Internal forces in beams – principles and calculations (2)

Welcome to the second part of the article on internal forces in beams. In this part, I will discuss the basic calculation principles and show different (but quite similar) approaches to calculating internal force values.

In this part of the article, I will cover the following topics:

• Basic calculation principles
• How engineers do it…
• How builders do it…

Basic principles – where do these signs come from?

As you already know from the previous part of the article (if you haven’t read it – I suggest you do), internal forces result from the equilibrium of forces in the material, acting due to applied loads. Since they result from equilibrium, guess what equations we can use to calculate them…

Yes, exactly, these will be the same equilibrium equations that were discussed in the topic related to statics principles. The only difference here is that we will analyze only one side of the beam—the division point is defined by the location of its imaginary cut. From this, three obvious but very important conclusions follow:

• Internal forces at a given point on the beam can be analyzed either from the left or from the right side of the imaginary cut.
• When analyzing from the left side, we remember all the loads (forces, moments, distributed  loads, support reactions, etc.) that are “before” the cut. When analyzing from the right side, we remember all the loads that are “after” the cut.
• Regardless of which side the internal forces are calculated from, they should have the same values at a given point.

These conclusions offer you significant opportunities to simplify your work in the field of internal force analysis.

Firstly (and this mainly applies to engineers—more on this in a moment), you can always choose the side of the analysis to minimize the number of elements in the equilibrium equation in the direction of a given internal force.

Secondly, if you’re unsure about the correctness of the result, you can calculate the values from the other side and verify. If you get two different results, it means that… something went wrong—in which case, you’re welcome to EquiBeam.

Moving on to the topic of signs—in the vast majority of cases, when analyzing internal forces in beams, we stick to the conventions shown in the diagram below. Why did I use the phrase “in the vast majority of cases”? Because, of course, you can find literature where these conventions are described differently, and there will be instructors who also present them differently. In that case, it’s better to stick to what the instructor says…

From the diagram above, we can automatically derive example equilibrium equations for the left side of the beam:

\sum_{} F_{ix}= ...+N(x)=0

\sum_{} F_{iy}= ...-T(x)=0

\sum_{} M_{iz}= ...+M_g(x)=0

\sum_{} M_{ix}= ...+M_s(x)=0

At this point, we can pause and ask ourselves whether, when writing internal force equations, we must always write full equilibrium equations and then additionally transform them?

The answer, of course, is NO. After all, many students will encounter situations where internal force equations are not written at all, and their values are not calculated in advance but their diagrams are drawn immediately. For this reason, instead of writing equations, we often use the rules described below regarding how the direction of a load affects the sign of the internal force. Below are the rules for determining positive signs—as you can easily guess, in other cases the sign will be negative.

normal forces

if the direction of the force or distributed  load is directed from the point of analysis to the considered side of the beam (stretching it), the normal force will be positive:

• when analyzing from the left side, these are forces acting to the left,
• when analyzing from the right side, these are forces acting to the right.

shear forces

the value of the shear force will be positive if:

• when analyzing from the left side, the forces and distributed  loads act upwards,
• when analyzing from the right side, the forces and distributed  loads act downwards.

bending moments

if a moment, force, or distributed  load bends the beam upwards in relation to the point of analysis, the value of the bending moment from this load will be positive:

• when analyzing from the left side, these are loads causing a clockwise moment in relation to the point of analysis,
• when analyzing from the right side, these are loads causing a counterclockwise moment in relation to the point of analysis.

torsional moments

if the moment direction is from the point of analysis towards the considered side of the beam, the torsional moment will be positive (similar to normal forces—you can use the right-hand rule to determine the direction of the torsional moment ☺):

• when analyzing from the left side, these are moments acting to the left,
• when analyzing from the right side, these are moments acting to the right.

You now know the methods for determining the sign of a given load when writing internal force equations (or when directly drawing a diagram). Fortunately, it’s not a lot to learn, and after completing a few exercises, it becomes second nature.

Now let’s talk about the two most common methods for drawing diagrams. Interestingly, the division between these methods is often linked to the type of study or, rather, the type of university department where mechanics or strength of materials classes are held—hence the names of the following sections.

How engineers do it… (and builders don’t bother…)

Let’s assume that the term “engineers” is a working nickname, but it has somehow become customary that in purely mechanical fields (e.g., Mechanics and Machine Construction, Mechatronics), to calculate internal force values, full equations are written.

Such an approach has many advantages but also some drawbacks… Let me list a few:

Advantages of writing full internal force equations:

• Describing internal forces using equations (derived from equilibrium equations, as you know) is closer to theory—it’s easier to explain where the signs come from (as I explained above) and where the behavior of the internal force diagrams comes from.
• Equations allow you to calculate values at any point on the beam fairly quickly (especially for systems with quadratic equations—represented as parabolas).
• Equations allow for quick transformations, enabling the calculation of the extremes of bending moments.
• They allow the use of analytical integration methods used in displacement calculations—analytical methods are insensitive to the types of integrated diagrams and do not require calculating internal force values for integration.

Disadvantages of writing full internal force equations:

• The main disadvantage of writing equations is the fact that you have to write them… It often takes a lot of time (especially for large systems).
• For large systems, the number of elements in the equation often increases significantly, meaning that later calculations of internal force values require a large number of mathematical operations (which is why calculations are done from both sides).

