Name: finding resultant loads on a beam
Uploaded: 2023-09-22T11:49:22.000Z
Duration: 1 min 29 s
Channel: JoVE

קרן באורך L הנתונה לעומס משתנה, שהוא שילוב של חלוקת עומס פרבולית וטרפזית לאורך ציר ה-x.
בעומס הפרבולי, שקול אלמנט דיפרנציאלי של כוח dR על אורך dx. העומס המתקבל מתקבל על ידי שילוב אלמנט הדיפרנציאל לאורך הטעינה. הוא ממוקם בצנטרויד של האזור הפרבולי, וקואורדינטת x של הצנטרויד נקבעת באמצעות עקרון המומנט.
ניתן לחלק את האזור הטרפזי לאזורים מלבניים ומשולשים, כאשר העומס המתקבל פועל בצנטרואידים שלהם. צנטרואידים אלה ממוקמים במחצית אורך המלבן ובשליש מאורך הבסיס מהצד האנכי של המשולש.
העומס המתקבל עבור האזור הטרפזי מתקבל על ידי הוספת העומסים המתקבלים הבודדים ועובר דרך הצנטרויד של האזור, שנקבע על ידי עקרון הרגע.
העומס המבוזר הכולל הוא סכום העומסים המתקבלים של האזורים הטרפזיים והפרבוליים, העוברים דרך הצנטרויד של השטח המשולב, שנקבע על ידי עקרון הרגע.

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	<li>מרים, י"ל; Kraige, L.G. ובולטון, J.N. (2020). מכניקה הנדסית - סטטיקה. הובוקן, ניו ג'רזי: ג'ון ווילי. עמ' 273</li>
	<li>היבלר, R.C. (2016). מכניקה הנדסית - סטטיקה ודינמיקה. הובוקן, ניו ג'רזי: אולם פירסון פרנטיס. עמ' EM-2</li>
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עומס לאורך ציר בודד

<div style="direction:rtl">בהנדסת מבנים, ניתוח קורות הנתונות לעומסים משתנים הוא היבט קריטי בהבנת ההתנהגות והביצועים של אלמנטים מבניים אלה. תרחיש נפוץ כולל קרן הנתונה לשילוב של התפלגות עומס שונים. חשבו על קורה באורך L הנתונה לעומס משתנה, שהוא שילוב של חלוקת עומס פרבולית וטרפזית לאורך ציר ה-x. במקרה זה, חיוני לקבוע את העומסים הנוצרים, את מיקומם ואת מרכז השטח המשולב כדי לחזות את תגובת הקורה בתנאי העמסה אלו. ראשית, בדוק את התפלגות העומס הפרבולי. חשבו על אלמנט דיפרנציאלי של כוח dR הפועל על אורך קטן dx. כדי לקבוע את העומס המתקבל עבור השטח הפרבולי, אלמנט הדיפרנציאלי dR משולב על פני כל אורך העומס. מיקומו של העומס הנוצר הזה הוא במרכז השטח הפרבולי. עיקרון המומנט מיושם כדי למצוא את קואורדינטת ה-x של המרכז, הקובע שהמומנט הראשון של השטח סביב ציר שווה למכפלת השטח ולמרחק המרכז שלו מהציר. לאחר מכן, חלוקת העומס הטרפז מנותחת על ידי חלוקתו לשני אזורים מלבניים ומשולשים. העומסים הנובעים עבור אזורים בודדים אלה פועלים במוקדים שלהם. עבור השטח המלבני, המרכז ממוקם במחצית מאורך המלבן. עבור השטח המשולש, המרכז נמצא במרחק שליש מאורך הבסיס מהצד האנכי של המשולש. על ידי הוספת העומסים הנוצרים הבודדים עבור האזורים המלבניים והמשולשים, ניתן לקבוע את העומס המתקבל עבור כל השטח הטרפז. מיקום העומס הנוצר עובר דרך מרכז השטח של הטרפז, אותו ניתן לקבוע גם באמצעות עקרון המומנט. בדומה, על ידי הוספת העומסים הנוצרים של האזור הפרבולי והטרפז כאחד, ניתן לקבוע את העומס הכולל הפועל על הקורה. המיקום של העומס הכולל הנוצר עובר דרך מרכז השטח המשולב, אשר ניתן לקבוע עוד יותר באמצעות עקרון המומנט.</div>

בהנדסת מבנים, ניתוח של קורות נתון עומסים משתנים הוא היבט קריטי של הבנת ההתנהגות והביצועים של אלה ...

Education

JoVE Core

Mechanical Engineering

הנדסת מכונות

Unit 1

יחידה 1

Force System Resultants

תוצאות מערכת כוחות

KEY TERMS AND CONCEPTS

מונחים ומושגים מרכזיים

Load along a Single Axis

finding resultant loads on a beam

Structural engineering often requires finding the resultant load on a beam with changing force patterns. Here, a beam of length L carries a varying load made from parabolic and trapezoidal load distributions along the x-axis. The goal is to find the total load, where it acts, and the centroid of the combined area so the beam’s response can be predicted.

The parabolic load is handled first. A small force element, dR, acts over a short distance, dx, and these elements are integrated across the full length of the load. The resultant load for the parabolic area acts at the centroid of that area. The x-coordinate of the centroid is found with the moment principle, which says the first moment of an area about an axis equals the area times the distance to its centroid.

The trapezoidal load is then split into two simpler shapes. One part is rectangular, and the other is triangular. Their resultant loads act at their own centroids. For the rectangle, the centroid lies halfway along its length. For the triangle, the centroid is one-third of the base length from the vertical side.

