Civil Engineering projects require precise quantification of materials and work done to ensure accurate cost estimation and efficient resource allocation. The process of abstracting measured quantities translates detailed measurement data from site drawings or bill of quantities into summarized forms for costing and project management. This chapter focuses on the mathematical methods used to prepare abstract sheets, organize measurements, apply standard measurement methods like SMM and CESMM, and review the accuracy of these abstractions in the Kenyan civil engineering context.

3.1 Preparation of Abstracting Sheet

Abstracting sheets consolidate quantities measured from site work or drawings into a format suitable for costing and tendering. The preparation involves selecting the correct format, organizing data logically, applying standard measurement rules, and verifying the accuracy of the abstracted quantities. Mastery of these processes ensures reliable cost estimates and reduces financial risks in construction projects.

3.1.1 Abstracting Formats

Abstracting formats define how quantities are summarized from detailed measurements. Common formats include tabular, columnar, and summary abstracts, each suited to different types of works and measurement complexity.

The general formula for total quantity abstracted is

$$ Q_{total} = \sum_{i=1}^{n} Q_i $$

where \(Q_i\) are individual measured quantities.

Worked Examples

Example 1: A contractor measures 5 beams each having a cross-sectional area of 0.15 m² and length 4 m. Find the total volume of concrete needed.

Given: Number of beams \(n=5\), Cross-sectional area \(A=0.15\, m^2\), Length \(L=4\, m\)

$$ Q_{total} = n \times A \times L $$

$$ Q_{total} = 5 \times 0.15 \times 4 $$

$$ Q_{total} = 3.0\, m^3 $$

Answer: 3.0 m³

Example 2: A wall is 12 m long, 3 m high, and 0.2 m thick. Calculate the total volume.

Given: Length \(L=12\, m\), Height \(H=3\, m\), Thickness \(T=0.2\, m\)

$$ Q_{total} = L \times H \times T $$

$$ Q_{total} = 12 \times 3 \times 0.2 $$

$$ Q_{total} = 7.2\, m^3 $$

Answer: 7.2 m³

Example 3: A slab of dimensions 8 m by 6 m and thickness 0.15 m requires concrete. Calculate the volume.

Given: Length \(L=8\, m\), Width \(W=6\, m\), Thickness \(T=0.15\, m\)

$$ Q_{total} = L \times W \times T $$

$$ Q_{total} = 8 \times 6 \times 0.15 $$

$$ Q_{total} = 7.2\, m^3 $$

Answer: 7.2 m³

Example 4: A contractor measures 10 columns each with a cross-sectional area of 0.25 m² and height 3.5 m. Find total concrete volume.

Given: Number of columns \(n=10\), Cross-sectional area \(A=0.25\, m^2\), Height \(H=3.5\, m\)

$$ Q_{total} = n \times A \times H $$

$$ Q_{total} = 10 \times 0.25 \times 3.5 $$

$$ Q_{total} = 8.75\, m^3 $$

Answer: 8.75 m³

Example 5: Calculate the volume of a circular manhole chamber with diameter 1.2 m and height 2.0 m.

Given: Diameter \(d=1.2\, m\), Height \(H=2.0\, m\), Radius \(r = \frac{d}{2} = 0.6\, m\)

$$ Q_{total} = \pi r^2 H $$

$$ Q_{total} = 3.142 \times 0.6^2 \times 2.0 $$

$$ Q_{total} = 3.142 \times 0.36 \times 2.0 $$

$$ Q_{total} = 2.26\, m^3 $$

Answer: 2.26 m³

3.1.2 Organizing Measurements for Clarity

Organizing measurements clearly involves grouping related quantities, using consistent units, and maintaining logical flow to facilitate accurate abstracting and cost analysis.

The total quantity for a group is the sum of individual quantities:

$$ Q_{group} = \sum_{i=1}^{m} Q_i $$

where \(Q_i\) are individual quantities in the group.

Worked Examples

Example 1: A contractor measures 3 slabs with volumes 7.2 m³, 6.5 m³, and 8.0 m³. Find the total concrete volume.

