Strength Of Material (shrinked)

Strength of Materials

Modulus Types •Modulus: Slope of the stress-strain curve –Initial Modulus: slope of the curve drawn at the origin. –Tangent Modulus: slope of the curve drawn at the tangent of the curve at some point. –Secant Modulus: Ratio of stress to strain at any point on curve in a stress-strain diagram. It is the slope of a line from the origin to any point on a stress-strain curve. Initial Modulus Tangent Modulus

Stress

Secant Modulus

Strain

Various regions and points on the stress-strain curve. ultimate tensile strength

3

yield strength

Strain Hardening

necking

Fracture 5

2 Elastic region slope=Young’s(elastic) modulus yield strength Plastic region ultimate tensile strength strain hardening fracture

Plastic Region Elastic Region 1

4 Strain (

) (e/Lo)

1.True elastic limit based on micro strain measurements at strains on order of 2 x 10-6 in | in. This elastic limit is a very low value and is related to the motion of a few hundred dislocations. 2.Proportional limit is the highest stress at which stress is directly proportional to strain. 3.Elastic limit is the greatest stress the material can withstand without any measurable permanent strain remaining on the complete release of load. With increasing sensitivity of strain measurement, the value of the elastic limit is decreased until at the limit it equals the true elastic limit determined from micro strain measurements. With the sensitivity of strain usually employed in engineering studies (10-4in | in), the elastic limit is greater than the proportional limit. 1.The yield strength is the stress required to produce a small-specified amount of plastic deformation. The usual definition of this property is the offset yield strength determined by the stress corresponding to the intersection of the stressstrain curve and a line parallel to the elastic part of the curve offset by a specified strain (Fig. 1).

Stress- strain diagrams for various materials • • • • • • • • • • • •

Stainless Steel Aluminum Brass Copper Molybdenum Nickel Titanium Tungsten Carbon fiber Glass Composites Plastics

E= 28.5 million psi (196.5 GPa) E= 10 million psi E= 16 million psi E= 16 million psi E= 50 million psi E= 30 million psi E= 15.5 million psi E= 59 million psi E= 40 million psi E= 10.4 million psi E= 1 to 3 million psi E= 0.2 to 0.7 million psi

Universal Testing Machine

Area in red indicates intensity of stress

• Equipment to measure Stress-Strain – Strainometers: measures dimensional changes that occur during testing • extensometers, deflectometers, and compressometers measure changes in linear dimensions. • load cells measure load • data is recorded at several readings and the results averaged, e.g., 10 samples per second during the test.

Material Properties There a 5 properties typically used to describe a materials behavior and capabilities:

1. Strength 2. Hardness 3. Ductility 4. Brittleness 5. Toughness

1. Strength The ability to resist deformation and maintain its shape -Given in terms of the yield strength, sy, or the ultimate tensile strength, sult

2. Hardness

The ability to resist indentation, abrasion, and wear - For metals, this is determined with the Rockwell Hardness or Brinell tests that measure indentation/ penetration under a load

STRENGTH and HARDNESS are related! A high-strength material is typically resistant to wear and abrasion...

A comparison of hardness of some typical materials: Material

Brinell Hardness

Pure Aluminum

15

Pure Copper

35

Mild Steel

120

304 Stainless Steel

250

Hardened Tool Steel

650/700

Hard Chromium Plate

1000

Chromium Carbide

1200

Tungsten Carbide

1400

Titanium Carbide

2400

Diamond

8000

Sand

1000

3. Ductility The ability to deform before ultimate failure Ductile materials can be pulled or drawn into pipes, wire, and other structural shapes Ductile materials include copper, aluminum,and brass

4. Brittleness The inability to deform before ultimate failure - The opposite of ductility, brittle materials deform little before ultimately fracturing - Brittle materials include glass and cast iron

Brittleness is the LACK of ductility...

5. Toughness The ability to absorb energy - Material Toughness (slow absorption) - not a readily observable property - Defined by the area under the stressstrain curve - Impact Toughness (rapid absorption) - Ability to absorb energy of an impact without fracturing Toughness and Ductility/brittleness are related! Brittle things….

