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Towline Pull Dynamic Requirements (English Units)

Description: This template computes the dynamic (moment area) towline pull requirements for vessels that are equipped to tow.  It is based on 46CFR173.095(b) through 46CFR173.095(f) of Subchapter S - Subdivision and Stability.  According to 46CFR173.095(a) evaluation is required for several conditions or drafts, over the vessel's range of operation.  Recommend using separate copies of this template for each condition or draft considered.  Minimum conditions to evaluate should include departure, arrival and light ship.  Refer to Reference D for additional details regarding these operating conditions.

As a free bonus this package also includes the static (metacentric height) towline pull requirements for vessels that are equipped to tow.  This free bonus is based on 46CFR173.090(b) of Subchapter S - Subdivision and Stability.  Several conditions or drafts, over the vessel's range of operation, are evaluated with this bonus template.  It is included because the static requirements must be computed first, to obtain input terms like s and P which are needed to compute the dynamic requirements.

According to 46CFR173.095(a) a vessel may comply with either the static or dynamic requirements.  Therefore if a vessel meets the normally more stringent static requirements, the dynamic requirements need not be computed.  However, a vessel will often not meet the static requirements.  So when this occurs, the dynamic requirements will require computation.  Fortunately vessels will often meet the less stringent dynamic requirements.

These calculations apply specifically to vessels equipped with Z-Drives.  This includes tractor tugs, OSVs, workboats and fish boats.  They do not apply to tugs, offshore supply vessels, workboats, fish boats or tractor tugs that are conventionally shafted, or with vertical axis propulsion (Voith-Schneider) tugs, or with paddle wheels.

There are several advantages and benefits of this calculative method.  The first is clear and neat documentation.  For each condition or draft evaluated all the inputs and outputs are identified in one document.  Second, once you get used to it, this approach is fast.  With some quick data inputting the template automatically and quickly generates the vessels "Curves of Statical Stability," determines the angle of the maximum righting arm, determines the vessel's limiting heel angle, computes the residual moment area available between the intercept angle and limiting angle, and then checks all these things to see if they meet the United States Coast Guard requirements.  Third, this approach is cost effective.  One could use Hullform, SHCP (Ship Hull Characteristics Program) or some other inexpensive hydrostatics program to generate input data required for this MathCAD document.  But this template can also be readily used in conjunction with the more expensive hydrostatics programs.  Alternatively and more economically a person, with the appropriate computational skill set, could easily use the "Curves of Form" and "Cross Curves of Stability" to generate the input for this MathCAD document.  So, this template clearly documents, saves time and it is economical.

Cost: $40.00

Electronic Document Types

  • Mathsoft MathCAD Version 6 or later (for dynamic template only)

  • Microsoft Internet Explorer or compatible web browser (for instruction of dynamic template only)

  • Microsoft Excel or compatible spreadsheet (for static template only)

  • zip file extraction utility (comes standard with Windows XP), required to unzip file that contains the dynamic and static templates.  

Vessel Input Data Required

  • Hydrostatic Input Information required:

    • Option One: A computer model of the hull envelope and a simple hydrostatics program are utilized.  With the following hydrostatic program inputs: Displacement, LCG & VCG (recommend setting VCG = 0).  The hydrostatic program should be able to output: VCB, BMT, or KMT (where KMT = VCB + BMT), Forward Draft, Mean (T) Draft, Aft Draft and Righting Arms in ten degree increments starting from zero and ending at 70.  This corresponds to INPUT = 1 in this template. 

    • Option Two: Involves usage of Curves of Form, Cross Curves of Stability and the Lines Drawing.  In the template this corresponds to INPUT=2.  This option usually means a draft with zero trim (level trim) is the selected condition under evaluation.  Otherwise it means the "Curves of Form" values correspond to a LCF draft based on the condition's input displacement.

  • General Arrangement Drawing, or some method of getting scaled configuration information (especially with respect to down flooding locations) about the vessel.

  • Lines Drawing, or some method of getting scaled hull envelope information about the vessel.

Number of Pages: three sheets when printed (for dynamic template)                         


  • Stars * are present next to variables, on this dynamic template, that quickly identify inputs that are required.

