20.3:
Deformations in a Symmetric Member in Bending
Consider the deformations of a symmetric prismatic member subjected to equal and opposite couples.
As the member bends uniformly, the lines on the wider faces of the members transform into a circle of constant curvature centered at point C.
By dividing the member into tiny cubic elements, it becomes apparent that the only non-zero stress component is the normal one, leading to uniaxial stress at any given point.
So, a neutral surface exists parallel to the member's upper and lower faces where longitudinal components of strain and stress are zero.
The deformation of an arc, located at a distance y above the neutral surface, can be expressed as the difference in the length of this arc and that of the neutral surface.
Expressing these lengths in terms of the radius and the angle subtended and dividing the deformation by the length of the neutral arc shows that the longitudinal normal strain varies linearly with the distance y from the neutral surface.
Assuming a positive bending, the negative sign indicates the beam's upward concavity.
20.3:
Deformations in a Symmetric Member in Bending
When analyzing the deformation of a symmetric prismatic member subjected to bending by equal and opposite couples, it becomes clear that as the member bends, the originally straight lines on its wider faces curve into circular arcs, with a constant radius centered at a point known as Point C. This phenomenon helps to understand the stress and strain distribution within the member more clearly.
When the member is segmented into tiny cubic elements, it is observed that the primary stress experienced within the member is normal stress, leading to uniaxial stress conditions at any point. This arrangement reveals the existence of a neutral surface, where both the strain and stress longitudinal components are zero. This surface runs parallel to the upper and lower faces of the member, and the distance from the neutral surface to point C is ρ
To explore the deformation of this member, consider an arc at a distance y from the neutral surface. The deformation is the difference in lengths from point C between the arc at y (L') and the neutral surface arc (L). Dividing the deformation δ = L' – L by the length of the neutral arc shows that the longitudinal normal strain varies linearly with the distance from the neutral surface. By applying Hooke's Law, which relates stress and strain in elastic materials, the stress can be determined at any point based on its distance from the neutral surface.