# Shear force diagram and bending moment diagram pdf

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Determining shear and moment diagrams is an essential skill for any engineer. This is a problem. Shear force and bending moment diagrams tell us about the underlying state of stress in the structure.

The quickest way to tell a great CV writer from a great graduate engineer is to ask them to sketch a qualitative bending moment diagram for a given structure and load combination! Your complete roadmap to mastering these essential structural analysis skills. Consider a simply supported beam subject to a uniformly distorted load. The beam will deflect under the load. In order for the beam to deflect as shown, the fibres in the top of the beam must contract or get shorter. The fibres in the bottom of the beam must get longer.

We can say the top of the beam is in compression while the bottom is in tension notice the direction of the arrows on the fibres in the deflected beam. Now, at some position in the depth of the beam, compression must turn into tension.

There is a plane in the beam where this transition between tension and compression occurs. This plane is called the neutral plane or sometimes the neutral axis. Imagine taking a vertical cut through the beam at some distance along the beam.

We can represent the strain and stress variation throughout the depth of the beam with a strain and stress distribution diagrams. Remember, strain is just the change in length divided by the original length. Compression strains above the neutral axis exist because the longitudinal fibres in the beam are getting shorter. Tensile strains occur in the bottom because the fibres are extending or getting longer. We can assume this beam is made of a linearly elastic material and as such the stresses are linearly proportional to the strains.

We know that if we multiply a stress by the area over which it acts, we get the resultant force on that area. The same is true for the stress acting on the cut face of the beam. The compression stresses can be represented by a compression force stress resultant while the tensile stresses can be replaced by an equivalent tensile force.

As a result of the external loading on the structure and the deflection that this induces, we end up with two forces acting on the cut cross-section. These forces are:. You might recognise this pair of forces as forming a couple or moment.

The bending moment diagram shows how and therefore stress varies across a structure. If we know the state of longitudinal or normal stress due to bending at a given section in a structure we can work out the corresponding bending moment. We do this using the Moment-Curvature equation a. Where is the second moment of area for the cross-section.

Building on our discussion of bending moments, the shear force represented in the shear force diagram is also the resultant of shear stresses acting at a given point in the structure. Consider the cut face of the beam discussed above. The shear stress, acting in this cut face is evenly distributed across the width of the face and acts parallel to the cut face. The average value of the shear stress, is simply the shear force at this point in the structure divided by the cross-sectional area over which it acts,.

However, this is just the average value of the shear stress acting on the face. The shear stress actually varies parabolically through the depth of the section according to the following equation,.

For the purposes of this tutorial, all we want to do is establish the link between the shear force we observe in the shear force diagram and the corresponding shear stress within the structure. Equations 4 and 5 do that for us. Based on this you should be comfortable with the idea that knowing the value of bending moment and shear force at a point are important for understanding the stresses in the structure at that point.

In reality, this is practically how we determine the shear force and bending moment at a point in the structure. Simple statics tell us that if the beam is in a state of static equilibrium, the left and right hand support reactions are,. If the structure is in a state of static equilibrium which it is , then any sub-structure or part of the structure must also be in a state of static equilibrium under the stabilising action of the internal stress resultants.

This is a key point! Imagine taking a cut through the structure and separating it into 2 sub-structures. This means, if we want to find the value of internal bending moment or shear force at any point in a structure, we simply cut the structure at that point to expose the internal stress resultants and. Then calculate what values they must have to ensure the sub-structure remains in equilibrium!

For example the sub-structure below must remain in equilibrium under the combined influence of:. This starts to make more sense when we plug some numbers into an example. The left hand reaction, is,. So, the internal bending moment required to maintain moment equilibrium of the sub-structure is kNm. Similarly, if we take the sum of the vertical forces acting on the sub-structure, this would yield kN. In the last section we worked out how to evaluate the internal shear force and bending moment at a discrete location using imaginary cuts.

But to draw a shear force and bending moment diagram, we need to know how these values change across the structure. What we really want is an equation that tells us the value of the shear force and bending moment as a function of.

