Bolt Group Calculator — Eccentric Load & Force Distribution

Enter a bolt pattern — a rectangular grid or individual coordinates — then apply a load with an eccentricity, and read the force in every single bolt. The tool finds the group’s elastic center, computes the polar moment of inertia, splits the load into a direct-shear share and a moment share per bolt, and flags the worst bolt with a pass/fail utilization verdict.

Grid & coordinate layouts Per-bolt force table Metric and imperial units

Input Parameters

Bolt pattern geometry, fastener, and the eccentric load.

Units

Layout Modehow the pattern is defined

Rows (Y)

Columns (X)

Spacing X

Spacing Y

Bolt Diameter

Grade

Shear Planesshear planes per bolt

Applied Load P

Load Anglefrom X-axis, °

Eccentricity e_xload position, mm

Eccentricity e_yload position, mm

e_x / e_y locate the applied load P in the same coordinate system as the bolt pattern. The moment about the elastic center is computed automatically from the offset between that point and the centroid — 90° keeps the load straight up.

PASS — worst bolt utilization 16.6%

The most-loaded bolt carries 5.008 kN against a grade-based allowable of 30.159 kN per bolt (single shear, threads included).

Group Properties

Bolts in group6

Elastic center (X, Y)80.00, 50.00 mm

I_p (polar moment of inertia)40,600 mm²

I_x (about horizontal axis)15,000 mm²

I_y (about vertical axis)25,600 mm²

Moment about elastic center800,000 N·mm

I_p = Σ(x_i² + y_i²) about the elastic center — the same unit-area convention the elastic center method uses, so it can be compared directly with hand calculations and textbook tables.

Force in Every Bolt (kN)

Bolt	F direct	F moment	F total	Utilization
#1	3.333	1.859	2.014	6.7%
#2	3.333	0.985	3.476	11.5%
#3worst	3.333	1.859	5.008	16.6%
#4	3.333	1.859	2.014	6.7%
#5	3.333	0.985	3.476	11.5%
#6	3.333	1.859	5.008	16.6%

Preliminary reference only. The elastic center method is a first-order working-load model; prying action, hole bearing, block shear, and connection slip are not evaluated here. Confirm the governing limit states against the applicable specification and have a qualified engineer review any safety-critical connection.

The Method

How the Elastic Center Method Distributes Load

When a load lands on a bolt group away from its center of symmetry, no single bolt simply carries P/n. The load does two things at once: it pushes the whole group in its own direction, and it twists the group around a point called the elastic center — the centroid of the bolt pattern. The elastic center method treats those two effects separately and adds the results bolt by bolt. Written as the classic one-line formula:

F_i = P/n + (M · r_i) / I_p

direct shear share + moment share, with r_i measured from the elastic center

The calculator implements this in vector form, which is what a real connection demands. The direct share pushes every bolt identically along the load’s direction — P_x/n and P_y/n. The moment share points perpendicular to each bolt’s radius vector, so a bolt to the left of the center gets pushed backward while a bolt to the right gets pushed forward. Adding the two vectors bolt by bolt produces the table above, and the magnitudes rarely repeat: two bolts at the same radius can still see different totals because their moment contributions point in different directions.

1 · Locate the elastic center

The centroid of the bolt coordinates — the point where a uniform load would balance. Every radius and every moment term is measured from here, not from the plate corner or the first bolt.

2 · Split the load

Direct shear divides P equally: each bolt takes P/n along the load direction. The eccentricity produces a moment M about the elastic center, computed as the cross product of the offset and the load.

3 · Add per bolt

Each bolt receives a moment force of (M/I_p)·r_i perpendicular to its radius. Sum the vectors with the direct share, take the magnitude, and compare against the allowable shear capacity.

One pattern, two different tools

The Bolt Circle Calculator gives you the hole coordinates; this page turns those coordinates plus a load into the force in each bolt; and the Bolt Shear Calculator checks a single bolt’s stress in detail. Feed a circular pattern’s coordinates into the coordinate mode above and the three tools chain into a complete eccentric-joint check.