To summarize, internal force equations often add more work, but it’s easier to use them for displacement calculations in large systems with complex loads…

If we assume that we want to use equations to describe internal forces (or if the instructor has decided that for us), let’s introduce a few general rules that should also streamline the approach to calculations:

• The basis for writing the equation is referencing the position of the analyzed point on the beam, usually recorded as the variable x.
• Internal force equations are written separately for each section—a section is a part of the beam where nothing changes (no new loads appear, distributed  loads do not end, etc.). Hence, in a given section, the variable x spans a certain range of values.
• The equation includes all loads that acted on the beam before the analyzed section and at the beginning of the analyzed section (yes, I’ve already mentioned this).
• Concentrated moments do NOT affect the equations for normal and shear forces.
• Forces and distributed  loads DO affect the equations for bending moments.

If you follow the rules presented, you will certainly not make mistakes when writing the equation. Finally, I’ll add a comment about a common student frustration—writing the equation for shear force and bending moment for a distributed  load.

When analyzing the section with a distributed  load, remember one thing: interpret the load as if only the part from the beginning to the point of analysis is acting. Thus, as the value of x changes, the total force from the load changes, as does the total bending moment generated by it—see the graphic below.

As a result, in internal force equations, the distributed  load can be described as:

T(x)= q\cdot (x-l), 

M_g (x)=q\cdot (x-l)\cdot \dfrac {(x-l)}2 = q \dfrac {(x-l)^2}2,

x∈[l,2l]

And if the distributed  load starts at the beginning of the beam, the equation simplifies:

T(x)= qx, 

 M_g (x)=q\cdot x\cdot \dfrac x2 = q \dfrac {x^2}2,

x∈[0,l]

Moreover, when the distributed  load ends, it stops generating a variable force—it then becomes a concentrated force at the location of its center of mass. This also affects the internal force equations!!!

Knowing the internal force equations, you can immediately calculate the values at the ends of the sections and draw the appropriate diagrams. If the equation describes a linear function, it’s a simple task; if it’s a quadratic (or higher) function, you have two options—either calculate values at several points in the section (at least 3), or use a very important piece of information…

The shear force function is the derivative of the bending moment. If we know the derivative diagram, it’s easy to draw the function diagram—we know where it increases, where it decreases, and if it has an extremum. I recommend reviewing your math—but let me remind you—if the shear force passes through 0 in its section (and not just touches it), the bending moment has its extremum there.

That’s pretty much everything you need to know about writing internal force equations. For examples with commentary, visit EquiBeam, and now let’s deal with a different approach to drawing diagrams—let’s say a freehand method (if you’re an engineer, it’s worth reading too).

How builders do it… (and engineers could use it too…)

Very often, in civil engineering departments, the topic of equations is completely omitted, and students are required to draw internal force diagrams “freehand”—preferably, the diagram should magically appear under the beam as soon as they glance at it ☺. And, of course—if you know the basic rules and have a lot of experience (yes, unfortunately, this requires calculating several—dozens of beams), you can create such a diagram very quickly even for complex systems. Below, I’ll list the advantages and disadvantages of this approach:

Advantages of drawing internal force diagrams without writing equations:

• Drawing internal force diagrams without writing equations significantly shortens the time for strength calculations.
• For complex systems (with many sections and loads), this method significantly reduces the number of mathematical operations necessary to create the diagram (making it much harder to make a mistake).
• For simple systems (e.g., without distributed  loads), it allows for rapid displacement calculations using graphical integration methods.

Disadvantages of drawing internal force diagrams without writing equations:

• Initially, it’s often difficult to understand why the diagrams behave the way they do, since we don’t see the equations.
• Displacement calculations using graphical integration methods often require performing additional operations, breaking diagrams into simpler sums, or using tables with graphical integral formulas—for some cases, these diagram formulas are often not available.

And, similar to the example for engineers—here too we can point out several general rules that enable efficient drawing:

• When drawing the diagram, assume that you start at the value of 0 and must end at 0. This virtual assumption allows you to quickly check if the diagram is correct because it balances the forces and moments acting in the system.
• The appearance of a concentrated force at a given point causes a jump in the normal or shear force diagram (a sudden change in value at that point) by the value of the force. Of course, the force affects the diagram for the direction in which it acts.
• The appearance of a concentrated moment at a given point causes a jump in the bending or torsional moment diagram by the value of the moment. Of course, as above, the moment affects the diagram for the direction in which it acts.
• A distributed load causes a change in the shear force by an amount equal to the area under its diagram—the change in the shear force in a given section is equal to the value of the area under the distributed  load diagram in that section.
• The shear force is the derivative of the bending moment, so it directly indicates how the bending moment should change. Automatically, the bending moment is the integral of the shear force—therefore, the change in the bending moment in a given section is equal to the value of the area under the shear force diagram.

And that’s basically it when it comes to drawing diagrams. In conclusion, I’ll point out a few details… When I said that we virtually start the diagram at 0, I meant that if a concentrated force acts at the initial point, there will be a jump in the force diagram from 0 to the value of that force—similarly with a concentrated moment for the bending moment diagram. Likewise, when finishing the diagram, if we don’t end at 0, it means that an appropriate concentrated force (or concentrated moment) should appear at that point, which could bring the diagram back to 0 if the beam were longer.

In the case of drawing the bending moment diagram when a distributed  load acts—the diagram will always be a segment of a parabola (or a higher polynomial if the load is variable). The shape of the parabola is easiest to read from the shear force diagram! If the shear force does not pass through 0—there is no extremum—meaning no vertex of the parabola. If the shear force is positive, the parabola rises. If at the beginning of the section the shear force has a small positive value and at the end a large positive value, the parabola rises slowly at first but then its rise accelerates—see the graphic below ☺.

To summarize—if we know the forces acting on the beam (including the support reactions), we can draw each internal force diagram freehand in a relatively short time. For calculation examples with commentary, visit EquiBeam, and below you’ll find a video where I discussed one of these examples.

If you’ve made it this far—thank you for reading the article. I am convinced it will clarify the basics of internal force calculations and diagram drawing for you.