Análisis Estructural (cálculo manual) /Aplicando los métodos.

Method of Moments Area in Structural Analysis

Overview of the Method

• The theory of the method of area moments has been discussed, focusing on two key theorems: one for calculating differences in slopes and another for determining tangential deviation. This deviation does not always align with deflection.

Application to Real Projects

• The course aims to apply theoretical knowledge to real projects through practical exercises. Participants have access to a project plan shared via Google Drive, which will be used for analysis.

Specific Project Details

• The focus is on analyzing a specific slab section within a larger structural model. A cutout will be made from this slab for detailed examination.

Structural Elements Description

• The analyzed beam is simply supported at one end and cantilevered at the other, serving primarily as support for closing off part of the slab rather than bearing significant loads itself. It is described as having minimal dimensions (15 cm width).

Load Calculations

• The objective is to determine ultimate load based on amplified loads, which leads to required resistance calculations involving bending moments and shear forces relevant to slabs. 1.4 times dead load and 1.7 times live load are specified according to Peruvian standards (Norma AC 318).

Load Distribution Methodology

Load Combinations According to Standards

• For load combinations, Peruvian norms suggest using 1.2 times dead load plus 1.6 times live load; comparisons with international standards like ACI 318 are also mentioned but not elaborated upon here.

Tributary Width Consideration

• A tributary width of 40 cm is established for distributing loads across the slab's length, calculated based on structural geometry (T-shaped cross-section). This width plays a crucial role in determining how distributed loads affect overall stability and performance of the structure.

Moment Area Method Insights

Understanding Moment Area Method Applications

• The moment area method serves as a tool for calculating slope differences and tangential deviations in beams under static conditions; it can also be adapted for hyperstatic beams using alternative methods such as Cross's method later discussed in class materials available online.

Hyperstatic Beam Analysis

• Future lessons will cover hyperstatic beam analysis up to ten spans using spreadsheet tools provided by the instructor, emphasizing practical applications alongside theoretical understanding in structural engineering contexts.

Dead Load Determination Process

Components of Dead Load Calculation

• Dead load includes self-weight and additional factors such as finished flooring and partition walls; these elements must be accounted for accurately when assessing total weight supported by structural components like slabs.

Additional Considerations

• There are discussions about how wall placements may influence loading scenarios; however, they can also be treated uniformly over square meter areas based on standard recommendations found in literature or university guidelines.

This structured approach ensures clarity while providing essential insights into each segment discussed during the lecture series related to structural analysis methodologies.

Load Calculations for Structural Design

Determining Thickness and Weight

• The initial thickness determined for the structure is 0.20 meters (20 centimeters), based on pre-dimensioning.

• According to Peruvian standards, a slab of this thickness has a self-weight of 300 kgf/m², which is crucial for load calculations.

Specific Weight Considerations

• The specific weight of concrete can also be used in calculations, measured in kgf/m³; however, additional analysis may be required if brick weight needs to be included.

• A minimum value of 300 kgf/m² is established by the standard for a slab thickness of 20 cm.

Area and Load Distribution

• For finished flooring, an additional load of 100 kgf/m² is considered, multiplied by the tributary width (0.40 m), resulting in a total load distribution clarity.

• It’s emphasized that loads should be distributed over lengths rather than areas to ensure accurate structural assessments.

Wall Loads and Dead Loads

• Wall loads are also calculated at 100 kgf/m²; multiplying this by the tributary width yields another load contribution of 40 kgf/m.

• The total dead load calculated so far amounts to 200 kgf/m after summing various contributions without amplification factors applied yet.

Live Load Considerations

• Live loads depend on usage as per norm N020; residential buildings typically have a live load requirement of 200 kgf/m².

• This live load must also be converted into linear terms by multiplying it with the tributary width (0.40 m), yielding an effective live load of 80 kgf per linear meter.

