deflection of beam

In engineering, deflection is the degree to which a structural element is displaced under a load. It may refer to an angle or a distance.

The deflection distance of a member under a load is directly related to the slope of the deflected shape of the member under that load, and can be calculated by integrating the function that mathematically describes the slope of the member under that load. Deflection can be calculated by standard formula (will only give the deflection of common beam configurations and load cases at discrete locations), or by methods such as virtual work, direct integration, Castigliano's method, Macaulay's method or the direct stiffness method, amongst others. The deflection of beam elements is usually calculated on the basis of the Euler–Bernoulli beam equation while that of a plate or shell element is calculated using plate or shell theory.

1. 1 . GENERAL THEORY  When a beam bends it takes up various shapes such as that illustrated in figure The shape may be superimposed on an x – y graph with the origin at the left end of the beam (before it is loaded). At any distance x meters from the left end, the beam will have a deflection y and a gradient or slope dy/dx and it is these that we are concerned with in this tutorial. We have already examined the equation relating bending moment and radius of curvature in a beam, namely M is the bending moment. I is the second moment of area about the centroid. E is the modulus of elasticity and R is the radius of curvature.  

2. Rearranging we have Figure 1 illustrates the radius of curvature which is defined as the radius of a circle that has a tangent the same as the point on the x-y graph. Mathematically it can be shown that any curve plotted on x - y graph has a radius of curvature of defined as =  

3. I n b ea m s , R is v e r y l a r g e a nd the e qu a t ion m a y be s implif ie d w ithout lo s s of acc u r a c y to = = M = EI T he produ c t E I is ca l l e d the f l e xur a l s tiff n e s s of t he b ea m. I n ord e r to s ol v e the s lope ( d y /dx) or t h e d e f le c tion ( y ) a t a n y point on the b ea m, a n e qu a tion for M in t e r m s of po s ition x mu s t be s ub s titut e d into e qu a t ion (1 A ). W e w ill now e x a mine this for t he 4 s t a nd a rd ca s e s .  

4. A cantilever beam with a point load at the end. A cantilever beam with a uniformly distributed load.  A simply supported beam with a point load at the middle. A simply supported beam with a uniformly distributed load.

5. C AN T I L E V ER W I TH P O I N T LO A D A T F R EE E N D D e f lec t ion a t fr e e e nd y= Slope at free end θ =  

6. C AN T I L E V ER W I TH A U NI F O R M LY DIS T RI B U T E D LO AD D e f lec t ion a t fr e e e nd y = Slope at free end θ =  

7. Cantilever Beam – eccentric load P at any point D e f lec t ion a t fr e e e nd Y = Slope at free end θ =  

8. Cantilever Beam – Couple moment M at the free end D e f lec t ion a t fr e e e nd Y = Slope at free end θ =  

9. S I MP LY SU PP O R TED B E A M W I TH P O IN T LO A D I N M I D D L E D e f lec t ion a t fr e e e nd y= Slope at free end θ =  

10. S I MP LY SU PP O R TED BE A M W I TH A U N I F O R M LY DIS T R I B U TED LO A D D e f lec t ion a t fr e e e nd y = Slope at free end θ =  

11. Beam Simply Supported at Ends – Couple moment M at the right end Slope at free end = = D e f lec t ion a t fr e e e nd y =  

12. Gandhinagar Institute of Technology : Department of Civil Engineering 14 THANK YOU