Title 22 Geometric Methods.pdf ✅

PSAD 22: Geometric Methods in Beam Deflection 1. Moment Diagram by Parts Moment Diagram by Parts is a method that geometrically shows the moments induced by various loadings at a specified part of a beam. For the beam deflection methods discussed in this topic, this is an essential step in their procedures. Step 1. Choose a reference point. For convenience, the reference point must be ideal such that it is easy to draw the corresponding moment diagrams of the loadings, and, perhaps, cancel certain loadings to reduce unknowns Step 2. Draw the moment diagrams with respect to the reference point. a. Moments that cause positive bending (concave up) about the reference point are positive, while those that cause negative bending (concave down) are negative. b. The ‘shape’ of the diagram is 2o (two degrees) more than the load that caused it. 2° load → 4° moment diagram 3° load → 5° moment diagram The moment diagrams will be analyzed as a whole and simultaneously.

3. Conjugate Beam Method This method relates slope and displacement of an actual beam with the shear and moment of a conjugate beam. The following are the steps and theorems involved with this method. Before applying the theorems of this method, the first step is to acquire the M EI diagram of the actual beam using moment diagram by parts for convenience. The moment diagrams will be treated as the loadings of the conjugate beam. The creation of conjugate beams will be discussed further later. Actual Beam Moment Diagram = Conjugate Beam Loading Diagram Sign Conventions: Downward Shear in Conjugate Beam = (CCW) Slope in Actual beam Upward Shear in Conjugate Beam = (CW) Slope in Actual beam Theorem 1. The slope at a point in the real beam is numerically equal to the shear at the corresponding point in the conjugate beam. Actual Beam Slope = Conjugate Beam Shear Note: the shear referred to here is internal shear. Thus, shear ≠ shear reaction Sign Conventions: Positive Bending in Conjugate Beam = Upward Displ. in Actual beam Negative Bending in Conjugate Beam = Downward Displ. in Actual beam Theorem 2. The displacement of a point in the real beam is numerically equal to the moment at the corresponding point in the conjugate beam Actual Beam Displacement = Conjugate Beam Moment Note: the moment referred to here is internal moment. Thus, moment ≠ moment reaction Conjugate Beam Supports are determined by applying theorems 1 and 2 on the support conditions of the actual beam. An example is shown below:
Other demonstrations of turning actual beams to conjugate beams are shown below: 1. Simply Supported Beam 2. Cantilever Beam 3. Beam with Internal Roller/Pin 4. Beam with Internal Hinge 5. Fully restrained Beam