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

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

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?

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!!!!!!

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

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.

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

AXIAL MEMBERS: Average Normal Stress (aka Direct Normal Stress):
Figure: 01-15

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:

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

t = P/As Average Shear Stress (AKA Direct Shear Stress):
Figure: 01-20a-c

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

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

Example: normal stress and shear stress:

Example 2 – direct normal and shear stress
DISCUSS ONLY! 1.119

= 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.

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

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

smax = Kt* savg smax = maximum stress savg = average stress (P/Amin)
Kt = stress concentration factor Figure: 04-21b,c

Figure: 04-25

smax = Kt* savg Figure: 04-22a

Figure: 04-24

Find: Max Stress:

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

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

Figure: 04-05b

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

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

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

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

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

Figure: UN

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

Angle of twist for Multiple Sections:
Figure: 05-19a

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.

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

How do we deal with stress concentrations??
Based on smaller of two connected shafts

Figure: 05-36

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

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

Shear and Moment diagrams:
Do not get actual equations, good if just after Vmax and Mmax

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 

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)

Example: Beam w/ Concentrated Moment:

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:

Examples: Find maximum moment Find area properties, I and c
Calculate stress See HO: Bending Stress Concepts

WHERE IS BENDING STRESS MAXIMUM???
Answer: Outer surface (furthest away from Neutral Axis) Value of x along length where moment is maximum!!

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)

Find: Max Shear and Bending Stress:
1.93” See HO: Stress Analysis 1 Examples, Steel Beam Selection, Steel W-Shape Selection Data

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

Example: Combined normal stress – find stress in horizontal portion:
Reduce to simple cantilever!