1. Copyright Joseph Greene 2003 All Rights Reserved 1 CM 197 Mechanics of Materials Chap 16: Deflections of Beams Professor Joe Greene CSU, CHICO Reference: Statics and Strength of Materials, 2 nd ed., Fa-Hwa Cheng, Glencoe/McGraw Hill, Westerville, OH (1997) CM 197

2. Copyright Joseph Greene 2003 All Rights Reserved 2 Design of Beams for Strength Topics –Introduction –Relationship Between Curvature and Bending Moment –The Formula Method –The Method of Superposition

3. Copyright Joseph Greene 2003 All Rights Reserved 3 Introduction Beam is subjected to transverse loads that produces bending moment. –Stiffer beams have less deflection Figure 16-1. Simple Beam in bending with concentrated load and uniform load deflects. –Beam designed for strength (Chap 15) won’t break or buckle, but it might deflect (bend) too far. Beams above a plastered ceiling might deflect too far and cause the plaster to crack. –Deflection calculation is needed to solve indeterminate beam problems (Chapter 17) –Bridge construction: If beams aren’t stiff enough then they will deflect and cause the road to have valleys and hills. Beams are made with a crown so that when they deflect the crown flattens out and the road surface is smooth.

4. Copyright Joseph Greene 2003 All Rights Reserved 4 Introduction Beam deflection methods –Formula method Easy to apply. Look up deflections in books and tables. Most common method used. Available for beams of uniform cross section. –Moment area method- more difficult. Assumptions –Beam is homogeneous –Beam obeys Hooke’s law having equal modulus of elasticity in tension and compression. –Beam is a vertical plane of symmetry on which the loads act. –Deflections are small and are caused by bending only. –Deflections due to shear are negligible.

5. Copyright Joseph Greene 2003 All Rights Reserved 5 Curvature and Bending Moment Relationship Consider beam with bending moment and resulting curvature upward due to positive bending moment. –Fig 16-2. Beam segment with a positive bending moment. Center of curvature, point O in Fig 16-2, is the distance, , rho from O to the beam neutral axis. –Radius of curvature is . –The beam axis is the neutral axis. Max Strain is ratio of distance from neutral axis to tension edge of beam to the the radius of curvature, . For elastic bending,  = E , (stress = modulus x strain) –Then, Equating flexure formula,  max = Mc/I –Then, radius of curvature is Eqn 16.2 Here, EI is constant –Radius of curvature, , of a beam at any section varies inversely with bending moment at that section. Higher Moment = smaller . –Example, 16-1 and 16-2 16-1 16-2

6. Copyright Joseph Greene 2003 All Rights Reserved 6 Formula Method Beam Deflection Formulas. Table 16-1 –Cantilever beams (Cases 1 through 4) –Simple beams (Cases 5 through 8) Formulas provide –Maximum deflection along beam. –Slope at the free end of the cantilever beams or at he ends of simple beams. –General deflection equation from which the deflections at any point along the beam can be calculated. Note: –Table lists absolute values of deflections. –Deflection and slope are inversely proportional to EI (Modulus*Moment of Inertia) = Flexural rigidity of beam. Higher flexural rigidity means less deflection. –Deflections are small. –Units must be consistent. Make sure you use consistent units. –Example 16-3 –Example 16-4

7. Copyright Joseph Greene 2003 All Rights Reserved 7 Method of Superposition From Table 16-1, –The load is related linearly to deflection and that deflection is small so that the geometry of the beam does not change significantly. –Deflections from several loadings can be superimposed since deflections are small. Beam deflection from each load can be separated. Algebraic summing of loads can be summed to represent the loads acting simultaneously. –If deflections were large then numerical methods are needed. –Method of superposition Use information from Table 16-1 for each loading condition and then combine the results. –Example 16-5 –Example 16-6 –Example 16-7 –Example 16-8.