# Bending moment diagrams with a hinge?

• Joelllllll
In summary: The second error is that you are neglecting the moment at C, which is the hinge. This is a very common mistake. The moment at C is equal to 30*4*6 - 120*8 = -240. The third error is that you are neglecting the moment at A, which is the start of the beam. This is also a common mistake. The moment at A is equal to 30*4*2 - 4*Cy.
Joelllllll

## Homework Statement

Hey guys I'm having trouble drawing the bending moment diagram for this beam with a hinge?
I'm not sure where to start? I've looked at stuff online and its completely different to the stuff in my textbook

## Homework Equations

equilibrium equations
ƩFx=0 ƩFy=0 ƩM=0
Am I right to say that Ay=0 because the moment at the hinge is 0?
ƩMb=0 so
4m*Ay=0 --> Ay=0?

## The Attempt at a Solution

I know you start with the equilibrium equations to find the reactions but I'm not sure that what I am doing is right?
Any help with drawing the BMD would be appreciated :)

#### Attachments

• photo.jpg
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The hinge is the key to the problem because the bending moment there is zero. You can take moments to the right of the hinge, or to the left. AY is not zero because you have ignored the moment at A. Have you drawn a free body diagram and put on it all the actions and reaction components?

Solving the forces gives
Ay=30kN
Cy=90kN
If I take the moment at B
4Ay+Ma=0
Ma=-120kNm??

There is a mistake in your latest post. You need to show your working. Taking moments about B to the right of B, what equation do you get? Can I ask you to state what definition of bending moment you are using (there are two possible)?

To the right of B
0=(30*4*2)-4Cy
Which means Cy=60kN??
Umm On the left side counterclock wise is positive and on the right clockwise is positive?

Yes Cy=60 upwards. Sign conventions vary. Just stick to one while you are learning. The sign is not related to clockwise or anticlockwise but relates to whether the beam is hogging (convex upwards) or sagging(convex downwards) from the point of view of an observer under the beam. Now you have Cy, you can get the reaction components at A because there are 3 unknowns and 3 equations. These statically determinate problems are all self-checking. You don't need to ask if you have got it right. For example, in this case, when you have got all the reaction components, you should be able to check vertical equilibrium and take moments about E for the whole thing to check the moments are zero there. I hope you are successful, because it's a good learning problem.

i calculated my reaction forces the vertical reaction at point a equal to 60kn and vertical reaction point at c is also 60kn
trying to calculate the bending moment at point A , i said sum of the moment at A equal to zero and my positive moment is clockwise. my equation is M(a)+(30*4*6)-(60*8)=0 so moment at A equal to -240knm. what confusing is how this beam is deflecting. do i take into account of the moment at point C when i am drawing my bending moment diagram?

will have a bending moment digram of

Your bending moment diagram is actually quite promising and is correct in the two end sections, but there are three errors. The first is that the gradient just to the left of B should be the same as the gradient just to the right of B. Not quite. The slope in BA is an asymptote to the diagram in BC. In BC. Your diagram in BC looks elliptical, perhaps restrained by the drawing program you are using. It should be a parabola with the 'sharpest point' at the top. If you extend the diagram's straight line in AB to meet the perpendicular bisector of BC, you will get a value twice the moment at that point. That gives you a correct scale to the parabolic part of the diagram. If this were an exam I would pass it with 56%.

## 1. What is a bending moment diagram with a hinge?

A bending moment diagram with a hinge is a graphical representation of the variation of bending moment along the length of a beam or structure that is supported by a hinge. It shows the internal forces acting on the beam due to applied loads and reactions, and helps to determine the maximum bending moment and shear force at any point on the beam.

## 2. How do you draw a bending moment diagram with a hinge?

To draw a bending moment diagram with a hinge, you first need to identify all the external loads and reactions acting on the beam. Then, you can apply the equations of equilibrium to determine the reactions at the hinge. Next, you can start plotting the bending moment diagram by taking a section of the beam and calculating the bending moment at that section. Repeat this process for different sections of the beam until you reach the end. Finally, connect the points to obtain the bending moment diagram.

## 3. What are the important features of a bending moment diagram with a hinge?

The important features of a bending moment diagram with a hinge are the maximum bending moment and shear force, which occur at the points of zero bending moment. The bending moment diagram also shows the regions of positive and negative bending moment, which correspond to tension and compression in the beam. Additionally, the slope of the bending moment diagram at any point represents the shear force at that point.

## 4. How is a bending moment diagram with a hinge useful in structural analysis?

A bending moment diagram with a hinge is useful in structural analysis because it helps to determine the maximum stresses and deflections in a beam or structure. It also provides information about the internal forces and reactions acting on the beam, which is crucial for designing and evaluating the strength and stability of a structure. In addition, the bending moment diagram can be used to identify critical points and sections where reinforcement may be needed.

## 5. Can a bending moment diagram with a hinge be used for beams with multiple hinges?

Yes, a bending moment diagram with a hinge can be used for beams with multiple hinges. The process for drawing the bending moment diagram is the same, except that you will need to calculate the reactions at each hinge and consider them when plotting the diagram. The maximum bending moment and shear force may occur at different points along the beam, but the overall procedure remains the same.

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