Mechanics of Materials: F1-1 (Hibbeler)
Summary
TLDRThis video script details a structural engineering problem involving a beam with specific support conditions and loadings. The focus is on calculating the internal normal force, shear force, and bending moment at Point C. The process involves drawing a free-body diagram, applying global equilibrium for overall system analysis, and determining reaction forces at supports A and B. The script then proceeds to find the internal forces at Point C by making an appropriate cut and analyzing the left side of the beam for normal force, shear force, and bending moment. The final results indicate a zero normal force, a shear force of 20 kN, and a bending moment of -40 kNm at Point C.
Takeaways
- 📏 The problem involves determining the normal force (N), shear force (V), and bending moment (M) at Point C of a beam.
- 🏗️ The beam is supported by a roller at A to the left of C and a pin at B to the right of C.
- ⚙️ A 60 kNm moment acts on the left end of the beam, and a 10 kN downward force acts on the right.
- 📈 The first step is to draw a free-body diagram of the entire system, including the forces and moments acting on the beam.
- 🔍 To find the reaction forces (Ay and By), moments are taken about point A, leading to the calculation of By as -10 kN.
- 📉 The normal force at C (NC) is found to be zero by considering the equilibrium in the X direction.
- 📊 The shear force at C (VC) is calculated to be 20 kN by summing forces in the Y direction.
- 📐 The bending moment at C (MC) is determined by summing moments about point C, resulting in -40 kNm.
- 🔧 The analysis uses the principles of statics to solve for the internal forces in the beam.
- 📋 The process involves making appropriate cuts in the beam to analyze the internal effects and applying the equations of equilibrium.
Q & A
What are the three internal forces to be determined in the beam at Point C?
-The three internal forces to be determined at Point C are the normal force (N), shear force (V), and bending moment (M).
What are the boundary conditions of the beam described in the script?
-The beam is supported by a roller at A to the left of C and a pin at B to the right of C.
What are the external forces and moments acting on the beam?
-There is a 60 kilonewton-meter moment acting on the left end of the beam and a downward 10 kilonewton force on the right.
How is the free-body diagram of the beam constructed?
-The free-body diagram includes the beam, the 10 kilonewton force, and the moment on the left, with vertical reaction forces at A and B, and dimensions noted.
Why are moments used to find the reaction forces instead of the sum of forces?
-Moments are used because there are two unknown reaction forces (Ay and By), and using moments allows solving for one of them without having two unknowns in a single equation.
What is the method used to find the reaction force By?
-The reaction force By is found by taking moments about point A and solving the equation for By.
What is the calculated value of By and why is it negative?
-The calculated value of By is -10 kilonewtons, which is negative because the force is acting downwards, opposite to the chosen positive direction.
How is the normal force at Point C determined?
-The normal force at Point C is determined to be zero by setting the sum of forces in the x-direction equal to zero, as there are no external forces acting in that direction.
What is the shear force VC at Point C?
-The shear force VC at Point C is 20 kilonewtons, found by setting the sum of forces in the y-direction equal to zero.
How is the bending moment MC at Point C calculated?
-The bending moment MC at Point C is calculated by summing moments about Point C, considering the applied moments and forces, resulting in -40 kilonewton-meters.
What is the significance of the negative bending moment MC?
-A negative bending moment MC indicates that the beam is bending in a clockwise direction when viewed from the left end of the beam.
Outlines
🔍 Analysis of Beam Forces and Moments
The paragraph introduces a structural engineering problem involving a beam with specific loading conditions. The goal is to determine the normal force, shear force, and bending moment at Point C. The beam is supported by a roller at A and a pin at B, with a 60 kNm moment at the left end and a 10 kN downward force at B. A free-body diagram is created, including reactions at A and B. The analysis begins with a global equilibrium approach to find the reaction forces Ay and By. Moments are taken about point A to solve for By, considering the clockwise and counterclockwise directions and the distances from A to B and B to the 10 kN force. The calculated By is found to be 20 kN downwards, and Ay is subsequently determined to be 20 kN upwards using the sum of forces in the y-direction. The paragraph concludes with the identification of all unknown forces acting on the beam.
