MARN-220 Lecture-2: Calculations of Area, Volume, Moment & Moment of Inertia

Areas

• Calculating areas of curvilinear surfaces and bodies is essential, including ship hulls.
• Trapezoidal Rule, Simpson's First Rule, and Simpson's Second Rule are used for area calculations.

Calculation of Areas using Trapezoidal Rule

• Divide the base into equal parts with length $h$.
• Draw perpendiculars (ordinates) to the base, meeting the curvature ($y<em>1, y</em>2, y_3,…$).
• Area of ABCD = $\frac{1}{2}(AD + BC) \times AB = \frac{1}{2}(y<em>1 + y</em>2)h$
• Area of BEFC = $\frac{1}{2}(y<em>2 + y</em>3)h$
• Area of (ABCD + BEFC) = $\frac{1}{2}h(y<em>1 + 2y</em>2 + y_3)$
• Area of AIJD = $\frac{1}{2}h(y<em>1 + 2y</em>2 + 2y<em>3 + 2y</em>4 + y<em>5) = h(\frac{y</em>1 + y<em>5}{2} + y</em>2 + y<em>3 + y</em>4)$
• Steps:
  1. Divide the base equally into several parts (e.g., $AI = 4 \times h$).
  2. Draw perpendiculars to the base meeting the curvature (e.g., $y<em>1, y</em>2, y_3…$).
  3. Sum the lengths of first and last perpendiculars and divide by 2 (i.e., $\frac{y<em>1 + y</em>5}{2}$).
  4. Add to the sum of the lengths of all other perpendiculars (i.e., $y<em>2 + y</em>3 + y_4$).
  5. Multiply the total sum by the base length (i.e., $h \times (\frac{y<em>1 + y</em>5}{2} + y<em>2 + y</em>3 + y_4)$).
• Example:
  • Half-ordinates: 1.78, 1.77, 1.67, 1.51, 1.25, 0.76, 0.09 m; $h$ = 1.27 m
  • Half-waterplane area: $1.27 \times (\frac{1.78 + 0.09}{2} + 1.77 + 1.67 + 1.51 + 1.25 + 0.76) = 10.03 m^2$
  • Whole waterplane area: $2 \times 10.03 = 20.06 m^2$

Calculation of Areas using Simpson’s First Rule

• Assumes the curved line is a portion of a parabola of the second order, $y = a<em>0 + a</em>1x + a<em>2x^2$, where $a</em>0, a<em>1, a</em>2$ are constants.
• Area of ABCD = $\frac{1}{3}h(y<em>1 + 4y</em>2 + y_3)$
• Area of ABCD = areas of (AGHD + GBCH)
• Combined area: $\frac{1}{3}h(y<em>1 + 4y</em>2 + y<em>3) + \frac{1}{3}h(y</em>3 + 4y<em>4 + y</em>5) = \frac{1}{3}h(y<em>1 + 4y</em>2 + 2y<em>3 + 4y</em>4 + y_5)$
• Steps:
  1. Divide the base equally into an even number of parts (e.g., $AB = 4 \times h$).
  2. Draw perpendiculars to the base meeting the curvature (i.e., $y<em>1, y</em>2, y_3…$).
  3. Sum the lengths of first and last perpendiculars (i.e., $(y<em>1 + y</em>5)$).
  4. Sum four times the lengths of even perpendiculars (i.e., $4y<em>2 + 4y</em>4$).
  5. Sum twice the lengths of odd perpendiculars (i.e., $(2y_3 + …)$).
  6. Multiply the total sum by one-third of the base length (i.e., $\frac{1}{3}h(y<em>1 + y</em>5 + 4y<em>2 + 4y</em>4 + 2y_3)$).
• Simpson’s First Rule Template:
  • Ordinates are multiplied by the following Simpson’s multipliers: 1 4 2 4 2 4 … 2 4 1
• Example:
  • Half-ordinates: 1.78, 1.77, 1.67, 1.51, 1.25, 0.76 and 0.09 m; $h$ = 1.27 m
  • Waterplane area, $A_w = 2 \times (\frac{1}{3}) \times 1.27 \times 23.87 = 20.14 m^2$
  • Functions for area: 1.78, 7.08, 3.34, 6.04, 2.50, 3.04, 0.09; SUM = 23.87
  • $A_w = 2 \times \frac{h}{3} \times \Sigma(y \times SM)$

