Statical Determinate Pin-Jointed Frame

1. In this post you will learn :

1. Calculate the member forces
2. Calculate the joint deflection

2. Method for calculating member forces

1. Method of Section

This method is using the section to compute the desired member force (member which is crossed by the section).

Let say we'll find the CD member force. The steps are :

• Find the support reactions
• Make a section that is crossed the CD member
• Find the CD member force using the available static equations :
• Σ H = 0   (The sum of all horizontal forces in structure system must be zero)
• Σ V = 0   (The sum of all vertical forces in structure system must be zero)
• Σ M = 0  (The sum of moment at any point in structure system must be zero)
• In this case the fastest way to calculate the CD member force is make an equation of bending moment in Joint K (Va.L1 + P1.L2 + CD.L2 = 0)
• Note : 
• For calculation assume all the member force is in tension (tie member, out from joint force, "+")
• Since the number of static equations are 3, the maximum number of unknown member forces in a section are also 3.

2. Joint Resolution

This method is developed based on the understanding of static equilibrium in all joints of a frame.

Let say we'll find the CD member force. The steps are :

• Find the support reactions
• Find the equation that consist CD member force using the available static equations :
• Σ H = 0   (The sum of all horizontal forces in structure system must be zero)
• Σ V = 0   (The sum of all vertical forces in structure system must be zero)
• Σ M = 0  (The sum of moment at any point in structure system must be zero)
• In this case form the Σ H = 0,  the CD member force will be equal to the BC member force.
• Use the same concept to find the BC member force
• Note : 
• For calculation assume all the member force is in tension (tie member, out from joint force, "+")
• Since the number of static equations are 3, the maximum number of unknown member forces in a choosen joint are also 3.

3. Tension Coefficient

Using the basic force concept, the AB force in X and Y direction will be :

Therfore :

This method is using the same concept with 2 methods above.

The different thing is only the form of the equation as shown in the formulas above.

3. Joint deflection

The Joint Deflection in a Pin-Jointed frame is calculated with using the 1st Castigliano's therorem.

1st Castigiliano's therorem :

"If the total of strain energy is partially differentiated with respect to an applied load the result is equal to the displacement of that load in its line of action"

What is Strain Energy?
Strain Energy is the energy that is stored in a member while it is experiencing deformation (strain). Strain Energy is the energy that will change a deformed member into its initial condition.

Let's say we have a member with initial length "L" and it's experiencing an axial tension load "P" as shown in following image :

The member length will be change into L + ΔL. This condition will be return into its initial condition if the axial tension load is removed. The energy to return the member into its initial condition is callled Strain Energy. 

Strain Energy :

Where :

W    = Total Strain Energy

P      = Member Axial Load

L      = Member Initial Length

E      = Member Modulus of Elasticity

A      = Member Section Area

Based on Castigliano's 1st Therorem, the joint deflection in a pin-jointed frame would be :

Where :

δ        = Total Joint Deformation (In the direction of d(Q))

d(Q)  = The Imaginer Force that acting in a joint to calculate joint deformation

W      = Total Strain Energy

u        = Member force coefficient due to imaginer force d(Q)

P       = Member Axial Load

L       = Member Initial Length

E       = Member Modulus of Elasticity

A       = Member Section Area

4. Example
(Will be shown in the next post!!!)