What is beam bending?

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 to resist the load. The bending stress varies linearly from zero at the neutral axis to maximum at the extreme fibers.

What is the difference between a simply supported beam and a cantilever?

A simply supported beam rests on two supports at its ends and is free to rotate. A cantilever beam is rigidly fixed at one end and free at the other. Cantilevers generally experience higher deflections and moments at the fixed support compared to simply supported beams of the same span and load.

How do I calculate beam deflection?

Beam deflection depends on the load type, span, elastic modulus (E), and moment of inertia (I). For a simply supported beam with central point load P: δ_max = PL³/(48EI). For UDL w: δ_max = 5wL⁴/(384EI). For cantilever with tip load P: δ_max = PL³/(3EI).

What is the moment of inertia in beam calculations?

The moment of inertia (I) is a geometric property of the beam cross-section that measures its resistance to bending. A larger I means less deflection and lower stress for the same load. For a rectangle: I = bh³/12. For a circle: I = πd⁴/64.

What is the maximum allowable deflection for a beam?

Common deflection limits are L/360 for floor beams under live load, L/240 for total load, and L/180 for roofs. Industrial applications may use L/500 or stricter. Always check the applicable building code or design standard for your specific application.

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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。

集中負荷均佈負荷懸臂樑 #164

負荷案例

樑長 L (m)

集中負荷 P (kN)

彈性模數 E (GPa)

慣性矩 I (cm⁴)

至最外層纖維距離 c (mm)

快速預設

結果

最大彎矩

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最大撓度

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最大彎曲應力

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跨距/撓度比

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負荷案例

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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。