ce 360 study guide

CE 360: Structural Mechanics IAnalysis of statically determinate structures and approximate

methods for indeterminate structures

by

Tonatiuh Rodriguez-Nikl

Spring 2015

California State University, Los Angeles

College of Engineering, Computer Science, and Technology

Department of Civil Engineering

Contents

Contents i

I Course Documents 1

1 Procedures 3

II Study Guide 9

1 Introduction and Moment Diagrams 11

2 Deflected Shapes of Beams 23

3 Beam Deflections by Virtual Work 27

4 Forces in Frames 43

5 Deflections in Frames 47

6 Deflections in Trusses 51

7 General Applications of Virtual Work 55

8 Approximate Analysis: Portal Method for Lateral Loading

of Moment Frames 59

9 Approximate Analysis: Moment Frames under Gravity Loads 63

10 Approximate Analysis: Braced Frames under Lateral Loads 67

11 Using Influence Lines 71

12 Drawing Influence Lines 75

13 Moving Point Loads and Maximum Absolute Response 79

i

ii CONTENTS

14 Cable Structures 83

III Exams 87

1 Fundamentals 89

2 Deflections & Approximate Analysis 91

3 Influence Lines & Cables 93

4 Comprehensive Final Exam 95

Part I

Course Documents

1

DOCUMENT 1Procedures

1.1 Shear and Moment Diagrams by Graphical Method

• Determine external reactions as needed by considering global equilibrium.• Redraw a free body diagram of the whole member. Draw actual dis-tributed loads (not resultants).

• Identify points of discontinuity (where there is any sudden change).• Between points of discontinuity consider the following properties ofslopes and areas:

– At any point in the diagram

⇤ The slope of the shear diagram = the value of distributed load⇤ The slope of the moment diagram = the value of shear force

– Between any two points

⇤ The change in shear = the area under the distributed load curve⇤ The change in moment = the area under the shear diagram

At points of discontinuity, use the following known results. Readingfrom left to right

– A concentrated force

⇤ Causes a jump in the shear diagram equal to the magnitude ofthe force and in the same direction as the force

⇤ Causes a kink in the moment diagram

– A concentrated moment

⇤ Has no e↵ect on the shear diagram⇤ Causes a jump in the moment diagram equal to the magnitudeof the moment (upward jump for clockwise moment, downwardfor counterclockwise)

If reading from right to left, all directions are reversed.

3

4 DOCUMENT 1. PROCEDURES

1.2 Drawing Deflected Shapes of Beams

• Obtain bending moment diagram (qualitative diagram is OK)• Draw deflected shape:

– Respect curvature implied by bending moment diagram,– Respect restraints,– Respect continuity (deflection and slope).

1.3 Sketching Deflected Shapes of Beams without theMoment Diagram

This procedure is highly iterative. You will make assumptions that may proveincorrect. At the end, you need to make sure that your solution is internallyconsistent. There is only one correct answer. This is like a di�cult Sudokupuzzle, where you sometimes have no choice but to make a guess and see howit works out.

• Leave space on your paper to draw a deflected shape, free body diagram,moment diagram, and shear diagram.

• Deflect the points directly under a load in the direction of the load (withmore complex loading the point may move against the load – you willcorrect for this later).

• Sketch the deflection and rotation boundary conditions imposed by re-straints.

• Sketch a smooth deflection curve that respects all points and slopes.Identify any points of inflection.

• Identify moment and shear boundary conditions. Draw a partial freebody diagram and include any known information on the moment andshear diagrams.

• Sketch the moment diagram. Be consistent with the moment boundaryconditions, the curvature of the deflected shape, locations of points ofinflection (zero moment), and the load (M 00 = !). Remember that themoment changes slope wherever there is a point load.

• Sketch the shear diagram. Be consistent with the shear boundary condi-tions, the moment diagram (M 0 = V ), and the load (V 0 = !).

• Complete the free body diagram. Jumps in the shear diagram representpoint forces. Jumps in the moment diagram represent point moments.

• Check that the deflected shape, free body diagram, and shear and mo-ment diagrams are consistent with each other. If not, adjust your sketchesiteratively until they are consistent.

