8.3Cantilevered Flange Joints

ECSS-E-HB-32-23A, Space Engineering — Threaded Fasteners Handbook

Last Updated 29 Oct 2026

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8.3.1 Overview

These joints have small contact areas at the interstice, leading to bending moments being transmitted through the flange. This results in the axis of the clamped parts being fixed rather than moving with change in external load FA (the case for joints with large areas of contact).

For this class of joints, the procedure for determining the flange compliance can be complex and time consuming. The theory presented in the following Sections applies to the special case of circular joints with cantilevered flanges. For more general cases with non-circular flanges it is recommended to use testing or FEM.

8.3.2 Compliance of Circular Flanges

8.3.2.1 Overview

References 8.3 and 8.5 give empirical relationships for joints with cantilevered circular flanges. This category of joints often occurs when gaskets or sealing rings are used.

8.3.2.2 Linear Superposition of Deformations

For cantilevered flange joints, three compliances are used in combination to describe the flanges’ bending displacements. Figure 8-14 indicates three modes of deformation that determine the deflections of the flange between the axis of the fastener and the axis of the clamped parts. These compliances are calculated as follows:

This is the compliance of the flanges under the preload taking into account the bending effects due to the eccentric clamping load. [8.3.1]

$\delta_{2} = \frac{\Delta L_{2}}{\Delta F_{b,A}}$ This is the bending compliance of the flanges due to the corresponding portion of the applied load seen by the fastener. In other words it is the flange bending due to the couple $\Delta F_{b,A}$ with forces acting at the axes of the fastener axis and externally applied load. [8.3.2]
$\delta_{3} = \frac{\Delta L_{3}}{\Delta F_{c,A}}$ This is the bending compliance of the flanges due to the load seen by the clamped parts and the corresponding portion of the applied load. In particular, it is the flange bending due to the couple $\Delta F_{c,A}$ with forces acting at the axes of the clamped parts and the externally applied load. [8.3.3]

Assuming no gapping occurs, the elongation of the flange due to the external load is the linear sum of the three extension modes of Figure 8-14, i.e.,

\Delta L_{c,A} = \Delta L_{3} + \Delta L_{2} + \Delta L_{b,A} \tag*{[8.3.4]}

Thus the compliance of the flange is,

\delta_{c} = \frac{\Delta L_{3} + \Delta L_{2} + \Delta L_{b,A}}{\Delta F_{c,A}} \tag{8.3.5}

Key Entities and Axes: Central Axis of Circular Flange: Vertical dash-dot line on the far left

Figure description:

Key Entities and Axes:

Central Axis of Circular Flange: Vertical dash-dot line on the far left.
Axis of applied load: Leftmost dashed vertical line passing through force vectors.
Axis of clamped parts contact area: Center dashed vertical line passing through force vectors.
Axis of fastener: Rightmost dashed vertical line passing through force vectors.

Variables and Forces:

$F_V$ : Upward force at contact area axis; downward force at fastener axis.
$F_{b,A}$ : Upward force at applied load axis; downward force at fastener axis.
$F_{c,A}$ : Upward force at applied load axis; downward force at contact area axis.

Equations for Displacement ( $\Delta L$ :

$\Delta L_1 = F_V \delta_1$
$\Delta L_2 = F_{b,A} \delta_{21}$
$\Delta L_3 = F_{c,A} \delta_{31}$

Visual Information:

The diagram shows three cross-sectional views of a flange segment undergoing bending under different loading configurations.
Displacement ( $\Delta L$ is measured as the vertical deflection between the horizontal plane and the deflected tip of the flange.

Figure 8-14 - Definition of Flange Compliance due to Bending Loads at Different Points

Two types of flanges are frequently employed for circular joints, fixed flanges and loose flanges. Fixed flanges are rigidly connected to the remaining structures. Loose flanges are separate annular rings that compress the mating parts together. The following two Sections provide analytical equations for the compliance of both types of flanges.

8.3.2.3 Compliance of Fixed Circular Flanges

Figure 8-15 shows a typical fixed flange with a small contact area such as a sealing ring. For this type of flange the three compliances of Figure 8-14 are given by the following equations (see Reference 8.5):

\delta_{1} = \psi a_{D}^{2} \tag{8.3.6}

\delta_{2} = \psi a_{D} a_{R} \tag{8.3.7}

\delta_{3} = \psi a_{D} (a_{R} - a_{D}) \tag{8.3.8}

where ψ is the inversion compliance (defined in Section 8.3.2.5 below).

