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T 'HE Engineering Experiment Station was established by act of the
: Board of Trustees, December 8, 1903. It is the purpose of the
Station to carry on investigations along various lines of engineer-
ing and to study problems of importance to proesional engineers and .
to the manufacturing, railway, mininig, constructional, and indutrial
interests of the State.
Thece EngineeringExperiment Station is vsted in the
head of the severaldepartments of the College of Engineering. These
consiftute the Station Staff and, with'the Director, determine the char-
acter of the investigtions to be undertaken. The work is carried on
under the supervision of the Staf, sometimes by research fellows as
gradiuate work, sometimes by members of the instructional staff of the
College of Engineering, but more frequent
to the Station corps.
',: .....The results of these investigations are published in the form of
bulletins, which record mostly the experiments of the Station's own staff
of investigators. There will also be issued from time to time in the
form of circular, compilations giving the results of the exprients of
engineers, industrial works, technical institutions, and governmental
testing departments.
The volume and number at the top of the title page of the cover
:are merely arbitrary numbers and refer to the general publications of
the University of Illinois; either above de title or below the seal is given-
the number of the Engineering Experiment Station'bulletin or circular
which should be used in referring to these publications.
For copies of bulletins, circulars, or other information address the
'n' g.ering Experiment Station, Urbana, Illinois.
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UNIVERSITY OF ILLINOIS
ENGINEERING EXPERIMENT STATION
BULLETIN No. 80 JUNE, 1915
WIND STRESSES IN THE STEEL FRAMES OF OFFICE
BUILDINGS.
BY W. M. WILSON, ASSISTANT PROFESSOR OF STRUCTURAL ENGINEERING,
AND G. A. MANEY, FORMERLY RESEARCH FELLOW IN DEPART-
MENT OF THEORETICAL AND APPLIED MECHANICS.*
CONTENTS.
Page
I. INTRODUCTION ... ............ . ........... . . 5
1. P relim inary .................................... 5
2. Acknowledgment ........... ................... 5
II. PRESENT METHODS OF CALCULATING WIND STRESSES IN
OFFICE BUILDINGS .... ...................... 5
3. Classification of Methods......................... 5
4. Approximate Methods .......................... 6
5. Exact M ethods ................... ............ . 7
III. OUTLINE OF THE PROPOSED ANALYSIS ................... 8
6. Outline of the Method........................... 8
IV. ASSUMPTIONS UPON WHICH THE ANALYSIS Is BASED ..... 9
7. Statement of Assumptions ....................... 9
V. FUNDAMENTAL EQUATION .......... ................. .. 10
8. Fundamental Proposition ....................... 10
9. Proof of the Proposition......................... 10
10. Derivation of Fundamental Equation.............. 11
VI. DERIVATION OF GENERAL EQUATIONS. ................... 14
11. N otation ...................................... 14
12. Derivation of Equations ......................... 15
*While the bulletin is the result of the joint efforts of the two writers, certain parts
are so distinctly the work of one that it seems desirable to make a statement relative to
the part which should be credited to each. The method of making the analysis involving
the use of the slope-deflection equations should be credited to G. A. Maney. The con-
ception of the bulletin as a whole, the method of presenting the results, and the authorship
of the text should be credited to W. M. Wilson. The numerical calculations of Tables 11
to 22, inclusive, were made by both writers.-The Editor.
ILLINOIS ENGINEERING EXPERIMENT STATION
Page
VII. NUMERICAL PROBLEM ............... ............. 19
13. Determination of the Stresses in a Symmetrical Three-
Span, Twenty-Story Bent .................... 19
VIII. APPROXIMATE METHODS ............ ................. 24
14. Nomenclature of Methods ........................ 24
15. Proposed Approximate Method ................... 25
16. Numerical Problem ............................. 28
17. Modifications of the Slope-Deflection Method....... 29
18. Application of the Proposed Approximate Method and
Modification of the Slope-Deflection Method .... 35
IX. COMPARISON OF THE APPROXIMATE METHODS WITH THE
SLOPE-DEFLECTION METHOD .................. 36
19. Symmetrical Three-Span Bent With Short Middle
Span .............. ...................... 36
20. Symmetrical Three-Span Bent With Long Middle
Span ................. . .................. . 36
21. Effect of the Proportions of a Bent Upon the Accuracy
of Method I................................ 37
22. Accuracy of the Approximate Methods When the Mo-
ment of Inertia of the Girders Is Proportional to
the Bending Moment ........................ 37
X. TEST OF A CELLULOID MODEL OF A BENT. ............... 40
23. Description of Tests ............................ 40
24. Results of Tests................................. 41
XI. DISCUSSION OF THE ASSUMPTIONS ...................... 42
25. Prelim inary ................................... 42
26. Assumption of Perfect Rigidity ................... 44
27. Assumption of the Unchanged Length............. 44
28. Assumption as to Length of Members.............. 46
29. Assumption as to Deflection Due to Shear .......... 46
30. Assumption as to Load .......................... 46
X II. CONCLUSIONS .................................. .... 47
WILSON-MANEY-WIND STRESSES
LIST OF TABLES.
Page
1. General Equations for a Symmetrical Single-Span Bent Any Number of Stories
H igh ..................................... ....... . ...................... 18
2. General Equations for a Symmetrical Two-Span Bent Any Number of Stories
High. (Insert) ................. .............. . ........................ 18
3. General Equations for a Symmetrical Three-Span Bent Any Number of Stories
H igh. (Insert) .............................................................. 1I
4. General Equations for a Symmetrical Four-Span Bent Any Number of Stories High.
(Insert) ............................. ....... ........................... 18
5. General Equations for a Symmetrical Five-Span Bent Any Number of Stories High.
(Insert) .............................. ................................. 18
6. General Equations for an Unsymmetrical Single-Span Bent Any Number of Stories
High. (Insert) ................... ........ ............................. 18
7. General Equations for an Unsymmetrical Two-Span Bent Any Number of Stories
High. (Insert) ........................ .................................... 18i
8. General Equations for an Unsymmetrical Three-Span Bent Any Number of Stories
High. (Insert) ................ . . ....................................... 18
9. General Equations for an Unsymmetrical Four-Span Bent Any Number of Stories
High. (Insert) ................. . .......... .... ......................... 18
10. General Equations for an Unsymmetrical Five-Span Bent Any Number of Stories
High. (Insert) .................. .... .......... . .... ..................... 18s
11. Properties of the Columns and Girders in the Symmetrical Three-Span Twenty-
Story Bent Shown in Fig. 5.................. . ......................... 48
12. Numerical Values of the Constants in the Equations of Table 3 for the Symmetrical
Three-Span Twenty-Story Bent Shown in Fig. 5 ............................... 49
13. General Equations for the Symmetrical Three-Span Twenty-Story Bent Shown
in Fig. 5............... . .. ...... .......... ......................... 50
14. Elimination of the Unknown Quantities in the Equations for the Symmetrical
Three-Span Twenty-Story Bent Shown in Fig. 5 .............................. 566
16. Determination of the Changes in the Slopes and the Ratios of Deflection to Story
Height in the Symmetrical Three-Span Bent Shown in Fig. 5.................. 66
16. Values of R and 0 for the Symmetrical Three-Span Twenty-Story Bent Shown in
Fig. 5, and the Functions of these Values that Occur in the Equations Used
to Determine the Moments in the Columns and Girders......................... 72
17. Values of K for the Columns and the Girders of the Symmetrical Three-Span
Twenty-Story Bent Shown in Fig. 5, and the Functions of these Values that
Occur in the Equations Used to Determine the Moments in the Columns and
Girders .............................................................. 73
18. Moments at the Ends of the Columns and Girders of the Symmetrical Three-Span
Twenty-Story Bent Shown in Fig. 5 ....................................... 74
19. Direct Stresses in the Columns, and the Shears in the Columns and Girders of
the Symmetrical Three-Span Twenty-Story Bent Shown in Fig. 5............... 75
20. Check on the Numerical Values of the Moments at the Ends of the Columns and
Girders of the Symmetrical Three-Span Twenty-Story Bent Shown in Fig. 5..... 76
ILLINOIS ENGINEERING EXPERIMENT STATION
21. Elimination of the Unknown Quantities in the Equations Used to Determine the
Slopes and the Deflections in the Bottom Story of the Symmetrical Three-Span
Twenty-Story Bent Shown in Fig. 5 by a Modification of the Slope-Deflection
M ethod ............. ....................................... . ..... ...... . 77
22. Determination of the Changes in the Slopes and the Ratio of the Deflection to
Story Height in the Bottom Story of the Symmetrical Three-Span Twenty-
Story Bent Shown in Fig. 5 by a Modification of the Slope-Deflection Method.... 78
23. Comparison of the Approximate Methods with the Slope-Deflection Method When
Applied to the Symmetrical Three-Span Twenty-Story Bent Shown in Fig. 5..... 79
24. Comparison of the Approximate Methods with the Slope-Deflection Method When
Applied to the Symmetrical Three-Span Twenty-Story Bent Shown in Fig. 5..... 80
25. Comparison of the Approximate Methods with the Slope-Deflection Method When
Applied to the Symmetrical Three-Span Twenty-Story Bent Shown in Fig. 13.... 81
26. Comparison of the Approximate Methods with the Slope-Deflection Method When
Applied to the Symmetrical Three-Span Twenty-Story Bent Shown in Fig. 13..... 82
27. Effect of the Proportion of a Bent Upon the Accuracy of Method I................ 85
28. Effect of the Proportions of a Bent Upon the Accuracy of Method I. All Girders
Proportional to the Bending M oments ........................ . .............. 84
29. Effect of the Proportions of a Bent Upon the Accuracy of Method II............... 85
30. Effect of the Proportions of a Bent Upon the Accuracy of Method III .............. 86
31. Effect of the Proportions of a Bent Upon the Accuracy of Method IV .............. 87
32. Log of the Test of Celluloid M odel No. 4 ................. ...... .. ............... 88
LIST OF FIGURES.
1. Proof of Fundamental Equation ................ . ............. .. ................ 11
2. Derivation of Fundamental Equation ............................ ............... 18
M
3. - D iagram .................................................................. 13
El
4. Bottom Five Stories of a Symmetrical Three-Span Bent .......................... 15
5. Symmetrical Three-Span Bent, Twenty Stories High................... ............ 20
6. Approximate Moment at the Top and Bottom of Column A. (Insert) .............. 26
7. Approximate Moment at the Top and Bottom of Column B. (Insert) ............... 6
8. Approximate Moment at the Ends of Girder b. (Insert).......................... 26
9. Moments Acting at Points A8 and B8 of a Symmetrical Three-Span Bent ............ 9
10. Diagram Showing Change in the Moment at the Top and the Bottom of Column A
of a Symmetrical Three-Span Bent Due to a Change of K of the Other Members 80
11. Diagram Showing Change in the Moment at the Top and the Bottom of Column B
of a Symmetrical Three-Span Bent Due to a Change of K of the Other Members 31
12. Diagram Showing Change in the Moment at the End of Girder b of a Symmetrical
Three-Span Bent Due to a Change in K of the Others Members................ 32
13. Symmetrical Three-Span Bent Twenty Stories High with Long Span at the Center.. 88
14. Celluloid M odel ..................... .................... .................. ... . 41
15. Diagram of Results of the Tests of Celluloid Model ............................... 43
WIND STRESSES IN THE STEEL FRAMES OF OFFICE
BUILDINGS
I. INTRODUCTION.
1. Preliminary.-The increase in the price of land in large cities
has made it necessary to build high buildings in order to get a large
rentable floor space on a small parcel of land. The type of building
generally used is known as the steel-skeleton building. In this type of
building the live and dead loads, including the weight of the walls, are
carried by a system of beams and girders to columns and are carried by
the columns to the footings.
In high buildings the horizontal shear due to the wind load is very
large; and, since it is usually imnpracticable to put diagonal braces be-
tween the columns, it is customary to make the steel frame rigid enough
to resist the horizontal shear by virtue of the stiffness of the columns
and girders. The exact determination of the stresses in a steel frame
due to a horizontal shear is one of the problems of structural engineering
which remains to be solved. While the writers realize that the method of
determining these stresses presented in this bulletin is based upon as-
sumptions which are not exactly true, they believe that the method is
more accurate than the methods ordinarily used.
2. Acknowledgment.-Messrs. Anderson, Becker, Gomez, and
Richart, graduate students in the College of Engineering of the Uni-
versity of Illinois, calculated the moments in Table 23, and the moments
as determined by methods I, II, and III in the keys to the diagrams
in Figs. 10, 11, and 12, and assisted with the calculations necessary to
determine the curves shown in these figures. Professor Ira 0. Baker
and Professor C. A. Ellis rendered valuable assistance, criticizing the
bulletin during its prepartion. The writers gratefully acknowledge in-
debtedness to these men.
II. PRESENT METHODS OF CALCULATING WIND STRESSES IN OFFICE
BUILDINGS.
3. Classification of Methods.-Methods of calculating wind stresses
in the steel frames of office buildings may be divided into two classes,
viz.: (1), those used in the actual design of buildings and (2), those
ILLINOIS ENGINEERING EXPERIMENT STATION
which have resulted from an effort to make an exact analysis of the
stresses. In the methods of the first class, accuracy has been sacrificed
to shorten and simplify the calculations. In the methods of the second
class, the aim has been to make an exact analysis rather than an analysis
that can be used in the actual design of a building.
For the sake of convenience in reference, methods of the first class
are designated as approximate methods, and those of the second class
are designated as exact methods.
4. Approximate Methods.-(a) Fleming's Methods. Mr. R.
Fleming, in an article in Engineering News, describes three methods,
which are in current use.* These methods are designated as methods I,
II, and III. The three methods, as applied to a building in which all
columns of a story have the same section, are based upon the following
assumptions:
Assumptions in Method I.
1. A bent of a frame acts as a cantilever.
2. The point of contra-flexure of each column is at mid-height of
the story.
3. The point of contra-flexure of each girder is at its mid-length.
4. The direct stress in a column is directly proportional to the
distance from the column to the neutral axis of the bent.
Assumptions in Method II.
1. A bent of a frame acts as a series of portals.
2. The point of contra-flexure of each column is at mid-height of
the story.
3. The shear is the same on all columns of a story.
4. Each pair of adjacent columns of a bent acts as a portal, and
each interior column is a member of two adjacent portals. The direct
stress in an interior column, when the column is considered as a mem-
ber of the portal on one side, is of opposite sign from the direct stress
in the same column when considered as a member of the portal on the
opposite side and the resultant direct stress is equal to zero.
*Wind Bracing without Diagonals for Steel-Frame Office Buildings, Engineering News,
March 13, 1913.
WILSON-MANEY-WIND STRESSES
Assumptions in Method III.
1. A bent of a frame acts as a continuous portal.
2. The point of contra-flexure of each column is at mid-height of
the story.
3. The direct stress in a column is directly proportional to the
distance from the column to the neutral axis of the bent.
4. The shear is the same on all columns of a story.
(b) Smith's Methods. Professor Albert Smith, in a paper before
the Western Society of Engineers, describes a method which he has
used in his classes in Structural Engineering at Purdue University.:
This method is here designated as Method IV.
Assumptions in Method IV.
1. The point of contra-flexure of each column is at mid-height of
the story.
2. The point of contra-flexure of each girder is at its mid-length.
3. The shears on the internal columns are equal and the shear on
each external column is equal to one-half of the shear on an interior
column.
If all of the assumptions of any of these methods are accepted, the
stresses in a frame may be determined by applying the fundamental
equations of static equilibrium. It is apparent from the assumptions
that the results obtained by these methods are radically different.
5. Exact Methods.-(a) Melick's Method. Dr. Cyrus A. Melick*
used a method which takes into account the form of the elastic curves
and the deflections and changes in length of the members. The method
is so long that, when applied to a building only four stories high, the
amount of work required is almost prohibitive. To apply it to a build-
ing twenty stories high would be impracticable if not, in fact, impossible.
(b) Jonson's Method. Mr. Ernest F. Jonson suggested a method
which takes into account the deflections of the columns and the changes
in the slopes of the tangents to the elastic curves of the columns and
girders at the points where they intersect.t
If the method which he suggests were used, it would give the
stresses with a fair degree of accuracy; but his method involves so many
unknowns that its use would not be practicable in the actual design of
buildings.
MWind Stresses in the Frames of Office Buildings, by Albert Smith, Journal Western
Society of Engineers, Vol. XX, No. 4, p. 341.
*Stresses in Tall Buildings, by Cyrus A. Melick. Bulletin No. 8, College of Engineering,
University of Ohio.
tThe Theory of Frameworks with Rectangular Panels and Its Application to Buildings
which Have to Resist Wind, by Ernst F. Jonson, Tran. Am. Soc. C. E., Vol. 55, p. 418.
