Internal Axial Force

students, imagine pulling on a rope in tug-of-war 💪. The rope is stretched, and every part of it is carrying the pull from one side to the other. In Solid Mechanics 1, that idea becomes more formal: we study the internal axial force inside a member, meaning the force acting along the member’s own length. This lesson explains what internal axial force is, how to identify it, and why it matters in structures like trusses, bars, and simple frameworks.

What you will learn

By the end of this lesson, students, you should be able to:

explain the main ideas and vocabulary of internal axial force
determine whether a member is in tension or compression
use free-body diagrams to find internal axial force in simple cases
connect internal axial force to structures and internal actions
recognize where axial force appears in real engineering examples 🏗️

What is internal axial force?

A structure is made of parts that carry loads. When a load is applied, the inside of each part must resist it. The force that acts along the long axis of a member is called the internal axial force.

If a member is being pulled, the internal axial force is tension. If it is being pushed, the internal axial force is compression.

A simple way to think about it is this: if you cut a bar somewhere in the middle, the two pieces would try to move apart or together. The force appearing at the cut that keeps the pieces in equilibrium is the internal axial force.

This is an example of an internal action, because it describes what happens inside the structure, not just at the outside supports.

Tension and compression

Internal axial force has two main forms:

Tension: the member is stretched, and the internal force pulls away from the cut section.
Compression: the member is squeezed, and the internal force pushes toward the cut section.

The direction is important, but the sign convention depends on the method used in class or in a textbook. A very common convention is:

positive $N$ for tension
negative $N$ for compression

Here, $N$ is the internal axial force.

For example, if a steel rod is hanging from a ceiling and holding a weight at its end, the rod is in tension. If a short column supports the roof of a building, the column is often in compression.

students, you can think of tension like a stretched rubber band and compression like a squashed sponge 🧽.

How internal axial force is found

The main method is to cut the member and look at one side of the cut. This is called the method of sections.

Step 1: Choose a cut

Pick the point where you want the internal axial force.

Step 2: Draw a free-body diagram

Remove the part of the structure on one side of the cut. Replace the cut with the internal force $N$ acting along the member.

Step 3: Apply equilibrium

Because the part you isolated is still in static equilibrium, you use force balance. For a straight member loaded only along its axis, the most useful equation is:

$\sum F_x = 0$

If the only internal action is axial force, this equation is usually enough.

Step 4: Solve for $N$

The internal axial force is found from the balance of forces along the member.

A simple example: a straight bar in tension

Suppose a bar is pulled by equal and opposite forces of $10\,\text{kN}$ at both ends. Since the forces pull outward, the bar is in tension.

If you cut the bar anywhere in the middle and isolate one side, the internal force $N$ must balance the external load. Therefore,

$N = 10\,\text{kN}$

This means every cross-section in the bar carries an internal axial force of $10\,\text{kN}$ in tension, assuming the bar is uniform and the load is applied axially.

This is a very important idea: in a simple axially loaded member, the internal axial force can be the same along the entire length.

A simple example: a bar in compression

Now imagine a short vertical column supporting a load of $20\,\text{kN}$ downward at the top. The column pushes back on the load, so the internal axial force inside the column is compressive.

If we cut the column near the middle, the upper part will show a downward external load of $20\,\text{kN}$. To keep equilibrium, the internal force at the cut acts upward on the upper free body. The magnitude is

$N = 20\,\text{kN}$

Since it is compression, the sign may be written as

$N = -20\,\text{kN}$

if compression is taken as negative.

This sign difference matters because it tells us whether the member is being pulled or pushed.

Internal axial force in structures

Internal axial force is everywhere in structures, even when it is not obvious.

Trusses

In an ideal truss, members are connected by pins and loads are applied at the joints. Each member is treated as carrying only axial force. That means a truss member is either in tension or compression, but not usually bending in the ideal model.

This makes trusses very efficient for bridges and roof frames 🌉. The top chord might be in compression, the bottom chord in tension, and the diagonal members may switch depending on the loading.

