Centers of Gravity and Geometric Formulas for Engineering Calculations

Centers of Gravity and Geometric Formulas for Engineering Calculations

Every aggregate-handling structure — a stockpile under a radial conveyor, a bolted hopper bracket, a tracked mobile crusher rolling onto a transport trailer — depends on the same calculation: the location of its center of gravity. The centroid governs static reactions in beams and trusses, dictates whether a stockpile angle of repose is preserved as the heap grows, and determines tip-over thresholds for self-erecting plants. This reference compiles the geometric formulas a process or structural engineer needs most often when sizing aggregate equipment: centroids of standard 2D and 3D shapes, volume relationships for conical and rotary stockpiles, and the dimensional tables that translate feeder length and angle of repose into deliverable tonnage. All formulas are reproduced from the MEKA Crushing, Screening and Mining Equipment Handbook (Section 9, pages 137–142) and supplemented with the standard solids commonly used in plant layout calculations.

Why Centers of Gravity Matter in Aggregate Engineering

The centroid is more than a textbook abstraction. In aggregate plant design, three classes of problem reduce to a single question: where is the resultant of the distributed weight applied? Each class is summarized below.

Stockpile Sizing and Capacity Estimation

A conical stockpile fed by a radial belt conveyor stores a volume that is a strict function of feeder length, discharge height, and the material's static angle of repose (α ≈ 16°–18° for most washed aggregate, higher for crushed stone with interlocking particles). The geometric center of mass of that cone — at h/4 from the base for a uniform-density solid — is the point at which the stockpile's weight acts on its support tunnel. Conical and elongated (rotary) stockpile volumes follow directly from the same geometry. See the reference table in the Stockpile Geometry section below.

Structural Design of Conveyor and Hopper Supports

Conveyor gallery trusses, surge hopper brackets, and screen support frames carry distributed dead loads whose resultant must be located accurately to size foundation bolts and check overturning moments. Composite bodies (an I-section bracket bolted to a cylindrical column, a hopper combining a frustum and a cylindrical extension) are resolved by the moment-summation form ȳ = ΣAᵢȳᵢ / ΣAᵢ — Pappus's principle applied to engineering practice.

Stability Analysis of Mobile Plants

Tracked and wheeled crushing/screening plants must satisfy three independent stability checks: (1) static tip-over on grade, (2) dynamic stability during transport on a low-bed trailer, and (3) operational stability under crusher unbalance loads. All three reduce to locating the combined centroid of the chassis, power module, and processing equipment, then verifying that the vertical projection of that point falls inside the support polygon under the worst-case loading.

Centers of Gravity for Common 2D Shapes

The following five surfaces appear most often in aggregate equipment cross-sections — chute walls, transition pieces, screen deck plates, and stockpile sectional analysis. All formulas are reproduced from the MEKA Handbook, p. 140.

Trapezoid

For a trapezoid with parallel sides a (top) and b (base), and height h, the centroid distances from the base (c) and from the top (d) and the centroid offset along the median (e) are:

c = h(a + 2b) / 3(a + b)

d = h(2a + b) / 3(a + b)

e = (a2 + ab + b2) / 3(a + b)

Circular Segment

For a segment of radius r, chord c, arc length l, and segment height h, the centroid distance a from the chord is:

a = (r · c) / l = c(c2 + 4h2) / 8lh

Triangular Sector (Filled Triangle on a Sector)

For a sector triangle with chord c, radius r, half-angle α and area A:

b = c3 / 12A = (2/3) · r3 sin3 α / A

Circular Sector

For a circular sector of radius r, chord c, arc length l, area A, and half-angle α (in degrees), the centroid distance b from the apex is:

b = 2rc / 3l = r2c / 3A = 38.197 · r sin α / α

Annular Sector

For an annular sector bounded by outer radius R, inner radius r, and half-angle α (degrees):

b = 38.197 · (R3 − r3) sin α / (R2 − r2) α

Centers of Gravity for 3D Solids

The 3D solids below cover the geometries most relevant to bins, hoppers, stockpiles, and bracket assemblies. Formulas are reproduced from the MEKA Handbook, pp. 141–142, and supplemented with standard textbook results for the hemisphere and conical shell.

