CiEn 361 Structural Analysis

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CiEn 361- Structural Analysis

• Lecture 1
• Fall 2000
• Huey K. Lawson

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Key Information
Instructor Huey Kenneth Lawson Website http//
blackboard.subr.edu Text Hibbeler, R. C.,
Structural Analysis, Fourth Edition,
Prentice Hall Office Suite 222, J. B. Moore
Hall
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Course Overview

• Introduction to Structures and Loads
• Review of Statically Determinate Structures
• Review of Statically Determinate Trusses
• Internal Loading in Beams and Frames
• Deflections
• Force Method of Analysis for Indeterminate
  Structures
• Displacement Methods of Analysis
• Influence Lines
• Analysis of Non-prismatic members

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Chapter 1 - Hibbeler

• Types of Structures and Loads

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A structure

• Refers to a system of connected parts used to
  support a load.

"A good structural engineering design must be
judged on three criteria minimum use of
materials, minimum cost, and maximum aesthetic
expression (with satisfactory performance)"
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The Design Process
To discuss the design process, consider the
following design problem
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Controlling Design Specification
A project of this type will generally be
constructed following the traffic, highway
alignment and bridge design specifications of the
American Association of State Highway and
Transportation Officials (ASSHTO). These will
specify, among many other requirements, the
number of lanes required for the expected
traffic, the sightlines and side distances
required for safety, and how the bridge will be
designed. Environmental and aesthetic concerns,
as well as construction process considerations
will weight heavily on many of the design choices
to be made.
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After a site has been selected and the main
logistical constraints established, typical steps
in the structural engineering design process are
as follows
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Selection of the Structural Material and Form
There are many options available to the
designer, and the choices will be based on
economics, client preferences, and the expertise
of the designer and construction company. In
general a single selection will be made for a
small project a large project will carry several
options through the preliminary design stages
(i.e., in sufficient detail to generate reliable
budget figures). The main idea for the structural
system (form) is to provide a complete load path
for vertical loads (gravity and snow), lateral
loads (wind, earthquake, earth and water
pressure), self-restraining loads (temperature,
shrinkage and creep), and support movements. The
structural form has to provide good performance
both under everyday loading (the service load
level) and under extreme overloads (the ultimate
strength load level). Ideally the form will
provide both ductile behavior (i.e., under an
extreme overload, it will undergo gross
deformations and give warning of impending
failure) and alternate load paths should one of
its main components fail. The need for alternate
load paths gives rise to indeterminate
structures. The next pages show some possible
alternatives and some pros and cons of each
option. The pros and cons are not meant to
provide an exhaustive list of the advantages and
disadvantages of each structural system. They are
given only to illustrate the fact that in
engineering design there are no unique solutions.

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Pros and Cons?
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Pros and Cons?
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Pros and Cons?
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List of Long Bridge Span for Each Type

• Beams 100 ft.- Rio-Niteroi (860 ft., Brazil)
• Truss cantilever 300 ft. Quebec (1793 ft.,
  Canada) Firth of Forth (1710 ft. Scotland)
• Arch rib - 400 ft. Kirk (1280 ft., Croatia)
  Fremont (1245 ft.)
• Arch -truss 800 ft. New River Gorge (1700
  ft.) Bayonne (1675 ft.)
• Cable-stayed 300 ft.- Tatara (2920 ft.,
  Japan) Normandie (2808 ft., France)
• Suspension - 1000 ft. - Akashi-Kaikyo (6529 ft.
  Japan) Storebelt (5328, Denmark)

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Determination of the external loads
The design loads will be derived primarily from
the clients needs and the location of the
structure. Since all the loads that will be
placed on a structure are difficult to determine
a priori, this step presents many problems to the
designer. Generally this step is handled with the
aid of load codes (legal documents adopted by
jurisdictions) such as ASCE 7-98 and the ASSHTO
bridge design specification. These load codes
provide minimum loads and are probabilistic in
nature. Examples of the probabilistic nature are
the wind velocity distributions assumed, the
models used to predict earthquake occurrences,
and the load combinations used. The load codes
assume that a small but finite probability of
failure is acceptable and the design process
implies a certain risk (risk probability of
failure x cost of failure).
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For the design of a bridge such as the one in our
example, the following loads will need to be
considered