Simple frameworks

In a simple framework, members can carry more than one type of internal action, such as axial force, shear force, and bending moment. But axial force is still one of the first things to check. It tells us whether a member is mainly stretched or compressed.

Columns and ties

A tie is a member designed to resist tension. A column is often designed to resist compression. These are common examples of axial loading in buildings and machines.

Why axial force matters

Internal axial force is important because materials behave differently in tension and compression.

A steel tie rod can carry large tensile force.
A slender column can fail by buckling if compression becomes too large.
A cable can carry tension but cannot resist compression effectively.

So, knowing the internal axial force helps engineers choose the right material and shape for a member.

For example, a bicycle spoke is mostly in tension, while the legs of a table are mostly in compression. These are everyday examples of axial action in structures 🚲.

Reading a free-body diagram correctly

When solving problems, it is easy to confuse external loads with internal forces. A good habit is to ask:

What part of the structure am I isolating?
What forces act on that isolated part?
What direction must the internal axial force act to satisfy equilibrium?

If the assumed direction gives a negative answer, that means the actual force acts in the opposite direction.

For a straight horizontal bar, the internal force usually acts along the bar’s axis. For a vertical member, it acts vertically. The key idea is always the same: the force is along the member’s length.

A worked reasoning example

Suppose a vertical rod is fixed at the top and supports a downward load of $5\,\text{kN}$ at the bottom. To find the internal axial force at a section halfway down the rod:

cut the rod at the chosen section
keep the lower part as the free body
the lower part has the $5\,\text{kN}$ downward load at the bottom
the cut must supply an upward internal force $N$ to balance it

Using equilibrium,

$\sum F_y = 0$

$N - 5\,\text{kN} = 0$

Thus,

$N = 5\,\text{kN}$

The member is in tension if the lower part is being pulled upward by the upper support, or in compression depending on the actual arrangement and sign convention. The important part is that the internal force is found by equilibrium after the cut.

Common mistakes to avoid

students, here are common errors students make:

forgetting to cut the member and instead trying to guess the force
mixing up internal axial force with external support reactions
using the wrong sign convention
drawing the internal force in the wrong direction on the free-body diagram
assuming all members in a structure carry the same axial force

A careful diagram and a clear sign convention prevent most of these mistakes.

How axial force fits into Structures and Internal Actions

This lesson is part of a bigger topic called Structures and Internal Actions. In that topic, the main internal actions are usually:

internal axial force $N$
shear force $V$
bending moment $M$

Internal axial force is the most direct one because it acts along the member’s axis. It often comes before studying shear force and bending moment, since it helps build the habit of cutting structures and using equilibrium.

In trusses and frameworks, axial force is often the first internal action to analyze. In beams, axial force may be smaller or even zero in many common loading cases, but it can still be important when loads act horizontally or when members are part of a frame.

Conclusion

Internal axial force tells us how a member carries load along its length. When a member is pulled, it is in tension; when it is pushed, it is in compression. The main tool for finding axial force is the method of sections, where a cut is made and equilibrium is applied to one side of the structure.

This idea is central to Solid Mechanics 1 because it connects the outside loads on a structure to the hidden forces inside it. Whether you are studying a truss, a cable, a column, or a simple bar, students, internal axial force helps explain how and why structures stay standing.

Study Notes

Internal axial force is the force acting along the length of a member.
It is an internal action, meaning it acts inside the structure.
Tension means the member is stretched; compression means it is squeezed.
A common sign convention is positive for tension and negative for compression.
The main method for finding axial force is to cut the member and use equilibrium.
For simple axial members, use $\sum F_x = 0$ or $\sum F_y = 0$ depending on the direction.
Truss members are idealized as carrying only axial force.
Columns usually carry compression, while ties and cables usually carry tension.
Internal axial force helps engineers design safe and efficient structures.
It is one of the main internal actions in the topic Structures and Internal Actions.