Fastened Joint Strength: Method

Shear-Out Length Derivation for Fastened Joints

Problem Statement

When a fastener transfers load near a free edge, the material between the fastener hole and the edge can fail in shear. This derivation determines the minimum shear-out length x for both straight-edge (sheet) and radius-edge (lug) configurations, accounting for the effective fastener diameter D_avg which is particularly important for countersunk fasteners.

Key Assumption: 40° Shear-Out Angle

Standard Practice: The shear-out failure plane is assumed to occur at 40° from the direction of load application. This is a well-established convention in aerospace structural analysis.

References for the 40° Angle:

• Bruhn, E.F. "Analysis and Design of Flight Vehicle Structures" (Chapter C9) - Documents the 40° shear-out angle as standard practice based on experimental testing
• ANC-5 Bulletin "Strength of Metal Aircraft Elements" (referenced in this tool's calculations) - Establishes empirical methods for joint analysis including shear-out at 40°
• MIL-HDBK-5J / MMPDS - Uses the 40° angle in bearing and shear-out strength calculations for mechanical joints

Physical Interpretation: The 40° angle represents the typical orientation of the critical shear plane observed in experimental testing of mechanically fastened joints. The actual failure plane can vary based on material properties, joint configuration, and loading conditions, but 40° provides a conservative and widely-validated design value.

Geometry Definitions

The following parameters define the joint geometry:

• D_avg = average effective fastener diameter (accounts for countersinking effects - see separate derivation)
• e = edge distance from hole center to nearest free edge
• c = offset distance from lug geometric center to hole center (lug geometry only)
• R = lug radius from geometric center to free edge
• θ = 40° = shear-out angle from load direction
• x = shear-out length (minimum distance along failure plane)
• t = sheet/plate thickness
• F_su = ultimate shear stress of the material

Why Use D_avg Instead of D?

For countersunk fasteners, the effective diameter varies through the thickness due to the conical countersink geometry. Using the nominal shank diameter D would overestimate the available material for shear-out resistance. The average diameter D_avg accounts for this variation and provides a more accurate (and appropriately conservative) representation of the effective fastener size for strength calculations.

For protruding head fasteners, D_avg = D, so the formulas apply universally to both configurations.

Case 1: Straight Edge (Sheet) Geometry

Consider a fastener in a sheet with a straight free edge perpendicular to the load direction.

Geometry Setup:

• Fastener center is at distance e from the free edge
• Hole edge is at distance D_avg/2 from fastener center
• Load is applied perpendicular to the free edge
• Critical shear plane runs at 40° from the load direction

Derivation:

The shear-out failure path begins at the edge of the fastener hole and extends to the free edge at a 40° angle from the load direction.

1. Distance from fastener center to free edge:

e

2. Distance from fastener center to hole edge (along load direction):

D_avg / 2

3. Projected component of hole radius in the load direction at 40°:

(D_avg / 2) × cos(40°)

4. Shear-out length from hole edge to free edge:

x = e - (D_avg / 2) × cos(40°)

x = e - D_avg × cos(40°) / 2

Final Formula (Straight Edge):

x = e - D_avg × cos(40°) / 2

Case 2: Radius Edge (Lug) Geometry

Consider a fastener in a lug with a circular free edge. The hole may be offset from the lug geometric center.

Geometry Setup:

• Lug has circular perimeter with radius R measured from the lug geometric center
• Fastener hole center is offset by distance c from the lug geometric center
• Edge distance e is the minimum distance from hole center to lug perimeter
• Relationship: R = e - c (radius equals edge distance minus offset)
• Critical shear plane runs at 40° from the load direction

Coordinate System:

• Place origin at lug geometric center
• Load direction along positive x-axis
• Hole center at distance c from origin (along load direction)
• Lug perimeter is a circle: x² + y² = R²

Derivation:

The shear-out failure path begins at the edge of the fastener hole (at 40° from load direction) and extends to the intersection with the lug perimeter.

1. Hole edge location at 40° angle:

The edge of the fastener hole at the critical 40° shear plane is located at:

x_hole = c + (D_avg / 2) × cos(40°)

y_hole = (D_avg / 2) × sin(40°)

2. Lug perimeter intersection along 40° line:

The line at 40° from the hole edge can be parameterized as:

x = c + (D_avg / 2) × cos(40°) + s × cos(40°)

y = (D_avg / 2) × sin(40°) + s × sin(40°)

where s is the distance along the line from the hole edge.

3. Intersection with lug circle (x² + y² = R²):

For the lug geometry, we need to find where the shear plane intersects the circular lug boundary. Using geometric analysis of the circle and the offset hole, the minimum shear length can be derived.

