Hi friends, 👋👋
Last week we calculated reaction forces. This week we’ll show how to calculate internal forces such as bending moments, normal forces and shear forces of statically determinate structures.
Calculating internal forces is one of the first things we learn in uni, which we need throughout our career.
Based on internal forces, structural elements such as timber beams, steel columns, and concrete slabs are designed and verified.
Here’s what we’ll cover:
What internal forces are
Sign convention
Procedure of calculating internal forces
So let’s get into it. 🚀🚀
What are internal forces?
Structural elements must resist forces within itself – the so-called internal forces – which are produced by external forces like dead load, snow load, wind load, etc.
If the structural elements don’t resist those internal forces, the elements break/fail.
In general, there is 3 different internal forces:
Normal force N
Shear force V
Moment M
What is a moment?
The moment or also called bending moment happens when a structural element bends. This can happen by loads or external moments. The unit is Kilonewton meter [kNm].
A positive Moment (+) leads to compression in the top part of the Cross-section of the structural member (e.g. beam) and tension in the bottom, while a negative Moment (-) turns this into compression in the bottom and tension in the top part of the Cross-section.
What is the normal force?
The Normal force N acts in the axis of the beam. The unit is Kilonewton [kN].
A negative Normal force (-) leads to compression in the structural member, while a positive Normal force (+) leads to tension in the member.
What is the shear force?
The Shear force V acts perpendicular to the beam axis. The unit is Kilonewton [kN].
The best definition of shear I have ever read comes from J. E. Gordon in his book Structures: Or Why Things Don’t Fall Down. I have written a whole article about the best takeaways of the book, and you can read about his explanation of shear here.
Before we can calculate the internal forces we need to understand the sign convention which explains the direction of the internal forces.
Sign Convention
To know the forces & moments at a specific point in a beam, we do a cut at that point perpendicular to its axis. The beam is separated into two parts and the internal forces are visualized.
The sign convention is implemented to know which force is positive and which force is negative.
The normal forces, shear forces and moments on both sides are equal in magnitude but are applied in opposite directions!
For example, the Shear force V(x) at beam cut 1 is applied downwards⬇️, while the Shear force V(x) at beam cut 2 is applied upwards⬆️. The same goes for the Moments M(x) ↪️↩️ and Normal forces N(x) ➡️⬅️.
Note that there is a variety of different sign conventions due to different coordinate systems. I was taught to use the force direction like in the picture above, but I am curious to know if in any other country, you apply the forces in a different direction. Let us know in the comments below. ✏️✏️
Procedure of calculating internal forces
Calculate the reaction forces
Cut the structural member perpendicular to its axis & Visualize internal forces on both cuts
Calculate internal forces from 3 equilibrium equations ∑H, ∑V and ∑M
Now, let’s apply this theory in an example. 👇👇
Example: Simply supported beam with line load
Parameters:
q = 2 kN/m
l = 5 m
Step #1: Calculate reaction forces
3 equilibrium equations
Due to symmetry, Av=Bv and the vertical equilibrium equation can be solved to
The horizontal equilibrium shows that the horizontal force Ah=0.
In the moment equilibrium equation, it can be checked if the results obtained for Av and Bv are correct by inserting the values.
Step #2: Cut the structural member perpendicular to its axis & visualize internal forces on both cuts
Cut at 2m from left support
Now, based on the reaction forces the internal forces can be calculated in accordance to a variable x or at specific points as in the picture (2m from left support).
Step #3: Calculate internal forces from 3 equilibrium equations ∑H, ∑V and ∑M
We first calculate the internal forces (moment, shear and normal force) in dependence of the variable x. Later on we replace x with the value of 2 m and 3 m respectively.
3 equilibrium equations for left cut
Because we already calculated the values of the reaction forces, we find the internal forces as
Now we set x = 2m to get the internal forces at that specific point.
To double-check if the internal forces are calculated correctly, the same can be done for the right cut and the internal forces must equal those from the left cut.
3 equilibrium equations for right cut
Internal forces are found as
Now we set x = 3m to get the internal forces at that specific point.
The internal forces at the left and right cut are equal, and the internal forces of the simply supported beam are successfully calculated. ✅✅
Conclusion
Calculating internal forces is one of the first things you get introduced to at uni, and you need to calculate them throughout your professional career. To speed things up structural engineers use software such as Autodesk Robot or Polybeam.
You can also create your own beam analysis program in Excel or Python. James O’Reilly shows step-by-step in his Substack guide how to analyse a simply supported beam in Python. I can recommend his newsletter if you want to get into Python. 👍👍
This was the 5th edition of the engineering mechanics series.
Hope to see you next Wednesday for another post. 😎😎
Have a great rest of the week. ✌️✌️
Cheers,
Laurin.