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Engineering LibreTexts

5.5: Method of Sections

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  • Page ID 52781

  • Jacob Moore & Contributors
  • Pennsylvania State University Mont Alto via Mechanics Map

The method of sections is a process used to solve for the unknown forces acting on members of a truss . The method involves breaking the truss down into individual sections and analyzing each section as a separate rigid body. The method of sections is usually the fastest and easiest way to determine the unknown forces acting in a specific member of the truss.

Using the Method of Sections:

The process used in the method of sections is outlined below.

A truss bridge with a 30-meter span consisting of members A, D and H (left to right), each 10 meters long. The span connects to the ground with a pin joint at A's left end and a roller joint at H's right end. The endpoints of member D are attached to the top ends of two vertical members: C (left) and G (right), with the lower ends of C and G connected by a horizontal member F. Diagonal members B and I, at 20° below the horizontal, connect the left end of F to the pin joint, and the right end of F to the roller joint, respectively. Downwards forces of 60 kN and 80 kN are applied at the top ends of C and G, respectively.

Figure \(\PageIndex{1}\): The first step in the method of sections is to label each member.

Free body diagram of the truss bridge from Fig. 1 above: in addition to the downwards applied forces, the upwards reaction forces of 66.7 kN at the left end of member A and 73.3 kN at the right end of member H are shown.

Figure \(\PageIndex{2}\): Treat the entire truss as a rigid body and solve for the reaction forces supporting the truss structure.

The free body diagram from Fig. 2 above is depicted with a vertical dotted line dividing the bridge down the middle.

  • Any external reaction or load forces that may be acting at the section.
  • An internal force in each member that was cut when splitting the truss into sections. Remember that for a two-force member, the force will be acting along the line between the two connection points on the member. We will also need to guess if it will be a tensile or a compressive force. An incorrect guess now, though, will simply lead to a negative solution later on. A common strategy then is to assume all forces are tensile; then later in the solution any positive forces will be tensile forces and any negative forces will be compressive forces.

Free body diagram of the left side of the bridge as divided by the dotted line in Fig. 3 above. In addition to the applied and reaction forces present on A, tension forces exerted by the halves of members D, E and F that were removed from the diagram by the cut on the halves present in the diagram are included.

Figure \(\PageIndex{4}\): Next, draw a free body diagram of one or both halves of the truss. Add the known forces, as well as unknown tensile forces for each member that you cut.

  • For 2D problems you will have three possible equations for each section: two force equations and one moment equation. \[ \sum \vec{F} = 0 \quad\quad\quad\quad \sum \vec{M} = 0 \] \[ \sum F_x = 0 \, ; \,\,\, \sum F_y = 0 \, ; \,\,\, \sum M_z = 0 \]
  • For 3D problems you will have six possible equations for each section: three force equations and three moment equations. \[ \sum \vec{F} = 0 \] \[ \sum F_x = 0 \, ; \,\,\, \sum F_y = 0 \, ; \,\,\, \sum F_z = 0 \] \[ \sum \vec{M} = 0 \] \[ \sum M_x = 0 \, ; \,\,\, \sum M_y = 0 \, ; \,\,\, \sum M_z = 0 \]
  • Finally, solve the equilibrium equations for the unknowns. You can do this algebraically, solving for one variable at a time, or you can use matrix equations to solve for everything at once. If you assumed that all forces were tensile earlier, remember that negative answers indicate compressive forces in the members.

Example \(\PageIndex{1}\)

Find the forces acting on members BD and CE. Be sure to indicate if the forces are tensile or compressive.

A truss bridge with a 30-meter span; the leftmost end, point A, connects to the ground with a pin joint and the rightmost end, F, connects to the ground with roller joint. The span is formed from 3 10-meter horizontal members: AB, BD, and DF. Two vertical members, BC and DE, attach to the endpoints of that span's central member and extend below it. Another horizontal member connects points C and E. The diagonal members AC and FE, each 20° below the horizontal, connect the ends of that lower horizontal member to the endpoints of the bridge span. A downwards force of 60 kN is applied at point B, and a downwards force of 80 kN is applied at point D.

Example \(\PageIndex{2}\)

Find the forces acting on members AC, BC, and BD of the truss. Be sure to indicate if the forces are tensile or compressive.

A tower composed of trusses arranged in a long rectangle composed of 4 identical smaller rectangles that are each 10 m in length and 6 m in height. The lowest points of the tower, A on the left and B on the right, are attached to the ground with a pin joint and a roller joint respectively. Points C and D are attached to A and B respectively by vertical members, and B and C are linked by a diagonal member. At each side of the lower horizontal member of the topmost rectangular subunit, a 10-meter member protrudes horizontally and a diagonal member links the free end of the protrusion to the corresponding upper corner of the topmost rectangle. The endpoints of the protruding members experience a force downwards and to the right, at 15° from the vertical. The magnitude of the left member's force is 40 kN and the magnitude of the right member's is 50 kN.

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Chapter 5: Trusses

5.3 Method of Sections

The method of sections uses rigid body analysis to solve for a specific member or two. Instead of looking at each joint, you make a cut through the truss, turning the members along that line into internal forces (assume in tension). Then you solve the rigid body using the equilibrium equations for a rigid body: [latex]\sum F_x=0\;\sum F_y=0\;\sum M_z=0[/latex]

is split into two to solve for F E .

For this example, you could choose the right half or left half. For some problems, being strategic is necessary otherwise you need to make multiple cuts. In this problem you had to solve for the reaction forces first, but that isn’t always the case as you can sometimes just make the cut (see example 2 below).

