22.1:
Shear on the Horizontal Face of a Beam Element
A prismatic beam's small element acted upon by vertical and horizontal shearing forces, along with normal forces, is considered for analyzing shear on a horizontal face.
Using the equilibrium equation, the balance of forces acting on this element is examined to determine the horizontal shearing force. This force is then equated to bending moments and the first moment of the cross-section.
Analyzing either the lower element or the upper element yields the same result, as the shearing forces exerted on each other are equal in magnitude but opposite in direction.
It's also observed that the first moment of the portion below a certain line in the beam is equal in magnitude but opposite in sign to the first moment of the portion above.
The first moment reaches its maximum when the distance from the neutral axis equals zero, as elements above the neutral axis positively contribute to the integration part of the horizontal shearing forces equation, while those below negatively contribute.
When horizontal shear is divided by the length of the element, the shear flow is obtained.
22.1:
Shear on the Horizontal Face of a Beam Element
To understand shear on the flat side of a prismatic beam element, consider the vertical and horizontal shearing forces, and the normal forces, acting on the element. The element's upper (U) and lower (L) sections, which are divided by the beam's neutral axis, are examined. The equilibrium of these forces is determined by applying the equilibrium equation, which helps identify the horizontal shearing force. This force is directly related to the bending moments and the cross-section's first moment of inertia. The analysis yields consistent outcomes whether examining the upper or lower part of the beam, as the shearing forces are equal in magnitude but opposite in direction.
It's important to note that the first moment of the beam's section below a specific line is the same as the first moment of the section above it, both in magnitude and direction. This first moment reaches its maximum at the beam's neutral axis. Here, the sections above the neutral axis contribute positively towards the calculation of horizontal shearing forces, while those below contribute negatively. The calculation of shear flow involves dividing the horizontal shear by the length of the beam element. Understanding this process is vital for providing a comprehensive view of how shear operates on a horizontal plane within the beam, crucial for assessing structural integrity and design requirements of beam elements in engineering projects.