Where is the position along the beam. Consider making an imaginary cut, just like above, except now we can make the cut at a distance along the beam. Now the internal shear force and bending moment revealed by the cut are functions of , the cut position.

But the procedure is exactly the same to determine. Now we can use equation 12 to determine the value of the internal bending moment for any value of along the beam. Plotting the bending moment diagram is simply a matter of plotting the equation. However this may not always be the case. In this example, the bending moment for the whole structure is described by a single equation…equation You might remember from basic calculus that to identify the location of the maximum point in a function we simply differentiate the function to get the equation for the slope.

In other words, at the location of the maximum bending moment, the slope of the bending moment diagram is zero. So we just need to solve for this location. Once we have the location we can evaluate the bending moment using equation Remember, equation 13 represents the slope of the bending moment diagram. So we now let it equal to zero and solve for. Surprise surprise, the bending moment is a maximum at the mid-span,. Now we can evaluate equation 12 at m.

There we have it; the location and magnitude of the maximum bending moment in this simply supported beam, all with some basic calculus. This example is an extract from this course. If you get a bit lost with this example, it might be worth your time taking a look at this DegreeTutors course. We want to determine the shear force and bending moment diagrams for the following simply supported beam.

You can continue reading through the solution below…or if you prefer video, you can watch me walk through the solution here. The first step in analysing any statically determinate structure is working out the support reactions. We can kick-off by taking the sum of the moments about point A, to determine the unknown vertical reaction at B, ,. Now with only one unknown force, we can consider the sum of the forces in the vertical direction to calculate the unknown reaction at A, ,.

Our approach to drawing the shear force diagram is actually very straightforward. The first load on the structure is acting upwards, this raises the shear force diagram from zero to as point A. The shear force then remains constant as we move from left to right until we hit the external load of acting down at D.

When we reach the linearly varying load at E, we make use of the relationship between load intensity, and shear force that tells us that the slope of the shear force diagram is equal to the negative of the load intensity at a point,.

This is telling us that the linearly varying distributed load between E and F will produce a curved shear force diagram described by a polynomial equation. In other words, the shear force diagram starts curving at E with a linearly reducing slope as we move towards F, ultimately finishing at F with a slope of zero horizontal.

When the full loading for the beam is traced out, we end up with the following,. This is obtained by subtracting the total vertical load between E and B from the shear force of at E. This is because we can make use of the following relationship between the shear force and the slope of the bending moment diagram,. Between D and E, the shear force is still constant but has changed sign.

This tells us the slope of the bending moment diagram has also changed sign, i. But the fact that the shear force changes sign at B, means the bending moment diagram has a peak at that point. Finally, the externally applied moment at F tells us that the bending moment diagram at this location has a value of. We can combine all this information together to sketch out a qualitative bending moment diagram, based purely on the information encoded in the shear force diagram.

Now we simply have to cut the structure at discrete locations indicated with red dashed lines above to establish the various key values required to quantitatively define the bending moment diagram. In this case three cuts are sufficient:. Then by considering moment equilibrium of the sub-structure we can solve for the value of. And finally for cut , this time considering equilibrium of the sub-structure to the right-hand side of the cut.

We can now sketch the complete quantitative bending moment diagram for the structure. In fact at this point we can summarise the output of our complete structural analysis. So there you have it. There is quite a lot more we could say about shear and moment diagrams. ## Bending moment

A cantilever beam is subjected to various loads as shown in figure. Draw the shear force diagram and bending moment diagram for the beam. Bending moment between C and A;. The sign of bending moment is taken to be negative because the load creates hogging. Draw the shear force and bending moment diagrams for the beam. Since, there is no load between points A and C; for this region Fx remains constant.

