Calculation of Internal Moments in Beams and Frames
30 Questions
0 Views

Choose a study mode

Play Quiz
Study Flashcards
Spaced Repetition
Chat to lesson

Podcast

Play an AI-generated podcast conversation about this lesson

Questions and Answers

What is required to sum the effects on all the elements along the beam?

  • Differentiation
  • Multiplication
  • Division
  • Integration (correct)
  • In the context of beam deflections, what method requires applying a virtual unit load?

  • Distributed load method
  • Integration using tables
  • Method of virtual work (correct)
  • Tabular method
  • How is the integral 1 mM dx determined when using the tabular method?

  • Matching moment diagrams (correct)
  • Applying differentiation
  • Choosing random x coordinates
  • Using a single integration
  • What do the definite integrals on the right side of the equations represent in beam analysis?

    <p>Amount of virtual strain energy</p> Signup and view all the answers

    Why can't a single integration be performed across the beam's entire length in certain cases?

    <p>Discontinuous distributed load</p> Signup and view all the answers

    What is used when a solution for displacement requires several integrations in beam deflection analysis?

    <p>Tabular method</p> Signup and view all the answers

    What is the formula provided for calculating the change in length of a member due to a change in temperature?

    <p>dL = a dT L</p> Signup and view all the answers

    What happens to the displacement of a selected truss joint due to a temperature change, according to Equation 8-13?

    <p>It can be determined</p> Signup and view all the answers

    How should the units be handled when applying the virtual-work equation?

    <p>Units should cancel from both sides</p> Signup and view all the answers

    For a truss member, how is an increase in length due to a temperature increase represented?

    <p>Positive</p> Signup and view all the answers

    What happens if a negative value is obtained when applying 1 # ∆ = Σn dL?

    <p>The displacement ∆ is opposite to the unit load</p> Signup and view all the answers

    What should be retained when substituting terms into the equation of virtual work?

    <p>The algebraic sign for each corresponding n and N forces</p> Signup and view all the answers

    What is the procedure used to determine the displacement and/or slope at a point on the elastic curve of a beam or frame?

    <p>Method of virtual work</p> Signup and view all the answers

    How is the internal moment 'M' caused by real loads represented?

    <p>Positive</p> Signup and view all the answers

    In the context of the text, what is the purpose of placing a unit couple moment at the direction of the desired displacement?

    <p>To determine the slope</p> Signup and view all the answers

    What is used to calculate the internal moment 'm or mu' as a function of each x coordinate?

    <p>Real loads</p> Signup and view all the answers

    What is the displacement of point B in meters based on the given calculation results?

    <p>$0.8544$ m</p> Signup and view all the answers

    Which equation represents the Virtual-Work Equation mentioned in the text?

    <p>$∆B = 1 kN # ∆ B = 1 kN # ∆ B = Virtual-Work Equation$</p> Signup and view all the answers

    What must be included for a more complete accountability of strain energy in a structure?

    <p>Shear, axial force, and torsion</p> Signup and view all the answers

    What is the modulus of elasticity represented by in the provided equations?

    <p>E</p> Signup and view all the answers

    In the L-shaped frame problem, what method is recommended for determining the horizontal displacement of end C?

    <p>Method of virtual work</p> Signup and view all the answers

    What does the internal moment M depend on in a beam or frame?

    <p>External force P</p> Signup and view all the answers

    For Prob. 8–57, what method is advised for determining the vertical displacement at point A?

    <p>Castigliano’s theorem</p> Signup and view all the answers

    How is the slope u at a point in a beam or frame determined?

    <p>By finding the partial derivative of internal moment M with respect to an external couple moment M′</p> Signup and view all the answers

    What is the recommended approach for solving Prob. 8–61 and finding the vertical deflection at point C?

    <p>Castigliano’s theorem</p> Signup and view all the answers

    What is the moment of inertia I calculated about in the provided equations?

    <p>The neutral axis</p> Signup and view all the answers

    In solving Prob. 8–59, what technique is recommended for calculating the slope at point A and the vertical displacement at point B?

    <p>Castigliano’s theorem</p> Signup and view all the answers

    What methodology is suggested for determining the horizontal displacement at point C in Prob. 8–55?

    <p>Method of virtual work</p> Signup and view all the answers

    Why is it generally easier to differentiate prior to integration when determining the slope at a point?

    <p>To make the process more manageable and less complicated</p> Signup and view all the answers

    For Prob. 8–56, which technique is recommended for solving the problem?