...are not tough!

Stiffness •Stiffness is a measure of the materials ability to resist deformation under load as measured in stress. –Stiffness is measures as the slope of the stress-strain curve

–Hookean solid: (like a spring) linear slope •steel •aluminum •iron •copper

•Stiffness is usually measured by the Modulus of Elasticity (Stress/strain) •Steel is stiff (tough to bend).

Analysis and Design of Beams for Bending

Introduction • Objective - Analysis and design of beams • Beams - structural members supporting loads at various points along the member • Transverse loadings of beams are classified as concentrated loads or distributed loads

• Applied loads result in internal forces consisting of a shear force (from the shear stress distribution) and a bending couple (from the normal stress distribution) • Normal stress is often the critical design criteria

Requires determination of the location and magnitude of largest bending moment

Classification of Beam Supports

Shear and Bending Moment Diagrams • Determination of maximum normal and shearing stresses requires identification of maximum internal shear force and bending couple. • Shear force and bending couple at a point are determined by passing a section through the beam and applying an equilibrium analysis on the beam portions on either side of the section. • Sign conventions for shear forces V and V’ and bending couples M and M’

Quantifying Bending Stress y

• Sagging condition Compression

A y

A B

B

Tension Neutral Axis Bending Stress :

M : Bending Moment I : 2nd Moment of area of the cross section y : Vertical distance from the neutral axis : tensile (+) or compressive(-) stress

Logitudinal Bending Stress Quantifying Bending Stress • Hogging condition

y

Tension

A

A

B B

Compression Neutral Axis

Neutral Axis : geometric centroid of the cross section or transition between compression and tension

Torsion

Contents Torsional Loads on Circular Shafts

Design of Transmission Shafts

Net Torque Due to Internal Stresses

Torsion of Noncircular Members

Axial Shear Components

Shaft Deformations Shearing Strain Stresses in Elastic Range

Normal Stresses Torsional Failure Modes Angle of Twist in Elastic Range

Stress Concentrations

Torsional Loads on Circular Shafts • Stresses and strains of circular shafts subjected to twisting couples or torques • Turbine exerts torque T on the shaft • Shaft transmits the torque to the generator • Generator creates an equal and opposite torque T’

Net Torque Due to Internal Stresses • Net of the internal shearing stresses is an internal torque, equal and opposite to the applied torque,

• Although the net torque due to the shearing stresses is known, the distribution of the stresses is not • Distribution of shearing stresses is statically indeterminate – must consider shaft deformations • Unlike the normal stress due to axial loads, the distribution of shearing stresses due to torsional loads can not be assumed uniform.

Axial Shear Components • Torque applied to shaft produces shearing stresses on the faces perpendicular to the axis. • Conditions of equilibrium require the existence of equal stresses on the faces of the two planes containing the axis of the shaft • The existence of the axial shear components is demonstrated by considering a shaft made up of axial slats.

The slats slide with respect to each other when equal and opposite torques are applied to the ends of the shaft.

Shaft Deformations • From observation, the angle of twist of the shaft is proportional to the applied torque and to the shaft length.

• When subjected to torsion, every cross-section of a circular shaft remains plane and undistorted. • Cross-sections for hollow and solid circular shafts remain plain and undistorted because a circular shaft is axisymmetric. • Cross-sections of noncircular (nonaxisymmetric) shafts are distorted when subjected to torsion.

Shearing Strain • Consider an interior section of the shaft. As a torsional load is applied, an element on the interior cylinder deforms into a rhombus. • Since the ends of the element remain planar, the shear strain is equal to angle of twist. • It follows that

• Shear strain is proportional to twist and radius

Stresses in Elastic Range • Multiplying the previous equation by the shear modulus, From Hooke’s Law,

, so

The shearing stress varies linearly with the radial position in the section. • Recall that the sum of the moments from the internal stress distribution is equal to the torque on the shaft at the section, • The results are known as the elastic torsion formulas,

Torsional Failure Modes • Ductile materials generally fail in shear. Brittle materials are weaker in tension than shear. • When subjected to torsion, a ductile specimen breaks along a plane of maximum shear, i.e., a plane perpendicular to the shaft axis. • When subjected to torsion, a brittle specimen breaks along planes perpendicular to the direction in which tension is a maximum, i.e., along surfaces at 45o to the shaft axis.

