Name: beam deflection and boundary conditions
Uploaded: 2024-05-23T13:00:02.000Z
Duration: 1 min 23 s
Channel: JoVE

يمكن التعبير عن انحناء منحنى المستوى عند نقطة على المنحنى باستخدام تعبير يتضمن المشتقات الأولى والثانية للمنحنى.
المنحدر ضئيل بالنسبة للمنحنى المرن للحزمة ، لذلك يتم التعبير عن المعادلة الحاكمة للمنحنى المرن كمعادلة تفاضلية خطية من الدرجة الثانية ، مع الأخذ في الاعتبار صلابة الانحناء.
إذا اختلفت صلابة الانحناء على طول الحزمة ، يتم التعبير عنها كدالة x. ومع ذلك ، بالنسبة للشعاع المنشوري ، تظل صلابة الانحناء ثابتة.
يوفر تكامل هذه المعادلة الزاوية التي يتكون منها المماس إلى المنحنى المرن عند نقطة مع الأفقي.
هذه الزاوية الصغيرة ، عند دمجها ، تعطينا انحراف الحزمة في أي وقت. تحدد الشروط الحدودية لدعم الحزمة الثوابت في المعادلات.
المدعومة ، المتدلية ، والناتئة هي الأنواع الثلاثة من الحزم التي يتم النظر فيها في المقام الأول. بالنسبة للحزم المدعومة والمتدلية ، يكون الانحراف عند نقاط الدعم صفرا.
كل من الانحراف والانحدار عند نقطة الدعم يساويان صفرا للحزم الكابولية. تتيح هذه الشروط حساب الثوابت.

معادلة المنحنى المرن

Equation of the Elastic Curve

<div style="direction: rtl;">
 إن مفهوم الانحناء في المنحنيات المستوية، وهو أمر بالغ الأهمية في الهندسة الإنشائية، يحدد مدى انحناء العارضة بشكل حاد تحت الحمل. يتم تحديد هذا الانحناء باستخدام المشتقتين الأولى والثانية للمنحنى.
 
 خذ بعين الاعتبار عارضة كابولية ذات حمل نقطي عند نهايتها الحرة (على سبيل المثال، لوح الغوص). عند تحليل انحراف العارضة مع المنحدرات الصغيرة، يصبح شكل المنحنى المرن للشعاع هو المفتاح. تتضمن المعادلة الحاكمة لهذا التحليل عزم الانحناء وصلابة الانحناء للعارضة، والتي هي نتاج معامل المرونة وعزم القصور الذاتي للمقطع العرضي للعارضة.
 
 <img alt="Equation 1" src="/files/ftp_upload/15667/15667_Equation_1.svg" style="text-align: center; height: 32px;">
 بالنسبة للعوارض المنشورية، حيث يظل المقطع العرضي ثابتًا، يتم تبسيط التحليل، مما يجعل صلابة الانثناء ثابتة على طول طول العارضة. يتيح دمج المعادلة الحاكمة حساب الزاوية التي يشكلها مماس المنحنى عند أي نقطة، والتي، عند مزيد من التكامل، تؤدي إلى انحراف العارضة عند تلك النقطة.
 
 <img alt="Equation 2" src="/files/ftp_upload/15667/15667_Equation_2.svg" style="text-align: center; height: 28px;">
 تعتبر الشروط الحدودية عند دعامات العارضة حيوية لاستكمال هذه الحسابات. تعتبر أنواع العوارض المدعومة، والمتدليّة، والكابولية من الأنواع الشائعة من العوارض، ولكل منها شروط حدودية مميزة. على سبيل المثال، يكون الانحراف والانحدار عند نقطة دعم العارضة الكابولية صفرًا، وهو أمر ضروري لحساب ثوابت معادلات الانحراف.
 
 يعد التنبؤ الدقيق بانحراف العارضة أمرًا بالغ الأهمية لضمان السلامة الهيكلية والأداء الوظيفي. يمكن أن يتسبب الانحراف المفرط في حدوث أعطال هيكلية أو مشكلات في قابلية الخدمة، مما يؤكد أهمية فهم سلوك العارضة تحت الحمل.
 
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يمكن التعبير عن انحناء منحنى المستوى عند نقطة على المنحنى باستخدام تعبير يتضمن المشتقات الأولى والثانية للمنحنى.
ال...

Education

JoVE Core

Mechanical Engineering

الهندسة الميكانيكية

Unit 1

الوحدة الأولى

Deflection of Beams

انحراف الكمرات

KEY TERMS AND CONCEPTS

المصطلحات والمفاهيم الأساسية

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 moment with flexural rigidity. Flexural rigidity is the product of the modulus of elasticity and the moment of inertia of the beam’s cross-section. For a prismatic beam, the cross-section stays constant, so the flexural rigidity is constant along the beam.

Integrating the governing equation gives the angle of the tangent to the curve at any point. A second integration gives the deflection at that point. Boundary conditions at the beam supports are needed to find the constants in these equations.

Supported, overhanging, and cantilever beams each have different boundary conditions. For a cantilever beam, the deflection and slope at the support point are zero. Predicting deflection is important because too much bending can lead to structural failure or serviceability problems.

Beam Deflection and Boundary Conditions

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 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

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 beam deflection, curvature, and boundary conditions work together in structural engineering, using cantilever and other beam types.

Beam Deflection and Boundary Conditions | 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

Beam Deflection and Boundary Conditions

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