Here are more examples of how to make a cut and showing the naming convention:

Source: Internal Forces in Beams and Frames, Libretexts. https://eng.libretexts.org/Bookshelves/Civil_Engineering/Book%3A_Structural_Analysis_(Udoeyo)/01%3A_Chapters/1.05%3A_Internal_Forces_in_Plane_Trusses

Here is a detailed explanation:

The  method of sections  is a process used to solve for the unknown forces acting on members of a  truss . The method involves breaking the truss down into individual sections and analyzing each section as a separate rigid body. The method of sections is usually the fastest and easiest way to determine the unknown forces acting in a specific member of the truss.

Using This Method:

The process used in the method of sections is outlined below:

  • Any external reaction or load forces that may be acting at the section.
  • An internal force in each member that was cut when splitting the truss into sections. Remember that for a two force member, the force will be acting along the line between the two connection points on the member. We will also need to guess if it will be a tensile or a compressive force. An incorrect guess now though will simply lead to a negative solution later on. A common strategy then is to assume all forces are tensile, then later in the solution any positive forces will be tensile forces and any negative forces will be compressive forces.
  • You will have three possible equations for each section, two force equations and one moment equation.$$\sum\vec F=0\; \; \sum\vec M=0\\\sum F_x=0\; \; \sum F_y=0\; \; \sum M_z=0$$
  • Finally, solve the equilibrium equations for the unknowns. You can do this algebraically, solving for one variable at a time, or you can use matrix equations to solve for everything at once. If you assumed that all forces were tensile earlier, remember that negative answers indicate compressive forces in the members.

Source:Engineering Mechanics, Jacob Moore, et al. http://mechanicsmap.psu.edu/websites/5_structures/5-5_method_of_sections/methodofsections.html

Additional examples from the Engineering Mechanics webpage :

Find the forces acting on members BD and CE. Be sure to indicate if the forces are tensile or compressive.

Source: Engineering Mechanics, Jacob Moore, et al. http://mechanicsmap.psu.edu/websites/5_structures/5-5_method_of_sections/pdf/MethodOfSections_WorkedExample1.pdf

Find the forces acting on members AC, BC, and BD of the truss. Be sure to indicate if the forces are tensile or compressive.

If we make a cut in the top section, we don’t need to solve for the reaction forces.

Source: Engineering Mechanics, Jacob Moore, et al.  http://mechanicsmap.psu.edu/websites/5_structures/5-5_method_of_sections/pdf/MethodOfSections_WorkedExample2.pdf

Even more examples are available at: https://eng.libretexts.org/Bookshelves/Civil_Engineering/Book%3A_Structural_Analysis_(Udoeyo)/01%3A_Chapters/1.05%3A_Internal_Forces_in_Plane_Trusses

In summary:

Key Takeaways

Basically : Method of sections is an analysis technique to find the forces in some members of a truss. It separates the truss into two sections then uses the rigid body equilibrium equations.

Application : To calculate the loads on bridges and roofs, especially if you need to know only one or two of the members.

Looking Ahead : The next section explores a trick that makes solving faster, especially for method of joints.

Engineering Mechanics: Statics Copyright © by Libby (Elizabeth) Osgood; Gayla Cameron; Emma Christensen; Analiya Benny; and Matthew Hutchison is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License , except where otherwise noted.

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Method of Sections

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The Method of Sections involves analytically cutting the truss into sections and solving for static equilibrium for each section.

Related Papers

International Journal of Computational Methods

Janusz Rębielak

The paper presents features of the two-stage method as an approximate method of calculations of statically indeterminate trusses. Values of forces determined in particular members are resultants of forces calculated in both stages for two intermediate statically determinate trusses. Each of the intermediate truss is defined by removing, from the statically indeterminate truss, a number of members, which equals the statically indeterminacy of the basic truss. In the paper are presented comparisons between values of forces calculated in members of selected type of statically indeterminate plane truss under load by forces applied in symmetrical way and in asymmetrical way.

methods of sections solved problems pdf

Journal of Civil Engineering and Architecture

Journal of Civil Engineering and Architecture DPC

The paper presents results of calculations of forces in members of selected types of statically indeterminate trusses carried out by application of the two-stage method of computations of such structural systems. The method makes possible to do the simple and approximate calculations of the complex trusses in two stages, in each of which is calculated a statically determinate truss being an appropriate counterpart of the basic form of the statically indeterminate truss structure. Systems of the statically determinate trusses considered in the both stages are defined by cancelation of members, number of which is equal to the statically indeterminacy of the basic truss. In the paper are presented outcomes obtained in the two-stage method applied for two different shapes of trusses and carried out for various ways of removing of appropriate members from the basic trusses. The results are compared with outcomes gained due to application of suitable computer software for computation of the same types of trusses and for the same structural conditions.

Sunny Marakana

The paper presents examples of approximate calculations of force values in members of selected types of trusses, which are at the same time an internal and external statically indeterminate systems. The two-stage method makes possible the approximate calculation of such trusses by help of, for example, the Cremona’s method. In each stage, a statically determinate truss is considered, pattern of which is defined by removing from the basic truss a suitable number of members. There are also presented results of calculations of the same trusses done by means of suitable computer software together with analyses and comparison of outcomes.

