Lesson: Columns of Calculation

Authors: Peter Anderson, Wade Goodridge, Sarah Lopez, Natalie Shaheen 

Class Size 

  • Written for 15 students, working in 4 groups of 3-4. For different class sizes, groups should still be 3-4 students. 

Lesson Structure 

150 minutes 

  • Balsa testing: 50 minutes 
  • Balsa book calculations and share out: 35 minutes 
  • Column plan calculations: 40 minutes 
  • Build time: 25 minutes 

Objectives 

Students will be able to: 

  • Test columns of balsa wood to determine their resistance to compression. 
  • Calculate a roof load for their model. 
  • Choose column number and material to resist a roof load. 
  • Assemble posts/columns for the student structure. 

Prerequisite Knowledge 

  • Calculating an average 
  • Using a table of values 
  • Footprint dimensions of student’s structure (from Forming Foundations) 

Accessibility 

  • Accessible measuring tools (e.g., click rule, talking tape measure) 
  • Accessible column tester 
  • Accessible calculators 

Materials 

  • Balsa pieces for testing: 
    • Balsa, 1/16” x 1/16” - (8 6” sticks, 8 8” sticks, 8 10” sticks) 
    • Balsa, 1/8” x 1/8” - (8 6” sticks, 8 8” sticks, 8 10” sticks) 
    • Balsa, 1/8” x 1/4” - (8 6” sticks, 8 8” sticks, 8 10” sticks) 
    • Balsa, 1/4” x 1/4” - (8 6” sticks, 8 8” sticks, 8 10” sticks) 
    • Balsa, 1/4” x 3/8” - (8 6” sticks, 8 8” sticks, 8 10” sticks) 
  • For building: 
    • Balsa, 4 36” sticks per student, one of each of the sizes chosen in the instructor preparation below. Note: There will be a significant amount of balsa left over, but there should be enough so that all students can freely choose any size that supports the requirements of their design. 
  • Column tester - 1 per group, + 1 extra 
  • Accessible calculator (device or app) - 1 per group 
  • Braille writers, 20/20 pens, paper (for recording calculations) 
  • Hot glue guns and glue - 1 gun and 10 sticks per 2 students 
  • Empty coffee cans (for glue guns that are not in use) - 1 per 2 students 
  • Safety glasses - 1 per student, + 6 extra 
  • Weights, 1 lb. - 4 per group 
  • Weights, 5, lb. - 3 per group 
  • Weights, 10 lb. - 1 per group, + 2 extra 
  • Weights, 25 lb. - 1 per group, + 1 extra 
  • Angle/length jig - 1 per 2 students 
  • C-clamp - 1 per student 
  • Razor saw - 1 per student 
  • Shears - 1 per 3 students 
  • Handout: Load Adjustment Factors (BRF)
  • Handout: Load Adjustment Factors (Word)
  • Handout: Column Plan (BRF) 
  • Handout: Column Plan (Word)
  • Handout: Bracing BRL (PDF for hardcopy production only)
  • Handout: Bracing LP (PDF for hardcopy production only)

Note: Refer to Accessible Lab Equipment & Instructional Materials for additional information regarding specialized tools/materials. 

