Example 1 – Check structural strength of Xi rocket nosecone for two loading conditions:

1. Handling load

2. Flight load at Mach 1.0 velocity at 10-degree angle-of-attack

Nosecone Description

The Xi nosecone has a tangent-ogive profile. The base diameter is 3.0 inches and length 7.56 inches giving a fineness ratio of 2.52 . The nosecone is CAD designed and 3D printed of grey PLA plastic. The nosecone houses a Big Red Bee BRB900 (900 MHz) GPS transmitter in a shock-resistant cradle. The nosecone is designed for minimum mass. Maximum expected velocity in flight is 1.0 Mach. The figure below illustrates the nosecone design, which incorporates a shoulder for attachment to the rocket body tube utilizing three #6-32 flat-head alloy-steel screws. A minimum Safety Factor of 2.0 is the requirement for acceptable strength.

Xi nosecone with cutaway view on right

1. Handling condition

The handling load condition is shown below. A force is applied at the tip of the nosecone, perpendicular to the long-axis. This lateral force (F _HANDLING) is reacted at the nosecone shoulder (R) by bearing against the body tube. The offset-moment (M) is reacted by the screws attaching the shoulder to the rocket body.

Examining the nosecone cross-section, it can be seen that there are two possible “critical sections” where fracture may occur, labeled as A-A and B-B. Section A-A is considered as this is the thinnest part of the nosecone wall at the furthest distance from the applied load. Section B-B is considered as this location is furthest from the applied load and where the reactions occur. Dimensions to these two critical sections are shown in the figure below:

Based on the dimensions of the nosecone cross-section at these two sections, the pertinent geometric property, section modulus, is calculated.

At section A-A:

Outer diameter (D_o) = 2.86 inch with wall thickness 0.050 inch, giving

Inside diameter (D_i)= 2.76 inch

Section modulus =

At section B-B: (location of screws attaching nosecone to rocket body)

Outer diameter (D_o) = 2.93 inch with wall thickness 0.080 inch, giving

Inside diameter (D_i)= 2.77 inch

Section modulus =

The magnitude of the handling load is next calculated, followed by the bending stress at each section, and the resulting minimum Safety Factor.

Mass of Xi rocket = 7.7 lbs (typical), apply 10% growth factor:

m = 1.10 × 7.7 = 8.48 lbs

Conservatively take handling load as 5×m:

F _HANDLING = 5 × 8.48 = 42.4 lbf

Bending moment at section A-A and B-B:

M _A-A = 42.4 × 5.71 = 242.1 lbf-in

M _B-B = 42.4 × 7.88 = 334.1 lbf-in

Bending stress at section A-A and B-B

where

f_{b A-A} = 242.1/0.305 = 794 lbf/in²

f_{b B-B} = 334.1/0.497 = 672 lbf/in²

Section A-A is more critical than section B-B for this loading condition.

The nosecone is 3D printed of PLA plastic. Properties are given below:

PLA

Ref.https://www.makeitfrom.com/material-properties/Polylactic-Acid-PLA-Polylactide

Glass transition temperature = 60C.

Max. temperature mechanical = 50C.

Melting onset = 160C.

Thermal conductivity = 0.13 W/m-K

Density = 1.3 g/cc

E = 0.51x10^6 psi

elogation at break = 6%

Flexural modulus = 0.58x10^6 psi

Flexural strength = 12 ksi

Shear modulus = 0.35x10^6 psi

Tensile strength = 7.3 ksi

The applicable property for this failure mode is tensile strength . The strength properties indicated are for virgin PLA filament. Three knockdown factors should be applied:

1. Printing process deficiencies. Loading direction is perpedicular to extrusion direction, relying upon heat fusion to bond layers. Layed-down material dimensions may not match blueprint. Apply knockdown factor of 20%

2. Environment. PLA aborbs moisture and may weaken in service. Apply knockdown factor of 20%

3. Pigment. Natural PLA (no dye added) has the highest strength. Pigments tend to reduce strength ^[1] . Apply knockdown factor of 5%.

