Cantilever Beams

Stress at any point:-

(1)

Deflection at any point:-

(2)

Stress at critical points:-

(3)

This is the maximum stress when the cross section is uniform.

Deflection at cr1tical points:-

Maximum ( At centre)

(4)

Stress between each support and the load

(5)

Stress at the Centre

(6)

This is the maximum stress when the cross section is constant.

The deflection between each support and the load

(7)

The maximum deflection occurs at the load

(8)

Stress for a portion of length a

(9)

Stress for a portion of length b

(10)

Stress at the point of load. This is the maximum stress if the cross section is constant.

(11)

Deflection for the portion of length a

(12)

Deflection for the portion of length b

(13)

Deflection at the load

(14)

When a is the length of the shorter portion and b the longer one, the maximum deflection is in the longer one at:-

(15)

And the deflection is:-

(16)

The stress in between each support and the adjacent load.

(17)

Stress at the load points and any point between.

(18)

Deflection between each support and the adjacent load

(19)

Deflection between loads

(20)

The maximum deflection is at the centre and is:-

(21)

Deflection at the loads is:-

(22)

Stress between each support and the end adjacent;-

(23)

Between the supports:-

(24)

Stress at each support:-

(25)

Stress at the centre:-

(26)

The greater of these is maximum stress when the section is constant.

Should l > 2c the stress is zero at points on each side of the centre. Should the section be constant and l = 2.828 c ,the stresses at the centre and at the supports are equal and opposite. They are:-

(27)

The Deflection between each support and the adjacent end:-

(28)

Deflection between the supports

(29)

Deflection at the ends:-

(30)

Deflection at the centre

(31)

When l is between 2c and 2.449c the maximum upward deflections occur at points on each side of the centre.the value of these are :-

(32)

Between load and adjacent support

(52)

Between supports

(53)

Between the unloaded end and the adjacent S = 0

Stress at the support adjacent to the load

(54)

If the cross saection is Constant, this is the maximum stress. The stress is zero at the other support.

The deflection between the load and th adjacent support

(55)

Deflection between the supports

(56)

Deflection between the loaded and adjacent support

(57)

Deflection at the load

(58)

The Maximum upwards deflection is at 0.42265 l and is:-

(59)

Deflection at the unloaded end

(60)