Name: shear change under distributed loading
Uploaded: 2025-05-22T09:06:04.000Z
Duration: 1 min 23 s
Channel: JoVE

Rozważmy belkę, w której występuje obciążenie rozłożone, dwa obciążenia skupione i kilka momentów. Ustal zależność między ścinaniem a rozłożonym obciążeniem.
Najpierw rozważ przekrój elementarny na belce wolny od jakiegokolwiek obciążenia skupionego lub momentu pary i narysuj wykres przekroju swobodnego ciała.
Aby zachować równowagę, siła tnąca działająca po prawej stronie przekroju powinna być zwiększana o małą i skończoną wartość.
Wypadkowa siła rozłożonego obciążenia działa w ułamkowej odległości od prawego końca przekroju.
Korzystając z równania równowagi dla siły pionowej, uzyskuje się zależność między ścinaniem a obciążeniem.
Następnie, dzieląc obie strony równania przez Δx i pozwalając, aby Δx zbliżyło się do zera, można określić nachylenie siły tnącej, które jest równe rozłożonemu natężeniu obciążenia.
Wreszcie, przegrupowanie równania i całkowanie obciążenia rozłożonego na przekroju elementarnym między dwoma dowolnymi punktami prowadzi do relacji między zmianą ścinania a obszarem pod krzywą obciążenia.

<ol>
	<li>Hibbeler, R.C. (2016). Mechanika inżynierska \u2012 Statyka i dynamika. Hoboken, New Jersey: Pearson Prentice Hall. str. 370-371.</li>
	<li>Piwo, F.P.; Johnston, E.R.; Mazurek, D.F.; Cromwell, PJ i Jaźń, BP (2019). Mechanika wektorowa dla inżynierów \u2012 Statyka i dynamika. Nowy Jork: McGraw-Hill. Strony 391-392.</li>
</ol>

relacja między rozłożonym obciążeniem a siłą poprzeczną

Zrozumienie relacji między rozłożonym obciążeniem a siłą poprzeczną jest kluczowe w analizie belek poddanych różnym warunkom obciążenia. Przyjrzyjmy się przypadkowi belki doświadczającej rozłożonego obciążenia, dwóch skupionych obciążeń i momentu skręcającego.

<img alt="Figure 1" src="/files/ftp_upload/14304/14304_Figure_1v2.svg" style="text-align: center; height: 300px;" />

Relację między siłą poprzeczną a rozłożonym obciążeniem dla danego przypadku można ustalić, stosując podaną procedurę. Po pierwsze, rozważmy fragment elementarny na belce, wolny od jakichkolwiek skupionych obciążeń lub momentu skręcającego. Możemy sporządzić diagram bieżący, aby przeanalizować siły działające na ten odcinek. Diagram przedstawia rozłożone obciążenie wzdłuż belki, siłę poprzeczną V(x) po prawej stronie odcinka oraz przyrost siły poprzecznej dV, zapewniający równowagę.

Aby utrzymać równowagę w kierunku pionowym, siła poprzeczna działająca po prawej stronie odcinka powinna być zwiększona o niewielką, skończoną wartość ΔV. Wypadkowa siła rozłożonego obciążenia, w(x)Δx, działa w pobliżu prawego końca odcinka.

Korzystając z równania równowagi siły w kierunku pionowym, można otrzymać następującą zależność między siłą poprzeczną a obciążeniem:

<img alt="Equation 1" src="/files/ftp_upload/14304/14304_Equation_1v2.svg" style="text-align: center; height: 13px;" />

Następnie podzielimy obie strony równania przez Δx i przybliżymy Δx do zera:

<img alt="Equation 2" src="/files/ftp_upload/14304/14304_Equation_2v2.svg" style="text-align: center; height: 32px;" />

To równanie pokazuje, że nachylenie siły poprzecznej jest równe intensywności rozłożonego obciążenia.

Wreszcie możemy przestawić równanie i zintegrować rozłożone obciążenie na odcinku elementarnym między dwoma dowolnymi punktami Q i R:

<img alt="Equation 3" src="/files/ftp_upload/14304/14304_Equation_3v2.svg" style="text-align: center; height: 50px;" />

To równanie pokazuje, że zmiana siły poprzecznej między punktami Q i R jest równa powierzchni pod wykresem funkcji obciążenia na tym odcinku. Różnica w sile poprzecznej między punktami Q i R jest równa powierzchni pod krzywą rozłożonego obciążenia między tymi dwoma punktami.