Matias Aleman

Dimas Satrio

Kristian Barrio

The paper presents a very simple method, which in two stages enables to calculate the plane statically indeterminate truss by the application of one of methods used for the force calculation in the statically determinate trusses [1]. The results are obtained in a very simple and quick way. Although the force values are approximated but they are relatively very close to those, which are determined by the exact methods. The point of the two-staged calculation process of the statically indeterminate trusses is to determine schemes of two independent and simple statically determined trusses, which after superposition of their patterns will give in the result a pattern of the initial, more complex form of the statically indeterminate truss. Each of the simple truss has to be of the same clear span, the load forces have to be of the half values and they have to be applied to the same nodes like in truss of the initial structural configuration. For example in the first stage, see Fig. 1, f...

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Tutorial: How to Solve a Truss Structure using Method of Sections

In this tutorial, we will explore and learn the benefits of using the Method of Sections to solve your truss structure. What are trusses? If you’re unsure about this, visit our What is a truss article. The method of sections is used to solve larger truss structures in a fast, simple manner. It involves taking a ‘cut’ through a number of members to evaluate their axial forces and use this as our basis to solve the rest of the truss structure.

The great thing is, SkyCiv Truss does this automatically for you. Model your own trusses and the software will show interactive step-by-step working out of the method of sections!

Watch the Video Tutorial

Sample Question

For our worked example, we’ll be looking at the following question:

Question: Using the method of sections, determine the forces in members 10, 11, and 13 of the following truss structure:

SkyCiv Truss Tutorial, method of sections

Step 1: Calculate the Reactions to the Supports

Like most static structural analyses, we must first start by locating and solving the reactions at supports . This will give us the boundary conditions we need to progress in solving the truss structure. Simplifying the structure to just include the loads and supports:

Analysis and Calculation using Method of Sections

Without spending too much time calculating the reactions, you generally start by taking the sum of moments about a point. Taking the sum of moments about the left support gets us:

Tutorial to Solve Truss by Method of Sections - 1

So the reaction at the right support (R B ) is 17.5 kN in an upward direction. Now, taking the sum of forces in the y gives us the reaction R A as 7.5kN in an upward direction:

Tutorial to Solve Truss by Method of Sections - 2

Step 2: Make a cut along the members of interest

Here comes the most important part of solving a truss using the method of sections. It involves making a slice through the members you wish to solve. This method of structural analysis is extremely useful when trying to solve some of the members without having to solve the entire structure using the method of joints. So, in our example here would be our slice:

SkyCiv Truss Tutorial, method of sections

Focussing on the left side only, you are left with the following structure:

SkyCiv Truss Tutorial, method of sections

Now think of this structure as a single-standing structure. The laws of statics still apply – so the sum of moments and forces must all equal zero. The members with arrows (F 13 , F 10 , F 11 ) are what stabilize the reaction and forces applied to the structure. Note that the sum of moments is taken about node 7 – as would exclude the forces of members 13 and 10 – leaving F 11 to be isolated.

Using the above Free Body Diagram, we can obtain the following formulae:

Sum of forces in the y-direction:

[math] \begin{align} +\uparrow \text{   } \sum{F_y} &= 0\\ 7.5\text{ kN} – 10 \text{ kN} – F_{10}sin(45^{\circ}) &= 0\\ F_{10} &= -3.536 \text{ kN} \end{align} [math]

Sum of moments about node 7:

[math] \begin{align} +\circlearrowleft \text{   } \sum{M_7} &= 0\\ -(15 \text{ m})(7.5 \text{ kN}) + (5 \text{ m})F_{11} &= 0\\ F_{11} &= 22.5 \text{ kN} \end{align} [math]

Sum of forces in the x-direction:

[math] \begin{align} +\rightarrow \text{   } \sum{F_x} &= 0\\ F_{13} + F_{11} + F_{10}cos(45^{\circ}) &= 0\\ F_{13} &= -F_{11} – F_{10}cos(45^{\circ}) \\ F_{13} &= – (22.5 \text{ kN}) – (-3.536 \text{ kN})cos(45^{\circ}) \\ F_{13} &= -22.5 \text{ kN} + (3.536 \text{ kN})cos(45^{\circ}) \\ F_{13} &= -20 \text{ kN} \end{align} [math]

Final Solution

We can use these results to solve the remaining members in the truss structure. We hope this truss calculation example has been useful and feel free to comment with your questions below. As a reference, the results for the entire Truss structure can be found below (using our Truss Calculator ) which is great for checking your answers!

Method of Sections

Summary of Steps

  • Always Start by calculating reactions at supports
  • Make a slice through the members you wish to solve
  • Treat the half structure as its own static truss
  • Solve the truss by taking the sum of forces = 0
  • Take the moment about a node of more than one unknown member

SkyCiv Truss Software

We hope that you found this tutorial useful for your projects. Visit our truss tutorials for more useful information about truss and don’t forget to check out our guide to solving truss by Method of Joints .

SkyCiv Truss can calculate the method of sections automatically for you. Or try our Free Truss Calculator which will give you the final answer (no hand calculations).

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The Method of Sections

The method of sections is a process used to solve for the unknown forces acting on members of a truss . The method involves cutting the truss into individual sections and analyzing each section as a separate rigid body. Where the method of joints is the fastest way to find the forces in all members, the method of sections is usually the fastest and easiest way to determine the unknown forces acting in a specific member of the truss.

Using the Method of Sections:

The method of sections and the method of joints both have the same initial steps, but will differ from one another once the external forces are found. The process used in the method of sections is outlined below.

