Glossary

12.1 Normal stresses in beams

4 min read•july 30, 2024

Normal stresses in beams are crucial for understanding how structures handle loads. These stresses vary linearly from the , with maximum values at the extreme fibers. Knowing their distribution helps engineers design safer, more efficient structures.

The flexure formula is key for calculating normal stresses in beams. It relates stress to , distance from the neutral axis, and moment of inertia. Understanding this relationship is essential for analyzing beam behavior under different loading conditions.

Normal Stress in Beams

Stress Distribution

Top images from around the web for Stress Distribution

  • Studying Stress Distribution Under Steel Base Plates Using IPE Profiles. View original

    Is this image relevant?

  • Frontiers | Flexural Behavior of Reinforced Concrete Beams Strengthened by Textile Reinforced ... View original

    Is this image relevant?

  • Studying Stress Distribution Under Steel Base Plates Using IPE Profiles. View original

    Is this image relevant?

1 of 3

Top images from around the web for Stress Distribution

  • Studying Stress Distribution Under Steel Base Plates Using IPE Profiles. View original

    Is this image relevant?

  • Frontiers | Flexural Behavior of Reinforced Concrete Beams Strengthened by Textile Reinforced ... View original

    Is this image relevant?

  • Studying Stress Distribution Under Steel Base Plates Using IPE Profiles. View original

    Is this image relevant?

1 of 3
  • Normal stress in a beam varies linearly from the neutral axis to the extreme fibers of the cross-section
  • The distribution of normal stress in a beam depends on the magnitude and direction of the applied bending moment
  • Compressive stresses develop on the side of the beam where the bending moment causes the fibers to shorten (top side in a with a downward load)
  • Tensile stresses occur on the side where the fibers elongate (bottom side in a simply supported beam with a downward load)
  • The magnitude of normal stress at any point in the beam cross-section is directly proportional to its distance from the neutral axis

Maximum Stress Location

  • The maximum normal stresses occur at the extreme fibers of the beam cross-section, which are furthest from the neutral axis
  • In a rectangular beam, the maximum normal stresses occur at the top and bottom surfaces
  • For an I-beam, the maximum normal stresses are located at the top and bottom flanges
  • The maximum tensile and compressive stresses have equal magnitudes but opposite signs
  • Identifying the location of maximum normal stress is crucial for determining the critical points in a beam design

Neutral Axis Location

Symmetric Cross-Sections

  • The neutral axis is the line in the beam cross-section where the normal stress is zero
  • For symmetric cross-sections, the neutral axis passes through the centroid of the cross-section
  • Examples of symmetric cross-sections include rectangular, circular, and I-shaped beams
  • In a rectangular beam, the neutral axis is located at the geometric center of the cross-section
  • For an I-beam, the neutral axis coincides with the horizontal centerline of the web

Unsymmetric Cross-Sections

  • In unsymmetric cross-sections, the neutral axis location can be determined using the first moment of area concept
  • Examples of unsymmetric cross-sections include T-shaped and L-shaped beams
  • The first moment of area is calculated by multiplying each by its distance from an arbitrary axis and summing the results
  • The neutral axis location is found by setting the first moment of area equal to zero and solving for the distance from the arbitrary axis
  • For a T-beam, the neutral axis is typically located below the geometric center due to the larger area of the flange compared to the web
  • In an L-shaped beam, the neutral axis is located closer to the corner with the larger moment of area

Maximum Normal Stress Calculation

Flexure Formula

  • The maximum normal stress in a beam can be calculated using the flexure formula: σ=My/I\sigma = My / I
  • σ\sigma represents the normal stress at a given point
  • MM represents the bending moment at the cross-section
  • yy represents the distance from the neutral axis to the point of interest
  • II represents the moment of inertia of the cross-section about the neutral axis
  • To find the maximum normal stress, substitute the maximum distance from the neutral axis (ymaxy_{max}) and the corresponding bending moment (MM) into the flexure formula

Moment of Inertia

  • The moment of inertia (II) is a geometric property that depends on the shape and dimensions of the beam cross-section
  • For common cross-sectional shapes, the moment of inertia can be found using standard formulas
  • Example formulas for moment of inertia:
    • Rectangle: I=bh312I = \frac{bh^3}{12}
    • Circle: I=πr44I = \frac{\pi r^4}{4}
    • I-shape: I=bh312−b1h1312I = \frac{bh^3}{12} - \frac{b_1h_1^3}{12}
  • For irregular cross-sections, the moment of inertia can be calculated using the parallel axis theorem or by integration
  • The parallel axis theorem states that the moment of inertia about any axis parallel to the centroidal axis is equal to the moment of inertia about the centroidal axis plus the product of the area and the square of the distance between the axes

Bending Moment and Normal Stress

Proportional Relationship

  • The bending moment in a beam is directly proportional to the normal stress at any given point in the cross-section
  • As the bending moment increases, the normal stress at any point in the beam cross-section also increases proportionally
  • Doubling the bending moment will result in a doubling of the normal stress at any given point
  • The proportional relationship between bending moment and normal stress is described by the flexure formula (σ=My/I\sigma = My / I)

Stress Distribution along the Beam

  • The direction of the bending moment (positive or negative) determines the nature of the normal stress (tensile or compressive) at a given point in the beam cross-section
  • A positive bending moment induces at the bottom fibers and at the top fibers of the beam
  • A negative bending moment induces compressive stress at the bottom fibers and tensile stress at the top fibers of the beam
  • The distribution of bending moment along the length of the beam influences the distribution of normal stress in the beam
  • Points of maximum bending moment in a beam correspond to the locations of maximum normal stress, which are critical for design considerations and
  • In a simply supported beam with a uniformly distributed load, the maximum bending moment and normal stress occur at the midspan of the beam
  • For a with a concentrated load at the free end, the maximum bending moment and normal stress are located at the fixed support