Relacja między rozłożonym obciążeniem a siłą poprzeczną

Zrozumienie zależności między obciążeniem rozłożonym a siłą tnącą w analizie statyczno-wytrzymałościowej ma kluczowe znaczenie dla analizy belek poddanych różnym ...

relation-between-the-distributed-load-and-shear

Education

JoVE Core

Mechanical Engineering

Inżynieria mechaniczna

Unit 1

Jednostka 1

Internal Forces

Siły wewnętrzne

KEY TERMS AND CONCEPTS

Kluczowe terminy i pojęcia

Relation Between the Distributed Load and Shear

shear change under distributed loading

Shear force changes along a beam under a distributed load. In structural analysis, this relationship helps explain how beams respond to loading conditions that also include concentrated loads and a couple moment.

To study the link, consider a small beam section that has no concentrated load or couple moment on it. Draw a free-body diagram for that elemental section. It includes the distributed load along the beam, the shear force V(x) on the right-hand side of the cut, and a small added shear force dV needed to keep the section in equilibrium.

The vertical forces must balance. For that reason, the shear force on the right-hand side changes by a small amount, ΔV. The distributed load over the small length Δx creates a resultant force of w(x)Δx. That force acts at a fractional distance from the right end of the section.

From the equilibrium equation for vertical forces, a relation between shear and load is obtained. When both sides are divided by Δx and Δx is allowed to approach zero, the slope of the shear force curve equals the distributed load intensity. This means the load level at a point controls how quickly shear changes along the beam.

The equation can also be rearranged and integrated over the elemental section between two points, Q and R. This integral form shows that the change in shear force equals the area under the distributed load curve between those points. In other words, the difference in shear force from Q to R is found by measuring the area under the load curve.

Shear Change Under Distributed Loading

sign convention

Beam Forces: Tension, Shear, and Bending

Beam analysis depends on the internal forces and moments inside a structure. These include normal force, shear force, and bending moment. Engineers use the method of sections to cut the beam and find these values. A sign convention helps show whether each result is positive or negative. Normal force acts perpendicular to the beam’s cross-section. It can pull the material in tension or push it in compression. Tension is taken as positive, while compression is taken as negative. This makes it...

normal and shear force

Beam Loads and Internal Forces

Beam loads create internal forces inside a structure. Weight, pressure, and other external forces can all act on a beam. These internal forces affect the beam’s stability and strength. Engineers study these forces to find their size and direction. A common approach is the method of sections. In this method, an imaginary cut is made at a chosen point on the beam, and the forces at that section are analyzed. One internal force is the normal force, also called the axial force. It acts...

bending and torsional moments

Forces That Bend and Twist Materials

Bending and torsional moments describe how materials respond to force. These ideas are important in structural engineering because they help explain how beams, rulers, and screws change shape under load. A moment depends on both the size of the force and the distance from the point of interest to the force’s line of action. A bending moment forms when an external force makes a structural element bend. If the element bends upward, the top side is in compression and the bottom side is in tension.

internal loadings in structural members: problem solving

internal-loadings-in-structural-members-problem-solving

Beam Internal Force Analysis at Point A

Beam internal force analysis shows how normal force, shear force, and bending moment act inside a structural member. These internal loadings help engineers check whether a beam can support the external forces applied to it. The example uses a beam OC that weighs 5 kN and is inclined at 53.13° with the horizontal. The beam is supported at both ends. The goal is to find the internal loadings at point A. The first step is to find the reaction force at point C. This is done by taking the moment...

beams

Beam Types and Support Conditions

Beams are key parts of structural engineering and construction. They support loads that act at different points along their length. Engineers use beam shape and support conditions to understand how a beam will perform. Beams can be classified by geometry. Straight beams are the most common and have a constant cross-section along their length. Tapered beams change in cross-section from one end to the other, while curved beams have a curved shape. Beam cross-section also matters. Common shapes...

