a 1,000-word essay. In the essay, you must summarize what you consider to be the key concepts from the chapters covered, and you must include at least three photographs that you've taken to help explain the concepts.
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6186695_67669608_engr213e_week5_chapter_0728129.pdf
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EQUILIBRIUM OF BEAMS
Chapter 7
Perspec.ve The study of beams (slender structural elements subjected to bending loads) is extremely important in both civil and mechanical engineering, and the first step is covered in this chapter. Learn to iden.fy and describe beams using the terminology described in the textbook. Pay aHen.on to the sign conven.ons used for internal shear force and bending moment. And most important of all, keep in mind that developing shear and bending moment diagrams is nothing but sta.cs applied to free-‐body diagrams of beams cut at key loca.ons along their length. Same as we have been doing, just a bit more intricate.
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INTRODUCTION 7.1
Terminology • Please pay aHen.on to all of the terms highlighted in the text:
– Beam – Pin support – Roller support – Simply supported – Can.lever – Overhanging – Concentrated load – Distributed load – Concentrated moment – etc.
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External Reac.ons – Simply Supported • Find external reac.ons first, as part of the beam analysis process
Pin support: ver.cal and horizontal reac.ons (horizontal is usually zero) and no moment
Roller support: ver.cal reac.on and no moment
External Reac.ons – Can.lever • Find external reac.ons first, as part of the beam analysis process
Built-‐in support: ver.cal and horizontal reac.ons (horizontal is usually zero) and a moment
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External Reac.ons • Use a Free Body Diagram of the beam to find the external reac.ons • A distributed load is modeled as an equivalent concentrated force
RA RB
F = wL This one is a bit trickier but the principal is exactly the same. Replace the regions of distributed load with sta.cally equivalent concentrated forces to find the external reac.ons.
Restore the Distributed Loads • Important point:
– Once the external reac.ons are found, treat the beam as if the distributed loads are once again there
– When you cut the beam to find internal forces and moments (see next sec.on) you will have por.ons of the beam with por.ons of the distributed loads ac.ng
– These will once again be replaced by sta.cally equivalent concentrated forces!
• Yes, it is back-‐and-‐forth, but that’s how it’s done.
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SHEAR AND MOMENT IN BEAMS
7.2
Varia.on Along the beam • The big difference between beams and other situa.ons we
have examined (axial loading, torsional loading) is the type and varia.on of internal reac.ons along the component
– We need to track two internal reac.ons for beams: • Shear force, V • Bending Moment, M
– We need to expect the internal reac.ons to vary along the length • Constant internal reac.ons are common in axial loading and torsion • Constant internal reac.ons are rare in bending
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V-‐M Diagrams • The Shear Force and Bending Moment Diagrams (V-‐M) are
simply a standardized way of presen.ng the internal reac.on informa.on for a beam
– Signs are standardized • We will in general work from the led
– Diagrams are standardized • Align ver.cally with the beam illustra.on
Transfer posi.ve V and M to this cut face and analyze the FBD: that’s all there is to it!
Cuts • Cut a beam at arbitrary loca.ons (x – measured from the led)
– Between concentrated loads – Within distributed loads
cut cut cut cut cut
cut cut cut cut cut cut
Yikes! That’s a lot of cuts. I wouldn’t ask you to analyze this beam in an exam, but do understand why the cuts are where they are.
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Internal Forces and Moments • V is simply the internal shear force ac.ng on a cut face • M is simply the internal moment ac.ng on a cut face
The distributed load por.on of the beam is represented by this Free Body Diagram.
To proceed: • Replace the distributed load with a
concentrated force • Sum ver.cal forces to find V • Sum moments about the cut to find M
Note how V and M are oriented. We will always use these as the posi.ve direc.ons.
Textbook Examples • The example problems in the textbook (7.1-‐7.5) are extremely
good and illustrated in great detail
• They cover all the basic loading and support situa.ons (simply supported and can.lever; concentrated forces, concentrated moments, distributed forces)
• If you understand these examples, and can generate V-‐M diagrams for beams of easy-‐to-‐moderate difficulty, you are in good shape
• Great aHen.on to detail is required! Make large, clear diagrams and label everything carefully
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GRAPHICAL METHOD FOR SHEAR AND
MOMENT DIAGRAMS
7.3
Graphical Check • I prefer to think of this as methods to check your equa.ons,
not as a way to generate the V-‐M Diagrams
– The equa.ons are required for deflec.on and buckling analysis – It is possible but awkward to go backwards from plots to equa.ons
• It is also good Sta.cs prac.ce to use the equilibrium method for genera.ng the V-‐M equa.ons
• I find many difficul.es among students rela.ng plots with equa.ons, please prac.ce this skill!
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The Basic Checks • These are the most helpful graphical checks:
– Where a concentrated force is applied, the V-‐plot jumps by that amount. This even applies to the points where external reac.ons act if you imagine star.ng from off the beam where everything is zero.
– Where a concentrated moment is applied, the M-‐plot jumps by that amount. This even applies to the points where external reac.ons act if you imagine star.ng from off the beam where everything is zero.
– In regions where no load is applied, or where a distributed load acts, the V-‐curve is the deriva.ve (slope) of the M-‐curve
• Look for these paHerns in all of the example problems
