Slope and deflection calculation by unit load method

Problem 7-3

Use unit load method to find the deflection at the center of the beam shown in figure 7-3(a). Take E= 200 GPa and I=400x10⁶ mm⁴

Figure 7-3(a)

Solution:

In the case of unit load method the deflection at a point of beam is given as

 

To write the equations of bending moment for different parts of the beam we have to first calculate the support reactions by applying the equations of equilibrium

(i) reactions due to actual loading

A_y = 13.75 kN;     D_y = 31.25 kN

write the equations of bending moment for different part of the beam (origin of axes at A).

M_x = 13.75x ;                              (0 ≤ x ≤ 2)

M_x = 13.75x - 5(x-2) ;                 (2 ≤ x ≤ 4)

M_x = 13.75x - 5(x-2) -10(x-4)(x-4)/2;     (4 ≤ x ≤ 8)

(ii) reactions due to virtual unit load at C

a_y = 0.5 kN;   b_y = 0.5 kN

m_x = 0.5x                                 (0 ≤ x ≤ 4)

m_x = 0.5x - 1(x-4)                     (4 ≤ x ≤ 8)

Depending upon the validity of different bending moment equations, the deflection at point C can be written as follows;

substituting the appropriate bending moment equations for M_x and m_x

 δ_c = 302.58 kN.m³/EI

substituting the values of E I in the above expression we get;

  δ_c = (302.58 kN.m³)/{(200*10⁶kN/m²)(400*10^-6m⁴)}

  δ_c = 0.0038 m = 3.8 mm

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