8.2Joints with Large Areas of Contact

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

Last Updated 29 Oct 2026

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8.2.1 Compliance of the Clamped Parts

8.2.1.1 Overview

The equations for calculating the compliance of the clamped parts in the concentric loading case were given in Section 7.6. The eccentric case is more complicated since, in addition to longitudinal deformation, the eccentricity of the external load produces bending deformations in the flanges leading to additional extension of the joint. Hence, the longitudinal compliance of the flanges in an eccentric joint is greater than that of a concentrically loaded joint. Also, for joints with large areas of contact, the distribution of the clamping pressure depends on the magnitude of the external load (as discussed in Section 8.1), producing further changes in compliance.

A relatively simple analysis (modified from Reference 8.4) of eccentric joint deformation is possible if the following assumptions are made:

• The clamped parts form a prismatic bending solid
• There is no one-sided gapping (i.e. the interface pressure is always positive)
• All the cross-sections of this prismatic solid remain flat under loading
• A linear stress distribution is established in these cross-sections
• The bending resistance of the fastener is much lower than that of the bending solid and can be neglected

These simplifying assumptions limit the joints that can be considered to those in which the interface area loaded by one fastener, Aj , is not much greater (3 to 5 times) the underhead area. Serious errors can occur if larger areas are assumed (see Reference 8.2). Additionally, it is important that the joint’s transverse width, C, and clamping eccentricity, s, satisfy the following relations:

C \leq (D_{uh,brg} + h_{\min}) \tag{8.2.1}

s \leq \frac{(D_{uh,brg} + h_{\min})}{2} \tag{8.2.2}

where C is the distance from the fastener joints axis to the flange edge as indicated in Figure 8-7. It is important that joints that do not satisfy these criteria are analysed using FEM or testing.

Entities: Bolt with a hexagonal head and nut, two clamped plates

Figure description:

Entities: Bolt with a hexagonal head and nut, two clamped plates (cross-hatched, and force vectors.
Forces: $F_A$ (applied tensile forces acting along axis A-A.
Dimensions/Variables:
- $a$ : Distance from axis O to axis A.
- $s$ : Distance from axis O to bolt centerline S.
- $h_{min}$ : Thickness of the top plate.
- $D_{uh,brg}$ : Bearing diameter under the bolt head.
- $u_B$ : Distance from left edge to bolt centerline S.
- $u$ : Distance from left edge to axis O.
- $C$ : Total width of the clamped interface.
Reference Axes:
- A: Axis of applied force.
- S: Centerline of the bolt.
- O: A reference axis located to the right of the bolt centerline.

Figure 8-7 - Limitations of the Eccentric Joint Analysis Method

In a multi-fastener joint such as Figure 8-8 the spacing between fasteners (fastener pitch) is defined as $t$ , and the flange length assumed to be loaded by each fastener is defined as $b$ , which is assumed to be given by,

b = \begin{cases} D_{uh,brg} + h_{\min} & \text{if } t > \left(D_{uh,brg} + h_{\min}\right) \\ t & \text{if } t \leq \left(D_{uh,brg} + h_{\min}\right) \end{cases} \tag*{[8.2.3]}

Entities and Dimensions: Assumed Interface Area AB for One Fastener: Indicated by the

Figure description:

Entities and Dimensions:

Assumed Interface Area $A_B$ for One Fastener: Indicated by the cross-hatched rectangular region.
$b$ : The width/length of the shaded interface area.
$t$ : The distance (pitch between the centerlines of two adjacent fasteners/holes.
Fasteners/Holes: Represented by two circular features with horizontal centerlines.

Figure 8-8 - The Interface Area for Multiple Fastener Joints

The clamped interface area shown in Figure 8-8 has an area, $A_{c} = b.C$ , and a radius of gyration,

G_{c} = \left[ \frac{I_{c}}{A_{c}} \right]^{\frac{1}{2}} \tag{8.2.4}

where $I_{c}$ is the second moment of area of the clamped interface (about its neutral axis), which is given by $bt^{3}/12$ for the interface shown in Figure 8-8.

