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### Strength of Materials

9/18/07

Topics to Cover

- Pressure
- Stress
- Strain
- Scale and Square-Cube Effect
- Tension
- Compression
- Shear

Effect of force on non-rigid objects

- Besides the force, what else is important?
- Examples to consider:
- Ice on a pond
- Hammering a nail into wood
- Pressure!
- Area
- The AREA over which it is applied; the larger the area, the lower the pressure.

Utilizing pressure

- Many simple devices
- knives, picks, snowshoes, bulletproof vests, all simply change the AREA over which a force is applied.
- Distribution of Force (http://www.physics.unl.edu/outreach/football.html)
- Bed of Nails (http://video.google.com/videoplay?docid=-1891333794864062153)
- Bed of Nails (wileyplus – ch. 6)

Pressure

- What are the units of pressure?
- Hint – force per unit area
- Pounds per square inch, abbreviated psi.
- Inflating a tire; a car tire typically takes ~ 30 psi.
- Pressure of the air around us is about 14 psi.

Pressure – con’t

- Pressure can take several forms in the material.
- Pulling the material apart, material is in TENSION, the pressure is called the stress.
- Pushing together, material is in COMPRESSION, and it’s again a stress.
- Examples
- strings can have tension, but not compression
- Springs or rigid objects can do both

Think – Pair - Share

- Consider a bed of nails with 600 nails
- A person weighs 150 lbs lies on it
- Will it puncture the skin? (It takes 2 lbs per nail to puncture the skin)
- How much weight must be applied to puncture the skin?

http://phun.physics.virginia.edu/demos/nail_bed.html

Think – Pair - Share

- High-heeled shoes can cause tremendous pressure to be applied to a floor.
- Suppose the radius of a heel is 6.00 × 10-3 m.
- At times during a normal walking motion, nearly the entire body weight acts perpendicular to the surface of such a heel.
- Find the pressure that is applied to the floor under the heel because of the weight of a 50.0 kg woman.

Think – Pair - Share

- What exerts more pressure-per-square inch when walking a 100 lb woman in high heels or a 6,000 lb elephant in bare feet?
- [At the moment when only the heel rests on the ground.]
- Stiletto heels have an area of about 1/16 of a square inch.
- Elephants, unlike humans, walk with two feet on the ground at a time.
- Each foot is about 40 square inches.

Answer

- Pressure
- Woman’s heels
- Elephant’s feet
- The heels exert more pressure!

Stress

- Internal distribution of forces within a body that balances and reacts to the loads applied to it
- Tension, compression, or shear

Strain

- The amount of deformation in a material. If we are stretching a bar, then the strain is the percent change in its length.
- Simple case of strain proportional to stress
- Small forces
- can’t model objects breaking, but it does work for the amount a bridge sags when a car drives on it

Equations

- E is the modulus of elasticity
- Units of pressure (N/m2 or psi)

Stress and Strain – con’t

- What is the amount a bar or wire can be stretched or compressed in one dimension? (Hooke’s Law).
- Table 2.1 on page 21 of our book has a list of constants for Moduli of Elasticity
- Conversion from N/m2 to Psi is 6830:1

Behavior of Materials

- Elastic Behavior
- Recovers shape – no deformation
- Plastic Behavior
- Range between elastic limit and breaking
- Strain is not proportional to stress
- The Ultimate Tensile (or Compressive) Strength is where a stretched material breaks.
- sample values in N/m2 are
- 4e7 for concrete or brick,
- 1e8 for Al,
- 2.3e8 copper,
- 5e8 for steel
- Bone 12e7 tension, 17e7 compression.