Calculation of Areas using Simpson’s Second Rule

• Assumes the curved line is a portion of a parabola of the third order, $y = a<em>0 + a</em>1x + a<em>2x^2 + a</em>3x^3$, where $a<em>0, a</em>1, a<em>2, a</em>3$ are constants.
• Area of ABCD = areas of (AIJD + IBCJ)
• Area of ABCD = $\frac{3}{8}h(y<em>1 + 3y</em>2 + 3y<em>3 + y</em>4) + \frac{3}{8}h(y<em>4 + 3y</em>5 + 3y<em>6 + y</em>7) = \frac{3}{8}h(y<em>1 + 3y</em>2 + 3y<em>3 + 2y</em>4 + 3y<em>5 + 3y</em>6 + y_7)$
• Simpson’s Second Rule Template:
  • Ordinates are multiplied by the following Simpson’s multipliers: 1 3 3 2 3 3 2 … 3 3 1
• $Area = \frac{3}{8}h(y<em>1 + 3y</em>2 + 3y<em>3 + 2y</em>4 + 3y<em>5 + 3y</em>6 + 2y<em>7 + \dots + 3y</em>{n-1} + y_n)$
• Example:
  • Half-ordinates: 1.78, 1.77, 1.67, 1.51, 1.25, 0.76 and 0.09 m; $h$ = 1.27 m
  • Waterplane area, $A_w = 2 \times (\frac{3}{8}) \times 1.27 \times 21.24 = 20.23 m^2$
  • Functions for area: 1.78, 5.31, 5.01, 3.02, 3.75, 2.28, 0.09; SUM = 21.24
  • $A_w = 2 \times \frac{3h}{8} \times \Sigma(y \times SM)$

Calculation of Areas using Five-Eight Rule

• Five-Eight rule calculates the curvilinear area between two consecutive ordinates
• Assumes the curved line is a portion of a parabola of the second order, represented by the equation $y = a<em>0 + a</em>1x + a_2x^2$
• Area of ABCD = areas of (AEFD + EBCF)
• Area of ABCD = $\frac{1}{12}h(5y<em>1 + 8y</em>2 - y<em>3) + \frac{1}{12}h(5y</em>3 + 8y<em>2 - y</em>1)$

Calculation of Wetted Surface Areas

• Denny's Formula: $Wetted Surface = 1.7LT + \frac{\nabla}{T}$
• Taylor's Formula: $Wetted Surface = 15.6\sqrt{\Delta L}$
• Kirk's Formula: $Wetted Surface = 2.0LT + \frac{\nabla}{T}$
• Froude's Formula: $Wetted Surface = \frac{\Delta^{2/3}}{3.3 + \frac{L}{2.09\nabla^{1/3}}}$
• Haslar Formula: $Wetted Surface = \frac{\nabla^{2/3}}{36.38 + 1.636L}$
• Where:
  • L = Length between perpendiculars
  • T = Draft up to top of the keel
  • $\nabla$ = Volume displacement
  • $\Delta$ = Weight displacement

Volume

• Simpson’s rules are used to calculate volumes of bodies with curved surfaces, such as ships.
• Requires sectional areas (vertical) or plane areas (horizontal).
• Simpson’s First Rule:
  • $Volume = \frac{1}{3}h(a<em>1 + 4a</em>2 + 2a<em>3 + 4a</em>4 + 2a<em>5 + 4a</em>6 + \dots + 4a<em>{n-1} + a</em>n)$
• Simpson’s Second Rule:
  • $Volume = \frac{3}{8}h(a<em>1 + 3a</em>2 + 3a<em>3 + 2a</em>4 + 3a<em>5 + 3a</em>6 + 2a<em>7 + \dots + 3a</em>{n-1} + a_n)$
• Example (Simpson’s First Rule):
  • Underwater portion areas: 1.09, 5.50, 11.06, 14.62, 17.12, 18.82, 20.21 $m^2$, $h$ = 0.152 m
  • $Volume = (\frac{1}{3}) \times 0.152 \times 233.42 = 11.83 m^3$
  • Functions for volume: 1.09, 22.00, 22.12, 58.48, 34.24, 75.28, 20.21; SUM = 233.42
  • $V = \frac{h}{3} \times \Sigma(a \times SM)$
• Example (Simpson’s Second Rule):
  • Underwater portion areas: 1.09, 5.50, 11.06, 14.62, 17.12, 18.82, 20.21 $m^2$, $h$ = 0.152 m
  • $Volume = (\frac{3}{8}) \times 0.152 \times 208.04 = 11.86 m^3$
  • Functions for volume: 1.09, 16.50, 33.18, 29.24, 51.36, 56.46, 20.21; SUM = 208.04
  • $V = \frac{3h}{8} \times \Sigma(a \times SM)$

Moment

• Moment is the measure of its tendency to cause rotation.