ILLINOIS ENGINEERING EXPERIMENT STATION
(c) Method of Least Work. Professor Albert Smith has deter-
mined the wind stresses in symmetrical bents having two, three, and
four spans by the method of least work.* This method is exact, but, of
course, is extremely long.
III. OUTLINE OF THE PROPOSED ANALYSIS.
6. Outline of the Method.-In making an analysis of the stresses,
the writers made certain assumptions and applied certain fundamental
principles of mechanics and obtained equations from which the stresses
in a frame can be determined. The assumptions which have been made
are stated in Section IV and the derivation of the equations is given in
Sections V and VI.
It can be proven that the moment at an end of a member of a
frame is a function of the changes in the slopes of the tangents to the
elastic curve of the member at its ends and of the deflection of one end
of the member relative to the other end (see equation A, page 13).
In the strained position, all the columns and girders which intersect
at one point have been subjected to the same change in slope (see as-
sumption 1, Section IV); the vertical deflections of the ends of all
girders are equal to zero; and the horizontal deflections of the tops of
all columns of a story are equal (see assumption 2, Section IV).
Consider any story of a bent. Take the point of intersection of
the neutral axes of a column and a girder as a free body. It is in equi-
librium under the action of the moments at the extremities of the col-
umns and girders which intersect at the point. Each of the moments
may be expressed in terms of the changes in the slopes at the extremities
of the member, and the deflection of one end of the member relative to
the other. A moment equation can therefore be written for each point
where the columns and girders intersect, and the only unknown quan-
tities will be the changes in the slopes at the extremities of the columns
and the horizontal deflections of the columns in a story.
If all the columns of a story be taken together as a free body, the
sum of the moments at the two extremities of all the columns will be
balanced by a couple whose moment is equal to the total shear on the
story multiplied by the story height. The shear and the height of the
story are known, and the moments in the columns can be expressed in
terms of the slopes and the deflections at their extremities the same as in
the previous equations. It is therefore possible to write as many equations
for each story as there are columns in the story, plus one. As the only
*Wind Stresses in the Frames of Office Buildings, by Albert Smith, Journal Western
Society of Engineers, Vol. XX, No. 4, p. 341.
WILSON-MANEY--WIND STRESSES
unknown quantities in these equations are the changes in the slopes at the
extremities of the columns and the deflection in a story common to all
columns, there are as many equations per story as there are unknowns.
By solving these equations the slopes and the deflections can be deter-
mined. Knowing the slopes and the deflections, the moments can be
computed.
The product of the shear on a member and the length of the mem-
ber is equal to the algebraic sum of the moments at the extremities of the
member. Since the moments and the length of the member are known
the shear can be computed.
With the shears in the girders known, the direct stress in any col-
umn can be determined by taking the column as a free body and equat-
ing the sum of the vertical forces to zero.
The direct stress in a girder may be determined in a similar manner.
The method just described is based upon the proposition in me-
chanics used by Mr. Jonson, but the method which the writers have
developed differs from the one used by him in that the changes
in the slopes and the deflections have been used as the unknown quanti-
ties instead of the direct stresses and the moments. Four members,
two columns and two girders, intersect in a point. Each member is
subjected to a different direct stress and a different moment, whereas all
of the members are subjected to the same change in slope, and all of
the columns in a story are subjected to the same deflection. It is there-
fore apparent that there are fewer unknown slopes and deflections than
moments and direct stresses. The large reduction in the number of
unknowns very much simplifies the solution of the equations.
IV. ASSUMPTIONS UPON WHICH THE ANALYSIS Is BASED.
7. Statement of Assumptions.-The proposed analysis is based
upon the following assumptions:
1. The connections between the columns and girders are perfectly
rigid.
2. The change in the length of a member due to the direct stress
is equal to zero.
3. The length of a girder is the distance between the neutral
axes of the columns which it connects and the length of a column is
the distance between the neutral axes of the girders which it connects.
4. The deflection of a member due to the internal shearing stresses
is equal to zero.
ILLINOIS ENGINEERING EXPERIMENT STATION
5. The wind load is resisted entirely by the steel frame.
These assumptions will be discussed in Section XI.
V. FUNDAMENTAL EQUATION.
"8. Fundamental Proposition.-The fundamental equation used in
this analysis is based upon the following proposition:
When a member is subjected to flexure, the deflection of any point
in the neutral axis from the tangent to the elastic curve at any other
M
point is equal to the moment of the area of the -A- diagram for the
portion of the member between the two points, about the point where the
deflection is measured.
9. Proof of the Proposition.-The line AB, Fig. 1, represents the
neutral axis of a member subjected to flexure. The deflection of the
member is very much exaggerated in the figure. The actual deflection
is so small that the length of the curve may be considered equal to its
horizontal projection. It is required to prove that the deflection of any
point P from the tangent to the line AB at any other point Q is equal to
I dxsx.
Extend the tangents at the extremities of an element of the curve
until they intersect the vertical line through P. The intercept on this
vertical line between two consecutive tangents is equal to xadO. The
total deflection of P from the tangent at Q is equal to the algebraic
sum of these intercepts for the elements of the elastic curve between
the points Q and P. That is, y= f xd0. The equation of the
dO M
elastic curve may be written - . M Substituting the value of dO
dx EI
from this equation in the expression for y, gives y = dx-x.
Q EI
M M
The quantity -- dx can be considered as an elementary area of the -
diagram. The quantity J --dx-x can therefore be considered as the
moment of the -E diagram about the point P.
momento
WILSON-MANEY-WIND STRESSES
10. Derivation of Fundamental Equation.-Consider a member
which is not acted upon by any external forces or couples except at the
ends. The line AB in Fig. 2 represents the neutral axis of such a
member. The moment at A is represented by MAB and at B by MBA.
The change in the slope of the elastic curve at A due to the external
forces is represented by OA, and at B by Os. The deflection of A from
its original position A', is d. The deflection of A from the tangent at
B is represented by (d- 1-0) ; and the deflection of B from the tangent
A
FIG. 1. PROOF OF FUNDAMENTAL EQUATION.
M
at A is represented by (d- 10A). The -l diagram is shown in Fig. 3.
The signs of the quantities are determined by the following rules:
The change in the slope, or the angular deformation, is positive
(+) when the tangent to the elastic curve of the member is turned in
a clockwise direction.
Distances and deflections are positive when they are measured in
the same direction from the base line as are positive slopes.
The moment acting upon a member at the section where the deflec-
tion is measured is positive (+) when it produces a clockwise rotation.
Substituting (d - 1-), the deflection of A from the tangent at B,
ILLINOIS ENGINEERING EXPERIMENT STATION
I M
for y in the equation y = l- dx.x, gives
BM
(d - -0B) = dxx.
Substituting the value of M from the equation
= Mo ÷F[BA- AD.
M = MAB +MBA M BA x, gives
(d - 10) B A -- xdx + L MBA- M xdx.
A El EI-1 J
If the material is homogeneous and the section uniform, E and
I are constants. Performing the indicated integration, gives
MAB + MBA 1 MA 12
El 2 El 3 EI 3'or
d= 10B + --[ 2MBA + IMA ............. (1)
Substituting (d -l 'A), the deflection of B from the tangent at A,
for y in the equation, y J= - dx-x, gives
(d - 1'0A)-/ M
The moment at any section in the member, when considering the
deflection at B, is of opposite sign from the moment at the same section
when considering the deflection at A. (See preceding rule for determin-
ing the sign of the moment.)
Substituting the value of M from the equation
M = - MBA [ Jx, gives
/M- f MBA- M AB M BA, gives
(d - 1A) = --MA xdx - x2dx or
J El
MBA 12 MA l2 MA 12
S E 2 El 3 El 3
d= A + [-MBA - 2MABI ................. (2)
Multiplying equation (2) by 2, gives
2d= 21-A + --- [ - 2MBA - 4MAB .............. (3)
_0EIl
WILSON-MANEY-WIND STRESSES
FIG. 2. DERIVATION OF FUNDA-
MENTAL EQUATION.
M
FIG. 3. - DIAGRAM.
El
Adding equations (1) and (3) gives
3d = 21-A + -A B +--2 [- 3M AB.
I d
Substituting K for and R for - and solving for MAD, gives,
MAB = 2 EK [2 0A + OB - 3R]................ (A)
When d= 0, equation (A) becomes
MAB = 2 EK (2 A + B)....................... (B)
Equation (A) is general and may be applied to any length of any
member in bending provided the length considered has no intermediate
external force applied to it. That is, one or more of the quantities
ILLINOIS ENGINEERING EXPERIMENT STATION
OA, OB, and d may be negative and equation (A) will still give the
moment at the point A in both magnitude and sign. Equation (A) is
the fundamental equation upon which the analysis which follows is based.
Equation (B) is a special form of equation (A). Equation (A) may
be expressed as follows:
The moment at the end of any member is equal to 2EK times the
quantity: Two times the change in the slope at the near end plus the
change in the slope at the far end minus three times the deflection
divided by the length. E is. the modulus of elasticity of the material and
K is the ratio of the moment of inertia to the length of the member.
VI. DERIVATION OF GENERAL EQUATIONS.
11. Notation.-The following notation has been used:
A, B, C, etc. = the columns of a bent, beginning at the right and
reading toward the left.
a, b, c, etc.= the bays of a bent, beginning at the right and reading
toward the left. The girder in bay a is designated as girder a, in
bay b as girder b, in bay c as girder c, etc.
A, A2, A3, etc. = the intersections of the neutral axes of the girders at
the tops of the first, second, third, etc., stories with the neutral axis
of column A.
B, B2, B3, etc. = the intersections of the neutral axes of the girders at
the tops of the first, second, third, etc., stories with the neutral axis
of column B.
d = deflection of the columns in a story height.
E = modulus of elasticity of the material.
h = length of. a column measured from neutral axis to neutral axis of
the girders.
I = moment of inertia of the girder and column sections.
J =2 + for all columns and girders which intersect at
a point.
I I
K = - for girders and - for columns.
l= length of a girder measured from neutral axis to neutral axis of
the columns.
M = bending moment.
N = 2 for all columns in a story.
WILSON-MANEY-WIND STRESSES
d
h
W = total horizontal shear in a bent at any story.
w increment of the horizontal shear in a bent in a story height.
0 = change in the slope of the tangent to the elastic curve.
Subscripts are added to the letters to indicate the particular part
of a bent to which a given symbol applies. For example, referring to
Fig. 4, girder b, is the girder in bay b at the top of the first story,
A1 is the intersection of the girder at the top of the first story with
FIG. 4. BOTTOM FIVE STORIES OF A SYMMETRICAL THREE-SPAN BENT.
column A, 0Ai is the slope of the tangent to the elastic curve at the
point A1; d. is the deflection of the columns in the third story; J4, is
the J at the point B4; KA2 is the K of column A in the second story.
The moment at the right end of the girder in bay a at the top of the
second story is designated as MAB, and the moment at the left end of
the same girder is designated as M2BA. The moment at the top of the
column B in the third story is designated as MB32, and the moment at
the bottom of the same column is designated as MB23.
12. Derivation of Equations.-Fig. 4 shows the bottom five stories
of a symmetrical three-span bent. It is required to find the stresses in
all of the members.
ILLINOIS ENGINEERING EXPERIMENT STATION
Consider the columns of the third story acting together as a free
body. (Any other story could have .been used.) The algebraic sum of
all of the moments at the tops and the bottoms of all the columns plus
the product of the total shear in the story and the story height, is equal
to zero. That is, 2 (MA2 + MA3 + M 2 + MB23) + Wh, = 0
Substituting the values of the moments as given by equations A and
B of Section V gives
2 [ 2EKA, (2A3, + OA - 3R,) + 2EKA, (20A2 + OA6 - 3R,) +
2EKBa (20B, + 0B2 - 3R,) + 2EKB, (202 + O,, - 3R,) ]
+ Wh, = 0.
Letting N = 2 1: ( ) for all of the columns in a story, and
collecting, gives
W hi
2KA3 0A2 + 2K3 0,B2 - NR, + 2KA3 A3 + 2K, O3 = - ...... (1)
Consider the point A,3 as a free body. Taking I M = 0, gives
MA34 + MA32 + M3AB = 0. Substituting the values of these moments as
given by equations A and B gives,
2EKA4 (20A, + OA4 -34) + 2EKA3 (20A. + 0,A - 3R,) +
2EK3a (20A3 + OB3) = 0.
Combining the coefficients of the unknowns and cancelling the common
factor 2E gives,
A2KA, + OA, (2KA4 + 2KA, + 2Ka,) + 0A4 KA4 + ?B3Kas -
Rs3KA, - R43KA4 = 0.
Substituting JA3 for 2 - ) of all the members that intersect
at A3, the equation becomes:
KA, 0A2 - 3KA R, + JA, OAs + Ka, OB0 - 3KA44 + KA40A4 = 0. .. (2)
The point B, is in equilibrium under the action of the four mo-
ments M3, MB , MBB4, and M32. Equating the sum of these four
moments to zero gives,
2EKa, (20B, + OA,) + 2EKb, (20B, + 0:) + 2EKs, (20, +
OB, - 3R4) + 2EKs, (20,3 + OB2 - 3R,) = 0.
Combining the coefficients for the unknowns and cancelling the common
factor 2E, gives,
20B3 (Ka, + Kba + KB, + Ks,) + OAKa, B+ OKb + OB4Ke4 +
OB2KB, - 3R4KB4 - 3RKBa = 0.
Substituting J1, for 2 ( -) of all the members intersecting at
B., the equation becomes,
WILSON-MANEY-WIND STRESSES
Ka, 0,B - 3KBR,3 + Ka. OA3 + (Kb, + Je,) 0.B - 3KB4R, +
K ,0B4 = 0 ................................................. (3)
The three equations, 1, 2, and 3, can be written for any story by
making the proper changes in the subscripts. As there are only three
unknowns, OA, 0B, and R for each story, as many equations can be writ-
ten for a bent as there are unknowns to be determined. It is possible to
solve these equations for the unknown quantities algebraically, but the
large number of equations involved makes the work very difficult. It
is simpler to substitute the numerical values of the coefficients in the
equations and solve for the numerical values of the unknown quantities
by the process of elimination explained in Section VII.
For convenience in the application of this process of elimination,
the equations are written in tabular form as shown in Table 3 in which
the unknown changes in the slopes and the ratios of deflection to story
height are written at the tops of the columns and the coefficients of
these unknowns are written below. For example, in writing equation
A of Table 3, which is - N1R, + 2KA19A1 + 2KB1,B - 6-h
- N,, the coefficient of R1, is placed in the column under R1; 2KA1, the
coefficient of 0A,, is placed in the column under OA1; 2KB, the
W h
coefficient of Os1, is placed in the column under OB,; and - W1
is placed in the column headed "Right-Hand Member of Equation."
Having the equations written in this form, it is unnecessary to repeat
the unknown quantities when eliminating them from the equations by
the method used in Table 14.
Table 3 contains the general equations to be used in determining
the slopes and the deflections in a symmetrical three-span bent any
number of stories high. The subscripts 1, 2, and 3 refer to the first,
second, and third stories respectively, and the subscripts x, y, and z refer
to the second from the top, the next to the top, and the top stories re-
spectively. The equations for the intervening stories are of the same
form, but have different subscripts.
By using the general method outlined above, that is, by writing
an equation similar to equation 1 for each story and an equation similar
to equations 2 and 3 for each intersection of a column with a girder
for each story, as many equations as there are unknowns can be written
for a bent containing any number of spans and any number of stories.
General equations for symmetrical bents of from one to five spans are
given in Tables 1 to 5 and similar equations for unsymmetrical bents
of from one to five spans are given in Tables 6 to 10.
In order to check the equations of Tables 1 to 10, a model of a
ILLINOIS ENGINEERING EXPERIMENT STATION
Bba
Z o
N
A
- C
0 - C )
o sCo 0
C-. C
C a ^C §C
.0 0CC-
0 6
I __ I
" o 0
0
Coo"
* CO
Co. ^
CO
uop!nbf
.~ H
C)
N
co C
CC^
1-4
H
z
z
p4
01
a .
HO
0~
P4
0
H0z
c o
P4
w<
Iz
I V XIk j N
t«i
I
I
I
TABLE 2.
GENERAL EQUATIONS FOR A SYMMETRICAL TWO-SPAN BENT
ANY NUMBER OF STORIES HIGH.