1.4 Beam Deflections by Virtual Work

• Draw moment diagram due to applied loads (real moments M).• For each desired deflection � (or rotation ✓)

1.5. REACTIONS AND INTERNAL FORCES IN FRAMES 5

– Apply a unit force (or moment) at the location and in the directionof desired deflection (or rotation)

– Draw moment diagram due to unit load (virtual moment m)

• Deflections (or rotations) are given by �(or ✓) =RL

mM/EI dx. Thisintegral can often be computed with the graphical integration tables.

1.5 Reactions and Internal Forces in Frames

• Draw a free body diagram of the entire structure (global FBD).• Separate the structure into individual pieces anywhere there is a hinge.Draw FBDs of each piece.

• Using the global FBD and the FBDs for individual members, solve forall reactions and forces at the hinges.

• Divide all pieces into straight segments. Place an unknown axial force,shear force, and bending moment at each cut.

• Solve for the unknown axial forces, shear forces, and bending moments.• Draw moment diagrams for each straight segment using the methods ofSection 1.1.

• Summarize the results on a sketch of the structure.

1.6 Deflections in Structures by Virtual Work

• Calculate bending moments (M) and axial forces (N) under real loads.Note that axial forces are usually neglected in frames and bending mo-ments do not exist in trusses.

• Apply a unit force (or moment) at location of and in direction of desireddeflection (or rotation)

• Calculate bending moments (m) and axial forces (n) due to unit load.• Deflections are given by

� =NbX

i=1

Z

Li

m

i

M

i

EI

i

+NaX

i=1

n

i

N

i

L

i

EA

i

where Nb

is the number of members with bending moment and N

b

is thenumber of members with axial force.

1.7 Sketching Deflected Shapes for Frames

• Sketch the joints in their displaced and rotated position. This will involvean educated guess that will be checked later.

• Sketch a trial deflected shape using the assumed joint displacement androtation. Respect

– The curvature implied by the moment diagram (if available)– The deflection imposed by restraints


– Continuity where appropriate (this includes maintaining the angleat rigid joints)

• Check the four items just listed and equilibrium at all joints. If any errorsor inconsistencies are notes, return to the first step and repeat the sketch.

• If the moment diagram was not available it can be sketched now basedon the curvature of the deflected shape.

1.8 Using Influence Lines

• Obtain influence line r(x) by either of previous methods. Be sure influ-ence line was obtained using a unit force or a unit beam deflection.

• Response R to various loads is as follows.

– Single load of magnitude P0 at a location x0:

R = P0 · r(x0)

– Multiple loads of magnitude P

i

at respective locations xi

:

R =X

i

P

i

· r(xi

)

The worst case for moment at a point is when one of the loads isabove the point. The worst case for shear at a point is when one ofthe loads is just left of just right of the point.

– Distributed load w(x) between two points A and B:

R =

ZB

A

w(x)r(x) dx

– Special case of a uniform load of magnitude w0:

R = w0

ZB

A

r(x) dx

| {z }Area under r(x)

1.9 Drawing Influence Lines (as function of x)

• Select a response (moment at midspan, reaction at a support, shear atend, etc.).

• Place a unit load at a distance x from the left end of the beam.• Solve for the response as a function of x. The result may be a piecewisefunction.

• Plot the resulting function

1.10. DRAWING INFLUENCE LINES (MULLER-BRESLAU PRINCIPLE)7

1.10 Drawing Influence Lines (Muller-Breslau Principle)

• Allow the force of interest to displace the beam a unit distance:

– Reaction: move support unit distance,– Shear force: displace ends apart (left end down) a unit distance

while maintaining beam segments parallel,– Bending moment: rotate right side of beam down a unit angle rela-

tive to left.

If the beam is statically determinate, all sections of the beam will remainstraight.

• If desired, obtain other values of influence line by small angle trigonom-etry.

1.11 Maximum Absolute Response

See class notes

1.12 Approximate Analysis of Indeterminate BuildingFrames

• Make reasonable assumptions to reduce structure to statically determi-nate.

