Ship contruction

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

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

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

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

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

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

7. STRESS in Bending:
 Remember: Stress = Force/Area (psi)
 And Stress (in bending) increases as we
move away from the neutral axis
NA
C T 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
c1
c2

8. Moment of Inertia:
 For rectangular beams, I = bh3/12
NA
b
h
 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!
 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 for I/c is Z (section modulus)
in units of in4/in = in3
 Thus, smax = M / Z (in psi—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)
Upright 2x4
I = 10.67 in4
c = 2 in
Z = 5.33 in3
smax = 2700 psi
Upright 2x12
I = 288 in4
c = 6 in
Z = 48 in3
smax = 300 psi
Flat 2x12
I = 8 in4
c = 1 in
Z = 8 in3
smax = 1800 psi
 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. Remember:
Load  Shear Bending Moment?
Distributed load: w lbs/ft
L
wL/2 wL/2
LOAD
diagram x
Cumulative Load:
Vx = ∫w dx
x
0
Cumulative Area:
x
SHEAR
diagram
x
BENDING
diagram
Mx = ∫Vx dx
x
0

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
yawing
racking hogging & sagging

17. And sometimes, stuff happens …