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**Modeling for Analysis CE 636 - Design of Multi-Story Structures**

T. B. Quimby UAA School of Engineering Modeling for Analysis

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**Levels of Analysis Preliminary Final Analysis Hybrid Analysis**

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

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**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.

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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: the shear and warping torque resistance of elevator, stair, and service shafts.

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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.

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**Effect of Shear Stiffness**

The stiffer the shear connection, the larger the proportion of external moment that is carried by external forces.

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**Mobilizing Orthogonal Bents**

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**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.

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**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.

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**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).

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**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 in text) Non rectangular bodies will require generation of a transitional mesh.

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**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.

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**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).

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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).

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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

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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.

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**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.

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**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.

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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

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**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

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