Lecture No. 4 - Shear force and bending moment diagram - Procedure for Determining Shear force and Bending Moment Diagrams

Oct 29, 2026

Lecture No. 4 explains beam internal forces and how to construct shear force and bending moment diagrams. It introduces beams under transverse loading, outlines support types (simply supported, cantilever, overhanging) and load types (concentrated, distributed, and line load). It derives shear-moment equations for a cantilever with an end load, presents sign conventions, and gives a step-by-step method for obtaining V and M for statically determinate beams.

LanguageEnglish

CategoryTechnology

FormatPDF

Pages21

Reading Time53 min

AccessFree to download

Shear force and bending moment diagram

Introduction

Supports, Types and Loads

Shear-Moment Equations and Shear-Moment Diagrams

Sign Conventions

Procedure for Determining Shear force and Bending Moment Diagrams

Lecture No. 4 Shear force and bending moment diagram 6.1 Introduction The term beam refers to a slender bar that carries transverse loading; that is, the applied forces are perpendicular to the bar. In a beam, the internal force system consists of a shear force and a bending moment acting on the cross section of the bar. The study of beams, however, is complicated by the fact that the shear force and the bending moment usually vary continuously along the length of the beam. 6.2 Supports, Types and Loads Beams are classified according to their supports and may be summarized as: 􀁸 A simply supported beam, Figure 6-1, has a pin support at one end and a roller support at the other end. The pin support prevents displacement of the end of the beam, but not its rotation. The term roller support refers to a pin connection that is free to move parallel to the axis of the beam; hence, this type of support suppresses only the transverse displacement. 􀁸 A cantilever beam is built into a rigid support at one end, with the other end being free, Figure 6-1 b. The built-in support prevents displacements as well as rotations of the end of the beam. 􀁸 An overhanging beam, illustrated in Figure 6-1 c is supported by a pin and a roller support, with one or both ends of the beam extending beyond the supports. Figure 5-1 types of beams. A concentrated load, such as P in Figure 1-a, is an approximation of a force that acts over a very small area. In contrast, a distributed load is applied over a finite area. If the distributed load acts on a very narrow area, the load may be approximated by a line load. The intensity w of this loading is expressed as force per unit length (N/m) . The load distribution may be uniform, as shown in Figure 1-b, or it may vary with distance along the beam, as in Figure 1-c. 6.3 Shear-Moment Equations and Shear-Moment Diagrams Consider the cantilever beam shown in Figure 6-2 a, which is subjected to a concentrated load P at the free end. If a cutting plane at C is drawn, a free body diagram through this section (Figure 6-2 b) shows a shear forces Vand bending moment Mat the cutting section. It is the objective in this section to determine the shear force Vand the bending moment Mat every cross section of the beam. To accomplish this task, we must derive the expressions for V and M in terms of the distance x measured along the beam. By plotting these expressions to scale, we obtain the shear force and bending moment diagrams for the beam. Figure 5-2 (a) Cantilever beam subjected to a concentrated load,(b) Section through C 6.4 Sign Conventions It is necessary to adopt sign conventions for applied loading, shear forces, and bending moments. We will use the conventions shown in Figure 4, which assume the following tobe positive. Shear forces that tend to rotate a beam element clockwise. Bending moments that tend to bend a beam element concave upward. Figure 5-3: Sign Conventions. Procedure for Determining Shear force and Bending Moment Diagrams The following is a general procedure for obtaining shear force and bending moment diagrams of a statically determinate beam: 􀁸 Compute the support reactions from the FBD of the entire beam. 􀁸 Divide the beam into segments so that the loading within each segment is continuous. Thus, the end-points of the segments are discontinuities of loading, including concentrated loads and couples. Perform the following steps for each segment of the beam: 􀁸 Introduce an imaginary cutting plane within the segment, located at a distance x from the left end of the beam, that cuts the beam into two parts. 􀁸 Draw a FBD for the part of the beam lying either to the left or to the right of the cutting plane, whichever is more convenient. At the cut section, show VandM acting in their positive directions. 􀁸 Determine the expressions for V and M from the equilibrium equations obtainable from the FBD. These expressions, which are usually functions of x, are

FAQs

What are shear force and bending moment in beam analysis?

For a beam under transverse loading, the internal force system consists of a shear force and a bending moment acting on the beam’s cross section. Their values typically vary continuously along the beam length.

How are concentrated loads and distributed loads distinguished?

A concentrated load acts over a very small area, while a distributed load acts over a finite area. If the distributed load is over a very narrow area, it can be approximated as a line load, with intensity expressed as force per unit length.

What is the standard procedure to determine shear force and bending moment diagrams?

Compute support reactions from the entire beam FBD, divide the beam into segments with continuous loading, then for each segment draw an imaginary cutting plane at distance x. Use equilibrium of the chosen side to derive V(x) and M(x), mark shear and moment in their positive directions, and plot the SFD (V-diagram) and BMD (M-diagram) accordingly.

Oct 29, 2026

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