Unit Ii.pptx

UNIT II Areas and volumes

Area

COMPUTATION OF AREAS OF IRREGULAR FIGURES

If the boundaries are irregular and curved, the area is determined by ordinates method. This method is suitable for long narrow strip, such as railway, roadway, drainage. Etc.,

From the lengths and their common interval the area may be computed by the following rules

1. The end ordinate rule 2. The mid ordinate rule 3. The average ordinate rule or mean ordinate rule 4. The trapezoidal rule 5. Simpson’s rule

1. End ordinate rule

A = d (O1 + O2 + O3 + O4 + O5 + O6)

2. Mid ordinate rule

A = d (M1 + M2 + M3 + M4+ M5)

3. Average ordinate rule (or) Mean ordinate rule

𝐀=(𝑺𝒖𝒎 𝒐𝒇 𝒂𝒍𝒍 𝒐𝒓𝒅𝒊𝒏𝒂𝒕𝒆𝒔 / 𝑵𝒐 𝒐𝒇 𝒐𝒓𝒅𝒊𝒏𝒂𝒕𝒆𝒔 ) x 𝑻𝒐𝒕𝒂𝒍 𝒍𝒆𝒏𝒈𝒕𝒉 𝒐𝒇 𝒃𝒂𝒔𝒆 𝒍𝒊𝒏𝒆 𝐀=(𝐎𝟏+𝐎𝟐+𝐎𝟑+𝐎𝟒+𝐎5)𝟓 x 𝑫 Where, O1, O2, O3….. = The ordinates of each division D = Total distance of base line

4. Trapezoidal rule

𝐀𝐫𝐞𝐚,𝐀= 𝒅/𝟐 ( (𝑶𝟏+𝑶𝟕 )+ 𝟐(𝑶𝟐+𝑶𝟑+𝑶𝟒+𝑶𝟓+𝑶𝟔))

5 SIMPSON’S RULE

𝐀= 𝒅/ 𝟑 ( 𝑶𝟏+𝑶𝟕)+𝟐 (𝑶𝟑+𝑶𝟓 )+ 𝟒 (𝑶𝟐+𝑶𝟒+𝑶𝟔)

Computation of volumes of irregular solids 1. End area rule 2. Mid area rule 3. Average area or Mean area rule 4. Trapezoidal rule 5. Simpson’s or Prismoidal rule.

1. End area rule End area rule • This is one of the approximate methods. Volume V = (Sum of all areas of cross section except last one) x Common interval V = d (A1 + A2 + A3 + ……….. A n-1)

2. Mid area rule Volume V = Common interval X (Sum of all mid sectional area) V = d (Am1 + Am2 + Am3 + ……….. A m-1)

3. Average area or Mean area rule Volume V = mean cross sectional area x Total base length V =(𝑨𝟏+𝑨𝟐+𝑨𝟑+⋯…𝑨𝒏)/ 𝐧 𝑿 𝑳 n = number of cross sections L = Total length

4. Trapezoidal rule • The trapezoidal rule gives correct volume of a solid. It gives fairly good result. Volume 𝐕𝐨𝐥𝐮𝐦𝐞 𝐕= 𝒄𝒐𝒎𝒎𝒐𝒏 𝒊𝒏𝒕𝒆𝒓𝒗𝒂𝒍 / 𝟐 ( 𝑺𝒖𝒎 𝒐𝒇 𝒂𝒓𝒆𝒂𝒔 𝒊𝒏 𝒇𝒊𝒓𝒔𝒕 𝒂𝒏𝒅 𝒍𝒂𝒔𝒕 𝒔𝒆𝒄𝒕𝒊𝒐𝒏 + 𝟐 𝑺𝒖𝒎 𝒐𝒇 𝒂𝒓𝒆𝒂𝒔 𝒐𝒇 𝒂𝒍𝒍 𝒐𝒕𝒉𝒆𝒓 𝒔𝒆𝒄𝒕𝒊𝒐𝒏𝒔 𝐕= 𝒅/𝟐 ((𝑨𝟏+𝑨𝒏)+ 𝟐 𝑨𝟐+𝑨𝟑+𝑨𝟒……..𝑨𝒏−𝟏 )

5. Prismoidal rule 𝐕= 𝒄𝒐𝒎𝒎𝒐𝒏 𝒊𝒏𝒕𝒆𝒓𝒗𝒂𝒍/𝟑 ( 𝑺𝒖𝒎 𝒐𝒇 𝒇𝒊𝒓𝒔𝒕 𝒂𝒏𝒅 𝒍𝒂𝒔𝒕 𝒔𝒆𝒄𝒕𝒊𝒐𝒏 𝒂𝒓𝒆𝒂 + 𝟐 (𝑺𝒖𝒎 𝒐𝒇 𝒂𝒍𝒍 𝒐𝒅𝒅 𝒂𝒓𝒆𝒂) + 𝟒 (𝑺𝒖𝒎 𝒐𝒇 𝒂𝒍𝒍 𝒆𝒗𝒆𝒏 𝒂𝒓𝒆𝒂 ) ) 𝐕= 𝒅/𝟑 ((𝑨𝟏+𝑨𝒏)+ 𝟐( 𝑨𝟑+𝑨𝟓+ ……. )+ 𝟒 ( 𝑨𝟐+𝑨𝟒+𝑨𝟔+⋯))

EMBANKMENTS AND CUTTINGS The method of the computation of the cross sectional area will depend upon the type of cross section. The following types of cross sections are generally in use 1. Level section 2. Two level section 3. Sidehill two-level section 4. Three level section 5. Multi-level section

1. Level section

Area of cross section = 𝑇𝑜𝑝 𝑤𝑖𝑑𝑡ℎ+𝐵𝑜𝑡𝑡𝑜𝑚 𝑤𝑖𝑑𝑡ℎ /2 𝑥 ℎ𝑒𝑖𝑔ℎ𝑡 Area of cross section A = (2𝑊+𝑏)2 X h = 𝑏+2𝑆ℎ+𝑏2 X h A = 2(𝑏+𝑆ℎ)2 X h A = (b + Sh) h

2. Two level section

3. Side hill two level section

Sectional area of cutting portion = 𝟏 𝟐 𝒓−𝒔𝟏 𝒃 𝟐+𝒓𝒉 2 Sectional area of filling portion = 𝟏 𝟐 𝒓−𝒔𝟏 𝒃 𝟐−𝒓𝒉 2

4. Three level section

A = h/2 (W1 + W2 + b) + b/4 (w1/ r1 + w2/ r2)