Subject to the above assumptions, clamped parts compliance values can be calculated for the two classes of eccentric joints shown in the following sections.

8.2.1.2 Eccentric Joints Loaded at the Fastener Axis

This class of joints has the external load is introduced at the fastener axis (i.e. $s = a$ ), as shown in Figure 8-9.

Key Entities: Bolt: Fastener with a hexagonal head

Figure description:

Key Entities:

Bolt: Fastener with a hexagonal head.
Nut: Fastener component at the bottom of the assembly.
Clamped Parts: Two cross-hatched blocks joined by the bolt.

Visible Details:

$F_A$ : Axial force vectors pointing away from the top and bottom of the fastener.
$s = a$ : Labeled horizontal distance between the center axis of the bolt and a parallel dashed vertical line.
Dashed Lines: Indicate the central axis of loading and a secondary vertical reference line.
Interface: A horizontal line dividing the two clamped blocks.

Figure 8-9 - An Eccentrically Loaded Joint where $s = a$

The compliance of the clamped parts, $\mathcal{D}_{c}^{*}$ , is higher than $\mathcal{D}_{c}$ and is calculated by multiplying the compliance of an equivalent concentric joint by an eccentricity factor, according to,

\left[ \frac{s^{2}}{\frac{G_{c}^{2} A_{c}}{A_{sub}}} \right] \tag*{[8.2.5]}

which can be written in a more simple form as,

\delta_{c}^{*} = \delta_{c} \left(1 + \lambda^{2}\right) \tag*{[8.2.6]}

where $\lambda$ is the length ratio given by,

\lambda = \frac{s / G_{c}}{\sqrt{A_{c} / A_{sub}}} \tag{8.2.7}

where $A_{sub}$ is the effective clamping area defined below in Section 8.2.1.4.

8.2.1.3 General Case of an Eccentric Joint

This (more general) class of joints is shown in Figure <mark>7.2.4</mark>, and has clamped parts compliance given by,

\delta_{c}^{**} = \delta_{c} \left( 1 + \frac{a}{s} \lambda^{2} \right) \tag*{[8.2.8]}

where; $\lambda$ is defined by equation [8.2.7].

The value of $a$ in equation [8.2.8] is always taken as positive. The value of $s$ is positive if S-S and A-A are on the same side of O-O or negative if they are on opposite sides. Thus, $\delta_{c}^{**}$ can be either greater than or less than $\delta_{c}$ .

Entities: Bolted joint assembly featuring a bolt with a hexagonal head and nut, two clamped plates,

Figure description:

Entities: Bolted joint assembly featuring a bolt with a hexagonal head and nut, two clamped plates, and a designated compression zone indicated by hatching.
Reference Lines:
- Line A-A: Axis for external force application.
- Line S-S: Central axis of the bolt.
- Line O-O: Fulcrum or pivot axis.
Dimensional Parameters:
- u: Distance from line A-A to line O-O.
- a: Distance from the force application point to line O-O.
- s: Distance between the bolt axis (S-S and the reference line O-O.
- d: Nominal diameter of the bolt shank.
- D_sub: Diameter or width associated with the effective clamping/compression zone.
Forces:
- F_A: Opposing external tension forces applied along axis A-A.

Figure 8-10 - The General Eccentrically Loaded Joint

8.2.1.4 The Effective Clamping Area

The effective clamping area, $A_{sub}$ , used in equation [8.2.7] is the cross-sectional area of the assumed compression zone (see Section 7.6.2) at its greatest extent, and is calculated as follows,

a. For a clamped sleeve with narrow flanges with $D_{avail} \leq D_{lim}$ ,

A_{sub} = \frac{\pi}{4} (D_{\text{lim}}^{2} - d) \tag{8.2.9}

b. For wide flanges with $D_{avail} > 3D_{lim}$ ;

A_{sub} = \frac{\pi}{4} \left[ \left[ D_{uh,brg} + \frac{L_{c}}{10} \right]^{2} - d^{2} \right] \tag{8.2.10}

c. For the intermediate case with $D_{\text{lim}} < D_{\text{avail}} \leq 3D_{\text{lim}}$

A_{sub} = \frac{\pi}{4} (D_{uh,brg}^{2} - d^{2}) + \frac{\pi}{8} \left[ \frac{D_{\text{lim}}}{D_{uh,brg}} - 1 \right] \left[ \frac{D_{uh,brg} L_{c}}{5} + \frac{L_{c}^{2}}{100} \right] \tag{8.2.11}

8.2.2 The Joint Diagram

The joint diagram developed for concentric joints in Section 7 can also be used to describe the characteristics of eccentric axially loaded joints.