– Portal method (lateral loading)

⇤ Point of inflection (zero moment) occurs at center of beams andat center of columns (except for pinned, ground level columns).

⇤ Interior columns carry twice the shear as exterior columns.

– Frames under gravity loading

⇤ Ends of beams have moment equal to 0.045wL2.⇤ Consider joint equilibrium. Divide moment imbalance evenlybetween all unloaded members (usually just columns).

⇤ If the base of of a bottom story column is fixed, the base car-ries half the moment as the top and the column is in doublecurvature.

– Braced frames (lateral loading)

⇤ Make an assumption about the force in the compression bracedepending on its slenderness. A slender brace will have zeroforce. A non-slender brace will divide force evenly with thetension brace.

⇤ There is no shear in the columns.

• Use equilibrium to solve for all quantities of interest.


1.13 Cable Structures with Point Loads

Solve like a truss with the method of joints with the di↵erence being that thegeometry is partially unknown. In general, this means solving a coupled systemof equations. The following procedure applies to book problems, which alwayshave a cable of known orientation.

• Use global equilibrium to find the force in the cable of known orientation.• Use method of joints to find the rest of the unknowns, including theunknown geometry.

Part II

Study Guide

9

LESSON 1Introduction and Moment

Diagrams

Lesson Objectives

• Classify structural systems• Draw moment diagrams using the graphical method

Related Reading

• Textbook

– Read §1.1 and 1.2 then skim the rest of chapter 1– §2.1 (skim pp. 40 – 45). The rest of chapter 2 is review. Read it in

as much depth as you need given the strength of your background.– §4.1 – 4.3

• Procedures §1.1

Preparation

State the calculus relationships between shear force, bending moment, anddistributed load.

11

STRUCTURAL ELEMENTS, SYSTEMS AND MODELS 13

Structural Elements, Systems and Models

The slides from class are included in the following pages.

Structural Elements (sec 1.2) • Rods: Straight, narrow, carry

load axially • Beams: Horizontal, carrying

vertical loads through bending

• Columns: Vertical, carrying vertical loads through compression

• Beam-Columns: Columns that also resist forces in bending

• Cables: Thin, carry load in tension, deform noticeably

Structural elements combine to form structural systems to carry loads

Moment Frame (steel / concrete)

• Forces resisted by large sidesway

• Statically indeterminate

http://www.eqclearinghouse.org/2011-03-11-sendai/2011/08/03/eeri-steel-structures-reconnaissance-group/dsc_0207/

All joints rigidly connected

Damaged in 2011 Tohoku Earthquake

14 LESSON 1. INTRODUCTION AND MOMENT DIAGRAMS

Concentric Braced Frame (Steel)

http://www.structuremag.org/article.aspx?articleID=717

Columns typically continuous (rigidly connected)

Braces and beams typically pin connected

Eccentrically-Braced Frame (Steel)

http://www.structuremag.org/images/0209-f2-6.jpg


Truss

http://www.past-inc.org/historic-bridges/Gloss-trussbridge.html

Floor System Bracing (another truss)

Bracing (another truss)

• All members pin connected

• Loading only at joints

Floor Systems

http://www.tatasteelconstruction.com/en/reference/teaching_resources/architectural_studio_reference/design/choice_of_structural_systems_for_multi/framing_schemati/

or girder

or beam

(may also rest on joists or stringers)


Suspension Bridge (Cable)

http://teachers.egfi-k12.org/wp-content/uploads/2010/03/800px-GoldenGateBridge-001.jpg

Note a truss supporting the roadway

Cable Stayed Bridge

http://1.bp.blogspot.com/-rUUFOFQX8Ro/TatjMDpPHrI/AAAAAAAAAyc/pJr6bQLv_pQ/s1600/pic2.jpg

Cable stays (support deck from main tower)


Structural Models (sec 2.1)

• No connection is perfectly rigid or flexible – choose the best fit

• Model neglects member size and uses centerline dimensions

What error is introduced? Why?