To start, if the joints are not already labeled, we will begin by labeling all the joints with a letter. The exact order you label the joints in are not important, so long as you are consistent in your work. We will refer to joints by their letter (A, B, C...) and we will refer to the members by the two joints they connect (AB, AC, BC...). As a note, this text will always arrange the member names alphabetically, in order to avoid confusion (member AB, rather than member BA)

The second step is to solve for the external reaction forces supporting the truss. We can sometimes skip this step if the reaction forces are not necessary to solve for the internal forces, but when in doubt it is better to solve for the external reaction forces up front. To do this, treat the whole truss as a single rigid body. Draw a free body diagram of the truss, including the load forces and the external reaction forces supporting the truss, write out your equilibrium equations (sum of forces in the x, sum of forces in the y, and sum of moments), and finally solve the equilibrium equations for the unknown reaction forces.

Next, to get to the heart of the method of sections, you will imagine cutting your truss into two separate sections. This cut should only travel through members (don't cut at a joint) and should cut through the member (or members) that you are trying to solve for. We should also try and minimize the number of members we cut though, limiting ourselves to a maximum of three members for a two dimensional problem.

After determining the cut you want to use, you will next draw a free body diagram of one half of the truss . This can be the half on either side of the cut you define, though choosing the side with fewer external reaction forces acting on it will generally make analysis easier. When drawing the free body diagram of this section be sure to include and label all known and unknown forces the forces acting on that section.

The known forces will be all load forces and external reaction forces acting on the half of the truss we are analyzing. Load or reaction forces acting on the other half of the truss should not be included on the free body diagram.

The unknown forces in the free body diagram should be the forces carried by the members that we cut through. As the members are all two force members, we will draw in one single tensile force for each member we cut through. This force will act along the line of the member itself. Just as with the method of joints, we do not know if the members are in tension or compression. However, if we assume tension then positive answers indicate the member is actually in tension while negative numbers indicate the member is in compression.

In the free body diagram, make sure you label all forces, and indicate important angles and dimensions.

Next you will want to write out the equilibrium equations for the section you just drew. These will be rigid bodies, so you will need to write out the force and the moment equations.

For 2D problems you will have three possible equations for each section, two force equations and one moment equation.

For 3D problems you will have six possible equations for each section, three force equations and three moment equations.

Finally, solve the equilibrium equations for the unknowns. You can do this algebraically, solving for one variable at a time, or you can use an equation solver to solve for everything at once. If you assumed that all forces were tensile earlier, remember that negative answers indicate compressive forces in the members.

Extending the Method of Sections:

In some cases, you will need to determine the forces acting in a few members of a truss, but you will find that there is no one cut you can make with the method of sections that goes through all the members you need to solve for. In cases such as this, the method of sections may not be enough, but solving for everything with the method of joints may be overkill. In these cases, we may resort to one of the two following strategies to extend the method of sections beyond it's base process.

The first strategy we can use to extend the method of sections is to simply make a second cut and employ the method of sections again. In this strategy, start by using the method of sections normally to solve for some of the forces we are looking for. After making this first cut, we will repeat the process with a second cut, making sure to cut through the rest of the members that we wish to solve for in this second cut. Additionally, this second cut can travel through more than three members, so long as it does not travel through three unsolved members. Anything we solved for in the first round can be treated as a known value, and does not need to be solved for again.

The second strategy to extend the method of sections, is to start with the method of sections, then to use the method of joints to work out from the initial cut. In this strategy, start by using the method of sections normally to solve for some of the forces we are looking for. After making the initial cut, we will switch to the method of joints, by picking a joint near the cut for analysis. For each joint we choose, just make sure there are no more than two unknowns to solve for. Continue analyzing joints until you have all the unknown forces you are looking for.

Video Lecture

Worked Problems:

Question 1:.

Use the method of sections to find the forces acting on members BD and CE. Be sure to indicate if the forces are tensile or compressive.

Question 2:

Use the method of sections to find the forces acting on members AC, BC, and BD of the truss. Be sure to indicate if the forces are tensile or compressive.

Question 3:

Use the method of sections to find the forces in members AB and DE. Be sure to indicate if the forces are tensile or compressive.

Question 4:

Use the method of sections to find the forces in members AC, BC, CD, and CE. Be sure to indicate if the forces are tensile or compressive.

Practice Problems:

Practice problem 1:, practice problem 2:, practice problem 3:.

methods of sections solved problems pdf

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Method of sections to solve a truss.

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methods of sections solved problems pdf

Contributed by: Rachael L. Baumann   (September 2017) Additional contributions by: John L. Falconer (University of Colorado Boulder, Department of Chemical and Biological Engineering) Open content licensed under CC BY-NC-SA

methods of sections solved problems pdf

The method of sections is used to calculate the forces in each member of the truss. This is done by making a "cut" along three selected members. First, calculate the reactions at the supports. Taking the sum of the moments at the left support:

methods of sections solved problems pdf

Begin solving for the forces of the members by making cuts. The order of the balances listed here is the order in which they should be solved. Force balances are done assuming we can figure out which members are under tension and which are under compression. A labeled truss is shown in Figure 1.

methods of sections solved problems pdf

Note that all the vertical members are zero members, which means they exert a force of 0 kN and are neither a tension nor a compression force; instead they are at rest.

methods of sections solved problems pdf

[1] SkyCiv Cloud Engineering Software. "Tutorial to Solve Truss by Method of Sections." (Aug 18, 2017) skyciv.com/tutorials/tutorial-to-solve-truss-by-method-of-sections .

Related Links

  • Cremona Diagram for Truss Analysis
  • Analysis of Forces on a Truss

Permanent Citation

Rachael L. Baumann "Method of Sections to Solve a Truss" http://demonstrations.wolfram.com/MethodOfSectionsToSolveATruss/ Wolfram Demonstrations Project Published: September 8 2017

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methods of sections solved problems pdf

Related Topics

  • Civil Engineering
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Engineering Statics: Open and Interactive

Daniel W. Baker, William Haynes

Section 6.5 Method of Sections

Key questions.