Figure description:

The image contains the text "ECSS" in a white, serif, italicized font. The letters are set against a purple parallelogram background, which is bordered by a thin black line. The letter "E" is separated from "CSS" by a vertical white gap.

The provided image is a cross-sectional diagram of a fixed circular flange with a weldneck, showing

Figure description:

The provided image is a cross-sectional diagram of a fixed circular flange with a weldneck, showing various dimensions and force vectors.

Key Entities and Labels:

Forces:
- $F_A$ : Upward external load vector acting along the axis of the pipe/neck.
- $F_b$ : Downward force vector acting through the bolt hole center (point S.
- $F_c$ : Upward reaction force vector acting at a contact point (point P.
Dimensions:
- $d_a$ : Outer diameter of the flange.
- $d_t$ : Pitch circle diameter of the fasteners.
- $d_i$ : Inner diameter of the flange/pipe.
- $d$ : Diameter of the bolt hole.
- $s_R$ : Thickness of the pipe wall at the top.
- $s_F$ : Thickness of the neck section.
- $t$ : Thickness of the flange plate.
- $a_D$ : Radial distance from the contact point (P to the bolt hole center (S.
- $a_R$ : Radial distance from the internal wall axis (A to the bolt hole center (S.
Reference Points:
- A: Point on the internal wall surface at the base.
- P: Point of contact/support on the flange face.
- S: Center point of the fastener/bolt hole.
Structural Features:
- Cross-hatched area representing the flange and weldneck body.
- Centerline (dashed vertical line on the left indicating axis of symmetry.

Figure 8-15 - Dimensions of a Fixed Circular Flange (with a Weldneck)

In general, the contacting interstice between the fixed flanges can be either a large area or a small area. Only the case of the small contact area is considered in this theory.

8.3.2.4 Compliance of Loose Circular Flanges

Figure 8-16 shows the bending dimensions and bending deformations of a loose circular flange. For this flange the joint axis (O-O) and the external load axis (A-A) are generally coincident and as a result δ 3 is zero. Thus, the bending deformation due to an external load FA is only due to the increase in fastener load ∆ b a and is given by,

\Delta L = \Delta F_{b,A} \psi a_{F}^{2} \tag{8.3.9}

This is rearranged to give the flange compliance,

= \frac{\Delta L}{F} = \psi a_{F}^{2} \tag{8.3.10}

where ψ is the inversion compliance (defined in Section 8.3.2.5 below).

The image illustrates the bending deformation of a circular loose flange through two

Figure description:

The image illustrates the bending deformation of a circular loose flange through two cross-sectional diagrams: one in an initial state and one in a deformed state.

Key Entities and Dimensions:

$d_a$ : Outer diameter of the flange.
$d_i$ : Inner diameter of the flange.
$t$ : Thickness of the flange.
$D_h$ : Diameter of the fastener hole.
$F$ : Applied forces causing bending.
$\Delta L$ : Vertical displacement/deformation at the inner edge.
$\alpha$ : Angle of rotation/deformation.
$a_F$ : Radial distance between the applied opposing forces $F$ .

Figure 8-16 - Bending Deformation of Circular Loose Flange

8.3.2.5 Inversion Compliance

The inversion compliance, $\psi$ , is given by,

\psi = \frac{d_{a} + d_{i}}{4 \pi E h W} \tag{8.3.11}

where $W$ is a factor accounting for the presence of the holes in the flange given by,

W = \frac{1}{12} (d_{a} - d_{i} - d^{*}) h^{2} \quad \text{(for loose flanges)} \quad [8.3.12]

or $\frac{1}{12}(d_{i} + S_{F})S_{F}^{2} + \frac{1}{12}(d_{a} - d_{i} - d^{*})h^{2} \quad \text{(for fixed flanges)} \tag{8.3.13}$

where $d_a$ and $d_i$ are flange annulus' outer and inner diameters (see Figure 8-15 and Figure 8-16), $S_F$ is the radial thickness of the weldneck (see Figure 8-15), $h$ is the thickness of the flange, and $d^*$ is defined by,

d^{*} = \begin{cases} \frac{d}{2} & \text{for} \quad W_{nom} \geq 500 \\ \left(1 - \frac{W_{nom}}{1000}\right) d & \text{for} \quad W_{nom} < 500 \end{cases} \tag{8.3.14}

where $d$ is the diameter of the fastener holes, and $W_{nom}$ is the flange nominal width specified in mm.

