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Questions and Answers
What is a beam in mechanics of deformable bodies?
What is the difference between statically determinate and statically indeterminate beams?
What is required to solve the reactions of a statically indeterminate beam?
What is the purpose of a shear and moment diagram?
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What type of beam can be solved using only the equations of static equilibrium?
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What is the result of applying a force or couple that lies in a plane perpendicular to the longitudinal of the bar?
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What is the main difference between a statically determinate beam and a statically indeterminate beam?
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What is required to determine the reactions of a statically determinate beam?
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What is the rate of change of the shearing force with respect to x equal to?
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What is the slope of the shear diagram at a given point equal to?
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What is the equation for the rate of change of the shear force with respect to x?
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What is the equation for the rate of change of the moment with respect to x?
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What is required to draw moment and load diagrams corresponding to a given shear diagram?
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What is required to draw a moment and load diagram?
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What is the purpose of specifying values at all points of zero shear?
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What is the significance of change of load positions in drawing moment and load diagrams?
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What is the result of not specifying values at all change of load positions and at all points of zero shear?
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What is the purpose of drawing moment and load diagrams?
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Study Notes
Mechanics of Deformable Bodies
- Shearing stress varies directly with shear.
Bending Moment
- Defined as the summation of moments about the centroidal axis of any selected section of all loads acting either to the left or to the right side of the section.
Shear and Moment Diagrams
- Equations for the beams are written, and the diagrams are drawn, specifying values at all change of loading positions and at points of zero shear.
- Neglect the mass of the beam in each problem.
Sample Problem #1
- Draw the shear and moment diagram for a beam loaded as shown in the figure.
Types of Loading
- Loads applied to the beam may consist of:
- Concentrated load (load applied at a point)
- Uniform load
- Uniformly varying load
- Applied couple or moment
Shear and Moment Diagrams: Simple Beam
- Consider a simple beam of length L with a uniform load w (N/m) throughout its length and held in equilibrium by reactions R1 and R2.
- The portion removed must then be replaced by:
- Vertical shearing force V
- Couple M to hold the left portion of the bar in equilibrium under the action of R1 and wx.
Beam Sign Convention
- The couple M is called the resisting moment or moment.
- The force V is called the resisting shear or shear.
- The sign of V and M are taken to be positive if they have the senses indicated above.
Stress Acting on a Beam
- 2 kinds of stress act over the transverse section of a beam:
- Bending stress (varies directly with the bending moment)
- Shear stress (varies directly with shear)
Types of Supports
- Review of types of supports.
Beams
- A beam is a bar subject to forces or couples that lie in a plane containing the longitudinal of the bar.
- Beams can be determinate or indeterminate according to determinacy.
Statically Determinate Beams
- Statically determinate beams are those beams in which the reactions of the supports can be determined by the use of the equations of static equilibrium.
Statically Indeterminate Beams
- If the number of reactions exerted upon a beam exceeds the number of equations in static equilibrium, the beam is said to be statically indeterminate.
- To solve the reactions of the beam, the static equations must be supplemented by equations based upon the elastic deformations of the beam.
Relations Among Load, Shear, and Moment
- The rate of change of the shearing force (V) with respect to x is equal to the load (w) at a given point.
- The rate of change of the moment (M) with respect to x is equal to the shear force (V) at a given point.
- The slope of the shear diagram at a given point equals the load at that point.
Sample Problems
- Sample Problem #1: Draw moment and load diagrams corresponding to the given shear diagrams. Specify values at all change of load positions and at all points of zero shear.
- Sample Problem #2: Draw moment and load diagrams corresponding to the given shear diagrams. Specify values at all change of load positions and at all points of zero shear.
- Sample Problem #3: Draw moment and load diagrams corresponding to the given shear diagrams. Specify values at all change of load positions and at all points of zero shear.
Note: The sample problems are repeated, but they all follow the same pattern of drawing moment and load diagrams from given shear diagrams and specifying values at certain points.
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Description
This quiz covers the concepts of bending moment and shearing stress in mechanics of deformable bodies, including signs of bending and related calculations.