Stiffness

- Ability to resist strain
- Steels
- Same E
- different yield points
- different ultimate strengths

Stress and Strain – example 1

- How far can you stretch a 20 ft steel cable before it snaps?
- E for steel is 2*1011
- Ultimate tensile strength for steel is 5.2*108

Materials

- Wood and metal pieces respond about the same to compression and tension.
- Stone pieces do NOT
- strong in compression (hard to crush)
- weak in tension (easy to pull apart)

Space Elevator

- Discussion of Space Elevator http://science.nasa.gov/headlines/y2000/ast07sep_1.htm
- Basically a cable goes up to geosynchronous orbit with ballast mass (asteroid) slightly above it.
- Best calculations are that a Tensile Strength of 6.2 x 1010 N/m2 = 9.1 x 106 psi (about 100 times stronger than steel) is required.
- Carbon nanotubes have a tensile strength 3 times this (Young’s Modulus is 1012 N/m2),
- Growing them long enough is a real concern.

How tall can a mountain on Earth be?

- Rock
- Compressive Strength of 2 x 108 N/m2= 3 x 104 psi
- Specific gravity ~ 3
- Pressure at the bottom of a column is therefore weight/area
- Weight = weight_water*specific gravity*height*area
- areas cancel
- Pressure at bottom = (0.036 pounds/in3)*(about 3)*height = Ultimate Strength.
- Height = (3 x 104 psi)/((.036 pounds/in3)*3) is about 3 x 105 inches, or 25,000 feet (over 4 miles)
- Height of Mt. Everest is 29,000 feet – pretty good guess
- If gravity was weaker, then the weight would be less, and mountains could grow taller.
- Mars has a gravity about 1/3 of Earth’s, and its tallest mountain (Mount Olympus) is about three times as big as Everest.

Scale

- Imagine a simple building, as a cube 10’ on a side.
- Volume is 10’ x 10‘ x 10’ = 1,000 ft3
- Area is 10’ x 10’ = 100 ft2
- If you double each dimension, what happens to the VOLUME and AREA of the cube?
- Volume is 20’ x 20‘ x 20’ = 8,000 ft3
- 8 fold increase in volume
- Area is 20’ x 20’ = 400 ft2
- 4 fold increase in area

Scale – con’t

- Weight of a structure scales with volume, and the strain on the supports scale with area
- Building larger and larger with the same materials leads to more and more strain and eventually collapse.
- Thus larger buildings need thicker supports:
- If you double the size, for example, then AFTER doubling you need to make the supports twice the area, or 40% larger in each direction.
- Of course eventually this becomes impractical, which is why one needs stronger materials to build larger structures: stone instead of wood, steel instead of stone.

Scale – con’t

- If you could somehow make a mouse as big as a gorilla, it would probably just collapse
- Its bones (the support system) would be too thin to support the increased weight at that size.
- Any sci-fi movie with a “shrinking ray” is nonsense.

Shear Modulus

- Shear Modulus (ripping apart)
- Shear refers to cases where we are not directly pushing in or pulling out on a material, but are applying forces perpendicular to the surface.
- (Example: thick book)
- Effectively shear is a combination of tension on one axis and compression on another, causing the material to deform or twist.

Stress/Area Relationship Problem - Compresion

- Given:a 6400# weight is supported by an

8" x 8" post.

Determine:a. the compression stress in the 8" x 8" post b. the bearing stress between the post and the steel plate c. the bearing stress between the steel plate and the concrete.

Solution:a. The compressive force in the post : 6400 pounds/ 64 in2 = 100 psi b. The stress at the point between the post and the steel bearing plate is dictated by the smallest area being loaded. The stress between the two is again 6400 pounds divided by 64 square inches for a total stress of 100 psi. c. The stress between the plate and the concrete floor is also determined by the smallest area. In this case it is the 10" x 10" plate: 6400 pounds / 100 in2. The resulting stress is 64 psi, a clear reduction in the stress.

Think – Pair - ShareStress/Area Relationship Problem - Tension

- Given:
- A 1000 lb load suspended from a ceiling by a 1"x 1" member.Determine:a. the stress in the member b. the stress if the size of the member is increased to 2" x 2".