Additional insight in section2.1


IN-CLASS EXERCISES 19

In-Class Exercises

During this lesson we will be working on shear and moment diagrams as a class.Some of you may already be very good at drawing them. If you are finding thislesson less challenging than you would like, start working on the AdditionalProblems listed in the next section. One page of the textbook is reproducedfor your convenience. Don’t follow the directions as written, just draw shearand moment diagrams of the entire beam.

FOLLOWUP PROBLEMS 21

Followup Problems

1. F4-15 (p 160)2. Draw shear and moment diagrams of the following beam

3. 4-33 (p 162)

Additional Practice: Using the graphical method, draw shear and momentdiagrams for all beams pictured in pages 144 to 149 and pages 160 to 162.

LESSON 2Deflected Shapes of Beams

Lesson Objectives

• Draw deflected shapes of beams using a moment diagram• Draw deflected shapes of beams (and shapes of shear and moment dia-grams) directly from loading

Related Reading

• Textbook §8.1 and 8.2• Procedures §1.2 and 1.3

Preparation

State the relation between moment and curvature (see §8.2)

23


Followup Problems

Draw deflected shapes for the beams pictured in the following problems. Useboth methods covered in class. First, attempt to draw the deflected shapewithout the moment diagram. Then, calculate the moment diagram and re-draw the shape.

1. Problem 8-30 (p 336)2. One more problem to be given in class

Additional Practice: Draw deflected shapes for all beams pictured on pages314 and 315. Verify your solutions in o�ce hours.

LESSON 3Beam Deflections by

Virtual Work

Lesson Objectives

• Compute deflections in beams by virtual work

Related Reading

• Textbook §9.1 – 9.3, and 9.7 (ignore frame examples)

• Textbook front cover, Table for EvaluatingRL

0 mm

0 dx• Procedures §1.4

27

28 LESSON 3. BEAM DEFLECTIONS BY VIRTUAL WORK

Preparation

Draw deflected shapes and moment diagrams for the three beams picturedbelow:

THEORY ON DEFLECTIONS BY VIRTUAL WORK 29

Theory on Deflections by Virtual Work

The slides from the lecture presentation on virtual work are included in thefollowing pages.

Background

Hooke’s Law

Bending Stress

Axial Stress

Moment of Inertia

Work-Energy Principle


External Work

External Work


External Work

Internal Strain Energy


Ui for Axial Forces

Ui for Beam (bending)


Summary

External Work (Force)

External Work (Moment)

Internal Energy (Axial)

Internal Energy (Bending)

Application

Calculate deflection at the tip of the beam


Limitations

Can only calculate deflections: • At the location of the

force (or moment) • In the direction of the

force (or moment) • Due to a single point

force (or moment)

Virtual Work

1. Apply virtual force Pv causing virtual stresses ʍv.

2. Apply real load F, causing real strains ɸ and real deflection ȴ at Pv.

F

Pv

ȴ


Virtual Work

3. External virtual work is ௩ ή ο

4. Internal strain energy is

න ௩ߪ ή ��ߝ

5. Find ȴ by equating external virtual work and internal virtual strain energy

F

Pv

ȴ

Conclusion

External Work (Force)

External Work (Moment)

Internal Energy (Axial)

Internal Energy (Bending)

௩ ൌ ௩ ή ο

௩ ൌ ௩ܯ ή ߠ

௩ ൌ ௩ ή ή ܮܣܧ

௩ ൌ න ௩ܯ ή ܯܫܧ ݔ��


INTEGRATIO

NTABLES:

39

Integra

tion

Tables:

Theintegration

table

fromthebook

isrep

roduced

here

foryou

rconven

ience.

40LESSON

3.BEAM

DEFLECTIO

NSBY

VIR

TUALWORK

This

integrationtab

leis

more

detailed

andis

need

edfor

someprob

lems

involvingparab

olaswith

upw

ardcon

cavity.


Followup Problems

1. Problem 9-29 (p 388)2. Problem 9-31 (p 388)

Additional Practice:All the odd problems from 9-21 to 9-33 (page 388).

LESSON 4Forces in Frames

Lesson Objectives

• Determine reactions and internal forces in statically determinate frames.

Related Reading

• Chapter 2• Procedures §1.5

43

44 LESSON 4. FORCES IN FRAMES

Preparation

1. Solve for Ax

, Ay

and P .

