CE 636 - Design of Multi-Story Structures
T. B. Quimby
UAA School of Engineering
 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
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.
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.
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.
The stiffer the
shear connection,
the larger the
proportion of
external moment
that is carried by
external forces.
Cross Beam Size: W10x19
deflection at roof:
0.0255 ft
Flange total
Cross Beam Size: W36x135
deflection at roof:
0.0247 ft
Flange total
Change in deflection
Simplify analysis by replacing complex structures with
“simple” structures having the same lateral
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
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.
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
Only include members that contribute/effect to
the lateral force system.
Include all gravity and lateral forces carried by
members in the model.
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).
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.
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.
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
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).
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
 can use 2d beam elements with 6 DOF
Lumping like bents together in a 2D model
Wide Column and Deep Beam analogies
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.
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
 Intersection columns are represented twice. Area is
assigned to representation in parallel frame. Other
representation gets zero area.
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
Technique is conceptually complex.
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
 Lateral loads lumped at same levels
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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