Angle of Twist in Elastic Range • Recall that the angle of twist and maximum shearing strain are related, • In the elastic range, the shearing strain and shear are related by Hooke’s Law, • Equating the expressions for shearing strain and solving for the angle of twist, • If the torsional loading or shaft cross-section changes along the length, the angle of rotation is found as the sum of segment rotations

Design of Transmission Shafts • Principal transmission shaft performance specifications are: - power - speed • Designer must select shaft material and cross-section to meet performance specifications without exceeding allowable shearing stress.

• Determine torque applied to shaft at specified power and speed,

• Find shaft cross-section which will not exceed the maximum allowable shearing stress,

Stress Concentrations • The derivation of the torsion formula,

assumed a circular shaft with uniform cross-section loaded through rigid end plates. • The use of flange couplings, gears and pulleys attached to shafts by keys in keyways, and cross-section discontinuities can cause stress concentrations • Experimental or numerically determined concentration factors are applied as

Torsion of Noncircular Members • Previous torsion formulas are valid for axisymmetric or circular shafts • Planar cross-sections of noncircular shafts do not remain planar and stress and strain distribution do not vary linearly • For uniform rectangular cross-sections,

• At large values of a/b, the maximum shear stress and angle of twist for other open sections are the same as a rectangular bar.

Plastics

Design Parts Life

for Plastics

Consolidation

Cycle Benefits

Surface

Design Possibilities

Why Plastics?

Why Plastics?

Plastics Usage in Automobiles (North America) Total Lb. Year Lb./Vehicle (Billion) 1970 70 1988 150 2.2 1999 257 4.0 2000 255 4.2 2005 279 4.8 2010Source: Automotive Plastics 307Report 5.4 1999, 2000 Market Search, Inc.

Plastics Applications 2000

2010

M Lb.

M Lb.

Δ M Lb.

Interior

1,688

2,021

+333

Body

1,181

1,601

+420

Underhood

388

627

+239

Chassis

961

1,195

+234

4,217

5,444

1,226

Segment

Total

Source: Automotive Plastics Report 2000

Specific Gravity Steel

7.8

Aluminum

2.6

Magnesium

1.75

Plastics

0.9 - 1.6

Automotive Plastics Basics for Exteriors

How Plastics are Classified

Selection Considerations Physical

Properties

Chemistry Process

Characteristics

Relative

Part or System Cost

Most Common of the Approx. 60 Commercial Families of Plastic Mat’ls • Acrylonitrile Butadiene Styrene (ABS) • Acetal (POM) • Acrylics (PMMA) • Fluoropolymer (PTFE)

• • • •

Ionomer Nylon (PA) Phenolic Polycarbonate (PC)

Most Common of the Approx. 60 Commercial Families of Plastic Mat’ls • Polyester (PBT, PET) • Polyester Thermoset (SMC, BMC) • Polyethylene (PE) • Polyphenyleneoxide (PPO) • Polypropylene (PP)

• Polystyrene (PS) • Polyurethane (PUR) • Polyvinylchloride (PVC) • Styrene Acrylonitrile (SAN) • Vinyl Ester

Most Commonly Used Plastics in Automotive Applications

Plastics Applications 2000

2010

M Lb.

M Lb.

Δ M Lb.

