Chap6_VABLE - M Vable Mechanics of Materials Symmetric...

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6 254 Mechanics of Materials: Symmetric Bending of Beams M. Vable Printed from: http://www.me.mtu.edu/~mavable/MoM2nd.htm January, 2010 CHAPTER SIX SYMMETRIC BENDING OF BEAMS Learning objectives 1. Understand the theory of symmetric bending of beams, its limitations, and its applications for a strength-based design and analysis. 2. Visualize the direction of normal and shear stresses and the surfaces on which they act in the symmetric bending of beams. _______________________________________________ On April 29th, 2007 at 3:45 AM, a tanker truck crashed into a pylon on interstate 80 near Oakland, California, spilling 8600 gallons of fuel that ignited. Fortunately no one died. But the heat generated from the ignited fuel, severely reduced the strength and stiffness of the steel beams of the interchange, causing it to collapse under its own weight ( Figure 6.1 a ). In this chapter we will study the stresses, hence strength of beams. In Chapter 7 we will discuss deflection, hence stiffness of the beams. Which structural member can be called a beam? Figure 6.1 b shows a bookshelf whose length is much greater than its width or thickness, and the weight of the books is perpendicular to its length. Girders, the long horizontal members in bridges and highways transmit the weight of the pavement and traffic to the columns anchored to the ground, and again the weight is perpendicular to the member. Bookshelves and girders can be modeled as beams —long structural member on which loads act perpendicular to the longitudinal axis. The mast of a ship, the pole of a sign post, the frame of a car, the bulkheads in an air- craft, and the plank of a seesaw are among countless examples of beams. The simplest theory for symmetric bending of beams will be developed rigorously, following the logic described in Figure 3.15, but subject to the limitations described in Section 3.13. 6.1 PRELUDE TO THEORY As a prelude to theory, we consider several examples, all solved using the logic discussed in Section 3.2. They highlight observations and conclusions that will be formalized in Section 6.2. Example 6.1, discrete bars welded to a rigid plate, illustrates how to calculate the bending normal strains from geometry. Example 6.2 shows the similarity of Example 6.1 to the calculation of normal strains for a continuous beam. Example 6.3 applies the logic described in Figure 3.15 to beam bending. Example 6.4 shows how the choice of a material model alters the calculation of the internal bending moment. As we saw in Chapter 5 for shafts, the material model affects only the stress distribution, leaving all other equations unaffected. Thus, the kinematic equation describing strain distribution is not affected. Neither are the static equivalency equations Figure 6.1 (a) I-80 interchange collapse. (b) Beam example. (b) (a)
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6 255 Mechanics of Materials: Symmetric Bending of Beams M. Vable Printed from: http://www.me.mtu.edu/~mavable/MoM2nd.htm January, 2010 between stress and internal moment and the equilibrium equations relating internal forces and moments. Although we
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