Deflection of Beams
METHODS FOR SLOPE AND DEFLECTION AT A SECTION
1. Double Integration method
After integrating end conditions are applied for determination of constants of integration.
2. Macaulay’s method
3. Moment Area Method
Note: In moment area method continuity of slope is assumed so it is not applied when internal hinges are present.
PROPERTIES OF PLANE AREAS
Notation : A = area
= distances of centroid C
Ix, Iy = moments of inertia with respect to the x and y axes respectively.
Ixy = moments of inertia with respect to the x and y axes
Ip = Ix + Iy = polar moment of inertia
IBB = moment of inertia with respect to axis B-B.
(a) Rectangle (Origin of axes at centroid)
A = bh
Ixy = 0
(b) Rectangle (Origin of axes at corner)
(c) Triangle (Origin of axes at centroid).
(d) Triangle (Origin of axes at vertex).
(e) Isosceles triangle (Origin of axes at centroid).
(f) Trapezoid (Origin of axes at centroid).
(g) Circle (Origin of axes at center)
(h) Quarter circle (Origin of axes at center of circle)
4. Conjugate Beam Method
Important points :
(i) A stable and statically determinate real beam will have a conjugate beam which is also stable and statically determinate.
(ii) A unstable real beam will have statically indeterminate conjugate beam, hence if a conjugate beam is found to be statically indeterminate, it is concluded that the real beam is unstable and further analysis is not appropriate.
(iii) Statically indeterminate real beam will have unstable conjugate beam hence its conjugate load must be such that it maintains equilibrium.
Summary of End Conditions of a Conjugate Beam
5. Method of Superposition It is suitable for cantilevers containing concentrated loads & concentrated moments. This method can also be used for non prismatic bars i.e. varying EI(Flexural rigidity). 6. Strain Energy method (Castigliano’s theorem) It is suitable for cantilevers and beams having varying EI or varying depth of beam. This methods is very useful in case of determinate frames and arches. This can also be used when internal hinge is provided.
7. Unit load method
8. Dummy load method Method used to find deflection of truss joints (perfect frames) :
(i) Unit load method (Maxwell’s method)
(ii) Castigliano’s theorem (Strain energy method)
(iii) Graphical method (Williot Mohr diagram) (This method is used for trusses only & cannot be used for beams)