**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
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 (d^{2}y/dx^{2})
= 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 cm^{4}

EI= (200x10^{9}
N/m^{2})x(29008.3
cm^{4} ) = 58016.6 kN m^{2}

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.

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