Lecture17_2023F_annotated_1-11 - Sign conventions for bending moments and shear forces

Oct 25, 2026

Lecture notes defining sign conventions for internal bending moments and shear forces in beams, including how positive moments and shears are assigned on left and right faces of a section and how these relate to clockwise/counterclockwise rotations and concave curvature. The notes also set sign conventions for external distributed loads, concentrated loads, and external couples, then derive core equilibrium relationships connecting loading p(x), shear force V(x), and bending moment M(x).

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a) Sign conventions for bending moments and shear forces

Sign conventions for external loadings on beams

Geometric meaning of the equilibrium relationships for beams

c) Bending-moment and shear-force diagrams

Three methods for determining internal shear force and bending moment resultants

Lecture Notes # a)Sign conventions for bending moments and shear forces Sign conventions to be used in this course for internal bending moments and shearforces(see following figure): ·A positive bending moment M on the left face (negative x-face)of a section isCW.A positive bending moment M on the right face(positive x-face)of asection is CCW.Such a positive bending moment creates a concave curvature inthe deflection of the beam. ·A positive shear force V on the left face (negative x-face)of a section is inpositive y-direction.A positive shear force V on the right face (positive x-face)ofa section is in negative y-direction. When making a cut through a cross section of the beam,the positive sign conventionsfor the bending moment and shear force are as shown below: positive bending moment andshear force at cut in beam Sign conventions for external loadings on beams: ● Positive EXTERNAL distributed loads p(x)and EXTERNAL concentrated loadsPo act in the"+"y-direction: ● Positive EXTERNAL couples are in the“+”z-direction (CCW by the right handrule): | applied loading | FBD | key relationship(s) | | --- | --- | --- | | p(x)
x | p(x)△r
M(x)
M(x+Ar) | | | x | P₀
V(x⁻)
V(x+)
EF | v(x+)=v(x-)+P₀ | | | | 1 | )+x-V+Ax===+)(三M:-M人)+MXx小母小一y)A、p4xA( ): | Geometric meaning of the equilibrium relationships for beams
The slope of the shear force diagram at any location x equals the value of the distributed
external loading p at that location.
(integral form of te above)
The shear force at point x₂is equal to the shear force at x₁plus the area under the external
loading curve between these two points.
the bending moment diagram at any location x equals the value of the shear force
at that location.
M(x₂)=M(x)+
(integral form of the above)
●
The bending moment at point x₂is equal to the bending moment at x plus the area under the
external loading curve between these two points.
·V(x+)=V(x⁻)+P₀
The shear force diagram has an upward step jump at location x where an external point force
is applied.The value of the shear force jump increase equals the value of the external point
force.
·M(x+)=M(x⁻)-M₀
The bending moment diagram has a downward step jump at location x where an external
point moment is applied.The value of the bending moment jump decrease equals value of
the external point moment.
—1. | | --- | ∫(M(x)dxvxAc:N (x # c)Bending-moment and shear-force diagrams Three methods for determining the internal shear force and bending moment resultants: ·Using free body diagrams with cut sections,as demonstrated in the earlierexamples of this section of notes5good fo andtag ● Using equilibrium relationships among applied loads,shear force and bendingmoments deriveddiscontinuities).● Using a graphical meon the integration and discontinuity equations good foy complexloagsfrom the equilibrium method.The description of this method follows. goodo ml e lodire olie mounls Graphical method for constructing shear force and bending momentdiagrams Sign conventions: positive bending moment Basic relationships (as derived via equilibrium relations): Concentrated shear force Vo applied at location x: Concentrated moment Mo applied at location x: M(x⁺)=M(x)-M₀(jump DOWNin moment) M₁Beams:Flexural and shear stressesMechanics of Materials ## 1.)F:nd rsulławts 3)mpwcal·Example 9.3 Two transverse forces and a couple are applied as external loads to the cantileveredbeam AC.Draw the shear force and bending moment diagrams in the plot axes below. BAC6fi6fiMA10 kips6kipsVwMe. 0)(巨N。一从I网)-Ayx-10x-6 :・6kp张A-10=0-15:0 BCN,{K)=a+Ax=-8+4x.Dk] M(6)=-8+24=16M)、二-MA+M,-A4x=0yv:0=y=Ag=kMp区)sAB 1M=-8kitMD 20 kip·fi.V(x) h X -6.MM·X Topic9:10Mechanics of MaterialsM₂k)=-8+4×-10x+60=52~6x.M(2)=52一收=-20nt(6=1 Example 9.3 Two transvers

FAQs

How are positive bending moments defined on the left and right faces of a beam section?

A positive bending moment on the left face (negative x-face) causes clockwise (CW) curvature, while a positive bending moment on the right face (positive x-face) causes counterclockwise (CCW) curvature.

How are positive shear forces defined relative to the left and right faces?

A positive shear force on the left face acts in the positive y-direction, and a positive shear force on the right face acts in the negative y-direction.

What equilibrium relationships connect the distributed load p(x) with the shear-force and bending-moment diagrams?

The slope of the shear-force diagram at any location x equals p(x). The bending moment at x equals the value of the shear force at that location, and changes in bending moment between two points equal the area under the external loading curve between them.

Oct 25, 2026

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