Analyse the statically indeterminate beam illustrated below by integrating the g

Question

Analyse the statically indeterminate beam illustrated below by integrating the governing differential equation, applying the boundary conditions appropriately. Find the shear force, bending moment and displacement at point A. The cross section of the beam is t with EI 1500 kN m a 1.5 m, b 2.5 m, c 2.5 m and d -10 m. w 4 kN/m, and P 17 kN. The bar is also subjected constant to a differential temperature increase at the time the load is applied. This temperature change is constant over the length of the beam, but varies linearly over the beam height (which is 0.125 m) and is TI 10 °C at the top of the beam and T2 80 °C at the bottom of the beam (the coefficient of thermal expansion a 8.0e-06°C-1). L/2 L/2 Answers Shear force at point A (kN, internal convention): Bending moment at point A (kN.m, internal convention) Displacement at point A (mm, positive upwards)

Explanation / Answer

total legnth of bar = a+b+c+d = 1.5+2.5+2.5+1.0 =7.5m

total load = 1.5x4+1.5x17x1.0 =44.25KN/m2

1. let the both end of beam be R1 (left)and R2(right)

R1 = ql/2 = 4x7.5/2

30/2 = 15

reaction at one end R1 = 15Kn

reaction at end(left side) ,Rr = ql/2

= 17x7.5/2 = 127.5/2

=63.75Kn

reaction at another end(right side)R2 = 63.75Kn

let the reaction at point A = R3

reaction at point A = ql/2

= 4 x4/2 = 8/2 (length,l = a+b)

reaction at A = 4Kn

shear force at point A = (ql/2) -qx (q=load,x=legnth of pointA ,l=total legnth)

shear force = (4x4.0/2) - 4x7.5

= (16/2) - 30

= 8-30

shear force at point A= -22Kn

2. bending moment,i = wl2 /8

i = 44.25x(7.5)2 /8

i = 44.25(56.25) /8

i = 44.25x7

i= 309.75

bending moment at point A,i = 309.75 Nm/m

3. displacement = wl3/48El

cross section of the beam = 1500Kn/m2

= 44.25(7.5)3 / 48x1500

= (44.25x421.8) / 72000

= 18664.6/72000

displacement at point A = 0.259mm

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