6.0 ELASTIC DEFLECTION OF BEAMS 6.1 Introduction 6.2 Double-Integration Method 6.3 Examples 6.4 Moment Area Method 6.5 Examples

Importance of Beam Deflections Introduction P x P y Elastic curve The deflection is measured from the original neutral axis to the neutral axis of the deformed beam. The displacement y is defined as the deflection of the beam. It may be necessary to determine the deflection y for every value of x along the beam. This relation may be written in the form of an equation which is frequently called the equation of the deflection curve (or elastic curve) of the beam Importance of Beam Deflections A designer should be able to determine deflections, i.e. In building codes ymax <=Lbeam/300 Analyzing statically indeterminate beams involve the use of various deformation relationships.

Methods of Determining Beam Deflections Double-Integration Method Moment-Area Method Elastic Energy Methods Method of singularity functions

Double-Integration Method The deflection curve of the bent beam is In order to obtain y, above equation needs to be integrated twice. y r Radius of curvature x An expression for the curvature at any point along the curve representing the deformed beam is readily available from differential calculus. The exact formula for the curvature is

The Integration Procedure Integrating once yields to slope dy/dx at any point in the beam. Integrating twice yields to deflection y for any value of x. The bending moment M must be expressed as a function of the coordinate x before the integration Differential equation is 2nd order, the solution must contain two constants of integration. They must be evaluated at known deflection and slope points (i.e. at a simple support deflection is zero, at a built in support both slope and deflection are zero)

Sign Convention Positive Bending Negative Bending Assumptions and Limitations Deflections caused by shearing action negligibly small compared to bending Deflections are small compared to the cross-sectional dimensions of the beam All portions of the beam are acting in the elastic range Beam is straight prior to the application of loads

Examples y L x x PL P P @ x Integrating once @ x = 0 Integrating twice @ x = L y = ymax

y W N per unit length x x L WL @ x Integrating once @ x = 0

Integrating twice @ x = 0 Max. occurs @ x = L

Since the beam is symmetric L x y x Example Integrating Since the beam is symmetric

Integrating @ x = 0 y = 0 Max. occurs @ x = L /2

Since the beam is symmetric L/2 x P Example Integrating Since the beam is symmetric

Integrating @ x = 0 y = 0 Max. occurs @ x = L /2

Moment-Area Method First Moment –Area Theorem r dq B A ds D q dx x M Tangent at B q dq x dx ds The first moment are theorem states that: The angle between the tangents at A and B is equal to the area of the bending moment diagram between these two points, divided by the product EI. Tangent at A D M The second moment area theorem states that: The vertical distance of point B on a deflection curve from the tangent drawn to the curve at A is equal to the moment with respect to the vertical through B of the area of the bending diagram between A and B, divided by the product EI.

The Moment Area Procedure The reactions of the beam are determined An approximate deflection curve is drawn. This curve must be consistent with the known conditions at the supports, such as zero slope or zero deflection The bending moment diagram is drawn for the beam. Construct M/EI diagram Convenient points A and B are selected and a tangent is drawn to the assumed deflection curve at one of these points, say A The deflection of point B from the tangent at A is then calculated by the second moment area theorem Comparison of Moment Area and Double Integration Methods If the deflection of only a single point of a beam is desired, the moment-area method is usually more convenient than the double integration method. If the equation of the deflection curve of the entire beam is desired the double integration method is preferable. Assumptions and Limitations Same assumptions as Double Integration Method holds.

Examples L P A B D = ? PL Tangent at A q P Tangent at B M PL

L A W N per unit length B D = ? Tangent A WL

Example P P a a D = ? A L Tangent A P P Pa a