Mechanics of Materials: F1-1 (Hibbeler)

Summary

This 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

00:00

🔍 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.

05:01

🛠 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.

Keywords

💡Normal Force (N)

Normal force, often denoted as N, is the force exerted by a surface that supports the weight of an object resting on it. In the context of the video, normal force is analyzed at Point C of the beam to determine the internal forces acting on the structure. The video explains that for this particular beam, the normal force at Point C is zero due to the absence of external forces acting in the horizontal direction, which is a key finding in understanding the beam's structural integrity.

💡Shear Force (V)

Shear force, represented by V, is the force that causes or tends to cause two adjacent parts of a material to slide relative to each other. In the video, the calculation of shear force at Point C is crucial for understanding how the beam resists deformation. The script describes a process where the sum of forces in the vertical direction is set to zero, leading to the conclusion that the shear force VC at Point C is equal to 20 kilonewtons, indicating the force required to prevent sliding failure at that point.

💡Bending Moment (M)

Bending moment, symbolized by M, is a measure of the rotational effect of an external load on a structure. In the video, the bending moment at Point C is calculated to understand the beam's ability to resist bending stresses. The script outlines a moment equilibrium around Point C, considering the applied moments and the reaction forces, resulting in a bending moment MC of negative 40 kilonewton meters, which signifies the extent of bending stress at that point.

💡Free Body Diagram

A free body diagram is a graphical representation of all the forces acting on a body in equilibrium. The video script emphasizes the importance of drawing a free body diagram for the entire beam and for the portions of the beam to the left and right of the cut at Point C. This visual tool is essential for performing a statics analysis and determining the unknown forces and moments acting on the beam.

💡Roller Support

A roller support, as mentioned in the script, is a type of support that allows a structure to rotate freely but restricts vertical movement. The beam is supported by a roller at Point A, which means it can rotate at this point but cannot move vertically. This support condition is critical for the analysis of the beam's reactions and internal forces.

💡Pin Support

A pin support, also known as a hinge or pivot, allows for rotation and prevents movement in the direction perpendicular to the pin's axis. The video describes the beam being supported by a pin at Point B, which is significant for determining the support reactions and understanding how the beam behaves under load.

💡Kilonewton (kN)

The kilonewton is a unit of force in the International System of Units (SI), where one kilonewton is equal to one thousand newtons. The script uses kilonewtons to quantify the forces and moments acting on the beam, such as the 10 kilonewton force and the 60 kilonewton meter moment. This unit is essential for accurately calculating the structural loads and responses.

💡Static Equilibrium

Static equilibrium refers to the state in which all forces and moments acting on a body are balanced, resulting in no acceleration. The video script discusses achieving global equilibrium of the entire system by applying the principles of statics to find the unknown forces ay and by. Understanding static equilibrium is fundamental to the analysis of the beam's structural behavior.

💡Moments

In the context of the video, moments refer to the turning effect of forces about an axis, which is crucial for determining the bending and twisting of the beam. The script describes how moments are calculated by taking the product of force and the perpendicular distance from the force to the axis of rotation. Moments are used to solve for unknown forces and to analyze the beam's response to external loads.

💡Coordinate System

A coordinate system is a system of mathematical coordinates that allows for the unique identification of points within a space. The video script mentions the use of an x-y coordinate system to analyze the forces acting on the beam. This system is essential for correctly applying the principles of statics and for determining the direction and magnitude of forces and moments.

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