Virtual Principles in Aircraft Structures

The basic partial differential equations for the stresses and displacements in clas­ sical three dimensional elasticity theory can be set up in three ways: (1) to solve for the displacements first and then the stresses; (2) to solve for the stresses first and then the displacements; and (3) to solve for both stresses and displacements simultaneously. These three methods are identified in the literature as (1) the displacement method, (2) the stress or force method, and (3) the combined or mixed method. Closed form solutions of the partial differential equations with their complicated boundary conditions for any of these three methods have been obtained only in special cases. In order to obtain solutions, various special methods have been developed to determine the stresses and displacements in structures. The equations have been reduced to two and one dimensional forms for plates, beams, and trusses. By neglecting the local effects at the edges and ends, satisfactory solutions can be obtained for many case~. The procedures for reducing the three dimensional equations to two and one dimensional equations are described in Chapter 1, Volume 1, where the various approximations are pointed out.

1 / The basic three, two, and one dimensional equations in structural analysis.- 1.1 Introduction.- 1.2 Three dimensional equations.- 1.3 The displacement method of solution.- 1.4 The stress method of solution.- 1.5 The combined method of solution.- 1.6 Two dimensional equations.- 1.7 Saint Venant’s principle.- 1.8 One dimensional beam equations.- 1.9 No shear stresses in the beam.- 1.10 Beam cross section of a thin plate with one shear stress.- 1.11 Thin web beams with large flange areas and one shear stress.- 1.12 Torsion of circular cross section and thin wall closed box.- 1.13 Thin web box beam with general loading.- 1.14 Inelastic effects in beams with temperature.- 1.15 Example of inelastic axial stresses and strains with temperature.- 1.16 Sequence loading and thermal cycling in beams.- 1.17 Load-strain design curves for beams.- 1.18 Problems.- References.- 2 / Virtual displacement and virtual force methods in structural analysis.- 2.1 Introduction.- 2.2 The principle of virtual displacements.- 2.3 The unit displacement theorem.- 2.4 The principle of virtual forces.- 2.5 The unit load theorem.- 2.6 The principle of mixed virtual stresses and virtual displacements.- 2.7 The mixed unit displacement and unit load theorem.- 2.8 Two dimensional form of the virtual principles.- 2.9 One dimensional forms of the virtual principles.- 2.10 The one dimensional virtual principles with temperature, inelastic and large displacement effects.- 2.11 Matrix forms of the virtual principles.- 2.12 Problems.- References.- 3 / The virtual principles for pin-jointed trusses.- 3.1 Introduction.- 3.2 The unit displacement theorem for trusses.- 3.3 The unit load theorem for trusses.- 3.4 Inelastic effects with temperature changes in trusses.- 3.5 Matrix equations for trusses from the unit displacement theorem.- 3.6 Matrix equations for trusses from the unit load theorem.- 3.7 Matrix equations for trusses from the mixed unit displacement and unit load theorem.- 3.8 Problems.- References.- 4 / The virtual principles for simple beams.- 4.1 Introduction.- 4.2 Principle of virtual displacements for beams.- 4.3 Point values for beam elements by the principle of virtual displacements.- 4.4 Principle of virtual forces for beams.- 4.5 Point values for beam elements by the unit load theorem.- 4.6 Principle of mixed virtual stresses and virtual displacements for beams.- 4.7 Inelastic and temperature effects in simple beams.- 4.8 Matrix equations for beams from the unit displacement theorem.- 4.9 Matrix equations for beams from the unit load theorem.- 4.10 Matrix equations for beams from the mixed unit displacement and unit load theorem.- 4.11 The beam column equations.- 4.12 Problems.- References.- 5 / Box beam shear stresses and deflections.- 5.1 Introduction.- 5.2 Shear stresses in beams.- 5.3 Torsional shear stresses in beams.- 5.4 Shear flows in open box beams.- 5.5 Shear flows in single cell box beams.- 5.6 Shear flows in multi-cell box beams.