Problem 46 - An Introduction to the Mechanics of Solids in SI Units

Question

Consider the problem of pure bending of the symmetrical composite beam which has been made by bonding together two materials of different elastic properties. Carry out a development parallel to that given in Secs \(7.2\) to \(7.5\) to obtain the deformation and the stresses in the composite beam. Show that the neutral surface is located by the distance \(y_{N}\) pictured, where (a) \(y_{N}=\frac{E_{1} \bar{y}_{1} A_{1}+E_{2} \bar{y}_{2} A_{2}}{E_{1} A_{1}+E_{2} A_{2}}\) and that the moment-curvature relation is (b) \(\frac{d \phi}{d s}=\frac{1}{\rho}=\frac{M_{b}}{E_{1}\left(I_{z z}\right)_{1}+E_{2}\left(I_{z z}\right)_{2}}\) where \(\left(I_{z z}\right)_{1}\) and \(\left(I_{z z}\right)_{2}\) are, respectively, the moments of inertia of the areas \(A_{1}\) and \(A_{2}\) about the neutral surface. Finally, show that the bending stress in the beam is given by (c) \(\quad\left(\sigma_{x}\right)_{i}=-E_{i} \frac{M_{b} y}{E_{1}\left(I_{z z}\right)_{1}+E_{2}\left(I_{z z}\right)_{2}}\) where \(i\) takes on the value of 1 or 2 , depending on which material we are interested in.

Step-by-Step Solution

Answer

The neutral surface location is found using the equation \(y_{N}=\frac{E_{1} \bar{y}_{1} A_{1}+E_{2} \bar{y}_{2} A_{2}}{E_{1} A_{1}+E_{2} A_{2}}\). The moment-curvature relation is found using \(\frac{d \phi}{d s}=\frac{1}{\rho}=\frac{M_{b}}{E_{1}\left(I_{z z}\right)_{1}+E_{2}\left(I_{z z}\right)_{2}}\). The bending stress in the beam can be found using \(\left(\sigma_{x}\right)_{i}=-E_{i} \frac{M_{b} y}{E_{1}\left(I_{z z}\right)_{1}+E_{2}\left(I_{z z}\right)_{2}}\).

Step 1: Determine the Neutral Surface

The location of the neutral surface can be identified by the following equation: \(y_{N}=\frac{E_{1} \bar{y}_{1} A_{1}+E_{2} \bar{y}_{2} A_{2}}{E_{1} A_{1}+E_{2} A_{2}}\). This is derived using the principle of strain compatibility, which states that at the neutral surface there is no stress or strain.

Step 2: Derive the Moment-Curvature Relation

The moment-curvature relation \( \frac{d \phi}{d s}=\frac{1}{\rho}=\frac{M_{b}}{E_{1}\left(I_{z z}\right)_{1}+E_{2}\left(I_{z z}\right)_{2}}\) can be obtained using bending theory, where \(\left(I_{z z}\right)_{1}\) and \(\left(I_{z z}\right)_{2}\) are the moments of inertia of the areas \(A_{1}\) and \(A_{2}\) about the neutral surface.

Step 3: Calculate the Bending Stress

The bending stress in the beam can be computed using this equation \(\left(\sigma_{x}\right)_{i}=-E_{i} \frac{M_{b} y}{E_{1}\left(I_{z z}\right)_{1}+E_{2}\left(I_{z z}\right)_{2}}\), where \(i\) is either 1 or 2, depending on which material we are referencing. The principle of superposition is applied, stating that the total stress is the sum of the stresses caused by the individual materials.

Key Concepts

Neutral Surface

The neutral surface in the context of composite beam bending is an essential concept to understand. It represents the imaginary line or surface within the beam where there is no stretching or compression when bending occurs. In simpler terms, it's a location inside the beam where the fibers neither extend nor compress.

• The position of the neutral surface is determined by the beam's geometry and material properties.
• The equation for finding the neutral surface in a composite beam is: \[y_{N}=\frac{E_{1} \bar{y}_{1} A_{1}+E_{2} \bar{y}_{2} A_{2}}{E_{1} A_{1}+E_{2} A_{2}}\] This formula helps locate the neutral surface by considering the elastic moduli \(E_1\) and \(E_2\), centroid locations \(\bar{y}_1\) and \(\bar{y}_2\), and areas \(A_1\) and \(A_2\) of the two materials.

Understanding the position of the neutral surface allows engineers to predict how a beam will behave under loads and ensure safety and structural integrity.
This position is crucial because it dictates where the maximum and minimum stresses occur within the beam structure.

Moment-Curvature Relation

The moment-curvature relation is a fundamental concept in bending theory. It describes how the curvature of a beam relates to the bending moment exerted on it. Simply put, it tells us how the beam bends under the influence of external forces.

• The moment-curvature relation can be expressed as: \[ \frac{d \phi}{d s}=\frac{1}{\rho}=\frac{M_{b}}{E_{1}\left(I_{z z}\right)_{1}+E_{2}\left(I_{z z}\right)_{2}} \] Here, \(\frac{1}{\rho}\) represents the curvature of the beam, \(M_b\) is the bending moment, and \(\left(I_{z z}\right)_{1}\) and \(\left(I_{z z}\right)_{2}\) are the moments of inertia for materials one and two respectively.
• This equation helps us understand how different materials and their geometric properties influence the bending of a beam.
• In a composite beam, the interaction of different materials means their elastic properties both influence how easily or reluctantly a beam bends.

Recognizing these relationships helps design structures that can withstand forces without undesirable deflections or failures.
By knowing the curvature, engineers can predict how a beam will deform under specific loads.

Bending Stress

Bending stress is another pivotal factor in analyzing beams subject to bending. It refers to the internal stress induced within a material when an external bending moment is applied.
This stress is crucial because it determines whether the material will yield or break under the load.

• The formula to calculate the bending stress in a composite beam is: \[ \left(\sigma_{x}\right)_{i}=-E_{i} \frac{M_{b} y}{E_{1}\left(I_{z z}\right)_{1}+E_{2}\left(I_{z z}\right)_{2}} \] Where \(i\) signifies the material being considered, and \(y\) is the distance from the neutral surface to the point where the stress is evaluated.
• The principle of superposition, which states that total stress is the cumulative effect of individual stresses from multiple forces or materials, applies here.
• Bending stress varies linearly from the neutral surface and is maximal on the outermost fibers of the beam.

By understanding bending stress, engineers can ensure that beams are designed efficiently, being neither too weak nor excessively overdesigned.
Accurate prediction of bending stresses is essential for material selection and to prevent structural failures.