Engineering guide
Flat ends and flat plates are the simplest way in pressure vessel engineering to close off a cylindrical vessel, a nozzle or a manhole cover pressure-tight. Unlike dished ends, they carry the pressure almost entirely in bending, which is why their required wall thickness grows much more steeply with diameter. If you want to calculate a flat plate or flat end to AD 2000 — the German pressure vessel code — the governing rules are found in AD 2000-Merkblatt B5.
The B5 module determines the required wall thickness of unstiffened and stiffened flat ends and plates — circular, oval or rectangular — including the associated anchorages. The design configuration (welded-on end, bolted plate, plate with peripheral moment, etc.) is selected through a guided decision process, which yields the design coefficient C. For bolted plates with a peripheral moment, the gasket factors to AD 2000 B7 are built in, so that the bolting-up and operating conditions are assessed consistently.
Typical applications are blind flanges and blind covers, flat vessel ends of smaller diameters, handhole and manhole closures, and wall plates stiffened by anchors or stay bolts. The module is frequently used together with B1 (cylindrical shell), B7 (bolts) and B8 (flanges) for the complete design of an apparatus.
Scope
- Flat ends and plates and anchorage
Typical workflow
Select the design configuration
The interactive decision process establishes the configuration: flat end welded onto or into the shell, bolted plate with an internal or full-face gasket, or stiffened plate with anchorage. The chosen configuration determines the design coefficient C according to the figures of Merkblatt B5.
Enter loads and material properties
Design pressure and design temperature are entered; from these follow the nominal design stress K and the safety factor S of the plate or end material. For plates with a peripheral moment, the bolt and gasket loads from AD 2000 B7 are added.
Determine the governing diameter
Depending on the configuration, the design diameter is the inside diameter, the gasket or bolt circle diameter, or — for rectangular plates — the shorter side length with an aspect-ratio correction. For plates with openings, a ligament efficiency is additionally taken into account.
Calculate the required wall thickness
The wall thickness follows from the plate formula with the coefficient C, the governing diameter and the ratio of pressure to allowable stress K/S. The allowance c1 for wall thickness undertolerance and the corrosion/erosion allowance c2 are then added.
Compare load conditions and verify the anchorage
For bolted plates with a peripheral moment, the wall thickness is determined separately for the bolting-up and operating conditions; the larger value governs. For stiffened plates, the anchors or anchorages are additionally verified for their load-bearing capacity.
Frequently asked questions
When does a flat end make sense compared with a dished end?
Flat ends are simple to fabricate and economical at small diameters or low pressures, for example for covers, handholes or nozzle closures. Since the wall thickness grows in proportion to the diameter and to the square root of the pressure, flat ends quickly become uneconomically heavy at large diameters and higher pressures — in that case torispherical (Klöpper) or semi-ellipsoidal (Korbbogen) ends to AD 2000 B3 are the better choice.
What does the design coefficient C describe?
C captures the edge restraint and support conditions of the plate as well as the configuration. A plate with a rigid, moment-carrying connection receives a smaller coefficient than a simply supported one, because the restraining edge moment reduces the bending stress at the plate center. The C values are defined per figure in Merkblatt B5 and must not be chosen freely.
Why must both the bolting-up and the operating condition be checked for bolted plates?
At bolting-up, the full bolt preload acts through the lever arm to the gasket as a peripheral moment on the unpressurized plate; in operation, internal pressure, residual gasket load and bolt load are superimposed. Depending on the gasket factors and lever arms, the bolting-up condition can require the greater plate thickness — skipping it is a typical source of error.
How are openings in flat plates taken into account?
Openings weaken the load-bearing cross-section and are captured through a ligament efficiency in the wall thickness formula; alternatively, reinforced nozzles or welded-on reinforcements are possible. For closely spaced openings, minimum ligament widths apply — otherwise a separate assessment is required.
Worked example
A circular flat end welded to the cylindrical shell of a vessel is to be designed. This worked example calculates the required wall thickness to AD 2000-Merkblatt B5 for a design pressure of 10 bar.
Given
| Design diameter D | 400 mm |
|---|---|
| Design pressure p | 10 bar |
| Nominal design stress K (at design temperature) | 170 N/mm² |
| Safety factor S | 1.5 |
| Design coefficient C (per configuration acc. to B5) | 0.45 |
| Allowance c1 (wall thickness undertolerance) | 0 mm |
| Allowance c2 (corrosion) | 1 mm |
Solution
Form the allowable stress ratio
The governing quantity is the ratio of pressure to allowable stress. With p in bar, the term under the square root is:
p · S / (10 · K) = 10 · 1.5 / (10 · 170) = 0.008824
Wall thickness without allowances
s0 = C · D · √(p · S / (10 · K))
s0 = 0.45 · 400 mm · √0.008824 = 0.45 · 400 · 0.0939 = 16.91 mm
Add the allowances
s = s0 + c1 + c2 = 16.91 mm + 0 mm + 1 mm = 17.91 mm
The next available plate thickness of 18 mm is selected.
Result
| Required wall thickness s | 17.91 mm |
|---|---|
| Selected plate thickness | 18 mm |
Technical content source: LV-Solved-Problems-Content 628d313247e7