# Bending Moment And Shear Force In Statically Determinate Beams Pdf

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Statics observes the effect of forces on a rigid body, ignoring any possible deformations which may occur in the process. The forces are in equilibrium.

## CHAPTER 3- SHEAR FORCE, BENDING MOMENT OF STATICALLY DETERMINATE BEAM

Chapter 9. Influence Lines for Statically Determinate Structures. Structures such as bridges and overhead cranes must be designed to resist moving loads as well as their own weight. Since structures are designed for the critical loads that may occur in them, influence lines are used to obtain the position on a structure where a moving load will cause the largest stress.

Influence lines can be defined as a graph whose ordinates show the variation of the magnitude of a certain response function of a structure as a unit load traverses across the structure. Response functions of a structure may include axial forces in members, support reactions, bending moments, shear forces, and deflection at specific points in the structure.

It is very important to emphasize the need for students to fully grasp the afore-stated definition, since most of the confusion and difficulty encountered when drawing influence lines stems from a lack of understanding of the difference between this topic and the bending moment and shearing force topics detailed in chapter four. A shearing force or bending moment diagram shows the magnitude of the shearing force or bending moments at different points of the structure due to the static or stationary loads that are acting on the structure, while the influence lines for certain functions of a structure at a specified point of the structure show the magnitude of that function at the specified point when a unit moving load traverses across the structure.

The influence lines of determinate structures can be obtained by the static equilibrium method or by the kinematic or Muller-Breslau method. Influence lines by the static equilibrium method are referred to as quantitative influence lines, as they require some calculations, while those by kinematic method are known as the qualitative influence lines, as the method enables the analyzer to obtain the correct shape of the influence lines without any quantitative efforts.

In the subsequent sections, students will consider how to construct the influence lines for beams and trusses using these two methods. To grasp the basic concept of influence lines, consider the simple beam shown in Figure 9.

Statics help to determine the magnitude of the reactions at supports A and B , and the shearing force and bending moment at a section n , as a unit load of arbitrary unit, moves from right to left. Taking the moment about B as the unit load moves a distance x from the right-hand end suggests the following:. Equation 9. The influence line for R A can be represented graphically by putting some values of x into the equation.

Since the equation is linear, two points should be enough. The graphical representation of the influence line for R A is shown in Figure 9. Similarly, the expression for the influence line for the reaction R B is found by taking the moment about A. Substituting some values for x into the equation helps to construct the influence line diagram for R B. The graphical representation of the influence line for R B is shown in Figure 9. When the unit load is on the right side of the section, the shear force at the section can be computed considering the transverse forces on the left side of the section, as follows:.

When the unit load is on the left side of the section, it is easier to compute the shear force in the section by considering the forces on the right side of section, as follows:. The graphical representation of the influence line for the shearing force at a section n of the simple beam is shown in Figure 9. When the unit load is on the right side of the section, the bending moment at the section can be computed as follows:.

When the unit load is on the left side of section, the bending moment at the section can be computed as follows:. The graphical representation of the influence line for the bending moment at a section n of the simple beam is shown in Figure 9. In practice, influence lines are mostly constructed, and the values of the functions are determined by geometry.

The procedure for the construction of influence lines for simple beams, compound beams, and trusses will be outlined below and followed by a solved example to clarify the problem. For each case, one example will be solved immediately after the outline. The procedures for the construction of the influence lines I.

Point B is at the position of support B. The idea here is that when the unit load moves across the beam, its maximum effect on the left-end reaction will be when it is directly lying on the left end support. As the load moves away from the left end support, its influence on the left end reaction will continue to diminish until it gets to the least value of zero, when it is lying directly on the right end support.

The explanation for the influence line for the right end support reaction is similar to that given for the left end support reaction. The maximum effect of the unit load occurs when it is lying directly on the right support. As the load moves away from the right end support, its influence on the support reaction decreases until it is zero, when the load is directly lying on the left support. Influence line for shear b and moment c at secton m.

For example, the distance a in Figure 9. Alternatively, ignore steps b , and c and d and go to step f. For example, the distance from the right end support to the section n denoted as point Z. For example, the distance from the left end support to the section n denoted point Y.

