Cantilever Beam Problem 4

Shubham Kola

Watch Cantilever Beam Problem 4 - Shubham Kola on Dailymotion

Transcript

00:00In this video, we are going to learn how to draw a shear force diagram and bending movement

00:12diagram for a cantilever beam as shown in figure.

00:16So, the statement is given as, Draw a shear force and bending movement diagram for a cantilever

00:23beam AB 1.5m long is loaded as shown in figure.

00:30So this is the cantilever beam AB of length 1.5m and carrying a uniformly distributed load

00:36of 1kN per meter or a length of 1m and there is one point load of 2kN acting on the beam

00:45at point B.

00:48So for this setup, we have to draw shear force and bending movement diagrams.

00:55So first of all, I will draw the free body diagram for this beam section.

00:59Here, first we have to convert this UDL into point load.

01:07So to convert this, I will multiply this UDL value i.e. 1kN per meter with the length over

01:15which the UDL act, that is 1 meter.

01:20So here, I will get the point load of 1kN.

01:24Now this calculated point load is act on the midpoint of length over which the UDL act.

01:30Now, this type of problem, we are going to solve in three steps.

01:35In the first step, we have to calculate the value of support reaction force Ra.

01:44So to calculate this value, I will use the condition of equilibrium i.e. summation of

01:51Fy equal to 0.

01:53That means addition of all forces in the vertical axis equal to 0.

01:58While doing the addition of all vertical forces, I will consider upward forces as positive and

02:07downward forces as negative.

02:09Here, Ra is the vertical reaction force.

02:12So I will add this force with plus sign.

02:16And the converted point load is acting on the beam in the downward direction.

02:19So as per the sign convention, I will add this 1kN force with negative sign.

02:26And also there is one point load of 2kN in the downward direction.

02:31So I will add this 2kN downward force with negative sign.

02:37So from this, I will get the value of reaction force Ra, 3kN.

02:44So now with the help of this calculated value of Ra, I will further calculate the values of

02:49shear forces at all the points of beam.

02:53So the next step is, Calculations of shear forces.

02:57And for shear force calculations, our sign convention is, upward forces are considered as positive

03:04and downward forces are considered as negative.

03:09And here I will start the shear force calculation from left hand side of the beam.

03:15For cantilever beam, while calculating the shear force at a particular point load, you should

03:21calculate shear force values for left side and right side of that particular point load.

03:28But while calculating the shear force at uniformly distributed load, i.e. UDL, you should calculate

03:34shear force values at start point and end point of UDL.

03:38That is shear force at point A and shear force at point C, we need to calculate.

03:45But in the problem, at point A, there is reaction force Ra and this is the point load.

03:52And as it is point load, here I will calculate the shear force values for left side and right

03:57side of point A.

04:00Therefore, first to calculate shear force at point A to its left, i.e. Sf at A to the left

04:08equal to.

04:09So, as you can see, there is no force acting at left side of point A. Therefore, Sf at A

04:16to the left equal to 0 kN.

04:19So, to draw a shear force diagram, I will draw a horizontal reference line of 0 kN shear force.

04:29So, here I will mark the point of 0 kN shear force on the reference line.

04:35Now, if I go to the section to the right of point A, then there is reaction force Ra in

04:41the upward direction.

04:42So, as per the sign convention, I will consider upward force as positive.

04:48So, here the shear force is plus 3 kN.

04:53Here as the shear force value is positive, hence I will mark the point of 3 kN shear force

04:59above the reference line of 0 kN shear force.

05:03And I will connect these two points with the vertical line.

05:06Now, the point C is the end point of UDL.

05:10So, I am taking section to the point C, that is Sf at point C.

05:18And here I will carry forward previous value of shear force up to point A to its side, which

05:23is 3 kN.

05:24And to the left side of point C, there is UDL of 1 kN per meter that we had converted into

05:31point load of 1 kN in the downward direction.

05:34So, as per the sign convention, I will consider downward force as negative.

05:39So, I will add this converted point load with negative sign.

05:44So, by calculating, this will get the shear force value at point C as 2 kN.

05:51Here as the shear force value is positive, hence I will mark the point of 2 kN shear force above

05:58the reference line of 0 kN shear force.

06:03And here the type of load is UDL over length 1 meter.

06:06Hence, to draw a shear force diagram, I will indicate UDL with an inclined line.

