Moment-area theorem

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Moment Area

Basic Concepts

The reactions of the beam shown in the figure consist of horizontal and vertical forces at end A, a moment at end A and a vertical force at end B.

Because there are only three independent equations of equilibrium for this beam, it is not possible to calculate all four of the reactions from equilibrium alone

The number of reactions in excess of the number of equilibrium is called the degree of static indeterminacy. Thus a propped cantilever beam is statically indeterminate to the first degree

Statically determinate structures When a beam is statically determinate, we can obtain all reactions, shear forces and bending moments from free-body diagrams and equations of equilibrium. There are different methods for analysing a statically indeterminate beam.

Basic concepts

The excess reactions are called statically redundant

Since this reaction is in excess of those needed to maintain equilibrium, it can be released from the structure by removing the support at B

When support B is removed, we are left with a cantilever beam. The structure that remains when the redundants are released is called the released structure or the primary structure.

Principle of Superposition • Selecting the redundant reactions.• write equations of equilibrium that relate the other unknown reactions to the redundants and the loads.

• Assume that both the original loads and the redundants act upon the released structure.

• Find the deflections in the released structure by superposing the separate deflections due to the loads and the redundants.

• Write equations of compatibility (or equations of superposition) expressing the fact that the deflections of the released structure (at the points where restraints were removed) are the same as the deflections in the original beam (at those same points)

• finally, with the redundants known, we can determine all other reactions from the equations of equilibrium. Furthermore, we can also determine the shear forces and bending moments from equilibrium.

Moment Area method

it is clear that the distance from D on the elastic curve intersected by the tangent drawn to this curve at another point C is the sum of the intercepts dt created

by tangents to the curve at adjacent points. Each of these intercepts may be considered as the arc of a circle of radius x subtended by an angle d theta

The difference in slope between any two points on the elastic line is numerically equal to the area of the bending moment diagram (M) divided by EI between the two points.

Second moment area theorem

The tangential deviation of point D from the tangent at point C is equal to the first moment of the M/EI diagram between C and D, evaluated with respect to D.