1. Hull Girder Response - Quasi-Static Analysis

2. Basic Relationships Model the hull as a Free-Free box beam.
Beam on an elastic foundation
Must maintain overall Static Equilibrium.
Force of Buoyancy = Weight of the Ship
LCB must be in line with the LCG

3. Basic Relationships
From Beam Theory – governing equation for bending moment:
Beam is experiencing bending due to the differences between the Weight and Buoyancy distributions
Where f(x) is a distributed vertical load.
Net Load
Buoyancy rg a(x)
Weight g m(x)

4. net load curve - f(x) = b(x) - w(x)
Basic Relationships
buoyancy curve - b(x)
weight curve - w(x)
net load curve - f(x) = b(x) - w(x)
Sign Convention
Positive
Upwards
+ f

5. Basic Relationships The solution for M(x) requires two integrations:
The first integration yields the transverse shear force distribution, Q(x)
Impose static equilibrium on a differential element f M Q
Q + dQ
M + dM dx
But ships are “Free-Free” Beams - No shear at ends! Q(0) = 0 and Q(L) = 0, so C = 0

6. Finding Shear Distribution
Net Load - f
Sign Convention
Positive
Upwards
+ f :
Shear Force - Q
+ Q
Positive
Clockwise
+ Q :
- Q

7. Basic Relationships
The second integration yields the longitudinal bending moment distribution, M(x):
Sum of the moments about the right hand side = 0 f M Q
Q + dQ
M + dM dx
Again, ships are “Free-Free” Beams - No moment at ends! M(0) = 0 and M(L) = 0, so D = 0

8. Finding Bending Moment Distribution
Shear Force - Q
Sign Convention
+ Q
Positive
Clockwise
+ Q :
- Q
Bending Moment - M
Positive
Sagging
+ M :
- M

9. Shear & Moment Curve Characteristics
Zero shear and bending moments at the ends.
Points of zero net load correspond to points of minimum or maximum shear.
Points of zero shear correspond to points of minimum or maximum bending moment.
Points of minimum or maximum shear correspond to inflection points on bending moment curve.
On ships, there is no shear or bending moments at the forward or aft ends.

10. Still Water Condition Static Analysis - No Waves Present
Most Warships tend to Sag in this Condition
Putting Deck in Compression
Putting Bottom in Tension

11. Quasi-Static Analysis
Simplified way to treat dynamic effect of waves on hull girder bending
Attempts to choose two “worst case”conditions and analyze them.
Hogging Wave Condition
Wave with crest at bow, trough at midships, crest at stern.
Sagging Wave Condition
Wave with a trough at bow, crest at midships, trough at stern.
Wave height chosen to represent a “reasonable extreme”
Typically:
Ship is “balanced” on the wave and a static analysis is done.

12. Wave Elevation Profiles
The wave usually chosen for this analysis is a Trochoidal wave. It has a steeper crest and flatter trough.
Chosen because it gives a better representation of an actual sea wave than a sinusoidal wave.
Some use a cnoidal wave for shallow water as it has even steeper crests.

13. Trochoidal vs. Sine Wave

14. Excess Weight Amidships - Excess Buoyancy on the Ends
Sagging Wave
Excess Weight Amidships Excess Buoyancy on the Ends
Compression
Tension

15. Excess Buoyancy Amidships - Excess Weight on the Ends
Hogging Wave
Excess Buoyancy Amidships Excess Weight on the Ends
Tension
Compression

16. Weight Curve Generation
The weight curve can be generated by numerous methods:
Distinct Items (same method as for LCG)
Parabolic approximation
Trapezoidal approximation
Biles Method (similar to trapezoidal)
They all give similar results for shear and bending moment calculations. Select based on the easiest in your situation.

17. Distinct Item Method
Each component is located by its l, t and v position and weight
Can be misleading for long components

18. Example Weight Curve

19. Weight Item Information
For each weight item, need W, lcg, fwd and aft W
fwd
lcg
aft FP

20. Trapezoid Method Models weight item as a trapezoid
Best used for semi-concentrated weight items
Need the following information:
Item weight – W (or mass, M)
Location of weight centroid wrt FP - lcg
Forward boundary wrt FP - fwd
Aft boundary wrt FP - aft
lcg must be in middle 1/3 of trapezoid

21. Trapezoid Method Find l and x
Solve for wf and wa so trapezoid’s area equals W and the centroid is at the lcg FP
lcg x wa G wf
fwd
l/2 l
aft

22. Biles Method
Used for weight items which are nearly continuous over the length of the ship.
Assumes that weight decreases near bow & stern.
Assumes that there is a significant amount of parallel middle body.
Models the material with two trapezoids and a rectangle.

23. Biles Method
lcg x G
1.2h wa wf
aft FP

24. The Three Types of Structure
Characteristics
Primary Structure
Secondary Structure
Tertiary Structure
In-plane rigidity
Quasi-infinite
Finite
Small
Loading
In-plane
Normal
Stresses
Tension, Compression and Shear
Bending and Shear
Bending, Shear and Membrane
Examples
Hull shell, deck, blkhd, tank top
Stiffeners on blkhd, shell
Unstiffened shell
Boundaries
Undetermined
Primary structure