Virtual Principles in Aircraft Structures: (6-7 Mechanics of Structural Systems)

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.

Table of Contents:
1 / The basic three, two, and one dimensional equations in structural analysis.- 2 / Virtual displacement and virtual force methods in structural analysis.- 3 / The virtual principles for pin-jointed trusses.- 4 / The virtual principles for simple beams.- 5 / Box beam shear stresses and deflections.- 6 / Shear lag in thin web structures.- 1 / Allowable stresses of flight vehicle materials.- 2 / Analysis and design of joints and splices.- 3 / Structural design of aircraft components.- 4 / Analysis and design of pressurized structures.- 5 / Approximate solutions using the virtual principles.- 6 / Dynamics of simple beams.- 7 / The plate equations.- 8 / Approximate matrix equations for plate finite elements.- 9 / Matrix structural analysis using finite elements.- 10 / Composite Materials.- 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.