Left-Hand Member of Equation
Story No. 1 Story No 2 Story No. 3 Intervening Story No. X Story No. Y Story No. Z
I Sto ries (Second from Top) (Next to Top) (Top)
Right-Hand
Member
of Equation
Rl 0Al lB1 I 2 A 2 OB IR3 A3 OB3
RX IAX AY Y IAY I BY RZ I AZ OBZ
Coefficients of Unknown Slopes and Ratios of Deflection to Story Height
Similar equations for intervening stories
TABLE 3.
GENERAL EQUATIONS FOR A SYMMETRICAL THREE-SPAN BENT
ANY NUMBER OF STORIES HIGH.
Left-Hand Member of Equation
Story No. 1 Story No. 2 Story No. 3 Intervening Story No. X Story No. Y Story No. Z
Stories (Second from Top) (Next to Top) (Top)
R1 0AI 0BI R2 0A2 0B2 R3 0A3 OB3 RX AX 0BX RY 0Ay 0BY Rz OAZ 8BZ
Coefficients of Unknown Slopes and Ratios of Deflection to Story Height
Right-Hand
Member
of Equation
Similar equations for intervening stories
TABLE 4.
GENERAL EQUATIONS FOR A SYMMETRICAL FOUR-SPAN BENT
ANY NUMBER OF STORIES HIGH.
Similar equations for intervening stories
TABLE 5.
GENERAL EQUATIONS FOR A SYMMETRICAL FIVE-SPAN BENT
ANY NUMBER OF STORIES HIGH.
Similar equations for intervening stories
TABLE 6.
GENERAL EQUATIONS FOR AN UNSYMMETRICAL SINGLE-SPAN BENT
ANY NUMBER OF STORIES HIGH.
Left-Hand Member of Equation
Story No. 1 Story No. 2 Story No. 3 Intervening Story No. X Story No. Y Story No. Z
Stories (Second from Top) (Next to Top) (Top)
R1 A1 I B1 R2 OA2 OB2 R3 OA3 OB3 X OAX BX Ry AY OBY RZ AZ BZ
Coefficients of Unknown Slopes and Ratios of Deflection to Story Height
Right-Hand
* Member
of Equation
Similar equations for intervening stories
TABLE 7.
GENERAL EQUATIONS FOR AN UNSYMMETRICAL TWO-SPAN BENT
ANY NUMBER OF STORIES HIGH.
Similar equations for intervening stories
S KAY KBY KCy -NY KAY KBY Key -Wy h y-6E
T KAY -3KAY JAY Kay -3KAz KAZ O
U KBy -3K y KaY KBy KbY --3KBZ KBZ 0
V KCy -3KOy KbY JCY -3KCZ KCZ O
W ___________KAZ KBZ KCZ -Nz KAZ KBZ KCZ -W h -6E
X KAZ -3KAZ JAZ Kaz 0
Y KBZ --3KB KaZ JBZ Kbz 0
Z KCZ -3KCZ KbZ JCZ O
TABLE 8.
GENERAL EQUATIONS FOR AN UNSYMMETRICAL THREE-SPAN BENT
ANY NUMBER OF STORIES HIGH.
Similar equations for intervening stories
TABLE 9.
GENERAL EQUATIONS FOR AN UNSYMMETRICAL FOUR-SPAN BENT
ANY NUMBER OF STORIES HIGH.
Story No. 3 Intervening Story No. X Story No. Y Story No. Z
Stories (Second from Top) (Next to Top) (Top)
R3 0A3 0B3 C3s D3 1 I 3 RX I AX OBX OCX I DX OEX Ry OAY IBY I CY 0DY 8EY RZ OAZ I BZ I CZ ODZ SEZ
Coefficients of Unknown Slopes and Ratios of Deflection to Story Height
Similar equations for intervening stories
TABLE 10.
GENERAL EQUATIONS FOR AN UNSYMMETRICAL FIVE-SPAN BENT
ANY NUMBER OF STORIES HIGH.
Similar equations for intervening stories
KBy
K1y4
KAY
JAY
Ka.y
KAZ
KAZ
K(~y
K~hY
JCY
Key
Kcz
KDY
K0Y
JoyT
KdY
KI,1
KDZ
KAZ
KAZ
JgAZ
K& .
K,1
JDZ
Kdz
-Wyhy+6E
0
0
0
0
0
O
0
0
0
0
0
0
O
I I
I I
I I
I I I
1--
-L-1
.L -I
K1Y
KDY
r=1 :
WILSON-MANEY-WIND STRESSES
bent was tested and the measured deflections and changes in the slopes
were compared with the same quantities calculated by the above equa-
tions. As the entire model was cut from a sheet of celliloid the joints
were perfectly rigid. The results of the tests are given in Fig. 15.
The fact that the measured and computed deflections and changes in
the slopes agree very closely indicates that the above analysis is correct.
A solution of a numerical problem illustrating the use of the
equations in Table 3 is given in Section VII.
VII. NUMERICAL PROBLEM.
13. Determination of the Stresses in a Symmetrical Three-Span
Twenty-Story Bent.-Fig. 5 shows a symmetrical three-span bent twenty
stories high. The bent resists a horizontal wind load of 30 lb. per sq. ft.
on a vertical strip one foot wide. It is required to find the moment and
the shear in the columns and girders and the direct stress in the columns.
The properties of the girder and column sections are shown in Table
11, page 48. The equations of Table 3 are applicable. The numerical
values of the constants in these equations are given in Table 12. Table
13 contains the equations of Table 3 with the substitution of the
numerical values of the constants given in Table 12. The figures given
in the right-hand column are coefficients of .0001 as indicated at the
head of the column; that is, the right-hand members of the equations
are equal to .0001 times the numbers in the right-hand column. This
method of writing the right-hand members of the equations obviates the
necessity of repeating a large number of ciphers.
The unknown quantities in the equations of Table 13 are eliminated
in Table 14. The quantity R1 appears in the first three equations of
Table 13 only. Dividing each of these equations by the coefficient of
R, gives the three equations A, B, and C of Table 14, in all of which
the coefficient of RB is equal to plus unity. Combining these latter
equations as indicated, that is, subtracting equation B from A to get
equation (A--B) and subtracting equation C from B to get equation
(B - C), gives two new equations from which R, has been eliminated.
Reducing the coefficients of the left-hand terms of equations (A - B)
and (B - C) to plus unity gives equations 1 and 2 of Table 14. Equa-
tions D and E of Table 13 have also been reduced so that the coefficients
of the left-hand term of each are equal to plus unity. They are re-
written in their new form as equations D and E in Table 14. Combin-
ing the four equations 1, 2, D, and E, as indicated in Table 14, eliminates
ILLINOIS ENGINEERING EXPERIMENT STATION
FIG. 5. SYMMETRICAL THREE-SPAN BENT, TWENTY STORIES HIGH.
WILSON-MANEY--WIND STRESSES
the left-hand term, OA1. By continuing this process all the unknown
quantities are eliminated except the last, or OB20. Its value thus
becomes known and is found to be .0338 X .0001 radians. With the
value of B,2o known, the value of 0A20 can be determined from equa-
tion 153. With 0820 and 6A20 both known, the value of R20 can be
determined from equation 150. The other values of 0 and R can be
determined in a similar manner.
The process of determining the values of 0 and R is given in
Table 15. The equations used are taken from Table 14.
As in Tables 13 and 14, the right-hand member is equal to the
number in the column at the right multiplied by .0001. To illustrate,
equation 150 is
R2o - .2702 0A20 - .2993 020 = .0735 X .0001.
The left-hand term of the left-hand member of each equation is
the unknown which is to be determined. The first line in each group
is the algebraic form of the equation; and the successive lines are the
numerical values of the corresponding terms. For example, equation
155 is
O20 = .0338 X .0001 or .00000338.
Again, equation 153 is:
0A2o - .0443 082o = .0502 X .0001.
From the preceding equation
.0443 B2o = .0443 X .00000338 = .0015 X .0001.
The sign of this quantity as a term of the left-hand member was
negative, and the term will, therefore, be positive after it has been
transferred.
Equation 153 now becomes
0A2o = .0502 X .0001 + .0015 X .0001= .0517 X .0001,
as indicated in the last line of the group containing equation 153.
Equation 150 contains R2o, A20, and 0820; but 0A20 and 0820 have
been determined by equations 153 and 155, respectively, therefore R20
can be determined by equation 150. Similarly, equation 147, contain-
ing OBs, and the three known quantities, R,20, A20, and OB20 can be
used to determine 0~B9. Equation 147 differs from the preceding equa-
tions in that it contains both positive and negative terms in the left-
hand member. The second term of the left-hand member, being nega-
tive, when transferred, is added to the right-hand member. The sum
of the two quantities is .0890 X .0001. Since the third and fourth terms
of the left-hand member are positive, their sum, when transferred, is
ILLINOIS ENGINEERING EXPERIMENT STATION
subtracted from the quantity above. The final form of the equation is:
0B1, =.0835 X .0001. In a similar manner the other slopes, 0, and
the ratios of the deflection to the story height, R, can be determined.
0 is expressed in radians, and R is an abstract quantity.
The tabulation of the calculations as shown in Tables 14 and 15
facilitates the solution of a large number of equations. There are how-
ever a number of practical considerations which should be borne in
mind when solving a problem by this method.
An error at any point affects all of the calculations which follow.
It is therefore desirable to have two computers carry on the work simul-
taneously and compare results at frequent intervals in order to avoid
the loss of time due to errors.
A number of combinations can be made from each group of equa-
tions. It is desirable to combine the equations so as to make the
coefficient of the left-hand term of the left-hand member of the resulting
equation as large as possible compared with the coefficients of the other
terms in the equation. To illustrate this point, consider equations 25,
26, and M of Table 14. If equation M had been subtracted from equa-
tion 26, the coefficient of 014 in the resulting equation would have been
.1463 and the coefficient of R5 would have been 2.9027. The latter is
about twenty times as great as the former. Any small error in the
actual value of the former coefficient would be a large percentage of
error. This same percentage of error would be introduced into the
latter coefficient and produce a large actual error. With the combination
of equations 25 and M as used, the coefficient of OB is 1.0695 and of
RS, 3.4313. The latter coefficient is only about three times as large as
the former and hence the effect of an inaccuracy in the value of the
former upon the latter is correspondingly smaller than in the first case.
It is possible in some cases to combine the equations in such a way
as to give equations which are algebraically independent, but which are
numerically nearly identical. Combining such equations gives results
which are likely to be inaccurate. Any tendency of the equations to
become identities can be avoided by changing the order in which they
are combined.
In getting the values of the remaining unknown quantities after
one has been determined, each unknown is expressed in terms of the
known quantities. By referring to Table 14 it is seen that in the first
equation of a group of equations, any one of which could be used to get
the value of an unknown, the coefficient of the quantity which is to be
determined is larger than the coefficients of the known quantities. If
WILSON-MANEY--WIND STRESSES
this equation is used, any inaccuracies in the known quantities will
affect the accuracy of the results less than they would if an equation had
been used in which the coefficient of the unknown is less than the co-
efficient of the known quantities. The various equations available should
be examined and the one used which causes the least accumulation of
inaccuracies.
In order to determine the extent to which inaccuracies in the cal-
culations accumulate, two independent sets of calculations were made;
one set was made on a 20-inch slide rule and the other on a Fuller's
cylindrical slide rule. The two sets of calculations were compared at
frequent intervals, and mistakes were corrected; but no adjustment of
inaccuracies was made. The maximum variation in the slopes and
deflections as determined in the two sets of calculations was very small.
This indicates that the calculations can be made with a slide rule without
greater inaccuracies than are permissible.
The moments at the ends of the columns and girders can be deter-
mined by substituting the values of the deflections and the slopes, given
in Table 15, in equations A and B of Section V. To facilitate this work,
the quantities which occur in these equations are given in Tables 16
and 17. The values of R and 0 taken from Table 15 are given in the
second, third, and fourth columns of Table 16; and the functions of
the values of R and 0 which occur in the equations are given in the
remaining columns of the same table. The values of K taken from
Table 12 are given in the second, third, fourth, and fifth columns of
Table 17; and the functions of the values of K which occur in the
equations are given in the remaining columns of the table. The moments
at the ends of the columns and girders given by the fundamental equa-
tion, MAB = 2EK (20A + OB - 3R), are the products of two factors,
one of which is given in Table 16 and the other in Table 17. The
values of the moments are given in Table 18.
The shear in any member is equal to the algebraic sum of the
moments at the two ends of the member divided by its length. The
direct stress at any section in column A is equal to the algebraic sum
of the shears in the girders in bay a above the section. The direct stress
at any section in column B is equal to the algebraic sum of the shears
in the girders in bay a plus the algebraic sum of the shears in the
girders in bay b above the section. The shears in the columns and
girders and the direct stresses in the columns are given in Table 19.
The sum of all of the moments at the top and the bottom of all
the columns in a story is equal to the total shear on the story multiplied
ILLINOIS ENGINEERING EXPERIMENT STATION
by the story height. The algebraic sum of the moments in the girders
on the two sides of a column is equal to the algebraic sum of the moments
in the columns on the two sides of the girder.
The total shear on a story multiplied by the story height and the
algebraic sum of all of the moments at the top and the bottom of all
columns of a story, as given in Table 18, are given in the second and
third columns of Table 20. The algebraic sum of the moments in
column A at sections immediately above and below the girders and the
moments in the girders in bay a at sections adjacent to column A, are
given in the fourth and fifth columns of Table 20. The algebraic sum
of the moments in column B at sections immediately above and below
the girders and the algebraic sum of the moments in the girders in bays
a and b at sections adjacent to column B, are given in the sixth and
seventh columns of Table 20. The accuracy of the computations can
be checked by comparing the two values for the same quantity as given
in adjacent columns of Table 20. To illustrate, for the sixth story, the
total shear in the story multiplied by the story height is 913,000 inch-
pounds, and the sum) of the moments at the tops and bottoms of all
columns in the story is 914,400 inch-pounds. For a perfect check, the
two quantities would be equal. An inspection of the table shows that
the values check very closely.
The moments given in Table 18 are due to a horizontal wind load
of 30 lbs. per sq. ft. acting on a vertical strip one foot wide. To obtain
the moments due to the wind load on any portion of the building, multi-
ply the moments in Table 18 by the width in feet of the portion of the
building .on which the wind load acts.
VIII. APPROXIMATE METHODS.
14. Nomenclature of Methods.-The method used in Section VII
to determine the stresses in a symmetrical three-span twenty-story bent
can be used in the actual design of a building, but a shorter method is
desirable. The writers propose an approximate method which is much
shorter than the method used in Section VII and which they believe
is sufficiently accurate to be used in the actual design of buildings. For
the sake of convenience in reference, this approximate method will be
designated as the "Proposed Approximate Method" as distinguished
from the method used in Section VII, which will be designated as the
"Slope-Deflection Method." The slope-deflection method can be modi-
fied to advantage under certain special conditions. Such modifications
WILSON-MANEY-WIND STRESSES
will be designated as "Modifications of the Slope-Deflection Method."
15. Proposed Approximate Method.-The proposed approximate
method is based upon the following assumptions in addition to those
used in the slope-deflection method.
1. The changes in the slope at the top of a column in the story
above and in the story below the one in which the stresses are to be
determined, are equal to the change in the slope at the top of the corre-
sponding column in the latter story.
2. The ratio of the deflection to the length of the columns in the
story above the one in which the stresses are to be determined, is equal
to the ratio of the deflection to the length of the columns in the latter
story.
In other words, in determining the stresses in the second
story, OA, and OAs are assumed to be equal to OA2; OB1 and OB, are as-
sumed to be equal to OB2; and Rs is assumed to be equal to R,. Also,
in figuring the stresses in the third story OA2 and 0A, are assumed to
be equal to 6A,; OB2 and 0B4 are assumed to be equal to OB3; and RB
is assumed to be equal to R,. This does not mean, however, that the
values of OA,2, 9B2, and R, used in determining the stresses in the sec-
ond story are equal to the values of OA,, OB3, and R, respectively, used
in determining the stresses in the third story.
An examination of the equations in Table 3 shows that, if assump-
tions 1 and 2 were true, three equations containing only three unknown
quantities could be written for each story of the bent. To illustrate,
consider the equations for the second story. In accordance with assump-
tions 1 and 2,
Al = A2 = 0A3
O 02 = O
R, = R,
Substituting 6A2 for OA, and OA,, 0B2 for OB, and Os, and R2
for R, in equations D, E, and F of Table 3 gives:
- N2 + 4KA2 OA2 + 2 4K B2 WA ....... W()
-NR, ± 4 + 6 = - •6E .................... (1)
- 3 (KA, + KA,) R, + (KA2 + JA, + KA,) A + Ka20sB = 0 ........ (2)
- 3 (KB2 + Ka) R, + Ka2OA, + (K., + J"2 + Kb, + KBs) 2 = 0. (3)
These equations have been written for the second story. Similar equa-
tions can be written for any other story by making the proper changes
in the subscripts.