Triangle $S_{1}$ , $V_{1}$ , $P_{1}$ of Figure 8-11 represents the preloaded condition where the fastener tension $F_{b}$ is equal and opposite to the compressive load on the clamped parts $F_{K}$ . As the external eccentric load $F_{A}$ increases from zero the fastener extends under the partial load $\Delta F_{b,A,1}$ . This causes the clamped parts to lose compression linearly until the point when the joint begins to separate at the side nearest the load $F_{A}$ . During this phase the straight line $V_{1}$ to $H_{1}$ represents the behaviour of the clamped parts. If $F_{A}$ is further increased, the joint characteristics change to that of a hinge and lever producing a nonlinear load-deflection response of the clamped parts. The fastener therefore feels a high proportion of the external load $\Delta F_{b,A,1}$ than predicted by extrapolation of the line $V_{1}H_{1}$ .

Schematic Diagram of Geometric Forces Force and Geometric Relationships Diagram Annotations

Figure description:

Schematic Diagram of Geometric Forces

Force and Geometric Relationships Diagram

Annotations

F_V1 (vertical_line - position: left side, between baseline and v1 level)
F_A (vertical_line - position: center, between top line and curve)
F_K1 (vertical_line - position: right side, from baseline up)
Delta F_b,A,1 (vertical_line - position: top right, height of pulse)
Pulse waveform (text_label - position: upper right)

Label	Description
S1	Origin point on the horizontal dashed baseline
P1	Secondary point on the horizontal dashed baseline to the right of S1
v1	Peak point at the intersection of a solid line from S1 and a horizontal line
H1	Intermediate point on the dashed segment connecting v1 and P1
J1	Intersection point of the dashed line v1-P1 and a lower solid ray from S1
F_V1	Vertical distance from the baseline to the horizontal level of v1
F_A	Vertical dimension between the upper slanted line and a curved segment
F_K1	Vertical height measured from the baseline to the lower solid ray
Delta F_b,A,1	Vertical magnitude of the pulse waveform shown on the right

Notes: The diagram shows a geometric construction involving several points and vertical measurements labeled as forces or distances (F_V1, F_A, F_K1, Delta F_b,A,1. It includes a pulse waveform on the far right with its vertical extent marked relative to the horizontal level established at point v1.)

Figure 8-11 - The Joint Diagram for Eccentrically Loaded Joints showing the Non- Linearity of the Clamped Parts

Figure 8-12 shows the load increment in the fastener due to the same dynamic external load $F_{A}$ , and illustrates the fact that higher preload (indicated by $F_{v2}$ ) delay the onset of separation and thus $(V_{1}-H_{1}) < (V_{2}-H_{2})$ . The figure also indicates that, after one-sided opening commences, the higher the preload reduces the load increment in the fastener, i.e. $\Delta F_{b,A,2} < \Delta F_{b,A,1}$ .

Technical Engineering Diagram Geometric construction showing force or displacement vectors

Figure description:

Technical Engineering Diagram Geometric construction showing force or displacement vectors

Technical Schematic Analysis

Legend

Solid line — color: black; symbol: Solid line style
Dashed line — color: black; symbol: Dashed line style
Dash-dot line — color: black; symbol: Dash-dot line style