    <p>Castigliano’s theorem</p> Signup and view all the answers

    Study Notes

    Summation of Effects

    • To sum the effects on all elements along a beam, a virtual unit load is applied.

    Virtual Work Method

    • The virtual work method involves applying a virtual unit load at the point and direction of the desired displacement.
    • The internal moment caused by real loads is represented by 'M' which is a function of the x-coordinate.
    • Internal moment caused by a unit load is represented by 'm or mu' which is also a function of the x-coordinate.
    • The virtual work equation is expressed as: 1 # ∆ = Σn dL, where:
      • ∆ is the displacement at the point of application of the virtual load.
      • n is the internal normal force in the member due to real loads.
      • dL is the change in length of the member due to real loads.
      • Σ indicates the summation over all members of the structure.
    • The change in length of a member due to a change in temperature is calculated as: ∆T * α * L, where:
      • ∆T is the change in temperature.
      • α is the coefficient of thermal expansion for the material.
      • L is the original length of the member.
    • The displacement of a selected truss joint due to a temperature change is determined by Equation 8-13.
    • When applying the virtual-work equation, units must be handled consistently.
    • An increase in length of a truss member due to a temperature increase is represented by a positive value.
    • Obtaining a negative value when applying 1 # ∆ = Σn dL indicates a shortening in the member and a negative displacement at the point of interest.
    • When substituting terms into the equation of virtual work, the sign of the virtual force must be retained.

    Beam Deflection Analysis

    • The displacement and/or slope at a point on the elastic curve of a beam or frame can be determined using the virtual work method.
    • Definite integrals on the right side of the equations represent the work done by the internal forces due to real loads.
    • The tabular method is used to determine the integral 1 mM dx, where:
      • 1 is a constant factor.
      • M represents the internal moment due to real loads.
      • m is the internal moment due to the virtual unit load.
      • dx represents the infinitesimal change in length along the beam.

    Limitations of Single Integration

    • In certain cases, a single integration cannot be performed across the beam's entire length because the equation for the internal moment may change depending on the section of the beam.

    Multiple Integrations

    • When a solution for displacement requires several integrations in beam deflection analysis, the method of superposition is used, where the contributions from each section of the beam are summed together.

    Solving Problems

    • For Prob. 8–57, the vertical displacement at point A can be determined using the virtual work method.
    • For Prob. 8–61, the vertical deflection at point C can be solved by applying the virtual work method and using the method of superposition when necessary.
    • For Prob. 8–59, the slope at point A and the vertical displacement at point B can be calculated using the virtual work method and integrating along the length of the beam.
    • For Prob. 8–55, the horizontal displacement at point C can be determined using the virtual work method and considering the internal forces and moments in the L-shaped frame.
    • For Prob. 8–56, the problem can be solved using the virtual work method and considering the internal forces and moments in the beam due to the applied loads.
    • The formula provided for calculating the change in length of a member due to a change in temperature is used in determining the displacement of a selected truss joint due to a temperature change.

    Key Concepts

    • The modulus of elasticity (E) in the provided equations represents the material's resistance to deformation.
    • The moment of inertia (I) is calculated about the neutral axis of the beam and reflects the beam's resistance to bending.
    • The internal moment (M) in a beam or frame depends on the applied loads and the geometry of the structure.

    Additional Notes

    • To have a more complete accountability of strain energy in a structure, the strain energy due to shear forces should be considered.
    • The internal moment 'M' caused by real loads is defined as the bending moment at a section of the beam.
    • Placing a unit couple moment at the direction of the desired displacement helps to determine the displacement or rotation at that point.
    • It is easier to differentiate prior to integration when determining the slope at a point because differentiation reduces the order of the equation, making integration simpler.
    • The slope 'u' at a point in a beam or frame is determined by integrating the bending moment equation.
    • The horizontal displacement of end C in the L-shaped frame problem can be determined using the virtual work method and considering the internal forces and moments in the frame.

    Studying That Suits You

    Use AI to generate personalized quizzes and flashcards to suit your learning preferences.

    Quiz Team

    Description

    This quiz focuses on calculating internal moments caused by real loads in beams and frames. It involves determining internal moments at different x coordinates using the conventional positive direction assumption. The quiz may include scenarios with virtual loads and the removal of real loads for specific beam or frame segments.

    More Like This

    Shear and Moment Diagrams in Beam Analysis
    10 questions
    Internal Anatomy of Crayfish
    10 questions
    Internal Parasites: Hookworm Flashcards
    10 questions
    Internal Audit Chapter 1 Flashcards
    13 questions
    Use Quizgecko on...
    Browser
    Browser