Name: beam slope and deflection by singularity
Uploaded: 2024-05-23T13:00:02.000Z
Duration: 1 min 19 s
Channel: JoVE

يمكن تحديد انحراف الشعاع في هيكل السقف باستخدام طريقة التكامل ، بشرط أن تمثل وظيفة تحليلية واحدة لحظة الانحناء.
ومع ذلك ، إذا كان تحميل الحزمة يتطلب وظائف متعددة لتمثيل عزم الانحناء ، فستكون هناك حاجة إلى ثوابت ومعادلات إضافية ، مما يؤدي إلى حسابات مطولة.
يمكن تبسيط هذا التعقيد باستخدام وظائف التفرد. ضع في اعتبارك شعاعا موشوريا مدعوما في النهايات يحمل حمولة غريب الأطوار.
يمكن تمثيل دالة قوة القص لهذه الحزمة باستخدام دالة التفرد المناسبة. يمكن اشتقاق وظيفة لحظة الانحناء من خلال دمج وظيفة قوة القص هذه.
يمكن الحصول على منحدر الشعاع وانحرافه من خلال دمج وظيفة اللحظة هذه. يمكن تحديد الثوابت في المعادلات من الشروط الحدودية.
يؤدي استخدام وظائف التفرد إلى التخلص من الحاجة إلى ثوابت ومعادلات إضافية ، مما يبسط الحساب. نتيجة لذلك ، توفر وظائف التفرد طريقة فعالة لحساب ميل وانحراف الحزمة تحت الأحمال المعقدة.

انحراف الشعاع

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 يعد التحديد الدقيق لانحراف العارضة وانحدارها في ظل ظروف التحميل المختلفة في الهندسة الإنشائية أمرًا بالغ الأهمية لضمان السلامة والسلامة الهيكلية. توفر وظائف التفرد طريقة مبسطة لتحليل العوارض، خاصة عندما تؤدي وظائف التحميل المتعددة إلى تعقيد معادلة عزم الانحناء.
 
 تعد دوال التفرد، الموضحة في درس سابق، أدوات رياضية قوية تمثل الانقطاعات داخل دالة شائعة في سيناريوهات التحميل الهيكلي. تساعد هذه الوظائف في التعبير عن معادلات قوة القص وعزم الانحناء بشكل مضغوط، حتى في ظل الأحمال المعقدة أو المتعددة.
 
 <img alt="Equation 1" src="/files/ftp_upload/15669/15669_Equation_1a.svg" style="text-align: center; height: 32px;">
 <img alt="Equation 2" src="/files/ftp_upload/15669/15669_Equation_1b.svg" style="text-align: center; height: 32px;">
 بالنسبة للعارضة المنشورية، التي تكون عادةً موحدًة على طولها ومدعومًة من كلا الطرفين، يمثل الحمل اللامركزي تحديات محددة. يمكن تشكيل قوة القص عند أي نقطة في مثل هذه العارضة باستخدام وظائف التفرد. تتعامل هذه الوظائف بسهولة مع حالات الانقطاع التي تحدثها الأحمال المطبقة عند نقاط محددة أو على فترات زمنية معينة، مثل الحمل الذي يتم تمثيله كدالة خطوة في مخطط قوة القص.
 
 عزم الانحناء، المستمد من دمج وظيفة قوة القص، أمر بالغ الأهمية لتقييم أداء الشعاع. تؤثر هذه الخطوة على توزيع إجهاد العارضة وانحرافها الإجمالي. يتم تحديد انحراف العارضة من خلال دمج وظيفة عزم الانحناء مرتين وتطبيق شروط حدود العارضة لحل ثوابت التكامل. إن استخدام وظائف التفرد لتشكيل قوى القص وعزوم الانحناء يلغي الحاجة إلى ثوابت إضافية متعددة ومعادلات معقدة، مما يؤدي إلى تبسيط العمليات الحسابية وتعزيز الكفاءة الحسابية. تسمح هذه الطريقة بالتقييم السهل لظروف التحميل المختلفة على انحراف العارضة وتوزيع الضغط، وهو أمر ضروري للتصميم الآمن وصيانة الأنظمة الهيكلية.
 
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يمكن تحديد انحراف شعاع في هيكل السقف باستخدام طريقة التكامل ، بشرط أن تمثل وظيفة تحليلية واحدة ...

Education

JoVE Core

Mechanical Engineering

الهندسة الميكانيكية

Unit 1

الوحدة الأولى

Deflection of Beams

انحراف الكمرات

KEY TERMS AND CONCEPTS

المصطلحات والمفاهيم الأساسية

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 beam example shown here, a prismatic beam, which means a beam with a uniform cross-section, is supported at both ends and carries an eccentric load, or a load placed off center.

The shear force at any point in the beam can be written with singularity functions. These functions handle the jump in the shear force diagram caused by the applied load. The bending moment is then found by integrating the shear force equation, and it is important because it affects the stress in the beam and the amount of deflection.

To find deflection, the bending moment equation is integrated twice. The beam boundary conditions are then used to solve for the integration constants. This approach keeps the equations compact, reduces the need for extra constants, and makes calculations more efficient for different loading conditions. It is a useful way to check beam performance and support safe structural design and maintenance.

Beam Slope and Deflection by Singularity

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 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 singularity functions simplify shear force, bending moment, slope, and beam deflection calculations for a prismatic beam with an eccentric load.

Beam Slope and Deflection by Singularity | 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 Slope and Deflection by Singularity

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