Can someone please help me with this problem?

In order to reduce the material cost, an engineer decides to use a hybrid beam instead of an all-carbon fiber beam. Both beams have the same overall dimensions with rectangular cross-sections. The hybrid beam contains carbon fibers in the outer layers and S-glass in the core. Costs of the materials are as follows:

Carbon/Epoxy: $ 25/lb
S-glass/Epoxy: $3.00/lb

The densities of carbon/epoxy and S-glass/epoxy composites are 1.70 g/cm3 (0.061 lb/in3) and 2.00 g/cm3 (0.072 lb/in3) respectively. The total carbon fiber thickness in the hybrid beam is equal to the core thickness. Compare the percentage weight penalty and cost savings for the hybrid beam over an all-carbon fiber beam. Do you expect the all-carbon and hybrid beams to have the same bending stiffness? If the answer is “no”, what can be done to make the two stiffness equal?

To compare the weight penalty and cost savings for the hybrid beam over an all-carbon fiber beam, we need to calculate the weights and costs of each beam.

Let's denote:
- W_total as the total weight of the beam
- W_carbon as the weight of the all-carbon fiber beam
- W_hybrid as the weight of the hybrid beam
- Cost_carbon as the cost of the all-carbon fiber beam
- Cost_hybrid as the cost of the hybrid beam

To calculate the weight and cost of the all-carbon fiber beam, we need to know the dimensions (length, width, and height) of the beam.

Once we have the dimensions, we can calculate the volume of the beam by multiplying the length, width, and height. Then, we can calculate the weight of the all-carbon fiber beam by multiplying the volume by the density of the carbon/epoxy composite (0.061 lb/in3). Finally, we can calculate the cost of the all-carbon fiber beam by multiplying the weight by the cost per pound of carbon/epoxy ($25/lb).

To determine the weight and cost of the hybrid beam, we need to know the thickness of the carbon fiber layers and the core. Let's denote the carbon fiber thickness as t_carbon (the same for both layers) and the core thickness as t_core.

To calculate the weight of the hybrid beam, we can divide it into three parts: the carbon fiber layers, the S-glass core, and the epoxy matrix. The weight of the carbon fiber layers is calculated by multiplying the length and width of the beam by the carbon fiber thickness (which is the total thickness in the hybrid beam) and the density of the carbon/epoxy composite. The weight of the S-glass core is calculated by multiplying the length and width of the beam by the core thickness and the density of the S-glass/epoxy composite (0.072 lb/in3). Lastly, the weight of the epoxy matrix is calculated by subtracting the weight of the carbon fiber layers and the S-glass core from the total weight of the hybrid beam.

To calculate the cost of the hybrid beam, we can multiply the weight of each component (carbon fiber layers, S-glass core, and epoxy matrix) by their respective costs per pound (carbon/epoxy is $25/lb, and S-glass/epoxy is $3.00/lb). The total cost is the sum of the costs of the three components.

Once we have the weights and costs of both beams, we can calculate the weight penalty as the percentage difference between the weight of the hybrid beam and the all-carbon fiber beam using the formula:

Weight penalty = (W_hybrid - W_carbon) / W_carbon * 100

Similarly, we can calculate the cost savings as the percentage difference between the cost of the hybrid beam and the all-carbon fiber beam using the formula:

Cost savings = (Cost_carbon - Cost_hybrid) / Cost_carbon * 100

To determine whether the all-carbon and hybrid beams have the same bending stiffness, we need to consider the material properties of each component (carbon fiber, S-glass, epoxy matrix) and their arrangement in the beam. The bending stiffness depends on the material modulus of elasticity, cross-sectional area, and moment of inertia of the beam.

If the all-carbon and hybrid beams do not have the same bending stiffness, we can adjust the dimensions and arrangement of the materials in the hybrid beam to achieve equal stiffness. For example, we can increase the thickness of the carbon fiber layers or adjust the arrangement of the carbon fiber and S-glass layers to achieve the desired stiffness. It would require further analysis and calculations based on the specific requirements and constraints of the beam design.