2 Strength of Materials can really be divided into 2 categories: Stress analysis:Structure exists, material and loading knownIS 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?
3 Possible modes of failure (mechanical)??? Fracture (s >> Su)Yield (s >> SyInstability (buckling)Fatigue and wearExcessive deformation (i.e. too soft)Creep or stress relaxation (polymers)1,2,4 – most important parameter? STRESS!!!!!!
4 Internal Force per unit area Definition of Stress:Figure: 01-10a-cDefinition of Stress:Internal Force per unit areaIntensity of internal force on a specific plane (area) passing through a point
5 Stress States:Preferred stress element for 2D stress:Figure: 09-01a-cb) 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.
6 Quick Review: Basic Types of Stress (ref: MCHT213)
7 AXIAL MEMBERS: Average Normal Stress (aka Direct Normal Stress): Figure: 01-15
8 3.4 Average Normal Stress: Requirements for Average Normal Stress, s = P/A:Member starts out straight and remains straight after loadingHomogenous, isotropicInvoke St. Venant’s PrincipalFigure: 01-14EXAMPLES of AVERAGE NORMAL STRESS:
9 Example Normal Stress: If P = 20K lbsand A = 2 in2s = ?Figure: 01-27a
10 t = P/As Average Shear Stress (AKA Direct Shear Stress): Figure: 01-20a-c
14 Example 2 – direct normal and shear stress DISCUSS ONLY!1.119
15 = St. Venant’s Principle and Stress Concentration Factors, Kt: Figure: 04-01aSt 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.
16 St. Venant’s Principal: Figure: UNNote, def’m of grid uniform at middle, therefore strain and stress will be uniform.
17 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: 04-21asmax > savg
18 smax = Kt* savg smax = maximum stress savg = average stress (P/Amin) Kt = stress concentration factorFigure: 04-21b,c
24 Or, for multiple sections: Deformation of Axial Member with Constant Load and Cross-Sectional Areas:Figure: 04-03Or, for multiple sections:
25 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 mmFigure: 04-05a
26 First, solve for internal loads: Figure: 04-05b
27 How would this answer change if aluminum instead of steel???? Figure: 04-05cHow would this answer change if aluminum instead of steel????
28 TORSION: Key points: Varies linearly with radius, r. Zero at center Max at outer fiber (r = c)Constant for given r.Solid vs. hollowTorsion of non-circular sections.Now how to calculate torque given power and rotational speed.Figure: 05-05
29 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: 05-06or
30 J = polar moment of inertia Solid shaft:Hollow shaft:WFor Design:r/sT = P/nT = 63,000 P/nN-mrpmlb-inhp
31 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
35 Angle of twist - For straight sections: TorqueLengthAngle of twist (rad)Modulus of Rigidity (Shear Modulus) – see back of bookFigure: 05-16Polar moment of inertia
36 Angle of twist for Multiple Sections: Figure: 05-19a
37 If the distance between gear E and the middle gear is 12 inches, find the angle of twist between the twogears. The shaft is steel and G = 11.5 x 106 psi.
38 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
39 How do we deal with stress concentrations?? Based on smaller of two connected shafts
43 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 cT = 4.1 N-m = 4,100 N-mma. Circular w/ dia = 10 mb. Square w/ side = 8.86 mmc. Hollow w/ od = 12.8 mm and id = 8 mmAlso see HO: Comparison of Torsion Elements, also overhead fabricated beam
44 Shear and Moment diagrams: Do not get actual equations, good if just after Vmax and Mmax
45 Example: Draw Shear & Moment diagrams for the following beam 12 kN8 kNACDB1 m3 m1 mRA = 7 kN RC = 13 kN
46 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)
47 Example: Beam w/ Concentrated Moment: Also see HO: Simple Beam with Uniform Load, Load-Shear-Moment Relationships
48 Beam Bending Stress: The Flexure Formula: Internal bending moment, lb-inMax bending stress, psiDistance from NA to outer fiber, inMoment of inertia, in4Or in general:
49 Examples: Find maximum moment Find area properties, I and c Calculate stressSee HO: Bending Stress Concepts
50 WHERE IS BENDING STRESS MAXIMUM??? Answer:Outer surface (furthest away from Neutral Axis)Value of x along length where moment is maximum!!
54 Beam Shear Stress: See HO: Shear Stress Calculations in Beams Internal Shear (lb)First Moment of area (in3) at point of interestFigure: dThickness of cross-section at point of interest (in)Moment of inertia of entire cross section (in4)