Name: moment-area method for cantilever deflection
Uploaded: 2024-05-23T13:00:03.000Z
Duration: 1 min 15 s
Channel: JoVE

Moment-alan yöntemi, eğimi ve sapmayı belirlemek için konsantre bir yük ve moment altında bir konsol kiriş üzerinde uygulanabilir.
İşlem, kirişin serbest cisim diyagramının çizilmesi ve sabit uçtaki reaksiyonların hesaplanması ile başlar.
Daha sonra, momentin sıfıra eşit olduğu noktayı belirlemek için eğilme momenti diyagramı çizilir.
Daha sonra, AO ve OB segmentlerine karşılık gelen alanları belirlemek için M/EI diyagramı çizilir ve x eksenine göre konumlarına göre pozitif veya negatif işaretler atar.
İlk Moment-alan teoremi, teğetler arasındaki açıyı hesaplamak için uygulanır ve ikinci Moment-alan teoremi, A ile B arasındaki toplam alanın B'den geçen dikey bir eksen etrafındaki ilk momente eşit olan teğetsel sapmayı belirlemek için kullanılır.
Son olarak, teğetsel sapmaya eşit olan B'deki sapma belirlenir. Kirişin saptırılmış şekli daha sonra görsel anlayış için çizilir.

 simetrik yüklü kirişler

Beams with Symmetric Loadings

Moment-alan yöntemi, çeşitli yükler altında kirişlerin eğimini ve sapmasını belirlemek için yapı mühendisliğinde kullanılan analitik bir araçtır. Serbest ucunda konsantre yük ve moment bulunan bir konsol düşünün. İlk adım, sabit uçtaki reaksiyonları hesaplamak için bir serbest cisim diyagramı oluşturmaktır. Daha sonra, bükülme momentinin sıfıra eşit olduğu noktalara odaklanarak, bükülme momentinin kirişin uzunluğu boyunca nasıl değiştiğini görselleştirmek için bükülme momenti diyagramı çizilir.

Daha sonra M/EI diyagramı çizilir; burada M bükülme momenti, E esneklik modülü ve I kiriş kesitinin atalet momentidir. Bu diyagram, bükülme momentinin sıfır olduğu noktalar arasındaki bölümler için eğrinin altındaki alanları tanımlar ve alanlara x eksenine göre konumlarına göre pozitif veya negatif işaretler atar.

Birinci moment-alan teoremi, M/EI diyagramı altındaki alanın iki nokta arasında integrali alınarak kirişin herhangi bir noktasındaki eğimin hesaplanması için uygulanır. İkinci moment-alan teoremi daha sonra kirişin sapmasını bulmak için kullanılır ve bunu M/EI diyagramı altındaki alanın uç noktadan geçen dikey bir eksene göre ilk momentine eşitler.

Son olarak, analitik bulguların görsel bir temsilini sağlayacak ve kirişin uygulanan yük koşulları altında bükülmesini gösterecek şekilde kirişin sapmış şekli çizilir. Bu yöntem, konsol kirişlerin yapısal performansına ilişkin kesin bilgiler sunar.

 Simetrik Yüklü Kirişler

Moment-alan yöntemi, eğimi ve sapmayı belirlemek için konsantre bir yük ve moment altında bir konsol kiriş üzerinde uygulanabilir.
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Education

JoVE Core

Mechanical Engineering

Makine Mühendisliği

Unit 1

Ünite 1

Deflection of Beams

Kirişlerin Sehimleri

KEY TERMS AND CONCEPTS

TEMEL TERİMLER VE KAVRAMLAR

moment-area method for cantilever deflection

The moment-area method is used in structural engineering to find the slope and deflection of a beam under load. It is especially useful for a cantilever beam with a concentrated load and a moment at the free end.

The analysis begins with a free-body diagram. This diagram helps calculate the reactions at the fixed end. After that, the bending moment diagram is drawn to show how the bending moment changes along the beam. Points where the bending moment is zero are important in this step.

Next, the M/EI diagram is constructed. In this diagram, M is the bending moment, E is the modulus of elasticity, and I is the moment of inertia of the beam cross-section. The areas under the curve are identified between the zero-moment points. Each area is given a positive or negative sign based on its position relative to the x-axis.

The first moment-area theorem is then used to find the slope at a point on the beam. It does this by using the area under the M/EI diagram between two points. The second moment-area theorem is used to find deflection. It equals the first moment of the area under the M/EI diagram about a vertical axis through the endpoint.