Annotations

Delta F_b,A,2 (text_label - position: Top right sinusoidal fluctuation)
FV2 (vertical_line - position: Leftmost vertical dimension line)
FV1 (vertical_line - position: Center-left vertical dimension line)
FA (vertical_line - position: Upper central vertical dimension line)
FK2 (vertical_line - position: Rightmost vertical dimension line)

| Label | Element | Description |\n| :--- | :--- | :--- |\n| S2 | Point | Base point located at the bottom left of the diagram. |\n| V2 | Point | Intersection vertex at the top of the primary diagonal from S2. |\n| H1 | Point | Upper point on the dashed curved path following V2. |\n| H2 | Point | Lower point on the dashed path below H1. |\n| J1 | Point | Reference point on an intermediate dash-dot line. |\n| J2 | Point | Upper point on the secondary steep diagonal line. |\n| P1 | Point | Intermediate point on the secondary steep diagonal line, defining the base of FV1. |\n| P2 | Point | Bottom point of the secondary steep diagonal line segment. |\n| FV2 | Dimension | Total vertical height from the S2 reference level to the top horizontal line through V2. |\n| FV1 | Dimension | Vertical height from the horizontal level through P1 to the top horizontal line. |\n| FA | Dimension | Vertical distance between the extension of the diagonal line and the curved path. |\n| FK2 | Dimension | Vertical height from the bottom reference plane to the level of point J2. |\n| Delta F_b,A,2 | Dimension | Amplitude of the periodic sinusoidal variation shown at the top right. |

Notes: The diagram illustrates a series of geometric relationships between various points (S2, V2, H1, H2, J1, J2, P1, P2 and their associated vertical dimensions (F. A periodic signal variation is also characterized at the top right.))

Figure 8-12 - Joint Diagram with a Higher Preload

8.2.3 The Force Ratio

For an eccentric axially loaded joint with a large contact area, assuming the clamped parts behave linearly, it can be shown that the force ratio, Φ e, n , prior to one-sided opening is given by,

\Phi_{e,n} = n_{ecc} \left[ \frac{\delta_{c}^{**}}{\delta_{b} + \delta_{c}^{*}} \right] \tag*{[8.2.12]}

where $\delta_{c}^{*}$ and $\delta_{c}^{**}$ are defined in Section 8.2.1, and $n_{ecc}$ is defined below in Section 8.2.4.

As was the case for concentric joints, the force ratio is used to determine the incremental loads experienced by the fastener and clamped parts due to the externally applied force according to the relations; $\Delta F_{b,A} = \Phi_{e,n} F_{A}$ and $\Delta F_{c,A} = (1 - \Phi_{e,n}) F_{A}$ .

The difference in the force ratio compared with that of the equivalent concentric joint is the quantitative consequence of the “prying” effect (see Section 8.1.2).

8.2.4 Loading Plane Factor

For eccentric axially loaded joints, the loading factor derived in Section 7.7 need to be corrected. The assumption is made that all cross-sections stay planar during deformations of the joint. First the loading plane factor has to be determined for a theoretical joint, where the fastener’s longitudinal axis is moved to a point that leads to a concentric axially loaded joint. Then the eccentric loading plane factor is calculated with,

n_{ecc} = \frac{n_{con} + a s \frac{L_{c}}{E_{c} I_{c}}}{1 + s^{2} \frac{L_{c}}{E_{c} I_{c}}} \tag{8.2.13}

where; $n_{con}$ is the concentric loading plane factor from Section 7.7.

8.2.5 The Interface Opening Limit

Figure 8-13 shows the limiting value of the external load, $F_{A,sep}$ , at which the interface begins to open. This is calculated by,

F_{A,sep} = \frac{F_{V}}{\left[1 + \frac{au}{G_{c}^{2}}\right]\left[1 + \frac{su}{G_{c}^{2}}\right] - \Phi_{e,n}} \tag{8.2.14}

where $u$ is the eccentricity f the point of first gapping (relative to the joint axis O-O).