Equations 1, 2, and 3, obtained from the equations in Table 3, can
be used in determining the stresses in a symmetrical three-span bent
ILLINOIS ENGINEERING EXPERIMENT STATION
any number of stories high. If similar changes are made in the equa-
tions of Tables 1 to 10 inclusive, groups of equations can be obtained,
one for each bent, which can be used to determine the stresses in any
story of symmetrical bents of from one to five spans and also of unsym-
metrical bents of from one to five spans.
The sum of the moments at the tops and bottoms of all columns of a
story is equal to the total shear in the story multiplied by the story
height. The distribution of this moment to the ends of the columns
depends upon: first, the ratio of the K of column A to the K of column
B; second, the ratio of the K of column A to the K of girder a; and third,
the ratio of the K of girder a to the K of girder b.
The distribution of the moment was determined in a number of
bents for which the ratios of the K of column A to the K of column B,
the K of column A to the K of girder a, and the K of girder a to the K
of girder b had different values. Diagrams showing the distribution of
the moment in these bents also give the distribution of the moment in
other bents. The curves in Figs. 6, 7, and 8 show the distribution of
the moment in symmetrical three-span bents. In Fig. 6, the curves in
group I show the moment at the top and the bottom of column A in
bents in which the K of column A and the K of column B are equal.
The abscissae represent the ratios of the K of girder a to the K of girder
b; and the ordinates represent the moment at the top and the bottom of
column A in per cent of W X h. Beginning at the top and reading down,
the curves are for bents for which the ratio of the K of column A to
the K of girder a equals 0.5, 1, 2, and 4 respectively. The moment in
column A of a bent for which the ratio of the K of column A to the K
of girder .a has any intermediate value can be determined by interpola-
tion.
The curves in groups II, III, IV, and V, of Fig. 6, when used
with the curves in group I, give the moment at the top and the bottom
of column A in bents in which the K of column A and the K of column
B are not equal. The curves in groups II, III, IV, and V are for bents
for which the ratio of the K for column A to the K for girder a equals
4, 2, 1, and 0.5, respectively. Beginning at the left of each group and
reading to the right, the curves are for bents for which the ratio of
the K for column A to the K for column B equals 0.5, 1, and 2, re-
spectively.
The moment at the top and the bottom of column A can be ob-
tained from the diagram in Fig. 6 in the following manner:
First consider the moment in a bent in which the K of column A is
6 ic~
K, 0.50
\4(40
»HI+tSII
ITti-'
Oroup L
S. . . T tr
i--T
KMTtifti
KA -
1 K.0 -B - '
l^ttnnA\h^
!p ý4ýH41i lgHT !Wi I PIXi HTR+.k -4.. 4i^fH4111111111i111 li H H
-:± :-- I I I I I I'* -A ;: ::rr :::±j:£j4::#--:T--- . : --I I-:-----
± S T - :- ^ -- > ..' . t -...- - : -- .- t 't ^ -- ± -- I -- - -- - 4 --- - --- -- - -.- --. ?- a,"- .+ - -U ^Tj J.- +1
fttTi T TI TT 0 TT 7
Group T.I
4-^|
It 4+I¶ H-M i t+ +i - rI U i+-hI-
i i i i i i I I i i i i i , i i . . . . , : . . . . . . . . . . . . . . . . i l i I i i l i ] 1 1 1 iI I T I I I I I II 1i1 1 1 1 1
4+4 HT-+
--1111
0.5 U
1.0
6roup 117.
K I.0
KA....20
K8
-KA
--KB
1.0-
~-20
K8 -
U !F!HA:144I',444 I 1111 ',II .I , l . ',I. H I', , . !L -Iw: HI i HI!I + + H
JTý114111TT 441if] 11111H':H!1111:11:
S(roup .
KA =0.5
•Inim
l I I I I 'll 111 i i I I1 1 111 11111 1[ IP I :H4+ 11"lll llllllll l :
0 I Z 3 4 ~10
-+4+--+ ++-H-+I-H4-IH4+-+t tI
lH- -i+it t-
-++-HA-I-÷--H-H-H--*----- +4-+ -i--H-+--t-H--H- t-t-IfI- I Iil± iiiiii
- tt f -I4f t Tt- -1n
]--R4--44HU .jj44 T44-444--fl4 fl 4-.l....I.I.... I l'.l.l..'-' - i
IZ 6 8 10 If 6
10 IZ
6 1 12 14
Moment at Top and ottom of Column A in percent of W-h.
FIG. 6. APPROXIMATE MOMENT AT THE TOP AND THE BOTTOM OF COLUMN A.
RZG
it 6
6 8 10
KA4 0 = .0
K-o 0.5
Ratio of K for Girder a to K for Oirder b.
t+jj(1-- 4 --~ j
SGroup l.
iL:z. oj
L+ t mlnm mig
i i i 1 1 1i i I i i i i
.Ff H ' ;,;i ;[ ;i [ ii iii ii tr iii ii iii ijtt t] ti i f' 1 i-fi11 t
inf444±
T-7774
4+t H++-++ t I 1 4 11 If +tf I I if f.',f ý V! , titjrý
Hiiffiii i iRi i i H i Hil!
14 14 tfm hltftf44m mttt- 4 tf-fh-.
- n sz- f
= P-x
- "A - i ^ ý I KR --*-
IILJllJlkJlllL I
mm
4l-U-U-Ll-tr
7 - -- - - lV-14
-X.= Kgo
ttllmt l l itt llt
+ I . . . . . . . . . . . . . ..
u11 - - l
-r7
a
i 1 1 1 1 44 1+
ý t44444+t f+
IT
I l J,.
am-
.41l.l.I
- \ U i 641WR 6C-1 1 1 11
M-fTfmf
IHE-11-
J- kL ,I i TT-ý--iT 4
III M 1,111111
TjTI1I11I11 4111 11 111111111 111 # # # # ' ' I ' I '+14.
tt ti4 --t lilt I14 1 4 11' Tt
I I I I r I T
1-4 .4R
4-u- i
T 14+ý44 14+ 4 H .... .
-I- +1-41+'-
[qh I T:1
f~±1+t
6 111; iý - !I [i 1H;4V i144 ;11.1 ý4 4
1 d
I- . 4 1 - ' -f-VII- I11 H f+ t --+ -I- -
r11 1 iT 4 TT 44- 4I L
16 vIj--
4I J llt+I t t
4# #f*4 11 /(A I . ++~
I\\
Ks
AM A# A ) 4
E) 1.0 '
KB
~ro~pff~~
H-, =.d -ni /I
flKa '
FTTTF2iITLUI ILLLIILLLLLILI hiLtttttttttLt lttitttttttttfttItt1tt1111-tt11111--i11111ttl11tt1.1j 1
14 4- + . ...f l , - . .
4 If 14 /16 18
II 1:11 II;iii :111
- = -.0
=10.5
+ Group ,V. 14
Grcvp
Ka
F 7=I1
0.: I
-A K8'
Ke =:0.5
II- 1 1111144-.4 1 iH Hi 1 1
T
r2 14 16 18 /I .14 /16 1
Momerrt cvf Top and Boffom of Column B in per cent of W-h.
Kz:
'IT4 Z 2'
10 I 1 /4 16
Ii;
44414 4 ;l llý 1 H44
FI(;. 7. APPROXIMATE MOMENT AT THE TOP AND THE BOTTOM OF COLUMN B.
ffi~iaff&T-g-ffff±
C
I-.
I4 iI17,Iff44~
0
-4..
-4-.
----- -----4 .I4jj1~ 1 }~
~GoP I.ýý71
z 0
R4-r' fA I 10.'
J+4, l-tFH4f++4-f+±f+H4A4411111 II 444-4Pý-~ &h1it 1 111111 1111. 1 -¾ ITU 44-------ý
I2 3
Rotjioof/for ~irder a toK for &Order b.
-141-H-till 1111-4 4 44+1 H M llllfll-14 4-H-H-ý44ý+444 ý ilý 14'1ý14i 1 1 11 1111 tj !!I! i I!!'!!!''!!!
1- l- t tt fl
Jf
I 1 1 1 1 1 l I t I l i I 1 1 1 1 1 1 1 1 1 1 I t J 1 1 1 1 1 1 1 [ I l i 1 1 1 1 1 ! 1 1 1 1 1 1 1 -
111 11 jlq ýT jj III'' Ili 1 1 i
1 1 1 1 1 W 1 1 1 1 1 1 1 1 1 ý- , ";
114,11tilý] U _44 I IN -- -1111111
W4i
i i i il i4i4 i i i i i i i i i i i i i i i H i i i i i i i ý i i i i i i i I i ; ý i i i i ! i . i ! i i i i i i i ! i i I i i i i i i i i I ! I ji i " H 1.1 ii ý 11 it, t i
H;H;Tt';H;-H44:H44444;44ý 4 ý444444:444:4+44+44-11 ý 1-114i 114i H44`4 11 H lHi H i. i i Hil i i ii fýfl fi if f i i Hi i i i H i i i i hi 4-4i
t ttT!+i-H fitti ý i I
T
i L I 1 1.1 R444
i i i i i i i i i i i i i i
L -- l I IIJllll IIllElll IIll¸
i t i i t i i ! ! i i i i i i i i i i i 3 t i i 2 i i i i 0 t i i i Fr . '. , ý . .-. V
-Tl-"-t -rl I I I r I 1-1-Trl-TI T-T T T I I I T I I TýT-,ýj A ýýT7-T-TýýJ;Jýj V,
J-1 f-j+++ftjtttjjj-f4,10,
!III!!! +-f++H-t+-H-fHii i H 1-i-ffill H It I
14 16 ve /Z '14 16 16
19 14 145 16
/Z /4
44-H4-++--444- +4-
+44-14+44+1+
- i-444 +t-
-- ++44+H+H++--
flUT U
Pt
4-4.t4 I
^
I
Tz[i[iL
t44414-+ 4-
f-f44##4
t1t tt1f i - 4- 1 ""t -d
Ms
-x
:t x
x
-zI I
-1ý-T
Aft
±Ht
-,. --.o^
*-ý Ya -'- -
i
-7n
f '.D H
:i: -rp = i n ý-
t- K/v - / tily -- I
no no
E
t-v / "U I
- - i - # I
-- /
-- I l, I
KA zZ.0 J KA :;05
A-LL- ý . ý - I ý , ''A
-- i I1r
t- KA
- I ý, ý 4-11 4
t-H tz fe
1II W. - M-
HI A . = M Ap i
20 30
_L' . . . -. . .. . . . p "
_IM
11 T1
LI,ý444-444 -A Lmi ii
K40
K0
I ' i i H i f1 i i!I H
L-N KJT
I44-
gitllhf
L Group L.]
AK=K8
tT11TTItTT 44 1 I l4tinn Ii [TT!W TT11TT1TTTrTT1T 141
ttt~
II
Group H. A
]40
Xa .0
t It
"A =2.o
K4=/.1O
Kfia
- 1111I1 iII TUIYV
Group .l
=Z-.0
0K5
11'N1 If}II
4 =0+4 .t54
fflN+Ht1TWWfl~fl1-7~I ~T-i P1~ThTh4Th~
Sroupl .-
KA
mm
roup
7K =0.5
^i;:;::|^Kf0=5]:
V iT FTTTTTVT~rHFVTTUTVPV1TFF1 TT 1T11 i i i iTnriT ITTnnTTI1TITFm i ITtTTTT 1: ltit' 1, Ti±49 P nTrmnHTU Ti T IlET4 1W
,, HH!i . + . .t 4 4 "! I" *T F-t+t4-H i H 1 H H 1I 1 i i iii ii i ! I i I 1 1 1 I 1 -l It 9 t- '' i l 1 +
h ti Whittith
tIttHHfi
ttfIt Ilf tt
4FTiA ''44R-Pl'ijfH 14iH [-''I .++ 44T-fýFIT ý -F:~ if4RI-4~~4:4:1 llt llUWl[ lU.[tf l- i t htt1i t-4t-Ht-+-HHH--i-tt-l-i-ti--ý-11:HF-ti-t 1-t 4R44 II
Ratio of K for Girder a to K for Oirder b
4 o
10 ZO 30
0 10 20 30
0 10 ZO
&I MI
titt utitta
0 o0 tO 30
Moment at End of 6irder b in per cent of W*h.
FIG. 8. APPROXIMATE MOMENT AT THE ENDS OF GIRDER b.
&. Lu
4m
i i VVY
i i i ý ý i i i ! i i ý ý i i 1 1 1 1 1 0 1 7 f , , ! 1 1-
iH+±±h
týl
T-TI T-T
f-t-t-a
4n n rmIH4t1
1 i 1 i i 1 i f i 14 1 i i i 4 1 i i ý 1 i i 1 i 11 i 111 44ý 4ýý4.
t it mt1 ttt t t t t i t- i t t fi F- - - . ttt... tt.. ..t.+... ....ttt t
4--+111 1 - H1 1 11VV4-4+4-
I4 0
0 I<
:L-
7 Kg 5-Z
?44^1
I i J I
I i I J [ I IiI I [ I J I I ]
V KA t0
w^ L0
4 -
? 'A
u lIl l I
I I liffil W.
TT l = /.0 -
i - -H
iW X.-
IAN' flA
im=0.5
rn iA =1.
Ka"
- W. A
ýd5a
4Wj--f
WILSON-MANEY-WIND STRESSES
Ka KA
equal to the K of column B. Determine the value of- K and of - .
Use the curves in group I. Trace a vertical line whose abscissa equals
Ka KA
until it intersects the curve corresponding to the value of --a
Kb Ka
Project the point of intersection horizontally to the left and read the mo-
ment in per cent of W X h from the vertical scale. As W and h are
known quantities the moment is determined.
Next consider the moment in a bent in which the K of column A
Ka KA
and the K of column B are not equal. Determine the value of - , KA
KA
and -K-. First use the curves of group I. Trace a vertical line whose
KB
abscissa equals -- until it intersects the curve corresponding to the value
KA
of --. Project the point of intersection horizontally to the right to the
KA
group of curves corresponding to the value of -- until it intersects the
KK
particular curve of this group corresponding to the value of- . Project
KB
this intersection point vertically downward and read the moment in per
cent of W X h from the horizontal scale.
Similarly, the moments at the top and the bottom of column B
can be obtained from Fig. 7, and the moment at the end of girder b
can be obtained from Fig. 8. It should be noted, however, that the moment
in girder b depends equally upon the W X h in the story above and in
the story below the girder. This being true, in getting the moment in
girder b, the average of the W X h for the two stories should be used.
The moment at the right end of girder a balances the sum of the
moments in column A just above and below the girder and is equal to
their algebraic sum. The moment at the left end of girder a balances
the sum of the moments at the right end of girder b together with the
moments in column B just above and below girder a and is equal to
their algebraic sum. It is therefore possible to obtain the moment at
the ends of all members in a bent from the curves in Figs. 6, 7, and 8,
subject, of course, to the error due to the use of assumptions 1 and 2
of this Section.
The curves in Figs. 6, 7, and 8 show that a large change in the
ratio of the K of one member to the K of another member causes a
ILLINOIS ENGINEERING EXPERIMENT STATION
relatively small change in the distribution of the moments in the bent.
16. Numerical Problem.-To illustrate the use of these curves the
solution of a problem is presented.
The seventh story of a symmetrical three-span bent is 20 ft. high,
and is subjected to a shear of 3,000 lbs. The eighth story of the same
bent is 20 ft. high, and is subjected to a shear of 2,500 lbs. The prop-
erties of the members in the seventh and eighth stories are as follows:
KA = 30 in.3; KB= 40in.; Ka = 20 in.3; and Kb = 16 in.3. It
is required to find the moments at the ends of all members in the sev-
enth story.
W X h = 3,000 X 20 X 12 = 720,000 in. lb. in the seventh story.
W X h = 2,500 X 20 X 12 = 600,000 in. lb. in the eighth story.
KA 30
-- _= .75
KsB 40 -
KA 30
. ..- = 1.5
Ka 20
_-^- =1.25
Kb 16
To get the moment in column A use Fig. 6. At the left of the
figure trace the ordinate whose abscissa is 1.25 to a point half way be-
tween the two middle curves ( Ka = 1.5, which is half way between 1
KKA
and 2), and project this point horizontally to a point half way ( K =
.75, which is half way between .5 and 1.0) between the two left-hand
curves of Group II, and also to a point half way between the two left-
hand curves of Group III. The abscissa of the former point is 9.35 per
cent, and of the latter point 9.15 per cent. As -A-is equal to 1.5 or the
Ka
average of 1 and 2, the moment at the top and the bottom of column A in
the seventh story, MA,, is the average of 9.35 per cent and 9.15 per cent or
9.25 per cent of W X h, that is, MA = .0925 X 720,000 = 66,700 in. lb.
