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Chapter: Mechanical - Strength of Materials - Deflection of Beams

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Solved Problems: Deflection of Beams

Mechanical - Strength of Materials - Deflection of Beams


Problem –1:

 

Determine the deflection of a given beam at the point loads. Take I = 64x10-4 mm4 & its Young’s   modulusN/mm(E).

 



 

Problem –2:

 

 

A steel cantilever beam of 6m long carries 2 point loads 15KN at the free end and 25KN at the distance of 2.5m from the free end. To determine the slope at free end & also deflection at free end I = 1.3x108mm4. E = 2x105 N/mm2

Solution:             

Given:                 

Length (l)    =       6m

loads (w1)   =       25KN

(w2)  =       15KN

I        =       1.3x108 mm4

          =       1.3x10-4m4

E       =       2x105 N/mm2

          =       2x108 KN/m2

 

Bending moment calculation:

Bending moment at C = 0                          

Bending moment at B = -(15x2.5) = - 37.5 KNm

Bending moment at A = -(15x6) –(25x3.5) = -177.5KNm      

To find Bending moment Area:                                    

Area of section (1)         =       ½ (bh)                           

a1      =       ½ (2.5x37.5)         =       46.875 m2  

Area of section (2)         =       lb                         

a2      =       3.5x37.5     =       131.25m2   

Area of section (3)         =       ½(bh)                            

          a3      =       ½(3.5x140) =       245m2              

Total bending moment area

A       =       a1 + a2 + a3                                     

          =       46.8 + 131.25 + 245      = 423.125m2       

To find the slope at free end:

According to moment area



 


 

Problem –3:

 

Determine the deflection under point load.

 

E = 2x105 KN/m2

 

I = 1x10-4 m4 .   Using moment area method.

 

Solution:

 

To find support reactions:

 

Taking moment about A,

 

 

 

RB x 4 –(10x 3) –10                =       0                                   

          4RB   =       40                                           

          RB     =       4                                             

          RA + RB =   20                                           

          RA     =       10                                           

RA     =       10KN                                                         

RB     =       10KN                                                         

Bending moment calculation:                                                 

Bending moment at        B       =       0                                   

Bending moment at        D       =       (10x1) =      10KNm               

Bending moment at        C       =       (10x3) - (10x2)     =       10KNm

Bending moment at        A       =       (10x4) - (10x3) –(10x1) = 0   

To find area of bending moment:                                                     

A1      =       ½(bh)                                               

          =       ½(1)x10      =       5m2                      

          A2      =                 1/2(bh)        =       5m2  

          A3      =                 lxb     = 10x2        =       20m2

To find slope at point load:

 




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