Amplification and Ultimate Load Calculation

• The combined loading will undergo amplification using factors such as 1.4 for dead loads and other relevant coefficients for live loads to determine ultimate loading conditions.

• Ultimate loads are critical as they inform moment diagrams necessary for structural design, leading to what is termed "required resistance."

Moment Calculations and Structural Analysis

• Required resistance or ultimate resistance must exceed design moments derived from both dead and live loads through appropriate amplification methods.

• For statically determinate structures like beams or slabs, static equations suffice for reaction calculations without needing advanced methods like moment area techniques initially.

Reaction Calculations

• Reactions at supports can be computed using static equilibrium principles; understanding these reactions is foundational before delving into more complex analyses later in the course.

Structural Analysis and Moment Calculations

Overview of Structural Analysis Process

• The discussion begins with an overview of structural analysis, focusing on calculating forces, including moment diagrams and shear forces.

• Emphasis is placed on determining resistance based on moments and shear, specifically using the method of areas for moment calculations.

Equilibrium and Reaction Forces

• The importance of equilibrium in structural analysis is highlighted, particularly when calculating reactions at supports.

• A review of static principles is conducted to ensure a solid understanding before proceeding with calculations.

Load Calculations

• The calculation process for ultimate load is initiated, leading to a value of 416 kgf/m for distributed loads.

• Conversion from distributed load to equivalent point load involves multiplying by the total length (3.15 m), resulting in necessary values for further calculations.

Moment Calculation Steps

• The next step involves calculating moments using established equations; this includes determining the reaction force needed for accurate moment evaluations.

• A summation approach is used to equate upward and downward forces, confirming calculated values align with static principles.

Segmenting the Structure into Tramos (Sections)

• Two segments (tramos) are identified within the structure for detailed moment analysis: one from 0 to 3.15 m and another working from right to left.

• Each segment's moments are defined mathematically, allowing for precise calculations based on applied loads.

Finalizing Moment Equations

• The first segment's moment equation incorporates both reaction forces and distributed loads, yielding a negative moment due to loading conditions.

• For the second segment, similar calculations are performed while noting that moments will be zero at fixed support points due to boundary conditions.

Understanding Structural Moments and Deflections

Moment Resistance and Calculations

• The discussion begins with the concept of angular displacement, noting that there is no moment resistance due to a lack of support, leading to zero values at the extremes.

• A mobile support is introduced, which generates a resistant moment. The speaker emphasizes that this moment typically does not yield negative results.

• The calculation of the moment at point D is explained using two segments; for segment 2, it’s calculated as -208 multiplied by 0.93, resulting in a negative value.

• The computed moment value is clarified as being in kilograms-force per meter (kgf·m), yielding -193.44 kgf·m for segment 2.

• A graphical representation of moments is discussed, indicating positive and negative moments based on structural deformation.

Maximum Moment Calculation

• The speaker mentions the need to find the maximum positive moment for potential design purposes but notes that this will be deferred for now.

• To calculate the maximum positive moment, one must derive the equation from segment one and set it equal to zero to find critical points.

• Once the position of maximum moment (e.g., x = 22 meters) is identified, it can be substituted back into the original equation to determine its value.

• Emphasis on using calculus techniques such as derivatives highlights their importance in structural analysis for finding maxima or minima in moments.

Methodology for Deflection Calculations

• The method of calculating deflections through differences in angles at various points (like point D and others along a cantilever beam).

• It’s noted that while calculating deflections geometrically can be complex due to parabolic shapes, it remains essential for accurate structural assessments.

Diagrams and Areas

• Discussion shifts towards creating diagrams representing moments by parts rather than attempting complex area calculations directly from parabolas.

• Simplifying calculations by focusing on areas under curves helps streamline understanding; however, care must be taken with equations involved.

Practical Application

• A practical example involving real project applications illustrates how theoretical concepts are applied in engineering contexts.

• Introduction of a segmented approach to drawing moment diagrams allows better visualization and understanding of load distributions across structures.

