Elastic Beams in Three Dimensions Lars Andersen and Søren R.K. Nielsen ISSN 1901-7286 DCE Lecture Notes No. 23 Department of Civil Engineering Aalborg University Department of Civil Engineering Structural Mechanics DCE Lecture Notes No. 23 Elastic Beams in Three Dimensions by Lars Andersen and Søren R.K. Nielsen August 2008 c Aalborg University Scientific Publications at the Department of Civil Engineering Technical Reports are published for timely dissemination of research results and scientific work carried out at the Department of Civil Engineering (DCE) at Aalborg University. This medium allows publication of more detailed explanations and results than typically allowed in scientific journals. Technical Memoranda are produced to enable the preliminary dissemination of scientific work by the personnel of the DCE where such release is deemed to be appropriate. Documents of this kind may be incomplete or temporary versions of papers—or part of continuing work. 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This includes the status of research projects, developments in the labora- tories, information about collaborative work and recent research results. Published 2008 by Aalborg University Department of Civil Engineering Sohngaardsholmsvej 57, DK-9000 Aalborg, Denmark Printed in Denmark at Aalborg University ISSN 1901-7286 DCE Lecture Notes No. 23 Preface This textbook has been written for the course Statics IV on spatial elastic beam structures given at the 5th semester of the undergraduate programme in Civil Engineering at Aalborg Uni- versity. The book provides a theoretical basis for the understanding of the structural behaviour of beams in three-dimensional structures. In the course, the text is supplemented with labora- tory work and hands-on exercises in commercial structural finite-element programs as well as MATLAB. The course presumes basic knowledge of ordinary differential equations and struc- tural mechanics. A prior knowledge about plane frame structures is an advantage though not mandatory. The authors would like to thank Mrs. Solveig Hesselvang for typing the manuscript. Aalborg,August2008 LarsAndersenandSørenR.K.Nielsen — i — ii DCE Lecture Notes No. 23 Contents 1 Beams in three dimensions 1 1.1 Introduction.................................... 1 1.2 Equationsofequilibriumforspatialbeams . .......... 1 1.2.1 Sectionforcesandstressesinabeam . .... 3 1.2.2 Kinematicsanddeformationsofabeam . .... 5 1.2.3 Constitutiverelationsforanelasticbeam . ........ 10 1.3 Differentialequationsofequilibriumforbeams . ............ 12 1.3.1 GoverningequationsforaTimoshenkobeam . ..... 13 1.3.2 GoverningequationsforaBernoulli-Eulerbeam . ........ 14 1.4 Uncouplingofaxialandbendingdeformations. .......... 15 1.4.1 Determinationofthebendingcentre . ..... 15 1.4.2 Determinationoftheprincipalaxes . ..... 21 1.4.3 Equations of equilibrium in principal axes coordinates .......... 25 1.5 Normalstressesinbeams . ... 28 1.6 Theprincipleofvirtualforces . ...... 29 1.7 Elasticbeamelements. ... 33 1.7.1 AplaneTimoshenkobeamelement . 34 1.7.2 Athree-dimensionalTimoshenkobeamelement . ....... 41 1.8 Summary ...................................... 44 2 Shear stresses in beams due to torsion and bending 45 2.1 Introduction.................................... 45 2.2 Homogeneoustorsion(St.Venanttorsion) . ......... 46 2.2.1 Basicassumptions ............................. 47 2.2.2 Solutionofthehomogeneoustorsionproblem . ....... 48 2.2.3 Homogeneous torsion of open thin-walled cross-sections......... 57 2.2.4 Homogeneous torsion of closed thin-walled cross-sections ........ 59 2.3 Shearstressesfrombending . .... 67 2.3.1 Shear stresses in open thin-walled cross-sections . ........... 69 2.3.2 Determinationoftheshearcentre . .... 75 2.3.3 Shearstresses inclosedthin-walledsections . .......... 82 2.4 Summary ...................................... 91 References 93 — iii — iv Contents DCE Lecture Notes No. 23 CHAPTER 1 Beams in three dimensions This chapter gives an introduction is given to elastic beams in three dimensions. Firstly, the equations of equilibrium are presented and then the classical beam theories based on Bernoulli- Euler and Timoshenko beam kinematics are derived. The focus of the chapter is the flexural de- formations of three-dimensional beams and their coupling with axial deformations. Only a short introduction is given to torsional deformations, or twist, of beams in three dimensions. A full de- scription of torsion and shear stresses is given in the next chapters. At the end of this chapter, a stiffness matrix is formulated for a three-dimensional Timosheko beam element. This element can be used for finite-element analysis of elastic spatial frame structures. 