Credit for image https://i.ytimg.com
Unit 4: Analysis of Beam and Plane Truss(9 hours)
-4.1 Introduction to Beam and Truss: Beams and trusses are common structural elements used in civil engineering and construction to support loads and distribute forces within a structure. They are essential components in buildings, bridges, and other structures, providing stability and structural integrity. Let's explore the basic concepts and characteristics of beams and trusses: 1. Beams: Beams are horizontal or inclined structural members that are designed to carry and transfer loads from one point to another. They are typically subjected to bending forces, which cause them to deform. Beams are characterized by their length, shape, and cross-sectional properties. Key features of beams: - They resist bending moments and shear forces. - Beams can be made of various materials, such as steel, reinforced concrete, or timber. - Common types of beams include simple beams, continuous beams, and cantilever beams. - Beams can have different cross-sectional shapes, such as rectangular, I-shaped (e.g., I-beams), T-shaped, or circular. - Beams are designed based on factors like the magnitude and distribution of loads, the span length, and material properties. Applications of beams: - Building structures: Beams support the weight of floors, roofs, and walls. - Bridges: Beams are used as girders to span gaps and carry vehicular and pedestrian loads. - Industrial structures: Beams are employed in manufacturing plants, warehouses, and other industrial facilities. 2. Trusses: Trusses are assemblies of interconnected members arranged in triangular patterns to create stable frameworks. They are designed to efficiently carry loads and provide rigidity to a structure. Trusses are primarily subjected to axial forces in their members. Key features of trusses: - Trusses consist of straight members connected by joints. - The triangular arrangement of members provides inherent stability and load-bearing capacity. - Trusses can be made of various materials, such as steel, timber, or aluminum. - Types of trusses include Pratt trusses, Howe trusses, Warren trusses, and many more, each with its own configuration and load-bearing characteristics. - Trusses are designed based on factors like the span length, applied loads, and desired structural efficiency. Applications of trusses: - Roof structures: Trusses are commonly used to support the roof of buildings, providing stability and spanning large distances. - Bridges: Trusses are used as the main load-bearing elements in bridge construction, enabling long spans with minimal materials. - Tower structures: Trusses form the framework of towers, such as transmission towers and observation towers, providing stability and strength. Both beams and trusses play crucial roles in structural engineering, offering different solutions for load-bearing and structural stability requirements. The selection of the appropriate type depends on factors like span length, load conditions, structural efficiency, and material availability. Engineers analyze and design beams and trusses using principles of structural mechanics to ensure their safety and performance within a given project.
4.2 Types of supports, loads and standard symbols
-4.2 Types of Supports, Loads, and Standard Symbols: In structural engineering, various types of supports and loads are encountered when analyzing and designing structures. Understanding these types and their standard symbols is crucial for effectively communicating and representing structural systems. Let's explore the common types of supports, loads, and their standard symbols: 1. Types of Supports: a) Roller Support: A roller support allows a structure to move horizontally along the support surface, preventing horizontal translation but allowing vertical movement. It provides a reaction force perpendicular to the surface. Symbol: A roller support is typically represented by a circle or a triangle with one side open, indicating freedom of horizontal movement. b) Pin Support: A pin support, also known as a hinged support, allows a structure to rotate around the pin axis. It restricts both horizontal and vertical movements and provides a reaction force perpendicular to the surface. Symbol: A pin support is represented by a small circle or dot at the connection point. c) Fixed Support: A fixed support, also called a built-in support, completely restrains both translation and rotation of a structure. It provides reaction forces and moments in all directions. Symbol: A fixed support is represented by a triangle, typically shaded, indicating full constraint. d) Guided Support: A guided support restricts translation in one direction but allows movement in the perpendicular direction. It provides a reaction force perpendicular to the surface and prevents movement along the guided direction. Symbol: A guided support is represented by a vertical line or double lines perpendicular to the surface, indicating constraint in one direction. 2. Types of Loads: a) Point Load: A point load is a concentrated load applied at a specific location or point on a structure. Symbol: A point load is represented by an arrow pointing downward or upward, indicating the direction and magnitude of the load. b) Uniformly Distributed Load: A uniformly distributed load is spread evenly over a specific length or area of a structure. Symbol: A uniformly distributed load is represented by a series of equally spaced arrows pointing downward or upward, indicating the direction and magnitude of the load. c) Concentrated Moment: A concentrated moment is a twisting force applied at a specific location on a structure, causing rotation. Symbol: A concentrated moment is represented by a curved arrow indicating the direction and magnitude of the moment. d) Torsional Load: A torsional load applies twisting forces to a structure, typically causing shear stress and deformation. Symbol: A torsional load is represented by two curved arrows in opposite directions, indicating the twisting effect. e) Thermal Load: A thermal load represents the effects of temperature changes on a structure, causing expansion or contraction. Symbol: A thermal load is represented by the letter "ΔT," indicating the change in temperature. Standard symbols for supports and loads may vary slightly based on regional or organizational conventions, but the basic representations described above are commonly used in structural engineering. These symbols allow engineers and designers to communicate the support conditions and loadings effectively, aiding in the analysis and design of structures.
