Aalto University School of Engineering Kul-24.4140 Ship Dynamics (P) Lecture 9 Loads
Where is this lecture on the course? Design Framework Lecture 5: Equations of Motion Environment Lecture 6: Strip Theory Manouvering Seakeeping Lecture 9: Loads Lecture 10: Control of Motions
Contents Calculation of response amplitude operator RAO Calculation of the spectral parameters and properties Short term load predictions Literature 1. Lewis, Principles of Naval Architecture Vol. III, SNAME, 1989 2. Korvin-Kroukovsky, B.V., Investigations of Ship Motions in Regular Waves, SNAME Transactions, Vol. 63, 1955. 3. Kukkanen, T. Specral Fatigue Analysis for Ship Structures. Uncertainties in Fatigue Actions. TKK, Konetekniikka, Lis.työ. 1996.
Weekly Exercise Exercise 9: Load Input for Structural Design - Given 08.03.2016 09:00, Return 14.03.2015 09:00 Define the design extreme bending moment and shear force for your ship using selected wave spectrum and Sealoads, Napa Justify the selection of wave spectrum that causes the maximum Discuss the non-linearities Report and discuss the work.
Motivation Hull Girder Load In the structural design of the ships, a common practice is to express the design loads by means of the sagging and hogging bending moments and shear forces. The sagging and hogging bending moments and shear forces are hull girder loads. The hull girder loads are balanced by internal forces and moments affecting at a crosssection of the ship hull (stress resultants) The accurate prediction of the extreme wave loads is important in the ultimate strength assessment of the hull girder. For ships in a heavy sea, the sagging loads are larger than the hogging loads. The linear theories cannot predict the differences between sagging and hogging loads, e.g. strip theory
Mass and hull/water interaction forces Important to know the internal forces and moments that act on a hull of ship operating in waves. These forces are primarily composed of mass and hull/water interaction forces. In still water mass force, or a distribution of it, is simply a distributed ship weight. Hull/water interaction force, in this case, is the buoyancy or it s distribution in a form of a hydrostatic pressure. In waves mass forces get an additional contributor associated with accelerations due to ship motions. As a result the inertia component is added to the weight. Moreover, hull/water interaction gets more complicated. Pressure acting on a hull surface in waves comprises apart hydrostatic part the radiation, Froude-Krylov and diffraction contributions. q W (x), q Δ (x) q(x) Q SW ( x) M SW (x) uppouman nostovoima q Δ ( x) laivan paino q W (x) kuorma tyynenveden leikkausvoima tyynenveden momentti L
Sagging and hogging The wave-induced primary stresses are important in the: ultimate strength assessment of the hull girder and plates Fatigue strength analysis of structural details Compression on deck and tension in bottom is called a sagging condition Tension in deck and and compression in a bottom is called hogging condition
Wave crests at bow and at stern In waves the sagging condition occurs if wave crests are at the bow and stern and hogging if a wave crest is at mid-ship. The sagging increases if the ship has large bow flare and the ship motions are large with respect to waves. The stern form of the ship can have the same effect if the ship has a flat bottom stern close to the waterline. Bow Waterline, sagging Waterline, hogging Force
Linear approach to the loads We shall adopt the body-fixed coordinate system, which is often called the seakeeping co-ordinate system For the sake of simplicity we shall limit the discussion to the global load in terms of vertical shear force and bending moment acting on a hull regarded as a beam resting on a flexible foundation.