After those parts are found, their loads are added to get the resultant load of the full trapezoidal area. That load passes through the centroid of the trapezoid, which can also be found with the moment principle. Finally, the resultant loads from the parabolic and trapezoidal areas are combined. Their sum gives the total load on the beam, and its location passes through the centroid of the combined area.

Finding Resultant Loads on a Beam

moment of a force: scalar formulation

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Torque, or the moment of a force, describes how a force can make an object turn about an axis. It is a vector quantity, so it has both size and direction. This idea is important in physics, mechanics, and engineering. A simple flywheel example shows how torque works. The flywheel rotates about point O when a force is applied to it. The moment arm is the perpendicular distance from point O to the line of action of the force. The moment, in newton-meter (N-m), is found by multiplying the force...

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Net Moment Sign Convention in 2D Forces

Net moment in two-dimensional force systems shows how multiple forces can cause rotation about a fixed point. The scalar formulation helps analyze equilibrium in structures that are acted on by several forces. It focuses on the x-y plane and uses a simple sign rule to describe the direction of rotation. To find the resultant moment, the moments from all forces in the system are considered about a chosen point A. Each moment is assigned a sign based on its sense of rotation. Positive moments...

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Moment of Force and Direction Rules

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cartesian form for vector formulation

Calculating Moment with Cartesian Vectors

Calculating moment with Cartesian vectors shows how position and force vectors work together. The moment of force is the cross product of these vectors, so it is a vector quantity. In Cartesian form, unit vectors help write the vectors in component form. The cross product can be arranged in determinant form. That determinant is then expanded to find the Cartesian vector form of the moment of force. This method gives a clear way to work through the calculation step by step. A revolving door...

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Resultant moment describes the total turning effect of multiple forces acting on an object. It shows how several forces can combine to make a body rotate about a point. In physics, moment is the tendency of a force to cause rotation. The resultant moment of a force system is a vector quantity. That means it can be added and subtracted like other vectors. For a system with two forces, F1 and F2, acting on a pole at points A and B, the resultant moment is written as MR = (rA x F1) + (rB x F2).

principle of moments

Resolving Forces in Moment Calculations

The principle of moments, also called Varignon's theorem, explains how a force creates rotation about a point. It is a key idea in physics and engineering. It also helps describe when a rigid body stays in equilibrium under external forces. The moment of a force is found by multiplying the force by the perpendicular distance from the point of application to the point being studied. A larger force or a larger distance gives a larger moment. This makes the idea useful for checking how strongly a...

principle of moments: problem solving

Torque About an Axis in 3D Systems

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Bicycle Torque and Moment About an Axis

The moment of a force about an axis describes how a force can make an object turn around a fixed line. In scalar analysis, the moment is found from the perpendicular distance between the axis of rotation and the line of action of the force. That distance is called the moment arm. A bicycle gives a clear example of this idea. When the cyclist pushes on the pedal, the chain applies a force to the wheel’s axle and makes it rotate. The axis of rotation passes through the axle, and the moment arm...

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Torque on a Bicycle Wheel: Vector Method

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couple

Couple Moment in Force Systems

A couple is a pair of parallel forces that are equal in magnitude and opposite in direction. The two forces are separated by a perpendicular distance called the couple arm. Together, they create a turning effect on a body instead of a net push. This turning effect is called the couple moment. It rotates the body about an axis that is perpendicular to the plane of the forces. The SI unit for a couple moment is the Newton-meter (N-m). A common example is tightening a bolt with a lug wrench. In...

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Couple Moments and the Right-Hand Rule

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equivalent couples

Equivalent Couples in Mechanical Systems

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moment of a couple: problem solving

Torque from a Couple: Solving Forces

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system of forces and couples

Equivalent Force and Moment Systems

Structural systems often experience several forces and couple moments at the same time. Engineers simplify these loading cases by replacing them with an equivalent force and moment system at a chosen point O. This makes the analysis easier while keeping the same external effects on the member. A key idea in this process is the principle of transmissibility. A force is a sliding vector, which means it can be moved anywhere along its line of action without changing how the body behaves...

simplification of a force and couple system i

simplification-of-a-force-and-couple-system-i

Resultant Forces in Force and Couple Systems

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simplification of a force and couple system: ii

simplification-of-a-force-and-couple-system-ii

Resultant Force and Wrench Motion

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distributed loads

Load Distribution on Shelves and Dams

Load distribution helps engineers describe how force spreads across a surface or structure. A distributed load is a continuous force per unit area, rather than a single point force. It is a common idea in many practical situations. A bookshelf full of stacked books is one simple example. The weight of the books is spread along the shelf, so the pressure changes across the surface. This pressure is measured in Newtons per square meter, or Pascals. A hydroelectric dam gives another clear...

load along a single axis

Structural engineering often requires finding the resultant load on a beam with changing force patterns. Here, a beam of length L carries a varying load made from parabolic and trapezoidal load distributions along the x-axis. The goal is to find the total load, where it acts, and the centroid of the combined area so the beam’s response can be predicted. The parabolic load is handled first. A small force element, dR, acts over a short distance, dx, and these elements are integrated across the...

distributed loads: problem solving

Equivalent Resultant Loads on Beams

Equivalent resultant loads on beams can be found by breaking a distributed load into smaller regions. This makes the beam easier to analyze. Each region can then be handled on its own before the results are combined. The magnitude of the equivalent resultant load for each region comes from the area of that region. In this setting, area represents the force applied by the distributed load over that part of the beam. After finding the force for each region, the position of each resultant load...

key terms and concepts

Learn how parabolic and trapezoidal loads on a beam are broken into simpler areas, then combined to find the resultant force and centroid.

Finding Resultant Loads on a Beam | 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

Finding Resultant Loads on a Beam

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