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**Chapter 3 – Stress and Deformation Analysis (ref MCHT 213!!)**

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**Strength of Materials can really be divided into 2 categories:**

Stress analysis: Structure exists, material and loading known IS IT SAFE????? Design: Determine geometry OR material based on an allowable stress (i.e. Sy/4). Generally certain aspects are fixed. Much more involved than analysis – why?? Ref Chapter 1 – start with design requirements/functions/evaluation criteria,etc. Show overhead – trailer – analysis or design? how would you analyze it?

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**Possible modes of failure (mechanical)???**

Fracture (s >> Su) Yield (s >> Sy Instability (buckling) Fatigue and wear Excessive deformation (i.e. too soft) Creep or stress relaxation (polymers) 1,2,4 – most important parameter? STRESS!!!!!!

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**Internal Force per unit area **

Definition of Stress: Figure: 01-10a-c Definition of Stress: Internal Force per unit area Intensity of internal force on a specific plane (area) passing through a point

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Stress States: Preferred stress element for 2D stress: Figure: 09-01a-c b) Stress state for plane stress can be summarized on a 2D element. a) In general, can have 6 independent stresses (3 normal and 3 shear) acting at a point. b) Many practical engineering problems involve only three independent stresses – called plane stress.

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**Quick Review: Basic Types of Stress (ref: MCHT213)**

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**AXIAL MEMBERS: Average Normal Stress (aka Direct Normal Stress):**

Figure: 01-15

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**3.4 Average Normal Stress:**

Requirements for Average Normal Stress, s = P/A: Member starts out straight and remains straight after loading Homogenous, isotropic Invoke St. Venant’s Principal Figure: 01-14 EXAMPLES of AVERAGE NORMAL STRESS:

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**Example Normal Stress:**

If P = 20K lbs and A = 2 in2 s = ? Figure: 01-27a

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**t = P/As Average Shear Stress (AKA Direct Shear Stress):**

Figure: 01-20a-c

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Average Shear Stress: Figure: 01-21a-d

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Example Shear Stress: If load = 2,000 lb and bolt diameter = ½”, Find shear stress in bolt. What if double shear, what would the new shear stress be?? Figure: 01-28a

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**Example: normal stress and shear stress:**

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**Example 2 – direct normal and shear stress**

DISCUSS ONLY! 1.119

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**= St. Venant’s Principle and Stress Concentration Factors, Kt:**

Figure: 04-01a St Venant’s: Stress profile, sufficiently removed from the local effect of loads will be uniform (i.e. = P/A) Stress and strain produced by statically equivalent load system will be the same.

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**St. Venant’s Principal:**

Figure: UN Note, def’m of grid uniform at middle, therefore strain and stress will be uniform.

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**smax > savg Example:**

Look at deformation in vicinity of hole. Is it uniform??? Stress profile is not uniform. smax occurs at area of discontinuity. Figure: 04-21a smax > savg

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**smax = Kt* savg smax = maximum stress savg = average stress (P/Amin)**

Kt = stress concentration factor Figure: 04-21b,c

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Figure: 04-25

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smax = Kt* savg Figure: 04-22a

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Figure: 04-24

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Find: Max Stress:

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**Or, for multiple sections:**

Deformation of Axial Member with Constant Load and Cross-Sectional Areas: Figure: 04-03 Or, for multiple sections:

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**Example: Multiple sections**

Example: Multiple sections. Find total deformation of end A with respect to D. Area = 20 mm2. Material is steel w/ E = 200 GPa = 200 x 109 Pa: = 100 mm = 150 mm = 200 mm Figure: 04-05a

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**First, solve for internal loads:**

Figure: 04-05b

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**How would this answer change if aluminum instead of steel????**

Figure: 04-05c How would this answer change if aluminum instead of steel????

