EAS 6939 Aerospace Structural Composites

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EAS 6939 Aerospace Structural Composites

Solution for Project Phase-I

Prepared By:

SameerLuthra and Raja Ganesh

Course Advisor

Dr. Raphael T. Haftka

Table of Contents

Contents

Inputs

Steps Followed and Formulas Used

Part 1(a): Using 1 Pair of orientation angles

Part 1(b): Using 2 Pairs of orientation angles

Part 2: Minimizing the Poisson’s ratio

Part 3: Lightest laminate of the form

Appendix 1: Excel File with answer reports

Appendix 2: Grader’s Feedback

Inputs

A Cylindrical Pressure vessel of is to be designed for .

Material Used:Graphite-Epoxy

Material Properties:

Other Inputs:

Steps Followed and Formulas Used

Minor Poisson’s Ratio for Ply is calculated as:

Matrix [Q] is calculated:

Where

Material Invariants U’s are then calculated as:

Lamination Parameters are then calculated as:

Matrix is calculated:

Some initial value of Laminate thickness, h is assumed which is iterated to find the optimal value.

Stresses in the Laminate due to Internal Pressure are calculated as:

Using these Stresses in the laminate due to Internal Pressure, the Laminate mid-plane Strains(Due to Pressure) for the Balanced Symmetric laminate are found as:

To find the average Thermal Loads in the Laminate:

Using these ThermalLoads in the laminate, the Laminate Non-Mechanical Strains for the Balanced Symmetric laminate are found as:

To transformstrains in a Laminate to strains in a ply with orientation angle:

To find Free strains in Laminate (x-y)co-ordinate systemin a ply with orientation angle:

So theNet thermalstrains in Laminate (x-y) co-ordinate system in a ply with orientation angle:

So the Effective Strains due to Internal Pressure& due to change in temperature ΔT in Laminate (x-y) co-ordinate system in a ply with orientation angle:

To transformEffective strainsfrom Laminate (x-y) co-ordinate system to Ply (1-2) co-ordinate system in a ply with orientation angle:

To find stresses in Ply (1-2) co-ordinate system in a ply with orientation angle:

Hoffman Criterion is applied separately to each pair of plies to design the lightest balanced and symmetric laminate:

Part 1(a): Using 1 Pair of orientation angles

Objective: Design of the lightest Balanced and Symmetric laminate that can carry safely all the loads. In this part we want to design the lightest angle-ply laminate (where n is a continuous variable) based on Hoffman Criterion.

Procedure to solve: First of all the design problem is formulated as:

Objective Function: Minimize the Thickness of the laminate i.e.

Design Variables:

Constraints:The design is based on the Hoffman Criterion:

Steps Followed: Formulas used and general steps followed have been discussed in the initial section. Initial values of & n are taken:

These values are then iterated to find the optimal solution.

Finally, n is rounded up to an integer value.

Results:The optimal values are:

Part 1(b): Using 2 Pairs of orientation angles

Objective: Design of the lightest Balanced and Symmetric laminate that can carry safely all the loads. In this part we want to design the lightest laminate with 2 pair of orientation angles (where are a continuous variables) based on Hoffman Criterion.

Procedure to solve: First of all the design problem is formulated as:

Objective Function: Minimize the Thickness of the laminate i.e.

Design Variables:

Constraints:The design is based on the Hoffman Criterion:

Steps Followed: Formulas used and general steps followed have been discussed in the initial section. Major steps followed till finding Laminate mid-plane strains are the same as the previous part.

Initial values of are taken:

These values are then iterated to find the optimal solution.

Results:The optimal values are:

Note: This is the same laminate that was obtained in section 1-a.

Part 2: Minimizing the Poisson’s ratio

Objective:Using the Balanced Symmetric laminate (with n = integer) designed in the previous part, design a Balanced Symmetric laminate with the same thickness (as calculated in part 1) that will not fail and has the lowest Poisson’s Ratio(considering the maximum of the two Poisson’s ratios)

Procedure to solve: First of all the design problem is formulated as:

Objective Function: Minimize the Poisson’s Ratio (Maximum of the two Poisson’s ratios).

Design Variable:

Constraints:Constraints:The design is based on the Hoffman Criterion:

Steps Followed: Most of the Formulas used and general steps followed have been discussed in the initial section. The n value (rounded to integer) obtained from Part 1 is kept constant.

Formulas for Poisson’s Ratios of the Laminate:

The value of Orientation angle θis then iterated to find the optimal solution (i.e. to Minimize(max(,)))

Results:The final values are:

Part 3: Lightest laminate of the form

Objective: Design of the lightest Balanced and Symmetric laminate that can carry safely all the loads. The laminate is designed based on Hoffman Criterion. In this part we want to design the laminate with Minimum thickness.

Procedure to solve: First of all the design problem is formulated as:

Objective Function: Minimize the Thickness of the laminate i.e.

Design Variables:

Constraints:The design is based on the Hoffman Criterion. There is another constraint that volume fraction of any orientation angle should not be less than 0.10 and should not be greater than 0.50.

Steps Followed: Formulas used and general steps followed have been discussed in the initial section. Some of the major steps followed are summarized as:

Initial values of are taken:

These values are then iterated to find the optimal solution.

The volume fractions of plies of each orientation are computed as follows:

Results:The optimal values are:

Appendix 1: Excel File with answer reports

Double-click to open

Appendix 2: Grader’s Feedback

Double-click to open