The moment, MAs, at the top and the bottom of column A in the eighth
story is .0925 X 600,000, or MAs = 55,500 in. lb. Similarly, the moment
at the top and bottom of column B, M-B, is 15.75 per cent of W X h, that
is, MB, = .1575 X 720,000 = 113,500 in. lb. in the seventh story; and
MBs = .1575 X 600,000 = 94,400 in. lb. in the eighth story. The moment
Mb7, at the end of girder b at the top of the seventh story is 13.85 per cent
of W X h, that is,
WILSON-MANEY-WIND STRESSES
94,400;a/6 ASooh/b
L
88 //6,400in./6 l22,200inb.
A8
113500in.lb 66I 700m/b\
(0) (b)
FIG. 9. MOMENTS ACTING AT POINTS A8 AND B8 OF A SYMMETRICAL THREE-
SPAN BENT.
Mb = .1385 ['20000 + 600000 = 91,500 in. lb.
The moment at the right end of girder a at the top of the seventh story
is equal to the sum of 66,700 in. lb., the moment at the top of column
A in the seventh story, and 55,500 in. lb., the moment at the bottom
of column A in the eighth story, or 122,200 in. lb. (see Fig. 9b). The
moment at the left end of girder a is equal to the sum of 113,500 in.
lb., the moment at the top of column B in the seventh story, and 94,400
in. lb., the moment at the bottom of column B in the eighth story, less
91,500 in. lb., the moment in the end of girder b; that is, the moment
at the left end of girder a is 116,400 in. lb. (see Fig. 9a).
A comparison of the moments in a bent as obtained by the pro-
posed approximate method and by the slope-deflection method is given
in Tables 23 to 26 inclusive.
17. Modifications of the Slope-Deflection Method.--The writers
made a study to determine the effect of a change in the section of one
member of a bent upon the moment in the other members. The mo-
ments were determined in a number of bents in which K = 1 for all
members except the one whose effect upon the distribution of the mo-
Wh
ment was to be studied. The quantity 6- was taken equal to 1 for all
stories and. for all bents. The K of column B was given successively
the values 0.5, 1, 2, and 4; and the corresponding moment at the ends
of columns A and B and at the ends of girders a and b was calculated
194, 400 /n, /. l S.. 5oo.;n./b
//$ $oo in.lb. 66 700jh/
ILLINOIS ENGINEERING EXPERIMENT STATION
-40 -30 -zo -/0 0 10 ZO 30 40 50
Change in the Moment at the Top or he Bot tom of Column A in 5tory
No. N in per cent of the Moment tohe Jame Point when for all//ember4 K'=.
FIG. 10. DIAGRAM SHOWING CHANGE IN THE MOMENT AT THE TOP AND THE
BOTTOM OF COLUMN A OF A SYMMETRICAL THREE-SPAN BENT DUE
TO A CHANGE OF K IN THE OTHER MEMBERS.
KEY TO THE DIAGRAM IN FIG. 10.
Member Changed
Girder a in Story No. N
No.(N-1)
Girder b " No. N
" No.(N-l1
Column A" " No. N
No.(N-1)
No. (N+1
ColumnB " No. N
No.(N-1)
No.(N+ll
Designation of the Curve Showing
the Change in the Moment at
Top of Bottom of
Column A Column A
1 2
2 1
3 4
4 3
5 5
6 7
7 6
8 8
9 10
10 9
WILSON-MANEY-WIND STRESSES
3 8 7 .4 6
^m\ r/t.¢.'-^
4 1 1 yt
3 ir-~.^ -rz
_____- _s ~ z^ _____
-5o -4 -3o -zo -/o o to zo ,So 40
Change /i the Momenf at the Top or the Bottom of Column b in 5/ory
No. N in per cent of the Moment aof the same point when for Hember. K= l
FIG. 11. DIAGRAM SHOWING CHANGE IN THE MOMENT AT THE TOP AND THE
BOTTOM OF COLUMN B OF A SYMMETRICAL THREE-SPAN BENT
DUE TO A CHANGE OF K IN THE OTHER MEMBERS.
KEY TO THE DIAGRAM IN FIG. 11.
Designation of the Curve Showing
the Change in the Moment at
Member Changed
Top of Bottom of
Column B Column B
Girder a in Story No. N - 1
" No.(N-1) 1 -
Girder b " No. N 2 -
S " No.(N-1) - 2
Coulmn A" No. N 3 3
" No.(N-1) 4 5
No.(N+1) 5 4
Column B " " No. N 6 6
No.(N-1) 7 8
No.(N+1) 8 7
Ji 1 1. 1S 1
U l ___ ] ____ ____ I ____ I ____ ___ I ___ '* 'l *- - - -i--- --- * --- * --- --- ' --- * --- -- -
ILLINOIS ENGINEERING EXPERIMENT STATION
4 4
S ------*-"*-I- --I----- "---------
3 5 CZ4
- I 0
-100 -o80 -60 -40 -o20 0 zO 40 (60 80 A
Change in /he Moment oat the End of 6rder b in7 J ary a No in percent of
the Moment ao the Jame PokIt when foral/l ember. K .
FIG. 12. DIAGRAM SHOWING CHANGE IN THE MOMENT AT THE END OF GIRDER
b OF A SYMMETRICAL THREE-SPAN BENT DUE TO CHANGE
OF K IN THE OTHER MEMBERS.
KEY TO THE DIAGRAM IN FIG. 12.
Designation of the Curve Showing
Member Changed the Change in the Moment at the
End of Girder b
Girder a in Story No. N 1
No.(N-1) 2
" " No.(N+1) 2
Girderb " " No. N 3
Column A " No. N 4
" " No.(N+1) 4
Column B" " No. N
" " No.(N-1) 6
" " No.(N+l) 5
WILSON-M-ANEY-WIND STRESSES
in the story in which the K of column B was changed, and also in the
stories immediately above and below that story. Similarly, the K of
column A, the K of girder a and the K of girder b were given successively
the values 0.5, 1, 2, and 4, and the corresponding values of the moment
at the ends of the girders and the columns were determined in the par-
ticular story and also in the stories immediately above and below that
story. The effect of the changes in the values of the K of the members
upon the moment is given in Figs. 10, 11, and 12.
Fig. 10 shows the changes in the moment at the top and the bottom
of column A. The abscissae are changes in the moment expressed in
per cent of the moment in a bent for which the K of all members is
equal to 1. Increases in the moment are laid off to the right of the
origin, and decreases are laid off to the left. The ordinates represent
the values of the K in the member changed. The story in which the
moment was calculated is designated as story No. N, the story above is
designated as story No. (N + 1), and the story below No. (N - 1). Each
curve shows the change in the moment at the top or the bottom of
column A due to a change in the value of the K of some member. The
number of the curve which shows the change in the moment in column
A due to a change in the K of any particular member can be obtained
from the key to the diagram in Fig. 10. For example, curve No. 3 shows
the change in the moment at the top of column A due to a change in the
K of girder b in story No. N. If the K of girder b is made equal to 2,
the moment at the top of column A is 8.9 per cent less than when the K
of girder b is equal to 1. Curve No. 3 also shows the change in the mo-
ment at the bottom of column A due to a change in the K of girder b in
story No. (N - 1). If the K for girder b in story No. (N - 1) is equal
to 2, the moment at the bottom of column A is 8.9 per cent less than when
the K for girder b is equal to 1.
Similarly, the changes in the moment in column B due to changes
in the K of the members of the bent are given in Fig. 11 and the
changes in the moment in girder b are given in Fig. 12.
Fig. 10 shows that the moment in column A of any story not only
depends upon the value of the K of the members in the same story but
depends also upon the value of the K of the members in the adjacent
stories. Figs. 11 and 12 show that the same statement is true relative
to the moment in column B and girder b. This being true, any approxi-
mate method which considers only the members in the story in which
the stresses are to be determined can not give accurate results except
ILLINOIS ENGINEERING EXPERIMENT STATION
in bents which have no very sudden changes in the sections of the
columns and girders.
Where there are sudden changes in the column and girder sections,
the proposed approximate method is not accurate. In such cases it is
sometimes possible to use a modification of the slope-deflection method
and obtain quite accurate results with a comparatively small expendi-
ture of labor. This modification of the slope-deflection method may be
made in either of two ways, as follows: (a), the bent may be divided
by a horizontal plane between any two adjacent floors, and the
lower part be treated independently of the upper part; or (b), the
bent may be divided by two horizontal planes any number of stories
apart, and the portion between these planes may be treated independent-
ly of the other portions of the building. These modifications will be
considered in detail.
(a) The wind stresses in the top stories of a bent are com-
paratively small and their exact determination is not important. An
examination of the equations of Table 15 shows that the coefficient of
the quantity to be determined is greater than the coefficients of the
quantities in which the unknown is expressed. This being true, errors
in the slopes and the deflection of one story have but little effect upon
the slopes and deflections of the succeeding stories. Further examina-
tion of the same table shows that there is a comparatively small differ-
ence between the change in the slope at the top of a column in one
story and the change in the slope at the top of the same column in the
story below. If the two changes in slope are assumed to be equal, the
error introduced is small. Bearing these facts in mind, consider again
the twenty-story bent of Section VII. If 6A1, and O,B are assumed
equal to OA,, and OBn1 respectively in equations f, g, and h of Table
13, the first 34 equations will contain 34 unknown quantities which
can be determined by the method used in Tables 14 and 15. That is,
assuming that the changes in the slopes of the columns in one story
are equal to the changes in the slopes of the same columns in the story
below, is equivalent to dividing the bent into two parts; and the
equations for the lower part can be solved independently of those of
the upper part. If the slopes which are assumed equal are not really
equal, the calculated slopes and deflections in the top story of the lower
part will not be exact, but the results of a number of calculations show
that the error in the slopes and the deflection in the next to the top
story is so small that it may be neglected. This being true, if, in the
actual design of the twenty-story bent of Section VII, it is con-
WILSON-MANEY-WIND STRESSES
sidered unnecessary to calculate the wind stresses in the top ten stories,
the bottom eleven stories may be treated as a complete bent and
the stresses determined by the method outlined above, thereby decreasing
the work. The stresses in the bottom ten stories will be quite exact.
The bent could have been divided at any other story without affect-
ing the accuracy of the results.
If there are sudden changes in the members the bent can be divided
above the story in which the change occurs, and the stresses in the lower
part can be determined by the method outlined above.
Table 23 shows that the largest errors in the proposed approximate
method are in the first story. The moments can be determined in the
first story by the modification of the slope-deflection method as follows:
Assume that the slopes in the third story are equal to the corre-
sponding slopes in the second story. The first seven equations of
Table 13 will then contain only seven unknown quantities. The solu-
tion of these equations is given in Tables 21 and 22. The changes in
the slopes and the deflection for the first story as given in Table 22
agree very closely with the values for the same quantities given in Table
15, and the moments, being functions of the slopes and deflection, will
also agree very closely with the moments given in Table 18.
(b) The moments in any particular story may be determined if
there is some story below the one in question in which it can be seen
by inspection that the changes in the slopes at the tops and the bottoms
of the columns are equal. Suppose the moments are to be determined
in the tenth story of the bent shown in Fig. 5, and that it is apparent
from inspection that OAg = A, and 0Bg= 0Bg. Then assume that
0A12= 6Ai, and 0B12 = Bll. The ten equations, Y to h inclusive
of Table 13, will contain ten unknown quantities. If the slopes in
the eighth and ninth stories, which have been assumed to be equal, are
equal then the slopes and the deflection, and therefore also the moments,
in the tenth story will be quite accurate, but if the slopes below the
story in question which are assumed to be equal, are not really equal,
the required moments will not be exact. That is, a difference between
the slopes below the story in question affects the results, whereas a differ-
ence between the slopes above the story does not materially affect the
results.
18. Application of the Proposed Approximate Method and Modifi-
cation of the Slope-Deflection Method.-To obtain the moment in a bent
in which there are sudden changes in the sections, the following com-
ILLINOIS ENGINEERING EXPERIMENT STATION
bination of methods can sometimes be used to advantage. Use the pro-
posed approximate method for obtaining the moment in the portion of
the bent in which there are no sudden changes in the sections of the
members. Use a modification of the slope-deflection method to get the
moments in the bottom story and in any intermediate stories in which
sudden changes in the members occur. The results thus obtained will
be sufficiently accurate to be used in the actual designs of buildings, and
the amount of work required will not be excessive.
IX. COMPARISON OF THE APPROXIMATE METHODS WITH THE SLOPE-
DEFLECTION METHOD.
19. Symmetrical Three-Span Bent with Short Middle Span.-For
comparison, the moments in the symmetrical three-span twenty-story
bent shown in Fig. 5 were calculated by the five following methods:
1. The slope-deflection method.
2. The proposed approximate method.
3. The three methods described by Mr. Fleming and known as
methods I, II, and III.
The moments as determined by these methods are given in Tables
23 and 24. For each story the moments in the upper line are in 1,000
in. lb., and those in the lower line are in per cent of the moments as
determined by the slope-deflection method. Tables 23 and 24 show in
the case of this bent: first, that the moments determined by methods II
and III are very seriously in error; second, that the moment determined
by method I and by the proposed approximate method agree very closely
with the moment determined by the slope-deflection method except at
points where there are sudden changes in the members of the bent;
third, that the errors in the moment determined by the proposed ap-
proximate method are less for the girders than for the columns.
20. Symmetrical Three-Span Bent with Long Middle Span.-The
distribution of the moment determined by the slope-deflection method
is affected by the ratio of the K of girder a, to the K of girder b. As
this ratio does not affect the distribution of the moment determined by
methods I, II, and III, the accuracy of the latter methods will depend
upon the relative values of K for the two girders.
In the bent shown in Fig. 5, the K is less for girder a than
for girder b, since the girders have substantially the same sections and
girder a is longer than girder b. In order to determine the effect of
the relative values of the K of girders a and b upon the accuracy of
the approximate methods, the moment in the bottom four stories of the
WILSON-MANEY-WIND STRESSES
bent shown in Fig. 13, a bent like the one shown in Fig. 5 ex-
cept that the long and short spans have been interchanged, was deter-
mined by each of the five methods mentioned in article 19. The results
are given in Tables 25 and 26. The moments determined by the pro-
posed approximate method are as accurate as those given in Tables 23
and 24; whereas the moments determined by method I are inaccurate,
and those determined by methods II and III are very inaccurate.
21. Effect of the Proportions of a Bent upon the Accuracy of
Method I.-In order to determine more fully the effect of the propor-
tions of a bent upon the accuracy of method I, the moments were deter-
mined in a number of bents having different proportions, by the slope-
deflection method and by method I. These moments are given in
Table 27.
In bents 1 to 9 of this table all girders and columns have the
same section. The ratio of the moment of inertia of the columns to the
moment of inertia of the girders and also the ratio of the moment of
inertia of the girder in bay a to the moment of inertia of the girder in
bay b, affects the moment determined by the slope-deflection method
but does not affect the moment as determined by method I. If the
sections of the columns and girders are not the same, the difference
between the moments determined by the two methods might vary even
more than Table 27 indicates. Any errors in the moment due to sudden
changes in the sections of the bent are in addition to the errors indicated
in Table 27.
The difference between the two methods is due to the fact that in
method I the direct unit stress in a column is proportional to the dis-
tance of the column from the neutral axis of the bent; whereas in the
slope-deflection method the stress in a column depends upon the shears
in the girders, and the shears in the girders depend upon the changes
in the slopes at the ends of the girders and upon the moment of inertia
of the girder sections.
22. Accuracy of the Approximate Methods when the Moment of
Inertia of the Girders is Proportional to the Bending Moment.-In the
comparison of the approximate methods with the slope-deflection method
given in Tables 23 to 27, the sections of the girders were chosen with-
out reference to the moment to which they were subjected, except the
girders in the bent shown in Fig. 5. The girders in this bent were
designed to resist the bending moment determined by method I. An-
other investigation was made to determine the accuracy of the approxi-
ILLINOIS ENGINEERING EXPERIMENT STATION
WI/nd laod detfo
a Presure of
Jo lf pers fft
ctrn on a
I ft. wde.
lal WtS
wab7 e-W
• _ -. -._ ---i.
ii
lwO I
3 - . ..-.--1--- S. -- ---_.
x ---- --I
/ AlU
145 - -
S -m -T--
S-- -..-.--
X0 IIz
V40 I? ll
SOW I l ff
sw 7 t
4m I 4 4m
L- n n r ri--
FIG. 13. SYMMETRICAL THREE-SPAN BENT TWENTY STORIES HIGH WITH LONG
SPAN AT THE CENTER.
