Axial Deformation - Theory & Concepts

Axial Deformation

Axial deformation refers to the change in length of a member subjected to axial loads (tension or compression). It represents how much a material elongates when pulled or shortens when compressed. This deformation is a direct consequence of the stress-strain relationship of the material and is critical for ensuring serviceability in design.

Fundamental Equations

Deformation Formula

For a homogeneous, prismatic bar subjected to a uniform axial load, the deformation (

\delta

) is given by the formula derived from Hooke's Law (

\sigma = E\epsilon

). By substituting

\sigma = P/A

and

\epsilon = \delta/L

into Hooke's law, we can derive the fundamental deformation formula:

\delta = \frac{PL}{AE}

where:

$P$ = internal axial force at the section (N).
$L$ = length of the member (mm).
$A$ = cross-sectional area (mm $^2$ ).
$E$ = Modulus of Elasticity (MPa).

The product

AE

is often referred to as the Axial Rigidity of the member. A higher axial rigidity means the member will deform less under a given load.

Varying Loads or Cross-Sections

If the load, area, or material varies along the length, the total deformation is the sum of the deformations of individual segments:

\delta = \sum \frac{P_i L_i}{A_i E_i}

For continuous variation (e.g., a tapered bar or distributed axial load):

\delta = \int_0^L \frac{P(x)}{A(x)E(x)} dx

Sign Convention

Tension (+): Causes elongation (positive $\delta$ ).
Compression (-): Causes contraction (negative $\delta$ ).

Deformation Due to Self-Weight

When a vertical bar is suspended from one end, it deforms not only from applied external loads but also under its own weight. The internal force

P(x)

varies linearly from zero at the free end to a maximum (the total weight) at the support.

For a prismatic bar with constant area

A

, length

L

, modulus

E

, and specific weight

\gamma

(weight per unit volume):

\delta_{weight} = \frac{\gamma L^2}{2E} = \frac{W L}{2AE}

where

W = \gamma A L

is the total weight of the bar. This indicates that a bar elongates half as much under its own self-weight as it would if the total weight

W

were applied entirely as a concentrated load at its free end.

Thermal Effects and Energy

Thermal Deformation

Changes in temperature cause materials to expand or contract. If this deformation is free to occur, no stress is induced.

\delta_T = \alpha L (\Delta T)

where:

$\alpha$ = Coefficient of Linear Thermal Expansion ( $/^\circ$ C).
$\Delta T$ = Change in temperature ( $T_{final} - T_{initial}$ ).

Thermal Stress

If the thermal deformation is restrained (prevented by supports), internal stresses develop. The stress is equivalent to the force required to push the member back to its original length.

\sigma_T = E \alpha (\Delta T)

Strain Energy from Axial Loads

Strain Energy ( $U$ ): When an axial load is applied, the member deforms and absorbs strain energy. For a gradually applied axial load, the strain energy within the proportional limit is given by the area under the load-deformation diagram.

U = \frac{1}{2} P \delta = \frac{P^2 L}{2 A E}

where

P

is the applied force,

\delta

is the deformation,

L

is the length,

A

is the area, and

E

is the modulus of elasticity.

Modulus of Resilience (U_r)

Modulus of Resilience ( $U_r$ ): The maximum strain energy density (strain energy per unit volume) a material can absorb without undergoing permanent plastic deformation.

U_r = \frac{\sigma_{yield}^2}{2E}

Impact Loading

Impact Loading: When an axial load is applied suddenly (dynamic load), the strain energy must absorb the kinetic energy of the impact, resulting in an internal force and deformation that can be twice as large as the static effect.

Statically Indeterminate Members

Statically Indeterminate Members

A structural member is considered statically indeterminate when the equations of static equilibrium (

\sum F = 0

\sum M = 0

) alone are insufficient to determine all the internal forces and support reactions. This typically happens when there are redundant supports—more constraints than necessary to maintain stability.

To solve these problems, we must supplement the equilibrium equations with Compatibility Equations. These equations relate the deformations of various parts of the structure to ensure geometric consistency (e.g., ensuring a continuous beam doesn't break apart at its joints).

Procedure

STEP-BY-STEP

Equilibrium Equations: Draw a Free Body Diagram (FBD) and write the static equilibrium equations (e.g., $\sum F_x = 0$ ).
Compatibility Equations: Formulate a relationship between the deformations of the members based on geometric constraints (e.g., $\delta_{total} = 0$ for fixed supports).
Force-Deformation Relations: Substitute $\delta = PL/AE$ into the compatibility equation.
Solve: Solve the system of simultaneous equations for the unknown forces.

Initial Gaps and Misfits

Sometimes, structural members are fabricated slightly too short or too long. If a member is too short by a gap distance

\Delta_{gap}

, and forces are applied to close the gap and connect it, internal stresses will develop.

The compatibility equation in such a scenario becomes:

\delta_{total} = \Delta_{gap}

This means the total elongation of the structure must equal the initial gap distance in order for the connection to be made.

Interactive Lab: Axial Deformation

Use the simulator below to explore how changes in Force, Length, Area, and Material stiffness affect the total axial deformation.

Axial Deformation & Stress-Strain Curve

L = 2000mm + 0.00mm

Applied Tensile Load (

P

): 0 kN

0Yield (125 kN)Ultimate (200 kN)

Normal Stress (

\sigma

)

0.0 MPa

Deformation (δ)

0.000 mm

Stress-Strain Diagram

Loading chart...

Key Takeaways

Axial Deformation ( $\delta = PL/AE$ ): The fundamental equation for members under axial load.
For varying cross-sections or loads, sum the deformations of individual segments ( $\delta = \sum PL/AE$ ) or integrate.
Deformation due to Self-Weight ( $\delta = WL/2AE$ ): A bar suspended vertically deforms half as much under its own weight as it would if the same weight were fully applied at the free end.
Thermal Expansion: Always consider temperature changes ( $\delta_T = \alpha L \Delta T$ ) in constrained structures, as they can generate significant stresses.
Strain Energy ( $U$ ) measures the potential energy stored due to deformation.
Indeterminate Structures: Equilibrium equations alone are not enough. You must use Compatibility (geometry of deformation) to generate extra equations.

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