Beams Fixed End

Overview

Engineering reference data for Beams Fixed End in mechanics.

Key Formulas

Newton's Second Law

$F = ma$

Force = mass × acceleration.

Work

$W = Fd\cos\theta$

Work = force × displacement × cos(angle).

Kinetic Energy

$E_k = \frac{1}{2}mv^2$

Energy of motion.

Potential Energy

$E_p = mgh$

Gravitational potential energy.

Variables

References

• Original Source

Beam Fixed at One End and Supported at the Other: Load Case Analysis

This section details the specific formulas and key parameters for analyzing a beam that is fixed at one end (A) and simply supported at the other end (B), under various loading conditions.

Single Point Load (F)

• Fixed End Moment (MA): The reaction moment at the fixed end is negative (hogging). $M_A = - \frac{F a b (L + b)}{2 L^2}$
• Deflection at Load Point (δF): Maximum deflection occurs at or near the point of load application. $\delta_F = \frac{F a^3 b^2 (3 L + b)}{12 L^3 E I}$
• Support Reactions: The fixed support (A) carries more load than the simple support (B) for most load positions. $R_A = \frac{F b (3 L^2 - b^2)}{2 L^3}$ $R_B = \frac{F a^2 (b + 2 L)}{2 L^3}$

Continuous Uniformly Distributed Load (q)

• Fixed End Moment (MA): $M_A = - \frac{q L^2}{8}$
• Maximum Bending Moment (M1): Occurs at a distance of x = 0.625 L from the fixed end. $M_1 = \frac{9 q L^2}{128}$
• Maximum Deflection (δmax): Occurs at x ≈ 0.579 L from the fixed end. $\delta_{max} = \frac{q L^4}{185 E I}$
• Mid-span Deflection (δ1/2): $\delta_{1/2} = \frac{q L^4}{192 E I}$
• Support Reactions: The fixed support (A) carries 5/8 of the total load. $R_A = \frac{5 q L}{8}$ $R_B = \frac{3 q L}{8}$

Continuous Declining (Triangular) Load (q)

• Fixed End Moment (MA): $M_A = - \frac{q L^2}{15}$
• Maximum Bending Moment (M1): Occurs at x = 0.553 L from the fixed end. $M_1 = \frac{q L^2}{33.6}$
• Maximum Deflection (δmax): Occurs at x ≈ 0.553 L. $\delta_{max} = \frac{q L^4}{419 E I}$
• Mid-span Deflection (δ1/2): $\delta_{1/2} = \frac{q L^4}{427 E I}$
• Support Reactions: $R_A = \frac{2 q L}{5}$ $R_B = \frac{q L}{10}$

Applied Moment (MB) at the Supported End

• Fixed End Moment (MA): A moment applied at the simple support induces a moment at the fixed end with the same magnitude but opposite sign. $M_A = -\frac{M_B}{2}$
• Maximum Deflection (δmax): Occurs at x = 2/3 L from the fixed end. $\delta_{max} = \frac{M_B L^2}{27 E I}$
• Support Reactions: The reactions are equal in magnitude and opposite in direction, creating a couple. $R_A = \frac{3 M_B}{2 L}$ $R_B = -\frac{3 M_B}{2 L}$

Single Point Load

For a beam fixed at end A and supported at end B with a single point load F applied at distance a from the fixed end and b from the supported end (L = a + b):

• Bending Moment at Fixed End (A):

$$M_A = -\frac{F a b (L + b)}{2 L^2}$$

• Bending Moment at Load Point:

$$M_F = R_B b$$

• Deflection at Load Point:

$$\delta_F = \frac{F a^3 b^2 (3 L + b)}{12 L^3 E I}$$

• Support Reactions:
  • At fixed end A:

$$R_A = \frac{F b (3 L^2 - b^2)}{2 L^3}$$

• At support B:

$$R_B = \frac{F a^2 (b + 2 L)}{2 L^3}$$

Continuous Uniform Load

For a beam fixed at end A and supported at end B with a continuous uniform load q over the entire length L:

• Bending Moment at Fixed End (A):

$$M_A = -\frac{q L^2}{8}$$

• Maximum Bending Moment:

$$M_1 = \frac{9 q L^2}{128} \quad \text{at} \quad x = 0.625 L$$

• Maximum Deflection:

$$\delta_{\text{max}} = \frac{q L^4}{185 E I} \quad \text{at} \quad x = 0.579 L$$

• Deflection at Midspan (x = L/2):

$$\delta_{1/2} = \frac{q L^4}{192 E I}$$

• Support Reactions:
  • At fixed end A:

$$R_A = \frac{5 q L}{8}$$

• At support B:

$$R_B = \frac{3 q L}{8}$$

Continuous Declining Load

For a beam fixed at end A and supported at end B with a continuous declining load q from the fixed end to zero at the supported end:

• Bending Moment at Fixed End (A):

$$M_A = -\frac{q L^2}{15}$$

• Maximum Bending Moment:

$$M_1 = \frac{q L^2}{33.6} \quad \text{at} \quad x = 0.553 L$$

• Maximum Deflection:

$$\delta_{\text{max}} = \frac{q L^4}{419 E I} \quad \text{at} \quad x = 0.553 L$$

• Deflection at Midspan (x = L/2):

$$\delta_{1/2} = \frac{q L^4}{427 E I}$$

• Support Reactions:
  • At fixed end A:

$$R_A = \frac{2 q L}{5}$$

• At support B:

$$R_B = \frac{q L}{10}$$

Moment at Supported End

For a beam fixed at end A and supported at end B with a moment $M_B$ applied at the supported end B:

• Bending Moment at Fixed End (A):

$$M_A = -\frac{M_B}{2}$$

• Maximum Deflection:

$$\delta_{\text{max}} = \frac{M_B L^2}{27 E I} \quad \text{at} \quad x = \frac{2}{3} L$$

• Support Reactions:
  • At fixed end A:

$$R_A = \frac{3 M_B}{2 L}$$

• At support B:

$$R_B = -\frac{3 M_B}{2 L}$$

Note: Negative reaction indicates direction opposite to the assumed positive direction.

Comparative Summary: Beam Fixed at One End and Supported at the Other

This section provides a summary table for quick reference across the different load cases analyzed on this page.

Structural Behavior Notes

The extracted data reveals important patterns in the structural response of a propped cantilever (fixed at one end, simply supported at the other).

Maximum Moment Location: For uniform and triangular distributed loads, the maximum bending moment does not occur at the fixed support. It shifts toward the simple support. For a continuous uniform load, it occurs at $x = 0.625L$, and for a triangular load, it occurs at $x = 0.553L$.

Maximum Deflection Location: Similarly, the point of maximum deflection is not at mid-span for these statically indeterminate beams. For a uniform load, maximum deflection occurs at $x = 0.579L$, and for a triangular load, it also occurs at $x = 0.553L$.

Reaction Distribution: The fixed support carries a larger share of the load than the simple support. For a uniform load, the fixed end reaction $R_A$ is $\frac{5}{8}qL$ (62.5% of total load), while $R_B$ is $\frac{3}{8}qL$ (37.5%). This ratio is even more pronounced for a triangular load ($R_A$ is 80% of the total load). An applied moment at the simple support creates an equal and opposite reaction pair that is inversely proportional to the beam length.