The still water condition uppouman nostovoima q Δ ( x) At each station, denoted by a position x, we have the vertical force per unit length given by a sum of weight and buoyancy at this section that is q W (x), q Δ (x) laivan paino q W (x) q(x) = m(x)g + ρga(x) q(x) kuorma With ship heaving η 3 and pitching η 5 motion we have to take additionally inertia and hydrodynamic F(x) loads as well. As a result the vertical force per unit length of a hull is getting a form q(x) = m(x)g + ρga(x) ( η 5 ) + F (x) m(x)!! η 3 x!! Q SW ( x) M SW (x) tyynenveden leikkausvoima tyynenveden momentti L
Bending moment and shear force at section x p Total vertical shear force and bending moment at section x p can be obtained by integrating load/ ship length along ship length from the stern up to the section x p as follows Q(x' p ) = x' p q(x') dx' M (x' p ) = x'q(x') dx' 0 the shear force and the bending moment are zero at the stem and at the stern. If we subtract from the above expressions the still water values of shear force and bending moment respectively we get a linear approximation of the internal load distribution along the ship length related to wave action. x' p 0
RAO of internal load As the model is linear, we can use the concept of RAO in order to relate the loads to the wave and ship operating condition (wave length, heading and ship speed). That is we can proceed similarly as we did with the other linear responses and derive a short term internal load prediction for a ship operating in irregular waves. The shortcoming of the linearity assumption is that the result does not distinguish between the sagging and the hogging condition except for the still water condition.
Response with Bonjean Curves
Strip Method Classical method to evaluate the loads on hull girder in regular waves is the Strip-method by Korvin-Kroukovsky (1955, 1957) The basic assumption is that the flow in length direction is neglible Then the hull can be divided into strips which are describing 2D phenomena Have straight sides (NOTE! Ships do not, but usually this assumption is sufficient due to small changes, hog. vs. sag.) Then the equations of motion can be solved 6 k = 1 H [( M + A ) η + B η + C η ] ( ω) jk 2 = jk Y A k ( ω) ( ω) 2 jk k jk k = F e j iωt, j = 1,2,...,6
Strip-Method 6 Σ k = 1 [(M jk + A jk ) η k + B jk η k + C jk η k ] = F j e œi ωt, j=1, 2,.., 6 Added mass - Froude-Krylof force - Diffraction force
The Components of Ship Equation of Motion In Length (x) direction Heave dynamical force components Hollow of wave at midship L BP = 520 feet, T = 10 s, H w = 20 feet
Calculation of the Shear Force and Bending Moment ( ) ( ) ( ) dx Q M dx f m x f x e Q x x V V x x w t i V f f = + = 0 0 0 3 2 3 5 3 ω η η ω
Example MV Arctic Basic Information L BP 196,59 m B 22,86 m T 10,97 m C B 0,76 Δ 38.030 ton DW 28.000 ton Allowed M SW 924,5 MNm (92.450 tonm) Section modulus: Z deck 12,982 m 3 Z bottom 14,627 m 3 Lloyd s 100 A1, Ice Class AC 2 NS Steel R e = 235 N/mm 2
MV Arctic (F n = 0,17, µ=180 o ) H(ω) [knm/m] Bending Moment RAO λ = 2πg ω 2 ω = 2πf = 2π T λ = gt2 2π
Mv Arctic Wave Bending Moment RAO µ = 180 ο, v = 15 knots Transfer function H(ω) [knm/m] Wave spectrum [m 2 s] T z = 8.5 s H s = 4.0 m Response spectrum S YY (ω) = H2 (ω) S XX (ω) [(knm/m) 2 m 2 s] ω
Example ship - RoPax D of Seatech programme (Mattila, Kukkanen, Matusiak)
Weight, shear force and bending moment
Response Amplitude Operators of the vertical shear force Q (station #6.5) and bending moment M (station #4) for RoPax in head seas at Fn=0.25.
Non-linearities disclosed by model tests; Kukkanen, Fn=0.25
Unsymmetry in sagging&hogging; Fn=0.25
Summary Hull girder loads are needed to have realistic estimate of the seaworthiness of the ship in terms of strength We need to consider Still water Wave bending Wave bending loads Can be assessed using the linear theory, e.g. strip Non-linear models, e.g. non-linear strip, panel method etc Large bow flare can cause high non-linearities in the hull