1ST THEOREM OF MOMENT AREA METHOD

1^ST THEOREM OF MOMENT AREA METHOD

Consider a simply supported beam of span L with supports at A and B, subjected to point load of magnitude W.

Fig 1

    Fig 2

   Fig 3

Consider figure 2, which indicates the deflected shape of the simply supported beam subjected to point load. Let C and D be the two points between the supports A and B in order to determine the slope for elemental length. Let dx be the elemental length between CD to determine the slope value which resembles shape of an arc and projected to meet at point O making an angle dϴ. Let R be the radius of arc. Let ϴ_CD be the angle between the tangents drawn from points C and D.Fig 3 represents M/EI diagram of the over all beam and shaded portion represent for points CD.

We know that from the bending equation,

M/I = E/R............. (1)

Referring Fig 2, we know that from property of Circle

dx =  R dϴ

Therefore, R = dx / dϴ............. (2)

Substituting (2) in (1)

M/I = E/ (dx / dϴ)

dϴ = (M/ E I) ( dx)

This is for elementary length dx

For CD portion

ϴ_CD = _C∫^D  (M/ E I) ( dx)

Therefore 1^st theorem of moment area states that 

“change in the slope of a beam between two points is equal to the area under the curvature diagram between those two points”.