• Dead load of structural components (slab, beams,
  girders and columns) and non-structural elements
  (barriers, illumination towers, utilities,
  wearing surface)
• Horizontal and vertical earth pressures
• Accumulated locked-in forces from the
  construction process
• Live loads due to vehicles (trucks), including
  friction, braking and centrifugal forces
• Wind and earthquake forces
• Vehicular and vessel collision forces
• Pedestrian live loads
• Creep, shrinkage and temperature effects
• Water and ice loads and stream pressures

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Selection of the limit states or performance
criteria

• Service vertical deflection (L/360 for buildings
  and L/800 for bridges, where L is the span),
  drift (sidesway, H/400 for buildings under
  everyday wind loads, where H is the height of the
  structure), vibration, cracking, long term (creep
  and shrinkage), temperature and other thermal
  stresses, fatigue, small earthquakes.
• Ultimate Strength the structure will yield and
  deform, but not collapse under the maximum design
  loads (say maximum credible earthquake).
• Overload for bridges, some overloads are
  expected (either legal or illegal) for medium
  size earthquakes for high winds some damage
  permitted.
• Durability, reparability, adaptation to new uses,
  real performance

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Creation of an analytical model
We will talk in more details about this step
latter, but this step is of concern in this
course. Numerous decisions need to be made here
about the complexity of the model to be used. The
level of complexity will depend on the importance
of the structure, the uncertainties of the loads
and material properties, and the availability of
solution tools (computer programs).
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The model has to have some physical attributes
(member sizes, strength of the materials, etc.)
before we can proceed to the next step. In
addition, the following remarks are in order

• The main advantage of the current
  state-of-the-art is that designers can develop a
  structural idealization that is very "close" to
  the true structure many different alternatives
  can be investigated many more loading types can
  be handled, and the problems can be approached in
  a systematic and consistent fashion. This is due
  mostly to the availability of computer programs.
• However, many important behavioral aspects (3D
  action, role of the floor slabs, torsion,
  semi-rigid connections, non-structural elements,
  etc.), are ignored even in the most
  "sophisticated" models used for preliminary
  design.
• The main disadvantage is that the approach can be
  turned into a "black box" (GIGO garbage in,
  garbage out). The output is only as good as the
  input, so the modeling is the key experience is
  needed, and the model and the real structure must
  correspond. Insuring the reasonableness of the
  results is still critical.

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Computations of stresses and strains
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This is the essence of this course!! once we
have a model and the loads, can we find all the
internal forces and external reactions ? Analysis
for stresses and deformations can usually be
accomplished with the aid of analysis charts and
tables for simple structures and preliminary
design purposes, and by computer for larger,
indeterminate structures. Note that step (4)
required us to guess the member sizes thus an
iterative procedure for steps (3)-(8) is
necessary in order to optimize the design.
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Evaluation of results
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- of concern in this course useless unless you
know how to compute well!!
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Design of members to satisfy (5).
The design of structural members is often
governed by design specifications developed by
industrial organizations. These are often related
to materials production ACI for concrete, AISC
and AISI for steel, MSI for masonry, etc.). These
design codes are adopted into regional building
codes such as the UBC (Uniform Building Code),
the Southern Building Code (SBC), and the
National Building Code. By the year 2001, we will
have the International Building Code (IBC) which
will unify all regional codes into a single
national one. Note that there is a difference
between a code, which is a legally binding
document, and a specification, which only
suggests rules for design. The ACI 318-95
documents, which govern the design of structural
concrete structures, are specifications they are
incorporated into the design codes by
organizations such as BOCA, and then become law
when adopted by -the local authorities.
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Reanalysis to ensure adequacy of the final design.
This step checks that, for the final design,
individual members and their connections, as well
as the overall structure, satisfy limit states.
Remember that limit states imply "failure to
perform as intended", not necessarily collapse!
cracking, creep and shrinkage of concrete
significantly affect its performance these are
not of concern with steel design. Another example
is durability characteristics- related mostly to
cracking resistance and permeability of the
concrete (chloride ion transfer). Further
examples include fire and fatigue resistance.
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Detailing
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the final design must be consistent with the
assumptions of the final model. Detailing is more
an art -than a precise science, although some
basic guidelines are available. All the hard work
of the other phases might be lost if detailing is
-not properly carried out.
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Objectives of the Design Process

