Capt. Pham Thanh Tuyen

**Prior to this, all bulk cargo ships across the world, when the Draft Survey be conducted to determine the weight of goods, the method “Quarter Mean Draft” has been used. Over time, with the continuous development of science and technology, today’s ships have a deadweight of up to ten thousand tons, a length of several hundred meters. Through research, I found that traditional methods of “Quarter Mean Draft” is ineffective, generating errors up to several hundred tons. This manuscript refers to the error of traditional methods, and propose new methods of consistent and accuracy, to contribute a scientific measures in the global shipping industry.**

**A. Method “Quarter Mean Draft”**

*1) Definition:*

- Fore Perpendicular- FP: a vertical line passing through the intersection of the loadline with the leading edge of the bow.
- After Perpendicular- AP: a vertical line passing through the axis of rudder.
- Length between Perpendicular- LPP: the distance between the Fore Perpendicular and the After Perpendicular.
- Draft: means the vertical distance from the waterline to the keel of ship (bottom of ship). At each location of ship longitudinally, draft be considered on both sides. However, the draft mentioned below is the average value of the starboard draft and port draft respectively. In other words, the starboard draft and port draft will be the same value if the ship does not skew horizontally. By taking the average value such, calculations in this manuscript can be considered in the state of ship no horizontal tilt.
- Fore Draft -df: the draft at the fore perpendicular.
- After Draft -da: the draft at the after perpendicular.
- Mid Draft – d¤: the draft at the middle of ship (the middle, equidistant the fore perpendicular and after perpendicular, see “Figure 1”).
- Mean Draft -dm: is the average value of the fore draft and the after draft, dm= (df+da)/2 ; in theory dm= d¤, but the actual ship was always sagging or hogging (sheer), so in general dm ≠ d¤.
- Displacement-D: is the weight of the water mass that be occupied by the ship, also the entire weight of the ship.
- TPC (Ton per centimeter): tons of cargo more loaded to increase the average draft 1 cm.
- Equivalent Draft –deqv: Assuming that the ship is balanced bow and stern, and not bent. Then, df= da= dm= d¤, this value is called the Equivalent Draft (deqv). There is a
*Hydrostatic Table*onboard, that is a database of records of the ship, were tabulated between Equivalent Draft deqv and Displacement D. In any state of ship in fact, the Surveyor will read the drafts at the scene, then adjust the values to obtain df, da, d¤. Taking the values of df, da, d¤ to synthesize a single draft of representatives, (known as the Final Mean Draft ) so that it acts as the Equivalent Draft deqv to investigate the Hydrostatic Table to obtain the displacement D, thereby determining the weight of goods, is the basis of formation of the Draft Survey methods with different accuracy refers to the manuscript. From now on, when considering the accuracy of the method is considered the accuracy of*dFinalMean*.

*2) Quarter Mean Draft*

In this method, the *Final Mean Draft *is calculated as follows:

From the formula (A.1) (A.2), (A.3), inferred

Since the average twice in the formula (A.2), (A.3), so *dFinalMean* is different d¤ a quarter of the (d¤ – dm), so it is called the Quarter Mean Draft .

“Figure 2” below will illustrate this more clearly

*3) Proof of method error*

*First way, with visual proof*: According to “A.1.4”, the ship can be considered without horizontal tilting, on the other hand, under “A.1.11”, the accuracy of the*dFinalMean*only fired when the ship get a sheer. So, no loss of generality, from now on we only need to consider the effect of the sheer on*dFinalMean*in the side projection area of the ship (“Figure 1” illustrates the side projection of the ship in a state not bent vertically) . Moreover, can assume the bow and the stern of ship are square and coincides with the fore perpendicular and after perpendicular as “Figure 2” below.

*Figure 2*

Region ABCKD denotes the ship below the waterline. Because the ship is curved along, the midpoint M of the ship’s bottom was sagging down into the K. The problem now becomes flat geometry problem. MK segment called deflection(sagging) Δd of ship.

Thus the MK (is also the deflection Δd) is divided into four equal parts, where NK is a quarter of Δd. Can be said that the draft *dFinalMean* is determined by one-quarter decrease deflection Δd, so *dFinalMean *called Quarter Mean Draft .

In “Figure 2”, *dQuarterMean* to be used as a Equivalent Draft to assume that the ship is not bent along. Thus, the assumed ship’s bottom is line EF passes through point N, we have the equivalent trapezoidal ABEF. At symmetrical half of the ship forward, it is easy to see that measures of equivalent assumption can only be acceptable when the NKP area equals the area of the DPF.

By visually, can immediately see the NKP area smaller than the area of DPF a lot. So error of method has been proven.

2. *Second way, proved by calculation:*

*Figure 3*

Call the ABCKD area is S1. Divide the length of Lpp into three equal parts by the points P, Q as “Figure 3”

According to the formula (A.8), (A.9), the area S1 and area S2 are equal only when d¤= d¤1.

“Figure 3” shows that the T is always above the point K, that means LT always less than LK. In other words, d¤1 is always smaller d¤.

Thus, the error of the method has been shown.

**B. The new methods, fix errors**

*1) Method to add the draft measures*

Let us install additional draft measures at the positions of third of her hull length from the bow and from the stern to be able to read and obtain the drafts df1 and da1 as “Figure 3” shown.

The formula (A.6) is the formula for calculating the original area exactly. Get this area divided by the length of the LPP will be *dFinalMean *as follows:

*Features: This method requires additional design of ship, but the advantage is simple to calculate.

*2) Method of Bending Moment*

*Figure 4*

Currently there have been software calculated Shearing Force and Bending Moment onboard.

Green curve in “Figure 4” denotes *bending moment M* along the longitudinal axis (*horizontal axis l* ) from the *After perpendicular* to the *Fore perpendicular*. The table to the left of the graph contains bending moment data have been calculated in the key frames numbered from zero starting at the *After perpendicular*. The adjacent frames coordinate with the horizontal axis and the moment curve to form the small trapezoids. By composite trapezoidal method, it is easy to calculate definite integrals of the *moment M* on the length of Lpp. In this example, Lpp = 132 (m), I calculated the results of integrals are:

* Features: This method requires Bending Moment data, but owns high precision and no need to add any draft measure.

* “Figure 4” is copied from a software named Loading Wizard 2011 which I’ve written.