🛠 Internal Forces Calculation in Beam
This paragraph continues the analysis by focusing on the internal forces within the beam at Point C. The process involves making a cut at Point C to isolate the left side of the beam for analysis. The internal loadings, including the normal force N, shear force V, and bending moment M, are introduced. The normal force at C is determined to be zero by setting the sum of forces in the x-direction to zero, as there are no external forces acting in this direction. The shear force at C, VC, is calculated by considering the sum of forces in the y-direction, resulting in 20 kN upwards. Finally, the bending moment at C, MC, is found by summing moments about point C, taking into account the 60 kNm moment, the reaction forces, and the distances involved. The moment equation yields an MC of -40 kNm, indicating a clockwise moment. The paragraph concludes with the determination of all internal forces for the beam at Point C.
Mindmap
Keywords
💡Normal Force (N)
💡Shear Force (V)
💡Bending Moment (M)
💡Free Body Diagram
💡Roller Support
💡Pin Support
💡Kilonewton (kN)
💡Static Equilibrium
💡Moments
💡Coordinate System
Highlights
Determine the results of normal force, shear force, and bending moment at Point C in the beam.
Beam is supported by a roller at A to the left of C and a pin at B to the right of C.
A 60 kilonewton meter moment acts on the left end of the beam.
A downward 10 kilonewton force is applied on the right.
Draw a free-body diagram of the entire system for statics analysis.
Identify vertical reaction forces at A and B, named as Ay and By.
Use moments to solve for unknown forces since there are two unknowns and one equation.
Sum moments about point A to find By, considering counterclockwise as positive.
Calculate By to be 40 kilonewtons using the moment equation.
Correct the direction of By force in the free-body diagram to downwards.
Use the sum of forces in the y direction to find Ay, which equals 20 kilonewtons.
Make appropriate cuts along the beam to analyze internal effects at Point C.
Analyze the left side of the beam to the left of the cut at Point C.
Set the sum of forces in the X direction to zero to solve for normal force NC.
Determine that there is no normal force created by the external forces at Point C.
Set the sum of forces in the y direction to zero to solve for shear force VC.
Calculate VC to be 20 kilonewtons.
Sum moments about Point C to find the bending moment MC.
Determine MC to be -40 kilonewton meters, indicating a clockwise bending moment.
Transcripts
problem f11 says determine the results
in the internal normal force Shear force
and bending moment at Point C in the
beam
so we are looking for the normal force n
Shear Force V and bending moment m
and taking a look at the beam it's
supported by a roller at a to the left
of c and a pin at B to the right of C
and there is a 60 kilonewton meter
moment acting on the Left End of the
beam and a downward 10 kilone even force
on the right
and now the first step for solving this
problem is of course drawing our free
body diagram of the entire system just
like we do in Statics
so here is the beam
the 10 kilonewton Force
and the moment on the left
and then point C
and at a we're going to have our
vertical reaction force and also at B
so I'll just name them a y and b y
considering an x y coordinate system
and then of course we need our
dimensions
and technically you can add the
horizontal Force at point B due to the
pin but in this case since there is no
external force acting in the X direction
we can just leave it out
so now this completes our free body
diagram of the system
so now we can move on to the global
equilibrium of the entire system which
is essentially our Statics analysis in
order to find the values of a y and b y
so now taking a look at the free body
diagram that we've drawn how can we find
the values of a y and b y
since a y and b y are two unknowns we
can't just go straight into using the
sum of forces in the y direction since
we'll have two unknowns and one equation
so in this case we can just simply use
moments and here we want to take a
moment about a point with an unknown
Force so either at point A or B that way
we can solve for either one
and I'm just going to call the Left End
D and the Right End e so now for example
I can start by summing up the moments
about point a
so we'll set the sum of moments at a
equal to zero and assuming
counterclockwise as positive
so starting off to the left of a of
course we have the 60 kilonewton meter
moment
