# Calculations and assumptions ### Building Elements The following building elements have been considered for structural calculations for your specific designs.
Building ElementOptions
Frame Material
  • Steel
  • Concrete
  • Timber
Floor Type
  • Composite beams with steel decking
  • Cross laminated timber - reinforced concrete composite slab (for steel)
  • Precast reinforced concrete units
  • One-way reinforced concrete slab
  • Two-way reinforced concrete Slab
  • One-Way cross-laminated timber slab
  • Two-Way cross-laminated timber slab
Foundation Types
  • Pad shallow foundation
  • Strip shallow foundation
  • Raft shallow foundation
  • Deep foundation
Lateral Loading Systems
  • Braced frames
  • Sheer walls
  • Concrete cores
### User Inputs #### Building Parameters * X dimension: Width of the building * Y dimension: depth of the building * Floor to Ceiling height : height from top of floor to bottom of the ceiling * Stories: number of floors * Occupancy: type of expected occupancy (commercial or residential) * imposedLoad (IL): expected live load per square meter * Superimposed deadLoad (DL): expected dead load from external/internal materials * Bearing Capacity (qₙᵤ): Bearing Capacity of the soil * Minimum Span: the minimum beam span to be considered * Maximum Span: the maximum beam span to be considered #### Emission Parameters * Steel GWP: GWP of steel in kgCO2e/kg, set to average of 1.59 * Concrete GWP: GWP of concrete in kgCO2e/m3, set to average 300 * Rebar GWP: GWP of rebar in kgCO2e/kg, set to average 1.08 * Timber GWP: GWP of timber in kgCO2e/m3, set to average 150 #### Cost Parameters * Steel Material Cost: in $/kg, set to average 1.43 * Steel Labour Cost: in $/kg, set to average 0.63 * Concrete Material Cost: in $/m3, set to average 123.54 * Concrete Labour Cost: in $/m3, set to average 298.97 * Timber Material Cost: in $/m3, set to average 747.51 * Timber Labour Cost: in $/m3, set to average 89.71 * Rebar cost: in $/kg, set to average 1.21 * Raft Foundation Cost: in $/m3, set to average 343.89 * Strip Foundation Cost: in $/m3, set to average 396.26 * Pad Foundation Cost: in $/m3, set to average 134.58 * Composite Floor Material Cost: in $/m3, set to average 62.58 * Composite Floor Labour Cost: in $/m3, set to average 42.92 ### Assumptions #### Material Types * All concrete is C30/37 has 2% reinforcements * Structural steel grade: S355 * Reinforcement steel grade: B500 #### Lateral Loading System compatibility The lateral loading systems have specified floor limits indicated below. Timber building are limited to 18 floors at this time. | | Steel | Concrete | Timber | | -------------- | ----- | -------- | ------ | | Braced frames | 1-20 | n/a | 1-4 | | RC shear walls | 4-20 | 2-20 | 4-18 | | RC Cores | 4-70 | 2-70 | 4-18 | #### Span, floor system, and material compatibility Timber | Floor Type | Maximum span (m) | Floor system load | | ---------------- | ---------------- | ---------------------- | | One-way CLT slab | 8 | Longest grid dimension | | Two-way CLT slab | 8 | Both directions | Concrete | Floor Type | Maximum Span (m) | Floor System Load Direction | | ---------------- | ---------------- | --------------------------- | | Precast RC units | 10 | Both directions | | One-way RC slab | 12 | Longest grid dimension | | Two-way RC slab | 12 | Both directions | Steel | Floor Type | Maximum Span (m) | Floor system load direction | | ---------------------------------- | ---------------- | --------------------------- | | Composite beams with steel decking | 18 | Shortest grid dimension | | CLT-RC composite slab | 8 | Shortest grid dimension | ### Variables * X Dimension (Lₓ) * Y Dimension (Lᵧ) * X Real Dimension (Lₓᵣ) * Y Real Dimension (Lᵧᵣ) * Span in X (Sₓ) * Span in Y (Sᵧ) * Number of bays in X (Bₓ) * Number of bays in Y (Bᵧ) * Area per floor (A) * Imposed Load (IL) * Dead Load (DL) * Stories (Nₓ) * Concrete Density (cρ) * Steel Density (sρ) * Timber Density (tρ) * Slab Depth (SD) * Number of Secondary Beams (Nₛ) * Number of columns (N꜀) * Beam Length (BL) * Primary Beam Length (BLₚ) * Secondary Beam Length (BLₛ) * Beam Width (W) * Beam Depth (D) * Beam face surface Area (Bₐ) * Slab Load (SL) * Load on Secondary Beam (UDLₛ) * Load on Primary Beam (UDLₚ) * Minimum moment of Inertia (Mₘ) * Beam Face Surface Area (BSA) * Minimum inertia (Iₘ) * Number of Lateral Loading Systems (Nₗₗₛ) * Floor to floor height (Fₕ) * Axial Load (NEd) * Foundation Area (Fₐ) * Founation Depth (Fd) * Soild Bearing Capacity (qₙᵤ) * Number of lateral loading systems in Y (Nₗₗₛᵧ) * Number of lateral loading systems in X (Nₗₗₛₓ) ### Structure Calculations #### General The number of bays in each direction is calculated as the rounded division (up) between the total dimension (Sₓ , Sᵧ), and the length of the bays. $$ B\_x = S\_x \cdot L\_x $$ $$ B\_y = S\_y \cdot L\_y $$ Due to the nature of the bays being an integer, the dimensions are the adjusted to account for the error between the target and real dimension: $$ L\_{xr} = B\_x \cdot S\_x $$ $$ L\_{yr} = B\_y \cdot S\_y $$ The area per floor is then calculated using the adjusted dimensions: $$ A = L\_{xr} \cdot L\_{yr} $$ #### Slabs For floor types ‘composite beams with steel decking’ and ‘cellular composite beams with steel decking’ the slab depth is 130mm
To determine the slab depth of concrete units, formulas [ACI CODE-318-19(22): Building Code Requirements for Structural Concrete and Commentary](https://www.concrete.org/store/productdetail.aspx?ItemID=318U19\&Language=English\&Units=US_Units) have been referenced. Precast RC units and two-way RC slabs: $$ SD = \frac{2S\_x+2S\_y}{180} $$ One-way RC slabs: $$ SD = \frac{min(S\_x,S\_y)}{40} $$ For timber slabs, the span is referenced against the a data table to determine the slab depth. Data taken from [*Structural Timber Elements A pre-scheme design guide*](https://www.istructe.org/sitefiles/handlers/DownloadFile.ashx?productId=889). In the case of CLT-RC composite slabs, a 50mm concrete surface is considered. The total load from the slab is calculated with a safety factor of 35% for the total dead load and 50% for the total imposed load: $$ SL=1.35(DL)+1.5(IL) $$ $$ SL=1.35(c\rho \cdot SD+DL)+1.5(IL) $$ #### Steel Beams In the case of composite slabs with steel decking, additional secondary beams are needed to support the slab load. Before calculating the beam sizes, the number of secondary beams is determined. $$ N\_s=\frac{BL\_p}{3} $$ The specific load is calculated on both primary and secondary beams. Load transferred to the secondary beams acting as one-way slab. Load transferred to the primary beams acting as point loads from the secondary beams. Loads from all adjacent slabs considered: $$ UDL\_s=SL \cdot \frac{BL\_p}{N\_s} $$ $$ UDL\_p=SL \cdot BL\_s $$ In the case of steel structures, the minimum moment and minimum inertia needed are calculated for each beam using the UDL: $$ M\_m = \frac{UDL \cdot L^2}{8} $$ $$ I\_m=\frac{5UDL \cdot L^4}{384 \cdot E \cdot \delta\_{max}} $$ Where 𝛿max represents the maximum displacement in beams for