Worked Example

A Five-Bolt Bracket Carrying an Eccentric Load

Take the classic cantilever bracket bolted calculation: a vertical plate fastened to a column flange by a five-bolt connection, supporting a downward load applied through a bracket arm. Say the bolts are M12 class 8.8 arranged as three on the top row and two on the bottom row, on an 80 mm horizontal pitch and a 100 mm vertical pitch, and the load of 15 kN acts 130 mm to the right of the group’s left edge and 50 mm below the top row — a textbook eccentrically loaded bolt group you can reproduce in the calculator in about thirty seconds.

Enter the five coordinates (0,0), (80,0), (160,0), (40,100), (120,100), set the load angle to 270° so the force points straight down, and place the load at e_x = 130 mm, e_y = 50 mm. The elastic center lands at (80, 60) mm; the polar moment of inertia of this pattern comes out at 48,000 mm². The bottom-right bolt at (160, 0) sits farthest from the center in the direction that adds to the direct shear, and it governs — its moment share reinforces the equal share every bolt already carries, while bolts on the opposite side of the center see their moment share cancel part of the direct shear.

That asymmetry is the whole point of an eccentric bolt group analysis. A designer who assumed uniform load distribution would size every bolt for P/5 and miss the governing bolt by a wide margin — often 30–60% on typical bracket proportions. The same logic extends to a flange carrying a pipe reaction: the pressure load may be centered, but the attached piping’s weight and thermal thrust rarely are, and the bolts nearest the reaction see the same amplification this example shows.

Reproduce this example

Layout mode: Coordinates — five bolts at (0,0), (80,0), (160,0), (40,100), (120,100) mm
Bolt: M12, Class 8.8, single shear — allowable ≈ 0.62 × 800 MPa on the 113 mm² shank
Load: 15 kN at 270°, positioned at e_x = 130 mm, e_y = 50 mm

Beyond First Order

The Instantaneous Center of Rotation Method

The elastic center method assumes every bolt behaves linearly: double the radius, double the force. That assumption holds comfortably at working loads, but as a highly eccentric group approaches failure, the bolts near the center unload into their yield plateau while the outer bolts keep picking up load. The group then no longer rotates about its centroid — it pivots about an instantaneous center whose position shifts as deformation accumulates. The AISC steel construction manual formalizes this as the IC method: guess the center’s position, compute each bolt’s deformation-consistent force from its nonlinear load-deformation curve, check moment equilibrium, and iterate until the assumed and equilibrium centers agree.

For practical purposes you need two takeaways. First, the IC method generally returns a higher capacity than the elastic center method for the same group — commonly 10–25% for heavily eccentric weld-and-bolt connections — because it credits the ductility the linear model ignores. Second, the elastic center result is therefore conservative for working-load sizing, which is exactly what you want from a preliminary tool. Where the two methods disagree materially, the connection is usually eccentric enough that a purpose-built analysis is warranted anyway.

Use the elastic center method when…

You are sizing bolts at working load, comparing layout options, or producing a number a reviewer can re-derive by hand. The linear model is transparent, conservative, and matches how most general engineering references present the eccentric bolt loading calculation.

Escalate to the IC method when…

The group is highly eccentric relative to its size, bolts approach their allowable at working load, or a specification explicitly requires limit-state verification. Published AISC tabulated coefficients cover common weld-group geometries; bolt groups follow the same iterative logic with the bolt load-deformation curve in place of the weld formula.

Quick Reference

Polar Moment of Inertia of Common Bolt Patterns

Because the elastic center method uses a unit area per bolt, I_p of a pattern depends only on geometry. The table below lists closed-form values for the layouts that cover most brackets, lugs, and mounting plates — multiply by your pitch (or radius) squared and you have the term the moment share divides by. Each value can be verified by summing x² + y² over the pattern, which is also exactly what the calculator reports in the Group Properties card.

Pattern	Geometry	Bolts	I_p (× pitch²)
2 in a line	spacing s, along one axis	2	2.000
3 in a triangle	equilateral, side s	3	2.000
4 square	square, side s	4	4.000
4 rectangle	a × b rectangle	4	2a² + 2b²
5 cross	4 corners s × s + 1 center	5	4.000
6 rectangle	2 rows × 3 cols, pitch a/b	6	6a² + 4b²
8 rectangle	2 rows × 4 cols, pitch a/b	8	10a² + 4b²
n on a circle	radius R, any n ≥ 2	n	n·R²
12 circle, 2 rings	6 at R + 6 at 2R	12	30R²

Reading the table: on a circle, every bolt sits the same distance from the center, so I_p is simply n·R² and the moment share is identical in magnitude for every bolt — only its direction differs. Rectangular patterns concentrate I_p along the long axis, which is why brackets orient their bolt rows parallel to the load’s eccentric arm whenever the detail allows it.

