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8/1 STRESS AND STRAIN IN BEAMS

8/1 STRESS AND STRAIN IN BEAMS. 8/2 BENDING BEAMS. LOADS ON BEAM PRODUCE STRESS RESULTANTS, V & M V & M PRODUCE NORMAL STRESSES AND STRAINS IN PURE BENDING V & M PRODUCE ADDITIONAL SHEAR STRESSES IN NON-UNIFORM BENDING. 8/3 TYPES OF BENDING.

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8/1 STRESS AND STRAIN IN BEAMS

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  1. 8/1 STRESS AND STRAIN IN BEAMS

  2. 8/2 BENDING BEAMS • LOADS ON BEAM PRODUCE STRESS RESULTANTS, V & M • V & M PRODUCE NORMAL STRESSES AND STRAINS IN PURE BENDING • V & M PRODUCE ADDITIONAL SHEAR STRESSES IN NON-UNIFORM BENDING

  3. 8/3 TYPES OF BENDING • PURE BENDING – FLEXURE UNDER CONSTANT M. i.e. V =0 = dM/dx • NON-UNIFORM BENDING – FLEXURE WHEN V NON-ZERO

  4. P Unit (small) Length P y z Pure bending region x 8/4 PURE BENDING a V P -P Non –uniform bending P.a M

  5. Unit (small) length Straight lines 8/5 PURE BENDING M M

  6. Straight lines remain straight Planes remain plane 8/6 RADIUS OF CURVATURE 

  7. 8/7 CURVATURE  = 1/ d  y ds x dx

  8. d= dx L1= (-y)d L1= dx - (y/)dx L1- dx = -(y/)dx NORMAL STRAIN x= -(y/)dx/dx=-.y L1 NEUTRAL AXIS

  9. 8/9 LONGITUDINAL STRAIN Strain x y 1

  10. Stress v Strain   8/10 LONGITUDINAL STRESS Strain Stress y

  11. y 8/11 NORMAL STRESS AND STRAIN Strain Stress x x = E.x 1 x = -y/ = -.y x = -E.y/ = -E..y

  12. 8/12 RELATION BETWEEN  & M • NEED TO KNOW WHERE NEUTRAL AXIS IS. FIND USING HORIZONTAL FORCE BALANCE • NEED A RELATION BETWEEN  & M. FIND USING MOMENT BALANCE –gives THE MOMENT CURVATURE EQUATION & THE FLEXURE FORMULA

  13. 8/13 NEUTRAL AXIS PASSES THROUGH CENTROID dA y x Compression c1 y x M z Tension c2 First moment of area

  14. 8/14 MOMENT-CURVATURE EQN dA c1 y z = moment of inertia wrt neutral axis c2

  15. 8/15 FLEXURE FORMULA FOR BENDING STRESSES

  16. 8/16 MAXIMUM STRESSES OCCUR AT THE TOP & BOTTOM FACES

  17. 8/17 MAXIMUM STRESSES y 1 C1 S1 = I/c1 Section modulus x M C2 2

  18. 8/18 I & S FOR BEAM y 0 h z b

  19. 8/19 BEAM DESIGN • SELECT SHAPE AND SIZE SO THAT STRESS DOES NOT EXCEED ALLOW • CALCULATE REQUIRED S=MMAX/ALLOW • CHOOSE LOWEST CROSS SECTION WHICH SATISFIES S

  20. 8/20 IDEAL BEAM • RECTANGULAR BEAM, SR=bh2/6=Ah/6 • CYLINDRICAL BEAM, S=0.85.SR • IDEAL BEAM, HALF THE AREA AT h/2 • IDEAL BEAM, S=3.SR • STANDARD I-BEAM, S=2.SR

  21. 8/21 STRESSES CAUSED BY BENDING P=50kN q=20kN/m 3.5m 2.5m RB RA h=0.7m b=0.22m

  22. 8/22 V & M DIAGRAMS V=89.2kN 39.2kN V -10.8kN -80.8kN M=160.4kNm M

  23. 8/23 MAX FOR BEAM y 0 z h MAXIMUM STRESSES b

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