  • The following inputs are required for each operating condition or draft (required for dynamic templates only):

    • filename, name of file for these computations, one file per condition or draft evaluated.  (example:

    • VCG, Vertical Center of Gravity, feet (this value depends on TYPE defined next).  This VCG value should already include a correction for free surface effects.  Sometimes this is referred to as the "virtual vcg" for the condition or draft being evaluated.

    • TYPE, Type of Analysis. 

      • TYPE = 1 when a specific VCG is being checked.  This type is usually selected when output from a hydrostatic program is available for a specific vessel condition. 

      • TYPE = 2 when the maximum allowable VCG is being sought.  In a TYPE 2 analysis the VCG value is iterated upward until the vessel no longer satisfies the requirements, then it is changed back to it's last criterion compliant value.  This type of analysis usually applies to a specific operating draft, and is normally based on "Curves of Form" and "Cross Curves of Stability" data.

    • VCG0, Vertical Center of Gravity used in making the "Cross Curves of Stability" calculations, feet above Baseline.  

      • When Type = 1, with output from a hydrostatics program for a specific condition, normally set VCG0 equal to zero.  If the assumed VCG0 is not equal to zero, set it equal to the actual value it was assumed for in the hydrostatics program.

      • When Type = 2, set equal to VCG value assumed when making the "Cross Curves of Stability."

    • qf, Downflooding Angle, degrees, based on the condition or draft being evaluated and the vessel's geometry.  This input is to be as per Part 46CFR173.095(e) which requires using the location, that causes the smallest heel angle, to where the hull does not close automatically.  This is usually at a coaming top at the bottom of a watertight door opening.  This value is further discussed in References D and E.

    • LCG, Longitudinal Center of Gravity, feet, where distances aft of amidships are defined as positive. 

      • For INPUT= 1, this is the LCG value that was used as input for the hydrostatics program generating output for the specific vessel condition.

      • For INPUT= 2, this is usually the level trim case where this value is equal to the LCB value.  Otherwise it is the value obtained from the Weights and Moments analysis for the vessel in the specific vessel condition.

    • INPUT, Input source for "Curves of Statical Stability" data.  This data is not corrected for free surface.  In this analysis the free surface effects are already compensated for through the use of virtual VCG values.

      • INPUT = 1, input data (heel angles with corresponding righting arm values) is from a hydrostatics program output for the specific vessel condition being evaluated.

      • INPUT = 2, input data (heel angles with corresponding righting arm values) are from the vessel's "Cross Curves of Stability" for the specific displacement under evaluation. 

    • qi, Heel Angles, degrees, based on the condition or draft being evaluated and the vessel's geometry.  Usually input in equal increments.  Example values would be increments of 10, starting with zero, then 10, 20, 30, 40, 50, 60 and 70.  Eight values are input to the hydrostatics program or "Cross Curves of Stability."  These eight values are also input into a table at the bottom of the first page of this dynamic analysis template.

    • GZi", Righting Arms, feet, hydrostatic program output or "Cross Curves of Stability" data corresponding to the input heel angles (qi).  Eight values are to be entered into the table at the bottom of the first page of this dynamic analysis template.

  • The following inputs are required for each operating condition or draft (required for both static and dynamic templates):

    • T, Draft, meters

      • When Type = 1, with hydrostatics program output for a specific condition, set equal to the program output's mean draft. 

      • When Type = 2, this usually means a level draft is being evaluated.  Otherwise it is the draft at the LCF.  This option indicates usage of "Curves of Form" and "Cross Curves of Stability."

    • D, Displacement, long tons

    • KMT, Height from Keel to Metacenter, feet, compute this value if required where KMT = VCB + BMT = verical center of buoyancy + metacentric radius.

  • Characteristics of Vessel inputs, irrespective of draft (for both static and dynamic templates):

    • h, Maximum vertical distance from center of propeller shaft to towing bitts, feet

    • D, propeller diameter, feet

    • N, Number of Propellers, non dimensional value

    • Name of Vessel

    • Type of Vessel (i. e. Z-Drive Tug, Vertical Axis Tug, Conventionally Propped Tug, etc.)

