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Module Content:
Module Reading, Problems, and Demo:
MAE 2310 Str. of Materials © E. J. Berger, 2010 12- 1
Module 12: Introduction to BendingFebruary 26, 2010
1. Shear and moment diagram construction is by far one of the most important skills for this course.2. If V & M is something you don’t get, you MUST practice until you are confident (and of course see me if you need help).3. There are many online resources available to support your learning of V & M.
Reading: Sections 6.1-6.2Problems: Prob. 6-9Demo: noneTechnology: http://pages.shanti.virginia.edu/som2010
MAE 2310 Str. of Materials © E. J. Berger, 2010 12-
Concept: Internal Forces• our paradigm: IF a structure is in
equilibrium, then any PIECE of that structure must also be in equilibrium (i.e., any method of sections cut we make)
• when we make a method of sections cut, what types of forces/moments represent the physical situation at that cut?
• support reactions (Statics, Ch. 5) allow certain types of motion , so we know what types of forces are present at the connections
• in the general case (Statics, Ch. 7), we make no assumptions about the types of internal forces present; we simply include ALL force/moment components in the equilibrium equations
• for a 3-D problem, we get 3 forces and 3 moments
• for 2-D problem, we get 2 forces and 1 moment
2
MAE 2310 Str. of Materials © E. J. Berger, 2010 12-
Concept: Internal Forces• our paradigm: IF a structure is in
equilibrium, then any PIECE of that structure must also be in equilibrium (i.e., any method of sections cut we make)
• when we make a method of sections cut, what types of forces/moments represent the physical situation at that cut?
• support reactions (Statics, Ch. 5) allow certain types of motion , so we know what types of forces are present at the connections
• in the general case (Statics, Ch. 7), we make no assumptions about the types of internal forces present; we simply include ALL force/moment components in the equilibrium equations
• for a 3-D problem, we get 3 forces and 3 moments
• for 2-D problem, we get 2 forces and 1 moment
2
Hibbeler, Statics (Fig. 7-2)
MAE 2310 Str. of Materials © E. J. Berger, 2010 12-
Theory: Shear and Moment
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MAE 2310 Str. of Materials © E. J. Berger, 2010 12-
Theory: Shear and Moment
3
MAE 2310 Str. of Materials © E. J. Berger, 2010 12-
Theory: Shear and Moment
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MAE 2310 Str. of Materials © E. J. Berger, 2010 12-
Theory: Shear and Moment
3
MAE 2310 Str. of Materials © E. J. Berger, 2010 12-
Theory: More V & M
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See also Ex. 6.1-6.6 in our text
MAE 2310 Str. of Materials © E. J. Berger, 2010 12-
Theory: More V & M
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See also Ex. 6.1-6.6 in our text
MAE 2310 Str. of Materials © E. J. Berger, 2010 12-
Theory: More V & M
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See also Ex. 6.1-6.6 in our text
MAE 2310 Str. of Materials © E. J. Berger, 2010 12-
Theory: A Graphical Approach• remember our observation and derivations from last semester:
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MAE 2310 Str. of Materials © E. J. Berger, 2010 12-
Theory: A Graphical Approach• remember our observation and derivations from last semester:
5
dV
dx= !w(x)
MAE 2310 Str. of Materials © E. J. Berger, 2010 12-
Theory: A Graphical Approach• remember our observation and derivations from last semester:
5
dV
dx= !w(x)
dM
dx= V
MAE 2310 Str. of Materials © E. J. Berger, 2010 12-
Theory: A Graphical Approach• remember our observation and derivations from last semester:
5
dV
dx= !w(x)
dM
dx= V
• in words:
• slope of shear diagram = -(distributed load intensity at each point)
• slope of moment diagram = shear at each point
MAE 2310 Str. of Materials © E. J. Berger, 2010 12-
Theory: A Graphical Example
6
MAE 2310 Str. of Materials © E. J. Berger, 2010 12-
Theory: A Graphical Example
6
• w(x) slope is always positive and increasing, so dV/dx is always negative and decreasing
MAE 2310 Str. of Materials © E. J. Berger, 2010 12-
Theory: A Graphical Example
6
• w(x) slope is always positive and increasing, so dV/dx is always negative and decreasing
• V(x) has a positive section (left side) and a negative section (right side)
MAE 2310 Str. of Materials © E. J. Berger, 2010 12-
Theory: A Graphical Example
6
