Name: beam deflection from central load
Uploaded: 2024-05-23T13:00:13.000Z
Duration: 1 min 14 s
Channel: JoVE

Merkezde bir F nokta yükü taşıyan, L uzunluğunda, basit bir şekilde desteklenen bir kiriş PQ'su düşünün. Kirişin merkezindeki sapma ne olacak?
Burada, kirişin her iki ucundaki reaksiyonlar eşittir ve her biri merkezi yükün yarısıdır.
PC segmentini düşünün ve P ucundan x mesafesindeki herhangi bir noktayı seçin. Bu noktada, P noktasındaki reaksiyondan kaynaklanan moment, P noktasındaki yükün mesafeyle çarpımıdır. Moment denkleminin P ucundaki yüke göre kısmi bir diferansiyonu x'in yarısıdır. 
Kirişin segment QC'si için de benzer bir analiz yapılabilir.
Castigliano teoremini kullanarak, C noktasındaki sapma, uygulanan yükten dolayı gerinim enerjisinin kısmi türevleri tarafından belirlenir.
Bu denklem, PC ve QC ışınının iki segmenti dikkate alınarak basitleştirilebilir.
Kirişin uzunluğunun yarısından fazla bir sürede entegrasyonun gerçekleştirilmesi, kirişin merkezindeki sapma için bir ifade verir.

castigliano teoremi: problem Çözme

Merkezi bir nokta yükü taşıyan basit destekli bir kirişin sapması, özellikle Castigliano teoremi uygulanarak yapısal mekanik prensipleri kullanılarak analiz edilebilir. Bu teorem, yük uygulama noktasındaki yer değiştirmeyi yapıdaki gerinim enerjisinin kısmi türevleriyle ilişkilendirir. Merkezinde nokta yükü bulunan basit destekli kiriş, desteklerde simetrik reaksiyon kuvvetlerine sahiptir ve her biri yükün yarısını taşır. Kiriş boyunca herhangi bir noktadaki bükülme momenti, en yakın desteğe olan mesafe üzerinden hesaplanan bu reaksiyonlardan elde edilir.

Castigliano teoremi, yükün uygulandığı noktadaki sapmanın, kirişin gerinim enerjisinin yüke göre farklılaştırılması ile belirlendiğini belirtir. Gerinim enerjisi, kirişin uzunluğu boyunca entegre edilen kiriş boyunca bükülme momentine dayalı olarak hesaplanır. Bu kiriş için bükülmeden kaynaklanan gerinim enerjisi, kiriş uzunluğunun yarısı boyunca entegre edilen bükülme momenti ifadesinin karesinden hesaplanır.

<img alt="Equation 1" src="/files/ftp_upload/15689/15689_Equation_1.svg" style="height: 106px;">

Kiriş simetrik olduğundan, kirişin tamamını hesaba katmak için bu değer iki katına çıkarılır ve kirişin merkezindeki sapma hesaplanır. Yük büyüklüğüne, kiriş uzunluğunun küpüne bağlıdır ve atalet momenti ile kiriş kesitinin esneklik modülünün çarpımı ile ters orantılıdır.

<img alt="Equation 2" src="/files/ftp_upload/15689/15689_Equation_2.svg" style="height: 42px;">

Castigliano Teoremi: Problem Çözme

Merkezde bir F nokta yükü taşıyan, L uzunluğunda, basit bir şekilde desteklenen bir kiriş PQ'su düşünün. Kirişin merkezindeki sapma ne olacak?
İşte, ...

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Castigliano&#39;s Theorem: Problem Solving

beam deflection from central load

Beam deflection from a central load can be found with Castigliano's theorem. The method applies to a simply supported beam with a point load at the center. It links the deflection at the load point to the strain energy stored in the beam.

For this beam, the supports share the load equally. Each support carries half of the force. The bending moment at any point along the beam comes from these reactions and is written using the distance from the nearest support.

Castigliano's theorem says the displacement at the load point is found by taking the partial derivative of the strain energy with respect to the load. Strain energy is the energy stored by the beam as it bends. It is calculated from the bending moment along the beam and integrated over the beam length.

Because the beam is symmetric, the strain energy is first found over half the beam and then doubled for the full length. The center deflection depends on the load, the cube of the beam length, and the beam's moment of inertia and elastic modulus. It is inversely proportional to the product of those two material and cross-section properties.

Beam Deflection from Central Load

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Using Strain Energy to Find Deflection

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castigliano&#39;s theorem: problem solving

Beam deflection from a central load can be found with Castigliano's theorem. The method applies to a simply supported beam with a point load at the center. It links the deflection at the load point to the strain energy stored in the beam. For this beam, the supports share the load equally. Each support carries half of the force. The bending moment at any point along the beam comes from these reactions and is written using the distance from the nearest support. Castigliano's theorem says the...

key terms and concepts

Learn how Castigliano's theorem is used to find the center deflection of a simply supported beam with a central point load using strain energy.

Beam Deflection from Central Load | 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 from Central Load

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