Preparation 

  1. Assemble the accessible column testers used to apply a force to the balsa column. 
    • Place the 1st (largest) bottom plate on table. 
    • The 2nd bottom plate and 4 dowels measuring 0.5” x 11.25” should be glued to this bottom plate and the dowels should extend vertically upward. 
    • Align the top plate system (1st, 2nd, 3rd, and top plate glued together) so that it slips over the 0.5” dowels vertically mounted in the bottom plate system. 
    • This top plate system should slide up and down a short distance on the dowels. This is to be able to fit test pieces that are placed vertically into the center holes of the top plate and bottom plate system. When a test piece is in the system, the weight of the top plate should rest on the test piece, not the top of the dowels. 
    • Weights will be placed on top of the top plate system.
    • Note: If you use sandbag style weights, it is recommended that you build the tester with a flat plate or shallow tray shape on top instead of the post shown. If you use round metal weights, the post on top will help hold the weights in place. 
  2. Characterize the balsa. Many factors can affect the strength of the balsa such as humidity, temperature, batch of balsa etc. Because of this, local testing should be done before teaching the lesson to determine what sizes of balsa will work best for students to test.  
    • Using a column tester and 2 of each size of balsa sticks, test the failure point of each stick. This will give you a rough idea of how much weight will be needed to fail each size.  
    • A column tester. The source files to produce a tactile graphic of the column tester are available in the Plans for Constructing NFB EQ Tools and Manipulatives section of the curriculum.
    • Based on your results, choose 4 of the 5 balsa widths to use as options for students. You want to choose sizes that fail at a point that is easily measured by the column tester apparatus. Roughly, this would mean a failure point greater than 1lb, but less than 70lbs. It is likely that either the smallest size or the largest size will fail outside this range. If only 2 or 3 sizes work well, just use those. 
  3. Prepare one table of the testing materials for each group. Each table should have paper and writing utensils, an accessible column tester, and a set of weights (four 1 lb., three 5 lbs., one 10 lbs., and one 25 lbs.). 

Formulas

Maximum_column_load=Average_column_failure_point2Maximum\_column\_load = \frac{Average\_column\_failure\_point}{2}

Roof_snow_load=0.7⋅Ground_snow_loadRoof\_snow\_load = 0.7 \cdot Ground\_snow\_load

Modified_snow_load=Importance_Factor⋅Heating_factor⋅Exposure_factor⋅Roof_snow_loadModified\_snow\_load = Importance\_Factor \cdot Heating\_factor \cdot Exposure\_factor \cdot Roof\_snow\_loadTotal_roof_load=Modified_snow_load+Dead_load=Modified_snow_load+15Total\_roof\_load = Modified\_snow\_load + Dead\_load = Modified\_snow\_load + 15

Structure_area=length⋅widthStructure\_area = length \cdot width

Model_area=Structure_area144Model\_area = \frac{Structure\_area}{144}

Total_model_roof_force=Model_area⋅Total_Roof_LoadTotal\_model\_roof\_force = Model\_area \cdot Total\_Roof\_Load

Force_per_column=Total_model_roof_forceNumber_of_columnsForce\_per\_column = \frac{Total\_model\_roof\_force}{Number\_of\_columns}

Procedure 

  1. Introduce the lesson.   
    • Tell. “This next session is about building columns to support your roof. We will figure out with math and experimentation how many posts of what thickness we need and where they could go in order to support the roof of our structure.” 
  2. Structural engineer process.   
    • Tell. “When a structural engineer designs a building, they start by determining what load they will need to support based on several factors about the building. Then, when they are ready to choose their beams they consult a steel book to determine how much weight each type of steel beam can support.” 
  3. Balsa testing why.   
    • Tell. “Since we don’t have a ‘balsa wood book’ we’re going to start by testing the balsa we have to determine how much force each column size can support.” 
  4. Testing activity: Materials.   
    • Do. Divide students into groups of 3-4 at the tables you have prepared for testing. 
      • Assign each group a few sizes of balsa to test. (Note: There are a total of 12 sizes at this point; 4 widths in 3 lengths each. The length of a stick will affect its failure point, hence why the different lengths are tested separately) Depending on how many groups you have, each group may have more or less to test. Try to combine sizes based on width, rather than length. For example, if you have 4 groups (which would be ideal!) one group will test all the 1/4” x 1/4” pieces (of all 3 lengths), another group would test all the 1/4” x 3/8” pieces, etc. Give each group 5 pieces of each size they are to test. 
      • Distribute the extra 10 lbs. and 25 lbs. weights to the groups testing thicker pieces. 
  5. Explain structural failure.   
    • Tell. “For each balsa stick your group has, you will test it to see how much weight it can support before it breaks.” 
      • Introduce the testing device. Tell. “You have a testing fixture on your table. This device has 4 posts sticking up with a plate that can slide up and down on the posts.” 
      • Testing setup. Tell. “To test a column, you will lift up the plate, stand the column up inside the fixture, and then let the plate rest on the column. Center the balsa test piece so it stands vertically in the center holes on the top and bottom plates.” 
      • Testing procedure. Tell. “Next, you will start adding weight to the top of the plate to see how much the column will hold. Add the weight 1lb. at a time, so you can find the exact point when the column fails.” 
      • Testing endpoint. Tell. “As soon as the column breaks, stop adding weight, and record how much weight your column held before it broke.” 
  6. Testing time.   
    • Do. Direct groups to conduct the first test all together. Circulate between the groups and help students use the apparatus correctly and find the endpoint. After that, they may proceed at their own pace.  
  7. Recording results.  
    • Students Do. Have each group record their results, and calculate an average value for each size that they tested. 
  8. Cleanup.   
    • Students Do. As students finish, direct them to dispose of any broken balsa and stack the weights, then move to the calculation tables. 

