# 统计代写|主成分分析代写Principal Component Analysis代考|Finite Element Analysis for Beam Bending

This chapter develops finite elements for beam bending based on elastic flexure theory from basic strength of materials. It culminates in a general three-dimensional beam element that accounts for axial, bending, and torsional effects. The chapter omits some aspects of structural behavior like stiffening from tensile loading, buckling under compression, and transverse shear effects, which can be optionally included in commercial finite element software packages as per user requirements.

Here’s a summary of solutions to selected problems:

In Problem P4.1, using Equation 4.32, you would show that normal stress σx\sigma_xσ x ​ is generally discontinuous at the common node (node 2) between two beam elements unless the bending moments are equal and opposite at that node.

Problem P4.2 asks to construct shear force and bending moment diagrams for a loaded beam, illustrating their relevance to the differential equations of equilibrium (Equations 4.10 and 4.17) and the shear-bending moment relationship V=dMdxV = \frac{dM}{dx}V= dx dM ​ from strength of materials.

For Problem P4.3, you’re asked to compare the maximum deflection from strength of materials theory with that obtained from a single finite element model for a uniformly loaded beam.

In Problem P4.4, you determine nodal forces and moments for a beam with a linearly varying load using the work-equivalence approach.

You use the results from Problem P4.4 to find the deflection at node 2 in Problem P4.5, treating the beam as a single finite element, and then compute the reaction force, moment, and maximum bending stress, comparing these values to those from strength of materials.

Problem P4.8 requires determining the minimum number of elements needed to model a given beam and constructing the global nodal load vector accordingly.

Problem P4.16 involves deriving the strain energy expression and the component k11 of the stiffness matrix for a tapered beam element.

You are requested to convert the equilibrium equations of a beam-axial element to the global coordinate system and verify Equation 4.65, plus express the strain energy and derive the global equilibrium equations using the principle of minimum potential energy.

Problems P4.21 and P4.21(b) ask to analyze two different 2D frame structures with various conditions at node 2, using beam-axial elements.

Problems P4.23 and P4.24 involve analyzing different frame structures supported by beam-axial elements with fixed and/or pinned joints.

Problem P4.26 deals with a cantilevered beam subjected to two-plane bending. You are asked to model the beam as a single element and compute deflections and maximum bending stress for both concentrated and uniformly distributed loads.

Each of these problems exercises various aspects of finite element analysis for beams, emphasizing the transition from classical methods in strength of materials to the numerical techniques employed in modern computational engineering. The reader is expected to apply the concepts introduced throughout the chapter to solve these practical engineering scenarios.

### MATLAB代写

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