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# 311 Ch14

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### 311 Ch14

1. 1. <ul><li>VI .) Stresses in Beams </li></ul><ul><li>A.) Bending Stresses </li></ul><ul><li>1.) Introduction </li></ul><ul><li>When a beam is subjected to positive moment, the top shortens and the bottom lengthens. </li></ul>
2. 3. <ul><li>The bending strains are zero at the neutral axis. </li></ul><ul><li>The bending strains increase proportionately with the distance from the neutral axis. </li></ul><ul><li> The maximum strains occur at the top and bottom of the beam (outer fibers) </li></ul>
3. 4. <ul><li>Since stress is directly related to strain </li></ul><ul><li> =  E), bending stress is also zero at the neutral axis and increases with the distance from the neutral axis. </li></ul><ul><li>The maximum bending stresses occur at the top and bottom of the beam (outer fibers) </li></ul>
4. 5. <ul><li>2.) The Flexure Formula </li></ul><ul><li> b = My </li></ul><ul><li> I </li></ul><ul><li>  b = Bending stress at a distance “y” from the neutral axis (ksi). </li></ul><ul><li>y = distance from the neutral axis (in). </li></ul><ul><li>M = Moment at a given cross-section of a beam (k-in). </li></ul><ul><li>I = Moment of inertia of the cross- section (in 4 ). </li></ul>N.A. y
5. 6. <ul><li>3.) Maximum Bending Stress </li></ul><ul><li> b,max = Mc </li></ul><ul><li> I </li></ul><ul><li> b,max = Maximum bending stress at a given cross-section (ksi). </li></ul><ul><li>c = Distance from N.A. to the outer fibers of cross-section (in) </li></ul><ul><li>M, I = as defined before. </li></ul>N.A. c top c bot y
6. 7. <ul><li>4.) Section Modulus (S) </li></ul><ul><li> b,max = Mc = M </li></ul><ul><li> I S </li></ul><ul><li>S = I_ </li></ul><ul><li> c </li></ul>N.A. c top c bot y NA c top c bot
7. 8. <ul><li>5.) Strong Axis vs. Weak Axis </li></ul>Strong Axis N.A. Weak Axis
8. 9. <ul><li>B.) Shear Stress - Average Web Shear </li></ul><ul><li> v = V_ </li></ul><ul><li>dt w </li></ul>d t w d t w d t w
9. 10. <ul><li>C.) The Beam Shear Stress Formula </li></ul><ul><li> v = VQ </li></ul><ul><li>Ib </li></ul><ul><li>V = Shear force (lb) </li></ul><ul><li>Q = First Moment of Area </li></ul><ul><li>I = 2 nd Moment of Inertia(in 4 ) </li></ul><ul><li>b = thickness (in) at the location where shear is being computed </li></ul>
10. 11. <ul><li>C.) The Beam Shear Stress Formula </li></ul><ul><li>Q = First Moment about the neutral axis of the area between the horizontal plane where the shear is to be calculated and the top or bottom of the beam. </li></ul>Plane where shear Is being computed Neutral axis y Q = Ay
11. 12. <ul><li>C.) The Beam Shear Stress Formula </li></ul><ul><li> v = 1.5V for solid rectangular cross-sections </li></ul><ul><li>A </li></ul><ul><li> v = 4V for solid circular cross-sections </li></ul><ul><li>3A </li></ul>