Calculating Moment of Waterplane Area about no.1 Ordinate

• Moment of load-water plane area about no.1 ordinate:
  • Half-ordinates: 1.78, 1.77, 1.67, 1.51, 1.25, 0.76, 0.09 m; $h$ = 1.27 m
  • $Moment = 2 \times \frac{1}{3} \times 1.27 \times 1.27 \times 57.62 \approx 62 m^3$
• Calculating Moment of Waterplane Area about middle Ordinate
  • Half Ordinates, y S M Functions for Area Number of Intervals Products for Moment from no1 Ordinate
  • The moment of load-water plane area about middle ordinate $<br /> = 2 \times \frac{1}{3} \times 1.27 \times 1.27 \times (22.84 − 8.85) \approx 15 m^3$

Calculation of Moment of Area between Two ordinates

• 3 +10 -1 rule is used to calculate the longitudinal moment of a curvilinear area between two consecutive ordinates
• For example: the moment of the area between $y<em>1$ and $y</em>2$ is
• $\frac{1}{24}h^2(3y<em>1 + 10y</em>2 - y_3)$

Moment of Inertia (of a plane area)

• Calculating Moment of Inertia using Simpson’s Rules
• To calculate the moment of inertia of a plane area ABCD about the base AB, we can use either -
  • Moment of Inertia of the strip about AB is $\int \frac{1}{3} . dx y^2 = \frac{1}{3} \int y^3 dx$
  • Moment of Inertia of the whole area about AB is
• Simpson’s First Rule Template
  • $\frac{1}{3} \times \frac{1}{3} h(y<em>1^3 + 4y</em>2^3 + 2y<em>3^3 + 4y</em>4^3 + 2y<em>5^3 + 4y</em>6^3 + \dots + 4y<em>{n-1}^3 + y</em>n^3)$
• Simpson’s Second Rule Template
  • $\frac{1}{3} \times \frac{3}{8} h(y<em>1^3 + 3y</em>2^3 + 3y<em>3^3 + 2y</em>4^3 + 3y<em>5^3 + 3y</em>6^3 + 2y<em>7^3 + \dots + 3y</em>{n-1}^3 + y_n^3)$
  • Moment of inertia about centre line = 2 x (1/3) x (1/3) x 1.27 x 56.57 = 15.97 m4
• Example (Simpson’s First Rule for Moment of Inertia
  • The half-ordinates of the load-water plane of a vessel are 1.78 , 1.77 , 1.67 , 1.51 , 1.25 , 0.76 and 0.09 m respectively. If the ordinates are 1.27 m apart find the moment of inertia of the load-water plane about the centre line.
• Simpson’s Multiplier y3 x SM (i.e. Half-ordinates3 x Simpson’s Multiplier) Σ(y3 x SM) = Why??
• Half Ordinates, y SM Functions for Moment
• Moment of inertia about centre line = 2 x (1/3) x (3/8) x 1.27 x 50.311 = 15.97 m4
• Example: Simpson’s Second Rule for Moment of Inertia
  • The half-ordinates of the load-water plane of a vessel are 1.78 , 1.77 , 1.67 , 1.51 , 1.25 , 0.76 and 0.09 m respectively. If the ordinates are 1.27 m apart find the moment of inertia of the load-water plane about the centre line.
• Simpson’s Multiplier y3 x SM (i.e. Half-ordinates3 x Simpson’s Multiplier) Σ(y3 x SM) = Why??
• Half Ordinates, y S M Functions for Moment

Approximate Formula for Moment of Inertia

• ${I<em>T} = 0.7(C</em>B - \frac{2}{3})L{B^3}$

Comparison of Calculations Methods