A beam 6 m long is simply supported at the
ends,and carries a uniform distributed load of
1500kg/m (including its own weight ) and three
concentrated loads of 1000,2000 and 3000 kg
acting respectively at the left quarter
point,centre point and right quarter point.
Draw the shear force ans bending moment
diagram and determine the maximum bending
moment.
To draw a shear force and bending moment diagram for the given beam and loading conditions, you can follow these steps:
Step 1: Identify the reactions at the support points.
Since the beam is simply supported, the reactions will be equal at both ends and can be calculated using the principles of static equilibrium. The total load acting on the beam is the sum of its own weight and the distributed load. Therefore, the total load on the beam can be calculated as:
Total Load = Weight of Beam + Distributed Load = 6m * 1500kg/m = 9000kg
Since the beam is symmetrical, each reaction at the support will be half of the total load, i.e., 9000kg / 2 = 4500kg.
Step 2: Create the shear force diagram.
The shear force diagram represents the varying magnitude of the shear force along the length of the beam. Start with the left end of the beam and work towards the right, taking into account the changes in load and support conditions.
- At the left end (x = 0), the shear force is equal to the left reaction (4500kg) since there are no loads to the left of it.
- At the left quarter point (x = 1.5m), there is a concentrated load of 1000kg acting downward. Therefore, the shear force decreases by 1000kg from the left end.
- At the center (x = 3m), there is a concentrated load of 2000kg acting downward. Therefore, the shear force decreases by an additional 2000kg from the left quarter point.
- At the right quarter point (x = 4.5m), there is a concentrated load of 3000kg acting downward. Therefore, the shear force decreases by an additional 3000kg from the center.
- At the right end (x = 6m), the shear force becomes zero as it is balanced by the right reaction (4500kg).
The shear force diagram would look like this:
Shear Force: |———————————|————————————————————————|
4500kg -1000kg -3000kg 0kg
Step 3: Create the bending moment diagram.
The bending moment diagram represents the varying magnitude of the bending moment along the length of the beam. Start with the left end of the beam and work towards the right, integrating the changes in the shear force.
- At the left end (x = 0), the bending moment is zero since there is no shear force acting on the beam.
- At the left quarter point (x = 1.5m), the shear force is -1000kg. Therefore, the bending moment starts increasing from zero at a linear rate.
- At the center (x = 3m), the shear force is -3000kg. Therefore, the bending moment increases further at an accelerated rate.
- At the right quarter point (x = 4.5m), the shear force is now zero, and the bending moment reaches its maximum value.
- At the right end (x = 6m), the bending moment decreases from its maximum value to zero.
The bending moment diagram would look like this:
Bending Moment: |-----\ /-----|
Max Moment
Step 4: Determine the maximum bending moment.
From the bending moment diagram, you can determine the maximum bending moment by observing the point of maximum deflection. In this case, the maximum bending moment occurs at the right quarter point (x = 4.5m), where the bending moment is the highest.
Therefore, the maximum bending moment can be determined by looking at the bending moment diagram at that point.
In this example, the maximum bending moment can be read from the diagram and will have a specific value associated with it (not provided in the question).
To draw the shear force and bending moment diagram, follow these steps:
Step 1: Calculate the reactions at the supports.
Since the beam is simply supported at the ends, the reactions at the supports can be determined using the principle of equilibrium. The total downward force acting on the beam can be calculated by multiplying the length of the beam by the weight per meter.
Total downward force = (weight per meter) * (length of beam)
= 1500 kg/m * 6 m
= 9000 kg
Since the beam is symmetrical, the reactions at both supports will be equal.
Reactions at each support = Total downward force / 2
= 9000 kg / 2
= 4500 kg
Step 2: Plot the concentrated loads on the beam.
The concentrated loads at the left quarter point, center point, and right quarter point can be represented by vertical downward arrows.
Step 3: Determine the position of the shear force and bending moment.
Starting from the left end of the beam, the shear force remains constant until a concentrated load is encountered. At that point, the shear force will change according to the magnitude and direction of the load.
At the points of concentrated loads, calculate the changes in the shear force and plot the values on the diagram. The magnitude and direction of the changes depend on whether the concentrated load is acting upward or downward.
The bending moment diagram is obtained by integrating the vertical shear force. Therefore, the bending moment at any point is the sum of all the shear forces to the left of that point.
Step 4: Plot the shear force diagram.
Starting from the left end of the beam, draw a horizontal line representing the initial shear force (equal to the left support reaction). Then, as you encounter concentrated loads, draw vertical lines indicating changes in shear force.
Step 5: Plot the bending moment diagram.
Starting from the left end of the beam, draw a vertical line representing zero bending moment. Then, at each concentrated load, calculate the change in bending moment and plot the values on the diagram.
Step 6: Determine the maximum bending moment.
The maximum bending moment occurs at the point where the shear force changes its sign. In this case, it will occur at the concentrated load with the highest magnitude, which is 3000 kg.
Here is a rough sketch of the shear force and bending moment diagram for the given beam:
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Shear Force | | | | | | | |
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|------------------------------------
Bending | | | | | | | |
Moment | | | | | | | | | | | | |
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1 2 3 4 5 6 7 8 9 10 11 12
The maximum bending moment occurs at the point where the shear force changes its sign, which is at the right quarter point.
Therefore, the maximum bending moment is at the right quarter point and is equal to the magnitude of the shear force at that point, which is 4500 kg.