Common Mistakes in Eccentric Bolt Group Calculations

Most errors in a bolted moment connection calculation trace back to one of five habits. Each one is avoidable with a checklist pass before you commit a bolt count.

Assuming every bolt carries the same force

P/n is only the direct-shear share. In a bracket with a moderate eccentricity the governing bolt can carry one and a half times the average — and the mistake hides because the pattern still looks symmetric.

Measuring eccentricity from the wrong point

Every moment term references the elastic center, not the plate edge, the weld line, or bolt #1. If your hand check and the calculator disagree, the coordinate origin is the first thing to compare.

Ignoring prying action

The elastic center method sees bolts as pure shear pins. Where the connected part is flexible and the load has a component normal to the plate, prying can add substantial tension to the end bolts — a separate check that this tool deliberately leaves out.

Forgetting which failure surface governs

Bolt shear is one limit state; plate bearing, block shear rupture, and net-section tear-out all act on the plate instead. A group that passes bolt-by-bolt can still fail through the plate if edge distances are tight.

Treating a tension case as a shear case

An overturning moment loads bolts in tension with a lever-model distribution — the outermost bolts pull, the compression zone bears. That is a different calculation from in-plane eccentric shear, and applying this page’s formula to it under-predicts the end-bolt forces.

Your Bolt Pattern Force Distribution Calculator

This bolt group calculator completes the load path between layout and tightening: once the per-bolt forces are known and the governing bolt passes, head back to the Bolt Torque Calculator on our homepage to set the tightening torque for those bolts, or browse the full toolset on the all calculators page — weight, length, shear, anchor, and bolt circle tools all share the same published engineering data this page uses.

FAQ

Bolt Group Analysis FAQ

How the method works, when to escalate beyond it, and what the calculator deliberately leaves out.

What is the elastic center method for a bolt group?

It is the standard first-order method for eccentrically loaded bolt groups. The applied load is split into a direct shear share, divided equally among the bolts, and a moment share about the group's elastic center, distributed in proportion to each bolt's distance from that center and the group's polar moment of inertia: F_i = P/n + (M·r_i)/I_p. This calculator implements the vector form, resolving both shares into X and Y components before combining them.

How is the polar moment of inertia of a bolt group calculated?

The polar moment of inertia of a bolt group is the sum of A·r² terms over every bolt, where r is each bolt's distance from the group centroid. Because every bolt is assigned the same unit area in the elastic-center method, the calculator uses I_p = Σ(x_i² + y_i²) measured from the centroid. Moving a bolt twice as far from the centroid quadruples its contribution — which is why bolts far from the center carry most of the moment load.

Which bolt in a group carries the most load?

The bolt farthest from the elastic center in the direction that opposes the direct shear. Direct shear divides equally, but the moment share grows linearly with radius, so the outermost bolts govern. The calculator's per-bolt table highlights the worst bolt and computes its utilization against the grade-based allowable shear capacity.

When is the instantaneous center (IC) method required instead?

The elastic center method assumes the group rotates about its centroid and that bolt force grows linearly with distance — accurate while bolts stay well below yield. The AISC instantaneous-center method accounts for the nonlinear force-deformation behavior of each bolt as the group approaches failure, and typically yields 10–25% higher capacity predictions for highly eccentric groups. Use the elastic center result for working-load sizing and treat the IC method as the limit-state refinement described on this page.

Does the calculator check tension in the bolts too?

This tool models in-plane eccentric shear — the classic bracket, lug, and flange-in-plane case where the load acts parallel to the plate and each bolt works in shear. When the load pulls perpendicular to the plate (an overturning moment), the bolts work in tension with a lever-model force distribution instead, which is a different check. Keep the two cases separate and verify the governing one for your connection.

How many bolts can I analyze?

Up to twelve bolts: either a rectangular grid of up to 6 rows × 6 columns with independent X and Y spacing, or a hand-entered coordinate list for circular patterns, staggered layouts, and irregular hole groups. For larger groups, analyze a representative symmetric unit or use a spreadsheet built on the same formulas printed on this page.