Name: beam deflection and level points
Uploaded: 2024-05-23T13:00:03.000Z
Duration: 1 min 17 s
Channel: JoVE

Simetrik olmayan yükler altında desteklenen bir kirişi analiz etmek, seviye noktasını tanımlamak için bilinen bir eğime sahip bir referans teğet gerektirir.
Referans tanjantın eğimi, uçlar arasındaki teğetsel kaymanın belirlenmesiyle hesaplanır.
Kirişin herhangi bir noktadaki eğimi, ilk moment alanı teoremi kullanılarak ve daha sonra bu eğim için bir denklem yazılarak belirlenebilir.
Bir noktadan referans teğete olan dikey mesafe, o noktanın desteğe göre teğetsel sapması olarak da bilinir, ikinci moment alanı teoremi kullanılarak belirlenebilir.
Bu sapma belirli bir segmente eşittir ve noktadan referans teğete olan dikey mesafeyi belirtir. Bir noktadaki sapma, yatay bir çizgiden dikey mesafesini gösterir.
Bu sapma, boyut olarak belirli bir segmente eşit olduğundan, iki segment arasındaki fark olarak tanımlanabilir.
Benzer üçgenleri inceleyerek, bu segmentlerden biri belirlenebilir. Sapmalar ve teğetsel sapmalar için işaret kuralları göz önüne alındığında, bir noktadaki sapma not edilir.

 simetrik olmayan yüklü kirişler

Desteklenen bir kirişin simetrik olmayan yükler altında analiz edilmesi, yapı mühendisliğinde kirişlerin çeşitli kuvvet dağılımlarına nasıl tepki verdiğini anlamak açısından önemlidir. Bu analiz, yapısal stabilite ve işlevsellik açısından çok önemli olan sapmanın hesaplanmasını ve kirişin eğiminin sıfır olduğu noktaların belirlenmesini içerir.

Birinci moment-alan teoremi kirişin herhangi bir noktasındaki eğimi belirler. Bu teorem, kiriş üzerindeki iki nokta arasındaki eğim değişiminin, bu aralıkta moment diyagramı altındaki alana karşılık geldiğini gösterir. Sapmaların ölçülmesine yardımcı olan referans teğeti, kirişin uçları arasındaki teğetsel kaymalardan hesaplanan bilinen eğimi ile tanımlanır.

İkinci moment-alan teoremi daha sonra herhangi bir noktanın bu referans teğetinden dikey sapmasını hesaplar; bu, teğetsel sapma olarak bilinir. Bu önlem, kirişin yük altındaki bükülme davranışını anlamak ve tasarım açısından kritik olan maksimum sapma noktalarını belirlemek için hayati öneme sahiptir.

Son olarak, birinci teoremi kullanarak istenen bir noktadaki eğimi belirledikten sonra ikinci teorem, bu noktanın referans çizgisinden dikey olarak ne kadar saptığını ölçer. Bu sapma, izin verilen maksimum sapmayı tanımlar ve yapısal arızaları veya aşırı deformasyonları önleyerek kirişin güvenlik ve operasyonel standartları karşılamasını sağlar. Bu tür analizler sayesinde operasyonel yükler altında bütünlüğü koruyan ve güvenlik gereksinimlerini karşılayan kirişler tasarlanabilir.

 Simetrik Olmayan Yüklü Kirişler

Simetrik olmayan yükler altında desteklenen bir kirişi analiz etmek, seviye noktasını tanımlamak için bilinen bir eğime sahip bir referans teğet gerektirir.
Eğimi ...

Education

JoVE Core

Mechanical Engineering

Makine Mühendisliği

Unit 1

Ünite 1

Deflection of Beams

Kirişlerin Sehimleri

KEY TERMS AND CONCEPTS

TEMEL TERİMLER VE KAVRAMLAR

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 that interval. A reference tangent is then chosen. This line has a known slope and is used to measure how the beam bends away from it.

The second moment-area theorem gives the vertical deviation from the reference tangent. This deviation is called the tangential deviation. It helps identify the level point, where the slope of the beam is zero, and it also shows the maximum deflection point.

Finding slope and tangential deviation is essential for checking structural stability and function. These values help engineers confirm that the beam stays within the maximum allowable deflection. That reduces the risk of failure or excessive deformation and supports safe design under operational loads.

Beam Deflection and Level Points

deformation of a beam under transverse loading

deformation-of-a-beam-under-transverse-loading

Beam Deflection from Loads and Supports

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...

equation of the elastic curve

Beam Deflection and Boundary Conditions

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...

elastic curve from the load distribution

Beam Deflection Under Distributed Load

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...

deflection of a beam

Beam Slope and Deflection by Singularity

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...

method of superposition

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...

moment-area theorems

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

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,...

beams with unsymmetric loadings

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...

maximum deflection

Locating the Beam’s Highest Deflection

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

Learn how moment-area theorems are used to find beam slope, tangential deviation, and level points for a supported beam under unsymmetrical loading.

Beam Deflection and Level Points | 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

Beam Deflection and Level Points

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