Given: \(Q_1=7.2\, m^3\), \(Q_2=6.5\, m^3\), \(Q_3=8.0\, m^3\)

$$ Q_{group} = Q_1 + Q_2 + Q_3 $$

$$ Q_{group} = 7.2 + 6.5 + 8.0 $$

$$ Q_{group} = 21.7\, m^3 $$

Answer: 21.7 m³

Example 2: Organize and sum measurements for 4 beams with volumes 1.2 m³, 1.5 m³, 1.3 m³, and 1.4 m³.

Given: \(Q_1=1.2\, m^3\), \(Q_2=1.5\, m^3\), \(Q_3=1.3\, m^3\), \(Q_4=1.4\, m^3\)

$$ Q_{group} = 1.2 + 1.5 + 1.3 + 1.4 $$

$$ Q_{group} = 5.4\, m^3 $$

Answer: 5.4 m³

Example 3: A wall has 3 sections with areas 15 m², 20 m², and 25 m². Calculate total area.

Given: \(A_1=15\, m^2\), \(A_2=20\, m^2\), \(A_3=25\, m^2\)

$$ A_{total} = A_1 + A_2 + A_3 $$

$$ A_{total} = 15 + 20 + 25 $$

$$ A_{total} = 60\, m^2 $$

Answer: 60 m²

Example 4: Group and sum excavation volumes of 4 pits: 3.5 m³, 4.0 m³, 5.5 m³, and 2.0 m³.

Given: \(Q_1=3.5\, m^3\), \(Q_2=4.0\, m^3\), \(Q_3=5.5\, m^3\), \(Q_4=2.0\, m^3\)

$$ Q_{group} = 3.5 + 4.0 + 5.5 + 2.0 $$

$$ Q_{group} = 15.0\, m^3 $$

Answer: 15.0 m³

Example 5: Calculate total length of 5 pipes measured as 12 m, 15 m, 10 m, 8 m, and 20 m.

Given: \(L_1=12\, m\), \(L_2=15\, m\), \(L_3=10\, m\), \(L_4=8\, m\), \(L_5=20\, m\)

$$ L_{total} = 12 + 15 + 10 + 8 + 20 $$

$$ L_{total} = 65\, m $$

Answer: 65 m

3.1.3 Applying SMM/CESMM Standards

The Standard Method of Measurement (SMM) and Civil Engineering Standard Method of Measurement (CESMM) provide standardized rules for measuring and abstracting quantities. Their application ensures uniformity and clarity in quantity take-off.

The total quantity follows the prescribed measurement rules:

$$ Q_{standard} = \sum_{i=1}^{n} Q_i^{(standard)} $$

where \(Q_i^{(standard)}\) are quantities measured according to SMM/CESMM rules.

Worked Examples

Example 1: Using SMM, measure the volume of a concrete slab 10 m long, 5 m wide, 0.2 m thick.

Given: Length \(L=10\, m\), Width \(W=5\, m\), Thickness \(T=0.2\, m\)

$$ Q = L \times W \times T $$

$$ Q = 10 \times 5 \times 0.2 $$

$$ Q = 10\, m^3 $$

Answer: 10 m³

Example 2: According to CESMM, the excavation volume for a trench 50 m long, 1.5 m wide, and 1.2 m deep is measured.

Given: Length \(L=50\, m\), Width \(W=1.5\, m\), Depth \(D=1.2\, m\)

$$ Q = L \times W \times D $$

$$ Q = 50 \times 1.5 \times 1.2 $$

$$ Q = 90\, m^3 $$

Answer: 90 m³

Example 3: Apply SMM to calculate the area of wall plastering for a wall 12 m long and 3 m high.

Given: Length \(L=12\, m\), Height \(H=3\, m\)

$$ A = L \times H $$

$$ A = 12 \times 3 $$

$$ A = 36\, m^2 $$

Answer: 36 m²

Example 4: Using CESMM, calculate the length of road resurfacing for a 500 m stretch with width 6 m.