Solution:a. The internal force in that member is 1000 lb tension and the tension stress (intensity of the force per unit of area) is 1000 pounds divided by the area of 1 square inch. This is a stress of 1000 psi. b. The 1000 lb load is now distributed evenly across an area of 4 square inches; thus, 1000 lb divided by 4 in2 is a stress of 250 psi. This clearly demonstrates the inverse relationship between stress and area.

Solution

- a. the stress in the member
- The internal force in that member is 1000 lb tension and the tension stress (intensity of the force per unit of area) is 1000 pounds divided by the area of 1 square inch. This is a stress of 1000 psi.
- b. the stress if the size of the member is increased to 2" x 2".
- The 1000 lb load is now distributed evenly across an area of 4 square inches; thus, 1000# divided by 4 in^2 is a stress of 250 psi. This clearly demonstrates the inverse relationship between stress and area.

Think – Pair - Share

- Given:
- A 5000 lb load on 1” diameter rod.Determine:a. Is the rod in tension or compression?
- B. What is the stress in the rod? b. What is the stress if the diameter of the rod is increased to 2".

Review

- How materials respond to applied forces
- When we apply a force over some area (a pressure), we create stress in the material, which then undergoes strain and changes shape in response (tension, compression, and shear).
- Scaling – pressure scales as area, weight as volume, so a larger object needs proportionally thicker supports

Lecture for 2/8/07

Deformations from Temperature

- Atomic chemistry reacts to changes in energy
- Solid materials
- Can contract with decrease in temperature
- Can expand with increase in temperature
- Linear change can be measured per degree

Thermal Deformation

- - the rate of strain per degree
- UNITS: °F, °C
- length change:
- thermal strain:
- - no stress when movement allowed

Changes in Size

- Similar to the stress-strain relationship, assume:
- temperature changes are not too extreme
- change in the object is linear with temperature
- Simple case is a solid bar of material with a length, width, and depth.
- When the object changes temperature by an amount T, the length of the bar becomes increases.

Expansion and Contraction

- Example
- a long bar of Aluminum 2” x 6” x 15 feet, with and = 1.3 x 10-5 per °F
- heat from room temperature of 70 F to 130 F, what are its new dimensions?
- Solution
- First step: T=130 – 70 = 60°F
- Multiply x T = 1.3 x 10-5 /°F x 60°F = 0.00078
- Consider each dimension
- 2” x 0.00078 = 2.002”
- 6” x 0.00078 = 6.005”
- 180” x 0.00078 = 180.14”
- expands a bit more than an eight of an inch length-wise.
- This is proportional to the length, so for a long bridge it could shift by an inch or even several inches – enough to make a fixed connection untenable.
- Hence a roller support can be an advantage for a material that expands and contracts – by allowing one side to translate, there is no compression in the horizontal direction for the bridge.
- Demo
- http://www.physics.umd.edu/lecdem/services/demos/demosi1/i1-11.htm

Superposition Method

- Can remove a support
- Replace the support with a reaction
- Enforce the geometry constraint

Superposition Method – con’t

- Total length change restrained to zero
- Constraint:

Masonry

- Like all masonry, bricks are strong in compression but weak in tension. You can see this with a single brick. How?

DemonstrationCompression and Tension

- DEMO: compressible foam
- A force pushing down on a broad surface will clearly cause compression beneath it.
- If the force is applied in the middle, then clearly everywhere will be compressed, though the sides less than the middle.
- However, if the force is applied at one edge, the top will “see-saw” upwards, so that as we compress one end we actually put the other side into tension.
- How far out can I go and not cause any tension?

Middle Third Rule

- Must stay within the “middle third” of the block to prevent the opposite side from going into tension
- this is the important MIDDLE THIRD RULE.
- Why is it important?
- Because bricks/stone/etc are much stronger in compression than in tension; need to keep them in compression or they can break!
- To get around this problem
- reinforce concrete with a steel bar
- the steel can take the tension when the concrete can’t.
- We’ll go over this in more detail in Chapter 8.

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