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**TORSION: Key points: Varies linearly with radius, r. Zero at center**

Max at outer fiber (r = c) Constant for given r. Solid vs. hollow Torsion of non-circular sections. Now how to calculate torque given power and rotational speed. Figure: 05-05

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**The torsion formula (see derivation):**

Torque (N-m, N-mm or lb-in, lb-ft, etc) Outer radius of shaft (m or in) Polar moment of inertia (m4 or in4) Max shear stress in shaft (MPa, psi/ksi, etc.) Figure: 05-06 or

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**J = polar moment of inertia**

Solid shaft: Hollow shaft: W For Design: r/s T = P/n T = 63,000 P/n N-m rpm lb-in hp

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**Stress Profiles: Shear stress profile – YOU MUST UNDERSTAND THIS!!!!**

Where is shear stress max? zero? How does it vary along the length and circumference? Figure: 05-07a

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Figure: UN

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**Angle of twist - For straight sections:**

Torque Length Angle of twist (rad) Modulus of Rigidity (Shear Modulus) – see back of book Figure: 05-16 Polar moment of inertia

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**Angle of twist for Multiple Sections:**

Figure: 05-19a

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If the distance between gear E and the middle gear is 12 inches, find the angle of twist between the two gears. The shaft is steel and G = 11.5 x 106 psi.

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**5.8 Stress Concentrations (last topic we’ll cover in Chapter 5)**

Consider the torsion member only (shaft) where do you think the stress concentrations are?? Again, stress concentrations occur where there’s an abrupt change in geometry! Figure: 05-35a-c

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**How do we deal with stress concentrations??**

Based on smaller of two connected shafts

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Figure: 05-36

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**Torsion of non-circular cross-sections:**

Where Q and K are determined based on cross-section from F3-10 Also, see equations for closed thin walled tubes! See HO: Stress Analysis 2 examples

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**c. Hollow w/ od = 12.8 mm and id = 8 mm**

Example: Torsion find max shear stress for the three cross-sections: a, b and c T = 4.1 N-m = 4,100 N-mm a. Circular w/ dia = 10 m b. Square w/ side = 8.86 mm c. Hollow w/ od = 12.8 mm and id = 8 mm Also see HO: Comparison of Torsion Elements, also overhead fabricated beam

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**Shear and Moment diagrams:**

Do not get actual equations, good if just after Vmax and Mmax

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**Example: Draw Shear & Moment diagrams for the following beam**

12 kN 8 kN A C D B 1 m 3 m 1 m RA = 7 kN RC = 13 kN

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**12 kN 8 kN 1 m 3 m 1 m 2.4 m 8 7 8 7 V -15 -5 7 M -8 A C D B (kN)**

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**Example: Beam w/ Concentrated Moment:**

Also see HO: Simple Beam with Uniform Load, Load-Shear-Moment Relationships

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**Beam Bending Stress: The Flexure Formula:**

Internal bending moment, lb-in Max bending stress, psi Distance from NA to outer fiber, in Moment of inertia, in4 Or in general:

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**Examples: Find maximum moment Find area properties, I and c**

Calculate stress See HO: Bending Stress Concepts

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**WHERE IS BENDING STRESS MAXIMUM???**

Answer: Outer surface (furthest away from Neutral Axis) Value of x along length where moment is maximum!!

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**Beam Shear Stress: See HO: Shear Stress Calculations in Beams**

Internal Shear (lb) First Moment of area (in3) at point of interest Figure: d Thickness of cross-section at point of interest (in) Moment of inertia of entire cross section (in4)

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**Find: Max Shear and Bending Stress:**

1.93” See HO: Stress Analysis 1 Examples, Steel Beam Selection, Steel W-Shape Selection Data

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Combined Loading: Look at each load individually and solve for stress at a given point due to that load. Repeat for all loads. Add like stresses Summarize stresses on an initial stress element.

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**Example: Combined normal stress – find stress in horizontal portion:**

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**Example: Combined normal stress – find stress in horizontal portion:**

Reduce to simple cantilever!

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**Ultimate Combined Loading Problem!!**

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**Final Concept: Beam Deflection – Superposition:**

No solution for this case Known case from App C Known case from App C Figure: 12-29a-cEx12.13

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STRUCTURAL MECHANICS: CE203

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