2. Draw moment diagrams for the beams shown.


Followup Problems

1. Draw moment diagrams for the frames and loading given in class.

Additional Practice: For problem F2-1 to F2-10 (p . 70 and 71) and 2-33,2-34, 2-36 – 2-44 (p. 75 to 77), draw axial, shear and moment diagrams for allmembers. A detailed solution will be provided for F2-7.

LESSON 5Deflections in Frames

Lesson Objectives

• Calculate deflections in frames using virtual work• Draw deflected shapes of frames

Related Reading

• Textbook §9.7• Procedures §1.7

47


Followup Problems

1. Problem 9-46 and 9-47 (p 390). Do the problems as stated and also drawthe deflected shape. Both problems refer to the same structure but askfor di↵erent displacements.

Additional Practice: Problems 9-42, 9-44, 9-51, 9-54, 9-58, and 9-60 (p. 389 –391). Also, draw deflected shapes.

LESSON 6Deflections in Trusses

Lesson Objectives

• Calculate deflections in trusses using virtual work

Related Reading

• §9.4

Preparation

Solve the problems given in class.

51


Followup Problems

1. Problem 9-15 (p 363)

Additional Practice: Do all problems from pages 361 to 363, ignoring thosefor Castigliano’s Theorem.

LESSON 7General Applications of

Virtual Work

Lesson Objectives

• Calculate deflections in any structure made from truss and frame mem-bers

• Draw deflected shapes for any structure made from truss and frame mem-bers

Related Reading

• Procedures §1.6

Preparation

Determine axial forces and bending moments for the two problems given inclass.

55


Followup Problems

For the structure shown below determine the vertical deflection at D (midwaybetween A & B). Use the following steps:

1. Find axial force in BC and moment diagram for AB due to applied loads2. Determine the virtual loading needed to find the vertical deflection at D3. Find axial force in BC and moment diagram in AB due to virtual load4. Use virtual work to find vertical deflection at D.

Use w = 0.5 kip/in, E = 29,000 ksi, IAB

= 600 in4, and A

BC

= 1.5 in2.

Additional Practice: In all problems use bending sti↵ness EI for bending mem-bers, and axial sti↵ness EA for members loaded only by axial forces (two forcemembers). Ignore axial deformations in bending members.

1. Use virtual work to find horizontal deflection at C.

Answer:

�C

=83.3 kip·ft

EA

+2016 kip·ft3

EI

(right)

58 LESSON 7. GENERAL APPLICATIONS OF VIRTUAL WORK

2. Calculate the vertical deflection at the right end. Use P = 200 lb, L =6 ft, a = 2 ft, and h = 1 ft.

Answer:

� =1.8 kip·ft

EA

+6.4 kip·ft3

EI

(down)

3. Use the structure and loading in F2-1. Ignore the directions in the textand find vertical and horizontal deflection at C. Answer: 208 kN ·m/EA

(up).

4. Use the structure and loading in F2-2. Ignore the directions in thetext. Find vertical deflections at B and exactly between A and B. An-swers: At B, 80 kN ·m/EA, and between A and B, 40 kN ·m/EA +33.3 kN ·m3

/EI.

5. Use the structure and loading in 2-38. Ignore the directions in the textand find vertical deflection at C. The cable is inextensible. Answer:27.25 kip·ft/EA+ 4.542 kip·ft3/EI.

6. Use the structure in problem 2-39. Ignore the distributed load. Placea load of 1 kip downward at C. Find vertical deflection at C. Answer:51.85 kip·ft/EA+ 29 kip·ft3/EI.

LESSON 8Approximate Analysis:

Portal Method for LateralLoading of Moment Frames

Lesson Objectives

• Use the portal method to analyze indeterminate moment frames underlateral load

Related Reading

• Textbook §7.1, 7.4 – 7.5• Procedures §1.11

Preparation

Read the assigned sections, then:

• State the assumed hinge locations for a portal frame with fixed bases andwith pinned bases.

59


Followup Problems

1. Problem 7-41 (p 294)

Additional Practice: All problems from pages 294 to 295, excluding prob-lems referring to the cantilever method.

