CIV 301 - THEORY OF STRUCTURES (3) Giza Higher Institute for Eng. & Tech. - Dr. M Abdel-Kader Methods Used for Calculating Deformations: 1. Double Integration Method 2. Moment-Area Method 3. Conjugate Beam Method 4. Virtual Work Method

CIV 301 - THEORY OF STRUCTURES (3) Giza Higher Institute for Eng. & Tech. - Dr. M Abdel-Kader Moment-Area Method It is based on moment-area theorems. It is a semi graphical method of dealing with deflection of beams subjected to bending. It is based on a geometrical interpretation of definite integrals. From the figure, since the curvature is small it can assume that ds = dx, hence,

CIV 301 - THEORY OF STRUCTURES (3) Giza Higher Institute for Eng. & Tech. - Dr. M Abdel-Kader Moment-Area Method Theorem I: If A and B are two points on a beam the change in slope between the tangents drawn to the elastic curve at points A and B is equal to the area of the moment diagram between these two points divided by the value of EI (the flexural rigidity).

CIV 301 - THEORY OF STRUCTURES (3) Giza Higher Institute for Eng. & Tech. - Dr. M Abdel-Kader Moment-Area Method

CIV 301 - THEORY OF STRUCTURES (3) Giza Higher Institute for Eng. & Tech. - Dr. M Abdel-Kader Moment-Area Method Theorem II: If A and B are two points on a beam the deviation of point B relative to the tangent drawn to the elastic curve at point A, in a direction perpendicular to the original position of the beam, tA/B is equal to the first moment of the area of the bending moment diagram between A and B about the ordinate through B divided by the value of EI.

CIV 301 - THEORY OF STRUCTURES (3) Giza Higher Institute for Eng. & Tech. - Dr. M Abdel-Kader Moment-Area Method

CIV 301 - THEORY OF STRUCTURES (3) Giza Higher Institute for Eng. & Tech. - Dr. M Abdel-Kader Moment-Area Method Sign convention The deviation at any point is positive if the point lies above the tangent, negative if the point is below the tangent. Measured from left tangent, if θ is anticlockwise, the change of slope (θBA =θB - θA) is positive, negative if θ is clockwise. Generally the tangential deviation is not equal to the beam deflection. However, in cantilever:

CIV 301 - THEORY OF STRUCTURES (3) Giza Higher Institute for Eng. & Tech. - Dr. M Abdel-Kader Moment-Area Method Example: Using the moment-area method, determine the slope and deflection at the free end. Solution: The B.M.D is as shown. Slope at A (A) Since the slope at B (B) is equal to zero, the change in slope between the tangents of the elastic curve at points A and B (BA) is

CIV 301 - THEORY OF STRUCTURES (3) Giza Higher Institute for Eng. & Tech. - Dr. M Abdel-Kader Moment-Area Method Example: Using the moment-area method, determine the slope and deflection at the free end. Solution: Deflection at A (dA) The deflection at A is equal to the deviation of point A relative to the tangent of the elastic curve at point B, tA/B. Applying the second moment-area theorem, then

CIV 301 - THEORY OF STRUCTURES (3) Giza Higher Institute for Eng. & Tech. - Dr. M Abdel-Kader Moment-Area Method Example: Using the moment-area method, determine the slope and deflection at the free end. Solution: The B.M.D is as shown. Slope at B (B) Since the slope at A (A) is equal to zero, the change in slope between the tangents of the elastic curve at points A and B (BA) is

CIV 301 - THEORY OF STRUCTURES (3) Giza Higher Institute for Eng. & Tech. - Dr. M Abdel-Kader Moment-Area Method Example: Using the moment-area method, determine the slope and deflection at the free end. Solution: Deflection at B (dB) The deflection at B is equal to the deviation of point B relative to the tangent of the elastic curve at point A, tA/B. Applying the second moment-area theorem, then

CIV 301 - THEORY OF STRUCTURES (3) Giza Higher Institute for Eng. & Tech. - Dr. M Abdel-Kader Moment-Area Method Deflections in simply supported beams: Mid-span Deflection If the simple beam is symmetrically loaded, the maximum deflection will occur at the mid-span. Therefore, the tangent drawn to the elastic curve at the point of maximum deflection is horizontal. It means that the deviation from the support to the horizontal tangent is equal to the maximum deflection.

CIV 301 - THEORY OF STRUCTURES (3) Giza Higher Institute for Eng. & Tech. - Dr. M Abdel-Kader Moment-Area Method Example: Using the moment-area method, calculate the mid-span deflection and end slope for the shown simply supported beam. E = 210 GPa and I = 834 cm4 Solution: Deflection at mid-span E = 210 GPa = 210 ×106 kN/m2 I = 834 cm4 = 834×10-8 m4  EI = (2.1×1011)( 8.34×10-6) = 1.7514×103 kN.m2 The deflection at the mid-span is equal to the deviation of the point A above the tangent to the elastic curve at B, then

CIV 301 - THEORY OF STRUCTURES (3) Giza Higher Institute for Eng. & Tech. - Dr. M Abdel-Kader Moment-Area Method Example: Using the moment-area method, calculate the mid-span deflection and end slope for the shown simply supported beam. E = 210 GPa and I = 834 cm4 Solution: Slope at the end A Since the slope at B (qB) is equal to zero, the change in slope between the tangents of the elastic curve at points A and B (qBA) is equal to

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