Conversely for a given external load, $F_{A}$ , there can be a value of preload, $F_{V,sep}$ , at which one-sided opening can just occur,

F_{V,sep} = F_{K,sep} + (1 - \Phi_{e,n})F_{A} \tag{8.2.15}

where $F_{K,sep}$ is the clamping force at the onset of interface gapping as given by,

F_{K,sep} = \frac{(a - s)u}{G_{c}^{2} + su} F_{A} \tag{8.2.16}

Geometric Analysis of Force Components Force and Slope Geometric Diagram Annotations

Figure description:

Geometric Analysis of Force Components

Force and Slope Geometric Diagram

Annotations

F_V (vertical_dimension - position: Left-hand side, vertical height indicator)
F_{A,sep} (vertical_vector - position: Middle-right, vertical distance between the n=1 line and the curved boundary)
n = 1 (line_label - position: Associated with the top-most dashed diagonal line)
1 > n > 0 (line_label - position: Associated with the second dashed diagonal line from the top)

Feature	Label	Description
Origin Point	-	Intersection point of the baseline and multiple sloped lines at the bottom left
Steep Sloped Line	n = 1	Dashed line extending from the origin to the top right
Shallow Sloped Line	1 > n > 0	Dashed line extending from the origin to the right with a lower slope than n=1
Dash-Dot Line	-	Line extending from the origin between the horizontal baseline and the 1 > n > 0 line
Baseline	-	Solid horizontal line extending from the origin to the right
Vertical Dimension	F_V	Double-headed vertical arrow indicating the height from the baseline to the top horizontal reference line
Separation Force	F_{A,sep}	Vertical arrow pointing upwards from a curved line to the n=1 line
Reference Points	-	Black dots marking specific locations: origin, baseline end, line intersections, and vector endpoints

Notes: The diagram illustrates a relationship between various sloped parameters (n=1, 1 > n > 0 and force components (F_V, F_{A,sep}. It includes a mix of solid, dashed, and dash-dotted lines with specific intersection points highlighted by dots.))

Figure 8-13 - An Eccentrically Joint Loaded to the Interface Opening Limit with the Loading Plane Inside the Joint

See Reference 8.4 for the derivation of the equations [8.2.14] and [8.2.15].

Figure description:

Text: ECSS

Key Entities:

Letter 'E'
Letters 'CSS'

Visual Details:

The letter 'E' is purple and italicized, set within a white parallelogram.
The letters 'CSS' are white and italicized, set within a purple parallelogram.
The overall shape is a combined parallelogram with a purple border.

8.2.6 Preload Considerations

8.2.6.1 Preload Scatter

The effects of preload scatter due to friction variables during assembly are the same for eccentric joints as for concentric joints, and are described in Section 6.

8.2.6.2 Preload Loss

Embedding, as described in Section 6.4.2, mainly occurs after preloading but before external loading is applied. The amount of embedding preload loss for an eccentric joint can be calculated as for a concentric joint from,

F_{z} = \frac{f_{z}(1 - \Phi_{e,n})}{\delta_{c}} \tag{8.2.17}

where fZ is the plastic deformation caused by embedding as given by Table 6-3, and Φ e, n is the force ratio.

8.2Joints with Large Areas of Contact

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.

8.2.1 Compliance of the Clamped Parts

8.2.1.1 Overview

A relatively simple analysis (modified from Reference 8.4) of eccentric joint deformation is possible if the following assumptions are made:

• The clamped parts form a prismatic bending solid
• There is no one-sided gapping (i.e. the interface pressure is always positive)
• All the cross-sections of this prismatic solid remain flat under loading
• A linear stress distribution is established in these cross-sections
• The bending resistance of the fastener is much lower than that of the bending solid and can be neglected

C \leq (D_{uh,brg} + h_{\min}) \tag{8.2.1}

s \leq \frac{(D_{uh,brg} + h_{\min})}{2} \tag{8.2.2}

where C is the distance from the fastener joints axis to the flange edge as indicated in Figure 8-7. It is important that joints that do not satisfy these criteria are analysed using FEM or testing.