    • Length x Beam x Depth of Vessel, feet

    • Name of Firm Operating Vessel

  • Characteristics of Vessel inputs, irrespective of draft (for static template only):

    • B, Moulded Beam, feet

    • d, Depth to top of Freeboard Deck, feet

    • HP, shaft power per shaft, at the engine, hp

    • h, efficiency of Z-Drive between engine and propeller, in range of 0.95 to 0.98 according to McGowen & Meyer paper listed in references below

    • q, Angle between lines shown in Figure F1A (plan view), degrees.  First line is drawn between vertical axes of Z-Drives.  Second line is drawn along the closest slipstream edge of the Z-Drive unit behind the unit facing directly athwartships (transverse).

  • Characteristics required irrespective of vessel (for static and dynamic criteria):

    • K, Formula Constant for English, 38.0, non dimensional value

    • AR, Residual Moment Area required for English units is equal to
      2.00 footdegrees.

  • Outputs from Static Analysis:  

    • f, Minimum Freeboard present, feet, for each draft specified

    • s, Modified Factor that applies specifically to Z-Drive Tugs, non
      dimensional value

    • P, shaft power per shaft, at the propeller, horsepower

    • GMr, Required Metacentric Height to meet static requirements, feet, for conditions or drafts  specified

    • KG, Maximum Allowable VCG (Vertical Center of Gravity) for static requirements, feet, for conditions or drafts specified

  • Outputs from Static Analysis Required as Inputs for Dynamic Analysis:

    • s, Modified Factor that applies specifically applicable to Z-Drive Tugs, non dimensional value

    • P, shaft power per shaft, at the propeller, horsepower

  • Outputs from Dynamic Analysis:  

    • GMa, Available Metacentric Height (corresponding to input VCG) to meet dynamic requirements, feet, for condition or draft specified

    • Curve of Statical Stability (Corrected Righting Arm Curve) with plot for the condition or draft under evaluation.  Corrected for VCG0 when a non-zero value is present.

    • Heeling Arm Curve with plot.

    • qm, Angle of Maximum Righting Arm, degrees.

    • qL, Limiting Angle, the lesser of the downflooding angle, 40 degrees or the angle or maximum righting arm, degrees.

    • qj, Intercept Angle (or Equilibrium Angle) between lines heeling arm and righting arm curves, degrees.

    • A2, Residual Moment Area Available, degreesfeet,  This is the area between the righting arm and heeling arm curves, in the region between the intercept angle to the limiting angle.

    • check, equal to one only if all of the dynamic criteria are met.  (i. e. the intercept angle must be less than the limiting angle (the lesser of 40 degrees, angle of maximum righting arm, or downflooding angle), and the residual moment area available must be more than 2.00 footdegrees)

    • KG, Maximum Allowable VCG (Vertical Center of Gravity) for dynamic requirements, feet, for condition or draft specified.  This applies if the TYPE=2 option is selected, otherwise (for the TYPE=1 option) you will know if the VCG input passes or fails the criterion's requirements.

  • Suggested Reading:

    • Reference A) Subchapter S - Subdivision and Stability, Title 46 Shipping, USCG, Washington, D. C.

    • Reference B) Procedure H1-04, MSC Guidelines for Review of Stability for Uninspected Tugboats (C), dated 3/21/00, USCG, Washington D. C.

    • Reference C) COMDTINST M16000.9, Marine Safety Manual, Volume IV, Sections 6.C.1, 6.E.1b. & 6.E.2, United States Coast Guard, Washington, D. C.

    • Reference D) McGowen, John F., & Meyer, Richard B., Has Stability Delayed the Delivery of Your Tug?, Marine Technology, January 1980, 6 pages, SNAME, Jersey City, N. J..  

    • Reference E) NAVIC 12-83, Intact Stability of Towing and Fishing Vessels, Research Results, dated 15 Nov. 1983, United States Coast Guard, Washington, D. C.

    • Reference F) Principles of Naval Architecture, by SNAME, Jersey City, N. J.

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Minimum System Requirements: Windows 95/98/NT/2000/XP/Vista/Windows7

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