AN APPROXIMATE METHOD FOR THE CALCULATION OFSHEAR FORCE AND BENDING MOMENT LOADING IN

TANKERS AND BULK CARRIERS.

Ertekin Bayraktarkatal,Yalçın Ünsan,

Naval Architecture and Ocean Engineering Faculty,Istanbul Technical University,

TURKEY

ABSTRACT

In this paper,a method for the calculation of the approximate values of shear force andbending moment loading in tankers and bulk carriers is proposed. The data is presented in graphicalform as well as practical equations to be used in the preliminary design stages. The method is basedon the data obtained from more than fifty ships and it is tested on various ships not included in theoriginal data base. It is demonstrated that the approximate results show satisfactory agreement withthe results of the exact calculations. The full load still water and wave crest - wave troughconditions are separately examined in the light of the classification society criteria and the resultsare presented in a comparative form.

1.INTRODUCTION

In the preliminary design stage, it isuseful to have as precise information aspossible on the magnitudes of maximum shearforce and bending moment loading. In thisstudy, a set of approximations to maximumshear force and bending moment loading ontankers and bulk carriers is presented in theform of graphs and equations. The longitudinalstrengths of more than 50 tankers and bulkcarriers are calculated to obtain the informationbase. Each ship is evaluated in still water aswell as in the crest and on the trough of a waveseparately. The maximum values of the shearforce and bending moment loads for each caseis obtained in the graphical form and thedistributions of the maximum bending momentvalues are depicted in comparison with thosecalculated by using Turkish and German

Lloyd's longitudinal strength criteria. The veryfew extreme cases which fall outside thegeneral character of the curves are omitted.

2. CALCULATION PROCEDURE ANDASSUMPTIONS

A computer program developed byI.T.U. Naval Architecture and OceanEngineering Faculty is utilized in thelongitudinal strength calculations. In thisprogram package, firstly the ship form isobtained, then the distributions of buoyancyfor the desired displacement for the still water,trimmed still water as well as the wave crestand trough conditions are calculated. Thecargo and other weights in the holds and tanksare treated as distributed loads taking into

107

consideration the volumetric variationsdictated by the geometry of the relevantsections of the ship hull. The trapezoidalintegration rule is used for the numericalintegration to carry out the calculations of theshear force and bending moment. The errorsinherent in the trapezoidal integration rule areminimized by choosing smaller intervals(about 120 intervals).

For the bending moments calculated inaccordance with the Turkish and GermanLloyd Rules, the hogging and saggingconditions are treated separately. The generalloading condition of the ships used in theanalysis is the full load departure condition.The hold and tank loads are assumed to havemostly homogeneous distributions along theship's length, therefore, the other possibleloading conditions where significantdiscontinuities may exist are not considered.

The range of the variation of the main shipdimensions of the tankers and bulk carriersused in the study is given below:

Lpp= 49.25 - 270.0 [m.]

B = 7.0 - 44.5 [m.]

H = 3.85 - 22.0 [m.]

T = 3.62 - 16.8 [m.]

CB = 0.575 - 0.842

The results obtained from the longitudinalstrength calculations are presented in the formof various graphs with the main variablesbeing the ship main dimensions. Some of thegraphs appropriate to the case are given in theAppendix. In these graphs, the maximum shearforce and bending moment values in stillwater, in wave crest and on wave trough arepresented with respect to the length betweenperpendiculars and the displacement (Fig. 1-6)

as the fundamental variables. In the last of thegraphs (Fig. 7), the maximum bendingmoment results found in this study arecompared with those calculated in accordancewith the Turkish and German Lloyd Rules. Allof the points in the graphs are approximated tosuitable curves by a fitting technique. Theexpressions obtained through this procedureare as follows:

Distribution of maximum bending moment instill water (Fig.1.A);

Msw = (Lpp)3.93675 * 4.49021E-5 [t.m]

Distribution of maximum bending moment inwave crest (Fig.1.B);

Mwc = (Lpp)4.55772 * 5.33522E-6 [t.m]

Distribution of maximum bending moment inwave trough (Fig.2.A);

Mwt =(Lpp)4.05716 * 9.95739E-5 [t.m]

Distribution of maximum bending moment instill water (Fig.2.B);

Msw = (Δ)1.3147 * 0.0240116 [t.m]

Distribution of maximum bending moment inwave crest (Fig.3.A);

Mwc = (Δ)1.50919 * 0.00872257 [t.m]

Distribution of maximum bending moment inwave trough (Fig.3.B);

Mwt = (Δ)1.36507 * 0.0583981 [t.m]

Distribution of maximum shear force in stillwater (Fig.4.A);

Tsw = (Lpp)2.95481 * 0.000308901 [ton]

108

Distribution of maximum shear force in wavecrest (Fig.4.B);

Twc = (Lpp)3.65893 * 1.45228E-5 [ton]

Distribution of maximum shear force in wavetrough (Fig.5.A);

Twt =(Lpp)2.9761 * 0.000600546 [ton]

Distribution of maximum shear force in stillwater (Fig.5.B);

Tsw = (Δ)0.991395 * 0.032944 [ton]

Distribution of maximum shear force in wavecrest (Fig.6.A);

Twc = (Δ)1.21026 * 0.00559035 [ton]

Distribution of maximum shear force in wavetrough (Fig.6.B);

Twt = (Δ)1.00842 * 0.0601052 [ton]

where,

Msw is the maximum bending moment in stillwater,

Mwc is the maximum bending moment in wavecrest,

Mwt is the maximum bending moment in wavetrough,

Tsw is the maximum shear force in still water,

Twc is the maximum shear force in wave crest,

Twt is the maximum shear force in wavetrough,

Lpp is the length between perpendiculars [m],

Δ is the displacement [ton].

