The following step-by-step procedure can be used to take advantage of structural symmetry in the analysis of structures. 1. Check the given structure for symmetry, as discussed in Sec- tion

10.1. If the structure is found to be symmetric, then proceed to step 2. Otherwise, end the analysis at this stage. 2. Select a substructure (half the structure) on either side of the axis of symmetry for analysis. The cross-sectional areas and moments of inertia of the members of the substructure, which are located along the axis of symmetry, should be reduced by half, whereas full values of these properties should be used for all other members. 3. Decompose the given loading into symmetric and antisym- metric components with respect to the axis of symmetry of the structure by using the procedure described in Section 10.2. 4. Determine the response of the structure due to the symmetric loading component as follows: a. At each joint and end of the substructure, which is located at the axis of symmetry, apply restraints to prevent rotation and deflection perpendicular to the axis of symmetry. If there is a hinge at such a joint or end, then only the deflection, but not rotation, should be restrained at that joint or end. b. Apply the symmetric component of loading on the sub- structure with the magnitudes of the concentrated loads at the axis of symmetry reduced by half. c. d. Analyze the substructure to determine its response. Obtain the symmetric response of the complete structure by reflecting the response of the substructure to the other side of the axis of symmetry. 5. Determine the response of the structure due to the antisym- metric loading component as follows: a. At each joint and end of the substructure located at the axis of symmetry, apply a restraint to prevent deflection in the direction of the axis of symmetry. In the case of trusses, the axial forces in members located along the axis of symmetry will be zero. Remove such members from the substructure. b. Apply the antisymmetric component of loading on the sub- structure with the magnitudes of the loads and couples, ap- plied at the axis of symmetry, reduced by half. Analyze the substructure to determine its response. Obtain the antisymmetric response of the complete structure by reflecting the negative of the response of the substructure to the other side of the axis of symmetry. c. d. 6. Determine the total response of the structure due to the given loading by superimposing the symmetric and antisymmetric re- sponses obtained in steps 4 and 5, respectively. The foregoing procedure can be applied to statically determinate as well as indeterminate symmetric structures. It will become obvious in subsequent chapters that the utilization of structural symmetry consid- erably reduces the computational effort required in the analysis of stat- ically indeterminate structures./n 7. Input STAAD SPACE START JOB INFORMATION ENGINEER DATE 27-Mar-22 END JOB INFORMATION INPUT WIDTH 79 UNIT FEET KIP JOINT COORDINATES 1000; 2096 0; 3 30 96 0; 4 30 0 0; 5 60 0 0; 6 60 96 0; 70 72 0; 8 30 72 0; 9 60 72 0; 10 0 48 0; 11 30 48 0; 12 60 48 0; 13 0 24 0; 14 30 24 0; 15 60 24 0; MEMBER INCIDENCES 11 13; 22 3; 3 3 8; 4 5 15; 5 6 3; 672; 78 11; 87 8; 996; 10 8 9; 11 10 7; 12 11 14; 13 10 11; 14 12 9; 15 11 12; 16 13 10; 17 14 4; 18 13 14; 19 15 12; 20 14 15; START USER TABLE TABLE 1 UNIT INCHES KIP PRISMATIC 1 30 2000 2000 2000 30 30 0 0 END UNIT INCHES KIP DEFINE MATERIAL START ISOTROPIC STEEL E 29000 POISSON 0.3 DENSITY 0.000283 ALPHA 6.5e-006 DAMP 0.03 TYPE STEEL STRENGTH FY 36 FU 58 RY 1.5 RT 1.2 END DEFINE MATERIAL MEMBER PROPERTY 1 TO 20 UPTABLE 1 1 CONSTANTS MATERIAL STEEL ALL UNIT FEET KIP SUPPORTS 145 FIXED UNIT INCHES KIP LOAD 1 LOADTYPE Dead TITLE LOAD CASE 1 JOINT LOAD 2 FX 25 7 FX 20/n Q; Analyse the shown structure using Symmetry and antisymmetry to det. Supports reactions and draw axial, Shear, moment diagram. lok 20k 20k J D A 2 12/1 31/1 3k/1 30 It 710 B * E = 29,000 kesi A = 30 in ² I = 2000 in 4 301 L FI AL TIM e * 24 24 24 model 1: analyze the entire structure Support reactions, axial, Shear, moment diagram. 2- analyze the structure as symetry and antisymetry at loading. symetry + anti symetric = Full structurey 3. Compare results from ) and (2)