1

Pre-Analysis

• Mathematical model: 3D Elasticity • Numerical solution strategy: Finite-element method• Hand-calculations of expected results/trends

Domain

• Model half of a bolt-and-nut assuming symmetry

• Four parts– Mid nozzle– Lower nozzle– Bolt– Nut

2

sx

sy

sz

tyz

tyx

txytxz

tzytzx

Mathematical Model: Governing Equations

����

��+

����

��+

���

��+ �� = 0

���

��+

����

��+

����

��+ �� = 0

����

��+

���

��+

����

��+ �� = 0

Physical principle: Equilibrium of infinitesimal element

�⃗ = � �⃗ or Σ�� = 0

3D Differential Equations of Equilibrium

• 3 eqs.: Force balance in x, y, z • 6 unknowns: ��, ��, ��, ���, ���, ���

Additional Equations: Constitutive Model

��

��

��

���

���

���

=�

(1 + �)(1 − 2�)

1 − ���000

�1 − �

�000

��

1 − �000

000

1 − 2�00

0000

1 − 2�0

00000

1 − 2�

��

��

��

���

���

���

−�

1 − 2�

������

000

− ������

��,����

��,����

��,����

000

3

Mathematical Model: Additional Equations

Strain-Displacement Relations

�� = ��

���� =

��

��

��� =��

��+

��

��

�� = ��

��

��� = ��

��+

��

����� =

��

��+

��

��

Mathematical Model: Governing Equations Summary

• Equations– 3: Force balance on infinitesimal element in x, y, z

directions – 6: Constitutive Model– 6: Strain-displacement relations– Total = 15

• Unknowns– 6 stress components: ��, ��, ��, ���, ���, ���

– 6 strain components: ��, ��, ��, ���, ���, ���

– 3 displacement components: �, �, �– Total = 15

4

Mathematical Model: Boundary Conditions

• At every point on the boundary, the traction or displacement has to be defined– Normal as well

as 2 tangential directions

Displacement or Essential Boundary Conditions (1/2)

• Symmetry condition from periodicity

5

Displacement or Essential Boundary Conditions (2/2)

• “Frictionless support” at top surface of mid nozzle – Normal displacement =

0– Tangential traction = 0

• Approximates connection to upper nozzle (which is not included in the model)

Traction or Natural Boundary Conditions (1/2)

• Pressure due to propellant – Calculated using 1D

gas dynamics– Varies in axial direction

(“z”)

• Force from regeneration channels– Pulls apart the mid and

lower nozzles

6

Traction or Natural Boundary Conditions (2/2)

• Traction at contact surfaces– It is not known a priori

where the parts are going to come into contact at the interfaces

– Traction is dependent on the displacement

• �⃗ = �⃗(�, �, �)• Highly nonlinear

Undeformed

Deformed

Numerical Solution Strategy

• System of algebraic equations are nonlinear � � = {�}

• Compare to linear case: � � = {�}

Mathematical Model (Coupled Boundary

Value Problems)

Set of algebraic equations in nodal

displacements

�(�) = {�}Piecewise polynomial

approximation for �, �, �

Source of nonlinearity:• Contact

7

Newton-Rhapson Method for Solving Nonlinear Algebraic Equations

• We need to find �such that � � − � = 0

• Scalar analog:� � − � = 0

Eg. �� − 20 = 0

Update eq. • �� �� �� =

�� �� �� − � �� + �

Newton-Rhapsonfor single nonlinear algebraic eq.

� � − �

�������

Newton-Rhapson Method for Solving Nonlinear Algebraic Equations

• Need to solve �(�) − � = 0• Initial guess ��

• Update using Newton-Rhapson to get ��

– Update eq.: �(��) �� = � + {� �̅� }

• Calculate residual:

– �(��) − {�} = ���������

• If residual is larger than tolerance, update guess and repeat– New update eq.: � �� �� = � + {� �̅� }– Solve to calculate ��

8

Hand Calculations of Expected Results

1. Reaction in axial (z) direction where mid nozzle is attached to top nozzle (not modeled)

2. Hoop stress ��

�

3. Thermal strain and deformation

4. Bolt preload check Δ� =� �

� �

Hand Calculations: Reaction in Axial (z) Direction

• Average gas pressure:

��.�����.��

�≈ 30 ���

• Top radius = 41.75 inchesBottom radius = 69.50 inches

• Projected area in z direction = �(69.50� −41.75�) = 9699 ���

• Net pressure force in z direction = 30 ∗ 9699 • Net reaction force in -z direction on 1/400th model

= ��∗����

���≈ 720 ���

9

Hand Calculations: Hoop Stress

• �� =��

�

• At exit:

• � = 12.17 ���

• � = 69.5 ��

• � = 0.5 ��

• ⇒ �� ∼ 1692 ���

Hand Calculations: Thermal Strain

• Thermal strain = � Δ� = �(700� − 70�)

• Recall that this term appears in the constitutive model

10

Hand Calculations: Bolt Preload

• Δ� =� �

� �

• � = 2320 ���

• � = 3.8 × 10�� ���

• � = 2.9 × 10����

• � ∼ 0.5 ��

• ⇒ Δ� = 0.001 ��

Δ�

Δ�

2

Δ�

2

F

F

FF

FF