Beam Fixed at Both Ends - Concentrated Load at Any Point
Beam Fixed at Both Ends - Concentrated Load at Any Point formulas |
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\(\large{ R_1 = V_1 \; \left(max.\; when \; a < b \right) \;\;=\;\; \frac {P\;b^2} {L^3} \; \left( 3\;a + b \right) }\) \(\large{ R_2 = V_2 \; \left(max.\; when \; a > b \right) \;\;=\;\; \frac {P\;a^2} {L^3} \; \left( a + 3\;b \right) }\) \(\large{ M_1 \; \left(max.\; when \; a < b \right) \;\;=\;\; \frac {P\;a\;b^2} {L^2} }\) \(\large{ M_2 \; \left(max. \;when \; a > b \right) \;\;=\;\; \frac {P\;a^2\;b} {L^2} }\) \(\large{ M_a \; \left(at \;point \;of \;load \right) \;\;=\;\; \frac {2\;P\;a^2\;b^2} {L^3} }\) \(\large{ M_x \; \left( x < a \right) \;\;=\;\; R_1\; x - \frac {P\;a\;b^2} {L^2} }\) \(\large{ \Delta_{max} \; \left(at \; x = \frac{2\;a\;L}{3\;a \;+\; b} \;when \;a > b \right) \;\;=\;\; \frac {4\;P\;a^3\;b^2} {3 \;\lambda\; I \;\left( 3\;a \;+ \;b \right)^2 } }\) \(\large{ M_a \; \left(at \;point \;of \;load \right) \;\;=\;\; \frac {2 \;P\;a^3\;b^3} {6\; \lambda\; I \;L^3} }\) \(\large{ \Delta_x \; \left( x < a \right) \;\;=\;\; \frac {2 \;P\;b^2\;x^2} {12\; \lambda \;I \;L^3} \; \left( 3\;a\;L - 3\;a\;x - b\;x \right) }\) \(\large{ x \; \left( point\; of\; contraflexure\;between\;supports \right) \;\;=\;\; \frac{a\;b^2\;P} {L^2\;R_1} }\) |
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| Symbol | English | Metric |
| \(\large{ \Delta }\) = deflection or deformation | \(\large{in}\) | \(\large{mm}\) |
| \(\large{ x }\) = horizontal distance from reaction to point on beam | \(\large{in}\) | \(\large{mm}\) |
| \(\large{ M }\) = maximum bending moment | \(\large{lbf-in}\) | \(\large{N-mm}\) |
| \(\large{ V }\) = maximum shear force | \(\large{lbf}\) | \(\large{N}\) |
| \(\large{ \lambda }\) (Greek symbol lambda) = modulus of elasticity | \(\large{\frac{lbf}{in^2}}\) | \(\large{Pa}\) |
| \(\large{ I }\) = moment of inertia | \(\large{in^4}\) | \(\large{mm^4}\) |
| \(\large{ R }\) = reaction load at bearing point | \(\large{lbf}\) | \(\large{N}\) |
| \(\large{ L }\) = span length under consideration | \(\large{in}\) | \(\large{mm}\) |
| \(\large{ a, b }\) = span length under consideration | \(\large{in}\) | \(\large{mm}\) |
| \(\large{ P }\) = total concentrated load | \(\large{lbf}\) | \(\large{N}\) |
diagrams
- Bending moment diagram (BMD) - Used to determine the bending moment at a given point of a structural element. The diagram can help determine the type, size, and material of a member in a structure so that a given set of loads can be supported without structural failure.
- Free body diagram (FBD) - Used to visualize the applied forces, moments, and resulting reactions on a structure in a given condition.
- Shear force diagram (SFD) - Used to determine the shear force at a given point of a structural element. The diagram can help determine the type, size, and material of a member in a structure so that a given set of loads can be supported without structural failure.
- Uniformly distributed load (UDL) - A load that is distributed evenly across the entire length of the support area.
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Tags: Beam Support Equations