4.3 Types of beams based on support condition and determinancy
-Based on the support conditions and determinancy, beams can be classified into different types in structural analysis and design. The classification is based on how the beams are supported and whether they have enough support conditions to determine their internal forces and deformations. Here are the common types of beams based on support condition and determinancy: 1. Simply Supported Beam: A simply supported beam is one of the most common types of beams. It is supported at both ends by roller supports or pin supports. The beam is free to rotate at the supports and undergoes vertical deflection under loading. A simply supported beam has two support reactions and is statically determinate. 2. Cantilever Beam: A cantilever beam is a beam that is fixed at one end and free at the other end. It is supported horizontally at one end, allowing it to resist vertical loads and moments. The fixed end prevents rotation and horizontal movement. A cantilever beam has one support reaction and is statically determinate. 3. Overhanging Beam: An overhanging beam is a beam that extends beyond its supports on one or both ends. It can have different support conditions at each end, such as simply supported, fixed, or a combination of both. An overhanging beam may have multiple support reactions and can be statically determinate or indeterminate, depending on the support conditions. 4. Continuous Beam: A continuous beam is a beam that spans over three or more supports. It is supported at multiple points along its length, creating a continuous load path. Continuous beams can have a combination of simply supported, fixed, or other support conditions. They have multiple support reactions and are usually statically indeterminate. 5. Fixed Beam: A fixed beam, also known as a restrained beam, is supported at both ends with fixed supports. The fixed supports prevent rotation, horizontal movement, and vertical deflection at the supports. A fixed beam has two support reactions and is statically determinate. 6. Propped Cantilever Beam: A propped cantilever beam is a beam that has one end fixed and the other end supported by a prop or a roller support. The propped end may have additional vertical support or moment restraint. A propped cantilever beam has one support reaction and is statically determinate. These are some of the common types of beams based on support condition and determinancy. Understanding the type of beam is essential for structural analysis and design, as it helps determine the support reactions, internal forces, and deformations of the beam under different loading conditions.
4.4 Relationship between load, shear force and bending moment
-In structural analysis, there is a relationship between the applied load, shear force, and bending moment along a beam or structural member. Understanding this relationship is crucial for analyzing and designing beams to ensure structural stability and integrity. Here's an explanation of the relationship between load, shear force, and bending moment: 1. Load: The load refers to the external forces or loads applied to a beam, such as point loads, distributed loads, or concentrated moments. Loads can include forces acting vertically, horizontally, or at an angle on the beam. 2. Shear Force: The shear force is the internal force that acts parallel to the cross-sectional area of a beam. It represents the tendency of the beam to shear or break along a particular section. The shear force varies along the length of the beam due to the applied loads and their distribution. - Shear force due to point loads: At a section where a point load is applied, the shear force experiences an abrupt change equal to the magnitude of the applied load. - Shear force due to distributed loads: For uniformly distributed loads, the shear force varies linearly along the length of the beam. At any section, the shear force is equal to the area of the load distribution on one side of the section. 3. Bending Moment: The bending moment is the internal moment that causes a beam to bend or deform. It is the result of the distribution of loads along the beam's length. The bending moment varies along the length of the beam and is responsible for inducing stresses and deflections in the beam. - Bending moment due to point loads: At a section where a point load is applied, the bending moment experiences a discontinuity. The magnitude of the bending moment at that section depends on the distance from the load and the load itself. - Bending moment due to distributed loads: For uniformly distributed loads, the bending moment varies parabolically along the length of the beam. The maximum bending moment occurs at the center of the distributed load. Relationship between Load, Shear Force, and Bending Moment: The shear force at any section of a beam is equal to the rate of change of the bending moment at that section with respect to the distance along the beam. In other words: Shear Force = d(Bending Moment) / dx This relationship can be expressed mathematically using differential equations and integrated to determine the shear force and bending moment diagrams along the beam's length. By analyzing these diagrams, engineers can assess the structural behavior of the beam, including areas of high stress or deflection, and design appropriate measures to ensure the beam's integrity. Understanding the relationship between load, shear force, and bending moment is fundamental in structural engineering as it enables engineers to analyze and design beams to withstand applied loads and ensure structural stability and safety.
