In the present contribution Castigliano’s theorem is extended to find more accurate results for the deflection curves of beam-type structures. The notion extension in the context of the second Castigliano’s theorem means that all stress components are included for the computation of the complementary strain energy, and not only the dominant axial stress and the shear stress. The derivation shows that the partial derivative of the complementary strain energy with respect to a scalar dummy parameter is equal to the displacement field multiplied by the normalized traction vector caused by the dummy load distribution. Knowing the Airy stress function of an isotropic beam as a function of the bending moment, the normal force, the shear force and the axial and vertical load distributions, higher-order formulae for the deflection curves and the cross section rotation are obtained. The analytical results for statically determinate and indeterminate beams for various load cases are validated by analytical and finite element results. Furthermore, the results of the extended Castigliano theory (ECT) are compared to Bernoulli–Euler and Timoshenko results, which are special cases of ECT, if only the energies caused by the bending moment and the shear force are considered. It is shown that lower-order terms for the vertical deflection exist that yield more accurate results than the Timoshenko theory. Additionally, it is shown that a distributed load is responsible for shrinking or elongation in the axial direction.

在本论文中，卡斯蒂利亚诺定理得到扩展，以找到梁型结构挠度曲线的更准确结果。在第二个Castigliano定理的上下文中，概念扩展意味着包括所有应力分量以计算互补应变能，而不仅是主要轴向应力和剪切应力。推导表明，相对于标量虚拟参数，互补应变能的偏导数等于位移场乘以虚拟负载分布引起的归一化牵引矢量。知道各向同性梁的艾里应力函数是弯矩，法向力，剪力以及轴向和垂直载荷分布的函数，得到挠曲曲线和截面旋转的高阶公式。通过分析和有限元结果验证了各种载荷工况下静定梁和不定梁的分析结果。此外，如果仅考虑由弯矩和剪切力引起的能量，则将扩展的Castigliano理论（ECT）的结果与Bernoulli–Euler和Timoshenko的结果进行比较，这是ECT的特殊情况。结果表明，垂直变形的低阶项比蒂莫申科理论产生的结果更准确。另外，示出了分布的载荷导致轴向方向上的收缩或伸长。通过分析和有限元结果验证了各种载荷工况下静定梁和不定梁的分析结果。此外，如果仅考虑由弯矩和剪切力引起的能量，则将扩展的Castigliano理论（ECT）的结果与Bernoulli–Euler和Timoshenko的结果进行比较，这是ECT的特殊情况。结果表明，垂直变形的低阶项比蒂莫申科理论产生的结果更准确。另外，示出了分布的载荷导致轴向方向上的收缩或伸长。通过分析和有限元结果验证了各种载荷工况下静定梁和不定梁的分析结果。此外，如果仅考虑由弯矩和剪切力引起的能量，则将扩展的Castigliano理论（ECT）的结果与Bernoulli–Euler和Timoshenko的结果进行比较，这是ECT的特殊情况。结果表明，垂直变形的低阶项比蒂莫申科理论产生的结果更准确。另外，示出了分布的载荷导致轴向方向上的收缩或伸长。如果仅考虑由弯矩和剪切力引起的能量。结果表明，垂直变形的低阶项比蒂莫申科理论产生的结果更准确。另外，示出了分布的载荷导致轴向方向上的收缩或伸长。如果仅考虑由弯矩和剪切力引起的能量。结果表明，垂直变形的低阶项比蒂莫申科理论产生的结果更准确。另外，示出了分布的载荷导致轴向方向上的收缩或伸长。