Enter Dome and Load Data
Use metres, kilopascals, and kilonewtons. The model assumes an externally loaded circular spherical-cap dome with equally shared supports.
Formula Used
a = D / 2The base radius a comes from base diameter D.
R = (a² + h²) / (2h)R is the radius of the parent sphere. h is actual dome height.
Aₛ = 2πRhAₛ is the curved surface area of the spherical cap.
Aₚ = πa²Aₚ is the horizontal projected base area.
Fᵥ = pAₚFᵥ is the idealised vertical pressure resultant. p is shell pressure.
W = qₙAₛ, Fᵈ = (Fᵥ + W)γW is surface dead-load force. γ is the entered design factor.
Pressure in kPa is numerically equal to kN/m². The calculator uses that unit relationship directly.
How to Use This Calculator
- Measure the circular base diameter and the crown height.
- Enter the maximum height allowed by your project requirement.
- Enter the uniform external shell pressure in kPa.
- Add surface dead load for the shell, finish, or roofing.
- Choose a design factor suitable for your preliminary load case.
- Enter the support count and allowable force per support.
- Calculate, then review height status and capacity status together.
- Export the values after checking units and assumptions.
Example Data
| Input | Example value | Unit |
|---|---|---|
| Base diameter | 10.00 | m |
| Actual dome height | 2.40 | m |
| Permitted dome height | 3.00 | m |
| Uniform shell pressure | 1.50 | kPa |
| Surface dead load | 0.60 | kN/m² |
| Design factor | 1.50 | ratio |
| Supports | 8 | count |
| Allowable force per support | 18.00 | kN |
Dome Height and Force Basics
A limited dome height calculation starts with geometry. The dome is treated as a spherical cap. Its base is circular. Height rises from the base plane to the crown. A permitted height is entered separately. The first check compares height with the limit.
Base diameter determines base radius. Base radius and actual height determine parent sphere radius. This geometry determines curved area, enclosed volume, and dome angle. A taller dome has more curved surface. It may therefore receive more surface loading when pressure is specified over the shell skin.
The calculator reports curved-surface and projected-area values. Each differs clearly. Curved area supports cladding, finish, and surface-load estimates. Projected area supports the vertical pressure resultant. Keeping them separate avoids misleading comparisons.
Pressure Model and Height Limit
Pressure is entered in kilopascals. A kilopascal equals one kilonewton per square metre. Shell pressure equals pressure times curved surface area. It measures pressure magnitude across dome skin. This compares finishes and shell demand.
Vertical pressure resultant equals pressure times horizontal projected base area. For circular domes, area is pi times base radius squared. Under this model, changing height does not alter vertical resultant with fixed diameter and pressure. Height still changes curved area, sphere radius, angle, volume, and surface demand.
Surface dead load can be added. It represents roofing, cladding, or a weight allowance per curved square metre. It adds to vertical pressure. The calculator then applies the selected design factor. This creates a screening force for shared supports.
Capacity and Support Checks
Design force is divided by supports. This produces shared support force. Real reactions may differ. Stiffness, ring-beam behaviour, openings, settlement, and uneven loading can change them. This is an estimate. It is not final structural analysis.
Each support has an allowable capacity. The page multiplies it by support count. It compares total capacity with design force. A positive margin means entered capacity exceeds demand. A negative margin means capacity is insufficient under stated assumptions. Capacity ratio gives comparison.
Height and capacity compliance are independent. A dome can meet height limit yet fail capacity comparison. It can have capacity while exceeding permitted height. Review status lines before using outputs. Changing height, diameter, pressure, support count, or load factor can change results.
Practical Use and Limits
Use units. Enter metres, kilopascals for pressure, kilonewtons per square metre for surface dead load, and kilonewtons for capacity. Do not mix millimetres with metres. Convert values before entry. Rounded inputs affect small or highly loaded domes.
This calculator does not check shell thickness, material strength, buckling, stress, wind, snow drift, seismic action, connections, ring tension, or foundations. It assumes a circular spherical cap and shared supports. A real dome may require finite-element analysis or a code-based shell procedure.
Treat outputs as an early design screen. Record pressure source and load combinations. Confirm whether pressure acts externally, internally, or directionally. Obtain a qualified engineer’s review before decisions. The height-limit check compares options. It does not certify a dome design.
Frequently Asked Questions
1. What is a limited dome height?
It is a maximum permitted crown height for a dome above its base plane. This calculator compares actual height with that entered limit and reports remaining allowance or required reduction.
2. What dome shape does this page assume?
It assumes a circular spherical cap. The dome is part of a sphere cut by a horizontal base plane. Other shapes require different geometry and load models.
3. Why are curved area and projected area both shown?
Curved area is useful for shell finishes and surface loading. Projected area is used for the idealised vertical resultant from uniform pressure acting normal to the spherical surface.
4. Does dome height always change vertical pressure force?
Not in this idealised model when base diameter and uniform pressure stay fixed. Height changes geometry and curved area, while the vertical pressure resultant uses the same horizontal projection.
5. What does uniform shell pressure mean?
It means the entered pressure is assumed constant over the dome surface. Real wind, snow, fluid, and impact loading can vary across the shell and need a more specific model.
6. Why add a surface dead load?
It allows a preliminary allowance for shell weight, cladding, roofing, or finishes. The page multiplies this value by curved dome area before adding it to the support demand.
7. What is the shell-pressure measure?
It is pressure multiplied by curved surface area. It summarises pressure magnitude over the shell skin. It should not be read as a complete structural stress, buckling, or connection calculation.
8. What does equal-share support force mean?
It divides the calculated design force by the entered support count. Actual reactions can differ because of stiffness, ring beams, openings, settlement, and non-uniform loading.
9. Can I enter millimetres?
Convert them to metres first. The formula and result labels assume metres. Mixing millimetres with metres can create force and volume results that are dramatically incorrect.
10. Does a positive capacity margin approve construction?
No. It only shows that entered capacity exceeds this simplified screening demand. Full design must consider materials, buckling, connections, foundations, load combinations, and governing codes.
11. What happens when the dome exceeds its height limit?
The result panel reports the exact reduction needed to meet the entered limit. It still calculates the remaining force outputs, allowing a direct comparison while you revise geometry.