Bolt Group Analysis Calculator
Find the maximum resultant force on a bolt in an eccentrically loaded rectangular bolt group using the elastic (vector) method.
🔩 What is Bolt Group Analysis?
Bolt group analysis finds the maximum force carried by any single bolt in a group of bolts sharing an eccentrically applied load, such as a bracket bolted to a column with the load applied some distance away from the bolt pattern's centroid. The most common approach, the elastic (vector) method, treats the bolt group like a rigid plate resisting torsion, combining a uniform direct shear force with a torsional force that grows with distance from the centroid.
Structural engineers use this analysis whenever a connection's load line does not pass through the bolt group's centroid, bracket connections, moment connections, and any bolted joint carrying an off-center shear load. The applied moment M = P x e (load times eccentricity) creates additional shear demand on the bolts farthest from the centroid, which is why corner bolts in a rectangular pattern are almost always the ones checked.
A common point of confusion is assuming every bolt in the group carries the same force. Under the elastic method, only the direct shear component (P divided by the number of bolts) is uniform, the torsional component grows linearly with each bolt's distance from the centroid, so the corner bolts farthest away carry substantially more force than bolts near the center.
This calculator computes the polar moment of inertia for a rectangular bolt grid, the direct shear per bolt, and the torsional force components at the critical corner bolt, then combines them vectorially to report the maximum resultant bolt force, with every step shown.
📐 Formula
📖 How to Use This Calculator
Steps
💡 Example Calculations
Example 1 — Bracket Connection, 2×3 Bolt Group
P = 80 kN, e = 150 mm, nx = 2, ny = 3, sx = sy = 75 mm
Example 2 — Larger Bracket, 2×4 Bolt Group
P = 120 kN, e = 200 mm, nx = 2, ny = 4, sx = sy = 80 mm
Example 3 — Small 2×2 Bolt Group
P = 60 kN, e = 100 mm, nx = 2, ny = 2, sx = sy = 60 mm
❓ Frequently Asked Questions
🔗 Related Calculators
What is the elastic (vector) method for bolt groups?
The elastic method analyzes an eccentrically loaded bolt group by treating it like a rigid plate that resists the applied moment through shear forces proportional to each bolt's distance from the group's centroid, similar to how torsional shear stress varies with radius in a solid shaft. It combines this torsional force with the group's direct (uniform) shear at each bolt to find the critical bolt's resultant force.
What is the formula for the maximum bolt force?
R = sqrt(Rx^2 + (Ry + Fd)^2), where Rx = My/J and Ry = Mx/J are the torsional force components at the critical bolt (x, y), Fd = P/n is the direct shear per bolt, M = P x e is the applied moment, and J is the bolt group's polar moment of inertia.
What is the polar moment of inertia J of a bolt group?
J = sum of (x^2 + y^2) for every bolt, measured from the group centroid. For a rectangular grid of nx columns and ny rows, this simplifies to J = ny x (sum of x-squared per column) + nx x (sum of y-squared per row), since bolt x-coordinates repeat identically down every row and y-coordinates repeat across every column.
Which bolt is the critical (most heavily loaded) bolt?
For a rectangular bolt group, the critical bolt is always at a corner, the point farthest from the group's centroid, since torsional shear force increases with distance from the centroid. This calculator automatically evaluates the resultant force at that corner bolt.
Is the elastic method conservative or exact?
The elastic method is a simplified, generally conservative approach, it assumes every bolt behaves as a linear elastic spring. The more advanced instantaneous-center-of-rotation method (used to generate AISC's published bolt group coefficient tables) accounts for the actual nonlinear load-deformation behavior of bolts and typically predicts a higher group capacity.
Why does increasing bolt spacing reduce the maximum bolt force?
Increasing bolt spacing increases the polar moment of inertia J, which appears in the denominator of the torsional force terms Rx = My/J and Ry = Mx/J. A larger J for the same applied moment directly reduces the torsional force share carried by the most heavily loaded bolt.
Does this calculator account for bolt shear capacity?
No, this calculator computes only the applied resultant force on the critical bolt. Comparing that force against the bolt's actual design shear capacity (based on its diameter, grade, and whether threads are in the shear plane) is a separate final check using standard bolt shear strength tables or formulas.
What load and eccentricity direction does this calculator assume?
This calculator assumes the applied shear load acts vertically (in-plane) with its eccentricity measured horizontally from the bolt group's centroid, the standard configuration for an eccentrically loaded bracket bolted to a supporting column or wall.
Can this calculator handle a non-rectangular bolt pattern?
No, this calculator is scoped to a rectangular grid of evenly spaced bolt columns and rows, the most common configuration in practice. An irregular bolt pattern requires computing each bolt's individual (x, y) coordinates and would need a dedicated bolt-by-bolt tool.
What units does this calculator use?
Load is entered in kilonewtons, eccentricity and bolt spacing in millimeters, with bolt columns and rows as plain counts. Results (direct shear, torsional components, and the resultant bolt force) share the same force unit as the applied load.