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Fixed Beam Deflection Formula (Both Ends Fixed)

Beam Fixed at Both Ends with Uniformly Distributed Load

Fixed–fixed beam with uniform load w acting from position a to position b along span L
Sign convention: w > 0 is downward over the segment [a, b]. Both a and b are measured from the left fixed end and must satisfy a ≤ b.

Convention: R₁ and M₁ act at the left fixed end; R₂ and M₂ act at the right fixed end. Positive reactions are upward and positive moments act clockwise on the beam.

Notation: \(\langle \cdot \rangle\) denotes a singularity (Macaulay) function: \(\langle x-a\rangle^n = (x-a)^n\) for \(x \ge a\), and \(0\) otherwise.

ParameterEquation
Reaction Force 1 \([R_1]\) \[ R_1 = w\!\left( \frac{(L-a)^{3}-(L-b)^{3}}{L^{2}} - \frac{(L-a)^{4}-(L-b)^{4}}{2L^{3}} \right) \]
Reaction Force 2 \([R_2]\) \[ R_2 = w(b-a) - R_1 \]
Reaction Moment 1 \([M_1]\) \[ M_1 = w\!\left( \frac{(L-a)^{4}-(L-b)^{4}}{4L^{2}} - \frac{(L-a)^{3}-(L-b)^{3}}{3L} \right) \]
Reaction Moment 2 \([M_2]\) \[ M_2 = M_1 + R_1 L - \frac{w}{2}\!\left[(L-a)^{2}-(L-b)^{2}\right] \]
Shear at distance \(x\) \([V]\) \[ V = R_1 - w\,\langle x-a\rangle^{1} + w\,\langle x-b\rangle^{1} \]
Moment at distance \(x\) \([M]\) \[ M = M_1 + R_1 x - \frac{w}{2}\,\langle x-a\rangle^{2} + \frac{w}{2}\,\langle x-b\rangle^{2} \]
Slope at \(x\) \([\theta]\) \[ \theta = \frac{M_1 x}{EI} + \frac{R_1 x^{2}}{2EI} - \frac{w}{6EI}\,\langle x-a\rangle^{3} + \frac{w}{6EI}\,\langle x-b\rangle^{3} \]
Deflection at \(x\) \([y]\) \[ y = \frac{M_1 x^{2}}{2EI} + \frac{R_1 x^{3}}{6EI} - \frac{w}{24EI}\,\langle x-a\rangle^{4} + \frac{w}{24EI}\,\langle x-b\rangle^{4} \]

Fixed Beam with Point Load (at any position)

Fixed–fixed beam with a point load P at distance a from the left support over span L
Sign convention: P > 0 acts downward; position a is measured from the left fixed end; reactions are positive upward.
Fixed–fixed beam with a concentrated point load at distance a from the left support.
ParameterEquation
Reaction Force 1 \([R_1]\)\[ R_1=\frac{P}{L^{3}}(L-a)^{2}(L+2a) \]
Reaction Force 2 \([R_2]\)\[ R_2=\frac{P a^{2}}{L^{3}}(3L-2a) \]
Shear at \(x\) \([V]\)\[ V=R_1-P\,\langle x-a\rangle^{0} \]
Reaction Moment 1 \([M_1]\)\[ M_1=-\frac{Pa}{L^{2}}(L-a)^{2} \]
Reaction Moment 2 \([M_2]\)\[ M_2=-\frac{P a^{2}}{L^{2}}(L-a) \]
Moment at \(x\) \([M]\)\[ M=M_1+R_1 x-P\,\langle x-a\rangle^{1} \]
Bending stress \([\sigma]\)\[ \sigma=\frac{Mc}{I} \]
End deflections\[ y_1=0,\quad y_2=0 \]
Deflection at \(x\) \([y]\)\[ y= y_1+\theta_1 x+\frac{M_1 x^{2}}{2EI}+\frac{R_1 x^{3}}{6EI} -\frac{P}{6EI}\,\langle x-a\rangle^{3} \]
End slopes\[ \theta_1=0,\quad \theta_2=0 \]
Slope at \(x\) \([\theta]\)\[ \theta=\theta_1+\frac{M_1 x}{EI}+\frac{R_1 x^{2}}{2EI} -\frac{P}{2EI}\,\langle x-a\rangle^{2} \]

Note: angled brackets \(\langle \cdot \rangle\) are singularity functions.

Fixed Beam with Applied Bending Moment

Fixed–fixed beam with a concentrated moment M applied at distance a along span L
Sign convention: M > 0 is clockwise as drawn (top fibers in compression). Use negative for counter-clockwise.
ParameterEquation
Reaction Force 1 \([R_1]\)\[ R_1=-\frac{6 M_o a}{L^{3}}(L-a) \]
Reaction Force 2 \([R_2]\)\[ R_2=\frac{6 M_o a}{L^{3}}(L-a) \]
Shear at \(x\) \([V]\)\[ V=R_1 \]
Reaction Moment 1 \([M_1]\)\[ M_1=-\frac{M_o}{L^{2}}(L^{2}-4aL+3a^{2}) \]
Reaction Moment 2 \([M_2]\)\[ M_2=\frac{M_o}{L^{2}}(3a^{2}-2aL) \]
Moment at \(x\) \([M]\)\[ M=M_1+R_1 x+M_o\,\langle x-a\rangle^{0} \]
Bending stress \([\sigma]\)\[ \sigma=\frac{Mc}{I} \]
End deflections\[ y_1=0,\quad y_2=0 \]
Deflection at \(x\) \([y]\)\[ y= y_1+\theta_1 x+\frac{M_1 x^{2}}{2EI}+\frac{R_1 x^{3}}{6EI} +\frac{M_o}{2EI}\,\langle x-a\rangle^{2} \]
End slopes\[ \theta_1=0,\quad \theta_2=0 \]
Slope at \(x\) \([\theta]\)\[ \theta=\theta_1+\frac{M_1 x}{EI}+\frac{R_1 x^{2}}{2EI} +\frac{M_o}{EI}\,\langle x-a\rangle^{1} \]

Supplements

References