WILSON-MANEY-WIND STRESSES
mate methods when applied to bents in which the girders were designed
to resist the bending moment due to the wind load as determined by the
method which is to be compared with the slope-deflection method. The
results of this investigation are given in Tables 28 to 31. In Table 28
the moments of inertia of the sections of the girders are proportional
to the bending moments determined by method I. The sections of the
columns are equal, and the moment of inertia of the sections of the
columns is equal to the moment of inertia of the section of girder a.
Table 28 shows that method I gives the moment in the columns and
girders quite accurately when the spans are equal and the story height
does not exceed the length of the span. This statement is true, in gen-
eral, only when the two column sections are equal and when the moments
of inertia of the sections of column A and girder a are equal.
In considering the merits of the approximate methods, it should
be noted that the moment which should be determined with the greatest
accuracy is the moment to which the joint that connects the girder to
the column is subjected. This moment is the moment at the ends of the
girders. If the moments of inertia of the sections are not made propor-
tional to the bending moment determined by method I, the method will
not be as accurate as Table 28 indicates.
The moments of inertia of the girder sections of Table 29 are
proportional to the bending moments in the respective girders as deter-
mined by method II. The moments of inertia of the girder sections of
Table 30 are proportional to the bending moments in the respective
girders determined by method III. It is apparent from Tables 29-30
and Tables 23-26 that methods II and III are so inaccurate that they
should never be used.
The moments of inertia of the girder sections of Table 31 are
proportional to the bending moments in the respective girders as deter-
mined by method IV. For bents having equal spans and equal column
sections this method gives quite accurate results.
While methods I and IV may be quite accurate in some cases, their
accuracy depends upon the proportions of the bent. For example, by
comparing Tables 27 and 28 it is apparent that if the moments of
inertia of the girder sections are not proportional to the bending moments
determined by method I, that method will not be as accurate as when
the girder sections are so proportioned. Again, in Tables 28 and 31
the sections of the columns are equal and the moments of inertia of
the sections of column A and girder a are equal; but if these relations
ILLINOIS ENGINEERING EXPERIMENT STATION
do not exist, the methods will not in general be as accurate as Tables
28 and 31 seem to indicate.
Any inaccuracy in methods I and IV due to sudden changes in the
members of the bent, are in addition to the inaccuracies shown in Tables
28 and 31.
For the bents shown in Tables 28 to 31, the moments determined
by the proposed approximate method are exactly the same as the moments
determined by the slope-deflection method.
X. TEST OF A CELLULOID MODEL OF A BENT.
23. Description of Tests.-In order to check the deflections and
the changes in the slopes calculated by the slope-deflection method, a
celluloid model of a bent was subjected to known shears, and the result-
ing deflections and changes in slopes were measured and compared with
the calculated values. The model was made of celluloid % inch thick,
and had the general dimensions shown in Fig. 14. A cord passing over
a pulley and attached to a weight at one end and to the top of the
model at the other, produced a uniform shear in all stories. Paper arms
were fastened to the model at points where the members intersect. The
movement of these arms indicated the changes in the slopes. One of
these arms, AD, is shown in Fig. 14. The external force caused the
point A2 to move in approximately a horizontal line and at the same
time to turn through a small angle, 0. The paper arm A2D has the
same motion as the point A,; thus the vertical displacement of D measures
the angle 0. The horizontal deflection of the model was obtained by
measuring the displacement of a point at the middle of a girder. Paper
arms were attached to all four columns at the tops of all stories simul-
taneously, and the changes in the slopes at all intersections were
measured for each application of the load. Similarly, the horizontal
deflection was measured at the middle of each of the three girders at
the top of each story.
In the original model, known as model No. 1, members No. 1, 2,
3, and 4 were /2 inch wide; all other members were 1/4 inch wide. After
this model had been tested, member No. 1 was reduced to 1/' inch; and
then the resulting model, now known as No. 2, was tested. Members No.
2, 3, and 4 were successively reduced to /4 inch; and the resulting models,
now known as models No. 3, 4, and 5, were also tested.
In testing model No. 1, observations were made when loads of
21/2 lb., 5 lb., 7½ lb., and 10 lb. were applied. The readings for the
two sides of the model agreed very closely in the bottom three stories;
WILSON-MANEY-WIND STRESSES
but in the fourth story they did not agree. Thinking that this discrep-
ancy might be due to excessive stresses, when testing model No. 2, a
maximum load of only 71/ lb. was used; but there was still the same
discrepancy in the readings. Thinking that the apparatus might be
out of order, the loads were removed and then applied a second time but
the second readings agreed very closely with the first. In testing models
No. 3, 4, and 5, loads of 11/2 lb., 3 lb., and 41/2 lb. were applied. The
loads were then removed and applied in the opposite direction. The
readings for the two sides of the model continued to agree very closely
in the bottom three stories; and in the fourth story they continued to
FIG. 14. CELLULOID MODEL.
disagree. It is therefore probable that either the material was not.
homogeneous or that there were internal stresses or local weaknesses in
the upper part of the model.
The observations made in the test of model No. 4 are given in
Table 32.
24. Results of Tests.-The results of the tests of fhe models are
given in Fig. 15. The models are shown by the sketches at the left
of the figure. In these sketches members represented by heavy lines
are 1/2 inch wide, and all other members are 1/4 inch wide. The upper
diagram of Fig. 15 shows the changes in the slopes and the lower
diagram shows the deflections.
In the upper diagram the first group of lines at the left represents
v_
I
ILLINOIS ENGINEERING EXPERIMENT STATION
the changes in the slope at the point A1, the second group represents
the changes at A,, and similarly the third and fourth groups represent
the changes at A3 and A4. The fifth group of lines from the left repre-
sents the changes in the slope at the point B,, the sixth group at B,, and
similarly for the other groups. In the left-hand group of lines the
change in the slope at the point A, of model No. 1 is laid off from the
origin on a horizontal line opposite the sketch of model No. 1; the
change in the slope at A1 of model No. 2 is laid off from the origin on
a horizontal line opposite model No. 2; the change in the slope at A , of
model No. 3 is laid off from the origin on the horizontal line opposite
model No. 3; and the change in the slope at A1 of model No. 4 is laid
off from the origin on the horizontal line opposite model No. 4. The
full lines connect points which represent the quantities -as measured,
and the dotted lines connect points which represent the same quantities
as computed by the slope-deflection method. The changes in the slopes
at A., A3, A, B,, B2, B3, and B4 are shown in a similar manner.
The lower diagram shows the ratios of the deflection to the story
height in the different stories of the models. The method of represent-
ing the ratios of deflection to story height is similar to the method used
to represent the changes in the slopes in the upper diagram.
It will be noticed that the calculated quantities are in general greater
than the observed quantities. The reason for this is: In the computa-
tions, the length of a member was taken equal to the distance between
center lines, whereas the length that is actually free to bend is the dis-
tance from outside to outside of the members. This accounts for the
difference between the observed and the computed values.
Fig. 15 shows that the changes in the slopes and the ratios of the
deflection to the story height as observed and as computed agree closely.
In other words, the tests support the theory upon which the slope-
deflection method is based.
XI. DISCUSSION OF THE ASSUMPTIONS.
25. Preliminary.-In making the analysis of the stresses the writers
made certain assumptions in regard to the action of the frame when
stressed. The effect of inaccuracies in these assumptions will now be
considered.
If all of the columns of a story are taken together as a free body,
the algebraic sum of the moments at the tops and bottoms of all the
columns is equal to the shear on the story multiplied by the story height.
As the moments in the columns are balanced by the moments in the
&NEY--WIND STRESSES
tt* "
4-;.
*c: 1
=N
__ __ __ __ = __ IF
-"P
l&N 06N CoM 6AN SoN
MOM
0
In
In
o
41
0
U)
'ZI0
Zl
4-
I* g
ILLINOIS ENGINEERING EXPERIMENT STATION
girders, the algebraic sum of the moments at the ends of the girders is
dependent upon the product of the shear in the story and the story
height. These facts should be borne in mind in discussing the effect
of inaccuracies in the assumptions upon the moments in a bent.
26. Assumption of Perfect Rigidity.-According to assumption No.
1, the connections between the columns and girders are perfectly rigid.
The truth of this assumption can be determined only by tests. As far
as the writers are aware no such tests have been made. While it is
undesirable to make a mathematical analysis which is based upon an
unverified assumption, some assumption relative to the rigidity of the
connections must be made before the stresses in a frame can be deter-
mined. The distribution of the stresses depends more upon the relative
stiffness than upon the actual stiffness of the connections. In view of
these facts, it seems that the assumption of perfect rigidity of the joints
is the most logical one that can be made.
27. Assumption of the Unchanged Length.-According to assump-
tion No. 2, the change in the length of a member due to the direct stress
is equal to zero.
If columns A and B change in length, the moment at the right-
hand end of girder a will be given by the equation
MAB = 2EKa (20A + OB - 3Ra),
in which Ra is equal to the difference between the changes in the lengths
of columns A and B divided by the length of girder a. In the derivation
of the general equations in Section VI, Ra was assumed to be equal to
zero. If the changes in the lengths of the columns do not alter the
values of 0A and 01, the change in the moment AMAB, at the right-
hand end of girder a due to the changes in the lengths of the columns is:
AMAB = 2EKa (- 3Ra).
The change in the moment at the left-hand end of girder a is equal
to the change at the right-hand end. The difference between the changes
in length of columns A and B, or the deflection of one end of girder a
relative to the other end, is given by the equation:
Pl
d - in which
d = the deflection,
P =the difference between the stress in column A and the stress in
column B,
I = the length of the columns,
A = the area of the column section,
E = the modulus of elasticity of steel.
WILSON-MANEY-WIND STRESSES
From Table 19, page 75, the compressive stresses in the first story
of columns A and B are 14,464 lb. and 4,587 lb. respectively.
Therefore P = 14,464 - 4,587 = 9,877 lb. From Table 11, page 48,
I = 264 in., Ka = 30.5 in.', and A = 112.8 sq. in. E = 29,000,000 lb.
per sq. in. Substituting these values in the above equation for d, gives:
d = .000798 in.
d
By definition, Ra - , therefore
.000798
Ra - ' 98- .00000303, and
264
AMAB = 2 X 29,000,000 X 30.5 (- 3 X .00000303) = - 16,050 in. lb.
This change in the moment is 5.6 per cent of the moment at the right-
hand end of girder a as given in Table 18, page 74. Likewise the
moment at the left-hand end of girder a is decreased by 16,050 in. lb.,
which is 6.6 per cent of the moment as given in Table 18.
The moment at the end of girder b is affected by the changes in the
lengths of two columns, B, one on each side of the center line of the
bent. One column is subjected to a tensile stress of 4,587 lb., and the
other to a compressive stress of 4,587 lb. The deflection of girder b
is given by the equation
2 X 4,587 X 264
d= 2 X 4,587 264 = .00074 inches.
112.8 X 29,000,000
00074
Rb- 4 - .00000343.
216
AMBB = 2 X 29,000,000 X 37.3 X 3 X .00000343 = 22,300 in. lb.,
which is 9.0 per cent of the moment at the end of girder b as given in
Table 18.
The change in the length of a column is a function of the unit
direct stress and also a function of the story height. The first story
of the building considered is much higher (22 ft.) than the other stories,
and the unit direct stress in the columns in the first story is greater
than in the stories above; hence the change in the length of the columns
is very much less in the other stories than in the first story. Therefore
the changes in the moments for the other stories will be less than those
in the first story computed above.
The changes in the moments in the columns have been deter-
ILLINOIS ENGINEERING EXPERIMENT STATION
mined and it remains to consider the effect of the direct stresses in the
girders. The direct stress in a girder is very small in comparison with
the direct stress in a column and therefore may be neglected.
If the changes in the slopes were not affected by the changes in
the lengths of the columns, the moments at the ends of girders a and b
would be decreased. As the sum of these moments is determined by
the product of the shear in the story and the story height, they can not
all be decreased simultaneously. Therefore all of the changes in the
slopes must be increased until the sum of the moments at the ends of
the girders will balance the moments in the columns. This will make
the moments at the ends of the girders practically the same as they
would have been if the columns had not changed in length. If the
moments at the ends of the girders are not materially affected, the
moments at the ends of-the columns will not be materially affected.
Therefore although the direct stresses in the columns do change the
lengths of the columns, they do not affect the stresses in the frame to any
great extent.
28. Assumption as to Length of Members.-In accordance with
assumption No. 3, the length of a member was taken as the distance
between the center lines of the members which it intersects. This makes
the changes in the slopes and deflections, as calculated, greater than the
actual values. The effect of the inaccuracy of this assumption upon the
distribution of the moments is, however, not so apparent.
The curves in Fig. 6, 7, and 8 show that the distribution of the
moment in a story depends upon the relative values of K of the mem-
bers, but that it takes a comparatively large change in the K of a mem-
ber to appreciably affect the distribution of the moments. The fact
that the length of a member has been taken equal to the distance between
center lines, has but little effect upon the relative values of K for the
members; and therefore does not materially affect the distribution of
the moments.
29. Assumption as to Deflection Due to Shear.-According to
assumption No. 4, the internal shearing stresses in a member produce no
deflection. The distribution of the stresses is dependent, not upon the
actual deflection of the columns due to shear, but upon the differences
between the deflections in the different columns. The shears on the
columns are small and the differences between the shears are still smaller;
and therefore the assumption that the deflection due to shear is equal to
zero will not cause any appreciable error.
30. Assumption as to Load.-According to assumption No. 5, the
WILSON-MANEY-WIND STRESSES
entire wind load is resisted by the steel frame. The walls of a building
no doubt help to resist the wind loads, but the resistance which they
exert is uncertain. If a portion of the load is considered as being re-
sisted by the walls the stresses in the steel frame are correspondingly
reduced; but the method of determining the stresses is not affected.
XII. CONCLUSIONS.
As a result of the investigation described in this bulletin the follow-
ing conclusions can be made relative to the methods used to determine
the wind stresses in the steel frames of office buildings.
a. Methods II and III of Section II are so inaccurate that they
should never be used; I and IV are quite accurate in some cases, but
they may give results which are seriously in error.
b. The method presented in Sections VI and VII, and designated
as the slope-deflection method, contains no approximations except those
in the assumptions. It has been shown that the inaccuracies in the
assumptions do not materially affect the results. Therefore the method
is very accurate.
c. While the slope-deflection method is long, it could be used in
the actual design of a building; but it has its greatest value as a standard
by means of which the accuracy of the approximate methods may be
determined.
d. The proposed approximate method is short; and, except at
points where there are very large changes in the size of the members,
gives results which are accurate enough to be used in the actual design
of a building.
48 ILLINOIS ENGINEERING EXPERIMENT STATION
TABLE 11.
PROPERTIES OF THE COLUMNS AND GIRDERS IN THE SYMMETRICAL THREE-
SPAN TWENTY-STORY BENT SHOWN IN FIG. 5.
Story
Number
1
2
3 and 4
5 and 6
"< 7 and 8
9 and 10
S11 and 12
13 and 14
15 and 16
17 and above
2
3 and 4
5 and 6
; 7and 8
0 9 and 10
o I and 12
13 and 14
15 and 16
17 and above
or 1
a 2and3
S 4 and 5
.- 6
i 7
S8 andabove
. 2
- 6
8 andabove
Section of Member
Web
Plate
17x4
17xj
17xi
17xj
17xj
17x
l7x
17xi
17xj
17x*
V7x.
17xi
17xi
17xj
17xk
17xi
17xj
17xi
17xj
17xi
42xR
36x|
36x#
30x#
30x9
24xi
42xA
36x4
36x4
36x8
30x8
30x#
24xg
4
Angles
8x8xj
8x8x0
8x8xj
8x8x
8x8xj
x88x
8x8x
8x6xf
86x64
8x6x3,x
8x8xj
8x8xj
88x8x
8x8xj
8x8xk
8x6x.
8x8xg
8x6xa
6x3jx8
6x3jx8
5x3jx8
6x3jxj
5x3jx$
5x3jx8
6x3jx8
6x34xx
5x3ixi
6x3jxj
5x3jx8
5x3jx8
Cover
Plate
2-181iU
2-18x1
2-18xl
a-l8xIj
2-18x14
2-18x1^'
2-18xl
2-18x1
2-18x,
2-18x*
2-18x1,
Moment of Inertia-(Inches)O
Area of
Section
Sq. In.