Understanding Support Moments in Structural Analysis

Key Concepts of Support and Moments

• The discussion begins with the calculation of a negative moment at a support, emphasizing the importance of understanding how loads affect structural integrity.

• Different types of supports are introduced, including fixed supports and movable supports, highlighting their roles in providing resistance to moments.

• The speaker illustrates how to distribute loads across beams, ensuring stability by using fixed supports effectively throughout the structure.

• Emphasis is placed on maintaining static and stable conditions in structures; mobile supports alone can lead to instability without proper reinforcement from fixed supports.

• The concept of reactions at points is discussed, comparing them to point loads that need careful consideration when analyzing beam behavior.

Load Distribution and Moment Diagrams

• A specific example is given where a distributed load is analyzed alongside point loads, stressing the significance of considering only relevant forces for accurate calculations.

• The necessity of removing certain loads from analysis when they do not contribute to the overall moment due to existing constraints like fixed supports is highlighted.

• The purpose behind creating moment diagrams is explained: simplifying calculations for areas and centroids associated with different loading scenarios.

• It’s noted that calculating areas under curves (like parabolas representing load distributions) can be complex but essential for accurate structural analysis.

• Techniques for transforming complicated diagrams into simpler forms are discussed, aiming for ease in area calculation without resorting to integration.

Simplifying Complex Structures

• Visual aids such as color coding are suggested as methods to differentiate between various components within moment diagrams for clarity during analysis.

• Formulas related to areas under parabolic curves are mentioned as tools that facilitate easier calculations when dealing with known shapes in structural design.

• Challenges arise when dealing with non-standard shapes or vertices that complicate area calculations; strategies must adapt accordingly based on shape characteristics.

• A focus on simplifying analyses through known figures helps avoid complications associated with integration methods typically used in more complex scenarios.

• The presence of multiple point loads introduces additional complexity into moment equations; thus, careful consideration must be taken when formulating these equations.

This structured approach provides an insightful overview while allowing easy navigation through key concepts discussed regarding support moments and load distribution in structural analysis.

Diagrams of Moments in Structural Analysis

Understanding Moment Diagrams

• The discussion begins with the introduction of a moment diagram, emphasizing its equivalence to previous diagrams. The speaker notes that this new configuration simplifies the process.

• The area of the new moment diagram is highlighted as being simpler than earlier versions, with specific dimensions (3.15 and 0.93) affecting the overall outcome.

• Importance is placed on maintaining clarity in drawings for accurate representation, indicating that well-done sketches prevent confusion during analysis.

Calculating Areas and Moments

• The speaker encourages participants to refer to their notes for formulas related to areas and distances from centroids, which are crucial for calculating moments accurately.

• A positive moment is discussed due to upward forces; the speaker illustrates how triangular shapes can be used in calculations, stressing the importance of understanding triangle properties.

Verifying Results

• Familiar geometric figures are referenced as tools for verification; participants are encouraged to calculate moments at various points and ensure consistency across results.

• A specific example is provided where a calculated moment at point D equals -179.90, reinforcing the need for accuracy in summing moments with correct signs.

Drawing Deformations

• Transitioning into drawing deformations based on previously established moment diagrams, it’s noted that determining deflections requires careful consideration of tangential deviations or slope differences.

• The method involves removing loads temporarily to visualize structural deformation accurately while ensuring clarity in representation.

Analyzing Concavity Changes

• Observations about concavity changes are made; these shifts indicate transitions between positive and negative moments within beam structures.

• A detailed explanation follows regarding how these changes affect beam behavior under load, highlighting critical points where concavity alters due to varying moments.

• Emphasis is placed on ensuring support points remain fixed during deformation analysis since they restrict vertical movement, guiding accurate sketching of potential deformations.

This structured approach provides a comprehensive overview of key concepts discussed in relation to moment diagrams and structural analysis techniques.

