Engineering task and calculation objective
The EC3Roof module designs circular conical roofs of tank and silo structures to DIN EN 1993-4-1, Section 7. Conical roofs close cylindrical tanks and silos at the top and are loaded by dead weight, snow, wind, imposed loads from access, and possibly internal vacuum. Two limit states govern the verification: the equivalent stress check of the roof shell and the buckling check under the greatest local external pressure.
The calculation covers the outside radius or outside diameter of the roof, the smallest wall thickness and the inclination angle of the cone relative to the horizontal. From the yield strength and modulus of elasticity of the steel and the partial safety factors for resistance, the equivalent stress resistance per condition 7.3.1 and the design value of the buckling resistance derived from the ideal external buckling pressure are determined. This makes it possible to calculate a conical roof without resorting to an elaborate numerical shell analysis.
The module complements the cylinder calculation EC3Cyl to DIN EN 1993-1-6, so that tank structures consisting of shell course, roof and hopper can be verified consistently to the Eurocode.
Standard and calculation basis: DIN EN 1993-4-1:2017-09 Abschnitt 7
Calculation workflow
- Record geometry and material: The inputs are the outside radius or outside diameter of the roof, the smallest wall thickness and the inclination angle of the cone relative to the horizontal, together with the yield strength and modulus of elasticity of the steel and the partial safety factors for resistance.
- Determine actions and local external pressure: From dead weight, snow, wind and imposed loads, plus any internal vacuum in the tank, the design value of the greatest local external pressure on the roof shell is formed – the governing action for the buckling check.
- Perform the equivalent stress check: The surface equivalent stresses of the conical shell are compared with the equivalent stress resistance per condition 7.3.1, which is derived from the yield strength and the partial safety factor.
- Perform the buckling check under external pressure: From the geometry and the modulus of elasticity, the ideal external buckling pressure of the conical roof is calculated and converted, via the factor for the buckling resistance, into the design value of the buckling resistance. Condition 7.3.1 (5) requires that the greatest local external pressure does not exceed this value.
- Evaluate the conditions and fix the wall thickness: The module checks the verification conditions of Section 7.3.1 and reports the utilization. If a condition is violated, the wall thickness is increased or the roof slope or a supporting structure is adjusted.
Input quantities
| Quantity | Symbol | Unit |
|---|---|---|
| Condition 7.3.1 (1) 2 equivalent stress resistance | (2) | – |
| Surface equivalent stresses | fe,Rd | N/mm² |
| Yield strength of the steel | fy | N/mm² |
| Partial safety factor 0 for resistance | γM0 | - |
| Ideal external buckling pressure | pn,Rcr | MPa(p) |
| Material | Wknr | – |
| Modulus of elasticity of the steel | E | N/mm² |
| Outside radius of the roof | r | mm |
| Smallest wall thickness | t | mm |
| Inclination angle of cone referred to horizontal line | Φ | ° |
| Reference value of buckling resistance | pn,Rd | MPa(p) |
| Partial safety factor 1 for resistance | γM1 | - |
| Factor for reference value of buckling resistance | αp | - |
| Reference value of greatest local external pressure | pn,Ed | MPa(p) |
| Condition 7.3.1 (5) | (5) | – |
| Condition 7.3.1 (1) 1 | 1 | – |
| Condition 7.3.1 (1) 2 | 2 | – |
| Outside diameter of the roof | da | mm |
Frequently asked questions
Which roofs does Section 7 of DIN EN 1993-4-1 cover?
The section deals with self-supporting circular roofs made of steel sheet on silos and tanks, here in the conical form. Roofs supported by rafters or grillages, where the sheet acts only as cladding, are not covered – there the supporting structures carry the loads and are verified according to the relevant structural steel parts of Eurocode 3.
Why is the buckling check often governing for a conical roof?
Conical roofs are very thin-walled relative to their span and are subject to distributed external pressure from snow, wind and vacuum. The shell then fails not by yielding but by loss of stability. That is why, in addition to the equivalent stress check, the standard explicitly requires comparing the greatest local external pressure with the design value of the buckling resistance derived from the ideal external buckling pressure.
What role does the inclination angle of the roof play?
The inclination angle relative to the horizontal determines how the area loads are converted into membrane forces of the conical shell: shallow roofs generate considerably higher meridional and circumferential forces for the same load and buckle earlier. The slope also influences the snow load assumptions. Very shallow cones quickly fall outside economical plate thicknesses and then require a supporting structure.
Must the vacuum load case in the tank also be considered?
Yes. In tanks with breathing processes (emptying, cooling) an internal vacuum can develop, which acts like additional external pressure on the roof and must be combined unfavourably with snow. The design of the venting system limits this vacuum; the value assumed must be consistent with the design value of the greatest local external pressure.