 CIV 301 - THEORY OF STRUCTURES )3( )3( -301 ) 3( 202 ( ( ) ) .) ( CIV 301 - Theory of Structures )3( )3 Credit Hours( Prerequisite CIV 202 Influence lines for statically determinate structures )beams, frames, arches, trusses(, deflection and slope for beams )double integration , conjugate beam, moment area(, deflection and slope using virtual work

)beams, frames, trusses(. Giza Higher Institute for Eng. & Tech. - Dr. M Abdel- CIV 301 - THEORY OF STRUCTURES )3( Deformations of Statically Determinate Structures Influence Lines Giza Higher Institute for Eng. & Tech. - Dr. M Abdel- CIV 301 - THEORY OF STRUCTURES )3( Deformations of Statically Determinate Structures

Large deflection of a beam under two concentrated loads Giza Higher Institute for Eng. & Tech. - Dr. M Abdel- CIV 301 - THEORY OF STRUCTURES )3( Deformations of Statically Determinate Structures Deflection of a frame under a concentrated load Giza Higher Institute for Eng. & Tech. - Dr. M Abdel- CIV 301 - THEORY OF STRUCTURES )3( Methods Used for Calculating Deformations: 1. Double Integration Method

2. Moment-Area Method 3. Conjugate Beam Method 4. Virtual Work Method Giza Higher Institute for Eng. & Tech. - Dr. M Abdel- CIV 301 - THEORY OF STRUCTURES )3( Double Integration Method The double integration (also called direct integration) method is a powerful tool in solving deflection and slope of a beam at any point because of the ability to get the equation of the elastic curve. In calculus, the radius of curvature (or R) of a curve y = f(x) is given by: 2 3/ 2 1 dy / dx

d 2 y / dx 2 1 1 EI '' 2 M d y / dx y 2 If EI is constant, the equation may be written as: EI y '' M

and M C1 x C2 EI y ' EI M C1 & EI y Giza Higher Institute for Eng. & Tech. - Dr. M Abdel- CIV 301 - THEORY OF STRUCTURES )3( Double Integration Method - Boundary Conditions: at x = 0 at x = L y=0

y=0 Giza Higher Institute for Eng. & Tech. - Dr. M Abdel- CIV 301 - THEORY OF STRUCTURES )3( Double Integration Method Example: Using the double integration method, determine the maximum deflection for the shown loaded cantilever. Take the origin at the free end Solution: Giza Higher Institute for Eng. & Tech. - Dr. M Abdel-

CIV 301 - THEORY OF STRUCTURES )3( Double Integration Method Example: Using the double integration method, determine the maximum deflection for the shown simply supported beam carrying a uniformly distributed load wo. Solution: ymax 5wo L4 384 EI

d max 5wo L4 384 EI which is equal to 0.625 dmax for equivalent concentrated load at the middle P = wo L Giza Higher Institute for Eng. & Tech. - Dr. M Abdel- CIV 301 - THEORY OF STRUCTURES )3( Double Integration Method Macaulay's Method If the loading conditions change along the span of beam, there is corresponding change in moment equation. This requires that a separate moment equation be written between

each change of load point and that double integration be made for each such moment equation. Evaluation of the constants introduced by each integration can become very difficult. These difficulties can be avoided by writing general moment equation for entire length of the beam in spite of the discontinuity of loading. Note that when either of the brackets (x-2) and (x-3) becomes negative means it is out of the region of boundary condition and not taken in the calculations (i.e.; put it equal zero). Giza Higher Institute for Eng. & Tech. - Dr. M Abdel- CIV 301 - THEORY OF STRUCTURES )3( Double Integration Method Macaulay's Method

For M, raise the bracket (x-a) to the power zero, i.e. M(x-a)0, this is because the term M(x-a)0 should have the units of moment. EIy '' M R A x |I M ( x a ) 0 EIy ' RA 12 x 2 |I M ( x a)1 C1 M 500 x |I 400 400 ( x 1) 2 |II ( x 4) 2 |III 1300( x 6) 2 2

EIy R A 16 x 3 |I 1 2 M ( x a ) 2 C1 x C2 Giza Higher Institute for Eng. & Tech. - Dr. M Abdel- CIV 301 - THEORY OF STRUCTURES )3( Double Integration Method Macaulay's Method ya from left = ya from right

y'a from left = ya from right ya from left = ya from right Giza Higher Institute for Eng. & Tech. - Dr. M Abdel- CIV 301 - THEORY OF STRUCTURES )3( Double Integration Method - Macaulay's Method Example: Determine the value of EIy at the middle of span between the supports for the beam loaded as shown. Solution:

Giza Higher Institute for Eng. & Tech. - Dr. M Abdel- CIV 301 - THEORY OF STRUCTURES )3( Thank You Very Welcome for Questions and Feedback cadkad.com [email protected] 01001212803 Facebook: Mohamed Abdel-Kader Giza Higher Institute for Eng. & Tech. - Dr. M Abdel-