4.5 Calculation of Axial Force, Shear Force and Bending Momentfor statically determinate beams
-To calculate the axial force, shear force, and bending moment in statically determinate beams, you need to follow a systematic approach using the principles of equilibrium and structural analysis. Here's a step-by-step process for calculating these internal forces: 1. Identify the Support Conditions and Beam Type: Determine the support conditions at each end of the beam (e.g., simply supported, fixed, etc.) and classify the beam type (e.g., simply supported, cantilever, etc.). This information is crucial for determining the reactions at the supports. 2. Determine Support Reactions: Use the principles of equilibrium (sum of forces and moments equal to zero) to determine the support reactions at each support. Consider both vertical and horizontal forces and moments, if applicable. 3. Draw the Free-Body Diagram: Sketch a clear and accurate free-body diagram of the beam, showing all external loads and support reactions. This will help visualize the forces acting on the beam. 4. Cut the Beam and Isolate a Section: Select a specific section along the beam for analysis. Cut the beam at that section and consider the forces and moments acting on either side of the cut. 5. Apply Equilibrium Conditions: Apply the equilibrium equations (sum of forces and moments equal to zero) to analyze the section and determine the internal forces. a) Axial Force: For a section subjected to axial forces, sum the vertical forces to find the axial force (N) at that section. It is equal to the algebraic sum of the forces acting in the vertical direction. b) Shear Force: For a section subjected to shear forces, sum the vertical forces to find the shear force (V) at that section. It is equal to the algebraic sum of the forces acting in the vertical direction. c) Bending Moment: For a section subjected to bending moments, sum the moments about a point to find the bending moment (M) at that section. It is equal to the algebraic sum of the moments acting about the section. 6. Repeat for Additional Sections: Repeat steps 4 and 5 for different sections along the beam until you have determined the axial force, shear force, and bending moment for the entire length of the beam. 7. Plot Shear Force and Bending Moment Diagrams: Once you have calculated the shear force and bending moment at different sections, plot the shear force diagram and bending moment diagram to visualize the variation of these forces along the length of the beam. It's important to note that this process assumes statically determinate beams, which have a sufficient number of support conditions to determine the internal forces and deformations. For statically indeterminate beams, additional analysis techniques (such as the slope-deflection method or moment distribution method) are required. By following these steps, you can accurately calculate the axial force, shear force, and bending moment in statically determinate beams and effectively analyze their structural behavior.
4.6 Drawing of Axial Force Diagram, Shear Force Diagram and Bending Moment Diagram for determinate beams with relevant examples
-To illustrate the process, let's consider a simply supported beam with a point load applied at the center. We will draw the axial force diagram, shear force diagram, and bending moment diagram for this determinate beam. Example: Simply Supported Beam with Point Load at the Center Given: - Length of the beam (L): 6 meters - Point load (P): 10 kN 1. Determine Support Reactions: Since the beam is simply supported, the reactions at the supports are equal. Reaction at each support = P/2 = 10 kN / 2 = 5 kN 2. Draw the Free-Body Diagram: Sketch a free-body diagram of the beam, showing the support reactions and the point load at the center. ``` 5 kN 5 kN <----->______________________________<-----> A B ``` 3. Cut the Beam and Isolate a Section: For simplicity, let's consider a section at a distance x from support A. ``` <----->___________x_______________<-----> A B ``` 4. Apply Equilibrium Conditions: a) Axial Force: The beam is subjected to only vertical forces, so the axial force (N) at any section is zero. b) Shear Force: The shear force (V) at any section is the sum of the vertical forces acting to the left or right of that section. - For section to the left of the point load: V = -5 kN (support reaction at A) - For section to the right of the point load: V = -5 kN (support reaction at B) + 10 kN (point load) c) Bending Moment: The bending moment (M) at any section is the sum of the moments about that section. - For section to the left of the point load: M = -5 kN × x - For section to the right of the point load: M = -5 kN × (L - x) + 10 kN × (L - x/2) 5. Plot Shear Force and Bending Moment Diagrams: Using the calculated values, plot the shear force and bending moment diagrams for the entire length of the beam (from x = 0 to x = L). Shear Force Diagram: ``` -5 kN 5 kN <--------------------------> A B ``` Bending Moment Diagram: ``` -5kN x 5kN x -5 kN x + 10 kN (L-x/2) <----------------------|-------------|-------------------------------------> A P/2 B ``` The shear force diagram represents the variation of the shear force along the beam's length, while the bending moment diagram represents the variation of the bending moment. The diagrams help visualize the internal forces and moments within the beam under the given loading conditions. It's important to note that the values in the diagrams are specific to this example and will vary depending on the beam's geometry, loading conditions, and support conditions in other cases.