Calculations and Share Out: 35 Minutes 

  1. Adjust for material variability.   
    • Tell. “Now that we know about what the failure point is for each balsa member, we need to decide how much force we can safely put on a post. Not all balsa sticks are the same, so even though the one you tested could support X pounds, the one you build with might support less. When engineering, we assume we are building exclusively with the weakest member. If the specific post is stronger, so much the better.” 
  2. Safety factor.   
    • Tell. “Also, if the column starts to bend or shorten, even just a tiny bit, that will shift and distort the wall, which may cause windows to crack, doors to stick, or throw off any other precise things you have in your wall. So, we don’t want to even push it close to the failure point since that’s not safe. Instead, we’ll use 50% of the breaking value as our maximum safe load.”   
    • Students Do. Direct students to divide all their averages by 2. 
  3. Data share-out.   
    • Students Do. Each group will share their results, so that each student can create a table showing the maximum safe load for each balsa size (their balsa book).  
    • Do. Hand out blank paper. Explain to students what the table needs to do (hold 12 pieces of labeled information). Orient the students briefly to the table, then conduct the share-out of the average values for each type of balsa. While one group speaks, others record their data in their balsa book. 

Column Plan Calculations: 40 Minutes  

  1. Roof load.   
    • Tell. “Now that we have a ‘balsa book’ ready to go, we need to calculate how much force our building needs to support. Our structure will have to support the roof, anything on top of the roof, such as the people and materials involved in roofing the structure, or any snow that falls on the roof.” 
  2. Ground snow load.   
    • Do. Pass out the column plan.   
    • Tell. “Of course, the amount of snow we need to design for will be different depending on where our building is. Thanks to observations, modern meteorology, and journals to collect these observations, we have clear knowledge of how much snow falls in almost any geographical location. Based on our project criteria, we must be able to withstand typical snowfall in Alaska, which can be up to 50 pounds per square foot (psf). This number is called the ground snow load.” 
  3. Roof snow load.   
    • Tell. “However, because of wind and gravity, not all of that snow that falls will stay on our roof, some blows off, some slides off. Because of this, the IBC (International Building Code) tells us that we only need to support 70% of the ground snow load at any given time. So, we can multiply our ground snow load (50 psf) by 0.7 to determine our roof snow load.”  
    • Students Do. Have students work through reducing the snow load. 50 * 0.7 = 35 psf. 
  4. Additional modifying factors.  
    • Do. Pass out the handout of load adjustment factors.   
    • Tell. “There are some other factors about your building that can affect how much snow you will need to design for.”
  5. Importance factor.  
    • Tell. “The first factor is importance, or risk. This is decided based on how much human life will be impacted if the structure fails. There are 4 categories of importance, but most buildings fall in category 2. Read over the first part of your hand out, and decide what category your building is. Write down the number that is listed with the category you decided.”  
    • Do. Check to make sure students write down the factor (0.8-1.2) not the category number. 
  6. Heating factor.  
    • Tell. “Next is the heating factor. If your building is unheated, less of the snow on the roof will melt, so more will stay on the roof. Read over the next part of your handout, and write down the number that corresponds to your heating.” 
  7. Exposure.  
    • Tell. “The last factor has to do with the environment around your building and whether snow is more likely to blow off or pile up on your roof. This factor is outside of your control and would be provided by the architect after they analyzed the site. You may assume that your structure is in a partially exposed environment. Write down the number for a partially exposed structure.” 
