Mechanics of Materials – MAE 243 (Section 002)

Spring 2008

Dr. Konstantinos A. Sierros

QUIZ PROBLEM

a) Associate the stress elements 1 and 2 (left hand side) with the appropriate Mohr’s circles (right hand side)

b) An element in biaxial stress is subjected to stresses σx = 8000 psi and σy = -1850 psi as shown in the figure below. Using Mohr’s circle determine the stresses acting on an element oriented at a counterclockwise angle θ = 68 degrees from the x-axis. Show result on sketch of properly oriented element

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PROB. 7.4-5

1850 psi

8000 psi

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FIG. 8-13

Stresses in a beam of rectangular cross section:(a) simple beam with points A, B, C, D, and Eon the side of the beam;(b) normal and shear stresses acting on stress elements at points A, B, C, D, and E;(c) principal stresses; and (d) maximum shear stresses

8.4: Maximum stresses in beams• Beams of rectangular cross-section

• Points A and E are at the top and bottom of the beam. Point C is at the midheight of the beam, and points B and D are in between

• If Hooke’s law applies, the normal and shear stresses at each of these five points can be readily calculated from the flexure and shear formulas (using the eqs 5-13 and 5-38 from Chapter 5)

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FIG. 8-14 Principal-stress trajectories for beams of rectangular cross section:(a) cantilever beam, and (b) simple beam. (Solid lines represent tensile principal stresses and dashed lines represent compressive principal stresses.)

8.4: Stress trajectories• By investigating the stresses at many cross sections of the beam, the principal stresses vary throughout the beam.

• Two systems of orthogonal curves, called stress trajectories, that give the directions of the principal stresses can be constructed

• The curves for tensile and compressive principal stresses always intersect at right angles, and every trajectory crosses the longitudinal axis at 45 degrees

• A stress contour (fig 8-15) is a curve connecting points of equal principal stress

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FIG. 8-15 Stress contours for a cantilever beam (tensile principal stresses only)

8.4:Wide flange beams

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FIG. 8-16 Stresses in a wide-flange beam

• Beams having other cross-sectional shapes, such as wide-flange beams, can be analyzed for the principal stresses in a manner similar to that described previously for rectangular beams using the shear and flexure formulae

• The largest principal stresses usually occur at the top and bottom of the beam (points A and E)

• The maximum shear stresses acting on a cross section of a wide flange beam always occur at the neutral axis. However, the max shear stresses occur either at points A and E or at points B and D

8.5: Combined loading

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FIG. 8-20 Examples of structures subjected to combined loadings: (a) wide-flange beam supported by a cable (combined bending and axial load), (b) cylindrical pressure vessel supported as a beam, and (c) shaft in combined torsion and bending

• In previous chapters we analyzed structural members subjected to a single type of loading

• However, in many structures the members are required to resist more than one kind of loading. This describes a situation of combined loading

• A structural member subjected to combined loadings can be analyzed by superimposing the stresses and strains caused by each load acting separately. Superposition is only permissible when stresses and strains are linear functions of the applied loads and when there is no interaction between the various applied loads

8.5: Combined loading – Method of analysis1. Select a point in the structure where the stresses and strains are to be

determined

2. For each load on the structure, determine the stress resultants at the cross-section containing the selected point. (The possible stress resultants can be an axial force, a twisting moment, a bending moment and a shear force)

3. Calculate normal and shear stresses at the selected point due to each of the stress resultants

4. Combine the individual stresses to obtain the resultant stresses at the selected point. (ie you must obtain the stresses σx σy τxy acting on a stress element at a point

5. Determine the principal stresses at the selected point, using either the stress transformation equations or Mohr’s circle

6. Using Hooke’s law determine the strains at the point

7. Select additional points and repeat the process. Continue until you have enough stress and strain info

Next time worked examples will be presented…