112.80
112.80
103.80
92.55
83.55
72.42
64.24
53.79
42.26
38.86
112.80
112.80
103.80
94.80
85.80
79.05
67.80
57.92
44.39
41.00
29.43
27.18
25.70
24.93
23.45
21.20
29.43
29.38
27.18
25.70
24.93
23.45
21.20
Cover
Plate
3950
3950
3080
2060
1266
3950
3950
3080
2240
1459
892
Total
6816
6816
5946
4926
4132
3036
2707
2634
2055
1891
6816
6816
5946
5106
4325
3758
2866
2684
2106
1896
8058
5641
5161
3717
3387
2025
8058
6303
5641
5161
3717
3387
2025
Length
of
Col-
imns =h
of
Gird-
ers=1
Inches
264
192
168
168
144
144
144
144
144
144
264
192
168
168
144
144
144
144
144
144
264
264
264
264
264
264
216
216
216
216
216
216
216
K for
Columns
=1
h
for
Girders
I
(Inches)3
25.8
35.6
35.4
29.4
28.7
21.1
18.8
18.3
14.3
13.1
25.8
35.6
35.5
30.4
30.0
26.1
19.9
18.6
14.6
13.2
30.5
21.4
19.5
14.1
12.8
7.7
37.3
29.2
26.2
23.8
17.2
15.7
9.4
Web
358
358
358
358
358
384
358
308
205
205
358
358
358
358
358
358
358
358
256
256
2315
1458
1458
844
844
432
2315
1458
1458
1458
844
844
432
Moment of Inertia--(Inches)4
4 Angles
Primary Second-
ary
318 2190
318 2190
318 2190
318 2190
318 2190
337 2315
299 2050
131 2195
105 1745
96 1590
318 2190
318 2190
318 2190
318 2190
318 2190
318 2190
318 2190
131 2195
1605 1745
96 1544
13 5730
13 4170
13 3690
13 2860
13 2530
13 1580
13 5730
15 4830
13 4170
13 3690
13 2860
13 2530
13 1580
WILSON-MANEY-WIND STRESSES
TABLE 12.
NUMERICAL VALUES OF THE CONSTANTS IN THE EQUATIONS OF TABLE
3 FOR THE SYMMETRICAL THREE-SPAN TWENTY-STORY BENT SHOWN
IN FIG. 5.
All quantities are expressed in inches3
Values of Values of Values of
K(= ) K(= ) =( )] 22
for for for all members
Sfor for intersecting at the
1 25.8 25.8 30.5 37.3 183.8 258 4 206.8
2 35.6 35.6 21.4 29.2 184.8 243.4 284.8
3 35.4 35.5 21.4 26.2 184.4 237.2 283.6
4 35.4 35.5 19.5 26.2 168.6 223.2 283.6
5 29.4 30.4 19.5 23.8 156.6 208.2 239.2
6 29.4 30.4 14.1 17.2 144.4 183.4 239.2
7 28.7 30.0 12.8 15.7 140.4 177.0 234.8
8 28.7 30.0 7,7 9.4 115.0 146.4 234.8
9 21.1 26.1 7.7 9.4 99.8 138.6 188.8
10 21.1 26.1 7.7 9.4 95.2 126.2 188.8
11 18.8 19.9 7.7 9.4 90.6 113.8 154.8
12 18.8 19.9 7.7 9.4 89.6 111.2 154.8
13 18.3 18.6 7.7 9.4 88.6 108.6 147.6
14 18.3 18.6 7.7 9.4 80.6 100.6 147.6
15 14.3 14.6 7.7 9.4 72.6 92.6 115.6
16 14.3 14.6 7.7 9.4 70.2 89.8 115.6
17 13.1 13.2 7.7 9.4 67.8 87.0 105.2
18 13.1 13.2 7.7 9.4 67.8 87.0 105.2
19 13.1 13.2 7.7 9.4 67.8 87.0 105.2
20 13.1 13.2 7.7 9.4 41.6 60.6 105.2
50 ILLINOIS ENGINEERING EXPERIMENT STATION
pujH-qm.i .
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P4
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WILSON-MANEY-WIND STRESSES 51
P4
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ILLINOIS ENGINEERING EXPERIMENT STATION
pu-nnba
jo iaqma
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eq OSc
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eq rt. eq ot
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WILSON-MANEY-WIND STRESSES
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54 ILLINOIS ENGINEERING EXPERIMENT STATION
UoqT~fbR --g 0400 a400 04004
uo squracp oo ° o0
P™H-lI I 1 I
ý4
H
z
z
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cc
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eq-0 eq 04( -
c-c- eq~t CO
cc4c cceq S
t04c0- COCO
CO0004 00.. -o<
uo lnba
jo -oN
I
WILSON-MANEY-WIND STRESSES 55
pToHltqnba "ig4 § I 1, Q1m(ýC
jo b E ' u ion C o 0 C
-I' I Ilr
>1
z
r4
o
i
E.
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CC
H
C
P4
01
z
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I -I
z
N
c
N
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1. a
0
~0
01
0
01
6
01
,
01
z
00
0
01
01
a1
01
01
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z
v
1.1
01
C
Co
0
01
C
01
0
C
,10
C
C
01
01
uolubnba
1o -ON
01 llC -0
01Cý t
Ci
ILLINOIS ENGINEERING EXPERIMENT STATION
TABLE 14.
ELIMINATION OF THE UNKNOWN QUANTITIES IN THE EQUATIONS FOR THE SYMMETRI-
CAL THREE-SPAN TWENTY-STORY BENT SHOWN IN FIG. 5.
Left-Hand Member of Equation
Story No. 1 Story No. 2 Story No. 3 Story No. 4
R1 I A1 0B1 R,2 A2 OB2 R3 IOA3 0B3 R4 0At 004
Coefficients of Unknown Slopes and Ratios of Deflection to Story Height
-0.2491
-2.3741
-0.394(
2.1252
-1.980(
1 000(
1.000(
1.000(
1 000(
0
0
0
-0.2491
-0.394(
-3.820d
0.144,
3.426'
0.068(
-1.730(
1.0000
1.798(
-2.730(
1.000(
1
1
1
1
0
0
0
' 1.3791
1.3790
-1.3790
0
-0.6494
-4.000(
-3,000(
--0.6494
4.0000
-1.000C
-0.3611
-1.4652
-1. 0000
-3.0000
1.1040
-0.4652
2.0000
1
1
1 -
0
0
-0.4599
0.4599
-0.4599
0.2164
0.2322
1.0000
5.1912
-0.0158
0.7678
-4.1912
-0.0088
0.2812
-4.1912
0.6012
-0.2900
4.4724
-4.7924
-0.2627
-9.6140
-2.3962
9.3513
-7.2178
1
1
1
1
0
0
0
-0.4591
0.4591
-0.232:
1.0001
0.601!
0.2321
-1.232!
0.3981
0.129!
0.4511
0.3981
7.6571
-0.322Z
0.052
-7.258W
-0.291V
-0.1128
-3.6292
-0.179(
3.5161
-0.0191
-0.4872
1.0028
0.4681
-1.490C
1.0028
1
1
1
1
0
0
0
-2.9834
2.9834
2.9834
-2.9911
-2,9834
5.974f
6.4127
2.9874
-6.4127
3.4252
-0.6857
-0.4746
-4.0055
-3.0000
-0.2111
3.5309
-1.0055
-0.4511
-2.3700
-1.0028
-3.0000
1.0189
-1.3672
1.9972
0.9944
-0.9944
-0.9941
0.9944
-0.9944
-2.1378
-0.4972
2.1378
-1.6406
0.2286
0.2273
1.0000
5.2091
0.0013
-0.7727
-4.2091
0.0027
0.5186
-4.1971
0.6028
-0.5159
4.7157
-4,7999
0.9972
-0.9972
-0.4986
0.4986
-0.0691
1.0028
0.6045
0.0691
-1.0719
0.3983
0.1476
0.7194
0.3971
7.4200
-0.5718
0.3223
-7.0230
-3.0000
3.0000
2.9915
-3.0000
-2.9915
5.9915
1.000C
-1.0000
-0.9972
0.9972
-0.9972
-1.0000
Co-
efficient
of .0001
0.5667
0.0000
0.0000
0.5667
0.0000
0.2667
0.0000
-1.1082
0.0000
0.2667
1.1082
-1.1082
0.1483
-0.4059
-1.1082
0.0000
0.5543
0.7023
-1.1082
0.5021
-1.5095
-0.5541
2.0116
-0.9554
0.2151
0.1324
-0.9110
0.0000
0.0827
1.0434
-0.9110
0.1768
-0.6990
-0.9084
0.0000
0.8758
0.2094
-0.9084
WILSON-MANEY-WIND STRESSES
TABLE 14.-(Continued).
ELIMINATION OF THE UNKNOWN QUANTITIES IN THE EQUATIONS FOR THE SYMMETRI-
CAL THREE-SPAN TWENTY-STORY BENT SHOWN IN FIG. 5.
Left-Hand Member of Equation
Story No. 3 Story No. 4 Story No. 5 Story No. 6
R3 0A3 1 B3 R4 A w 01 I dB4 R Io A5 B R6 I A6 B6
Coefficients of Unknown Slopes and Ratios of Deflection to Story Height
-0.2681
-3.449(
-2.4033
3.1802
2.1344
1
1
1
1
0
0
0
-0.298C
-0.2357
-3.517E
-0.0623
3.2199
-0.0196
1.5086
1.0028
-1.5282
1.5086
1.0028
1
1
1
1
0
0
0
2.1881
2.9999
-2.1881
-2.9999
-0.6881
-1.4057
-4.0057
-3.0000
0.7176
1.5943
-1,0057
-0.4696
1.0568
-1.0028
-3.0000
-1.5264
2.0596
4.0568
1
1
1
0
0
-0.7294
-0.4993
0.7294
0. 4993
0.2294
0.233C
1.000(
4.7628
-0.0046
-4.529C
-3.7628
0.003C
-3.001H
-3.752C
0.5496
3.0048
0.7501
-3.5515
-1.9687
0.3642
-0.8754
-2.3329
1.2397
1
1
1
1
0
0
0
-0.5007
0.5007
0.2346
1.0028
0.5508
-0.2346
-0.3162
0.4520
0.1535
-0.2096
0.4507
7.0255
0.3631
-0.6603
-7.2351
-0.2379
-0.3206
-1.7836
0.0827
1.4630
-0.0355
1.1803
1.0340
-1.2158
-1.0695
1.0340
1
1
1
1
0
0
0
-2.4914
2.4914
2.4914
1.6514
2.4842
-2.5691
-1.6514
-0.8328
4.2205
1.0819
-0,4044
1.0404
1.4863
-1.4440
-0.6371
-1.1657
-4.0684
-3.000(
0.5286
3.4313
-1.0684
-0.4348
-3.2083
-1.0333
-3.0000
2.7735
-2.175C
1.9667
1
1
1
0
0
0.8302
-0.8302
-0.8302
-0.5503
-0.8279
0.5503
0.2776
-0,5503
-0.3605
0.1348
-0.1357
-0.4953
0.2705
0.2123
0.2182
1.0000
5.3260
-0.0059
-0.7877
-4.3260
0.0048
0.7365
-4.1842
-0.6415
-0.7316
4.9207
-4.8257
-0.2638
-2.2623
-2.4540
1.9985
2.1902
0.8564
-0.8564
-0.2111
0.2111
0.1703
1.0340
0.6632
-0.1703
-1.0340
0.3708
0.1401
0.9668
0.3586
7.6321
-0.8267
0.6082
-7.2735
-0.2981
-0.2796
-3.6986
-0.0185
3.4005
-3.0000
3.0000
2.9016
-3.0000
-2.9016
5.9016
1.3341
3.0004
-1.3341
-3.0004
1.0000
-1.0000
--0.9672
0.9672
-0.9672
-0.4447
-0.4918
0.4447
0.4918
1.0000C
-1.000(1
-0.5085
0.5085
.2
oas
Co-
efficient
of .0001
0.4564
-0.1532
-0.4548
0.6096
0.9112
0.1917
0.4270
-0.8546
0 0000
-0.2353
0.4270
-0.8546
0.1540
0 2830
-0.8521
0.0000
-0.1290
1.1352
0.2830
0.0846
0.5512
0.0697
-0.4666
0.4815
0.2000
0.3884
-0.9609
0.0000
-0.1884
1.1609
-0.9609
0.1550
-1.0855
-0.9294
0.0000
1.2405
-0.1561
-0.9294
0.4473
0.0718
-0.4726
0.3755
0.9199
ILLINOIS ENGINEERING EXPERIMENT STATION
TABLE 14.-(Continued).
ELIMINATION OF THE UNKNOWN QUANTITIES IN THE EQUATIONS FOR THE SYMMETRI-
CAL THREE-SPAN TWENTY-STORY BENT SHOWN IN FIG. 5.
Left-Hand Member of Equation.
Story No. 5 Story No. 6 Story No. 7 Story No. 8
R5 I A5 0B5 R6 I A6 0B6 R7 0A7 0B7 8 R A B8
Coefficients of Unknown Slopes and Ratios of Deflection to Story Height.
-0.0092
1.5526
1.0341
-1.5618
-1.0433
1.0341
1
1
1
1
0
0
0
-0.6671
-1.3691
-4.068&
-3.000(
0.702Z
3.400(
-1.0681
-0. 4497
-3.2592
-1.033(
-3.000(
2.8091
-2.226&
1.967(
1
1
1
0
0
0
Sa
0.232Z
1.0341
0.479(
-0.2322
-1.0341
0.5542
0.1487
0.9911
0.5362
6.5993
--0.8425
0.4549
-6.0631
-0.2998
-0.2044
-3.0827
-0.0955
2.7828
-0.0540
1.4695
1.0453
-1.5235
-1.0993
1.0453
1
1
1
1
0
0
0
-2.928&
2. 9281
2.832(
-2.960(
-2.832(
5.792C
1.2721
2.9451
-1.2721
-2.9451
-0.7192
-1.5553
-4.0909
-3.0000
0.8361
3.3717
-1.0909
--0.5489
-3.0672
-1.0436
-3.0000
2.5183
-2.0236
1.9564
1
1
1
0
0
0.976
-0.976
-0.944
0.944
-0.944'
-0.4241
-0.479,
0.4241
0.4799
0.239'
0.2534
1.0001
4.8921
-0.0131
-0.760:
-3.8924
0.009(
0.6914
-3.7234
0.426Z
-0.682(
4.415(
-4.1501
-0.2711
-2.181E
-2.121,1
1.910E
1.850M
0.9870
-0.9870
-0.5018
0.5018
0.2650
1.0453
0.4460
-0.2650
-1.0453
0.5993
0.1739
0.9510
0.5733
6.4233
-0.7781
0.3777
-5.8500
-0.3090
-0.1867
-2.9903
-0.1223
2.6813
0.2221
0.224f
1.000(
4.912(
-0.002(
-0.7771
-3.912(
0.001t
0.7451
-3.7810
0.463E
-0.7430
4.527C
-4.245C
-0.2648
-2.0335
-2.1586
1.7687
1.8938
1
1
1
1
0
0
0
1.0001
-1.000(
-0.956i
0.9561
-0.956j
-0.472E
-0.489(
0.4724
0.489(
1.0000
-1.0000
-0.5111
0.5111
Co-
efficient
Of .0001
0.1879
0.4200
-0.8929
0.0000
-0.2321
1.0808
-0.8929
0.1486
-1.0359
-0.8634
0.0000
1.1845
-0.1724
-0.8634
0.4216
0.0775
-0.4390
0.3441
0.8606
0.1946
0.4544
-0.7281
0.0000
-0.2599
0.9227
-0.7281
0.1706
-0.8394
0.6964
0.0000
1.0100
-0.1430
-0.6964
0.4011
0.0706
-0.3560
0.3304
0.7570
-3.0000
3.0000
2.8700
-3.0000
-2.8700
5.8700
1.4184
3.0006
-1.4184
-3.0006
WILSON-MANEY-WIND STRESSES 59
TABLE 14.-(Continued).
ELIMINATION OF THE UNKNOWN QUANTITIES IN THE EQUATIONS FOR THE SYMMETRICAL
THREE-SPAN TWENTY-STORY BENT SHOWN IN FIG. 5.
