Theory of Structure Asst. Prof Dr. Laith Sh. Rasheed Lec Theory of Structure Asst.Prof Dr.Laith Sh. Rasheed Lec.18 (Deflection of Beams)

Karbala University College of Engineering Department of Civil Eng. Introduction: Computation of deflection is of great importance. Codes usually specify permissible deflection in structural elements like beams and slabs. Large deflection in beams or slab gives an unseemly sight and the occupants will feel uncomfortable. large deflection in beams may affect the attached elements and in slabs will cause cracking of the plaster in the ceiling.

Karbala University College of Engineering Department of Civil Eng. A large number of methods have been developed to calculate deflections of structures. The deflections to be considered in this text apply to structures having linear elastic material response. The methods can be broadly classified as: 1. Geometrical Methods 2. Energy Method

Karbala University College of Engineering Department of Civil Eng. In the geometrical methods, the basic equations of equilibrium, compatibility, boundary conditions, and the material stress-strain relations are used to generate the governing differential equations. These equations are then solved by analytic, graphic or combined method. The double integration method (Studied in last stage) Moment area method (Studied in last stage) The conjugate beam method √

Karbala University College of Engineering Department of Civil Eng. Conjugate Beam Method:- The conjugate beam method was developed by H. Muller-Breslau in 1865. The method is based on the relations between the derive shearing force and bending moment diagrams from the load diagrams and the operations required to compute the slopes and deflections from the curvature diagrams. From strength of materials it will be recalled that the differential equation of the elastic curve of a beam under loading is:

Karbala University College of Engineering Department of Civil Eng.  

Karbala University College of Engineering Department of Civil Eng. -------------------2 -------------------3 Where for a beam under distributed load:- And,

Karbala University College of Engineering Department of Civil Eng.  

Karbala University College of Engineering Department of Civil Eng.  

Karbala University College of Engineering Department of Civil Eng.  

Karbala University College of Engineering Department of Civil Eng. Theorem 1:- The shearing force at any point on the conjugate beam is equal to the slope at the corresponding point on the real beam. Theorem 2:- The bending moment at any point on the conjugate beam is equal to the deflection at the corresponding point on the real beam.

Karbala University College of Engineering Department of Civil Eng. When drawing the conjugate beam it is important that the shearing force and bending moment developed at the supports of the conjugate beam account for the corresponding slope and deflection of the real beam at its supports. The various real support conditions and their corresponding conjugate substitutes are shown in Fig. (1).

Karbala University College of Engineering Department of Civil Eng. Real beam Conjugate Beam

Karbala University College of Engineering Department of Civil Eng. Real beam Conjugate Beam

Karbala University College of Engineering Department of Civil Eng. Procedure for Analysis:- Draw the conjugate beam for the real beam. This beam has the same length as the real beam and has corresponding supports as listed in Fig(1). • In general, if the real support allows a slope, the conjugate support must develop a shear; and if the real support allows a displacement, the conjugate support must develop a moment. •

Karbala University College of Engineering Department of Civil Eng.  

Karbala University College of Engineering Department of Civil Eng.  

Karbala University College of Engineering Department of Civil Eng. Sol.: The conjugate beam is shown below The M/EI diagram is negative, so the distributed load acts downward, i.e., away from the beam.

Karbala University College of Engineering Department of Civil Eng. The negative signs indicate the slope of the beam is measured clockwise and the displacement is downward