  • How do we determine an appropriate section to cut through a truss?
  • How are equilibrium equations applied to a section?

Subsection 6.5.1 Procedure

  • Determine if a truss can be modeled as a simple truss .
  • Identify and eliminate all zero-force members . Removing zero-force members is not required but may eliminate unnecessary computations.
  • Solve for the external reactions, if necessary. Reactions will be necessary if the reaction forces act on the section of the truss you choose to solve below.
  • Use your imaginary chain saw to cut the truss into two pieces by cutting through some or all of the members you are interested in. The cut does not need to be a straight line. Every cut member exposes an unknown internal force, so if you cut three members you’ll expose three unknowns. Exposing more than three members is not advised because you create more unknowns than available equilibrium equations.
  • Include all applied and reaction forces acting on the section, and show known forces acting in their known directions.
  • Draw unknown forces in assumed directions and label them. A common practice is to assume that all unknown forces are in tension and label them based on the endpoints of the member they represent.
  • Write out and solve the equilibrium equations for your chosen section. If you assumed that unknown forces were tensile, negative answers indicate compression.
  • If you have not found all the required forces with one section cut, repeat the process using another imaginary cut or proceed with the method of joints if it is more convenient.

Instructions .

methods of sections solved problems pdf

Chapter 1: An Introduction to Statics

Chapter 2: force vectors, chapter 3: equilibrium of a particle, chapter 4: force system resultants, chapter 5: equilibrium of a rigid body, chapter 6: structural analysis, chapter 7: internal forces, chapter 8: friction, chapter 9: center of gravity and centroid, chapter 10: moment of inertia, chapter 11: virtual work.

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methods of sections solved problems pdf

Consider a symmetrical roof truss structure comprising the vertical, diagonal and horizontal members.

A free-body diagram is drawn to analyze the forces on DC and HC members using the method of section.

Here, the loads and the lengths of the horizontal and vertical members are the known parameters.

First, summing the moments about point A , the reaction force at E is calculated.

Further, using the force equilibrium condition for the truss, the reaction force at A is determined.

The symmetry of the truss ensures that both the reaction forces are equal.

Now, a cut is made along a plane intersecting DC, HC and HG members, and a free-body diagram of the smaller section is drawn.

Taking the summation of the moments about H gives the force along DC . The positive sign indicates the tensile force.

The force along HC is resolved into its sine and cosine components, and trigonometry is used to estimate the angle theta.

The moment equilibrium condition at E yields the force on HC , with the negative sign indicating the compressive force.

6.8: Method of Sections: Problem Solving I

Consider a symmetrical roof truss structure, composed of vertical, diagonal, and horizontal members. The length of each horizontal member is 4 m. The lengths of the vertical members FB and HD are 4 m, while the length of member GC is 6 m. The loads acting at joints F , G , and H are 2 kN, while those at joints A and E are 1 kN.

The method of sections is employed to calculate the forces acting on members DC and HC . The moment equilibrium condition is applied to point A , and the known values of forces and distances are substituted into the moment equation.

This results in an estimated reaction force of 4 kN at point E . Subsequently, the vertical force equilibrium condition at point A reveals that the reaction force at A is also 4 kN.

Due to the symmetry of the truss, the reaction forces at points A and E are equal.

The forces acting on members DC and HC can be obtained upon determining the reaction forces. A sectional cut is made along a plane intersecting members DC , HC , and HG , and a free-body diagram of the smaller section is considered.

By summing the moments about point H , the force along DC is calculated to be a positive 3 kN, indicating a tensile force.

The force along CH is resolved into its sine and cosine components. Trigonometry is used to find the angle between member CH and the horizontal axis to be 45°. Finally, applying the moment equilibrium condition at point E yields a force of -1.41 kN for F CH .

In this case, the negative result signifies a compressive force acting on member CH .

  • R. C., Hibbeler  Engineering Mechanics Statics, Pearson. Pp. 291-293
  • F.P. Beer, E.R. Johnston, D.F. Mazurek, P.J. Cornwell, B.P. Self, Vector Mechanics For Engineers Statics and Dynamics Engineering Mechanics Statics, Mc Graw-Hill Education. Pp. 217

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  • Problem 004-ms | Method of Sections

Problem 004-ms For the truss shown in Fig. T-05, find the internal fore in member BE.  

Solution 003-ms

$\Sigma M_F = 0$

$6R_A = 2(120)$

$R_A = 40 \, \text{ kN}$  

From section to the left of a-a  

$\Sigma M_A = 0$

$F_{BE} = 0$           answer

  • Add new comment
  • 113287 reads

why section left of a-a? not

why section left of a-a? not right?

more forces to consider,

more forces to consider, harder to solve

can any one explain please

can any one explain please why the Fbe is zero

More Reviewers

Engineering mechanics.

