Simple beam - Couple moment Mo at right end Calculator

Deflection Yab:

Slope θab:

Slope θa:

Slope θb:

Moment Mab:

Shear Vab:

Reactions Ra:

Reactions Rb:

Simple Beam - Couple Moment Mo at Right End

A simple beam is supported at both ends, with a couple moment \( M_0 \) applied at the right end of the beam. The couple moment creates rotational effects at the right end, but does not generate any shear force. The bending moment along the beam remains constant except at the right end, where the couple moment is applied, causing the bending moment to change.

Key Concepts

Simple Beam: A beam supported at both ends, with no intermediate supports.
Couple Moment at Right End: A couple moment applied at the right end of the beam, creating a rotational effect and bending moment along the beam.
Shear Force: A couple moment does not generate shear force, as it does not produce a translational effect on the beam.
Bending Moment: The bending moment remains constant along the beam's length, except at the right end where the couple moment is applied, which causes a shift in the internal bending moment.
Deflection: The deflection caused by the couple moment depends on the magnitude of \( M_0 \), the beam's length, and material properties, with the maximum deflection occurring near the right end.

Behavior of the Simple Beam

Reaction Forces:
- The reaction forces at the supports are calculated using equilibrium equations. The couple moment at the right end does not affect the shear forces but influences the bending moment at the right support.
Shear Force Diagram:
- Since a couple moment does not create shear force, the shear force diagram is zero throughout the length of the beam.
Bending Moment Diagram:
- The bending moment diagram is constant along the length of the beam, with the magnitude of the moment equal to \( M_0 \) at every point, except at the right end, where the applied couple moment causes a change in the internal bending moment.
Deflection: The deflection caused by the couple moment depends on the magnitude of \( M_0 \), the beam's length, and the material's properties. The deflection is maximum near the right end, where the couple moment is applied.

Applications

Structural Engineering: Couple moments applied at the right end of a simple beam are used in structural analysis to model rotational effects and bending in components subjected to torsion or rotation.
Construction: This loading condition is used in analyzing beams subjected to rotational loads, such as beams in cantilevered or framed structures under torsion.
Mechanical Systems: Couple moments at the right end are common in mechanical systems that experience torsional forces, such as shafts and other rotating components.

Formula

Deflection (AB)	\( y_{\mathrm{AB}} = \frac{-M_0 x}{6 L E I}\left(L^2-x^2\right) \)
Slope (AB)	\( \theta_{\mathrm{AB}} = \frac{-M_0}{6 L E I}\left(L^2-3 x^2\right) \)
Slope at A	\( \theta_{\mathrm{A}} = \frac{-M_0 L}{6 E I} \)
Slope at B	\( \theta_{\mathrm{B}} = \frac{M_0 L}{3 E I} \)
Moment (AB)	\( M_{\mathrm{AB}} = \frac{M_0 x}{L} \)
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