Entities: Bolt with a hexagonal head and nut, two clamped plates

Figure description:

Entities: Bolt with a hexagonal head and nut, two clamped plates (cross-hatched, and force vectors.
Forces: $F_A$ (applied tensile forces acting along axis A-A.
Dimensions/Variables:
- $a$ : Distance from axis O to axis A.
- $s$ : Distance from axis O to bolt centerline S.
- $h_{min}$ : Thickness of the top plate.
- $D_{uh,brg}$ : Bearing diameter under the bolt head.
- $u_B$ : Distance from left edge to bolt centerline S.
- $u$ : Distance from left edge to axis O.
- $C$ : Total width of the clamped interface.
Reference Axes:
- A: Axis of applied force.
- S: Centerline of the bolt.
- O: A reference axis located to the right of the bolt centerline.

Figure 8-7 - Limitations of the Eccentric Joint Analysis Method

b = \begin{cases} D_{uh,brg} + h_{\min} & \text{if } t > \left(D_{uh,brg} + h_{\min}\right) \\ t & \text{if } t \leq \left(D_{uh,brg} + h_{\min}\right) \end{cases} \tag*{[8.2.3]}

Entities and Dimensions: Assumed Interface Area AB for One Fastener: Indicated by the

Figure description:

Entities and Dimensions:

Assumed Interface Area $A_B$ for One Fastener: Indicated by the cross-hatched rectangular region.
$b$ : The width/length of the shaded interface area.
$t$ : The distance (pitch between the centerlines of two adjacent fasteners/holes.
Fasteners/Holes: Represented by two circular features with horizontal centerlines.

Figure 8-8 - The Interface Area for Multiple Fastener Joints

The clamped interface area shown in Figure 8-8 has an area, $A_{c} = b.C$ , and a radius of gyration,

G_{c} = \left[ \frac{I_{c}}{A_{c}} \right]^{\frac{1}{2}} \tag{8.2.4}

where $I_{c}$ is the second moment of area of the clamped interface (about its neutral axis), which is given by $bt^{3}/12$ for the interface shown in Figure 8-8.

Subject to the above assumptions, clamped parts compliance values can be calculated for the two classes of eccentric joints shown in the following sections.

8.2.1.2 Eccentric Joints Loaded at the Fastener Axis

This class of joints has the external load is introduced at the fastener axis (i.e. $s = a$ ), as shown in Figure 8-9.

Key Entities: Bolt: Fastener with a hexagonal head

Figure description:

Key Entities:

Bolt: Fastener with a hexagonal head.
Nut: Fastener component at the bottom of the assembly.
Clamped Parts: Two cross-hatched blocks joined by the bolt.

Visible Details:

$F_A$ : Axial force vectors pointing away from the top and bottom of the fastener.
$s = a$ : Labeled horizontal distance between the center axis of the bolt and a parallel dashed vertical line.
Dashed Lines: Indicate the central axis of loading and a secondary vertical reference line.
Interface: A horizontal line dividing the two clamped blocks.

Figure 8-9 - An Eccentrically Loaded Joint where $s = a$

\left[ \frac{s^{2}}{\frac{G_{c}^{2} A_{c}}{A_{sub}}} \right] \tag*{[8.2.5]}

which can be written in a more simple form as,

\delta_{c}^{*} = \delta_{c} \left(1 + \lambda^{2}\right) \tag*{[8.2.6]}

where $\lambda$ is the length ratio given by,

\lambda = \frac{s / G_{c}}{\sqrt{A_{c} / A_{sub}}} \tag{8.2.7}

where $A_{sub}$ is the effective clamping area defined below in Section 8.2.1.4.

8.2.1.3 General Case of an Eccentric Joint

This (more general) class of joints is shown in Figure <mark>7.2.4</mark>, and has clamped parts compliance given by,

\delta_{c}^{**} = \delta_{c} \left( 1 + \frac{a}{s} \lambda^{2} \right) \tag*{[8.2.8]}

where; $\lambda$ is defined by equation [8.2.7].

Entities: Bolted joint assembly featuring a bolt with a hexagonal head and nut, two clamped plates,

Figure description:

Entities: Bolted joint assembly featuring a bolt with a hexagonal head and nut, two clamped plates, and a designated compression zone indicated by hatching.
Reference Lines:
- Line A-A: Axis for external force application.
- Line S-S: Central axis of the bolt.
- Line O-O: Fulcrum or pivot axis.
Dimensional Parameters:
- u: Distance from line A-A to line O-O.
- a: Distance from the force application point to line O-O.
- s: Distance between the bolt axis (S-S and the reference line O-O.
- d: Nominal diameter of the bolt shank.
- D_sub: Diameter or width associated with the effective clamping/compression zone.
Forces:
- F_A: Opposing external tension forces applied along axis A-A.