  • Principles of Statics
  • Equilibrium of Force System
  • Method of Joints | Analysis of Simple Trusses
  • Problem 001-ms | Method of Sections
  • Problem 002-ms | Method of Sections
  • Problem 003-ms | Method of Sections
  • Problem 005-ms | Method of Sections
  • Problem 417 - Roof Truss by Method of Sections
  • Problem 418 - Warren Truss by Method of Sections
  • Problem 419 - Warren Truss by Method of Sections
  • Problem 420 - Howe Truss by Method of Sections
  • Problem 421 - Cantilever Truss by Method of Sections
  • Problem 422 - Right-triangular Truss by Method of Sections
  • Problem 423 - Howe Roof Truss by Method of Sections
  • Problem 424 - Method of Joints Checked by Method of Sections
  • Problem 425 - Fink Truss by Method of Sections
  • Problem 426 - Fink Truss by Method of Sections
  • Problem 427 - Interior Members of Nacelle Truss by Method of Sections
  • Problem 428 - Howe Truss by Method of Sections
  • Problem 429 - Cantilever Truss by Method of Sections
  • Problem 430 - Parker Truss by Method of Sections
  • Problem 431 - Members in the Third Panel of a Parker Truss
  • Problem 432 - Force in Members of a Truss by Method of Sections
  • Problem 433 - Scissors Truss by Method of Sections
  • Problem 434 - Scissors Truss by Method of Sections
  • Problem 435 - Transmission Tower by Method of Sections
  • Problem 436 - Howe Truss With Counter Braces
  • Problem 437 - Truss With Counter Diagonals
  • Problem 438 - Truss With Redundant Members
  • Method of Members | Frames Containing Three-Force Members
  • Centroids and Centers of Gravity
  • Moment of Inertia and Radius of Gyration
  • Force Systems in Space

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methods of sections solved problems pdf

Resources for Structural Engineers and Engineering Students

The cantilever method is very similar to the portal method. We still put hinges at the middles of the beams and columns. The only difference is that for the cantilever method, instead of finding the shears in the columns first using an assumption, we will find the axial force in the columns using an assumption.

The assumption that is used to find the column axial force is that the entire frame will deform laterally like a single vertical cantilever. This concept is shown in Figure 7.8 . When a cantilever deforms laterally, it has a strain profile through its thickness where one face of the cantilever is in tension and the opposite face is in compression, as shown in the top right of the figure. Since we can generally assume that plain sections remain plane (see Chapter 5 ), the strain profile is linear as shown. The relative values of the tension and compression strain are dependant on the location of the neutral axis for bending, which is in turn dependant on the shape of the cantilever's cross-section.

methods of sections solved problems pdf

The cantilever method assumed that the whole frame will deform laterally in the same way as the vertical cantilever. The location of the neutral axis of the whole frame is found by considering the cross-sectional areas and locations of the columns at each storey:

\begin{equation} \boxed { \bar{x}= \frac{\sum_i (A_{i} x_{i})}{\sum_i A_{i}} } \label{eq:frame-neutral-axis} \tag{1}\end{equation}

where $\bar{x}$ is the horizontal distance between the location of the neutral axis and the zero point, $A_{i}$ is the area of column $i$, and $x_i$ is the horizontal distance between column $i$ and the zero point. The location zero does not matter, but is commonly set as the location of the leftmost column.

Once we know the location of the neutral axis, using the assumption that the frame behaves as a vertical cantilever, we know that the axial strain in each column will be proportional to that column's distance from the neutral axis, just like the strain in any fibre a distance $x$ away from the neutral axis of a cantilever is proportional to the distance $x$. Since we are assuming that all of our materials are linear (stress is linear to strain), then this also means that the axial stress in each column is proportional to it's distance from the neutral axis of the frame. Also, columns on one side of the neutral axis will be in tension, and columns on the other side of the neutral axis will be in compression. The linear axial stress profile for a sample structure is shown at the bottom of Figure 7.8 . If we assume an unknown value for the stress in the left column ($\sigma_1$ in the figure) then the cantilever method can be used to find the stress in the other two columns as a function of their relative distance from the neutral axis as shown in the figure. From these relative stresses, we can determine the force in each column as a function of stress $\sigma_1$. Then, using a global moment equilibrium, we can solve for $\sigma_1$, and therefore for the axial force in each column. From this point, the structure is again broken into separate free body diagrams between the hinges as was done for the portal method and all of the remaining unknown forces at the hinges are found using equilibrium.

Since this method relies on the frame behaving like a bending cantilevered beam, it should generally be more accurate for more slender or taller structures, whereas the portal method may be more accurate for shear critical frames, such as squat or short structures.

The details of the cantilever method process will be illustrated using the same example structure that was used for the portal method (previously shown in Figure 7.4 ).

The most important part of the cantilever method analysis is to find the axial forces in the columns at each storey. We will start with the top story as shown at the top of Figure 7.9 .

methods of sections solved problems pdf

First, we must find the location of the neutral axis for the frame when cut at the top story using equation \eqref{eq:frame-neutral-axis} (the column cross-sectional areas are the same for both storeys and are shown in Figure 7.4 ):

\begin{align*} \bar{x} &= \frac{\sum_i (A_{i} x_{i})}{\sum_i x_{i}} \\ \bar{x} &= \frac{{10\,000}(0) + {20\,000}(5) + {15\,000}(10)}{ {10\,000} + {20\,000} + {15\,000}} \\ \bar{x} &= 5.555\mathrm{\,m} \end{align*}

where the location of the left column is selected as the zero point.

Knowing the neutral axis location (as shown in the top diagram of Figure 7.9 ), we can determine the axial stress in all of the columns in the top storey. We will do this in terms of the stress in the left column, which we will call $\sigma_2$ as shown. The stress in the middle column will be equal to $\sigma_2$ multiplied by the ratio of the distance from the second column to the neutral axis to the distance from the first column to the neutral axis:

\begin{align*} \left( \frac{0.56}{5.56} \right) \sigma_2 = 0.1\sigma_2 \end{align*}

Likewise, the stress in the right column will be:

\begin{align*} \left( \frac{4.44}{5.56} \right) \sigma_2 = 0.8\sigma_2 \end{align*}

From these stresses, we can determine the force in the columns by multiplying the stress in each column by it's cross-sectional area as shown in the top diagram of Figure 7.9 . Also, the left and middle columns are on the tension side of the neutral axis, so the column axial force arrows will point down as shown (pulling on the column) and the right column is on the compression side of the neutral axis, so the column axial force arrow for that column will point up as shown.