- 5.7 Shear center for closed box beams.- 5.8 Shear flows in tapered box beams.- 5.9 Inelastic and buckling shear stresses in beams.- 5.10 Axial and bending deflections of box beams with inelastic and temperature effects.- 5.11 Shear deflections of beams.- 5.12 Torsional rotation of beams.- 5.13 Rotation of swept wings.- 5.14 Spanwise airload distribution and static wing divergence under rotation.- 5.15 Static aileron effectiveness and reversal speed under wing rotation.- 5.16 Problems.- References.- 6 / Shear lag in thin web structures.- 6.1 Introduction.- 1. Solutions for determinate cases.- 6.2 Shear flows due to concentrated loads into thin webs.- 6.3 Shear flows around cut-outs in thin web beams.- 6.4 Cut-outs in box beams.- 6.5 Shear flows in ribs and bulkheads.- 6.6 Forces on ribs due to airloads and taper effects.- 2. Solutions for redundant cases.- 6.7 Restraint effects in thin web structures.- 6.8 Shear flows in redundant beams in one plane.- 6.9 Deflections of thin web structures.- 6.10 Flexibility matrices for shear web elements and stiffener elements.- 6.11 Matrix solutions for thin web beams in one plane.- 6.12 Matrix solutions for box beams.- 6.13 Load redistribution in swept back wings.- 6.14 Problems.- References.- 1 / Allowable stresses of flight vehicle materials.- 1.1 Introduction.- 1.2 Tension, shear, and bearing allowable stresses.- 1.3 Temperature effects on allowable stresses.- 1.4 Allowable compression stresses.- 1.5 Allowable combined stresses.- 1.6 Creep effects on allowable stresses.- 1.7 Room temperature fatigue effects upon allowable stresses.- 1.8 Temperature effects upon allowable fatigue stresses.- 1.9 Crack effects upon allowable fatigue stresses.- 1.10 Problems.- References.- 2 / Analysis and design of joints and splices.- 2.1 Introduction.- 2.2 Analysis of plate splices with axial tension forces.- 2.3 Multi-row tension splices.- 2.4 Joints with eccentric loading.- 2.5 Minimum weight design of splice for beam with rectangular cross section.- 2.6 Design of splices for I-beams and thin shear webs.- 2.7 Deflection effects on load distribution in splices.- 2.8 Temperature and inelastic effects on load distribution in splices.- 2.9 Welded joints.- 2.10 Bonded joints.- 2.11 Problems.- References.- 3 / Structural design of aircraft components.- 3.1 Introduction.- 3.2 Design of minimum weight columns without local buckling.- 3.3 Design of minimum weight sections with local buckling and crippling.- 3.4 Design of minimum weight columns with local buckling.- 3.5 Minimum weight design for stiffened panels in compression.- 3.6 Effective areas for stiffened panels.- 3.7 Effect of load intensity on wing design.- 3.8 Design of box beam cross sections with four spar caps.- 3.9 Analysis of diagonal tension beams.- 3.10 Problems.- References.- 4 / Analysis and design of pressurized structures.- 4.1 Introduction.- 4.2 Membrane stresses in thin shells.- 4.3 Cut-outs in thin shells with membrane stresses.- 4.4 Bending in circular cylindrical shells with axially symmetric loading.- 4.5 Bending in pressurized aircraft fuselages from stringers and frames.- 4.6 Bending of non-circular cross sections with internal pressure.- 4.7 Bending of non-circular fuselage rings with internal pressure.- 4.8 Bending of non-circular fuselage rings with point loads.- 4.9 Effect of internal pressure on buckling of cylindrical shells.- 4.10 Pressure stabilized structures.- 4.11 Problems.- References.- 5 / Approximate solutions using the virtual principles.- 5.1 Introduction.- 5.2 Approximate solutions for beams using the principle of virtual displacements.- 5.3 Approximate solutions for columns.- 5.4 The tapered cantilever beam with numerical integration.- 5.5 Tapered beam finite element matrices for columns.- 5.6 The unit load theorem and numerical integration.- 5.7 Approximate solutions for beams using the mixed virtual principle.- 5.8 Problems.- References.- 6 / Dynamics of simple beams.- 6.1 Introduction.- 6.2 Bending vibrations of simple beams.- 6.3 Forced motion of uniform beam.- 6.4 Approximate solutions for frequencies and mode shapes.- 6.5 Torsional vibrations of simple beams.- 6.6 Finite element matrices for beam frequencies.