• Protect life safety when the structure is
  subjected to the maximum expected loads
• The primary concern, and the only one actually
  addressed directly by building codes, is
  protection of human life. It overrides any other
  consideration that the engineer might have.
• Hammurabi's Code - "an eye for an eye".
• Buildings must not endanger their occupants even
  under extreme loads if hurricanes tornadoes or
  earthquakes hit, we want the building to survive.
  It may not be of much use afterwards, but it must
  not to collapse on its occupants.
• Insure satisfactory performance under everyday
  loads- does the building satisfy the requirements
  that its owner demanded? This is primarily a
  matter for the architect and design managers, but
  the structural engineer must insure certain
  things, i.e., that the building does not sway
  significantly under moderate winds (humans are
  very sensitive to motion, an even small
  accelerations can make people uncomfortable), or
  if the building sways enough, excessive
  deformations may cause doors and windows to jam,
  facade panels to crack (humans are also very
  sensitive to visual stimuli, the building may be
  structurally safe, but people will go by
  appearances).

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Objectives of the Design Process

• Provide economy of construction - always an
  important factor - mostly a matter of the correct
  selection of the materials and the expertise of
  the construction firm some of these choices are
  tied to geographical availability.
• Provide ease of construction In RC
  construction, for example, ease of fabrication
  and placement of the steel reinforcement and
  formwork (lack of steel congestion). Speed of
  construction how many floors per week?. The
  design must not be confusing or require
  significant changes from the design office to the
  erection stage (Kansas City Hyatt) easy to
  avoid errors.
• Provide ease of maintenance, repair, and
  upgrading new concern tied to emerging concerns
  with sustainable development.
• Deliver all of the above and simultaneously
  provide aesthetic value this refers to the
  appropriateness of the structure both in terms of
  its exterior appearance and impact and the
  interior comfort and "user-friendliness."

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CiEn 361- Structural Analysis

• Lecture 2
• Fall 2000
• Huey K. Lawson

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Gravity Loads

• Dead Loads
• Long term stationary forces including self weight
  of the structure and weight of permanent
  equipment.
• How much do various building materials weigh?
• For example see appendices A and B in Design and
  Wood Structures by D. Breyer.
• Weights are given in lb/ft2 (also written as psf)
  
• How do you use this information?
• Consider the following flat roof

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• How much of this load does the typical 2 X 10
  carry?
• It is a function of the member's tributary area.
• The area assumed to load any given member is
  called tributary area span X tributary width
  where tributary width is often on center,
  (o.c.), spacing.

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• For a typical 2 X 10, the tributary area 20 X 8
  160 ft2
• For an exterior 2 X 10, tributary area 20 X 4
  80 ft2
• Model the 2 X 10 with basic roof load as

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with w uniformly distributed line, load unit
psf load X tributary width.For this example
then, w 9.4 lb/ft2 (8ft) 75.2 lb/ft.
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But is this the only dead load carried by this
beam? Need to consider the beam self weight and
the weight of the 2 X 4 members framing into the
beam. There are a number of ways to calculate a
member's self weight, but all methods Are a
function of wood species, moisture content, and
dimensions. And you must be careful to properly
track units. For this example problem,
self-weight can be found as Assume dry Douglas
Fir-Larch, (DF-L), with specific weight of 35
lb/ft3. 2" nominal Þ 1.5" actual. 10" nominal Þ
9.25" actual.
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Calculate the dead weight of 2 X 4 at 2' o.c.
spanning into 2 X 10. Assume DF-L again One 2 X
4 member weighs 4" nominal Þ 3.5" actual.
                                     