Interior

1,688

2,021

+333

Body

1,181

1,601

+420

Underhood

388

627

+239

Chassis

961

1,195

+234

4,217

5,444

1,226

Segment

Total

Source: Automotive Plastics Report 2000

Primary Processing Methods • • • • • •

Blow Molding • Calendaring • Casting Compression Molding• • Extrusion • Reaction Injection Molding • • Injection Molding

Powder or Slush Molding Thermoforming Filament Winding Pultrusion Resin Transfer Molding Rotational Molding

Fatigue (Failure under fluctuating / cyclic stresses)

Fatigue occurs when a material is subjected to alternating stresses, over a long period of time. Under fluctuating / cyclic stresses, failure can occur at loads considerably lower than tensile or yield strengths of material under a static load. Estimated to causes 90% of all failures of metallic structures (bridges, aircraft, machine components, etc.)

Fatigue failure is brittle-like (relatively little plastic deformation) even in normally ductile materials. Thus sudden and catastrophic!

Examples: springs, turbine blades, airplane wings, bridges and bones

Fatigue does not always lead to failure

 Failure can occur if the stress surpasses the endurance limit of the material [Endurance Limit (Sn): Is the stress value below which an infinite number of cycles will not cause failure]

Steel will not fail if the endurance limit is not passed

Aluminum will eventually fail regardless of the endurance limit

Cyclic Stresses There are three common ways in which stresses may be applied: axial (tension or compression), torsional (twisting),or flextural (bending) Examples of these are seen in Fig. 1.

Figure 1 Visual examples of axial stress, torsional stress, and flexural stress.

Crack Initiation and Propagation:

Fatigue failure proceeds in three distinct stages: 1) Crack initiation in the areas of stress concentration (near stress raisers), 2) Incremental crack propagation, 3) Final catastrophic failure.

The S-N Curve : A very useful way to visualize time to failure for a specific material is with the S-N curve. The "S-N" means stress v/s cycles to failure, which when plotted uses the stress amplitude, sa plotted on the vertical axis and the logarithm of the number of cycles to failure. An important characteristic to this plot as seen in Fig. 2 is the fatigue limit

Figure 2 A S-N Plot for an aluminum alloy

Significance of the fatigue limit : If the material is loaded below this stress, then it will not fail, regardless of the number of times it is loaded. Material such as aluminum, copper and magnesium do not show a fatigue limit, therefore they will fail at any stress and number of cycles. Other important terms are - fatigue strength and fatigue life. The stress at which failure occurs for a given number of cycles is the fatigue strength. The number of cycles required for a material to fail at a certain stress in fatigue life.

S-N Curve for Ferrous v/s non-ferrous metals

Creep Creep is a time-dependent and permanent deformation of materials when subjected to a constant load at a high temperature (> 0.4 Tm).

Examples: turbine blades, steam generators.

Stages of creep

1. Instantaneous deformation: mainly elastic.

2. Primary/transient creep: Slope of strain vs. time decreases with time:

work-hardening

3. Secondary/steady-state creep: Rate of straining is constant: balance of work-hardening and recovery.

4. Tertiary: Rapidly accelerating strain rate up to failure: formation of internal cracks, voids, grain boundary separation, necking, etc.

Parameters of creep behavior The stage secondary/steady-state creep is of longest duration and the steady-state creep rate is the most important parameter of the creep behavior in long-life applications. Another parameter, especially important in short-life creep situations, is time to rupture, or the rupture lifetime, tr. t / s . e . = e&

Creep: stress and temperature effects With increasing stress or temperature: The instantaneous strain increases The steady-state creep rate increases The time to rupture decreases

Alloys for high-temperature use (turbines in jet engines, hypersonic airplanes, nuclear reactors, etc.)

Creep is generally minimized in materials with:  High melting temperature  High elastic modulus  Large grain sizes (inhibits grain boundary sliding)

Following materials are especially resilient to creep:  Stainless steels  Refractory metals (containing elements of high melting point, like Nb, Mo, W, Ta)  “Superalloys” (Co, Ni based: solid solution hardening and secondary phases)

5.6 Design Philosophy Design

• Loading known and geometry specified – Specify factor of safety, N, and determine material. • Loading known and material specified – Specify factor of safety, N, and determine required geometry. • Loading known and material and geometry specified – Determine factor of safety – Is it safe?? Analysis

Also check deflection!!