LESSON 9Approximate Analysis:Moment Frames under

Gravity Loads

Lesson Objectives

• Conduct approximate analysis of indeterminate moment frames undergravity loads

Related Reading


Preparation


• State the distance from the columns that the textbook suggests placinghinges for gravity loading.

63


Followup Problems

TBD

LESSON 10Approximate Analysis:Braced Frames under

Lateral Loads

Lesson Objectives

• Conduct approximate analysis of braced frames under lateral loads

Related Reading


67


Followup Problems

1. Problem 7-11 (p. 269)2. Problem 7-12 (p. 269)

Additional Practice: Problems 7-9 and 7-10, then problems 7-1 to 7-8.

LESSON 11Using Influence Lines

Lesson Objectives

• Define influence lines• Use influence lines for static loads

Related Reading

• Textbook §6.1 – 6.2• Procedures §1.10

Preparation


• State what is represented by the horizontal axis of an influence line.• State what is represented by the vertical axis of an influence line.

71


Followup Problems

You are given the following influence line

Location of Load (ft)

Rea

ctio

n at

A (u

nitle

ss)

0 5 10 15 20 25 30 35 40−1

−0.5

0

0.5

1

Determine Reaction at A for:

a) A load of 10 kips placed at 10 ftb) A load of 10 kips placed at 30 ftc) A series of concentrated loads placed as follows:

• 4 kips at 15 ft• 2 kips at 18 ft• 2 kips at 21 ft

d) A uniform load of 0.4 kips/ft placed over the whole beame) The worst possible placement of a uniform load of 0.6 kips/ft (first determine

the worst possible location, then determine the resulting reaction at A)

Additional Practice: In example 6.3 (p. 209) what is the shear at C for (i)a point load of 4 kips placed at midspan, (ii) point loads of 3 and 5 kips placedat 5’ and 7.5’, respectively, and (iii) a distributed load placed over the wholebeam. Also, (iv) where should the load be placed to obtain the worst shear.

LESSON 12Drawing Influence Lines

Lesson Objectives

• Draw influence lines by determining equation• Draw influence lines using the Muller-Breselau Principle

Related Reading

• Textbook §6.1 & 6.3 (p. 435 to 439)• Procedures §1.8 and 1.9

Preparation

1. Plot the following functions from x = 0 to x = L.

• y = x/L

• y = 1� x/L

• y = (L� x)/2• y = �x/L

• y = x/2

2. Read the assigned sections, then for each of the following cases, state themotion that is applied to the beam to find qualitative influence lines: (a)reaction at a support, (b) shear force at a point, and (c) bending moment.

3. Also, for the triangle that will be sketched in class, solve for h in termsof L and ✓. Use the following small angle approximations: tan a ⇡ a andtan b ⇡ b.

75


Followup Problems

1. Problem 6-1 and 6-2 (p. 225)2. Problem 6-13 and 6-14 (p. 226)3. Problem 6-24 (p. 227, use the Muller-Breselau Principle to draw influence

lines)

Additional Practice: Problems 6-3, 6-5, 6-7, 6-9, and 6-11 (p. 225 to 226).Problems 6-4, 6-6, 6-8, 6-10, and 6-12 (p 225 to 226). Then, to practice bothdrawing and using influence lines, do 6-15 to 6-19 (p. 226).

LESSON 13Moving Point Loads and

Maximum AbsoluteResponse

Lesson Objectives

• Use influence lines for moving series of point loads• Find the maximum absolute shear and moment in a beam

Related Reading

• Textbook §6.6 & 6.7

79


Followup Problems

1. Problem 6-71 (p. 257).2. Problems 6-74 and 6-75 (p. 257).

Additional Practice: Problems 6-59, 6-60, 6-61, 6-62, 6-63, 6-65, 6-66, and 6-72(p. 255 to 257). Problems 6-73, 6-76, 6-77, 6-78, 6-79, and 6-80 (p. 257 to258).

LESSON 14Cable Structures

Lesson Objectives

• Determine forces and geometry in cable structures with concentratedloads.

