Nothing Special   »   [go: up one dir, main page]
Scribd Logo
Upload

Welcome to Scribd!

  • 1 views

    Lecture No. 32 - Rotation and Deflection by Double Integration Method

    Uploaded by

    saftybla
    Useful

    Copyright:

    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd

    Lecture No. 32 - Rotation and Deflection by Double Integration Method

    Uploaded by

    saftybla
    1 views14 pages
    Useful

    Original Title

    Lecture No. 32_Rotation and Deflection by Double Integration Method

    Copyright

    © © All Rights Reserved

    Available Formats

    PDF, TXT or read online from Scribd

    Did you find this document useful?

    Is this content inappropriate?

    Useful

    Copyright:

    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd
    Download as pdf or txt
    1 views14 pages

    Lecture No. 32 - Rotation and Deflection by Double Integration Method

    Uploaded by

    saftybla
    Useful

    Copyright:

    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd
    Download as pdf or txt
    of 14

    MIRPUR UNIVERSITY OF SCIENCE AND TECHNOLOGY (MUST), MIRPUR

    DEPARMENT OF CIVIL ENGINEERING


    THEORY OF STRUCTURES-I (TOS-I)
    CIV-2441

    Lecture No. 32: ROTATION AND DEFLECTION BY DOUBLE INTEGRATION METHOD

    Engr. Israr ul Haq


    Assistant Professor

    Date: May 10, 2020


    Double Integration Method
    𝑑𝑣
    Slope 𝜃 ≈ 𝑡𝑎𝑛𝜃 = 𝑑𝑥 = (𝑀 𝐸𝐼)𝑑𝑥

    Deflection 𝑣=𝑓 𝑥 = (𝑀 𝐸𝐼)𝑑𝑥

    Sign Convention

    When applying above equations, it is important to use the


    proper sign for Mas established by the sign convention that
    was used in the derivation of this equation, Fig. 8–9a.

    THEORY OF STRUCTURES-I 3
    Double Integration Method
    Boundary and Continuity Conditions

    • The constants of integration are determined by evaluating the functions for slope or
    displacement at a particular point on the beam where the value of the function is known.

    • These values are called boundary conditions.

    • For example, if the beam is supported by a roller or pin, then it is required that the
    displacement be zero at these points.

    • Also, at a fixed support the slope and displacement are both zero.

    • If a single x coordinate cannot be used to express the equation for the beam’s slope or the
    elastic curve, then continuity conditions must be used to evaluate some of the integration
    constants.

    THEORY OF STRUCTURES-I 4
    Double Integration Method
    Procedure for Analysis

    • The following procedure provides a method for determining the slope and deflection of a
    beam (or shaft) using the method of double integration.

    • It should be realized that this method is suitable only for elastic deflections for which the
    beam’s slope is very small.

    • Furthermore, the method considers only deflections due to bending.

    • Additional deflection due to shear generally represents only a few percent of the bending
    deflection, and so it is usually neglected in engineering practice.

    THEORY OF STRUCTURES-I 5
    Double Integration Method
    Procedure for Analysis

    Elastic Curve

    • Draw an exaggerated view of the beam’s elastic curve. Recall that points of zero slope and
    zero displacement occur at a fixed support, and zero displacement occurs at pin and roller
    supports.

    • Establish the x and v coordinate axes. The x axis must be parallel to the undeflected beam
    and its origin at the left side of the beam, with a positive direction to the right.

    • If several discontinuous loads are present, establish x coordinates that are valid for each
    region of the beam between the discontinuities.

    • In all cases, the associated positive v axis should be directed upward.

    THEORY OF STRUCTURES-I 6
    Double Integration Method
    Procedure for Analysis

    Load or Moment Function

    • For each region in which there is an x coordinate, express the internal moment M as a
    function of x.

    • Always assume that M acts in the positive direction when applying the equation of moment
    equilibrium to determine M = f (x).

    THEORY OF STRUCTURES-I 7
    Double Integration Method
    Procedure for Analysis

    Slope and Elastic Curve

    • Provided EI is constant, apply the moment equation EI d2v/dx2 = M(x), which requires two
    integrations. For each integration it is important to include a constant of integration. The constants
    are determined using the boundary conditions for the supports and the continuity conditions that
    apply to slope and displacement at points where two functions meet.

    • Once the integration constants are determined and substituted back into the slope and deflection
    equations, the slope and displacement at specific points on the elastic curve can be determined.
    The numerical values obtained can be checked graphically by comparing them with the sketch of
    the elastic curve.

    • Positive values for slope are counterclockwise and positive displacement is upward.

    THEORY OF STRUCTURES-I 8
    Double Integration Method
    Example 8.4

    The cantilevered beam shown in Fig. 8–12a is subjected to a couple moment Mo at its end. Determine
    the equation of the elastic curve. EI is constant.

    THEORY OF STRUCTURES-I 9
    Double Integration Method
    Solution

    THEORY OF STRUCTURES-I 10
    Double Integration Method
    Solution

    THEORY OF STRUCTURES-I 11
    Double Integration Method
    Solution

    THEORY OF STRUCTURES-I 12
    Double Integration Method
    Solution

    THEORY OF STRUCTURES-I 13
    THANKS

    You might also like