Calculation of Internal Moments in Beams and Frames
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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.

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    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.

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