Calculations and assumptions

Building Elements

The following building elements have been considered for structural calculations for your specific designs.

Building Element	Options
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 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. 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). 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.