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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

(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:

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: Study Material, Lecturing Notes, Assignment, Reference, Wiki description explanation, brief detail

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