5.7 Design Factors, N (a.k.a. Factor of Safety)

FOR DUCTILE MATERIALS: •N = 1.25 to 2.0

Static loading, high level of confidence in all design data

•N = 2.0 to 2.5

Dynamic loading, average confidence in all design data

•N = 2.5 to 4.0

Static or dynamic with uncertainty about loads, material properties, complex stress state, etc…

•N = 4.0 or higher

Above + desire to provide extra safety

5.8 Failure Theories 1. Maximum Normal Stress 2. Modified Mohr Static Loading

3. Yield strength

4. Maximum shear stress 5. Distortion energy 6. Goodman 7. Gerber 8. Soderberg

Fatigue Loading

Theory to use depends on: or Uniaxial:

Ductile or Brittle Dynamic or Static

Bi-axial:

Failure Theory:

When to Use?

1. Maximum Normal Stress

Brittle Material/ Uniaxial Static Stress

2. Yield Strength (Basis for MCH T 213)

Ductile Material/ Uniaxial Static Normal Stress

3. Maximum Shear Stress (Basis for MCH T 213)

Ductile Material/ Biaxial Static Stress

4. Distortion Energy (von Mises)

Ductile Material/ Biaxial Static Stress

5. Goodman Method

Ductile Material/ Fluctuating Normal Stress (Fatigue Loading)

Failure When:

Design Stress:

Ductile Material/ Failure Theories for STATIC Loading Fluctuating Shear Stress (Fatigue Loading)

Uniaxial:

Ductile Material/ Fluctuating Combined Stress (Fatigue Loading)

or Bi-axial:

Failure Theory:

When Use?

1. Maximum Normal Stress

Brittle Material/ Uniaxial Static Stress

2. Yield Strength (Basis for MCH T 213)

Ductile Material/ Uniaxial Static Normal Stress

Failure When:

Failure Theories for FATIGUE Loading

3. Maximum Shear Stress (Basis for MCH T 213)

Ductile Material/ Biaxial Static Stress

4. Distortion Energy (von Mises)

Ductile Material/ Biaxial Static Stress

5. Goodman Method

a. Ductile Material/ Fluctuating Normal Stress (Fatigue Loading) b. Ductile Material/ Fluctuating Shear Stress (Fatigue Loading) c. Ductile Material/ Fluctuating Combined Stress (Fatigue Loading)

Design Stress:

Failure Theory:

When Use?

1. Maximum Normal Stress

Brittle Material/ Uniaxial Static Stress

2. Yield Strength (Basis for MCH T 213)

Ductile Material/ Uniaxial Static Normal Stress

3. Maximum Shear Stress (Basis for MCH T 213)

Ductile Material/ Biaxial Static Stress

4. Distortion Energy (von Mises)

Ductile Material/ Biaxial Static Stress

5. Goodman Method

a. Ductile Material/ Fluctuating Normal Stress (Fatigue Loading) b. Ductile Material/ Fluctuating Shear Stress (Fatigue Loading) c. Ductile Material/ Fluctuating Combined Stress (Fatigue Loading)

Failure When:

Design Stress:

Comparison of Static Failure Theories:

Shows “no failure” zones

Maximum Shear – most conservative

The Goodman Diagram - note difference between “no failure zone” and “safe zone”

sa

sm

General Comments: 1.

Failure theory to use depends on material (ductile vs. brittle) and type of loading (static or dynamic). Note, ductile if elongation > 5%.

2.

Ductile material static loads – ok to neglect Kt (stress concentrations)

3.

Brittle material static loads – must use Kt

4.

Terminology: •

Su (or Sut) = ultimate strength in tension

•

Suc = ultimate strength in compression

•

Sy = yield strength in tension

•

Sys = 0.5*Sy = yield strength in shear

•

Sus = 0.75*Su = ultimate strength in shear

•

Sn = endurance strength = 0.5*Su or get from Fig 5-8 or S-N curve

•

S’n = estimated actual endurance strength = Sn(Cm) (Cst) (CR) (Cs)

•

S’sn = 0.577* S’n = estimated actual endurance strength in shear

5.9 What Failure Theory to Use:

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