4. Pin Jointed Trusses, Methods of Sections

Question 1

How do we do the Method of Sections?

Can use this example to test yourself!

Answer 1

1. Work out the external forces on a FBD, such as the reaction forces on the supports. Using Vertical, Horizontal and rotational equilibrium equations.
2. Cut the between three bars, so you have three unkowns and it must be through the bars, not the joints.
3. Draw another FBD of the new isolated parts, with the three unkown and the one known force worked out from step 1.
4. Now once again use the Vertical, Horiztonal and rotational equilibrium equations to find an equation with only one of the unkown forces with the known force, so you can find them individually. Repeat until all forces are known. Remember that if you do moments about a point, if there is no distance, then you can ignore those forces, and that vertical and horizontal can often only have unkown force, particularly vert.
5. Draw a final FBD of the bars with there known forces now, noticing which bars are compressive or tensile.

Question 2

What are the three different classes of pin-jointed systems?

Answer 2

1. Mechanism: more equilibrium equations than unkown forces or reactions - unstable system. Good for machines
2. Statically determinate: same number of equilibrium equations as unkown forces - can work out using methods of sections or joints.
3. Statically indeterminate: more unkown forces than equilibrium equations.

Question 3

Describe a Mechanism pin-jointed system.

Answer 3

• The bars are free to rotate. In general, system is incapable of supporting static loads.

Question 4

Describe a Statically determinate pin-jointed system.

Answer 4

• All joint positions prescribed by bars and supports, no movement possible without deforming the bar.

Question 5

Describe a Statically indeterminate pin-jointed system.

Answer 5

• The joints are over-prescribed.
• In assemberly of the system, its a real struggle to add the pins in to make the red-bar in the image below fit.
• The bar is also redundant, as the structure stands without it.
• Furthermore, the static equilibrium equations are no longer enough to determine the internal forces of each bar. We must take another approach then method of section or Joints.

Question 6

What is the general formal test we can do to classify a pin-jointed system?

Answer 6

• 2j = r + m
• j = number of joints
• r = number of reaction forces at supports (vert and horizontal forces)
• m = number of bars

Note: that this formal test doesn’t always work, so that we must still look out for these classes.

Question 7

How can we calculate Elongation / Contraction, ΔL of a bar using the bar forces we found using Methods of Sections or Joints?

Answer 7

• ΔL = FL/EA
• Where F is bar force, L is bar length, E is young’s modulus of bar, A is the cross section of the bar.

Question 8

Using this example, How do you work out the truss deformation and (general) position of the bars now?

Answer 8

• Use the formula, ΔL = FL / EA for each bar, and their force. Try creating a table of resulst if it helps
• Then you can roughly sketch how the elognation or contraction looks on the FBD, then using an arc, find the point where the two arcs touch, as shown below.

Question 9

How do we accurately work out the new position relative to the origional position after Truss deformation?

Answer 9

• Draw a displacement diagram
• Work out the horiztonal displacement, which might be easy if it only has one horizontal deformation.
• Work out the vertical displacement, using this formula: ΔY = (cos(θ)ΔLac + ΔLbc) / sin(θ)