Solid Cone and Solid Pyramid

For a uniform solid right circular cone of height h or a solid pyramid of height h, the centroid lies on the axis at:

a = h / 4 (from the base)

This is the single most-used result in stockpile and hopper analysis. For a conical shell — the lateral surface only, treated as an open cone of zero wall thickness — the centroid shifts to h/3 from the base.

Truncated Cone (Frustum)

For a frustum with bottom radius R, top radius r, and height h:

a = h(R2 + 2Rr + 3r2) / 4(R2 + Rr + r2)

Volume of the frustum (used for hopper transitions and partial-fill calculations):

V = (πh / 3) · (R2 + Rr + r2)

Truncated Pyramid

For a frustum with bottom area A1 and top area A2:

a = h(A1 + 2√(A1 · A2) + 3A2) / 4(A1 + √(A1 · A2) + A2)

Wedge

For a wedge with base b, top c, and height h:

a = h(b + c) / 2(2b + c)

Solid and Hollow Hemisphere

For a uniform solid hemisphere of radius R, the centroid lies on the axis of symmetry at 3R/8 from the flat base. For the hemispherical shell (thin hollow surface), the centroid is at R/2. For a hollow hemisphere (a thick spherical shell of outer radius R and inner radius r):

a = 3(R4 − r4) / 8(R3 − r3)

Spherical Segment

For a spherical segment of sphere radius r and segment height h:

a = 3(2r − h)2 / 4(3r − h)

b = h(4r − h) / 4(3r − h)

Spherical Sector

a = (3/8)(1 + cos α) · r = (3/8)(2r − h)

Composite Bodies and Pappus's Theorems

For composite bodies, the centroid is found from the sum of moments divided by the sum of areas (or volumes):

ȳ = Σ (Ai · ȳi) / Σ Ai and x̄ = Σ (Ai · x̄i) / Σ Ai

Pappus's first theorem: the surface area generated by revolving a plane curve about an external axis equals the curve length times the path length traveled by the curve's centroid (2π · d̄). Pappus's second theorem: the volume generated by revolving a plane area equals the area times the path length of its centroid (2π · d̄). Both theorems collapse cumbersome integrals to a single multiplication and are routinely used to size cylindrical-to-conical transition pieces.

Stockpile Geometry: Practical Application

This section combines the centroid formulas above with the dimensional data tabulated in the MEKA Handbook to deliver immediately usable stockpile sizing calculations.

Conical Stockpile Volume and Center of Gravity

A radial stockpile belt conveyor of feeder length L m discharges material that piles at its static angle of repose α (typically 16°–18° for graded aggregate). The pile geometry depends on whether the discharge tunnel is fully buried, partially exposed, or supports multiple feeders:

• 1 Feeder: single cone, peak directly above the head pulley. Net volume ≈ 25 % of gross.
• 2 Feeders: ridged double cone (kissing cones). Net volume ≈ 30 % of gross.
• 3 Feeders: triple cone with two valleys. Net volume ≈ 35 % of gross.
• 4 Feeders: four cones, near-rectangular footprint. Net volume ≈ 38 % of gross.

For tunnels below ground level (handbook p. 137), gross volume follows:

Vgross = 1.4873 · h3 (m³, with h in m, valid for α = 40° static slope)

For tunnels above ground level with a 2.5 m exposed wall (handbook p. 138):

Vgross = 1.4873 · (h + 2.5)2 (m³)

Stockpile Volume Reference Table — Tunnels Below Ground Level

Reproduced directly from the MEKA Handbook, p. 137. Feeder lengths from 18 m to 100 m; angles of 18° (graded sand) and 16° (crushed stone); net volume reflects live (drawable) tonnage by feeder configuration.

Note: gross volume values shown × 100 represent the total enclosed cone volume; multiply by 100 to obtain m³, or interpret directly as the listed integer × 100 m³. Net (live) volume is the drawable fraction at each feeder configuration.

Rotary (Radial) Conveyor Stockpile Geometry

A rotary stockpile conveyor pivots about a fixed point at the tail (or head) and sweeps an arc of angle θ, building a kidney-shaped pile rather than a single cone. The total stock volume combines a base cone (from the partial slewing) with a swept frustum of revolution. Per MEKA Handbook p. 139:

Vtotal = 1.4873 · h3 + (π · r · h2 / tan α) · (θ / 360°)

where h is the discharge height, r is the radial reach (typically L · cos β, with β the conveyor inclination), and θ is the slew angle (0°–180°). At θ = 0° the result reduces to a single cone; at θ = 180° the pile becomes a half-kidney with maximum capacity per unit feeder length.

