4.1 Analysis of trusses using method of joints and method of sections

Trusses are key structures in engineering, using interconnected members to support loads efficiently. This section explores two powerful analysis techniques: the method of joints and the method of sections. Each approach offers unique advantages for solving truss problems and determining member forces.

Understanding these methods is crucial for structural analysis. The method of joints systematically examines force equilibrium at each connection, while the method of sections cuts through the truss to analyze larger portions. Mastering both techniques equips engineers to tackle various truss configurations effectively.

Method of Joints vs Sections

Differences in Analysis Techniques

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• Method of joints is a truss analysis technique that considers the equilibrium of forces at each joint in the truss, while method of sections analyzes the equilibrium of a portion of the truss created by an imaginary cut
• In the method of joints, equilibrium equations are written for each joint, considering forces in the members connected at that joint and any external forces or reactions applied there
• In the method of sections, equilibrium equations are written for forces and moments acting on one section of the truss

Differences in Problem Solving Approaches

• The method of joints solves for unknown forces in members by progressing in a logical order from joint to joint where there are at most two unknown member forces at each
• The method of sections can solve for forces in members that may not be solvable by inspection or the method of joints alone
• The method of joints is systematic and can be applied repeatedly until all member forces are known, while the method of sections is a more direct approach for determining the force in a specific member
• The method of joints requires that all external forces, including reactions, are known in advance, while the method of sections can be applied even if some reaction forces are unknown

Method of Joints Application

Determining External Reactions

• Draw a free body diagram of the entire truss, replacing all supports with the forces and moments that they exert on the truss
• Determine the external reactions at the supports using the equations of equilibrium (ΣFx = 0, ΣFy = 0, and ΣM = 0)

Solving for Unknown Member Forces

• Identify a joint with at most two unknown member forces, starting at a joint where most information is known (usually where external forces are applied or at the supports)
• Draw a free body diagram of the joint, representing all known and unknown forces as vectors
  • Unknown forces should be assumed to act in a direction away from the joint if the member is in tension and towards the joint if in compression
• Write and solve the equilibrium equations (ΣFx = 0 and ΣFy = 0) for the joint to determine the unknown forces
  • If one of the unknown forces cannot be found, move to the next joint and repeat the process until you can solve for that unknown
• Proceed to the next joint where there are at most two unknown forces, repeating the process of drawing the free body diagram, writing equilibrium equations, and solving for the unknown forces
• Continue this process, moving systematically from joint to joint until all member forces have been determined
  • It may be necessary to begin at another area of the truss where additional information is known

Verifying Results

• Verify the results by checking that the equilibrium equations are satisfied at each joint
• Ensure that the determined forces do not violate the properties of the truss or its supports (e.g., a member in tension should not have a negative force value)

Method of Sections Application

Selecting the Section Cut

• Identify the member whose force needs to be determined, ensuring that this member intersects the section cut at some point along its length
• Make an imaginary cut through the truss, dividing it into two sections
  • The cut should pass through no more than three members whose forces are unknown, including the desired member
• Choose one of the two sections to analyze and indicate the portion of the truss that is being removed
  • It is often best to select the section that has more known forces or will result in simpler calculations

Analyzing the Section

• Draw a free body diagram of the section of the truss that remains after the cut, showing all external forces, reactions, and unknown member forces exposed by the cut
  • Represent the unknown member forces as vectors, assuming they act in the positive direction (tension) until calculations reveal otherwise
  • If the calculated force is negative, the member is in compression
• Write and solve the equilibrium equations (ΣFx = 0, ΣFy = 0, and ΣM = 0) for the section of the truss
  • If needed, take the moment about a point where the lines of action of the other unknown forces intersect to eliminate them from the equation
• Repeat the process with additional sections and cuts as necessary to determine all required member forces

Verifying Results

• Verify the results by checking that the equilibrium equations are satisfied for all sections
• Ensure that the determined forces are consistent with the properties of the truss and its supports (e.g., a member in compression should not have a positive force value)

Truss Analysis Method Selection

Advantages of the Method of Joints

• The method of joints is most efficient for trusses with a high degree of static indeterminacy, where there are many more members than equilibrium equations available
  • It is systematic and can be used to find forces in all members
• The method of joints is preferred when the external reactions can be easily determined and when there are joints with only two unknown member forces
• It is also useful for verifying the results of the method of sections

Advantages of the Method of Sections

• The method of sections is most effective for determining the force in a specific member or a small group of members without solving for all member forces
  • It is particularly useful when the force in a member cannot be found by inspection or the method of joints alone
• The method of sections is advantageous when the external reactions are not easily determined, as it can be applied without knowing all reaction forces
• It is also helpful for analyzing compound trusses or trusses with complex geometries (e.g., a Pratt truss or a Howe truss)

Combining Methods

• In some cases, a combination of both methods may be necessary to fully analyze a truss
  • The method of joints can be used to determine as many member forces as possible, and the method of sections can be employed to find the remaining unknown forces
• The choice of method may also depend on the specific problem requirements, such as the need to find the force in a particular member (e.g., a critical tension member) or to verify the results using a different approach