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ShearMoment-in-Beams m Mechanics of .ppt
- 1. SHEAR & MOMENTIN BEAMS SHEAR & MOMENT IN BEAMS
- 2. BEAMS BEAMS Definition of aBeam A beam is a bar subject to forces or couples that lie in a plane containing the longitudinal section of the bar. According to determinacy, a beam may be determinate or indeterminate. Statically Determinate Beams Statically determinate beams are those beams in which the reactions of the supports may be determined by the use of the equations of static equilibrium. The beams shown below are examples of statically determinate beams.
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- 4. BEAMS BEAMS Statically Indeterminate Beams Ifthe number of reactions exerted upon a beam exceeds the number of equations in static equilibrium, the beam is said to be statically indeterminate. In order to solve the reactions of the beam, the static equations must be supplemented by equations based upon the elastic deformations of the beam. The degree of indeterminacy is taken as the difference between the number of reactions to the number of equations in static equilibrium that can be applied. In the case of the propped beam shown, there are three reactions R1, R2, and M and only two equations (ΣM = 0 and ΣFv = 0) can be applied, thus the beam is indeterminate to the first degree (3 - 2 = 1).
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- 6. LOADINGS LOADINGS Types of Loading •Concentrated Load – acts at a point on the beam •Distributed Load – acts over a length of the beam •Concentrated Moments - tends to bend and rotate the beam
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- 8. SHEAR AND MOMENTDIAGRAM - Graphical representation of shear and moment throughout the span of the beam Sign Convention We assume the following to be positive : - Shear forces that tend to rotate a beam element clockwise - Bending moments that tend to bend a beam element concave upward(the beam “smiles”)
- 9. SHEAR AND MOMENTDIAGRAM Method of Section – V and M as functions of x for each segment of the beam between a change in loading point or discontinuity in loading. Method of Area – change in shear equals the area under loading diagram and change in moment equals the area under the shear diagram.
- 10. METHOD OF SECTION 1.Compute the support reactions. 2. Divide the beam into segments so that loading in each segment is continuous. 3. Introduce a cutting plane within the segment, located at x distance from the left end of the beam, that cuts the beam into two parts. 4. Choose a section to consider, whichever is more convenient. At the cut section, show V and M acting in their positive directions. 5. Determine the expressions for V and M as a function of x using equations of equilibrium. 6. Plot the expressions for V and M for the segment. It is visually desirable to draw the V-diagram below the FBD of the entire beam, an then draw the M-diagram below the V-diagram.
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- 14. Sample Problems 1. Drawthe shear-force and bending-moment diagrams for the simply supported beam shown. Determine the maximum shear and bending moment that occur in the span.
- 15. Sample Problems 2. Drawthe shear-force and bending-moment diagrams for the simply supported beam. Determine the maximum shear and bending moment that occur in the span.
- 16. Sample Problems Draw theshear-force and bending moment diagrams for the beam shown. Determine the maximum positive bending moment and the maximum negative bending moment that occur in the beam.
- 17. Sample Problems 4. Drawthe shear-force and bending moment diagrams for the cantilever beam shown. Determine the maximum shear and bending moment that occur in the span.
- 18. Sample Problems 5. Drawthe shear-force and bending moment diagrams for the overhanging beam shown. Determine the (a) maximum shear, (b) maximum positive and maximum negative bending moment that occur in the beam.
- 19. Sample Problems 6. Drawthe shear-force and bending –moment diagrams for the simply supported beam shown. Determine the maximum shear and bending moment that occur in the span.