In solid mechanics , a bending moment is the reaction induced in a structural element when an external force or moment is applied to the element, causing the element to bend. The diagram shows a beam which is simply supported free to rotate and therefore lacking bending moments at both ends; the ends can only react to the shear loads. Other beams can have both ends fixed; therefore each end support has both bending moments and shear reaction loads. Beams can also have one end fixed and one end simply supported. The simplest type of beam is the cantilever , which is fixed at one end and is free at the other end neither simple or fixed. In reality, beam supports are usually neither absolutely fixed nor absolutely rotating freely. The internal reaction loads in a cross-section of the structural element can be resolved into a resultant force and a resultant couple.

Determining shear and moment diagrams is an essential skill for any engineer. This is a problem. Shear force and bending moment diagrams tell us about the underlying state of stress in the structure. The quickest way to tell a great CV writer from a great graduate engineer is to ask them to sketch a qualitative bending moment diagram for a given structure and load combination! Your complete roadmap to mastering these essential structural analysis skills. ## Draw The Shear And Moment Diagrams For The Beams Shown Below

Ninth Edition in SI Unit. McKenzie Examples in Structural Analysis. Draw shear force diagram and bending moment diagram 3. Show the critical values on the diagrams. Span length is 9 m. Ninth Edition in SI Unit.

### Shear Force and Bending Moment Diagrams Notes for Mechanical Engineering

Then the vertical components of forces and reactions are successively summed from the left end of the beam to preserve the mathematical sign conventions adopted. The shear at a section is simply equal to the sum of all the vertical forces to the left of the section. The shear force curve is continuous unless there is a point force on the beam. When the successive summation process is used, the shear force diagram should end up with the previously calculated shear reaction at right end of the beam. No shear force acts through the beam just beyond the last vertical force or reaction. If the shear force diagram closes in this fashion, then it gives an important check on mathematical calculations. The shear force will be zero at each end of the beam unless a point force is applied at the end.

We have shown these in Diagram 2b, and labeled them V shear force and M bending moment. Now, lets draw the shear and moment diagram remember to draw the Beams Solution. Determine the reactions at support A and B for the overhanging beam subjected to the loading as shown. Skip navigation Sign in. Tutorials on how to calculate the force reactions at supports, shear force diagrams and bending moment diagrams at supports. Draw the shear and moment diagram due to dead load. Note the magnitude and location of the maximum bending moment, MD.

#### Lesson 19. SOLVED EXAMPLES BASED ON SHEAR FORCE AND BENDING MOMENT DIAGRAMS

For the beam and loading shown, a Draw the shear and bending-moment diagrams, b Determine the maximum absolute values of the shear and bending moment. Step 2: Draw the Moment diagram, M. Draw the shear and moment diagrams for the cantilevered beam. View Answer Find the exact value of each of the remaining trigonometric functions of?. Solution: A Cantilever of length l carries a concentrated load W at its free end. Label the value of the. Ex; Draw the shear force and bending moment diagrams for the beam shown; Ex; Stresses in Beams Forces and couples acting on the beam cause bending flexural stresses and shearing stresses on any cross section of the beam and deflection perpendicular to the longitudinal axis of the beam.

Ninth Edition in SI Unit. McKenzie Examples in Structural Analysis. Draw shear force diagram and bending moment diagram 3. Show the critical values on the diagrams. Span length is 9 m. The beam shown in the photo is used to support a portion of the overhang for the entranceway of the building. The idealized model for the beam with the load acting on it is shown in Fig.

Plot the expressions for V and M for the segment. For the beam and loading shown, a Draw the shear and bending-moment diagrams, b Determine the maximum absolute values of the shear and bending moment. Shear and Moment Diagrams Calculate and draw the shear force and bending moment equations for the given structure. Solution for The beam whose support and loading condition is given, a Draw the shear force diagram b Show the bending moment diagram. An influence line for a given function, such as a reaction, axial force, shear force, or bending moment, is a graph that shows the variation of that function at any given point on a structure due to the application of a unit load at any point on the structure.

Given below are solved examples for calculation of shear force and bending moment and plotting of the diagrams for different load conditions of simply supported beam, cantilever and overhanging beam. All the steps of these examples are very nicely explained and will help the students to develop their problem solving skills. Moment of Inertia Calculator Calculate moment of inertia of plane sections e.