8.3.3 Force Ratio for Circular Cantilever Flanges

8.3.3.1 Single Flange

Consider a weld neck flange such as the one illustrated in Figure 8-15, fastened to an infinitely stiff surface such that only the flange undergoes bending deflections. During preloading the fastener extends by $\Delta L_{b}$ and the flange deflects by,

\Delta L_{1} = \delta_{1} F_{v} \tag*{[8.3.15]}

When the external load $F_{A}$ is added the fastener is stretched by,

\Delta L_{b,A} = \delta_{b} \Delta F_{b,A} \tag*{[8.3.16]}

The force ratio for the cantilevered flange joint, $\Phi_{C}$ , can be defined as follows,

\Phi_{C} = \frac{\Delta F_{b,A}}{\Delta F_{b,A} + \Delta F_{c,A}} = \frac{1}{\left(1 + \frac{\Delta F_{c,A}}{\Delta F_{b,A}}\right)} \tag{8.3.17}

Using Equations [8.3.1], [8.3.2], [8.3.3] and [8.3.5],

\Delta F_{b,A} (\delta_{b} + \delta_{2}) = \Delta F_{c,A} (\delta_{c} - \delta_{3}) \tag{8.3.18}

Therefore, combining Equations [8.3.17] and [8.3.18] the force ratio expressed as a function of the compliances is,

\Phi_{c} = \left[ 1 + \frac{\delta_{b} + \delta_{2}}{\delta_{c} - \delta_{3}} \right]^{-1} \tag{8.3.19}

For loose flanges where the clamping load and applied load are coincident $\mathcal{D}_{3} = 0$ and Equation [8.3.19] is simplified to,

\Phi_{c} = \left[ 1 + \frac{\delta_{b} + \delta_{2}}{\delta_{c}} \right]^{-1} \tag{8.3.20}

8.3.3.2 Twin Flanges

The compliances derived in sections 8.3.2.2, 8.3.2.3, 8.3.2.4 and 8.3.2.5 are for individual cantilevered flanges. In many cases, the joint has two symmetric cantilevered flanges. For such a joint, with fixed flanges and $m$ bolts the force ratio is given by,

\Phi_{c} = \left[ 1 + \frac{\delta_{b}/m + 2\delta_{2}}{\delta_{c} - 2\delta_{3}} \right]^{-1} \tag{8.3.21}

where $\mathcal{\delta}_{b}$ and $\mathcal{\delta}_{c}$ include the effects of both flanges and the increased $L_{c}$ (unlike those values used in the above sections 8.3.2.2 to 8.3.2.5).

For loose flanges, $\delta_{3} = 0$ , and Equation [8.3.21] becomes simplified to,

\Phi_{c} = \left[ 1 + \frac{\delta_{b}/m + 2\delta_{2}}{\delta_{c}} \right]^{-1} \tag*{[8.3.22]}

where $\mathcal{D}_{b}$ and $\mathcal{D}_{c}$ are redefined to account for both flanges, as in equation[8.3.21].

8.3.4 Simplified Joint Diagram for Eccentric Joints

8.3.4.1 Overview

Consider a weld neck type flange, as shown in Figure 8-15 with m bolts. If the fasteners are equally preloaded by FV /m and consequently stretched by ( δ b ) m , the two flanges can bend by 2δ 1 F V while also being compressed by ( δ c ) m . This preloading is represented in the joint diagram of Figure 8-17 by the triangle SPV.

Figure 8-17

Figure description:

Title: Figure 8-17 - Simplified Joint Diagram for Cantilevered Flanged Joints
Axes/Dimensions:
- Vertical axis labeled $F_V$
- Horizontal base dimension (left of O: $\frac{F_V}{m} \delta_B + 2\delta_1 F_V$
- Horizontal base dimension (right of O: $F_V \delta_P$
Key Entities/Points:
- S: Origin/starting point of the left triangle base.
- P: Endpoint of the right triangle base.
- V: Peak point of the joint diagram, forming triangle SPV.
- O: Point on the base directly below V.
- M, N, K, S, H: Points located on or near the extension lines to the right of peak V.
Graphic Elements:
- Solid lines forming a main triangle (SPV with a vertical divider (VO.
- An ascending solid line extending from V through K and M.
- A descending solid line from V through H to P.
- Dashed lines indicating projections for points M, N, and S.