Now, we can use a moment equilibrium on the top story free body diagram in Figure 7.9 to solve for the unknown stress. We will use the moment around point f:

\begin{align*} \curvearrowleft \sum M_f &= 0 \\ -100\mathrm{\,kN} ( 2\mathrm{\,m} ) - A_{col2} (0.1 \sigma_2) (5\mathrm{\,m}) + A_{col3} (0.8 \sigma_2) (10\mathrm{\,m}) &= 0 \\ -100\mathrm{\,kN} ( 2\mathrm{\,m} ) - (0.02\mathrm{\,m^2}) (0.1 \sigma_2) (5\mathrm{\,m}) + (0.015\mathrm{\,m^2}) (0.8 \sigma_2) (10\mathrm{\,m}) &= 0 \\ \sigma_2 = 1818.2\mathrm{\,kN/m^2}& \end{align*}

This resulting stress in the left column may be subbed back into the equations for the force in each column shown in the figure to get forces of $18.2\mathrm{\,kN}\downarrow$ in the left column, $3.6\mathrm{\,kN}\downarrow$ in the middle column, and $21.8\mathrm{\,kN}\uparrow$ in the right column.

For the lower story, the column areas are the same, so the neutral axis will be located in the same place as shown in the lower diagram in Figure 7.9 . This means that the relative stresses will also be the same. To solve for the stresses in the left column again for the lower storey ($\sigma_1$), we need to take a free body diagram of the entire structure above the hinge in the middle of the lower column (as shown in the figure). We should cut the lower storey at the hinge location because that way we do not have any moments at the cut (since the hinge is, by definition, a location with zero moment). If we chose to cut the structure at the base of the columns instead, we would have additional point moment reaction at the base of each column which would have to be considered in the moment equilibrium (which are unknown). Such moment reactions at the base of the columns are shown in Figure 7.8 . These extra moments would make it impossible to solve the equilibrium equation for $\sigma_1$. So, taking the cut at the lower hinges as shown in the lower diagram in Figure 7.9 , we can solve for $\sigma_1$ using a global moment equilibrium about point a:

\begin{align*} \curvearrowleft \sum M_a &= 0 \\ -100(6) - 50(2) - (0.02)(0.1 \sigma_1)(5) + (0.015)(0.8 \sigma_1)(10) &= 0 \\ \sigma_1 = 6363.6\mathrm{\,kN/m^2}& \end{align*}

This resulting stress in the left column may be subbed back into the equations for the force in each column shown in the figure to get forces of $63.6\mathrm{\,kN}\downarrow$ in the left column, $12.7\mathrm{\,kN}\downarrow$ in the middle column, and $76.4\mathrm{\,kN}\uparrow$ in the right column.

From this point forward, the solution method is the same as it was for the portal method. Split each storey free body diagram into separate free body diagrams with cuts at the hinge locations, and then work methodically through using equilibrium to find all of the unknown forces at the hinge cuts. This process is illustrated in Figure 7.10 .

methods of sections solved problems pdf

Like the portal frame example, the free body diagrams in Figure 7.10 are annotated with numbers in grey circles to show a suggested order for solving all of the unknown forces. Of course, as before, step 0 and step 1 consist of known values, either caused by external forces or the previous storey (for step 0) or the column axial forces that were solved using the cantilever method assumptions (for step 1). The rest of the unknowns are solved for using vertical, horizontal or moment equilibrium.

Once all of the unknown forces at the hinges are found, the shear and moment diagrams for the frame may be drawn using the same methods that were used for the previously described portal method analysis example. The final shear and moment diagrams for this analysis are shown in Figure 7.11 . This figure shows both the values from this cantilever method analysis compared with the previous portal method analysis example results (in square brackets). This shows that with a significantly different set of assumptions for this example frame, we get similar shear and moment diagrams using the two different methods.

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  1. Solving A Truss Using The Method Of Sections

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  2. Solved Topic 4-2 The Method of Sections To apply the method

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  3. Solved (2) Truss Analysis Method of Sections Determine the

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  4. Solved Use the method of sections as described below. You

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  5. Truss Analysis by Method of Sections Solved Example, Engineers Academy

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  1. Ch:6 Method of sections

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  5. Method of Sections

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COMMENTS

  1. 5.5: Method of Sections

    The method of sections is a process used to solve for the unknown forces acting on members of a truss. The method involves breaking the truss down into individual sections and analyzing each section as a separate rigid body. The method of sections is usually the fastest and easiest way to determine the unknown forces acting in a specific member ...

  2. PDF Method of Sections

    n The method of sections depends on our ability to separate the truss into two separate parts, hence two separate FBDs, and then perform an analysis on one of the two parts. 4. Method of Sections. n The method of sections utilizes both force and moment equilibrium. n The method of sections is often utilized when we want to know the forces in ...

  3. PDF Chapter 6: Analysis of Structures

    6.7 Analysis of Trusses: Method of Sections The method of joints is good if we have to find the internal forces in all the truss members. In situations where we need to find the internal forces only in a few specific members of a truss , the method of sections is more appropriate. For example, find the force in member EF:

  4. Problem 003-ms

    Engineering Economy. Geotechnical Engineering. Advance Engineering Mathematics. Problem 003-ms The truss in Fig. T-04 is pinned to the wall at point F, and supported by a roller at point C. Calculate the force (tension or compression) in members BC, BE, and DE. Figure T-04.