Factored tensile strength:

S _tf = 7300 × (1.0 – 0.20) × (1.0 – 0.20) × (1.0 – 0.05) = 4438 lbf/in²

Minimum Safety Factor based on Handling Load condition:

S.F. = 4438/794 = 5.59 > 2.0

[1] Ref. “The Effects of PLA Color on Material Properties of 3-D Printed Components”, Ben Wittbrodt and Joshua M. Pearce, Michigan Technological University, Houghton, MI 49931, USA

2. Flight Condition: Mach 1.0 velocity at 10-degree angle-of-attack

The Xi nosecone subjected to this flight condition is illustrated below. The nosecone is subjected to drag load (F_D) assumed acting at the tip, and a lateral load (N_Nose) acting at the nosecone Centre of Pressure. The angle-of-attack (a) is indicated. Assume this condition occurs at an altitude of 2500 feet ASL.

The drag load is calculated first:

Air density (r) is found from the following table:

https://www.engineeringtoolbox.com/standard-atmosphere-d_604.html

By linear interpolation, at 2500 feet ASL, r = 0.0022 slugs/ft³

The critical flight condition is 1.0 Mach. From the following table, the velocity is found:

https://www.engineersedge.com/physics/speed_of_sound_13241.htm

By linear interpolation, at 2500 feet ASL, V = 754.5 mph = 1107 ft/sec.

The drag coefficient, at small angle-of-attack, for a tangent-ogive nosecone with fineness ratio 2.52 is found using AeroLab software:

To be conservative, the maximum value (at 1.1 Mach) is taken:

C_dn = 0.26 frictional drag is neglected as tiny

The base area of the nosecone is:

A = ¼ p (3.0/12)² = 0.049 ft²

The drag force is calculated:

The next step is to calculate the normal force acting on the nosecone, given by:

N_NOSE = q A a (C_Na)_N

The dynamic pressure (q) is calculated:

q = ½ (0.0022) (1107)² = 1348 lbf/ft²

The angle-of-attack of 10° is converted to radians:

a = 10/180 × p = 0.175 radian

The slope of the normal force coefficient with respect to angle-of-attack (C_Na)_N is obtained from Figure 7 which is reproduced below:

The peak value at Mach 1.1 is conervatively chosen:

(C_Na)_N = 2.3 per radian, therefore:

N_NOSE = 1348 (0.049) 0.175 (2.3) = 26.6 lbf

The location of the nosecone C.P. is found using Figure 8, reproduced below:

At Mach 1.0, for a fineness ratio of 2.5, X_CP = 1.2 calibres aft of the nosecone tip. As the calibre of the nosecone is 3.0 inches, the C.P. location is 1.2 × 3.0 = 3.6 inches aft of the tip.

These two loading conditions are applied to the nosecone as shown in the figure below. Critical sections A-A and B-B are checked for bending stress and resulting Safety Factor determined.

To simplify the analysis, the drag force is resolved into two components, one axial and one lateral:

F_{D ax} = 17.2 cos 10° = 16.9 lbf

F_{D lat} = 17.2 sin 10° = 3.0 lbf

The axial component of the drag force is conservatively neglected, as it tends to put the critical sections in compression, which is not a critical failure mode.

Bending moment at section A-A and B-B:

M _A-A = 3.0 × 5.71+ 26.6 × 2.11 = 73.3 lbf-in

M _B-B = 3.0 × 7.88+ 26.6 × 4.28 = 137.5 lbf-in

Bending stress at section A-A and B-B

where

f_{b A-A} = 73.3/0.305 = 240 lbf/in²

f_{b B-B} = 137.5/0.497 = 277 lbf/in²

Both Section A-A and section B-B are less critical for this loading condition than the Handling condition.

Important – consistent units must be used in all calculations. For example, when calculating F_D, q, and N_NOSE , imperial units or SI (metric) should be:

Mass slugs (kilograms)

Force lbf (Newtons)

Length feet (metres)

Angle radians (radians)

Time seconds (seconds)

Other consistent units may be used; however, the above systems are recommended.

Rev.2023/07/21