If accurately drawn, with the right sense of proportionality, the intersection Q should lie directly on a vertical line passing through the section n. For the double overhanging beam shown in Figure 9. Step 1. Step 2. Step 3. Continue the straight line in step 2 until the end of the overhangs at both ends of the beam. The influence line for B y is shown in Figure 9.

Step 4. Determine the ordinates of the influence line at the overhanging ends using a similar triangle, as follows:. Continue the straight line at C until the end of the overhang at end D. At the position of support C point C , plot an ordinate —1. Draw a straight line connecting the plotted point —1 to the zero ordinate at the position of support B. Continue the straight line at B until the end of the overhang at end A. Step 5. Draw a vertical passing through the section whose shear is required to intersect the lines in step 2 and step 3.

Step 6. Connect the intersections to obtain the influence line, as shown in Figure 9. Step 7. Determine the ordinates of the influence lines at other points by using similar triangles, as previously demonstrated.

Draw a straight line connecting the plotted ordinate in step 1 to the zero ordinate in support C. Draw a straight line connecting the plotted ordinate in step 3 to the zero ordinate at support B. Continue the straight lines from the intersection of the lines drawn in steps 2 and 4 through the supports to the overhanging ends, as shown in Figure 9.

Determine the values of the influence lines at other points using similar triangles, as previously demonstrated. For the beam with one end overhanging support B, as shown in Figure 9. The influence lines in example 9. To correctly draw the influence line for any function in a compound beam, a good understanding of the interaction of the members of the beam is necessary, as was discussed in chapter 3, section 3. The student should recall from the previous section that a compound beam is made up of the primary structure and the complimentary structure.

The two facts stated below must always be remembered, since the extent of the spread of the influence line of compound beams depends on them. Remembering these facts will also serve as a temporary check to ascertain the correctness of the drawn influence line.

The moving unit load will have an effect on the functions of the primary structure when it is located at any point, not only on the primary structure but also on the complimentary structure, since the latter constitutes a loading on the former. The moving unit load will have effect only on the functions of the complimentary structure when it is located within the complimentary structure; it will not have an effect on any function of the complimentary structure when it is at any point on the primary structure.

For the compound beam shown in Figure 9. Prior to the construction of the influence lines for desired functions, it is necessary to first observe the extent of the influence lines through the schematic diagram of member-interaction, as shown in Figure 9.

The reaction A y is a function in the primary structure, so the unit load will have influence on this function when it is located at any point on the beam, as was previously stated in section 9.

With this understanding, construct the influence line of A y , as follows:. Draw a straight line connecting the plotted ordinate in step 1 to the zero ordinate in support B and continue this line until the end of the overhanging end of the primary structure, as shown in the interaction diagram.

Draw a straight line connecting the ordinate at the end of the overhang to the zero ordinate at support D. The influence line is as shown in Figure 9. The influence line for this reaction will cover the entire length of the beam because it is a support reaction in the primary structure. With this knowledge, construct the influence line for B y , as follows:.

Draw a straight line connecting the plotted ordinate in step 1 to the zero ordinate in support A. Continue the line in support B until the end of the overhanging end of the primary structure, as shown in the interaction diagram. Draw a straight line connecting the ordinate at the overhanging end to the zero ordinate at support D. Use a similar triangle to determine the values of the ordinate of the influence line. The reaction D y is a function in the complimentary structure and will be influenced when the unit load lies at any point along the complimentary structure.

It will not be influenced when the unit load transverses the primary structure, as was stated in section 9. Thus, the extent of the influence line will be the length of the complimentary structure. Knowing this, draw the influence line for D y. Draw a straight line connecting the plotted ordinate in step 1 to the zero ordinate at hinge C. The influence line for D y is as shown in Figure 9. The influence lines for the moment at B and the shear C are shown in Figure 9.

Shown in Figure 9. The schematic diagram of the member interaction shown in Figure 9. Construction of the influence lines follows the description outlined in the previous sections.

