling or collapse.
- Even if P/A stress may be well below the elastic limit of the
material.
5.10 Euler Column Buckling – Elastic Instability (Pg 227)
Deflections within elastic range Varies directly with Load
Exceptions:
Relatively long, thin portions of material to Compressive
Stress.
Examples:
Piston Connecting Rods,
Columns in buildings
Coil springs in compression
Jack screws
Euler assumed ideal case:
• Perfectly straight column
• Precisely axial load
• Homogenous Material
• And Stresses within Elastic range
If such Column is loaded below a certain value, Pcr,
Any slight lateral displacement given to column
Internal elastic storing moment when lateral displacing
force is removed.
So, for Load < Pcr the column is considered elastically stable.
When Load > Pcr
Slightest lateral displacement Eccentric load greater than Pcr
Column is elastically unstable.
Euler’s classical equation:
Pcr = π2 E I
-------
Le2
Where E = Modulus of Elasticity
I = Moment of Inertia of section
w.r.t. Buckling bending axis
Le = Equivalent length of column
Le = L for hinged end connections
I = A ϱ2 (i.e., Moment of inertia = Area times Radius of gyration2)
Substituting I = A ϱ2 in above equation (i.e., Moment of inertia = Area times Radius
of gyration2)
Pcr = π2 E A ϱ2
-------
Le2
Pcr = π2 E ϱ2
----- -------
A Le2
Pcr = π2 E
----- -------
A (Le/ ϱ) 2
Pcr = π2 E
----- -------
A (Le/ ϱ) 2
Note that this equation gives the value
of the P/A stress at which the column
becomes elastically unstable.
It has nothing to do with the yield
strength or ultimate strength of the
material.
Le / ϱ = Slenderness Ratio of the column
A demonstration model illustrating the different "Euler" buckling
modes. The model shows how the boundary conditions affect the
critical load of a slender column. Notice that each of the columns are
identical, apart from the boundary conditions.
(Pic courtesy: Grahams Child, Wiki Images)
SAMPLE PROBLEM 5.11
Determine the Required Diameter of a Steel Connecting Rod
An industrial machine requires a solid, round connecting rod 1 m
long (between pinned ends) that is subjected to a maximum
compressive force of 80,000 N. Using a safety factor of 2.5, what
diameter is required if steel is used, having properties of Sy = 689
MPa, E = 203 GPa?
5.16 Finite Element Analysis
The determination of stress and strains in machines and
structures is critical to design
The finite element method will solve problems when the
component geometry is complex and cannot be modeled
accurately with standard strength of materials analyses
Machine components can involve complicated geometric parts
fabricated from different materials.
The basic philosophy of the finite element method is
discretization and approximation.
Finite element method is a numerical approximation
technique that divides a component or structure into discrete
regions (the finite elements) and the response is described
by a set of functions that represent the displacements or
stresses in that region.
The finite element method requires formulation, solution
processes, and a representation of materials, geometry,
boundary conditions, and loadings.
Deformation, stress, plasticity, stability, vibration, impact,
fracture, etc. can be analyzed using FEA / FEM.
Figure: Types of Finite Elements
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