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樑彎曲計算器
計算簡支樑與懸臂樑在各種負荷類型下的最大彎矩、撓度與應力。
結果
簡支樑 — 中心集中負荷
簡支樑 — 均佈負荷
懸臂樑 — 端部集中負荷
彎曲應力
其中 σ 為彎曲應力， M 為彎矩， c 為至最外層纖維距離，且 I 為慣性矩。
| 負荷案例 | M_max | δ_max |
|---|---|---|
| SS — 中心集中 | PL/4 | PL³/(48EI) |
| SS — a 處集中 | Pa(L−a)/L | 複雜公式 |
| SS — 均佈負荷 w | wL²/8 | 5wL⁴/(384EI) |
| 懸臂樑 — 端點集中載荷 | PL | PL³/(3EI) |
| 懸臂樑 — 均佈載荷 w | wL²/2 | wL⁴/(8EI) |
⚠️ 注意： 這些公式假設線性彈性行為、微小撓度以及等截面（常數截面）樑。對於複雜載荷，請使用疊加法。
Beam bending occurs when a transverse load is applied to a structural member, causing it to deflect. The beam develops internal bending moments and shear forces. Bending stress varies linearly across the section — zero at the neutral axis and maximum at the extreme fibers.
A simply supported beam rests on two supports at its ends and is free to rotate at those points. A cantilever beam is rigidly fixed at one end and free at the other.
Cantilevers have larger deflections and moments at the fixed support than equivalently loaded simply supported beams.
Deflection depends on load type, span length (L), elastic modulus (E), and moment of inertia (I):
- SS center point load: δ = PL³/(48EI)
- SS UDL: δ = 5wL⁴/(384EI)
- Cantilever tip load: δ = PL³/(3EI)
The second moment of area (I) is a geometric property that measures the cross-section's resistance to bending. Larger I means less deflection and stress for the same load. For a rectangle: I = bh³/12. For a circle: I = πd⁴/64.
Common limits per building codes:
- L/360 — Floor beams (live load)
- L/240 — Total load
- L/180 — Roof beams
- L/500 — Precision machinery supports
Always check applicable design standards.
本頁所述的操作流程由 Vibromera 團隊使用 Balanset-1A（一款用於現場轉子平衡的便攜式雙通道動平衡儀及振動分析儀）開發並經過實地測試。
專業現場動平衡校正儀器與軟體。已在 50 多個國家使用。