Figure 8-10 - The General Eccentrically Loaded Joint

8.2.1.4 The Effective Clamping Area

The effective clamping area, $A_{sub}$ , used in equation [8.2.7] is the cross-sectional area of the assumed compression zone (see Section 7.6.2) at its greatest extent, and is calculated as follows,

a. For a clamped sleeve with narrow flanges with $D_{avail} \leq D_{lim}$ ,

A_{sub} = \frac{\pi}{4} (D_{\text{lim}}^{2} - d) \tag{8.2.9}

b. For wide flanges with $D_{avail} > 3D_{lim}$ ;

A_{sub} = \frac{\pi}{4} \left[ \left[ D_{uh,brg} + \frac{L_{c}}{10} \right]^{2} - d^{2} \right] \tag{8.2.10}

c. For the intermediate case with $D_{\text{lim}} < D_{\text{avail}} \leq 3D_{\text{lim}}$

A_{sub} = \frac{\pi}{4} (D_{uh,brg}^{2} - d^{2}) + \frac{\pi}{8} \left[ \frac{D_{\text{lim}}}{D_{uh,brg}} - 1 \right] \left[ \frac{D_{uh,brg} L_{c}}{5} + \frac{L_{c}^{2}}{100} \right] \tag{8.2.11}

8.2.2 The Joint Diagram

The joint diagram developed for concentric joints in Section 7 can also be used to describe the characteristics of eccentric axially loaded joints.

Schematic Diagram of Geometric Forces Force and Geometric Relationships Diagram Annotations

Figure description:

Schematic Diagram of Geometric Forces

Force and Geometric Relationships Diagram

Annotations

F_V1 (vertical_line - position: left side, between baseline and v1 level)
F_A (vertical_line - position: center, between top line and curve)
F_K1 (vertical_line - position: right side, from baseline up)
Delta F_b,A,1 (vertical_line - position: top right, height of pulse)
Pulse waveform (text_label - position: upper right)

Label	Description
S1	Origin point on the horizontal dashed baseline
P1	Secondary point on the horizontal dashed baseline to the right of S1
v1	Peak point at the intersection of a solid line from S1 and a horizontal line
H1	Intermediate point on the dashed segment connecting v1 and P1
J1	Intersection point of the dashed line v1-P1 and a lower solid ray from S1
F_V1	Vertical distance from the baseline to the horizontal level of v1
F_A	Vertical dimension between the upper slanted line and a curved segment
F_K1	Vertical height measured from the baseline to the lower solid ray
Delta F_b,A,1	Vertical magnitude of the pulse waveform shown on the right

Figure 8-11 - The Joint Diagram for Eccentrically Loaded Joints showing the Non- Linearity of the Clamped Parts

Technical Engineering Diagram Geometric construction showing force or displacement vectors

Figure description:

Technical Engineering Diagram Geometric construction showing force or displacement vectors

Technical Schematic Analysis

Legend

Solid line — color: black; symbol: Solid line style
Dashed line — color: black; symbol: Dashed line style
Dash-dot line — color: black; symbol: Dash-dot line style

Annotations

Delta F_b,A,2 (text_label - position: Top right sinusoidal fluctuation)
FV2 (vertical_line - position: Leftmost vertical dimension line)
FV1 (vertical_line - position: Center-left vertical dimension line)
FA (vertical_line - position: Upper central vertical dimension line)
FK2 (vertical_line - position: Rightmost vertical dimension line)

Figure 8-12 - Joint Diagram with a Higher Preload

8.2.3 The Force Ratio

For an eccentric axially loaded joint with a large contact area, assuming the clamped parts behave linearly, it can be shown that the force ratio, Φ e, n , prior to one-sided opening is given by,

\Phi_{e,n} = n_{ecc} \left[ \frac{\delta_{c}^{**}}{\delta_{b} + \delta_{c}^{*}} \right] \tag*{[8.2.12]}

where $\delta_{c}^{*}$ and $\delta_{c}^{**}$ are defined in Section 8.2.1, and $n_{ecc}$ is defined below in Section 8.2.4.