The critical insight is that the shear plane must span from the hole edge to the lug perimeter. For a hole centered at distance c from the lug center, with lug radius R, the geometry yields:

4. Geometric solution:

Using the constraint that the shear plane intersects both the hole edge and lug perimeter, and applying the Pythagorean theorem to the circular geometry:

x = c + R × [√(1 - (D_avg/(2R))² × sin²(40°)) - (D_avg × cos(40°))/(2R)]

This can be rewritten as:

x = c + R × √(1 - (D_avg/(2R))² × sin²(40°)) - D_avg × cos(40°) / 2

Final Formula (Radius Edge / Lug):

x = c + R × [√(1 - (D_avg/(2R))² × sin²(40°)) - (D_avg × cos(40°))/(2R)]

where R = e - c

Special Case: Centered Hole (c = 0)

When the hole is centered in the lug (c = 0), then R = e, and the formula simplifies to:

x = e × [√(1 - (D_avg/(2e))² × sin²(40°)) - (D_avg × cos(40°))/(2e)]

Implementation Notes

In the code implementation:

1. The 40° angle is converted to radians:

40 × π / 180 = 0.6981 radians

2. For straight edge:

x = e - D_avg * cos(40*pi/180) / 2

3. For radius edge (lug):

R = e - c

x = c + R*[(1-(D_avg/(2*R))^2 * sin(40*pi/180)^2)^0.5 - D_avg * cos(40*pi/180)/(2*R)]

Note: The code uses the square root form [...]^0.5 which is equivalent to √[...] in mathematical notation.

Application to Shear-Out Strength

The shear-out length x is used to calculate the ultimate shear-out strength:

P_sou = 2 × x × t × F_su

Where:

• P_sou = ultimate shear-out strength
• x = shear-out length (derived above)
• t = sheet/plate thickness
• F_su = ultimate shear stress allowable
• Factor of 2 = two shear planes (one on each side of the fastener)

Physical Interpretation

Effect of Edge Distance (e):

• Larger edge distance e increases shear length x, improving strength
• Minimum edge distance requirements (typically e/D ≥ 1.5 to 2.0) ensure adequate shear-out resistance
• For very small edge distances, shear-out becomes the critical failure mode

Effect of Fastener Diameter (D_avg):

• Larger fastener diameter reduces available shear length x
• For countersunk fasteners, D_avg > D, further reducing shear length
• This creates a trade-off between bearing strength (favors larger D) and shear-out strength (favors smaller D)

Effect of Lug Offset (c):

• For lug geometry, offset c toward the load direction increases shear length
• Offset away from load direction (negative c) decreases shear length
• Optimum lug design balances bearing, shear-out, and net section failures

Two Shear Planes:

The factor of 2 in the shear-out strength formula (P_sou = 2 × x × t × F_su) accounts for shear planes on both sides of the fastener hole. Each plane contributes equally to the total shear-out resistance.

Design Recommendations

Critical Edge Distance: The minimum edge distance should satisfy e/D_avg ≥ 1.5 to 2.0 to prevent premature shear-out failure. Values below 1.5 may require special analysis or testing.

Countersunk Fasteners: When using countersunk fasteners, remember that D_avg > D, which reduces the effective shear-out length. This effect becomes more pronounced for deeper countersinks and thinner sheets (higher t_cs/t ratios).

References

1. Bruhn, E.F. (1973). "Analysis and Design of Flight Vehicle Structures." Tri-State Offset Company, Chapter C9.
2. ANC-5 Bulletin (1951). "Strength of Metal Aircraft Elements." Army-Navy-Civil Aircraft Design Committee.
3. MIL-HDBK-5J (2003). "Metallic Materials and Elements for Aerospace Vehicle Structures." U.S. Department of Defense.
4. MMPDS (Current Edition). "Metallic Materials Properties Development and Standardization." Battelle Memorial Institute.

Important Note

Design Validation: These formulas provide analytical estimates based on the 40° shear-out angle assumption. For critical applications, design allowables should be validated through testing of actual fastener/material combinations and joint configurations. Local stress concentrations, fastener installation quality, and load distribution effects may influence actual failure modes and strengths.

Equivalent Bearing Diameter for Countersunk Fasteners

Problem Statement

When a fastener is countersunk into a sheet or plate, the bearing width varies through the thickness. The straight shank bears over its full diameter, but the conical head sits in a flared hole and is only partly effective in bearing. This derivation gives the equivalent bearing diameter D_avg,br that the tool uses in place of D in the bearing strength, and the average hole diameter D_avg it uses for the edge-distance ratio.