06:11So, I will connect these two points with the inclined line.

06:15Now, at point B, there is point load.

06:19Therefore, first to calculate shear force at point B to its left, i.e. SF at B to the left

06:27equal to, so here I will carry forward previous value of shear force up to point C, which is

06:352 kN.

06:36And when we go to the left of point B, then there is no forces acting at the left of point

06:42B.

06:43Therefore, SF at B to the left equal to 2 kN.

06:46Here, as you can see, there is no variation in shear force values, hence I will make the

06:52horizontal line with shear force value as 2 kN.

06:56Now, next to calculate shear force at point B to its right, i.e. SF at point B to its right

07:04equal to, so here I will carry forward previous value of shear force up to point B to its left,

07:13which is 2 kN.

07:15And when we go to the right side of point B, then there is one point load in the downward

07:20direction.

07:21So, as per the sign convention, I will consider downward force as negative.

07:27So, here I will add this point load of 2 kN with negative sign.

07:32So, here, plus 2 kN minus 2 kN gives me the value of shear force as 0 kN.

07:40That is, SF at point B to its right equal to 0 kN.

07:45And I will connect these two points with the vertical line.

07:49And here in shear force diagram, whatever the portion drawn above the reference line, I

07:54will show this by plus sign.

07:56So, here I have completed the shear force diagram.

08:00Now, the next step is, Calculations for bending moment.

08:05The bending moment at a section of beam is calculated as the algebraic sum of movement

08:11of all the forces acting on one side of the section.

08:16So, to calculate bending moment, we can start either from left end of beam or from right

08:22end of beam.

08:24Here, I will start from the right side.

08:27So, whenever you are calculating the bending moments, you should remember these conditions.

08:33So, here for cantilever beam, the condition is, at the free end, the bending moment will

08:40be 0.

08:41That is, BM suffix B equal to 0 kN.

08:45So, to draw bending moment, firstly, I will draw the reference line of 0 kN bending moment,

08:54so I will mark this value with a point on the reference line.

08:59So, now we have to calculate bending moment at point C.

09:04Here, in case of cantilever beam, while we are doing the calculations for bending moment,

09:11at a particular point, you should always add movement of all the forces present from the

09:18free end of cantilever beam up to that particular point at which we are calculating the bending

09:24moment.

09:26So, for bending moment calculations, our sign convention is, for sagging effect of beam,

09:31the force is considered as positive and for hogging effect of beam, the force is considered

09:38as negative.

09:39So, for point load of 2 kN, due to this, the beam shows hogging effect and for hogging effect

09:48of beam, I will consider this force as negative.

09:52So, I will add this point load of 2 kN with negative sign and I will multiply this point load

10:00with the distance from point of action of force, that is 0.5 m.

10:06So, by calculating, this will get the value minus 1 kNm.

10:11So, as it is negative value of bending moment, hence I will mark this point below the reference

10:17line of bending moment 0 kNm.

10:21And to draw bending moment, I will join these two points with the inclined line.

10:28Now, next we have to calculate the bending moment at point A.

10:34Here, the right-hand side of point A, there is UDL of 1 kNm per meter that we had converted

10:41into point load of 1 kNm.

10:43Due to this, the beam shows hogging effect and for hogging effect of beam, I will consider

10:50this converted point load as negative.

10:52So, I will add this converted point load with negative sign and I will multiply this converted

10:59point load with the distance from point of action of force, that is 0.5 m.

11:06And also there is one point load of 2 kNm, so due to this, the beam shows hogging effect

11:13and for hogging effect of beam, I will consider this point load as negative.

11:18So, here I will add this point load with negative sign.

11:22And I will multiply this point load with the distance from point of action of force, that

11:27is 1.5 m.

11:31So by calculating, this will get the value minus 3.5 kNm.

11:37So as it is negative value of bending moment, hence I will mark this point of bending moment

11:43below the reference line of bending moment 0 kNm.

11:46And to draw bending moment, I will indicate UDL with a parabolic curve.

11:52Hence, I will join these two points with a parabolic curve.

11:57Now, since I can see this bending moment diagram is drawn below the reference line of 0 kNm bending

12:05moment, hence I will show this portion by minus sign.

12:08So, here I have completed the shear force diagram and bending moment diagram for this cantilever

12:15beam.

12:16So, here I have a

Category

Transcript

Recommended