1. CIV 301 - THEORY OF STRUCTURES (3)
Giza Higher Institute for Eng. & Tech. - Dr. M Abdel-Kader
همد نظرية الإنشاءات (3) (3 ساعات معتمدة)
المتطلب السابق همد 202
خطوط تأثير القوي للمنشآت المحددة استاتيكاً (كمرات – أطر – جمالونات – أقواس)، حساب الترخيم وميل المماس في الكمرات (بالتكامل المزدوج – مساحة منحنى عزم الانحناء – الكمرة المرافقة )، حساب الترخيم وميل المماس بالشغل الافتراضي (كمرات، اطر، جمالونات).
CIV 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).

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

3. CIV 301 - THEORY OF STRUCTURES (3)
Giza Higher Institute for Eng. & Tech. - Dr. M Abdel-Kader
Deformations of Statically Determinate Structures
Large deflection of a beam under two concentrated loads

4. CIV 301 - THEORY OF STRUCTURES (3)
Giza Higher Institute for Eng. & Tech. - Dr. M Abdel-Kader
Deformations of Statically Determinate Structures
Deflection of a frame under a concentrated load

5. 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

6. CIV 301 - THEORY OF STRUCTURES (3)
Giza Higher Institute for Eng. & Tech. - Dr. M Abdel-Kader
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:
If EI is constant, the equation may be written as:
and

7. CIV 301 - THEORY OF STRUCTURES (3)
Giza Higher Institute for Eng. & Tech. - Dr. M Abdel-Kader
Double Integration Method
- Boundary Conditions:
at x = y =
at x = L y =

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

9. CIV 301 - THEORY OF STRUCTURES (3)
Giza Higher Institute for Eng. & Tech. - Dr. M Abdel-Kader
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:
which is equal to dmax for equivalent concentrated load at the middle P = wo L

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

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

12. CIV 301 - THEORY OF STRUCTURES (3)
Giza Higher Institute for Eng. & Tech. - Dr. M Abdel-Kader
Double Integration Method
Macaulay's Method
ya from left = ya from right
y'a from left = y’a from right
ya from left = ya from right

13. CIV 301 - THEORY OF STRUCTURES (3)
Giza Higher Institute for Eng. & Tech. - Dr. M Abdel-Kader
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: 

14. Thank You CIV 301 - THEORY OF STRUCTURES (3)
Giza Higher Institute for Eng. & Tech. - Dr. M Abdel-Kader
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