Quick Reference Tables

Centroid Locations for Standard Shapes

Volume Formulas for Aggregate Stockpiles

Frequently Asked Questions

How do I calculate the center of gravity of a trapezoid?

Use the formula c = h(a + 2b) / 3(a + b), where a is the top parallel side, b is the base, and h is the perpendicular height. The result is the perpendicular distance from the base to the centroid. This is the formula to use whenever you have a transition piece with two parallel edges of different length — chute side walls, hopper end plates, or screen deck taper pieces.

What's the volume of a conical stockpile?

For a uniform cone, V = ⅓ · π · r2 · h, where r is the base radius and h is the height. In real practice, the geometry is constrained by the angle of repose and the feeder discharge height, so the working formula is V = 1.4873 · h3 (MEKA Handbook p. 137, valid for α = 40° static slope). For configurations of 1–4 feeders, multiply gross volume by the live-fraction factor (25 %, 30 %, 35 %, or 38 % respectively) to obtain net deliverable volume.

How tall can a single-feeder stockpile go?

The maximum stockpile height h is limited by feeder belt length L and the material's static angle of repose α: h = L · sin β, where β is the conveyor inclination (typically 18°–22° for radial stackers). From the MEKA Handbook reference table (p. 137): a 50 m feeder at 18° gives h = 15.4 m and 1369 m³ live volume; a 100 m feeder at 16° gives h = 27.6 m and 7775 m³. Higher stockpiles increase capacity but require corresponding tunnel depth and reclaim feeder rating.

What is a rotary conveyor stockpile?

A rotary (or radial) stockpile conveyor pivots about a fixed point — usually under the feed end — and slews through an arc of 0° to 180°, depositing material in a kidney-shaped pile rather than a single cone. The geometry is the union of a partial cone of revolution and a longitudinal ridge of constant cross-section, allowing 3–6× the storage of a static stockpile of equivalent feeder length. Volume is calculated from MEKA Handbook p. 139.

Why does center of gravity matter for mobile plants?

Three reasons. (1) Static stability: the plant must not tip when parked on an incline; the vertical projection of the centroid must remain inside the support polygon. (2) Transport regulations: loading height and CG affect classification under road transport directives (e.g., EU Directive 96/53/EC, US DOT Bridge Formula). (3) Operational stability: crusher unbalance forces and screen vibration at full load shift the resultant; a CG too high or too far from the support centerline causes premature track wear or trailer dynamic instability above 50 km/h.

Appendix A — JSON-LD Schema Markup

Insert the following JSON-LD block in the page <head> element. Combines TechArticle, FAQPage, and BreadcrumbList schemas as recommended by Google's structured data guidelines.

<script type="application/ld+json">

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"description": "Reference formulas for centers of gravity and centroids of trapezoids, circular sectors, cones, and stockpile geometries.",

"author": {

"@type": "Organization",

"name": "MEKA Global",

"url": "https://www.mekaglobal.com/en"

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"name": "MEKA Global",

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"about": [

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}

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"text": "For a uniform cone V = (1/3)πr²h. In radial stockpile applications V = 1.4873·h³ for tunnels below ground level (MEKA Handbook, valid for α = 40° static slope). Net live volume is 25%, 30%, 35%, or 38% of gross for 1, 2, 3, or 4 feeders respectively."

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Appendix B — Cone Stockpile Volume Calculator (Vanilla JavaScript)

Self-contained widget. User selects feeder length, angle of repose, and number of feeders; output is interpolated from the MEKA Handbook p. 137 reference table. Drop the HTML block into any page; no dependencies required.