Figure 8-17 - Simplified Joint Diagram for Cantilevered Flanged Joints

8.3.4.2 Onset of Separation

Separation occurs when the preload reduces to zero, which is represented by the line PM in Figure 8-17. At the onset of separation FA = PM (where MN is equal to ∆ A and NP is equal to ∆ A ). Thus,

[8.3.23]

8.3.5 Joint Diagram for Eccentric Nonlinear Joints

In the case of symmetrically flanged joint with a gasket or seal which has non-linear deformation characteristics, the clamped parts compliance δ is dependent upon the load. c

Assuming m fasteners, Equation [8.3.23] can be rearranged to give;

\Delta F_{b,A} = \left( \frac{\delta_{c} - \delta_{3}}{\delta_{b} / m + \delta_{2}} \right) \Delta F_{c,A} = \frac{\Delta L_{c,A} - \Delta F_{c,A} \delta_{3}}{\delta_{b} / m + \delta_{2}} \tag*{[8.3.24]}

Since $\Delta L_{c,A}$ and $\Delta F_{c,A}$ are nonlinear, the joint diagram is also linear. To construct the joint diagram, a range of values of $\Delta L_{c,A}$ are assumed and using the gasket characteristics, the corresponding values of $\Delta F_{c,A}$ can be determined. Hence, the values of $\Delta F_{b,A}$ can be calculated and used to construct the joint diagram.

Technical Diagram with Geometric Labels Geometric and Algebraic Parameter Relationship Diagram

Figure description:

Technical Diagram with Geometric Labels

Geometric and Algebraic Parameter Relationship Diagram

Annotations

F_V (dimension_line - position: Vertical distance from horizontal baseline to point V)
F_A (dimension_line - position: Vertical distance between the upper and lower curves starting at V)
S, O, P, V, N, M (text_label - position: Key vertices of the geometric construction)
Dashed lines (reference_line - position: Represent linear theoretical relationships or coordinate projections)

Element	Type	X-Coordinate / Range	Y-Coordinate / Range	Label
Point S	Point	-((F_V / m * δ_B + 2 * δ_1 * F_V))	0	S
Point O	Point	0	0	O
Point P	Point	F_V * s_P	0	P
Point V	Point	0	F_V	V
Point N	Point	F_V * s_P	F_V	N
Point M	Point	F_V * s_P	y_M (where y_M > F_V)	M
Line SV	Straight Line	From S to V	[0, F_V	-
Curve VM	Curve (Convex)	[0, F_V * s_P	[F_V, y_M	-
Curve VP	Curve (Concave)	[0, F_V * s_P	[F_V, 0	-
Dashed Line VM	Straight Line	[0, F_V * s_P	[F_V, y_M	-
Dashed Line VP	Straight Line	[0, F_V * s_P	[F_V, 0	-
Dashed Line VN	Straight Line	[0, F_V * s_P	F_V	-
Dashed Line MP	Straight Line	F_V * s_P	[0, y_M	-
Dimension F_V	Vertical	Left of S	[0, F_V	F_V
Dimension F_A	Vertical	~0.6 * F_V * s_P	Between Curves VM and VP	F_A
Dimension SO	Horizontal	[-d_SO, 0	Below baseline	(F_V / m * δ_B + 2 * δ_1 * F_V)
Dimension OP	Horizontal	[0, F_V * s_P	Below baseline	F_V * s_P

Notes: The chart is a technical geometric diagram representing physical parameters. Horizontal distances are defined as expressions involving F_V, m, δ_B, δ_1, and s_P. Curves indicate non-linear variations relative to the dashed linear approximations.

Figure 8-18 - The Joint Diagram for Non-Linear Gasket Joints

8.3Cantilevered Flange Joints

ECSS-E-HB-32-23A, Space Engineering — Threaded Fasteners Handbook

Last Updated 29 Oct 2026

Hover any section to copy its link or text, ask Copilot about it, or bookmark it.

8.3.1 Overview

8.3.2 Compliance of Circular Flanges

8.3.2.1 Overview

References 8.3 and 8.5 give empirical relationships for joints with cantilevered circular flanges. This category of joints often occurs when gaskets or sealing rings are used.