The difference in the force ratio compared with that of the equivalent concentric joint is the quantitative consequence of the “prying” effect (see Section 8.1.2).

8.2.4 Loading Plane Factor

n_{ecc} = \frac{n_{con} + a s \frac{L_{c}}{E_{c} I_{c}}}{1 + s^{2} \frac{L_{c}}{E_{c} I_{c}}} \tag{8.2.13}

where; $n_{con}$ is the concentric loading plane factor from Section 7.7.

8.2.5 The Interface Opening Limit

Figure 8-13 shows the limiting value of the external load, $F_{A,sep}$ , at which the interface begins to open. This is calculated by,

F_{A,sep} = \frac{F_{V}}{\left[1 + \frac{au}{G_{c}^{2}}\right]\left[1 + \frac{su}{G_{c}^{2}}\right] - \Phi_{e,n}} \tag{8.2.14}

where $u$ is the eccentricity f the point of first gapping (relative to the joint axis O-O).

Conversely for a given external load, $F_{A}$ , there can be a value of preload, $F_{V,sep}$ , at which one-sided opening can just occur,

F_{V,sep} = F_{K,sep} + (1 - \Phi_{e,n})F_{A} \tag{8.2.15}

where $F_{K,sep}$ is the clamping force at the onset of interface gapping as given by,

F_{K,sep} = \frac{(a - s)u}{G_{c}^{2} + su} F_{A} \tag{8.2.16}

Geometric Analysis of Force Components Force and Slope Geometric Diagram Annotations

Figure description:

Geometric Analysis of Force Components

Force and Slope Geometric Diagram

Annotations

F_V (vertical_dimension - position: Left-hand side, vertical height indicator)
F_{A,sep} (vertical_vector - position: Middle-right, vertical distance between the n=1 line and the curved boundary)
n = 1 (line_label - position: Associated with the top-most dashed diagonal line)
1 > n > 0 (line_label - position: Associated with the second dashed diagonal line from the top)

Feature	Label	Description
Origin Point	-	Intersection point of the baseline and multiple sloped lines at the bottom left
Steep Sloped Line	n = 1	Dashed line extending from the origin to the top right
Shallow Sloped Line	1 > n > 0	Dashed line extending from the origin to the right with a lower slope than n=1
Dash-Dot Line	-	Line extending from the origin between the horizontal baseline and the 1 > n > 0 line
Baseline	-	Solid horizontal line extending from the origin to the right
Vertical Dimension	F_V	Double-headed vertical arrow indicating the height from the baseline to the top horizontal reference line
Separation Force	F_{A,sep}	Vertical arrow pointing upwards from a curved line to the n=1 line
Reference Points	-	Black dots marking specific locations: origin, baseline end, line intersections, and vector endpoints

Figure 8-13 - An Eccentrically Joint Loaded to the Interface Opening Limit with the Loading Plane Inside the Joint

See Reference 8.4 for the derivation of the equations [8.2.14] and [8.2.15].

Figure description:

Text: ECSS

Key Entities:

Letter 'E'
Letters 'CSS'

Visual Details:

The letter 'E' is purple and italicized, set within a white parallelogram.
The letters 'CSS' are white and italicized, set within a purple parallelogram.
The overall shape is a combined parallelogram with a purple border.

8.2.6 Preload Considerations

8.2.6.1 Preload Scatter

The effects of preload scatter due to friction variables during assembly are the same for eccentric joints as for concentric joints, and are described in Section 6.

8.2.6.2 Preload Loss

F_{z} = \frac{f_{z}(1 - \Phi_{e,n})}{\delta_{c}} \tag{8.2.17}

where fZ is the plastic deformation caused by embedding as given by Table 6-3, and Φ e, n is the force ratio.