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Therefore total dead weight supported by a
typical 2X10 is due to a summation of the basic
roof (and ceiling, if directly attached to
underside of roof members), self-weight, and
self-weight of any supporting members.
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Live Loads, L or Lr

• Live Loads, L
• Short duration forces which change in location
  and magnitude.
• Include people and furniture.
• Based upon intended use of building occupancy.
• In example, see UBC Table 16-A
• floor uniform loads are a function of
  use/occupancy.
• an example, the uniform load for
• residential 40 psf
• offices 50 psf
• light storage 125 psf

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Roof Live Loads, Lr

• UBC recognizes that roofs carry lower loads than
  floors, since roofs are generally not occupied.
• UBC specified roof loads account for
  miscellaneous loads like roofing, equipment,
  servicing.

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Roof Live Loads Lr (Contd)

• Obtain minimum unit roof loads from 1994 or 1997
  UBC table 16-C
• These live loads are assumed to act vertically
  open the area projected on a horizontal plane.
• f (roof slope, tributary load area) with smaller
  unit loads for steeper members with large
  tributary areas
• Flat roofs - higher probability that high unit
  live loads could occur.
• Tributary are a, (t.a), of member under
  consideration
• "If a member has a small t.a., it is likely that
  a fairly high unit live load could be imposed
  over that entire small surface area... It is less
  likely that a large t.a. will be uniformly loaded
  by the same high unit load considered for a small
  t.a. member." (Design of Wood Structures by D.
  Breyer)

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Procedure

• Calculate roof slope (member slope)
• Calculate t.a. of member.
• Choose Method 1 or Method 2 to obtain unit load.
• Method 1 is straightforward, but incremental.
• Method 2 provides a continuous range of loads,
  but requires the checking of two equations which
  can be found in '94 UBC 1606 or '97 UBC 1607.5.
• Apply load on the horizontal plane (see upcoming
  example under snow loads).

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Introduction to Snow Loads

• Often established by local building official as
  they can vary greatly over relatively small
  geographic areas.
• Appendix, Chapter 16 covers snow loads in much
  more detail than '94 UBC 1605.4 or '97 UBC 1614
  sections.
• This appendix provides detailed information to
  calculate
• Roof snow load as a function of ground snow load,
  building exposure, and importance.
• Unbalanced snow loads.
• Drift potential.

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Snow Load Intro. Contd

• Design snow load f (roof slope)
• If basic roof snow load is greater than 20 psf
  and if roof slope ³ 20, then Ra S/40 - 1/2,
  where Ra reduction in S in psf per degree gt
  20.
• Flagstaff's basic S 40 psf.
• Snow load is given along the horizontal plane.
• Other facts
• 1" newly fallen snow ² .5 psf.
• 1" packed snow ² 1 psf.

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Snow Load Example
Determine the load, V and M diagrams for a
typical 6 X 12 rafter using both the sloping roof
method and the horizontal plane method. The basic
roof S is given as 35 psf.
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Solution
1. Roof Live load Determination


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Roof live for typical

• Determine tributary width within horizontal
  plane t.a. 24(10) 240 ft2
• Roof slope tan q 1/2, q 26.5 or 1/2 6/12
  Þ 612
• Therefore according to UBC Table 16-C, basic Lr
  14 psf.