8.3.2.2 Linear Superposition of Deformations

This is the compliance of the flanges under the preload taking into account the bending effects due to the eccentric clamping load. [8.3.1]

$\delta_{2} = \frac{\Delta L_{2}}{\Delta F_{b,A}}$ This is the bending compliance of the flanges due to the corresponding portion of the applied load seen by the fastener. In other words it is the flange bending due to the couple $\Delta F_{b,A}$ with forces acting at the axes of the fastener axis and externally applied load. [8.3.2]
$\delta_{3} = \frac{\Delta L_{3}}{\Delta F_{c,A}}$ This is the bending compliance of the flanges due to the load seen by the clamped parts and the corresponding portion of the applied load. In particular, it is the flange bending due to the couple $\Delta F_{c,A}$ with forces acting at the axes of the clamped parts and the externally applied load. [8.3.3]

Assuming no gapping occurs, the elongation of the flange due to the external load is the linear sum of the three extension modes of Figure 8-14, i.e.,

\Delta L_{c,A} = \Delta L_{3} + \Delta L_{2} + \Delta L_{b,A} \tag*{[8.3.4]}

Thus the compliance of the flange is,

\delta_{c} = \frac{\Delta L_{3} + \Delta L_{2} + \Delta L_{b,A}}{\Delta F_{c,A}} \tag{8.3.5}

Key Entities and Axes: Central Axis of Circular Flange: Vertical dash-dot line on the far left

Figure description:

Key Entities and Axes:

Central Axis of Circular Flange: Vertical dash-dot line on the far left.
Axis of applied load: Leftmost dashed vertical line passing through force vectors.
Axis of clamped parts contact area: Center dashed vertical line passing through force vectors.
Axis of fastener: Rightmost dashed vertical line passing through force vectors.

Variables and Forces:

$F_V$ : Upward force at contact area axis; downward force at fastener axis.
$F_{b,A}$ : Upward force at applied load axis; downward force at fastener axis.
$F_{c,A}$ : Upward force at applied load axis; downward force at contact area axis.

Equations for Displacement ( $\Delta L$ :

$\Delta L_1 = F_V \delta_1$
$\Delta L_2 = F_{b,A} \delta_{21}$
$\Delta L_3 = F_{c,A} \delta_{31}$

Visual Information:

The diagram shows three cross-sectional views of a flange segment undergoing bending under different loading configurations.
Displacement ( $\Delta L$ is measured as the vertical deflection between the horizontal plane and the deflected tip of the flange.

Figure 8-14 - Definition of Flange Compliance due to Bending Loads at Different Points

8.3.2.3 Compliance of Fixed Circular Flanges

\delta_{1} = \psi a_{D}^{2} \tag{8.3.6}

\delta_{2} = \psi a_{D} a_{R} \tag{8.3.7}

\delta_{3} = \psi a_{D} (a_{R} - a_{D}) \tag{8.3.8}

where ψ is the inversion compliance (defined in Section 8.3.2.5 below).

Figure description:

The provided image is a cross-sectional diagram of a fixed circular flange with a weldneck, showing

Figure description:

The provided image is a cross-sectional diagram of a fixed circular flange with a weldneck, showing various dimensions and force vectors.

Key Entities and Labels:

Forces:
- $F_A$ : Upward external load vector acting along the axis of the pipe/neck.
- $F_b$ : Downward force vector acting through the bolt hole center (point S.
- $F_c$ : Upward reaction force vector acting at a contact point (point P.
Dimensions:
- $d_a$ : Outer diameter of the flange.
- $d_t$ : Pitch circle diameter of the fasteners.
- $d_i$ : Inner diameter of the flange/pipe.
- $d$ : Diameter of the bolt hole.
- $s_R$ : Thickness of the pipe wall at the top.
- $s_F$ : Thickness of the neck section.
- $t$ : Thickness of the flange plate.
- $a_D$ : Radial distance from the contact point (P to the bolt hole center (S.
- $a_R$ : Radial distance from the internal wall axis (A to the bolt hole center (S.
Reference Points:
- A: Point on the internal wall surface at the base.
- P: Point of contact/support on the flange face.
- S: Center point of the fastener/bolt hole.
Structural Features:
- Cross-hatched area representing the flange and weldneck body.
- Centerline (dashed vertical line on the left indicating axis of symmetry.

