Shear-Force & Bending-Moment Diagrams - Graphical Methods

Oct 25, 2026

Learning notes on evaluating shear-force and bending-moment diagrams for beams with discrete loads. The material establishes sign conventions for distributed loads, positive internal shear, and positive bending moments, then provides a step-by-step procedure: find support reactions, define section origins between loads, sum vertical forces for V, and sum moments to determine M while eliminating V from moment equations. It includes graphical methods and worked examples for beams with a couple and point load, and for simply supported beams under trapezoidal loading, deriving reactions and expressions for V(x) and M(x).

LanguageEnglish

CategoryTechnology

FormatPDF

Pages23

Reading Time58 min

AccessFree to download

Learning Objectives

Beam Sign Convention

Procedure

Shear-Force & Bending-Moment DiagramsGraphical MethodsExample 1

Shear-Force & Bending-Moment DiagramsGraphical MethodsExample 2

# Shear-Force &Bending-Moment DiagramsGraphical Methods use ttltion ## Learning Objectives 1)To evaluate the shear-force and bending-moment diagrams for systemswith discrete loads. 2)To do an engineering estimate of these quantities. Beam Sign Convention Distributed load - An upward load is positive Shear Force -A positive internal shear force causes a clockwiserotation of beam segment.(i.e.,it pushes a left-facing cross-section upward or a right-facing cross-section downward). Bending Moment -A positive internal moment causes compression inthe top fibers of the segment (i.e.,clockwise on aleft-facing cross-section or counter-clockwise on aright-facing cross-section). ## Procedure 1.Determine support reactions 2.Specify beam sections origin (left end)to between each discrete load(force or moment).Be sure V and M are shown acting in the positivesense. 3.Sum forces vertically to determine V 4.Sum moments at sectioned end to determine M.(This eliminates Vfrom the moment equation). Left-facingcross section Right-facingcross sectionPositive shear FIGURE 7a Positive bending moment FIGURE 7b Free-body diagramof segment△x(b) △X-(V+oV=0 Zy=0 ∑M= : △x→ M =MGncendrzted Momtnt:Mx=: M+-M +M一V_(之×-V+(主)=0长 X-0》从-m=-△X2V+X V-=V Fy=:从MatV+M+-V-(1)]-V(0x)=三mX=0:△X→0V+-V=0 sx-F=0V-V+F+W(X。)x=M+Concentroted oad.Fa change inarea undersheardistributed loading change inarea undermomentshear diagram (p) ## Shear-Force &Bending-Moment DiagramsGraphical MethodsExample 1 Given:A simply-supported beam is loaded with a 2 kN-m couple and a4kN load as shown. Find:Using graphical methods,draw the shear-force and bending-moment diagrams. =3KN eacdion: 三MA=0:-2 -4(4)十By(6)=。Ау 48;— 4ва>Ay=1KNZF=:By= 3KN ## Shear-Force &Bending-Moment DiagramsGraphical MethodsExample 2 Given:A simply supported beam is supporting a trapezoidal loading as shown and is in staticequilibrium. Find: a)Determine the reactions at supports A and B. b)Write an algebraic expression for the trapezoidal loading shown(w(x)). ),determine an expression for the shear force as a function ofc)Using the relationshipx(i.e.,V(x)). ,determine an expression for the bending moment as ad)Using the relationship

FAQs

What is the sign convention for distributed load, shear force, and bending moment in these notes?

A distributed load is positive when the load acts upward. A positive internal shear force produces a clockwise rotation of the beam segment. A positive internal bending moment compresses the top fibers of the segment.

How is shear force V determined for a sectioned beam?

First determine support reactions, then choose beam sections between discrete loads. Sum vertical forces for the section to solve for V, using the requirement that V is shown in its positive sense.

How are bending moment M expressions obtained using the graphical methods procedure?

Take moments at the sectioned end to determine M. This approach eliminates V from the moment equation, so M is found by applying the summed moment condition with M shown in its positive sense.

Oct 25, 2026

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