1.1 Introduction In what follows, the theory of three-dimensional beams is outlined. 1.2 Equations of equilibrium for spatial beams An initially straight beam is considered. When the beam is free of external loads, the beam occupies a so-called referential state. In the referential state the beam is cylindrical with the length l, i.e. the cross-sections are everywhere identical. The displacement and rotation of the beam is described in a referential (x,y,z)-coordinate system with base unit vectors {i, j, k}, the origin O placed on the left end-section, and the x-axis parallel with the cylinder and orientated into the beam, see Fig. 1–1. For the time being, the position of O and the orientation of the y- and z-axes may be chosen freely. The beam is loaded by a distributed load per unit length of the referential scale defined by the vector field q = q(x) and a distributed moment load vector per unit length m = m(x). A differential beam element of the length dx is then loaded by the external force vector qdx and external moment vector mdx as shown in Fig. 1–1. The length of the differential beam element may change during deformations due to axial strains. However, this does not affect the indicated load vectors which have been defined per unit length of the referential state. Measured in the (x,y,z)-coordinate system, q and m have the components qx mx q = q , m = m . (1–1) y y qz mz — 1 — 2 Chapter 1 – Beams in three dimensions qdx mdx F + dF M + dM idx −M − y F j x i k dx z l Figure 1–1 Beam in referential state. As a consequence of the external loads, the beam is deformed into the so-called current state where the external loads are balanced by an internal section force vector F = F(x) and an internal section moment vector M = M(x). These vectors act on the cross-section with the base unit vector i of the x-axis as outward directed normal vector. With reference to Fig. 1–2, the components of F and M in the (x,y,z)-coordinate system are: N Mx F = Q , M = M (1–2) y y Qz Mz Here, N = N(x) is the axial force, whereas the components Qy = Qy(x) and Qz = Qz(x) sig- nify the shear force componentsin the y- and z-directions. The axial component Mx = Mx(x) of the section moment vector is denoted the torsional moment. The components My = My(x) and Mz = Mz(x) in the y- and z-directions represent the bending moments. The torsional moment is not included in two-dimensional beam theory. However, in the design of three-dimensional frame structures, a good understanding of the torsional behaviour of beams is crucial. Assuming that the displacements remain small, the equation of static equilibrium can be established in the referential state. With reference to Fig. 1–1, the left end-section of the element is loaded with the section force vector −F and the section moment vector −M. At the right end-section, these vectors are changed differentially into F + dF and M + dM, respectively. Force equilibrium and moment equilibrium formulated at the point of attack of the section force vector −F at the left end-section then provides the following equations of force and moment DCE Lecture Notes No. 23 1.2 Equations of equilibrium for spatial beams 3 Qy y My x Mx N z Mz Qz Figure 1–2 Components of the section force vector and the section moment vector. equilibrium of the differential beam element: −F + F + dF + qdx = 0 ⇒ dF + q = 0 (1–3a) dx −M + M + dM + idx × (F + dF)+ mdx = 0 ⇒ dM + i × F + m = 0 (1–3b) dx From Eqs. (1–1) and (1–2) follows that Eqs. (1–3a) and (1–3b) are equivalent to the following component relations: dN dQ dQ + q =0, y + q =0, z + q =0, (1–4a) dx x dx y dx z dM dM dM x + m =0, y − Q + m =0, z + Q + m =0. (1–4b) dx x dx z y dx y z At the derivation of Eq. (1–4b), it has been utilised that i × F = i × (Ni + Qyj + Qzk)= Ni × i + Qyi × jQzi × k =0i − Qzj + Qyk. (1–5) Hence, i×F has the components {0, −Qz,Qy}. It is noted that a non-zero normal-forcecompo- nent is achieved when the moment equilibrium equations are formulated in the deformed state. This may lead to coupled lateral-flexural instability as discussed in a later chapter. 1.2.1 Section forces and stresses in a beam On the cross-section with the outward directed unit vector co-directional to the x-axis, the normal stress σxx and the shear stresses σxy and σxz act as shown in Fig.