Figure 8-15 - Dimensions of a Fixed Circular Flange (with a Weldneck)

In general, the contacting interstice between the fixed flanges can be either a large area or a small area. Only the case of the small contact area is considered in this theory.

8.3.2.4 Compliance of Loose Circular Flanges

\Delta L = \Delta F_{b,A} \psi a_{F}^{2} \tag{8.3.9}

This is rearranged to give the flange compliance,

= \frac{\Delta L}{F} = \psi a_{F}^{2} \tag{8.3.10}

where ψ is the inversion compliance (defined in Section 8.3.2.5 below).

The image illustrates the bending deformation of a circular loose flange through two

Figure description:

The image illustrates the bending deformation of a circular loose flange through two cross-sectional diagrams: one in an initial state and one in a deformed state.

Key Entities and Dimensions:

$d_a$ : Outer diameter of the flange.
$d_i$ : Inner diameter of the flange.
$t$ : Thickness of the flange.
$D_h$ : Diameter of the fastener hole.
$F$ : Applied forces causing bending.
$\Delta L$ : Vertical displacement/deformation at the inner edge.
$\alpha$ : Angle of rotation/deformation.
$a_F$ : Radial distance between the applied opposing forces $F$ .

Figure 8-16 - Bending Deformation of Circular Loose Flange

8.3.2.5 Inversion Compliance

The inversion compliance, $\psi$ , is given by,

\psi = \frac{d_{a} + d_{i}}{4 \pi E h W} \tag{8.3.11}

where $W$ is a factor accounting for the presence of the holes in the flange given by,

W = \frac{1}{12} (d_{a} - d_{i} - d^{*}) h^{2} \quad \text{(for loose flanges)} \quad [8.3.12]

or $\frac{1}{12}(d_{i} + S_{F})S_{F}^{2} + \frac{1}{12}(d_{a} - d_{i} - d^{*})h^{2} \quad \text{(for fixed flanges)} \tag{8.3.13}$

d^{*} = \begin{cases} \frac{d}{2} & \text{for} \quad W_{nom} \geq 500 \\ \left(1 - \frac{W_{nom}}{1000}\right) d & \text{for} \quad W_{nom} < 500 \end{cases} \tag{8.3.14}

where $d$ is the diameter of the fastener holes, and $W_{nom}$ is the flange nominal width specified in mm.

8.3.3 Force Ratio for Circular Cantilever Flanges

8.3.3.1 Single Flange

\Delta L_{1} = \delta_{1} F_{v} \tag*{[8.3.15]}

When the external load $F_{A}$ is added the fastener is stretched by,

\Delta L_{b,A} = \delta_{b} \Delta F_{b,A} \tag*{[8.3.16]}

The force ratio for the cantilevered flange joint, $\Phi_{C}$ , can be defined as follows,

\Phi_{C} = \frac{\Delta F_{b,A}}{\Delta F_{b,A} + \Delta F_{c,A}} = \frac{1}{\left(1 + \frac{\Delta F_{c,A}}{\Delta F_{b,A}}\right)} \tag{8.3.17}

Using Equations [8.3.1], [8.3.2], [8.3.3] and [8.3.5],

\Delta F_{b,A} (\delta_{b} + \delta_{2}) = \Delta F_{c,A} (\delta_{c} - \delta_{3}) \tag{8.3.18}

Therefore, combining Equations [8.3.17] and [8.3.18] the force ratio expressed as a function of the compliances is,

\Phi_{c} = \left[ 1 + \frac{\delta_{b} + \delta_{2}}{\delta_{c} - \delta_{3}} \right]^{-1} \tag{8.3.19}

For loose flanges where the clamping load and applied load are coincident $\mathcal{D}_{3} = 0$ and Equation [8.3.19] is simplified to,

\Phi_{c} = \left[ 1 + \frac{\delta_{b} + \delta_{2}}{\delta_{c}} \right]^{-1} \tag{8.3.20}

8.3.3.2 Twin Flanges

\Phi_{c} = \left[ 1 + \frac{\delta_{b}/m + 2\delta_{2}}{\delta_{c} - 2\delta_{3}} \right]^{-1} \tag{8.3.21}

where $\mathcal{\delta}_{b}$ and $\mathcal{\delta}_{c}$ include the effects of both flanges and the increased $L_{c}$ (unlike those values used in the above sections 8.3.2.2 to 8.3.2.5).