serviceability limit state and has been set to 𝛿𝑚𝑎𝑥 = L/250, and E is the module of elasticity set to 210000 MPa. The two values are checked against an IPE profile database. The smallest section that verifies both Iₘ and Mₘ is selected for both primary and secondary beams. #### Concrete Beams All concrete beams are assumed to have a width of 300mm. The depth of the beams is calculated: $$ D = \frac{BL}{12} $$ #### Timber Beams The slab load is transferred to the beams and a reference table is used to determine the beam sized. In the case of one-way CLT slabs the following is used: $$ UDL\_p=SL \cdot BL\_p $$ And for two-way CLT slabs: $$ UDL\_p=SL \cdot \frac{BL\_s}{2} \cdot (BL\_p - \frac{BL\_s}{2}) $$ $$ UDL\_s=SL \cdot (\frac{BL\_s}{2})^2 $$ #### Columns To determine the column size, the axial load needs to be calculated. Formulas adapted from [Eurocode 2 (2020)](https://www.slideshare.net/imyusyil/economic-concrete-frame-elements-to-eurocode-2#20). The calculation considers the total dead and live load for each floor. A 35% safety factor is added to the dead load (DL) and 50% to the live load (IL). Prior to calculating the load, the floor to floor height and number of columns is calculated.
A raised floor has been assumed in all the configurations with target depths of 150mm, and 150mm allocated for ceiling and lighting: $$ F\_h=Raised Floor + Ceiling/Lighting + SD + W\_p+W\_s $$ The number of columns is calculated as follows: $$ N\_c = (B\_x + 1)(B\_y + 1) $$ To determine the size of the column, the axial load is calculated as follows: NEd = {Slab Self Weight+ Beams Self Weight + Columns Self Weight +DL}DL +{IL}ILn $$ N\_{Ed}=1.35 \sum \[(\rho \cdot L\_{xr} \cdot L\_{xr}) + (\rho \cdot BL\_p \cdot B\_a) (B\_x+1)+(\rho \cdot BL\_s \cdot B\_a) (B\_y+1)+(\rho \cdot F\_h \cdot 0.45^2) + (DL \cdot A)] + 1.5\sum \[(IL \cdot A)(a\_n)] $$ A reduction factor (aₙ) is also introduced and calculated as such based on Eurocodes 2: $$ a\_n= \begin{cases} {1.1-\frac{N\_s}{10}},& \text{if } N\_x\leq 5\\ 0.6, & \text{if } 5< N\_x \leq 10\ 0.5 & \text{if } N\_x > 10 \end{cases} $$ #### Foundation Once the above-ground structure is calculated, the foundation area can be determined using the total axial load and the soil bearing capacity: $$ F\_a = \frac{N\_{Ed}}{q\_{nu}} $$ The foundation depth is dependent on the foundation type:
| Pad Foundation | Strip Foundation | Raft Foundation | Deep Foundation | | --------------- | ---------------- | --------------- | --------------- | | Fd = sqrt(Af)/2 | Fd = 300mm | Fd = 300mm | Fd = 4.5m | #### Loading System Once the choice of the stability system is made, determining the number of bays requiring reinforcement involves estimating based on height-to-width ratios. For concrete cores and shear walls, a recommended ratio of 4:1, as suggested by Norman et al. (2020), is utilized. Additionally, a target depth of 200 mm has been established for these structural walls. In the case of braced frames, a ratio of 8:1 is applied, following the reference from Narayanan (2018). It's noteworthy that the preliminary bracing size is initially set to 1/4 of the column size, assuming an X-bracing pattern. Location of the lateral loading systems is not considered as it does not significantly impact the cost or carbon. $$ N\_{llsx}=\frac{N\_x \cdot F\_h/ r}{S\_x} $$ $$ N\_{llsy}=\frac{N\_x \cdot F\_h/ r}{S\_y} $$ ### Cost and Embodied Carbon The overall cost and the embodied carbon of the project are calculated using the weights or quantities of each construction material.