<!-- Cone Stockpile Volume Calculator -->

<div id="meka-stockpile-calc" style="font-family:system-ui,sans-serif;max-width:560px;

padding:20px;border:1px solid #d0d7de;border-radius:8px;background:#f6f8fa">

<h3 style="margin-top:0;color:#1f4e79">Cone Stockpile Volume Calculator</h3>

<div style="margin:10px 0">

<label>Feeder length (m):

<input id="msc-len" type="number" min="18" max="100" value="50"

style="width:80px;margin-left:8px"></label>

</div>

<div style="margin:10px 0">

<label>Angle of repose:

<select id="msc-ang" style="margin-left:8px">

<option value="18">18° (washed sand/gravel)</option>

<option value="16">16° (crushed stone)</option>

</select></label>

</div>

<div style="margin:10px 0">

<label>Number of feeders:

<select id="msc-fed" style="margin-left:8px">

<option value="1">1 feeder (25% net)</option>

<option value="2">2 feeders (30% net)</option>

<option value="3">3 feeders (35% net)</option>

<option value="4" selected>4 feeders (38% net)</option>

</select></label>

</div>

<button id="msc-go" style="background:#1f4e79;color:#fff;border:none;padding:10px 20px;

border-radius:4px;cursor:pointer;font-weight:600">Calculate</button>

<div id="msc-out" style="margin-top:18px;padding:14px;background:#fff;border-radius:4px;

border:1px solid #d0d7de;display:none"></div>

</div>

<script>

(function() {

// MEKA Handbook p. 137 — Conical Stock Dimensions, tunnels below ground level

// [feederLen, angleDeg, h, grossVolX100]

var TBL = [

[18,18,5.6,256],[20,18,6.2,351],[22,18,6.8,467],[25,18,7.7,685],[28,18,8.6,962],

[30,18,9.3,1183],[35,18,10.8,1879],[40,18,12.4,2805],[45,18,13.9,3993],[50,18,15.4,5478],

[55,16,15.2,5174],[60,16,16.5,6718],[65,16,17.9,8541],[70,16,19.3,10668],[80,16,22.0,15924],

[90,16,24.8,22672],[100,16,27.6,31101]

];

var NET = {1:0.25, 2:0.30, 3:0.35, 4:0.38};

function interp(L, ang) {

var rows = TBL.filter(function(r){ return r[1] === ang; });

if (rows.length === 0) return null;

if (L <= rows[0][0]) return {h: rows[0][2], gross: rows[0][3]*100};

if (L >= rows[rows.length-1][0]) {

var last = rows[rows.length-1];

return {h: last[2], gross: last[3]*100};

}

for (var i = 0; i < rows.length - 1; i++) {

if (L >= rows[i][0] && L <= rows[i+1][0]) {

var f = (L - rows[i][0]) / (rows[i+1][0] - rows[i][0]);

return {

h: rows[i][2] + f*(rows[i+1][2] - rows[i][2]),

gross: (rows[i][3] + f*(rows[i+1][3] - rows[i][3]))*100

};

}

}

return null;

}

document.getElementById('msc-go').addEventListener('click', function() {

var L = parseFloat(document.getElementById('msc-len').value);

var ang = parseInt(document.getElementById('msc-ang').value, 10);

var fed = parseInt(document.getElementById('msc-fed').value, 10);

var res = interp(L, ang);

var out = document.getElementById('msc-out');

if (!res) {

out.innerHTML = '<strong style="color:#d73a49">Outside reference range. ' +

'Use the angle that corresponds to feeder length: 18° for 18–50 m, 16° for 55–100 m.</strong>';

out.style.display = 'block';

return;

}

var net = res.gross * NET[fed];

var netYd3 = (net / 0.7646).toFixed(0);

out.innerHTML =

'<table style="width:100%;border-collapse:collapse">' +

'<tr><td style="padding:4px 0">Stockpile height <em>h</em></td>' +

'<td style="padding:4px 0;text-align:right;font-weight:600">' + res.h.toFixed(1) + ' m</td></tr>' +

'<tr><td style="padding:4px 0">Gross volume</td>' +

'<td style="padding:4px 0;text-align:right;font-weight:600">' +

Math.round(res.gross).toLocaleString() + ' m³</td></tr>' +

'<tr><td style="padding:4px 0;border-top:1px solid #d0d7de"><strong>Net (live) volume</strong></td>' +

'<td style="padding:4px 0;text-align:right;font-weight:700;color:#1f4e79;border-top:1px solid #d0d7de">' +

Math.round(net).toLocaleString() + ' m³ (' + Number(netYd3).toLocaleString() + ' yd³)</td></tr>' +

'</table>' +

'<p style="font-size:13px;color:#586069;margin:10px 0 0">' +

'Values interpolated from MEKA Handbook p. 137. Verify on-site against material angle of repose.</p>';

out.style.display = 'block';

});

})();

</script>