and that is already counterclockwise so
it's positive
and then to the right we have the force
b y
so in this case for my drawing this is
also positive following the right hand
rule so we have plus b y times the
distance between A and B which is 2
meters
and finally of course on the far right
we have the 10 kilonewton Force which
creates a negative moment since the
moment is clockwise
so that is going to be minus 10 times
the distance which is
4 meters
so there we have our completed moment
equation
and as you can see we can now easily
solve for b y
so isolating b y we have b y equals
40 which is the negative 10 times 4
added to the other side minus the 60
and then divide it by two
which is equal to negative 10 kilo
newtons
so that is the value of b y
and now notice carefully that in my free
body diagram I had drawn the force Arrow
v y as upwards when it should actually
be downwards since we got a negative
value for b y
but since this was just an incorrect
guess we can go ahead and leave it like
that to avoid any confusion
and it's good to remember that even if
you draw it incorrectly on your free
body diagram the math will sort itself
out
and now that we know the value of b y we
can go ahead and simply use the sum of
forces in the y direction equal to zero
to find a y
so zooming up as positive this leaves us
with a y
minus the 10 kilonewtons again even if
your drawing is upwards you should still
leave the negative sign you get from
your calculation
and then minus the 10 kilonewtons that
acts at Point e
so solving for a y we get a y equals 20
kilonewtons
and so now we have found all the unknown
forces that are acting on the beam which
means that we are now able to finally
start solving for the internal forces
so first of course we have normal force
and again for this problem we are
interested in point C
and now of course the first step when
solving for internal forces is to make
the appropriate Cuts along the beam
typically between any changes in forces
in order to analyze the internal effects
on the beam
so for example here I can make a cut
between points A and B at Point C
just like so
and now for this case we can analyze the
left side of the beam
so here I'll be drawing the portion of
the beam that's to the left of the cut
along with the respective applied forces
and moments and also the dimensions
and now of course on the right we add in
the internal loadings so this is the
normal force n which acts in the X
Direction
and then we have Shear Force V
which acts in the y direction and also
the bending moment M around the z-axis
so now that we have our internal
loadings we can go ahead and solve for n
by setting the sum of forces in the X
Direction equal to zero
so the only Force we have here is NC
which acts to the right which is
positive
of course considering the x y coordinate
and like I mentioned previously in this
problem we don't have any applied forces
that are acting in the X Direction so
simply the normal force at Point c will
equal to zero so here we see that there
is essentially no normal force created
by the external forces
so now moving on to Shear Force
for this I'll just be using the same
diagram I drew above
here we're now looking for VC which acts
in the vertical Direction so we'll be
setting the sum of forces in the y
direction equal to zero to solve for VC
so we have of course the 20 kilonewton
force and then simply minus the VC
hence the shear Force VC will be equal
to 20 kilonewtons
so now finally we need to find the
bending moment
so of course for this we'll be summing
up our moments
but first we need to pick a point in
which we want to take our moments about
so here on the drawing this is point C
and since we're trying to find MC we can
go ahead and take the moments about that
point
so we can write the sum of moments about
Point C with counterclockwise as
positive
and this is around the z-axis
so setting this equal to zero we have a
60 kilonewton meter moment on the left
which is again positive since we are
considering counterclockwise as positive
so now taking the moment from C to a
this is going to be negative since it's
clockwise so that will simply be
negative 20 kilo newtons times the 1
meter
and then finally plus MC since MC is
going in the counterclockwise Direction
so that completes Our Moment equation
and we can now solve for MC
which ends up being negative 40
kilonewton meters
so that is our results and spending
moment
and so now we have found all the
internal forces for this question
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