- 6.7 Flutter of wing segment with one degree of freedom.- 6.8 Flutter of wing segment with two degrees of freedom.- 6.9 Dynamic loads on beams.- 6.10 Problems.- References.- 7 / The plate equations.- 7.1 Introduction.- 7.2 The plate inplane case using the principle of virtual displacements.- 7.3 The plate inplane case using the principle of virtual forces.- 7.4 The plate inplane case using the mixed virtual principle.- 7.5 The plate bending case using the principle of virtual displacements.- 7.6 The plate bending case using the principle of virtual forces.- 7.7 The plate bending case using the mixed virtual principle.- 7.8 Combined inplane and lateral forces.- 7.9 Combined forces with large bending deflections.- 7.10 Buckling of plates.- 7.11 Plate vibrations.- 7.12 Problems.- References.- 8 / Approximate matrix equations for plate finite elements.- 8.1 Introduction.- 8.2 The point unknowns for the matrices.- 8.3 The methods to obtain the matrix equations.- 8.4 Inplane plate element matrices from the principle of virtual displacements.- 8.5 Inplane plate element matrices from the principle of virtual forces.- 8.6 Inplane plate element matrices from the mixed virtual principle.- 8.7 Bending plate element matrices from the principle of virtual displacements.- 8.8 Bending plate element matrices from the principle of virtual forces.- 8.9 Bending plate element matrices from the mixed virtual principle.- 8.10 Matrices for constant stress triangular elements.- 8.11 Problems.- 9 / Matrix structural analysis using finite elements.- 9.1 Introduction.- 9.2 General beam elements in local coordinates.- 9.3 General beam elements in datum coordinates.- 9.4 Triangular plate elements with inplane forces.- 9.5 Assembly of finite elements by the virtual principles.- References.- 10 / Composite Materials.- 10.1 Introduction.- 10.2 Stress-strain equations for nonisotropic materials.- 10.3 Stress-strain equations for plane stress in an orthotropic material.- 10.4 Forces and moments in laminated plates.- 10.5 Stresses in laminated plates.- 10.6 Allowable stresses for laminated plates.- 10.7 Interlamina stresses.- 10.8 Joints in laminated plates.- 10.9 Bending deflections of laminated plates.- 10.10 Buckling loads for laminated plates.- 10.11 Vibrations of laminated plates.- 10.12 Problems.- References.- Appendix A / Notes on matrix algebra.- A.1 Definition of matrices.- A.2 Addition, subtraction, multiplication of matrices.- A.3 Determinants.- A.4 Matrix inversion.- A.5 Solution of systems of simultaneous equations by matrices.- A.6 Solution of systems of simultaneous equations by tri-diagonal matrices.- A.7 Solution of systems of equations by Jordan successive transformations.- A.8 Matrix representations.- A.9 Orthogonal matrices.- A.10 Eigenvalues and eigenvectors of matrices.- A.11 Note on matrix notation.- References.- Appendix B / External forces on flight vehicles.- B.1 Introduction.- B.2 Inertial forces for rigid body translation and rotation in a vertical plane.- B.3 Air forces on airplane wing.- B.4 Airplane equilibrium equations in flight. Load factors.- B.6 Wing spanwise lift coefficient distribution.- B.7 Spanwise lift coefficient distribution on twisted wings.- B.8 Spanwise airload, shear, and moment distributions on wing.- B.9 Distribution of inertia forces on wing and fuselage.- B.10 Forces and moments on landing gear structures.- B.11 Thermal forces.- B.12 Miscellaneous forces.- B.13 Deflection effects on the external forces.- B.14 Criteria for the structure to support the external forces.- B.15 Problems.- References.- Appendix C / Derivation of the strain energy theorems from the virtual principles.- C.1 Work and strain energy.- C.2 Maximum and minimum strain energy and total potential energy.- C.3 Theorem of minimum total potential energy.- C.4 Theorem of minimum strain energy.- C.5 Castigliano’s theorem (Part I).- C.6 Hamilton’s principle.- C.7 Theorem of minimum total complementary potential theory.- C.8 Theorem of minimum complementary strain energy.- C.9 Castigliano’s theorem (Part II).- C.10 Reissner’s variational principle.- C.11 Comparison of the virtual principles and the strain energy theorems.- References.