Key Assumption

Partly effective head height: only the lower part of the countersunk head, a fraction K_H,eff of its height measured from the shank, is credited with bearing. The tool uses K_H,eff = 0.50 (lower half of the head). This is a conservative engineering approximation; a value near 0.62 fits countersunk rivet tests better. Because the hole is tapered, half the height is less than half the head's projected area.

Geometry

• D = fastener shank diameter
• t = sheet or plate thickness
• H = head height (from the shank to the top of the head)
• t_sf = sub-flush depth (positive below the surface, negative if the head is proud)
• t_cs = countersink depth = H + t_sf, or (D_cs − D) / (2 tan(θ/2)) when a measured countersink diameter D_cs is entered
• θ = countersink included angle (90°, 100° or 130°)

Measured from the bottom of the countersink, the hole width at a height z inside the cone is

D(z) = D + 2z tan(θ/2)

Bearing Area

The projected bearing area is the bearing width integrated through the thickness. Two regions contribute:

Straight shank

Over the length below the countersink, t − t_cs, the full shank diameter bears:

A_shank = D (t − t_cs)

Countersunk head

The head fills the cone over its height H, but only its lower K_H,eff H is credited. The projected bearing area is the hole width integrated over that effective height:

A_head = ∫₀^{K_H,eff H} [D + 2z tan(θ/2)] dz

A_head = [D z + z² tan(θ/2)]₀^{K_H,eff H}

A_head = K_H,eff D H + K_H,eff² H² tan(θ/2)

The effectiveness factor appears squared in the taper term because the credited part of the cone is both shorter and narrower than the whole head. (Crediting a fraction of the whole head's area instead would give K_H,eff H² tan(θ/2), a larger value; that is not the model used.) The upper part of the head and any sub-flush depth above it bear nothing; the sub-flush depth enters only through t_cs, which shortens the shank region.

Equivalent Bearing Diameter

Adding the two regions and dividing by the thickness gives the equivalent diameter:

D_avg,br = (A_shank + A_head) / t

D_avg,br = [D (t − t_cs) + K_H,eff D H + K_H,eff² H² tan(θ/2)] / t

Factoring out D/t:

D_avg,br = (D / t) [t − t_cs + K_H,eff H + (K_H,eff² H² / D) tan(θ/2)]

This is the expression printed by the tool, with K_H,eff = 0.50 and the half-angle written as θ × π/360 (the included angle in degrees converted to the half-angle in radians). With K_H,eff = 0.50 the head terms are H/2 and H²/(4D) tan(θ/2). Dropping the taper term leaves t − t_cs + H/2, the common "half the countersink depth is effective" rule of thumb.

Example

Tool defaults with plate 1 countersunk 100°: D = 0.2495 in, t = 0.188 in, H = 0.107 in, and D_cs = 0.510 in, so t_cs = (0.510 − 0.2495) / (2 tan 50°) = 0.1093 in:

D_avg,br = (0.2495 / 0.188) [0.188 − 0.1093 + 0.5 × 0.107 + (0.5² × 0.107² / 0.2495) tan 50°] = 0.1936 in

The bearing diameter is 78 percent of the shank diameter: the countersink takes 0.109 in of the 0.188 in thickness out of full shank bearing, and the lower half of the head gives back the equivalent of 0.067 in.

Where This Sits Among Common Practice

When no tested joint allowable exists for a countersunk installation, these are the usual approaches, from most to least conservative, with the equivalent bearing diameter each gives for the example above:

Method	Davg,br
Straight shank only, t − tcs (tension-head bolts, fatigue-critical or knife-edge joints)	0.1044 in
Half the countersink depth effective, t − tcs/2, no taper credit (company stress-manual rule of thumb)	0.1769 in
Lower half of the head with taper credit (this tool, KH,eff = 0.50)	0.1936 in
Half of the head's projected area (KH,eff H² taper term, not used)	0.2117 in

Tested joint allowables (MIL-HDBK-5J Chapter 8, MMPDS) govern whenever they exist for the fastener and sheet combination; the handbook publishes flush-head joint strengths from tests precisely because the sheet bearing allowables of Chapter 3 do not apply directly to a countersunk hole. A fourth option, scaling the protruding-head bearing strength by the countersunk-to-protruding ratio of the nearest tested combination, is common for bolts and lockbolts that the handbook tests only as rivets.

Average Hole Diameter for e/D

The edge-distance ratio uses the hole, not the fastener. The hole is D wide below the countersink and widens to D + 2 t_cs tan(θ/2) at the surface. Averaging the hole width over the thickness:

D_avg = [D (t − t_cs) + ∫₀^t_cs (D + 2z tan(θ/2)) dz] / t = D + t_cs² tan(θ/2) / t

For the example, D_avg = 0.2495 + 0.1093² tan 50° / 0.188 = 0.3252 in. The tool divides the edge distance by this value, so a countersunk hole needs more edge distance than a straight one for the same e/D.

