a 1m long cantilevered beam of rectangular cross section carries a uniform load

Question

a 1m long cantilevered beam of rectangular cross section carries a uniform load of (w = 15 kN/m). The design specification calls for a 5-mm maximum deflection of the end of the beam. The beam is to be constructed of fir (E = 13 GPa). By analysis, determine at least 5 combinations of beam height h and beam width b that meet the specidication. What design conclusions can you draw about the influence of beam height and width on the maximum deflection? is the deflection more sensitive to h or b? If the bem were constructed with a different material, how would the delfection change?

Use the equation- Ymax= ((wL^4)/(8EI)).

Ymax - deflection of the end beam (m)
w - uniform loading (N/m)
L = beam length (m)
E - modulus of elasticitt of beam (Pa)
I = (b(h^3))/12 - moment of inertia of beam cross-section (m^4).

Explanation / Answer

Ymax = wL4/8EI

Ymax = 5 mm; w= 15 kN/m; L= 1 m; E= 13GPa; I= bh3/12

put all values in the formula, we get,

I = (15*103*14)/(8*13*109*5*10-3) = 2.8846*10-5 m4

I= bh3/12

bh3 = 2.8846*10-5*12= 3.46152*10-4 m4

case 1:

let b=h;

b=h= 0.1364 m = 136.4 mm

case:2

b= 150 mm, then calculated h = 132.14 mm

case:3

b= 130 mm, then calculated h= 138.6 mm

case:4

h=150, then calculated b= 102.56 mm

case:5

h=130, then calculated b= 157.55 mm

i will consider case 2 (i.e. b= 150mm and h=132.14mm)

beacause for maximum deflection of the beam in this is more as compared to others 'b' and 'h' values.

due to more surface area of the beam there will be more pressure on beam, then deflection will be more.

height, h is more sensitive as compared to width, b. because height is varying with power.

compared,

case 2 and case 4

case 3 and case 5

from these two comparision we conclude that height of the beam is more sensitivity.

deflection of the beam is also depend on modulus of elasticity..different materials have different modulus of elasticity. therefore deflection must be varying with modulus of elasticity of materials.

deflection is inversaly proportional to modulus of elasticity.

that means, higher value of modulus of elasticity, deflection will be less and vice versa.

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