Modeling for Analysis

1. CE 636 - Design of Multi-Story Structures T. B. Quimby UAA School of Engineering Modeling for Analysis

2. Levels of Analysis • Preliminary • Very rapid and simple approximate analysis • Gross approximations are made • Deflections and member forces should be within 15% of what a final analysis would give. • Final Analysis • Gives accurate deflections and member forces • Hybrid Analysis • Combines both preliminary and final analysis

3. Assumptions • Necessary to reduce the problem to a viable size • Materials of the structure and components are linear. • Only the primary structural components participate in the overall behavior. • Floor slabs are assumed to be rigid in plane. • Component stiffness of relatively small magnitude are assumed to be negligible. • Deformations that are relatively small, and of little importance, are neglected. • The effects of cracking in reinforced concrete members due to flexural tensile stresses are assumed to be representable by a reduced moment of inertia.

4. General Behavior • Loads from gravity forces result from tributary areas supported by members. • Resistance to external moment is provided by flexure of the vertical components and their axial action acting as the chords of a vertical truss. (See next slide) • Horizontal shear is resisted by: • Shear in the vertical components • the horizontal components of axial force in diagonal braces • Torsion on a building is resisted mainly by: • Shear in the vertical components • the horizontal components of axial force in diagonal braces • the shear and warping torque resistance of elevator, stair, and service shafts.

5. More General Behavior • Resistance to bending and torsion can be significantly influenced by the vertical shearing action between connected orthogonal bents or walls. (Flange action) • Horizontal force interaction occurs when a horizontally deflected system of vertical components with dissimilar lateral deflection characteristics is connected horizontally.

6. Effect of Shear Stiffness • The stiffer the shear connection, the larger the proportion of external moment that is carried by external forces.

7. Mobilizing Orthogonal Bents

8. Approximate Analysis Modeling • Simplify analysis by replacing complex structures with “simple” structures having the same lateral characteristics. • Shear Walls and Braced Frames (deflection controlled by flexure) can be modeled with an “equivalent beam”. • Multibay Frames can be represented by a single bay Frame. • More complex coupled systems can be represented by assemblies of simple structures that each represent a particular type of bent. May need to include “rigid” arms to account for geometric bent width. • Nonplanar assemblies can be represented by a column located at the shear center.

9. Accurate Analysis Modeling • Model needs yield accurate deflections and member forces. • Current computer analysis techniques use finite elements (stiffness method) and are capable of solving large, complex problems. • Input actual members, not simplified approximations. • Only include members that contribute/effect to the lateral force system. • Include all gravity and lateral forces carried by members in the model.

10. Typical Finite Elements • See text Figure 5.12 • Truss element. 2 DOF (one translational at each node). • Beam element. 12 DOF (three translational and three rotational at each node). • Quadrilateral membrane element: 8 DOF (two translational at each node). • Quadrilateral plate bending element. 12 DOF (1 translational, 2 rotational at each node).

11. Using Membrane Elements • Used for modeling of shear walls. • Only translational DOF • All DOF are in one plane • Cannot apply moment at the nodes • Need to add a fictitious element to approximate a rigid connections (see Figure 5.17 in text) • Non rectangular bodies will require generation of a transitional mesh.

12. Three Dimensional Frame Systems • Use beam elements for frames • Deform axially, in shear and bending in two transverse directions, and twist • Need area, two shear areas, two moments of inertia, and torsional constant. • omitting or using large values for member properties can simplify the problem.

13. Three Dimensional Shear Wall Systems • Use plane stress membrane elements since shear and bending are in-plane. • Story height, wall width elements are generally suitable. They give shear and chord forces at the node. • Rigidly connected links to other systems require the use of fictitious beams in the wall that are very rigid. (See text Figures 5.19 and 5.20).

14. P-Delta Effects • The text shows two methods for modeling for P-Delta effects if your program does not include analysis of P-Delta effects. • Use either negative shear area or negative moment of inertia to simulate the “softening” effects of gravity loads (i.e. P-delta effects).

15. Reduction Techniques • Large buildings can result in very complex models. • Reduced models must have the same deflections and member forces as the full model. • Use of symmetry and antisymmetry simplify the model and you get two benefits: • Reduced computer time and size requirements. • Less chance for error when adjusting member sizes. • Two dimensional models of three dimensional systems. • can use 2d beam elements with 6 DOF • Lumping like bents together in a 2D model • Wide Column and Deep Beam analogies

16. Symmetry • Must have symmetry or antisymmetry in both structure and loading. • Model 1/2 of the building with 1/2 the loads. • Be careful with the restraints at the “cut”. They must cause the restraint that the other half of the structure would provide. • See text Figures 5.23 and 5.24. • If a building is doubly symmetric (both symmetric and antisymmetric) you can model only one quarter of the structure.

17. 2D Models of 3D Systems (nontwisting structures) • put all bents in the same plane with axially rigid truss element links to represent the connecting rigid slab (see text Figure 5.25) • If orthogonal frames are mobilized (via stiff shear elements) they can also be included with a little work. • Connection to orthogonal frames is shear only so connecting link must be very rigid in shear and flexure while not transferring any axial force. (see text Figure 5.26). • Intersection columns are represented twice. Area is assigned to representation in parallel frame. Other representation gets zero area.

18. 2D Models of 3D Systems (twisting structures) • Translation in two orthogonal directions with twist is same a twist about a point somewhere else in the plane. (see text Figure 5.27) • Twisting generally occurs in asymmetric structures. • Technique is conceptually complex.

19. Lumping • Combination of several of a structure's similar, and similar behaving, components or assemblies of components into an equivalent single component or assembly. • Lateral Lumping • combining similar bents (text Figure 5.29) • lumped assembly's behavior must be the sum of all the represented assemblies' behaviors. • Vertical Lumping • Can be used in structures having repetitive beam sizes and story heights. • Combine 3-5 levels of beams together at the middle beam location. • Lateral loads lumped at same levels

20. Wide-Column and Deep-Beam Analogies • Shear walls can be modeled by “wide columns”. This gives you shear and moment at top and bottom of wall. • Must use “rigid” links for connections to beams or beams will be longer than they really are, increasing deflections and resulting in an oversized beam. • Same ideas hold true for deep beams connected to columns. • See text figures 5.32 through 5.34