The shear force at the section cut is considered positive if it causes clockwise rotation of the selected beam section, and it is considered negative if it causes counter-clockwise rotation. The sign is determined after a section cut is taken and the reactions are solved for the portion of the beam to one side of the cut. The signs of the shear and moment are important. The reactions are calculated such that the section of beam being considered is in static equilibrium. The internal reactions at the section cut are shown with blue arrows. The selected side is shown as the blue section of beam, and section shown in grey is ignored. In the figure above, the side of the beam to the right of the section cut was selected. The side that is selected does not affect the results, so choose whichever side is easiest. When the beam is cut at the section, either side of the beam may be considered when solving for the internal reactions. For example, the cantilever beam below has an applied force shown as a red arrow, and the reactions are shown as blue arrows at the fixed boundary condition. To find the shear force and bending moment over the length of a beam, first solve for the external reactions at each constraint. A boundary condition indicates the fixed/free condition in each direction at a specific point, and a constraint is a boundary condition in which at least one direction is fixed. This highlights the subtle difference between a constraint and a boundary condition. Therefore, a constraint does not exist at this point. This boundary condition indicates that the beam is free to move in every direction at that point (i.e., it is not fixed or constrained in any direction). Notice the Free boundary condition in the table above. Likewise, we see that a pinned boundary condition can develop axial and transverse reaction forces, but it cannot develop a reaction moment. Likewise, if the beam is fixed against rotation at a specific point, then an external reaction moment may develop at that point.īased on the above discussion, we can see that a fixed boundary condition can develop axial and transverse reaction forces as well as a moment. For example, if a beam is fixed in the y-direction at a specific point, then a transverse (y) external reaction force may develop at that point. If the boundary condition indicates that the beam is fixed in a specific direction, then an external reaction in that direction can exist at the location of the boundary condition. For each boundary condition, the table indicates whether the beam is fixed or free in each direction at the point where the boundary condition is defined. For a constraint to exist at a point, the boundary condition must indicate that at least one direction is fixed at that point.Ĭommon boundary conditions are shown in the table below. For a 2-dimensional beam, the directions of interest are the x-direction (axial direction), y-direction (transverse direction), and rotation. The boundary condition indicates whether the beam is fixed (restrained from motion) or free to move in each direction. Constraints are defined at single points along the beam, and the boundary condition at that point determines the nature of the constraint.
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