4.7 Analysis of member force for determinate truss by method of joints
-The method of joints is a common technique used to analyze the member forces in determinate trusses. Trusses are structural frameworks composed of interconnected members, typically in a triangular pattern, designed to carry loads primarily along their axial directions. The method of joints involves analyzing the equilibrium of individual joints to determine the forces in the truss members. Here's a step-by-step process for analyzing member forces in a determinate truss using the method of joints: 1. Identify the Truss and Support Conditions: Determine the truss structure and its support conditions (e.g., pinned, roller, or fixed supports). This information is essential for determining the reactions at the supports and simplifying the analysis. 2. Assign Joint Labels: Label each joint of the truss with a unique identifier. Start by labeling the external joints and then move towards the internal joints. 3. Analyze Joints: Begin with a joint that has only two unknown member forces. This is typically an external or a pin-connected joint. 4. Draw the Free-Body Diagram of the Joint: For the selected joint, draw a free-body diagram that includes all the forces acting on that joint. Include the known external loads, support reactions, and the unknown member forces at that joint. 5. Apply Equilibrium Conditions: Apply the principles of equilibrium (sum of forces equal to zero) to analyze the joint. Resolve the forces into their horizontal and vertical components and write the equilibrium equations for each direction. 6. Solve for Unknown Member Forces: With the equilibrium equations, solve for the unknown member forces at the joint. The forces can be tension (T) or compression (C), depending on their direction relative to the joint. 7. Move to the Next Joint: After determining the forces at one joint, move on to the next joint that has only two unknown member forces. Repeat steps 4 to 6 for this joint. 8. Continue Analyzing Joints: Proceed systematically, analyzing each joint with two unknown member forces until all the member forces in the truss are determined. It is important to maintain consistency in labeling and keep track of the forces already determined. 9. Verify the Solution: After determining all the member forces, check the equilibrium of the entire truss. Ensure that the sum of forces at each joint is zero, and the truss remains in a state of static equilibrium. By following these steps, you can systematically analyze the member forces in a determinate truss using the method of joints. This method simplifies the analysis by considering the equilibrium of individual joints and is suitable for trusses that can be treated as pin-jointed and subjected to axial forces only. For more complex trusses or those with additional constraints, other methods such as the method of sections or the method of tension coefficients may be required.
4.8 Analysis of member force for determinate truss by method of sections
-The method of sections is another commonly used technique to analyze the member forces in determinate trusses. It involves cutting the truss along a section and analyzing the equilibrium of the cut section to determine the forces in the desired members. Here's a step-by-step process for analyzing member forces in a determinate truss using the method of sections: 1. Identify the Truss and Support Conditions: Determine the truss structure and its support conditions (e.g., pinned, roller, or fixed supports). This information is essential for determining the reactions at the supports and simplifying the analysis. 2. Select a Section: Choose a section of the truss where you want to determine the member forces. The section should ideally intersect a maximum of three members at a time to simplify the analysis. 3. Cut the Truss: Cut the truss along the selected section. This will create a free body of the portion of the truss on one side of the section. 4. Draw the Free-Body Diagram of the Cut Section: Draw a free-body diagram of the portion of the truss that has been cut. Include all the forces acting on the cut members and the external loads applied to that portion of the truss. 5. Apply Equilibrium Conditions: Apply the principles of equilibrium (sum of forces equal to zero and sum of moments equal to zero) to analyze the cut section. Resolve the forces into their horizontal and vertical components and write the equilibrium equations for each direction. 6. Solve for Unknown Member Forces: With the equilibrium equations, solve for the unknown member forces in the cut section. The forces can be tension (T) or compression (C), depending on their direction relative to the section. 7. Repeat for Additional Sections: If necessary, repeat steps 2 to 6 for other sections of the truss to determine the forces in the remaining members. 8. Verify the Solution: After determining all the member forces, check the equilibrium of the entire truss. Ensure that the sum of forces at each joint is zero and the truss remains in a state of static equilibrium. By following these steps, you can analyze the member forces in a determinate truss using the method of sections. This method simplifies the analysis by considering the equilibrium of cut sections of the truss, allowing for the determination of forces in specific members. It is important to select sections strategically to minimize the number of unknown forces and simplify the calculations. For more complex trusses or those with additional constraints, other methods such as the method of joints or the method of tension coefficients may be required.
