## Calculation of Unbraced Lengths (Bracing Points) for Tower Legs

Deliverables (individual submission required):

1. Tower Leg Forces: Submit a hard copy of Table 1 (shown below) and a graph of the compression force in the right tower leg, C, vs. x for each of the four tower base widths (four curves).  Plot the compression force on the y-axis and the distance “x” on the x-axis.  You can plot all four curves on a single graph with a legend (better) or create four separate graphs (acceptable).  Answer the discussion question:  What happens to the tower leg force as the tower base width is increased?
2. Based on your calculations, recommend a tower base width for the tower you will build.
3. Tower Leg Bracing: Submit a hard copy of Table 2 (shown below) and a graph of the maximum permissible unbraced lengths Lb (in.) as a function of compressive load C (lbf) in a tower leg.   Answer the discussion questions:  How does Lb  vary with C?  For the tower base width you recommended and the corresponding compressive loads, what should be Lb at the bottom part of the tower?  How about in the middle section?  Towards the top?

1. Calculation of Internal Forces in Tower Legs (Square Tower) and Selection of Base Tower

Width

Tower Construction Requirements:

 Tower Height 24 in. Tower Width, WTOP 2 in. square Tower Width Base, WBASE 2 in. (minimum) 8 in. (maximum) Braces As necessary for stability of tower legs Materials As provided in reference section Connections Push-pins Tower to Base Connection Push-pins and angle brackets

Use static equilibrium equations from your reference section to calculate the maximum tension and compression forces in the tower legs.  Calculate the forces in the legs at the tower base and at the ¼, ½ , and ¾ points along the height of the tower.  Use Excel to perform these calculations for five tower geometries:  a tower top width of 2 in. square with a base width of 2 in., 3 in., 4 in., 5 in., and 6 in.

From the static equilibrium equations of the tower:

Solve for F2 using Eq. 1 and for F1 using Eq. 2.  Remember that for a four leg tower, the calculated forces F1 and F2 are divided by two (two tower legs on the left and two on the right of the tower).

Eq. 1.                                         2

Eq. 2.                                        ∑ = 0 = 2 − + 1

Use Excel to calculate F1 and F2  values for tower sections located  x = 6 in., 12 in., 18 in., and 24 in. from the top of the tower.  Note that the distance between the tower legs, W, will change based on the tower’s base width as shown below.

NOTE:   A negative sign for Force F1 indicates that the force in the leg is in tension and the direction of the force vector on the figure is downward instead of upward as shown.

 Tower base width ≠ tower top width Calculate width of the tower at the ¼ points. Ex.  For a 6 in. wide base, x = 0 in.,               W = 2 in. x = 6 in.,  W = 3 in. x = 12 in.,          W = 4 in. x = 18 in.,          W = 5 in. x = 24 in.,          W = 6 in.

Format your calculations to appear in a Table in Excel as follows:

Table 1 – Tower geometries and associated tower leg forces.

 Tower Base Width (in.) Distance from top of the tower, x (in.) W, Tower Width at “x” (in.) F1 (lbs) F2 (lbs) Compression / Tension Force in Left Tower Legs, T = F1/2 Compression Force in Right Tower Legs, C = F2/ 2 2 0 (top) 6 12 18 24 (base) 4 0 (top) 6 12 18 24 (base) 6 0 (top) 6 12 18 24 (base) 8 0 (top) 6 12 18 24 (base)

Graph the compression force in the right tower leg, C, vs. x for each of the four tower base widths (four curves).  Plot the compression force on the y-axis and the distance “x” on the x-axis.  You can plot all four curves on a single graph with a legend (better) or create four separate graphs (acceptable).  What happens to the compression force in the tower leg as the tower base width is increased?

Based on your calculations, choose a tower base width.  Note that while wider bases will reduce the overall tension and compression forces in the tower legs, they require more material and longer members.

Tower Base Width:

1. Calculation of Unbraced Lengths (Bracing Points) for Tower Legs

The Reference section discusses various modes of tower failure:  due to exceeding tensile strength, compressive strength and shear strength of the balsa wood, and due to buckling.  Here we are going to assume that the tower legs are strong enough to withstand the loading for all failure modes except buckling.  The goal of this part is to estimate the separation distance between so called bracing, which will protect the tower legs from buckling.  See Reference section for more detail.

Instructions:

Use the Euler buckling equation with a factor of safety = 3 to calculate the maximum permissible unbraced lengths, Lb, for axial loads of 1 lb to 10 lb in 1 lb increments.  Tabulate and graph the results.

Eq. 3.                                                                   2

Table 2 – Maximum permissible unbraced lengths, Lb, to prevent buckling.

 Compression Force in Tower Leg (lbs) Maximum Permissible Unbraced Length, Lb (in.) 1 2 3 4 5 6 7 8 9 10

Now consider the calculated unbraced lengths, Lb, when designing your tower bracing.  For your selected base width and the corresponding compressive loads (from Part A), what should be Lb at the bottom part of the tower?  How about in the middle section?  And how about towards the top of the tower?