Limits

• Protruding head (H = 0, t_cs = 0): both expressions reduce to D.
• Sub-flush head (t_sf > 0): t_cs grows, the shank region shrinks and D_avg,br falls; the unfilled cone above the head is not credited.
• Proud head (t_sf < 0): t_cs is less than H; the expression still credits the lower K_H,eff H of the head, which is inside the sheet as long as the head is not proud by more than half its height. Keep proud heads small.
• Knife edge (t_cs/t → 1): the shank region vanishes and only the reduced head area carries bearing; the tool warns above t_cs/t = 0.67.

Application to Bearing Strength

P_bru = K_wp F_bru D_avg,br t,   P_bry = K_wp F_bry D_avg,br t

where K_wp is the wet-pin factor and F_bru, F_bry are the bearing allowables at the e/D in use. Since D_avg,br t is the effective bearing area, the product is the bearing load the countersunk sheet can carry.

Important Note

This is an approximation. Countersunk bearing depends on the head and sheet material, hole preparation, installation and the load distribution along the cone. Use joint allowables from tests of the actual fastener and sheet combination (MIL-HDBK-5J Chapter 8) when they exist; the effectiveness factor is a design-guidance substitute, not a replacement.

Average Diameter for Shear-Out Strength with Countersunk Fasteners

Problem Statement

When a fastener is countersunk into a sheet or plate, the effective diameter that resists shear-out varies through the thickness. Near the surface, the countersink removes material, reducing the effective fastener diameter. This derivation determines an average diameter D_avg to use in shear-out strength calculations.

Geometry

Consider a countersunk fastener with:

• D = fastener shank diameter (constant below countersink)
• t = total sheet thickness
• t_cs = countersink depth (depth from surface to where full shank diameter begins)
• θ = countersink head angle (e.g., 100°)

At the sheet surface, the countersink hole has a larger diameter. As we go deeper into the sheet, the diameter decreases linearly until reaching the shank diameter D at depth t_cs.

Diameter Variation with Depth

Define a coordinate z measured from the sheet surface (z = 0 at surface, z = t at bottom of sheet).

Within the countersink region (0 ≤ z ≤ t_cs), the hole diameter at depth z is:

D(z) = D + 2 × z × tan(θ/2)

where tan(θ/2) is the taper rate. The factor of 2 accounts for both sides of the countersink cone.

Below the countersink (z > t_cs), the diameter is constant:

D(z) = D

Average Diameter Derivation

For shear-out calculations, we need an average effective diameter weighted by the sheet thickness. The shear-out load path goes through the entire thickness, so we integrate:

D_avg = (1/t) ∫₀^t D(z) dz

Split the integral into two regions:

D_avg = (1/t) [∫₀^t_cs (D + 2z tan(θ/2)) dz + ∫_tcs^t D dz]

Evaluate the first integral (countersink region):

∫₀^t_cs (D + 2z tan(θ/2)) dz = [D×z + z² tan(θ/2)]₀^t_cs

= D × t_cs + t_cs² tan(θ/2)

Evaluate the second integral (shank region):

∫_tcs^t D dz = D × (t - t_cs)

Combine the results:

D_avg = (1/t) [D × t_cs + t_cs² tan(θ/2) + D × (t - t_cs)]

D_avg = (1/t) [D × t + t_cs² tan(θ/2)]

D_avg = D + (t_cs²/t) × tan(θ/2)

Implementation Note

In the code, the countersink angle is stored as the full angle (e.g., 100°), so we use:

D_avg = D + (t_cs²/t) × tan(θ × π/360)

where θ × π/360 converts the full angle in degrees to the half-angle in radians.

Physical Interpretation

The average diameter equals the shank diameter D plus a correction term proportional to:

• The square of the countersink depth (t_cs²)
• The inverse of the sheet thickness (1/t)
• The taper rate tan(θ/2)

For protruding head fasteners (t_cs = 0), D_avg = D as expected. For deeper countersinks in thinner sheets, the average diameter increases more significantly.

Application to Shear-Out Strength

The shear-out strength is calculated using this average diameter:

P_sou = 2 × x × t × F_su

where x is the shear length that depends on edge distance e and the average diameter:

x = e - D_avg × cos(40°) / 2

Using D_avg instead of D accounts for the reduced material in the countersink region, providing a more accurate (and conservative) estimate of shear-out strength.