Given: Length \(L=500\, m\), Width \(W=6\, m\)

$$ A = L \times W $$

$$ A = 500 \times 6 $$

$$ A = 3000\, m^2 $$

Answer: 3000 m²

Example 5: Measure the volume of concrete for a circular column according to SMM with diameter 0.5 m and height 3 m.

Given: Diameter \(d=0.5\, m\), Radius \(r=0.25\, m\), Height \(H=3\, m\)

$$ Q = \pi r^2 H $$

$$ Q = 3.142 \times 0.25^2 \times 3 $$

$$ Q = 3.142 \times 0.0625 \times 3 $$

$$ Q = 0.589\, m^3 $$

Answer: 0.589 m³

3.1.4 Reviewing Abstracting Accuracy

Accuracy review involves verifying calculations and cross-checking measurements to avoid errors that cause cost overruns or delays.

The accuracy percentage is calculated as:

$$ Accuracy = (1 - \frac{|Q_{measured} - Q_{checked}|}{Q_{checked}}) \times 100\% $$

Worked Examples

Example 1: Abstracted quantity is 100 m³; checked quantity is 98 m³. Calculate accuracy.

Given: \(Q_{measured} = 100\, m^3\), \(Q_{checked} = 98\, m^3\)

$$ Accuracy = (1 - \frac{|100 - 98|}{98}) \times 100 $$

$$ Accuracy = (1 - \frac{2}{98}) \times 100 $$

$$ Accuracy = (1 - 0.0204) \times 100 = 97.96\% $$

Answer: 97.96% accuracy

Example 2: Abstracted quantity is 250 m²; checked quantity is 260 m².

Given: \(Q_{measured} = 250\, m^2\), \(Q_{checked} = 260\, m^2\)

$$ Accuracy = (1 - \frac{|250 - 260|}{260}) \times 100 $$

$$ Accuracy = (1 - \frac{10}{260}) \times 100 $$

$$ Accuracy = (1 - 0.0385) \times 100 = 96.15\% $$

Answer: 96.15% accuracy

Example 3: Quantity measured is 75 m length; checked length is 80 m.

Given: \(Q_{measured} = 75\, m\), \(Q_{checked} = 80\, m\)

$$ Accuracy = (1 - \frac{|75 - 80|}{80}) \times 100 $$

$$ Accuracy = (1 - \frac{5}{80}) \times 100 $$

$$ Accuracy = (1 - 0.0625) \times 100 = 93.75\% $$

Answer: 93.75% accuracy

Example 4: Abstracted volume is 150 m³; checked volume is 148 m³.

Given: \(Q_{measured} = 150\, m^3\), \(Q_{checked} = 148\, m^3\)

$$ Accuracy = (1 - \frac{|150 - 148|}{148}) \times 100 $$

$$ Accuracy = (1 - \frac{2}{148}) \times 100 $$

$$ Accuracy = (1 - 0.0135) \times 100 = 98.65\% $$

Answer: 98.65% accuracy

Example 5: Quantity measured is 60 m²; checked quantity is 58 m².

Given: \(Q_{measured} = 60\, m^2\), \(Q_{checked} = 58\, m^2\)

$$ Accuracy = (1 - \frac{|60 - 58|}{58}) \times 100 $$

$$ Accuracy = (1 - \frac{2}{58}) \times 100 $$

$$ Accuracy = (1 - 0.0345) \times 100 = 96.55\% $$

Answer: 96.55% accuracy

Chapter Summary

This chapter focused on the preparation of the abstracting sheet, highlighting the importance of choosing appropriate abstracting formats and organizing measurements clearly to facilitate accurate quantity takeoff. It emphasized applying SMM and CESMM standards to maintain consistency and reviewing the abstracting sheet to ensure precision. The process of transferring descriptions from booked items to the abstracting sheet was covered, stressing the need for accuracy, consistency, and compliance with project specifications and guidelines. Calculating and documenting squared quantities accurately using standard formats was explained, including methods to double-check calculations for reliability. Techniques for calculating net quantities were introduced, with attention to SMM and CESMM methods and adjustments required due to project variations. Finally, the chapter discussed running through all dimensions to identify and correct errors, ensuring that measurements are precise and correspond correctly with the project drawings.