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Ship Strength

Ship Strength. Stress & Strain Bending & Shear Moment of Inertia & Section Modulus. STRESS: Force/unit area (lbs/sq.in.). Load F. Load F. Compressive Stress s c = F/A. Tensile Stress s t = F/A. Cross-section area A. Cross-section area A. Reaction = F. Reaction = F.

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Ship Strength

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  1. Ship Strength Stress & Strain Bending & Shear Moment of Inertia & Section Modulus

  2. STRESS:Force/unit area (lbs/sq.in.) Load F Load F Compressive Stress sc = F/A Tensile Stress st = F/A Cross-section area A Cross-section area A Reaction = F Reaction = F Tension(e.g., bridge suspension cable) Compression(e.g., pier pylon)

  3. STRAIN:deformation in length Load F Load F d d STRAIN:d/L (dimensionless) L L Reaction = F Reaction = F Tensile Strain (Strain) positive Compression Strain negative)

  4. Hooke’s Law: Stress ∝ Strain Ultimate strength STRESS: F/A Fracture point Structural steel Yield point Factor of Safety Proportional Limit • Re: to lower stress: • Reduce LOAD, or • Increase AREA y = mx s = E d/L Slope, E = Modulus of Elasticity Maximum working stress STRAIN: d/L

  5. C T NA C T NA BENDING HOGGING: Top side in TENSION Bottom side in COMPESSION NA Top side in COMPESSION SAGGING: Bottom side in TENSION

  6. x x x Cumulative Load: Cumulative Area: x 0 x 0 Vx = ∫wdx Mx = ∫Vxdx Bending Moments: Since there is both tension & compression in bending …we need to look at L LOAD diagram Distributed load: w lbs/ft wL/2 wL/2 SHEAR diagram BENDING diagram Note: Bending is maximum at the location where Shear is zero

  7. T C c1 NA c2 STRESS in Bending: • Remember: Stress = Force/Area (psi) • And Stress (in bending) increases as we move away from the neutral axis Still need to know: • Distance from NA (c) • Something about how the cross-sectional area is distributed N & S of the neutral axis … • Moment of Inertia, I

  8. Moment of Inertia: • For rectangular beams, I = bh3/12 • Thus for a 2x4: I = bh3/12 = 2 x 43/12 = 32/3 = 10.67 • And for a 2 x 12: I = 2 x 123/12 = 288! NA h b • Note that a 2x12 lying “flat” (b=12; h=2) I = bh3/12 = 12 x 23/12 = 8 (less than a 2x4 standing up)

  9. Moment of Inertia: (cont’) • In general, the more area farther from the NA, the greater the moment of inertia • Distributing the same area at a greater distance from the NA produces a stronger beam of the same weight

  10. STRESS in Bending: (reprise) • Maximum Stress (in bending), smax = Mc/I Where: M = maximim bending moment unit: lb-in (lb-ft x 12) c = maximum distance from NA unit: inches I = moment of inertia of beam cross-section unit: inches4 • Shorthand notation forI/c is Z (section modulus) in units of in4/in = in3 • Thus, smax = M / Z (inpsi—lb-in/in3 = lb/in2)

  11. STRESS in Bending: (reprise) • Consider the sections examined under a bending moment of 1200 ft-lb (14400 in-lb) • If the maximum allowable working stress for this material were 3000 psi, which would you choose?

  12. Moment of Inertia: (cont’) • There are tables of I (and Z) for standard steel “sections” I-beam Tee Channel L • And I for any shape can (with some effort) be calculated

  13. How about this shape? • Ship sections can be reduced to an equivalent (if irregular) I-beam • The section modulus, Z, can then be calculated • … and the I-beam examined under different bending load conditions • Yes, it is complicated

  14. x x x SHEAR diagram Cumulative Area: Cumulative Load: x 0 x 0 Mx = ∫Vxdx Vx = ∫wdx BENDING diagram Remember: Load  Shear Bending Moment? L LOAD diagram Distributed load: w lbs/ft wL/2 wL/2

  15. How Complicated? Consider … • Irregular loading: downward forces of cargo & structure • Buoyant (upward) forces proportional to underwater volume of hull • … and that’s in still water • Shear forces & bending moments vary as wave crests amidships or fore & aft

  16. If fact, stresses are dynamic … rolling pitching hogging & sagging racking yawing

  17. And sometimes, stuff happens …

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