The final step is to sketch the deflected shape of the beam. This gives a clear picture of how the beam bends under the applied load and moment. The method provides precise insight into the structural behavior of cantilever beams.

Moment-Area Method for Cantilever Deflection

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Beam deflection depends on the loads on a beam and the support conditions at its ends. For indeterminate beams, especially those with overhanging parts and multiple concentrated loads, careful analysis is needed to predict how the beam will bend. The first step is to draw an accurate free-body diagram. A free-body diagram shows the forces and moments acting on the beam. It helps reveal how the bending moment changes along the beam length. The bending moment diagram also shows how the beam...

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Beam deflection and boundary conditions help describe how a beam bends under load. In structural engineering, the curvature of a plane curve shows how sharply the beam bends. That curvature is found from the curve’s first and second derivatives. A cantilever beam with a point load at its free end is a common example. A diving board works like this type of beam. When slopes are small, the beam’s elastic curve becomes the main shape used in the analysis. The governing equation connects bending...

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Beam deflection under distributed load is a key part of beam analysis in engineering. It helps predict how a beam bends and reacts when a load is spread along its length. Engineers use this relationship to judge whether a structure will stay safe and functional. The analysis starts with the link between load, shear force, and bending moment. These quantities can be written in differential form. When beam flexibility is treated as constant, further differentiation leads to a higher-order...

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Beam slope and deflection can be found with singularity functions when beam loading is hard to write as one simple equation. This method is useful in structural engineering because it helps describe beams under several loads without breaking the problem into many separate cases. Singularity functions are mathematical tools for showing discontinuities in a load, shear force, or bending moment equation. They are especially helpful when a load acts at a single point or over a set interval. In the...

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Beam Deflection Under Combined Loads

Beam deflection under combined loads can be found with the method of superposition. This structural engineering method is used to analyze how multiple loads affect a beam. It works by finding the slope and deflection from each load separately, then adding those results together. The method applies only when the beam stays within its elastic limit. In that range, the material deforms in a linearly elastic way. That makes it valid to combine the effects of separate loads without changing the...

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Beam Deflection Using Moment-Area Rules

The moment-area theorems help describe beam bending in structural engineering. They are often used for building floor supports and other beams under load. The theorems use the elastic curve, which is the shape a beam takes as it deforms. The first theorem links slope change to area on an M/EI diagram. Here, M is the bending moment, E is the modulus of elasticity, and I is the moment of inertia. When M/EI is plotted along the beam length, the area under the curve gives the total rotation...

beams with symmetric loadings

The moment-area method is used in structural engineering to find the slope and deflection of a beam under load. It is especially useful for a cantilever beam with a concentrated load and a moment at the free end. The analysis begins with a free-body diagram. This diagram helps calculate the reactions at the fixed end. After that, the bending moment diagram is drawn to show how the bending moment changes along the beam. Points where the bending moment is zero are important in this step. Next,...

beams with unsymmetric loadings

Beam Deflection and Level Points

Beam deflection and level points can be found for a supported beam under unsymmetrical loadings using moment-area theorems. This type of analysis helps show how a beam bends when forces are spread unevenly along it. It is an important part of structural engineering because it links load patterns to beam response. The first moment-area theorem is used to find the slope at any point on the beam. It states that the change in slope between two points equals the area under the moment diagram over...

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Maximum deflection in a supported beam depends on where the load is placed. With an unsymmetrical load, such as a train moving across a bridge, the beam may bend most at a point that is not the midpoint. Finding that point is important for judging stress and deflection in the structure. The point of maximum deflection is called point O. At point O, the tangent to the deflection curve is horizontal. That means the slope is zero there. To locate point O, the slope at any point X along the beam...

key terms and concepts

Use the moment-area method to analyze a cantilever beam with a load and end moment. Learn how free-body diagrams, M/EI diagrams, slope, and deflection are found.

Moment-Area Method for Cantilever Deflection | JoVE K12

JoVE Core Makine Mühendisliği, denge, kuvvetler, kafes kiriş analizi ve yapısal mekanik gibi temel konuları kapsayan kısa, anlaşılması kolay animasyonlu videolar sunar. Gerçek dünyadaki uygulamalarını keşfedin ve bilim insanlarının önderlik ettiği deneylerle bu kavramların uygulanmasına tanık olun.

mechanical engineering

Moment-Area Method for Cantilever Deflection

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