Rigid Force Alpha System

Engineering mechanics 'Static' lecture 1. Force System. Before dealing with a group or system of forces, it is necessary to examine the properties of a single force in some detail, A force has been define as an action of one body on another. In dynamics we will see that a force is defined as an action which tends to cause acceleration of a body.

Stress and force when thermal expansion is restricted

Linear expansion due to change in temperature can be expressed as

Lies of astaroth all cards. dl = α l_o dt (1)

where

dl = elongation (m, in)

α = temperature expansion coefficient (m/mK, in/in ^oF)

l_o = initial length (m, in)

dt = temperature difference (^oC, ^oF)

The strain - or deformation - for an unrestricted expansion can be expressed as

ε = dl / l_o (2)

where

ε = strain - deformation

The Elastic modulus (Young's Modulus) can be expressed as

E = σ / ε (3)

where

E = Young's Modulus (Pa (N/m²), psi)

σ = stress (Pa (N/m²), psi)

Thermal Stress

When restricted expansion is 'converted' to stress - then (1), (2) and (3) can be combined to

σ_dt = E ε

= E dl / l_o

= E α l_o dt / l_o

= E α dt (4)

where

σ_dt = stress due to change in temperature (Pa (N/m²), psi)

Axial Force

The axial force acted by the restricted bar due to change in temperature can be expressed as

F = σ_dt A

= E α dt A (5)

where

F = axial force (N)

A = cross-sectional area of bar (m², in²)

Example - Heated Steel Pipe - Thermal Stress and Force with Restricted Expansion

A DN150 Std. (6 in)steel pipe with length 50 m (1969 in) is heated from 20^oC (68^oF) to 90^oC (194^oF). The expansion coefficient for steel is 12 10^-6 m/mK (6.7 10^-6 in/in^oF). The modulus of elasticity for steel is 200 GPa (10⁹ N/m²) (29 10⁶ psi (lb/in²)).

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Expansion of unrestricted pipe:

dl = (12 10^-6 m/mK) (50 m) ((90^oC) - (20^oC))

= 0.042 m

If the expansion of the pipe is restricted - the stress created due to the temperature change can be calculated as

σ_dt = (200 10⁹ N/m²) (12 10^-6 m/mK) ((90^oC) - (20^oC))

= 168 10⁶ N/m² (Pa)

= 168 MPa

Note! - if there is pressure in the pipe - the axial and circumferential (hoop) stress may be added to restricted temperature expansion stress by using vector addition.

The outside diameter of the pipe is 168.275 mm (6.63 in) and the wall thickness is 7.112 mm (0.28 in). The cross-sectional area of the pipe wall can then be calculated to

A = π ((168.275 mm) / 2)² - π ((168.275 mm) - 2 (7.112 mm)) / 2)²

= 3598 mm²

= 3.6 10^-3 m²

The force acting at the ends of the pipe when it is restricted can be calculated as

F = (168 10⁶ N/m²) (3.6 10^-3 m²)

= 604800 N

= 604 kN

The calculation in Imperial units

Expansion of unrestricted pipe:

dl = (6.7 10^-6 in/in^oF) (1669 in) ((194^oF) - (68^oF))

= 1.4 in

Stress in restricted pipe:

σ_dt = (29 10⁶ lb/in²) (6.7 10^-6 in/in^oF) ((194^oF) - (68^oF))

= 24481 lb/in² (psi)

Cross sectional area:

A = π ((6.63 in) / 2)² - π ((6.63 in) - 2 (0.28 mm)) / 2)²

= 5.3 in²

Axial force acting at the ends:

F = (24481 lb/in²) (5.3 in²)

= 129749 lb

Example - Thermal Tensions in Reinforced or Connected Materials

The lego ninjago movie video game. When two materials with different temperature expansion coefficients are connected - as typical with concrete and steel reinforcement, or in district heating pipes with PEH insulation etc. - temperature changes introduces tensions.

This can be illustrated with a PVC plastic bar of 10 m reinforced with a steel rod.

The free expansion of the PVC bar without the reinforcement - with a temperature change of 100 ^oC - can be calculated from (1) to

dl_PVC = (50.4 10^-6 m/mK) (10 m) (100 ^oC)

= 0.054 m

The free expansion of the steel rod with a temperature change of 100 ^oC - can be calculated from (1) to

dl_steel = (12 10^-6 m/mK) (10 m) (100 ^oC)

= 0.012 m

If we assume that the steel rod is much stronger than the PVC bar (depends on the Young's modulus and the areas of the materials) - the tension in the PVC bar can be calculated from the difference in temperature expansion with (4) as

σ_PVC = (2.8 10⁹ Pa) (0.054 m - 0.012 m) / (10 m)

= 11.8 10⁶ Pa

= 11.8 MPa

The Tensile Yield Strength of PVC is approximately 55 MPa.

Thermal Expansion Axial Force Calculator

This calculator can be used to calculate the axial force caused by an object with restricted temperature expansion. The calculator is generic and can be used for both metric and imperial units as long as the use of units are consistent.

Length of restricted object (m, inches)

Area of restricted object (m², in²)

Temperature difference (^oC, ^oF)

Young's modulus (GPa, 10⁹ psi)

Expansion coefficient (10^-6 m/mK, 10^-6 in/in^oF)

Related Topics

• Temperature Expansion - Thermal expansion of pipes and tubes - stainless steel, carbon steel, copper, plastics and more
• Mechanics - Forces, acceleration, displacement, vectors, motion, momentum, energy of objects and more
• Statics - Loads - force and torque, beams and columns

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