Chapter 3: Analysis of Statically Determinate Structures

This chapter delves into the analysis of structures that can be analyzed solely using the equations of static equilibrium. The focus is on determining support reactions, internal forces, and member forces without the need for additional compatibility equations or advanced methods. Here, I will provide a comprehensive breakdown and in-depth explanation of the chapter, covering the following key topics:

Analysis of Beams & Trusses

1. Introduction to Statically Determinate Structures
2. Analysis of Beams
3. Analysis of Trusses
4. Method of Joints for Trusses
5. Method of Sections for Trusses
6. Analysis of Frames and Machines
7. Properties of Statically Determinate Structures
8. Internal Loadings in Structural Members
9. Shear and Moment Diagrams
10. Applications and Examples

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1. Introduction to Statically Determinate Structures

Statically determinate structures are those in which all internal forces (such as bending moments, shear forces, and axial forces) and support reactions can be determined solely using the three equations of equilibrium:

• Sum of all horizontal forces, 
  $\Sigma F_x = 0$
• Sum of all vertical forces, 
  $\Sigma F_y = 0$
• Sum of all moments about any point, 
  $\Sigma M = 0$

For a structure to be statically determinate, the number of unknown reactions and internal forces must equal the number of available equilibrium equations. This condition ensures that the structure can be fully analyzed without requiring compatibility equations or deformation considerations. Common examples of statically determinate structures include simple beams, simple trusses, and some types of frames.

2. Analysis of Beams

Beams are horizontal structural members that support loads perpendicular to their longitudinal axis. The analysis of beams involves determining the reactions at the supports and the internal shear forces and bending moments at various points along the length of the beam.

Types of Beams:

• Simply Supported Beam: A beam with pin and roller supports that allow rotation but restrict translation.
• Cantilever Beam: A beam that is fixed at one end and free at the other.
• Overhanging Beam: A beam that extends beyond its support on one or both ends.
• Continuous Beam: A beam that spans over more than two supports.

Steps to Analyze Beams:

1. Free-Body Diagram (FBD): Draw the FBD of the entire beam, including all applied loads (point loads, distributed loads, and moments) and support reactions.
2. Support Reactions: Use equilibrium equations (
   $\Sigma F_x = 0$$\Sigma F_y = 0$$\Sigma M = 0$
3. Internal Forces: Calculate internal shear forces and bending moments at critical points along the beam. This may involve breaking the beam into sections and analyzing each section separately.

Common Loading Scenarios:

• Concentrated Point Load: A single force acting at a specific point.
• Uniformly Distributed Load (UDL): A load spread evenly along a length of the beam.
• Uniformly Varying Load (UVL): A load that varies linearly along the length of the beam.

3. Analysis of Trusses

A truss is a structure composed of members connected at their ends to form a rigid framework. Trusses are typically used in bridges, roofs, and towers. The primary assumption in truss analysis is that members are connected by pin joints, and loads are applied only at the joints, causing the members to experience axial forces (either tension or compression) without bending.

Types of Trusses:

• Simple Trusses: Composed of basic triangular units that form a stable framework.
• Compound Trusses: Formed by joining two or more simple trusses together.
• Complex Trusses: Have more members and joints than simple or compound trusses and often require more advanced analysis methods.

4. Method of Joints for Trusses

The method of joints is a technique used to find the internal forces in each member of a truss.

Steps to Use the Method of Joints:

1. Calculate Support Reactions: Use equilibrium equations for the entire truss to determine the external reactions at the supports.
2. Isolate a Joint: Start with a joint that has at most two unknown member forces. Draw the FBD of the joint, showing all forces acting on it, including the axial forces in the members.
3. Apply Equilibrium Equations: Use 
   $\Sigma F_x = 0$$\Sigma F_y = 0$
4. Move to Adjacent Joints: Repeat the process for the next joint with known forces, moving systematically through the entire truss until all member forces are determined.

5. Method of Sections for Trusses

The method of sections is another powerful tool for truss analysis. It involves cutting the truss into sections to directly solve for forces in specific members, rather than solving joint by joint.