  8. Modified snow load.  
    • Tell. “To determine your final modified snow load, you need to multiply the 3 factor numbers you wrote down by the roof snow load you calculated previously.”  
    • Do. Have students perform this calculation, and check that they have set it up correctly.
    • Note: it is not unusual for the modified snow load to be the same as the unmodified number. 1 is the most common value for all 3 factors. If this is the case, students should have a value of 35 psf. 
  9. Live and dead loads.   
    • Tell. “In addition to snow on the roof, we also need to support the weight of the roof itself, which we call the dead load, and the weight of any people or materials that might have to go on our roof, which we call the live load. The IBC tells us that we can assume a maximum live load of 20 pounds per square foot.” 
  10. Live vs. snow.   
    • Tell. “Do you think you will have people up working on your roof when it’s covered in snow?  Probably not. Because of this, we don’t have to support both the people and the snow at the same time, but instead we just have to support whichever is the biggest load. In your case, which one is bigger?” The snow load. “So as long as we can support the snow load, we will also be able to support our live load just fine.” 
  11. Dead load.  
    • Tell. “However, we do have to support the weight of the roof at the same time as the snow, so we need to add the dead load to the modified snow load to get the total load we need to support. The dead load for a shingled roof is 15 lbs. / ft2. Add this number to your latest modified snow load.”
    • Note: Make sure students use their modified number, not the original 50 or 35. 
  12. Note on recordkeeping.   
    • Tell. “From here on out, we need to write our calculations out as if it was math class. Every calculation needs a title. Be sure to flag your final results in some way, so you can easily find them later. Record your last calculation under the title of ‘Total Roof Load.’” 
  13. Calculate square footage of student’s footprint.   
    • Tell. “Multiply the length by the width of your structure to calculate the total area. Label this ‘Structure Area.’”   
    • Do. Work students through this calculation individually. Particularly help any students with non-rectangular structures. 
  14. Scaling factor.   
    • Tell. “Since we are focused on the design of our model, we actually want to know the total area of our scaled model. Because we are working at 1/12 scale, the total area will be 1/144th the area of the full size structure. Divide by 144 to calculate model area. Label this ‘Model Area’.” 
  15. Calculate the total roof force.   
    • Tell. “Multiply the square footage of your model roof by your Total Roof Load. This will tell you how much the whole model roof has to hold. Label this ‘Total Model Roof Force.’” 
  16. Calculate the force per column.   
    • Tell. “Divide this result by the number of columns you want to have. Label this ‘Force per Column.’ There is a tradeoff to be made here. More columns means they can be thinner, fewer means they must be thicker. Think about the placement of these columns and make sure they don’t interfere with your door.”  
  17. Column choices.   
    • Tell. “Now you can check the ‘Table of Wood Strengths’ (our notes from our balsa column tests) from earlier to find what size columns you need to be able to support that much weight. If none of the columns can support what you need, you may need to rethink how many columns you will use.” 
  18. Work time.   
    • Do. Circulate through students at this time, to check their progress and look over their work. Talk through any difficult choices without guiding them yourself. 
  19. Draw the column plan.   
    • Tell. “Now you will draw a column plan to document the choices you have made. Use your base template (Created in Forming Foundations) to trace a footprint of your structure onto paper. Mark the location of your columns onto the paper. You can draw a box with an ‘X’ inside for each column. Label each column with its size. Also include your ‘Force per Column’ number somewhere on this plan.”  