Understanding Deformation in Structural Analysis

Analyzing Deformation and Moments

• The discussion begins with the concept of deformation, highlighting that certain parts may not align as expected due to variations in length, leading to potential downward bending.

• A visual representation is created to illustrate the deformed shape coinciding with the moment diagram, emphasizing the challenges of accurately drawing exaggerated deformations.

• The objective is set to calculate specific angles (e.g., tita 6) and deflections at designated points, indicating a focus on understanding angular relationships in structural elements.

Calculating Tangents and Angles

• A practical tip is provided: when dealing with cantilever sections, it’s crucial to draw tangents at support points for accurate calculations. This sets up a methodical approach for further analysis.

• The tangent line's slope is determined based on concavity from the deformation diagram, which will aid in identifying critical angles related to structural behavior.

Identifying Key Angles and Deviations

• An angle corresponding to tita is identified at point C, establishing a reference for subsequent calculations regarding deflection and deviation from original positions.

• The discussion transitions into calculating tangential deviation from point C relative to its tangent line. It emphasizes that both deformed and undeformed states must be considered for accuracy.

Utilizing Trigonometry for Angle Calculation

• The need arises to calculate an angle using known lengths within a right triangle formed by the deformation. This highlights the application of trigonometric principles in structural analysis.

• Specific dimensions are referenced (e.g., 3.15 meters), reinforcing how established theories guide these calculations while ensuring clarity about what measurements are being used.

Applying Elasticity Principles

• Once tangential deviation is calculated, it can be linked back through trigonometric functions (like tangent), allowing for precise angle determination relevant to structural integrity assessments.

• Emphasis is placed on using second theorem applications rather than first theorem approaches since they provide more straightforward solutions for specific angles rather than differences between them.

Finalizing Calculations with Area Considerations

• A distinction between methods indicates that while first theorem applications deal with slopes or differences, second theorem applications focus on direct measurements like tangential deviations derived from geometric considerations.

• To compute tangential deviation effectively, an area calculation between two points (point 6 and point D), multiplied by distance factors divided by elasticity modulus and inertia values, becomes essential in deriving accurate results.

This structured approach ensures clarity in understanding complex concepts related to deformation analysis within structural engineering contexts.

Calculating Areas and Distances in Geometry

Defining Areas and Points

• The discussion begins with the identification of three areas labeled as Area 1, Area 2, and Area 3. The speaker emphasizes the need to calculate distances from specific points.

• The focus shifts to determining the distance from point C to point D, highlighting the importance of these measurements in geometric calculations.

Calculating Distances and Areas

• A method for calculating distances within a triangle is introduced, specifically using two-thirds of the base measurement (3.15 meters).

• The area of a parabola is discussed, with variables defined as x-ray and height (h). The formula for area is presented as b cdot h/3 .

Understanding Triangular Areas

• The calculation involves determining heights based on angles; specifically, it mentions that heights are derived from a right angle.

• An equation for Area 1 is established: textArea 1 = 3 cdot 3.15/4 = 2.36 , indicating an important step in area calculation.

Significance of Positive and Negative Areas

• It’s noted that some areas yield negative values due to their positions; thus, they must be marked accordingly while positive areas remain unchanged.

• A formula for calculating moments based on base and height is introduced: textMoment = fractextBase cdot textHeight2 .

Verifying Calculations

• Verification methods are discussed where calculated moments should match when taken from different sides (left vs. right).

• Emphasis on ensuring consistency between left-side and right-side moment calculations reinforces accuracy in geometry.

Finalizing Area Calculations

• Further calculations lead to refined values for areas based on previously established formulas.

• Results are summarized with specific numerical outputs for each area calculated, emphasizing precision in mathematical expressions.

Analyzing Tangential Deviations

• Discussion concludes with analyzing tangential deviations relative to absolute values; understanding signs indicates whether points lie above or below tangent lines.