Thus far, the examples and text have only considered cases where the moving unit load is applied directly to the structure. But, in practice, this may not always be the case. For instance, sometimes loads from building floors or bridge decks are transmitted through secondary beams, such as stringers and cross beams to girders supporting the building or bridge floor system, as shown in Figure 9.

## types of loads on beams pdf

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Objectives To understand the structural behaviour of beams. To determine the effects of external loads such as axial. To derive the differential relationships between the load intensity shear force and bending moment. Bending moment: The algebric sum of the moments of. Shear force diagram: A diagram in which ordinate represents shear force and abscissa represents the position of the section is called shear force diagram. Bending moment diagram: A diagram in which ordinate represents bending moment and abscissa represents the position of the section is called bending moment diagram. Cantilever subjected to: a concentrated load at free end uniformly distributed load over entire span uniformly varying load over entire span.

Figure (a) Statically determinate beams. Supports and Loads. ❑Beams are classified according to their supports. A simply supported beam.

## 1.9: Influence Lines for Statically Determinate Structures

Shear and bending moment diagrams are analytical tools used in conjunction with structural analysis to help perform structural design by determining the value of shear force and bending moment at a given point of a structural element such as a beam. These diagrams can be used to easily determine the type, size, and material of a member in a structure so that a given set of loads can be supported without structural failure. Another application of shear and moment diagrams is that the deflection of a beam can be easily determined using either the moment area method or the conjugate beam method. Although these conventions are relative and any convention can be used if stated explicitly, practicing engineers have adopted a standard convention used in design practices.

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### 1.9: Influence Lines for Statically Determinate Structures

When the beam is statically determinate, the external and internal forces of the beam can be Definition of Statically analysed using the statics Determinate Beam. Roller support Similar as pinned support but it cannot resist forces either in horizontal OR vertical direction. Exert forces in both horizontal axial or normal force and vertical forces shear force and resist any rotations of the beam bending moments. Cable Joint Exert forces in its axis only axial or normal force Hinge Joint Exerts forces in both horizontal axial or normal force and vertical force shear force directions. Simplfying equation 2. Manipulating eqn 2. The increase in bending moment between two sections is given by the area of the Vshear force between two section.

This type of beam may be used when the designer wants to control the deflection at the mid-span because the two fixed supports prevent rotation. You should judge your progress by completing the self assessment exercises. There are different types of beams are available along with the different applied loads. Roller Supports. A cantilever beam is fixed at one end and free at other end. The beam end resists to take any kind of translation or bending moment.

#### Equipment for Engineering Education

A Beam is defined as a structural member subjected to transverse shear loads during its functionality. Due to those transverse shear loads, beams are subjected to variable shear force and variable bending moment. Shear force at a cross section of beam is the sum of all the vertical forces either at the left side or at the right side of that cross section. Bending moment at a cross section of beam is the sum of all the moments either at the left side or at the right side of that cross section. A beam is said to be statically determinate if all its reaction components can be calculated by applying three conditions of static equilibrium. When the number of unknown reaction components exceeds the static conditions of equilibrium, the beam is said to be statically indeterminate. Shear force: If moving from left to right, then take all upward forces as positive and downward as negative.

When the beam is statically determinate, the external and internal forces of the beam can be Definition of Statically analysed using the statics Determinate Beam. Roller support Similar as pinned support but it cannot resist forces either in horizontal OR vertical direction. Exert forces in both horizontal axial or normal force and vertical forces shear force and resist any rotations of the beam bending moments. Cable Joint Exert forces in its axis only axial or normal force Hinge Joint Exerts forces in both horizontal axial or normal force and vertical force shear force directions. Simplfying equation 2. Manipulating eqn 2. The increase in bending moment between two sections is given by the area of the Vshear force between two section.

ME Mechanical Team; 5 years ago ; Views; 0 0. The diagrams show the way that point loads and uniform loads are illustrated.

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05.03.2021 at 15:16

1. Harcourt C.

Chapter 9.

2. Ronan P.

loads, shear force and bending moment at any section of the beam. To derive the Classification of statically determinate beams. Cantilever beams.