Build Time: 25 Minutes 

  1. Post assembly.   
    • Students Do. Direct students to begin assembling their posts. They will need to find the appropriate width of balsa stick, cut it into sections matching their column height, and glue the columns onto their footprint box from Forming Foundations. For posts that aren’t corner posts, it is useful to measure where to place the middle of a glue dot, score that location, then put the dot in place. 
  2. Check post angles.  
    • Tell. “It is important as you are building to make sure that your posts do not lean. On the construction site, engineers visit and check the worker’s work for accuracy in this and other ways.” 
  3. Bracing.   
    • Do. After most students get 2 columns up, stop the class and discuss bracing. Pass out Bracing Handout. 
    • Tell. “Real post and beam structures use bracing to strengthen and stabilize the structure. Braces are diagonal pieces that attach to a column on one end and a beam on the other, and create a triangle shape at the corner of the structure. Your handout shows where you might put braces in your structure. Adding braces to your model will help support the joints, so that the connection points are less likely to fail, and can keep your columns going straight up and down. At this stage, you might add some lower braces to help your columns stand up straight, but once you add your roof in a later lesson, you may also want to add some upper braces. If you are wondering whether you should brace, put a piece of chipboard on top of the columns. If the columns don’t lay flat against it, consider bracing the ones that are uneven.” 
  4. Limits of bracing.   
    • Tell. “The only real limit to this kind of bracing is how close the columns are together and how close together you can cut safely. Your columns have been calculated to be strong enough on their own, so you can use the thinnest, smallest braces possible. In addition to providing stabilization, braces do also strengthen your columns by shortening the unsupported length, but to explain or calculate those details we would need to teach many other things.” 
  5. Technique of bracing.   
    • Tell. “A brace makes a right triangle with the bottom of your structure and the column.  The actual piece of wood is cut into a trapezoid with 45° angles on either side that go inward.  When actually making these, it is useful to make them in pairs, since that is frequently how they are used.” 
    • Teach. 
      • Cut a piece of balsa wood at a 45° angle. The wood will need to be held down with more force than for a straight cut. Keep both pieces. 
      • Decide on an approximate length of the brace. Cut a 2nd 45° angle inward toward the other.  
      • Using the other piece of wood, match the 1st brace. It can be useful to put the finished brace on top of another one and score the new piece to get the lines in the right spot. 
      • If bracing multiple columns, put 1 of these 2 braces into place to be sure it works. If it does, use that brace as a template for the others. 
      • A closeup of a braced corner of a partially constructed balsa model. The trapezoidal-shaped brace, the corner column, and the base of the model form an isosceles right triangle.
  6. Work time.   
    • Students Do. Direct students to continue working. Students who finish quickly should be encouraged to brace their columns or work on other deliverables. 
  7. Looking forward.   
    • Tell. “In our next phase, we will put a roof on these columns.  If the columns don’t come out perfectly, there are some ways we can improve that later, but do your best to make them straight now.” 

End of Lesson Check 

At the end of this lesson, students should have a rectangular box with columns of the same length coming out of it. The columns may not be perfectly straight; that is okay, but they must be the same length. Inspect these columns with a T-square and check to ensure their structure continues to align with the provided criteria and constraints. 

Standards Alignment

NGSS Standards Alignment:

  • SEP 3 - Planning and carrying out investigations 
  • CCC 6 - Structure and function 
  • HS-ETS1-2 

CCSS Standards Alignment:

  • CC.9-10.R.ST.3, CC.9-10.R.ST.4, CC.9-10.R.ST.5, CC.9-10.R.ST.7, CC.11-12.R.ST.3, CC.11-12.R.ST.4, CC.11-12.R.ST.5, CC.11-12.R.ST.7 
  • CC.9-12.G.GMD.4, CC.9-12.G.MG.1, CC.9-12.G.MG.2, CC.9-12.G.MG.3 

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