Steps to Use the Method of Sections:

1. Calculate Support Reactions: Determine the reactions at the supports using equilibrium equations.
2. Cut Through the Truss: Make a "cut" through the truss that passes through no more than three unknown members whose forces need to be determined.
3. Draw the FBD of One Section: Isolate one side of the cut truss and draw the FBD, including all external forces and the forces in the cut members.
4. Apply Equilibrium Equations: Use 
   $\Sigma F_x = 0$$\Sigma F_y = 0$$\Sigma M = 0$

6. Analysis of Frames and Machines

Frames and machines consist of beams, columns, and other members connected together. They are capable of carrying loads that produce bending moments, shear forces, and axial forces.

Frames:

• Rigid Frames: Have rigid connections that can resist moments, leading to bending in the members.
• Pin-Connected Frames: Have members connected with pins, which means the members only carry axial forces.

Machines:

• Structures designed to transmit and alter forces. Unlike trusses and frames, machines often have moving parts.

Analysis Procedure:

1. Determine Support Reactions: Analyze the entire structure to calculate the support reactions.
2. Analyze Each Member Separately: Use equilibrium equations on individual members to determine the internal forces and moments.
3. Check for Equilibrium: Ensure that the entire frame or machine satisfies the conditions for static equilibrium.

7. Properties of Statically Determinate Structures

Statically determinate structures have certain advantages:

• Predictable Behavior: Internal forces and moments can be determined using equilibrium alone.
• Simpler Analysis: Easier to analyze compared to statically indeterminate structures.
• Material Economy: Since the analysis is straightforward, the design can be optimized efficiently.

However, they also have limitations:

• Lack of Redundancy: If a member fails, the entire structure may collapse.
• Sensitivity to Settlements: They cannot accommodate differential settlements without developing significant stresses.

8. Internal Loadings in Structural Members

The analysis of internal loadings involves determining the axial force, shear force, and bending moment at any point along the length of a member. This is crucial for designing structural components that can safely withstand the applied loads.

Procedure to Determine Internal Loadings:

1. Section the Member: Cut the member at the point of interest and draw the FBD of one side.
2. Apply Equilibrium Equations: Use 
   $\Sigma F_x = 0$$\Sigma F_y = 0$$\Sigma M = 0$
3. Construct Shear and Moment Diagrams: These diagrams graphically represent the variation of shear force and bending moment along the length of the beam.

9. Shear and Moment Diagrams

Shear and moment diagrams are essential tools in structural analysis. They help visualize how shear forces and bending moments vary along the length of a beam.

Steps to Draw Shear and Moment Diagrams:

1. Calculate Support Reactions: Start by finding the reactions at the supports.
2. Cut the Beam at Various Sections: Analyze sections of the beam between points where loads are applied or where the loading condition changes.
3. Plot Shear Forces: For each section, calculate the shear force and plot its value. The diagram typically starts from zero at a free end or a known reaction.
4. Plot Bending Moments: Calculate the bending moment for each section and plot its value. The moment diagram typically starts from zero at a free end or a hinge.

Key Points:

• Sign Convention: A standard sign convention must be used for shear forces and bending moments to maintain consistency.
• Relationship between Load, Shear, and Moment:
  • The slope of the shear diagram at any point is equal to the intensity of the distributed load.
  • The slope of the moment diagram at any point is equal to the shear force at that point.
  • Points of zero shear force correspond to maximum or minimum bending moments.

10. Applications and Examples

Chapter 3 provides numerous examples to reinforce the concepts covered. These examples typically involve real-world scenarios, such as analyzing:

• A simply supported beam with various loading conditions.
• A truss bridge with multiple point loads and the determination of member forces.
• A frame with both vertical and horizontal loads, where the internal bending moments, shear forces, and axial forces need to be calculated.

Each example is presented with step-by-step solutions that guide the reader through the process of structural analysis, including:

1. Drawing free-body diagrams.
2. Applying equilibrium equations.
3. Calculating internal forces.
4. Constructing shear and moment diagrams.
5. Checking the results for accuracy and consistency.

Conclusion

Chapter 3 of "Structural Analysis" by R.C. Hibbeler is a comprehensive guide to analyzing statically determinate structures, including beams, trusses, frames, and machines. It provides the foundational tools and methods necessary for understanding and solving real-world structural problems. Mastering the concepts and techniques covered in this chapter is essential for students and professionals in civil and structural engineering, as they form the basis for more advanced topics in structural analysis and design.