Key Terms to Review (17)

Bending Moment: A bending moment is a measure of the internal moment that induces bending in a beam or structural element when external loads are applied. It reflects how much a beam wants to bend in response to these loads, which is crucial in understanding how structures respond to forces and maintaining their integrity.
Cantilever beam: A cantilever beam is a structural element that is anchored at one end while the other end extends freely without support. This configuration creates a moment about the fixed end when loads are applied to the free end, which leads to specific shear and bending moment characteristics crucial for understanding beam behavior under various loads.
Compressive Stress: Compressive stress is the internal resistance offered by a material when it is subjected to an axial load that tends to compress or shorten it. This stress is crucial for understanding how materials respond under load and is directly related to other fundamental concepts like stress-strain relationships, particularly Hooke's law, which describes the linear elastic behavior of materials.
Cross-sectional area: The cross-sectional area is the area of a particular slice of a three-dimensional object taken perpendicular to its longest dimension. In the context of normal stresses in beams, this area is crucial because it directly influences how loads applied to the beam are distributed and ultimately affects the stress experienced by the material. Understanding this concept helps in evaluating how different shapes and sizes of beams can bear loads without failing.
Equilibrium: Equilibrium refers to a state in which all forces and moments acting on a system are balanced, resulting in no net force or acceleration. This balance is crucial in analyzing structures and mechanical systems, as it ensures stability and prevents motion. Understanding equilibrium allows for the application of various methods to solve problems related to forces, energy, and material behavior.
Failure analysis: Failure analysis is the process of investigating the reasons behind the failure of a material or component to determine its causes and to prevent future occurrences. This involves assessing how materials respond under stress and strain, understanding their limits through stress-strain diagrams, and applying concepts like Hooke's Law to predict and analyze failure points. It plays a crucial role in ensuring the safety and reliability of structures and materials by identifying critical stress levels that lead to failure.
Flexural analysis: Flexural analysis is the process of determining how beams respond to bending forces, focusing on the normal stresses developed within the material. This involves examining the internal moment, shear forces, and resulting stress distributions to ensure that the beam can withstand applied loads without failing. Understanding flexural analysis is crucial for designing safe and efficient structures that can handle various loading conditions.
M/i = σ/y: The equation $$\frac{m}{I} = \frac{\sigma}{y}$$ relates bending moment, moment of inertia, normal stress, and distance from the neutral axis in the context of beams. It highlights how the internal stress in a beam due to bending is directly proportional to the distance from the neutral axis and inversely proportional to the beam's moment of inertia. This relationship is fundamental in understanding how beams resist bending forces and helps in designing structural components that can withstand various loads.
Neutral Axis: The neutral axis is an imaginary line within a beam or structural member where the material experiences no longitudinal stress during bending. It is significant because it separates the areas of compression from those of tension, helping to identify how a beam will deform under load and where stress concentrations will occur.
Shear and Moment Diagrams: Shear and moment diagrams are graphical representations used in structural engineering to illustrate how shear forces and bending moments vary along a beam. These diagrams are crucial for understanding how external loads affect internal stresses, allowing engineers to determine safe and efficient designs for structural elements. By visualizing shear forces and bending moments, these diagrams help in identifying critical points where maximum stresses occur, which is essential for analyzing normal stresses in beams and formulating the elastic curve equation.
Simply Supported Beam: A simply supported beam is a type of structural member that is supported at both ends, allowing it to freely rotate and translate without any moment resistance at the supports. This basic configuration is crucial in analyzing how loads affect the beam, as it simplifies calculations for shear forces, bending moments, normal stresses, shear stresses, combined loading scenarios, and deflection.
Structural design: Structural design is the process of designing the physical components of a structure to ensure it can safely withstand the loads and forces it will encounter during its use. This involves analyzing materials, loads, and structural behavior to create safe and efficient structures that meet both functional and aesthetic requirements. Understanding how normal stresses in beams play a crucial role in this process is essential, as beams are fundamental elements that support loads and transfer forces within structures.
Superposition Principle: The superposition principle states that in a linear system, the response caused by multiple loads acting simultaneously is equal to the sum of the responses that would be caused by each load acting independently. This concept is essential for analyzing structures under various forces, making it easier to understand how different loads interact and affect overall behavior.
Tensile stress: Tensile stress is defined as the internal force per unit area experienced by a material when it is subjected to a pulling or stretching force. This concept is crucial for understanding how materials behave under loads, as tensile stress directly relates to the material's ability to withstand deformation without failure. By analyzing tensile stress, one can connect its implications to various material behaviors, deformation characteristics, and the overall structural integrity of axially loaded members and beams.
Yield Strength: Yield strength is the stress at which a material begins to deform plastically, meaning it will not return to its original shape after the load is removed. This concept is crucial as it helps determine the limits of material performance under various loading conditions, affecting design and safety in engineering applications.
Young's Modulus: Young's Modulus is a measure of the stiffness of a material, defined as the ratio of stress (force per unit area) to strain (deformation per unit length) within the linear elastic region of the material. It helps quantify how much a material will deform under an applied load, playing a crucial role in determining both elastic and plastic behavior, as well as in analyzing stress and strain in various structural applications.
σ = p/a: The equation σ = p/a defines normal stress, where σ (sigma) represents stress, p is the applied force or load, and a is the cross-sectional area. This fundamental relationship shows how stress is distributed across an area when a force is applied, which is crucial for understanding material behavior under load. By analyzing normal stress, engineers can determine how materials will react when subjected to different types of forces, helping in design and safety assessments.
© 2024 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.
Back
Practice QuizGlossary
Practice QuizGlossary
Next guide