Calculating Shear and Bending forces in a Cantilever Beam.

In summary, the homework statement is trying to find the Shear force and bending moments in a simple cantilever bean with a point load and a U.D.L. The equations for equilibrium must be used, and the attempt at a solution provides a summary of how to find the reactions and points at which distributed loads act.
  • #1
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Homework Statement



I am trying to work out the Shear force and bending moments in a simple cantilever bean with a point load and a U.D.L but am not sure of the equations or how to go about starting it. Its a 2 m long cantilever beam built into a wall at the right-hand end, Point C. A UDL is applied with a downward force of 4 kN/m at Point B (which is 0.8 m from the free end) to the wall. An upward force of 2 kN is applied to the free (left-hand) end, Point A.

Homework Equations



I presume you have to use equations for equilibrium but am not sure how to apply them.

The Attempt at a Solution

 
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  • #2
mm391 said:

Homework Statement



I am trying to work out the Shear force and bending moments in a simple cantilever bean with a point load and a U.D.L but am not sure of the equations or how to go about starting it. Its a 2 m long cantilever beam built into a wall at the right-hand end, Point C. A UDL is applied with a downward force of 4 kN/m at Point B (which is 0.8 m from the free end) to the wall. An upward force of 2 kN is applied to the free (left-hand) end, Point A.

Homework Equations



I presume you have to use equations for equilibrium but am not sure how to apply them.

The Attempt at a Solution


first of all, you need to determine the reactions at the fixed end. what shape is the UDL? (rectangle, triangle etc). 4kn/m doesn't specify this.

a fixed end has 3 reactions. moment, force in y direction, force in x direction.

to find reactions, you need to sum the moments about a point (which will be equal to zero) then sum the forces in the y direction and the forces in the x direction. the sums will all be equal to zero. (you have no forces acting in the x direction, so just scratch that).

if you sum the moments about point A, you will have:

moment caused by shear force + moment caused by UDL + reaction moment of fixed end
support = 0

then

sum of forces in y direction:

- shear force - UDL + reaction y direction force of fixed end support = 0

notice the positive and negatives. the shear force and UDL are acting DOWN, the reaction of the fixed end has to COUNTERACT these forces so it points UP.


do you know how to find the moments about a point and how to find the points at which distributed loads act?
 
  • #3
Am i righy in saying to find a moment you multiply the loads by the distance from the moment adding them togther and they should all = 0 (following sign convention).

To find the point at which the UDL acts you multiply the force by the length of the UDL and that force acts at the centre point of the UDL.
 
  • #4
mm391 said:
Am i righy in saying to find a moment you multiply the loads by the distance from the moment adding them togther and they should all = 0 (following sign convention).

To find the point at which the UDL acts you multiply the force by the length of the UDL and that force acts at the centre point of the UDL.

yes and no. it depends on what shape the UDL is. is it a rectangular shape? then yes, the UDL "acts" at the midpoint of the UDL. however, if the UDL is the shape of let's say a triangle, it "acts" 1/3 of the distance from the bigger end of the triangle. the SUM of all the forces in the y direction will equal zero, the SUM of all the forces in the x direction will equal zero, and the SUM of all the moments about a POINT will equal zero.

basically for a UDL, you find the area of the UDL, then that number is the total force it is inducing on the structure. You just have to find where its acting as a point load, then take the distance from where its acting to the point you're using as a reference (sum of moments about a point = 0) and multiply distance * force = moment. make sure you include all forces in the summations.

its easier if i just draw this real quick. here's an example of calculating the reactions of a fixed end cantilever beam with JUST a UDL. i won't do your work for you, but if you calculate your reactions and post back here ill help more.

UDLHelp.jpg
 
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  • #5


I would suggest starting by drawing a free body diagram of the cantilever beam and labeling all the known and unknown forces. This will help you visualize the problem and determine which equations to use.

To calculate the shear force, you can use the equation V = ∑Fy, where V is the shear force and ∑Fy is the sum of all the vertical forces acting on the beam. In this case, the only vertical forces are the 2 kN upward force at Point A and the 4 kN/m downward UDL, which can be represented as a single force of 4 kN acting at the midpoint of the UDL (0.4 m from Point B). Therefore, V = 2 kN + 4 kN = 6 kN.

To calculate the bending moment, you can use the equation M = ∑(Fy)(y), where M is the bending moment, ∑(Fy) is the sum of all the vertical forces acting on the beam to the left or right of a particular point, and y is the distance from that point to the force. Again, the only vertical forces acting on the beam are the 2 kN upward force at Point A and the 4 kN/m downward UDL. Using Point A as the reference point, the bending moment can be calculated as M = (2 kN)(2 m) + (4 kN/m)(0.8 m)(0.4 m) = 4.8 kNm.

To check your calculations, you can also use the equations for equilibrium, ∑Fx = 0 and ∑Fy = 0, to make sure that the sum of all the forces and moments acting on the beam is equal to zero. This will help ensure that your calculations are correct.

I hope this helps guide you in solving your problem. Remember to always start by drawing a free body diagram and labeling all known and unknown forces, and then use the appropriate equations to calculate the shear and bending forces.
 

What is a cantilever beam and what forces does it experience?

A cantilever beam is a structural element that is supported at only one end and is free to deflect under load. It typically experiences a combination of shear and bending forces due to external loads.

How do you calculate shear force in a cantilever beam?

Shear force is calculated by taking the algebraic sum of all the vertical forces acting to the left or right of a particular point on the beam. It can also be calculated using the equation V = dM/dx, where V is shear force, M is bending moment, and x is the distance from the point of interest.

What is the formula for calculating bending moment in a cantilever beam?

The bending moment at any point on a cantilever beam can be calculated using the equation M = F * d, where M is bending moment, F is the force acting on the beam, and d is the perpendicular distance from the point of interest to the line of action of the force.

What are the assumptions made when calculating shear and bending forces in a cantilever beam?

The main assumptions made are that the beam is rigid, the external loads are applied at discrete points, and the beam is loaded within its elastic limit. Other assumptions may include neglecting the effects of self-weight and assuming the beam is a straight line under load.

How do you interpret the results of shear and bending force calculations in a cantilever beam?

The results of shear and bending force calculations can provide insight into the structural integrity of the beam and can be used to determine the maximum load the beam can withstand. They can also be used to design appropriate support structures for the beam.

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