• Determine forces and geometry in cable structures with uniform loads.

Related Reading

• Textbook §5.1 – 5.3• Procedures §1.12

83


Followup Problems

1. Problem 5-22. Problem 5-8

Additional Practice: Problems 5-3, 5-4, and 5-5. Problems 5-7, 5-9, 5-10, 5-11,5-16 and 5-17.

Part III

Exams

87

EXAM 1Fundamentals

Note: The objectives listed here may di↵er slightly. An update will be pro-vided prior to the exam.

Instructions

You’ll be allowed one 8-1/2”⇥11” sheet of notes and a calculator. All necessarytables will be provided for you. Bring a blue book or looseleaf paper.

Lesson Objectives Tested on Exam

• Draw moment diagrams using the graphical method• Draw deflected shapes of beams using a moment diagram• Draw deflected shapes of beams (and shapes of shear and moment dia-grams) directly from loading

• Compute deflections in beams by virtual work• Determine reactions and internal forces in statically determinate frames.

89

EXAM 2Deflections & Approximate

Analysis


Instructions



• Calculate deflections in any structure made from truss and frame mem-bers

• Draw deflected shapes for any structure made from truss and frame mem-bers

• Use the portal method to analyze indeterminate moment frames underlateral load

• Conduct approximate analysis of indeterminate moment frames undergravity loads

• Conduct approximate analysis of braced frames under lateral loading

91

EXAM 3Influence Lines & Cables


Instructions



• Use influence lines for static loads• Draw influence lines by determining equation• Draw influence lines using the Muller-Breselau Principle• Use influence lines for moving series of point loads• Find the maximum absolute shear and moment in a beam• Determine forces and geometry in cable structures with concentratedloads.

• Determine forces and geometry in cable structures with uniform loads.

93

EXAM 4Comprehensive Final Exam


Instructions

You’ll be allowed three 8-1/2”⇥11” sheet of notes and a calculator. All neces-sary tables will be provided for you. Bring a blue book or looseleaf paper.


Problems will be divided into the following four groups. Approximately onequarter of the points will be allocated to each group. Minimal partial creditwill be given (in e↵ect the lower level problems are partial credit for the higherlevel problems). The exam will be scored as follows: 0% to 20% is the F range,20% to 40% is the D range, 40% to 60% is the C range, 60% to 80% is the Brange, and 80% to 100% is the A range.

D-Level

• Draw moment diagrams using graphical method• Compute deflections in beams by virtual work (with moment diagramsgiven)

• Determine reactions and internal forces in statically determinate frames.• Use influence lines for point loads and distributed loads at specific loca-tions

95

Kevin Morales

Kevin Morales

Kevin Morales

Kevin Morales

96 EXAM 4. COMPREHENSIVE FINAL EXAM

C-Level

• Calculate deflections in frames using virtual work (moment diagramsgiven)

• Calculate deflections in trusses using virtual work (truss forces given)• Determine column shears for the portal method• For approximate analysis of moment frames under gravity loads, deter-mine the moments at the ends and center of loaded beams

• Use influence lines for a movable uniform load (determine worst case)

B-Level

• Draw deflected shapes of beams and frames (with moment diagram given)• Calculate deflections in frames using virtual work (moment diagrams notgiven)

• Calculate deflections in trusses using virtual work (truss forces not given)• Use the portal method to determine column moments• Conduct approximate analysis of braced frames under lateral loading(brace forces only)

• Draw influence lines by determining equation• Use influence lines for a moving series of point loads• Find the maximum absolute shear and moment in a cantilever or simplysupported beam

A-Level

• Draw deflected shapes of beams directly from loading• Draw deflected shapes of frames directly from loading• Calculate deflections in any structure made from truss and frame mem-bers

• Draw deflected shapes for any structure made from truss and frame mem-bers (no moment diagram given)

• Use the portal method to analyze indeterminate moment frames underlateral load (all member forces)

• Conduct approximate analysis of indeterminate moment frames undergravity loads (all member forces)

• Conduct approximate analysis of braced frames under lateral loading (allmember forces)

• Draw influence lines using the Muller-Breselau Principle