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CiEn 361- Structural Analysis

- Lecture 1
- Fall 2000
- Huey K. Lawson

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

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

Chapter 1 - Hibbeler

- Types of Structures and Loads

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)"

The Design Process

To discuss the design process, consider the

following design problem

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.

After a site has been selected and the main

logistical constraints established, typical steps

in the structural engineering design process are

as follows

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.

1

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Pros and Cons?

Pros and Cons?

Pros and Cons?

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)

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

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

3

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).

4

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.

Computations of stresses and strains

5

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.

Evaluation of results

6

- of concern in this course useless unless you

know how to compute well!!

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.

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).

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."

CiEn 361- Structural Analysis

- Lecture 2
- Fall 2000
- Huey K. Lawson

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.

- 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

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.

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.

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.

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

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.

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)

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).

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.

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.

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.

Solution

1. Roof Live load Determination

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.

UBC Table 16-FWind Stagnation Pressure (qs) at

Standard Height of 33 Feet

1Wind speed from Section 1615.

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

Chapter 2

Support Connections in Actual and Ideal Structures

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

Superposition Principle

Equations of Equilibrium

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.

The Ideal Cut SectionInternal Loadings

V Shear Force N Normal Force M Bending

Moment

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.

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.

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.