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UBC Table 16-FWind Stagnation Pressure (qs) at
Standard Height of 33 Feet
1Wind speed from Section 1615.
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1The limitation of Ip for panel connections in
Section 1631.2.4 shall be 1.0 for the entire
connector.2Structural observation requirements
are given in Sections 108, 1701 and 1702.3For
anchorage of machinery and equipment required for
life-safety systems the value of Ip shall be
taken as 1.5.
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Live Load Reduction
FPS Units
L reduced design live load per square foot of
area supported by the member L0 the unreduced
live load per square foot of area supported by
the member AI influence area in square feet,
equal to four times the tributary or effective
load-carrying floor area for a column, and two
time the tributary or effective load-carrying
floor area for a beam
50 max reduction for members supporting one
floor 40 max reduction for members support more
than one floor
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Chapter 2
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Support Connections in Actual and Ideal Structures
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Principle of Superposition
The total displacement or internal loadings
(stress) at a point in a structure subjected to
several external loadings can be determined by
adding together the displacements or internal
loadings (stress) caused by each of the external
loads acting separately.
Requirements

• The material must behave in a linear-elastic
  manner, so that Hookes law is valid, and
  therefore the load will be proportional to the
  displacement.
• The geometry of the structure must not undergo
  significant change when the loads are
  applied,I.e., small displacement theory applies.
  Large displacements will significantly change the
  position and orientation of the loads

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Superposition Principle
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Equations of Equilibrium
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For Most Structures Idealized in 2 Dimensions

• Requires that a free body diagram be constructed
  of the structure or its members.
• All forces and couple moments must be shown that
  act on the member
• Method of sections will be used to determine the
  internal loadings at a specific point with a cut
  perpendicular to the axis of the member at that
  point.

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The Ideal Cut SectionInternal Loadings
V Shear Force N Normal Force M Bending
Moment
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Determinacy
A structure is statically determinate when the
equilibrium equations can be utilized to
determine all the forces in the structure. A
structure is statically indeterminate when there
are more unknown forces than available
equilibrium equations. The additional equations
needed to solve for the unknown reactions are
obtained by relating the applied loads and
reactions to the displacement or slope at
different points on the structure. These
equations, which are referred to as compatibility
equations, must equal the degree of indeterminacy.
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Stability

• Two situations of stability are also required to
  maintain equilibrium by providing that members
  are properly held or constrained by their
  supports.
• Partial Constraints a structure having fewer
  reactive forces than equations of equilibrium.
• Improper Constraints - a structure having
  reactions that are concurrent at a point.

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Procedure for the Application of The Equilibrium
Equations

• Disassemble the structure and draw the free body
  diagram for each member. Also, It may be
  convenient to supplement a member free body
  diagram with a free body diagram of the entire
  structure. Some or all of the support reactions
  can be determined using this diagram.
• Recall that reactive forces common to two members
  act with equal magnitudes but opposite directions
  on the respective free-body diagrams of the
  members.
• All two-force members should be identified. These
  members, regardless of their shape, have no
  external loads on them, and therefore their free
  body diagrams are represented with equal but
  opposite collinear forces acting on their ends.
• In many case it is possible to tell by inspection
  the proper arrowhead sense of direction of an
  unknown force or couple moment however, if this
  seems difficult, the directional sense can be
  assumed.

• Count the total number of unknowns to make sure
  that an equivalent number of equilibrium
  equations can be written for solution. Except
  for two-force members, recall that in general
  three equilibrium equations can be written for
  each member.
• Many times the solution for unknowns will e
  straightforward if the moment equation ?Mo 0 is
  applied about point (O) that lies at the
  intersection of the lines of action of as many
  unknown forces as possible.
• When applying the force equations ??Fx 0 and
  ??Fy 0, orient the x and y axes along lines
  that will provide the simplest reduction of the
  forces into their x and y components.
• If the solution of the equilibrium equation
  yields a negative magnitude for a unknown force
  or couple moment, it indicates that its arrowhead
  sense of direction is opposite to that which was
  assumed on the free-body diagram.