  5. 5.3 Method of Sections

    5.3 Method of Sections. The method of sections uses rigid body analysis to solve for a specific member or two. Instead of looking at each joint, you make a cut through the truss, turning the members along that line into internal forces (assume in tension). Then you solve the rigid body using the equilibrium equations for a rigid body: ∑F x ...

  6. PDF Example Method of Sections

    Title: 201709201003.pdf Created Date: 9/20/2017 5:30:29 PM

  7. PDF Engineering Mechanics

    As we are solving the problem using the method of joints, we take equilibrium at each point. As we have assumed the forces in all the members are tensile, the direction of the reaction force they exert on the hinges are as shown. FF AF kN EquilibriumatA FFF F kN FF BF kN EquilibriumatB y AC y AC x ABBC AB y BC y BC 0sin 0 2.13 0 cos0 8.84 0sin ...

  8. (PDF) Method of Sections

    A section has finite size and this means you can also use moment equations to solve the problem. This allows solving for up to three unknown forces at a time. fMethod of Sections The Method of Sections involves analytically cutting the truss into sections and solving for static equilibrium for each section. The sections are obtained by cutting ...

  9. 3.6 The Method of Sections

    Select a section cut. The obvious choice for a cut is section a-a as shown in Figure 3.9, because it cuts through all of the members that we are trying to find axial forces for; however, the problem is that this section has four members, which we cannot calculate directly using only our three equilibrium equations.So, we need a way to cut down the number of unknown forces at section a-a from ...

  10. PDF THE METHOD OF SECTIONS

    GROUP PROBLEM SOLVING Given: Loading on the truss as shown. Find: The force in members BC, BE, and EF. Plan: a) Take a cut through the members BC, BE, and EF. b) Analyze the top section (no support reactions!). c) Draw the FBD of the top section. d) Apply the equations of equilibrium such that every equation yields answer to one unknown.

  11. Solving Truss by Method of Sections

    Step 2: Make a cut along the members of interest. Here comes the most important part of solving a truss using the method of sections. It involves making a slice through the members you wish to solve. This method of structural analysis is extremely useful when trying to solve some of the members without having to solve the entire structure using ...

  12. 3.7 Practice Problems

    3.7 Practice Problems. Selected Problem Answers. For each truss below, determine the forces in all of the truss members using the method of joints. For each truss below, determine the forces in all of the members marked with a checkmark ( ) using the method of sections. 3.7a Selected Problem Answers. 3.6 The Method of Sections. Up.

  13. Mechanics Map

    The Method of Sections. The method of sections is a process used to solve for the unknown forces acting on members of a truss.The method involves cutting the truss into individual sections and analyzing each section as a separate rigid body. Where the method of joints is the fastest way to find the forces in all members, the method of sections is usually the fastest and easiest way to ...

  14. Method of Sections to Solve a Truss

    The method of sections is used to calculate the forces in each member of the truss. This is done by making a "cut" along three selected members. First, calculate the reactions at the supports. Taking the sum of the moments at the left support: . Next do a force balance of the forces: , where and are the reaction forces, and and are the point ...

  15. Method of Sections

    Method of Sections. In this method, we will cut the truss into two sections by passing a cutting plane through the members whose internal forces we wish to determine. This method permits us to solve directly any member by analyzing the left or the right section of the cutting plane. To remain each section in equilibrium, the cut members will be ...

  16. Statics: Method of Sections

    The method of sections is used to solve for the unknown forces within specific members of a truss without solving for them all. The method involves dividing the truss into sections by cutting through the selected members and analyzing the section as a rigid body. The advantage of the Method of Sections is that the only internal member forces exposed are those which you have cut through, the ...

  17. (PDF) Who Needs the Method of Sections and the Method of Joints? Just

    Statics courses can sometimes give students the misconception that there are many different approaches to solving statics problems, such as the "method of sections" or the "method of joints."

  18. Method of Sections: Problem Solving II

    6.9: Method of Sections: Problem Solving II. Consider an arbitrary truss structure composed of diagonal, vertical, and horizontal members fixed to the wall. To calculate the force acting on members CB, GB, and GH, method of sections can be used. The loads and lengths of the horizontal and vertical members are known parameters, as shown in the ...

  19. Method of Sections: Problem Solving I

    6.8: Method of Sections: Problem Solving I. Consider a symmetrical roof truss structure, composed of vertical, diagonal, and horizontal members. The length of each horizontal member is 4 m. The lengths of the vertical members FB and HD are 4 m, while the length of member GC is 6 m. The loads acting at joints F, G, and H are 2 kN, while those at ...

  20. 6-8 Structural Analysis Chapter 6 Method of Sections Hibbeler ...

    SUBSCRIBE my Channel for more problem Solutions!Engineering Statics by Hibbeler 14th EditionChapter 6: Structure Analysis (Method of Joints)Determine the for...

  21. Problem 004-ms

    Differential Calculus. Fluid Mechanics and Hydraulics. Elementary Differential Equations. Timber Design. Engineering Economy. Geotechnical Engineering. Advance Engineering Mathematics. OK, I agree. Problem 004-ms For the truss shown in Fig. T-05, find the internal fore in member BE.

  22. 7.4 The Cantilever Method

    Like the portal frame example, the free body diagrams in Figure 7.10 are annotated with numbers in grey circles to show a suggested order for solving all of the unknown forces. Of course, as before, step 0 and step 1 consist of known values, either caused by external forces or the previous storey (for step 0) or the column axial forces that were solved using the cantilever method assumptions ...