# Fundamental Calculation
The fundamental principle for calculating embodied carbon is to scale the physical quantity of structural material by the carbon factor of a given set of life cycle modules.
$Embodied \: Carbon = physical \: quantity ∗ carbon \: factor$
Physical quantities of mass or volume are typically used, in both metric or imperial units:
$EC_{A1-A3} [kgCO_2 e]=mass [kg] * CF_{A1-A3} \: [(kgCO_2 e)/kg]$
$EC_{A1-A3} [kgCO_2 e]=volume [yd^3] * CF_{A1-A3} \: [(kgCO_2 e)/yd^3]$
Carbon factors are developed by performing detailed life cycle assessments (LCAs) (see [[How to Perform an LCA]] for further details). These product-level LCAs are typically reported in [[Environmental Product Declarations]] (EPDs).
# Example
It's best to see this through an example. Consider the following quantity takeoff for a structural system for a 6-storey, 160,000sf composite steel frame with drilled pier foundations, and a reinforced concrete shear wall lateral system.
| Component | Material | Quantity | Units |
| -------------- | -------------------- | -------- | -------------- |
| Foundation | 4000 psi Concrete | 896 | $yd^3$ |
| | Rebar | 145 | $short \: ton$ |
| Lateral System | 4000 psi Concrete | 166 | $yd^3$ |
| | Rebar | 449 | $short \: ton$ |
| Floor | 3000 psi LW Concrete | 1782 | $yd^3$ |
| | Steel Deck | 190 | $short \: ton$ |
| Beam | Hot Rolled Steel | 1923 | $short \: ton$ |
| Column | Hot Rolled Steel | 198 | $short \: ton$ |
To calculate the A1-A3, “cradle-to-gate” embodied carbon, we must identify carbon factors for each material. See [[Life cycle stages and modules]] for details about A1-A3.
Let’s just consider the rebar for a sample calculation. From the QTO, the total rebar quantity can be summed:
Foundation rebar: 145 short tons
Lateral system rebar: 449 short tons
Total qty of rebar: 594 short ton
Now, the carbon factor for rebar for life cycle modules A1-A3 needs to be determined.
We use data from [[Environmental Product Declarations]]. For a "[Steel Reinforcement Bar](https://www.crsi.org/wp-content/uploads/CRSI_Industry-Wide_EPD_Sep2022.pdf)" industry average value published by the Concrete Reinforcing Steel Institute, we get:
$CF_{A1-A3} = 854 kgCO_2 e /metric \: tonne$
But the units of our QTO is in short tons. So, a unit conversion is needed.
$CF_{A1-A3} = 774 kgCO_2 e /short ton$
With the material quantity and the carbon factor, we can calculate the A1-A3 embodied carbon:
$EC_{rebar,A1-A3}=quantity \: [short \: ton] * CF \: [\frac{kgCO_2 e}{short \: ton}]$
$EC_{rebar,A1-A3}=594*774$
$EC_{rebar,A1-A3}=459,756 kgCO_2 e$
We then repeat this for each material identified as part of the quantity take-off.
| Component | Material | Quantity | Units | $CF_{A1-A3}$($kgCO_2 e/unit$) | $EC_{A1-A3}$ ($kgCO_2 e$) |
| -------------- | -------------------- | -------- | -------------- | ----------------------------- | ------------------------- |
| Foundation | 4000 psi Concrete | 896 | $yd^3$ | 251 | 224,896 |
| | Rebar | 145 | $short \: ton$ | 774 | 112,230 |
| Lateral System | 4000 psi Concrete | 166 | $yd^3$ | 251 | 41,666 |
| | Rebar | 449 | $short \: ton$ | 774 | 347,526 |
| Floor | 3000 psi LW Concrete | 1782 | $yd^3$ | 367 | 654,994 |
| | Steel Deck | 190 | $short \: ton$ | 2105 | 399,950 |
| Beam | Hot Rolled Steel | 1923 | $short \: ton$ | 1107 | 2,128,761 |
| Column | Hot Rolled Steel | 198 | $short \: ton$ | 1107 | 219,186 |
| | | | | Total: | 4,129 tonnes |
We have computed the total embodied carbon of the structural system from the QTO to be 4,129 tonnes of $CO_2 e$. This value can be normalized by the floor area of the building, resulting in $278 kgCO_2 e / m^2$.
These results can be broken down by material, building system, or lifecycle stage/module to provide additional insights to ultimately make reductions. This breakdown is called a 'contribution analysis'. Below is a contribution analysis separated by material for the example.
![[Contribution Analysis.png]]
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