Slope and deflection Calculation by Macaulay's method

Problem 6-2

Use Macaulay's method to determine the values of slope and deflection at 5m from the left support due to imposed load as shown in figure 6-2(a). The material of beam has modulus of elasticity as 200 GPa. The beam cross-section is I-shaped with top and bottom flange width as 250 mm and thickness 20 mm, web height as 300 mm and web thickness as 15 mm.

Figure 6-2(a)

Solution:

The loading is not same throughout the span, therefore we use Macaulay's Method for finding slope and deflection of the given beam.

The given beam has two unknown reaction components which are calculated in Prob 4-1 as  A_y = 66 kN and B_y = 24 kN;

The equivalent loading is shown in figure 6-2(b).

Figure 6-2(b)

The differential equation of elastic curve for this  beam can be written as follows;

 EI (d²y/dx²) = 66[x] - 20[x]² /2 +20[x-4]²/2 - 10[x-8]         Eq. 1

Integrating Eq. 1 we get;

EI(dy/dx) = 66[x]²/2 - 20[x]³ /6 +20[x-4]³/6 - 10[x-8]²/2 +C1     Eq.2

We do not have any information about slope at the ends,

We continue integrating Eq. 2 for finding deflection;

EIy = 66[x]³/6 - 20[x]⁴ /24 +20[x-4]⁴/24 - 10[x-8]³/6 +C1x + C2

Now apply the condition for deflection at the supports;

at x=0, y=0 (neglect the terms which become negative with x=0)

we get C2=0;

Further at x=10, y=0; we get

0=11000 - 200000/24 + 1080 - 80/6 + 10C1

C1= -374.66

Therefore the equation for slope can be written as

EI(dy/dx) = 66[x]²/2 - 20[x]³ /6 +20[x-4]³/6 - 10[x-8]²/2 - 374.66

and the equation for deflection would be

EIy = 66[x]³/6 - 20[x]⁴ /24 +20[x-4]⁴/24 - 10[x-8]³/6 - 374.66x

Slope at x=5;

   dy/dx = (37)/EI

Deflection at x=5;

   y = (- 1018)/EI

The value of moment of inertia can be calculated by using moment of inertia calculator

  I_xx = 29008.3 cm⁴

  EI= (200x10⁹ N/m²)x(29008.3 cm⁴ ) = 58016.6 kN m²

substituting the value of EI in the expressions for slope and deflection we get;

  dy/dx = 37/58016.6 = 0.00064 rad.

   y = - 1018/58016.6 = - 0.0175 m (negative sign indicates that the deflection is downward)

You can also use our Slope deflection calculator for different combinations of load on Simple beam.