strain buckling

• 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