For loose flanges, $\delta_{3} = 0$ , and Equation [8.3.21] becomes simplified to,

\Phi_{c} = \left[ 1 + \frac{\delta_{b}/m + 2\delta_{2}}{\delta_{c}} \right]^{-1} \tag*{[8.3.22]}

where $\mathcal{D}_{b}$ and $\mathcal{D}_{c}$ are redefined to account for both flanges, as in equation[8.3.21].

8.3.4 Simplified Joint Diagram for Eccentric Joints

8.3.4.1 Overview

Figure 8-17

Figure description:

Title: Figure 8-17 - Simplified Joint Diagram for Cantilevered Flanged Joints
Axes/Dimensions:
- Vertical axis labeled $F_V$
- Horizontal base dimension (left of O: $\frac{F_V}{m} \delta_B + 2\delta_1 F_V$
- Horizontal base dimension (right of O: $F_V \delta_P$
Key Entities/Points:
- S: Origin/starting point of the left triangle base.
- P: Endpoint of the right triangle base.
- V: Peak point of the joint diagram, forming triangle SPV.
- O: Point on the base directly below V.
- M, N, K, S, H: Points located on or near the extension lines to the right of peak V.
Graphic Elements:
- Solid lines forming a main triangle (SPV with a vertical divider (VO.
- An ascending solid line extending from V through K and M.
- A descending solid line from V through H to P.
- Dashed lines indicating projections for points M, N, and S.

Figure 8-17 - Simplified Joint Diagram for Cantilevered Flanged Joints

8.3.4.2 Onset of Separation

[8.3.23]

8.3.5 Joint Diagram for Eccentric Nonlinear Joints

In the case of symmetrically flanged joint with a gasket or seal which has non-linear deformation characteristics, the clamped parts compliance δ is dependent upon the load. c

Assuming m fasteners, Equation [8.3.23] can be rearranged to give;

\Delta F_{b,A} = \left( \frac{\delta_{c} - \delta_{3}}{\delta_{b} / m + \delta_{2}} \right) \Delta F_{c,A} = \frac{\Delta L_{c,A} - \Delta F_{c,A} \delta_{3}}{\delta_{b} / m + \delta_{2}} \tag*{[8.3.24]}

Technical Diagram with Geometric Labels Geometric and Algebraic Parameter Relationship Diagram

Figure description:

Technical Diagram with Geometric Labels

Geometric and Algebraic Parameter Relationship Diagram

Annotations

F_V (dimension_line - position: Vertical distance from horizontal baseline to point V)
F_A (dimension_line - position: Vertical distance between the upper and lower curves starting at V)
S, O, P, V, N, M (text_label - position: Key vertices of the geometric construction)
Dashed lines (reference_line - position: Represent linear theoretical relationships or coordinate projections)

Element	Type	X-Coordinate / Range	Y-Coordinate / Range	Label
Point S	Point	-((F_V / m * δ_B + 2 * δ_1 * F_V))	0	S
Point O	Point	0	0	O
Point P	Point	F_V * s_P	0	P
Point V	Point	0	F_V	V
Point N	Point	F_V * s_P	F_V	N
Point M	Point	F_V * s_P	y_M (where y_M > F_V)	M
Line SV	Straight Line	From S to V	[0, F_V	-
Curve VM	Curve (Convex)	[0, F_V * s_P	[F_V, y_M	-
Curve VP	Curve (Concave)	[0, F_V * s_P	[F_V, 0	-
Dashed Line VM	Straight Line	[0, F_V * s_P	[F_V, y_M	-
Dashed Line VP	Straight Line	[0, F_V * s_P	[F_V, 0	-
Dashed Line VN	Straight Line	[0, F_V * s_P	F_V	-
Dashed Line MP	Straight Line	F_V * s_P	[0, y_M	-
Dimension F_V	Vertical	Left of S	[0, F_V	F_V
Dimension F_A	Vertical	~0.6 * F_V * s_P	Between Curves VM and VP	F_A
Dimension SO	Horizontal	[-d_SO, 0	Below baseline	(F_V / m * δ_B + 2 * δ_1 * F_V)
Dimension OP	Horizontal	[0, F_V * s_P	Below baseline	F_V * s_P

Figure 8-18 - The Joint Diagram for Non-Linear Gasket Joints