3.CONCLUSIONS

All the expressions derived from the graphsare tested against other tankers and bulkcarriers which are not used in this analysis.These tests revealed surprisingly good resultsand the expressions promise to be a valuableapproximation for the maximum shear forceand maximum bending moment in thepreliminary design stage. It must be kept inmind that the ships which constitute theinformation base in this analysis are assumedto be fully loaded and the loads are assumed tobe distributed homogeneously without anymajor discontinuity along the ship length.

In the wave crest situation, if the shiplength is greater than 180 m., the maximumbending moment value calculated from therelevant classification society rule is lowerthan that suggested by this study. It should beremembered that the wave crest condition isdefined by the Turkish and German LloydRules with an additional wave bendingmoment distribution (Fig.7.A). On the otherhand, in the wave trough condition the twodistributions show good agreement (Fig.7.B).

References

[1] INAN, M., Strength of Materials (inTurkish), Doyuran Matbaasì,(1981)

[2] LEWIS, E. V. (Ed.), Principles of NavalArchitecture, VolumeI, Stability and Strength,Second Revision, SNAME Pub., (1988).

[3] SAVCI,M.,Longitudinal Strength of Ships(in Turkish), I.T.U. Pub., (1988)

[4] German Lloyd, Rulesfor the Classificationof Steel Ships (1993)

[5] Turkish Lloyd, Rules for the Classificationof Steel Ships (1993)

109

Appendix

0.00 100.00 200.00 300.00

0.00

50000.00

100000.00

150000.00

Lpp ( m )

STILL WATER

0.00 100.00 200.00 300.00

0.00

200000.00

400000.00

600000.00

Lpp ( m )

WAVE CREST

Y = pow(X,3.93675) * 4.49021E-005

Max

. Ben

ding

Mom

ent (

t.m

)

Y = pow(X,4.55772) * 5.33522E-006

Max

. Ben

ding

Mom

ent (

t.m

)

(A)

(B)

Figure 1

110

0.00 100.00 200.00 300.00

0.00

400000.00

800000.00

Lpp ( m )

WAVE TROUGH

0.00 50000.00 100000.00 150000.00 200000.00

0.00

100000.00

200000.00

Displacement ( t )

STILL WATER

Max

. Ben

ding

Mom

ent (

t.m

)M

ax. B

endi

ng M

omen

t ( t.

m )

Y = pow(X,4.05716) * 9.95739E-005

Y = pow(X,1.3147) * 0.0240116

(A)

(B)

Figure 2

111

0.00 50000.00 100000.00 150000.00 200000.00

0.00

400000.00

800000.00

Displacement ( t )

WAVE CREST

0.00 50000.00 100000.00 150000.00 200000.00

0.00

400000.00

800000.00

Displacement ( t )

WAVE TROUGH

Max

. Ben

ding

Mom

ent (

t.m

)M

ax. B

endi

ng M

omen

t ( t.

m )

Y = pow(X,1.50919) * 0.00872257

(A)

(B)

Y = pow(X,1.36507) * 0.0583981

Figure 3

112

0.00 100.00 200.00 300.00

0.00

2000.00

4000.00

6000.00

Lpp ( m )

STILL WATER

0.00

0.00

4000.00

8000.00

12000.00

WAVE CREST

Max

. She

ar F

orce

( t )

Max

. She

ar F

orce

( t )

Y = pow(X,2.95481) * 0.000308901

Y = pow(X,3.65893) * 1.45228E-005

(A)

113

100.00 200.00 300.00

Lpp ( m )

(B)

Figure 4

0.00 100.00 200.00 300.00

0.00

5000.00

10000.00

Lpp ( m )

WAVE TROUGH

0.00 50000.00 100000.00 150000.00 200000.00

0.00

2000.00

4000.00

6000.00

Displacement ( t )

STILL WATER

Max

. She

ar F

orce

( t )

Max

. She

ar F

orce

( t )

Y = pow(X,2.9761) * 0.000600546

Y = pow(X,0.991395) * 0.032944

(A)

(B)

Figure 5

114

0.00 50000.00 100000.00 150000.00 200000.00

0.00

4000.00

8000.00

12000.00

Displacement ( t )

WAVE CREST

0.00 50000.00 100000.00 150000.00 200000.00

0.00

5000.00

10000.00

Displacement ( t )

WAVE TROUGH

Max

. She

ar F

orce

( t )

Max

. She

ar F

orce

( t )

Y = pow(X,1.21026) * 0.00559035

Y = pow(X,1.00842) * 0.0601052

(A)

(B)

Figure 6

115

0.00 100.00 200.00 300.00

0.00

400000.00

800000.00

Lpp ( m )

Max

. Ben

ding

Mom

ent (

t .m

)

0.00 100.00 200.00 300.00

0.00

400000.00

800000.00

Lpp ( m )

Max

. Ben

ding

Mom

ent (

t .m

)WAVE CREST

- - - - Rules

_______ Present Study

WAVE TROUGH

(A)

(B)

- - - Rules

_____ Present Study

Figure 7

116