Outline: Review External Effects on Beams Beams Internal Effects Sign Convention Shear Force and Bending Moment Diagrams (text method) Relationships between Loading, Shear Force and Bending Moments (faster method for SFD & BMD) 1
Beams are bars of material that support Beams are common structural members Beams can support both concentrated and distributed loads We will only analyze statically determinate beams in this class 2
Typical Beam Configurations 3
Beams can resist: Tension / Compression (happy face / frown face) Shear force Bending moment Torsional moment The amount of each kind of internal load can change throughout the length of the beam depending on external loads 4
We can cut the beam at any point along its length to analyze the internal forces at that point It is often difficult to tell the direction of the shear and moment without calculations so represent V y and M bz in their positive directions and let the result tell you (+/-) whether you drew it the right way CONVENTION 5
In North America, if the moment tends to cause the beam to curve it is positive; if the moment tends to cause it to curve it is negative. 6
The bending moment is positive if the upper side is in it it s a A positive shear force at the cut causes the beam to rotate clockwise and matches the smiley face shape 7
Positive Shear and Moment on a whole Beam Positive Shear and Moment on a cut-out Beam Note: Coming from the left or the right makes a difference, so always start from left. 8
The maximum bending moment (magnitude could be negative or positive) is often the primary consideration in the design of a beam Variations in shear and moment are best shown graphically 9
Use equilibrium of whole beam to find external reactions and supports Then cut each section of the the beam and replace any distributed loads with equivalent concentrated loads and solve for unknown axial, shear and moment at point x (your cut) Plot V y (shear force vs x) and M bz (bending moment vs x) along the beam 10
Solve for R 1 and R 2 Isolate left section and solve for V y and M bz between the left support and 4kN load Isolate right section and solve for V y and M bz between the right support and 4 kn load Isolate a section between every change in an external load, but do not cut the section at the concentrated load 11
Draw the shear and bending moment diagrams for the diving board, which supports the 80 kg man. Determine the location and magnitude of the maximum bending moment. 12
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Draw the shear and bending moment diagrams for the beam and loading shown and determine the location and magnitude of the maximum bending moment. 15
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Draw the shear and bending moment diagrams for the beam and loading shown. 17
Diagrams illustrate the value of the internal forces (axial/shear/moment) that occur at each point along a structure (beam). Additionally, 18
The shear diagram is the graphic representation of the shear force at successive points along the beam. The shear force (V y ) at any point is equal to the algebraic sum of the external loads and reactions to the left of that point. Since the entire beam must be in equilibrium (the sum of F y = 0), the shear diagram must close at zero. 19
The moment diagram is the graphical representation of the magnitude of the bending moment at successive points along the beam. The bending moment for the moment diagram (M bz ) at any point equals the sum of moments of the forces on the beam to the left about that point. Since the entire beam is in equilibrium (Sum of M = 0), the bending moment diagram must close to zero at right side. 20
Cut the beam between each concentrated load. For each section solve for the unknown shear force and bending moment Equation for each in terms of x (the distance along the beam Sub in endpoint values of x to get numerical values of V y and M bz at each cut Plot V y (shear force vs x) and M bz (bending moment vs x) along the beam 21
The text method works every time, but it is time consuming You can draw the diagrams from the FBD if you know the relationships between w (load intensity), V (shear force) and M (bending moment). the negative slope of the shear diagram at a given point equals w the load at the point dv dx dm dx the slope of the moment diagram at the given point is shear at the point 22 V
If there is no change in the load along the length under consideration, the shear curve is a straight horizontal line (or a curve of zero slope). If a concentrated load exists, then there is a vertical jump in the SFD ( force = drop down on SFD, force = step up on SFD) If a load exists, and is uniformly distributed, the slope of the shear curve is constant and nonhorizontal. If a load exists, and increases in magnitude over successive increments, the slope of the shear curve is positive (approaches the vertical); if the magnitude decreases, the slope of the shear curve is negative (approaches the horizontal). 23
If the slope of the SFD is zero, then the moment curve has a constant slope that is equal to the value of the shear for that increment. If the slope of the SFD is positive, then the slope of the moment curve is getting steeper. If the slope of the SFD is negative, then slope of the moment curve is getting flatter. Changes in the shear diagram will produce changes in the shape of the moment curve. The area under the shear curve between two points is equal to the change in bending moment between the same two points. 24
The area of the shear diagram to the left or to the right of the section is equal to the moment at that section. The slope of the moment diagram at a given point is the shear at that point. The maximum moment occurs at the point of zero shear. When the shear (also the slope of the moment diagram) is zero, the tangent drawn to the moment diagram is horizontal. 25
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Draw the shear and bending moment diagrams for the beam and loading shown and determine the location and magnitude of the maximum bending moment. 28
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Draw the shear and bending moment diagrams for the beam and loading shown and determine the location and magnitude of the maximum bending moment. 31
ENGR 1205 Chapter 10 32
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Draw the shear and bending moment diagrams for the beam and loading shown. 35