Simple beam - Couple moment Mo at left end I Calculator

Results:

Deflection Yab:

Deflection Ymax at x = :

Slope θab:

Slope θa:

Slope θb:

Moment Mab:

Shear Vab:

Reactions: Ra:

Reactions: Rb:

Simple Beam - Couple Moment Mo at Left End I

A simple beam is supported at both ends, with a couple moment \( M_0 \) applied at the left end of the beam. The application of the couple moment at this point induces internal rotational effects along the beam. This type of load does not produce any shear force but creates a bending moment that is constant along the beam, except at the left end, where the couple moment is applied.

Key Concepts

Simple Beam: A beam supported at both ends, with no intermediate supports.
Couple Moment at Left End: A couple moment applied at the left end of the beam, creating a rotational effect without causing any shear force.
Shear Force: No shear force is generated by the couple moment, as it does not translate the beam.
Bending Moment: The bending moment is constant along the length of the beam. However, at the left end, where the couple moment is applied, the bending moment is influenced by the magnitude of \( M_0 \).
Deflection: The deflection caused by the couple moment depends on the magnitude of \( M_0 \), the length of the beam, and the material properties of the beam.

Behavior of the Simple Beam

Reaction Forces:
- The reaction forces at the supports are calculated using equilibrium equations, considering the couple moment's effect on the bending moment. The couple does not affect the shear forces but induces bending along the beam.
Shear Force Diagram:
- Since a couple moment does not create shear forces, the shear force diagram remains zero across the beam's length.
Bending Moment Diagram:
- The bending moment diagram is constant along the entire length of the beam, with the magnitude of the moment equal to \( M_0 \) everywhere, except at the left end where the couple is applied.
Deflection: The deflection caused by the couple moment is calculated using beam deflection formulas, which depend on \( M_0 \), the beam's length, and the material properties. The deflection will be uniform along the length of the beam.

Applications

Structural Engineering: Couple moments applied at the left end of a beam are used in structural analysis to model rotational effects, such as in systems with supports subjected to torsion or rotation.
Construction: This loading condition is used in analyzing beams under torsional loading, particularly in applications where beams experience rotational forces at the end, like in cantilevered structures or frames.
Mechanical Systems: Couple moments at the left end of a beam are encountered in mechanical systems that experience torsional forces, such as shafts, gears, and other rotating components.

Formula

Deflection (AB)	\( y_{\mathrm{AB}} = \frac{M_0 x}{6 L E I}\left(2 L^2-3 L x+x^2\right) \)
Maximum Deflection	\( y_{\mathrm{MAX}} = \frac{M_0 L^2}{9 \sqrt{3} E I} \quad \text{at} \quad x = \left(\frac{3-\sqrt{3}}{3}\right) L \)
Slope (AB)	\( \theta_{\mathrm{AB}} = \frac{M_0}{6 L E I}\left(2 L^2-6 L x+3 x^2\right) \)
Slope at A	\( \theta_{\mathrm{A}} = \frac{M_0 L}{3 E I} \)
Slope at B	\( \theta_{\mathrm{B}} = \frac{-M_0 L}{6 E I} \)
Moment (AB)	\( M_{\mathrm{AB}} = \frac{-M_0}{L}(L-x) \)
Shear (AB)	\( V_{\mathrm{AB}} = \frac{M_0}{L} \)
Reactions	\( R_{\mathrm{A}} = \frac{M_0}{L} \quad R_{\mathrm{B}} = \frac{-M_0}{L} \)

Definitions

Symbol	Physical quantity	Units
E·I	Flexural rigidity	N·m², Pa·m⁴
y	Deflection or deformation	m
θ	Slope, Angle of rotation	-
x	Distance from support (origin)	m
L	Length of beam (without overhang)	m
M	Moment, Bending moment, Couple moment applied	N·m
P	Concentrated load, Point load, Concentrated force	N
w	Distributed load, Load per unit length	N/m
R	Reaction load, reaction force	N
V	Shear force, shear	N