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Shear force and bending moment diagrams are analytical tools used in conjunction with
structural analysis Structural analysis is a branch of solid mechanics which uses simplified models for solids like bars, beams and shells for engineering decision making. Its main objective is to determine the effect of loads on physical structures and their c ...
to help perform
structural design Structural engineering is a sub-discipline of civil engineering in which structural engineers are trained to design the 'bones and joints' that create the form and shape of human-made structures. Structural engineers also must understand and c ...
by determining the value of
shear force In solid mechanics, shearing forces are unaligned forces acting on one part of a Rigid body, body in a specific direction, and another part of the body in the opposite direction. When the forces are Collinearity, collinear (aligned with each ot ...
s and
bending moment In solid mechanics, a bending moment is the Reaction (physics), reaction induced in a structural element when an external force or Moment of force, moment is applied to the element, causing the element to bending, bend. The most common or simplest ...
s at a given point of a
structural element In structural engineering, structural elements are used in structural analysis to split a complex structure into simple elements (each bearing a structural load). Within a structure, an element cannot be broken down (decomposed) into parts of dif ...
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 Structural integrity and failure is an aspect of engineering that deals with the ability of a structure to support a designed structural load (weight, force, etc.) without breaking and includes the study of past structural failures in order to ...
. Another application of shear and moment diagrams is that the
deflection Deflection or deflexion may refer to: Board games * Deflection (chess), a tactic that forces an opposing chess piece to leave a square * Khet (game), formerly ''Deflexion'', an Egyptian-themed chess-like game using lasers Mechanics * Deflection ...
of a beam can be easily determined using either the moment area method or the conjugate beam method.


Convention

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.


Normal convention

The normal convention used in most engineering applications is to label a positive shear force - one that spins an element clockwise (up on the left, and down on the right). Likewise the normal convention for a positive bending moment is to warp the element in a "u" shape manner (Clockwise on the left, and counterclockwise on the right). Another way to remember this is if the moment is bending the beam into a "smile" then the moment is positive, with compression at the top of the beam and tension on the bottom. This convention was selected to simplify the analysis of beams. Since a horizontal member is usually analyzed from left to right and positive in the vertical direction is normally taken to be up, the positive shear convention was chosen to be up from the left, and to make all drawings consistent down from the right. The positive bending convention was chosen such that a positive shear force would tend to create a positive moment.


Alternative drawing convention

In
structural engineer Structural engineers analyze, design, plan, and research List of structural elements, structural components and structural systems to achieve design goals and ensure the safety and comfort of users or occupants. Their work takes account mainly of ...
ing and in particular
concrete Concrete is a composite material composed of aggregate bound together with a fluid cement that cures to a solid over time. It is the second-most-used substance (after water), the most–widely used building material, and the most-manufactur ...
design the positive moment is drawn on the tension side of the member. This convention puts the positive moment below the beam described above. A convention of placing moment diagram on the tension side allows for frames to be dealt with more easily and clearly. Additionally, placing the moment on the tension side of the member shows the general shape of the deformation and indicates on which side of a concrete member
rebar Rebar (short for reinforcement bar or reinforcing bar), known when massed as reinforcing steel or steel reinforcement, is a tension device added to concrete to form ''reinforced concrete'' and reinforced masonry structures to strengthen and aid ...
should be placed, as concrete is weak in tension.


Relationships among load, shear, and moment diagrams

Since this method can easily become unnecessarily complicated with relatively simple problems, it can be quite helpful to understand different relations between the loading, shear, and moment diagram. The first of these is the relationship between a distributed load on the loading diagram and the shear diagram. Since a distributed load varies the shear load according to its magnitude it can be derived that the slope of the shear diagram is equal to the magnitude of the distributed load. The relationship, described by Schwedler's theorem, between distributed load and shear force magnitude is: :\frac = -q Some direct results of this is that a shear diagram will have a point change in magnitude if a point load is applied to a member, and a linearly varying shear magnitude as a result of a constant distributed load. Similarly it can be shown that the slope of the moment diagram at a given point is equal to the magnitude of the shear diagram at that distance. The relationship between distributed shear force and bending moment is: :\frac = Q A direct result of this is that at every point the shear diagram crosses zero the moment diagram will have a local maximum or minimum. Also if the shear diagram is zero over a length of the member, the moment diagram will have a constant value over that length. By calculus it can be shown that a point load will lead to a linearly varying moment diagram, and a constant distributed load will lead to a quadratic moment diagram.


Practical considerations

In practical applications the entire stepwise function is rarely written out. The only parts of the stepwise function that would be written out are the moment equations in a nonlinear portion of the moment diagram; this occurs whenever a distributed load is applied to the member. For constant portions the value of the shear and/or moment diagram is written right on the diagram, and for linearly varying portions of a member the beginning value, end value, and slope or the portion of the member are all that are required.


See also

*
Bending In applied mechanics, bending (also known as flexure) characterizes the behavior of a slender structural element subjected to an external Structural load, load applied perpendicularly to a longitudinal axis of the element. The structural eleme ...
*
Euler–Bernoulli beam theory Euler–Bernoulli beam theory (also known as engineer's beam theory or classical beam theory) is a simplification of the linear elasticity, linear theory of elasticity which provides a means of calculating the load-carrying and Deflection (engine ...
*
Bending moment In solid mechanics, a bending moment is the Reaction (physics), reaction induced in a structural element when an external force or Moment of force, moment is applied to the element, causing the element to bending, bend. The most common or simplest ...
* Singularity function#Example beam calculation


References


Further reading

* Cheng, Fa-Hwa. "Shear Forces and Bending Moments in Beams" Statics and Strength of Materials. New York: Glencoe, McGraw-Hill, 1997. Print. * Spotts, Merhyle Franklin, Terry E. Shoup, and Lee Emrey. Hornberger. "Shear and Bending Moment Diagrams." Design of Machine Elements. Upper Saddle River, NJ: Pearson/Prentice Hall, 2004. Print.


External links

{{DEFAULTSORT:Shear And Moment Diagram Beam theory Continuum mechanics Diagrams Moment (physics) Structural analysis