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Indeterminate Structure- solution for propped cantilever

Problem 7-1

The propped cantilever with applied loading is shown in figure 7-1(a). Find the support reactions and draw  Bending moment diagram.

Figure 7-1(a)

Solution:

The free body diagram of this structure in fig 7-1(b) shows that  the given propped cantilever has 3 support reactions (Ay, By, MB)  whereas there are only 2 equations of equilibrium (ΣFy = 0 and ΣMz = 0) available for this structure. Therefore it is statically indeterminate of degree one.

Figure 7-1(b)

To solve for the unknown reaction of this structure we need one more equation which can be obtained by compatibility. It is evident that deflection at A is zero. Considering Ay as redundant R and applying principle of superposition we get;

Total deflection at A is equal to the sum of deflection due to applied loading and deflection due to redundant R (refer to figure 7-1(c).

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Figure 7-1(c)

Deflection at A due to applied loading = wL4 /8EI = (10 x 44)/8EI

Deflection at A due to redundant R = (R x 43)/3EI

Taking downward deflection as negative and upward as positive and applying principle of superposition.

ΣδA = (R x 43)/3EI  - (10 x 44)/8EI

But ΣδA = 0 as the support A is a rigid support.

Therefore (R x 43)/3EI  - (10 x 44)/8EI  = 0

R = {(10 x 44)/8EI} / { ( 43)/3EI}

Hence R = 15 kN

Now we can easily determine the other reactions

By  = 10 x 4 - 15 = 25 kN

MB= 15x4 - 10x4x2 = - 20 kNm               (clockwise)

The bending moment diagram is shown in figure 7-1(d)

Fig 7-1(d|)

The resultant BMD shows that this beam will have a point of contra-flexure which can be determined in a simple way by writing the equation of bending moment at a distance 'x';

Mx = 15x -10x2 /2 = 0

Therefore x = 3m from the prop A,

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

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 Last updated on Thursday January 31, 2013