shear diagram

Beam Shear Force Diagram

Beam shear force diagrams show how shear changes along the length of a supported beam. In beam mechanics, they help students track the internal shear force caused by perpendicular loads. The beam AB in this example is supported at both ends, so the first step is to draw a free-body diagram that shows the external forces and internal reactions. The reaction forces at supports A and B are found with equilibrium equations for force and moment. For the beam shown, the vertical reaction at A is 24...

bending moment diagram

Beam Load Analysis with Moment Diagrams

Beam load analysis uses a bending moment diagram to show how bending changes along the length of a beam. This diagram helps engineers and designers check whether a structure can handle applied forces. It is also useful for parts such as shelving arms, which must resist forces and moments from a load. To build the diagram, start by finding the reactive forces and any couple moments on the beam. In some cases, especially when a beam is inclined, these forces must be resolved into components that...

relation between the distributed load and shear

Shear force changes along a beam under a distributed load. In structural analysis, this relationship helps explain how beams respond to loading conditions that also include concentrated loads and a couple moment. To study the link, consider a small beam section that has no concentrated load or couple moment on it. Draw a free-body diagram for that elemental section. It includes the distributed load along the beam, the shear force V(x) on the right-hand side of the cut, and a small added shear...

relation between the shear and bending moment

relation-between-the-shear-and-bending-moment

Shear and Moment Diagrams in Beam Loads

Shear and moment diagrams help show how a beam responds to loads. A beam can carry a distributed load, concentrated loads, and a couple moment. These loads create internal shear forces and bending moments. The beam is examined by isolating a small elemental section and drawing a free-body diagram. For that section to stay in equilibrium, the moment on the right side must be slightly larger than the moment on the left side. The distributed load also has a resultant force that acts at a...

shear and bending moment diagram: problem solving

shear-and-bending-moment-diagram-problem-solving

Beam Load Analysis with Shear Diagrams

Beam load analysis uses shear and bending moment diagrams to track the internal forces in a beam. These diagrams are important when a beam carries concentrated loads and a distributed load. They help show how the beam responds so structures can be designed safely and efficiently. The first step is to draw a free-body diagram of the whole beam. A free-body diagram shows the concentrated loads, the distributed load, and the reaction forces at the supports. Use the equilibrium equations, where...

cable subjected to concentrated loads

Finding Tension in a Loaded Cable

Flexible cables can carry vertical point loads between two fixed supports. The cable is treated as flexible, inextensible, and nearly weightless. Under these conditions, it changes shape to form straight-line segments, and each segment carries a constant tensile force. To analyze the cable, a free-body diagram is drawn for the whole cable. This helps identify the reaction forces at the supports. In many cases, the number of unknown reaction components is greater than the available equilibrium...

cable subjected to a distributed load

Finding Cable Shape Under Distributed Load

Cable shape under a distributed load is found by analyzing the forces along a small piece of the cable. This method is important in suspension bridge design because the main cables must carry changing loads safely. The load causes the tensile force in the cable to change, and it also creates cable deformation. The analysis starts with a free-body diagram of a short cable segment. The diagram shows the distributed load and the change in tensile force across that segment. Equilibrium equations...

cable subjected to its own weight

Cable Sag Under Its Own Weight

Overhead power transmission cables sag under their own weight, and that shape affects how they carry electricity across long distances. To study this behavior, the cable is analyzed with a generalized loading function. This function treats the load along the cable's arc length instead of its projected length, which gives a more accurate picture of the cable's response. The analysis then focuses on a small cable segment. A free-body diagram is drawn to show the forces acting on that segment.

cable: problem solving

Cable Tension Under Uniform Loading

Cable tension under uniform loading is found by analyzing a cable fixed between two supports. The problem starts with the support conditions and the load on the cable. From there, the cable’s shape must be described with an equation that matches the curve formed by the load. The next step is to integrate the shape equation to get the cable’s shape function. This makes it possible to solve for constants in the equation. The boundary condition at the origin gives one constant. The slope, which...

key terms and concepts

Learn how distributed load affects shear force in beams, from a free-body diagram to the slope relation and the area under the load curve.

Shear Change Under Distributed Loading | JoVE K12

JoVE Core Mechanical Engineering offers concise, easy-to-understand animated videos covering key topics such as equilibrium, forces, truss analysis, and structural mechanics. Explore their real-world applications, and witness these concepts in action through scientist-led experiments.

mechanical engineering

Shear Change Under Distributed Loading

Description

Transcript

Tags