• w(x) slope is always positive and increasing, so dV/dx is always negative and decreasing
• V(x) has a positive section (left side) and a negative section (right side)
• so, M(x) has a positive slope on the left side, negative slope on the right, and a zero slope when V(x) = 0
MAE 2310 Str. of Materials © E. J. Berger, 2010 12-
Need a Reminder? (Table 6-1, p. 283)
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MAE 2310 Str. of Materials © E. J. Berger, 2010 12-
An Example: Ex. 6.9
8
MAE 2310 Str. of Materials © E. J. Berger, 2010 12-
An Example: Ex. 6.9
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① w is constant: w(x) = wo
MAE 2310 Str. of Materials © E. J. Berger, 2010 12-
An Example: Ex. 6.9
8
① w is constant: w(x) = wo
② shear is linear:dV
dx= !wo " V (x) = !wox + C
MAE 2310 Str. of Materials © E. J. Berger, 2010 12-
An Example: Ex. 6.9
8
① w is constant: w(x) = wo
② shear is linear:dV
dx= !wo " V (x) = !wox + C
③ moment is quadratic:
dM
dx= V ! M(x) = "wox
2 + Cx + D
MAE 2310 Str. of Materials © E. J. Berger, 2010 12-
An Example: Ex. 6.9
8
V (x) = woL ! wox
MAE 2310 Str. of Materials © E. J. Berger, 2010 12-
An Example: Ex. 6.9
8
V (x) = woL ! wox
M(x) = !
wox2
2+ woxL !
woL2
2
M(x) = wo
!
!
L2
2+ x(L !
x
2)
"
MAE 2310 Str. of Materials © E. J. Berger, 2010 12-
Let’s play a game...
9
Statics Concept Review and ReminderCE/MAE 231, Spring 2008
The intent of this brief exercise is to understand your preparation for studying strength of materials. The exercise contains some statics review problems and is design to let us know how prepared you are for this course, and also for you to self-assess your readiness (and highlight those subjects from statics that you should review on your own).
1. We want to determine the reaction forces at all the pin joints in this machine. To do this, we need to draw FBDs of each piece of the component. Circle the FBD which is incorrect.
2. For the beam and loading shown, is the shear and moment diagram correct? If not, explain which parts of the diagram are incorrect.
MAE 2310 Str. of Materials © E. J. Berger, 2010 12-
Let’s play a game...
9
Statics Concept Review and ReminderCE/MAE 231, Spring 2008
The intent of this brief exercise is to understand your preparation for studying strength of materials. The exercise contains some statics review problems and is design to let us know how prepared you are for this course, and also for you to self-assess your readiness (and highlight those subjects from statics that you should review on your own).
1. We want to determine the reaction forces at all the pin joints in this machine. To do this, we need to draw FBDs of each piece of the component. Circle the FBD which is incorrect.
2. For the beam and loading shown, is the shear and moment diagram correct? If not, explain which parts of the diagram are incorrect.
dV
dx= !w(x)
dM
dx= V
plus boundary conditions
MAE 2310 Str. of Materials © E. J. Berger, 2010 12-
Statics Concept Review and ReminderCE/MAE 231, Spring 2008
The intent of this brief exercise is to understand your preparation for studying strength of materials. The exercise contains some statics review problems and is design to let us know how prepared you are for this course, and also for you to self-assess your readiness (and highlight those subjects from statics that you should review on your own).
1. We want to determine the reaction forces at all the pin joints in this machine. To do this, we need to draw FBDs of each piece of the component. Circle the FBD which is incorrect.
2. For the beam and loading shown, is the shear and moment diagram correct? If not, explain which parts of the diagram are incorrect.
Let’s play a game...
9
Statics Concept Review and ReminderCE/MAE 231, Spring 2008
The intent of this brief exercise is to understand your preparation for studying strength of materials. The exercise contains some statics review problems and is design to let us know how prepared you are for this course, and also for you to self-assess your readiness (and highlight those subjects from statics that you should review on your own).
1. We want to determine the reaction forces at all the pin joints in this machine. To do this, we need to draw FBDs of each piece of the component. Circle the FBD which is incorrect.
2. For the beam and loading shown, is the shear and moment diagram correct? If not, explain which parts of the diagram are incorrect.
dV
dx= !w(x)
dM
dx= V
plus boundary conditions
