Fastened Shear Joint - Ultimate and Yield Strength Check

Shear-Out Length

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

Equivalent Bearing Diameter for Countersunk Fasteners

Problem Statement

When a fastener is countersunk into a sheet or plate, the bearing area varies through the thickness. The full shank provides bearing along its length, but the countersunk head only contributes partial bearing due to its conical shape. This derivation determines an equivalent average diameter D_avg to use in bearing strength calculations, accounting for the reduced effectiveness of the countersunk head region.

Key Assumption

Half-Effective Head: The countersunk head is assumed to be only 50% effective for bearing. This is a conservative engineering approximation recognizing that the conical head surface does not provide the same bearing support as the cylindrical shank.

Geometry

Consider a countersunk fastener with:

• D = fastener shank diameter (constant cylindrical section)
• t = total sheet thickness
• t_cs = countersink depth = H + t_sf
• H = fastener head height (from shank to outer surface of head)
• t_sf = subflush depth (positive if countersunk below surface, negative if proud)
• θ = countersink head angle (e.g., 100°)

Bearing Area Components

The bearing area can be divided into three regions through the sheet thickness:

Region 1: Full Shank (Below Countersink)

Length: L₁ = t - t_cs

In this region, the full shank diameter D bears against the sheet material. The bearing area contribution is:

A₁ = D × (t - t_cs)

Region 2: Head Below Shank Level (Lower Half of Head)

Length: L₂ = H / 2 (half the head height)

This is the portion of the countersunk head between the shank and the midpoint of the head. Since we assume the head is 50% effective, and this is the lower half where bearing support is better, the bearing area contribution is:

A₂ = D × (H / 2) × 0.5 = D × H / 4

The 0.5 factor accounts for the half-effectiveness assumption.

Region 3: Head Above Midpoint (Upper Half of Head)

Length: L₃ = H / 2

In this region, the head diameter varies from D at the midpoint to a larger diameter at the top. The diameter at distance z from the midpoint is:

D(z) = D + 2z × tan(θ/2)

The average diameter in this region can be found by integrating:

D_avg,3 = (2/H) ∫₀^H/2 [D + 2z × tan(θ/2)] dz

Evaluating this integral:

D_avg,3 = (2/H) × [D×z + z² tan(θ/2)]₀^H/2

D_avg,3 = (2/H) × [D×(H/2) + (H/2)² tan(θ/2)]

D_avg,3 = D + (H/2) × tan(θ/2)

Accounting for the half-effective assumption, the bearing area contribution from this region is:

A₃ = D_avg,3 × (H/2) × 0.5

A₃ = [D + (H/2) × tan(θ/2)] × (H/2) × 0.5

A₃ = D×H/4 + (H²/4) × tan(θ/2) × 0.5

A₃ = D×H/4 + (H²/8) × tan(θ/2)

Total Equivalent Bearing Area

The total bearing area is the sum of all three regions:

A_total = A₁ + A₂ + A₃

A_total = D×(t - t_cs) + D×H/4 + D×H/4 + (H²/8) × tan(θ/2)

A_total = D×(t - t_cs) + D×H/2 + (H²/8) × tan(θ/2)

Factor out common terms:

A_total = D×(t - t_cs + H/2) + (H²/8) × tan(θ/2)

Equivalent Average Diameter

The equivalent average diameter is found by dividing the total bearing area by the sheet thickness:

D_avg = A_total / t

D_avg = [D×(t - t_cs + H/2) + (H²/8) × tan(θ/2)] / t

Rearranging:

D_avg = (D/t) × (t - t_cs + H/2 + (H²/4) × tan(θ/2) / D)

Or equivalently:

D_avg = (D/t) × [t - t_cs + H/2 + (H²/4D) × tan(θ/2)]

Implementation Note

In the code, since we're already working with diameter D, the formula simplifies to:

D_avg = (D/t) × [t - t_cs + H/2 + (H²/4) × tan(θ × π/360)]

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

Note that the (H²/4) term in the code absorbs the division by D since it's multiplied back in the overall expression.

Physical Interpretation

The equivalent diameter consists of several contributions:

• Base shank contribution: D × (t - t_cs)/t - Full shank below countersink
• Half-effective head contribution: D × (H/2)/t - Lower portion of head at 50% effectiveness
• Tapered head contribution: (H²/4t) × tan(θ/2) - Additional bearing from upper tapered portion at 50% effectiveness

For a protruding head fastener (H = 0, t_cs = 0), the formula correctly reduces to D_avg = D.

Subflush Depth Effect

The subflush depth t_sf affects the countersink depth:

• t_sf > 0: Fastener is subflush (countersunk below surface) → increases t_cs
• t_sf = 0: Fastener is flush with surface
• t_sf < 0: Fastener is proud (head protrudes above surface) → decreases t_cs

A greater subflush depth reduces the effective bearing area by increasing t_cs, which decreases the full-bearing shank length.

Application to Bearing Strength

The bearing strength is calculated using this equivalent diameter:

P_bru = K_wp × F_bru × D_avg × t

where:

• K_wp = wet pin factor (accounts for moisture effects)
• F_bru = ultimate bearing allowable stress (from material data)
• D_avg = equivalent bearing diameter
• t = sheet thickness

Using D_avg instead of D provides a more accurate (and appropriately conservative) estimate of bearing strength for countersunk fasteners, accounting for the reduced bearing effectiveness in the countersunk region.

Important Note

This is an approximation! The actual bearing behavior of countersunk fasteners is complex and depends on many factors including installation quality, hole preparation, and load distribution. The 50% effectiveness assumption is a simplified engineering approach. For critical applications, use bearing allowables from actual fastener/material combination testing when available.

Shear-Out Diameter

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.