ZCc
49
50
V
W
49-50
49-V
V-W
51
52
53
X
51-52
52-53
53-X
54
55
56
54-55
54-56
57
58
Y
57-58
57-Y
Y-Z
59
60
61
a
59-60
60-61
61-a
62
63
64
62-63
62-64
Left-Hand Member of Equation
Story No. 7 Story No. 8 Story No. 9 Story No. 10
R7 OA7 I B7 R8 OAS OB8 R9 I A9I B9 RO OA10 OB10
Coefficients of Unknown Slopes and Ratios of Deflection to Story Height
1
1
1
1
0
0
0
-0.0640
1.4491
1.0453
-1.5131
-1.1093
1.0453
1
1
1
1
0
0
0
-0.7425
-1.6215
-4.090C
-3.000(
0.8797
3.3487
-1.0909
-0.5814
-3.018f
-1.043(
-3.000
2.4372
-1.975C
1.9564
1
1
1
0
0
0.2474
0.2643
1.0000
4.0073
-0.0169
-0.7526
-3.0073
0.0112
0.6784
-2.8769
0.2567
-0.6672
3.5553
-3.1336
-0.2738
-1.8002
-1 6019
1.5265
1.3281
1
1
1
1
0
0
0
0.2763
1.0453
0.2683
-0.2763
-1.0453
-0.7770
0.1826
0.9423
0.7433
5.1933
-0.7598
0.1990
-4.4500
--0.3117
-0.1008
-2.2747
-0.2110
1.9630
-0.1382
1.4779
1.2371
-1.0161
-1.3753
1.2371
1
1
1
1
0
0
0
-2.2056
2.2056
2.1100
-2.6100
-2.1100
4.7200
1.0685
2.4127
-1.0685
-2.4127
-0.7000
-1.8165
-4.4743
-3.0000
1.1165
3.7743
-1.4743
-0.6908
-2.7443
-1.1918
-3.0000
2.0535
-1.5525
1.8082
1
1
1
0
0
0.7352
-0.7352
-0.7036
0.7036
-0.7036
-0.3563
-0.3597
0.3563
0.3597
0.2334
0.2708
1.0000
4.7299
-0.0374
-0.7666
-3.7299
0.0231
0.5574
-3.0151
0.2950
-0.5342
3.5725
-3.3102
-0.2602
-2.3013
-1.8308
2.0411
1.5706
0.8700
-0.8700
-0.4447
0.4447
0.3348
1.2371
0.3649
-0.3348
-1.2371
0.8722
0.2072
0.8995
0.7051
5.6705
-0.6924
0.1945
-4.9654
-0.3372
-0.1253
-2.7462
-0.2119
2.4090
-3.0000
3.0000
2.4253
-3.0000
-2.4253
5.4253
1.5624
3.0000
-1.5624
-3.0000
1.000(
-1.000(
-0.8084
0.8084
-0.8084
-0.5207
-0.4471
0.5201
0.4471
1.0000
-1.0000
-0.5531
0.5531
1.000O
--1.000
-0.5531
0.5531
-0
Co-
efficient
of .0001
0.1729
0.4093
-0.6741
0.0000
-0.2363
0.8470
-0.6741
0.1562
-0.7636.
-0.6449
0.0000
0.9198
-0.1187
-0.6449
0.3774
0.0601
-0.3296
0.3173
0.7070
0.2079
0.5323
-0.8484
0.0000
--0.3245
1.0562
-0.8484
0.2008
-0.7681
-0.6859
0.0000
0.9689
-0.0822
-0.6859
0.4719
0.0529
-0.3793
0.4189
0.8512
60 ILLINOIS ENGINEERING EXPERIMENT STATION
TABLE 14.-(Continued).
ELIMINATION OF THE UNKNOWN QUANTITIES IN THE EQUATIONS FOR THE SYMMETRICAL
THREE-SPAN TWENTY-STORY BENT SHOWN IN FIG. 5.
.0
o
65
66
b
c
65-66
65-b
b-c
67
68
69
d
67-68
68-69
69-d
70
71
72
70-71
70-72
73
74
fC
73-74
73-e
e-f
75
76
77
g
75-76
76-77
77-g
78
79
80
78-79
78-80
Left-Hand Member of Equation
Story No. 9 Story No. 10 Story No. 11 Story No. 12
R 0OA9 OB9 R10 A10 0B10 R11 0All 0Bll R12 A12 0B12
Coefficients of Unknown Slopes and Ratios of Deflection to Story Height
1
1
1
1
0
0
0
-0.1039
1.534(
1.2371
-1.637S
-1.340S
1.2371
1
11
1
10
0
0
0
-0.7655
-1.9103
-4.4743
-3.000O
1.1448
3.7088
-1.4743
-0.6991
-2.7660
-1.1918
-3.0000
2.0668
-1.5742
1.8082
1
1
1
0
0
o. 2847
1.0000
4.6120
-0.0296
-0.7449
-3.5120
0.0180
0. 5555
-2.8392
0.2950
-0.5375
3.3947
-3.1342
-0.2601
-2.1566
-1.7333
1.8865
1.4732
1
1
1
1
0
0
0
0.3522
1.2371
0.3649
-0.3522
-1.2371
0.8722
0.2151
0.9226
0.7050
5.1954
-0.7075
0.2176
-4.4904
-0.3423
-0.1382
-2.4833
-0.2041
2.1310
-0.1082
1.4464
1.0585
-1.5547
-1.1668
1.0585
1
1
1
1
0
0
0
-2.6730
2.6730
2.1610
-2.2873
-2.1610
4.4483
1.3728
2.4602
-1.3728
-2.4602
-0.72771
-1.6700
-4.1170
-3.0000
0.9422
3.3893
-1.1170
-0.6061
-2.9051
-1.0553
-3.0000
2.2990
-1.8498
1.9447
1
1
1
0
0
0.8911
-0.8911
-3.7204
0.7204
-0.7204
-0.4576
-0.3984
0.4576
0.3984
0.2426
0.2705
1.0000
4.8195
-0.0278
-0.7574
-3.8195
0.0179
0.6492
-3.6083
0.3869
-0.6313
4.2575
-3.9952
-0.2746
-2.3018
-2.0547
2.0272
1.7801
0.7624
-0.7624
--0.4216
0.4216
0.2862
1.0585
0.4096
-0.2862
-1.0585
0.6499
0.1841
0.9070
0.6140
6.1913
-0.7235
0.2936
-5 5773
-0 3147
-0.1587
-2.8682
-0.1560
2.5535
-3.0000
3.0000
2.8343
-3.0000
-2.8343
5.8343
1.5323
3.0003
-1.5323
-3.0000
1.000(
-1.000(
-0.944E
0.9445
-0.9445
-0.5105
-0.4855
0.5109
0.485t
1.0000
-1.0000
-0.5143
0.5143
Co-
efficient
of .0001
0.2053
0.5420
-0.7771
0.0000
-0.3367
0.9824
-0.7771
0.2056
-0.7327
-0.6283
0.0000
0.9383
-0.1044
-0.6283
0.4540
0.0663
-0.3474
0.3877
0.8014
0.2055
0.5441
-0.7926
0.0000
-0.3385
0.9981
-0.7926
0.2178
-0.8555
-0.7488
0.0000
1.0733
-0.1067
-0.7488
0.4669
0.0577
-0.3851
0.4092
0.8519
--0.1038
1.534C
1.2371
--1.6378
--1.340
1.2371
1
1
1
1
0
0
0
0.25511
i. 000@
--1.0000
-0.5143
0.5143
WILSON-M.ANEY-WIND STRESSES
TABLE 14.-(Continued).
ELIMINATION OF THE UNKNOWN QUANTITIES IN THE EQUATIONS FOR THE SYMMETRICAL
THREE-SPAN TWENTY-STORY BENT SHOWN IN FIG. 5.
Left-Hand Member of Equation
Story No. 11 Story No. 12 Story No. 13 Story No. 14
R, 0All B11 R12 0A12 B B12 13 0A13 OBI3 R14 A14 OB14
Coefficients of Unknown Slopes and Ratios of Deflection to Story Height
-0.0769
1.4344
1.0585
-1.5113
-1.1355
1.0585
1
1
1
1
0
0
0
-0.7559
-1.6854
-4.1170
-3.0000
0.9294
3.3611
-1.1170
-0.6150
-2.9602
-1.0553
-3.0000
2.3452
-1.9049
1.9447
1
1
1
0
0
0.252(
0.273(
1.000(
4.766(
-0.021(
-0.748(
-3.766(
0.013C
0.6587
-3.5583
0.386E
-0.6449
4.217C
-3.9452
-0. 275C
-2.2137
-2.028E
1.9387
1. 7539
1
1
1
1
0
0
0
0.2889
1.0585
0.4096
-0.2889
-1.0585
0.6489
0.1911
0.9323
0.6131
6 0609
-0.7412
0.3192
-5.4478
-0.3160
-0.1676
-2.8015
-0.1485
2.4855
-0.0766
1.4173
1.0164
-1.4939
-1.0930
1.0164
1
1
1
1
0
0
0
-2.7592
5.5632
1.4484
2.8609
-1.4484
-2.8609
-0.7471
-1.6314
-4.0329
-3.0000
0.8842
3.2857
-1.0329
-0.5920
-3.0063
-1.0163
-3.0000
2.4143
-1.9900
1.9837
1
1
1
0
0
-0.9197
0.9197
-0.9197
-0.4828
-0.473(
0.4828
0.473(
0.249(
0.2697
1.000(
4.8417
-0.0207
-0.751(
-3.8417
0.013E
0 6871
-3.7802
0.414C
-0.6733
4.4673
-4.1942
-0.2789
-2.2413
-2.1143
1.9624
1.8354
-0.9347
-0.4807
0.4807
0.2741
1.0164
0.4208
-0.2741
-1.0164
0,5956
0.1835
0.9299
0.5861
6.3443
-0.7464
0.3438
-5.7582
-0.3092
-0.1728
-2.9030
-0.1364
2.5938
-3.0000
3.0000
2.9520
-3.0000
-2.9520
5.9520
1.4834
3.0000
-1.4834
-3.0000
1.0000
-1.0000
-0. 9840
0.9840
-0.9840
-0.4945
-0.4961
0.4945
0.4961
1.0000
-1.0000
-0.5042
0.5042
Co-
efficient
of .0001
0.2018
0.4786
-0.7128
0.0000
-0.2768
0.9147
-0.7128
0.1831
-0.8055
-0.6734
0.0000
0.9886
-0.1321
-0.6734
0.4216
0.0693
-0.3463
0.3522
0.7679
0.1816
0.4378
-0.6503
0.0000
-0.2562
0.8320
-0.6503
0.1715
-0.7612
-0.6399
0.0000
0.9327
.-01213
-0.6399
0.3863
0.0610
-0.3226
0.3254
0.7089
ILLINOIS ENGINEERING EXPERIMENT STATION
TABLE 14.-(Continued).
ELIMINATION OF THE UNKNOWN QUANTITIES IN THE EQUATIONS FOR THE SYMMETRICAL
THREE-SPAN TWENTY-STORY BENT SHOWN IN FIG. 5.
g1
97
98
n
o0
97-98
97-n
n-o
99
100
101
p
99-100
100-101
101-p
102
103
104
102-103
102-104
105
106
q
r
105-106
105-q
q-r
107
108
109
107-108
108-109
109-8
110
111
112
110-111
111-112
Left-Hand Member of Equation
Story No. 13 Story No. 14 Story No. 15 Story No. 16
R13 0A13 0B18 R14 I A14 I B14 R15 0A15 I B15 R16 A16 1 B16
Coefficients of Unknown Slopes and Ratios of Deflection to Story Height
-0.0695
1.4135
1.0164
-1.4830
-1.0859
1.0164
1
1
1
1
0
0
0
-0.7559
-1.6349
-4.0329
-3.0000
0.8790
3.2769
-1.0329
-0.5928
-3.0177
-1.0163
-3.0000
2.4249
-2.0014
1.9837
1
1
1
0
0
0.2520
0.2704
1.0000
4.4044
-0.0184
-0.7480
-3.4044
0.0124
0.6888
-3.3497
0.4140
-0.6765
4.0385
-3.7637
-0.2790
-2.0179
-1.8974
1.7389
1.6184
1
1
1
1
0
0
0
0.2747
1.0164
0.4208
-0.2747
-1.0164
0. 5956
0.1853
0.936C
0.586C
5.9148
-0.7507
0.3499
-5.3287
-0.3096
-0.1748
-2.6867
-0.1348
2.3771
--0.0775
1.4690
1.0210
-1.5465
-1.0985
1.0210
1
1
1
1
0
0
0
-2.3443
2.3443
2.3067
-2.3551
-2.3067
4.6618
1.1525
2.3505
-1.1525
-2.3505
-0.6624
-1.4525
-4.0419
-3.0000
0.7901
3.3795
-1.0419
--0.5109
-3.0762
-1.0206
-3.0000
2.5653
-2.0556
1.9794
1
1
1
0
0
0.7814
-0.7814
-0.7689
0.7689
-0.7689
-0.3842
-0.3877
0.3842
0.3877
0.2209
0.2396
1.0000
5.0771
--0.0186
-0.7790
-4.0771
0.0120
0.7092
-3.9933
0.5274
-0.6972
4.7025
-4.5207
-0.2718
-2.2875
-2.2836
2.0157
2.0118
-0.785C
-0.3953
0.39W5
0.244(
1.021C
0.5381
-0.2446
-1.021(
0.482S
0.1581
0.9294
0.472C
6.9832
-0.7713
0.4568
-6.5106
-0.300C
-0.2222
-3.2886
-0.0784
2.9882
-3.000C
3.0000
2.9383
-3.0000
-2.9383
5.9383
1.4294
3.0000
-1.4294
-3.000O
1.0000
-1.0000
-0.9795
0.9795
-0.9795
-0.4765
-0.4946
0.4765
0.4946
1.000(
-1.000C
-0.5052
5052
Co-
efficient
of .0001
0.1658
0.3863
-0.5711
0.0000
-0.2205
0.7369
-0.5711
0.1487
--0.6786
-0.5619
0.0000
0.8274
-0.1167
-0.5619
0.3412
0.0583
-0.2833
0.2829
0.6246
0.1627
0.3859
-0.6258
0.0000
-0.2232
0.7886
-0.6258
0.1444
-0.7178
-0.6130
0.0000
0.8622
-0.1048
-0.6130
0.3361
0.0510
-0.3097
0.2851
0.0457
WILSON-MANEY-WIND STRESSES 63
TABLE 14.-(Continued).
ELIMINATION OF THE UNKNOWN QUANTITIES IN THE EQUATIONS FOR THE SYMMETRICAL
THREE-SPAN TWENTY-STORY BENT SHOWN IN FIG. 5.
o
0
z;4a
113
114
t
11
113-114
113-t
t-u
115
116
117
v
115-116
116-117
117-v
118
119
120
118-119
118-120
121
122
w
x
121-122
121-w
w-x
123
124
125
y
123-124
124-125
125-y
126
127
128
126-127
126-128
1
1
1
1
0
0
0
-0.0381
1.4854
1.0211
-1.5243
-1.0591
1.0211
1
1
1
1
0
0
0
-0.7090
-1.4913
-4.0411
-3.000(
0.7821
3.332=
-1.0410
-0.513,
-3.1444
-1.020(
-3.000(
2.6312
-2.123E
1.9794
1
1
1
0
0
0.2364
0.2456
1.000(
4.9092
-0.009R
-0.763C
-3.9092
0.0062
0.7204
-3.8290
0.5281
-0.7142
4.5494
-4.3571
-0.2714
-2.1422
-2.2013
1.8708
1.9299
1
1
1
1
0
0
0
0.2511
1.0210
0.5385
-0.2511
-1.0210
0.4825
0.1647
0.9633
0.4726
6.8055
-0.7986
0.4907
-6.3329
-0.3035
--0.2310
-3.1993
-0.0724
2.8958
-0.0387
1.5027
1.0092
-1.5414
-1.0479
1.0092
1
1
1
1
0
0
0
-2.748(
2.748(
2.691(
-2.7161
-2.691(
5.4077
1.2673
2.731E
-1.2673
-2.7318
-0.6774
-1.4177
-4.0214
-3.0000
0.7403
3.344C
-1.0214
-0.4810
-3.1958
-1.0136
-3.0000
2.7148
-2.1822
1.9864
1
1
1
0
0
0.9161
-0.9161
-0.8973
0.8973
-0.8973
-0.4225
-0.4533
0.4225
0.4533
0.2259
0.2353
1.0000
5.1753
-0.0094
-0.7741
-4.1755
0.0061
0.7398
-4.1437
0.5833
-0.7337
4.8835
-4.7270
-0.2703
-2.2380
-2.3797
1.9677
2.1094
0.9054
-0.9054
-0.4574
0.4574
0.2374
1.0092
0.5878
-0.2374
-1.0092
0.4214
0.1542
0.9630
0.4182
7.3021
-0.8088
0.5448
-6.8839;
-0.2979
-0.2496
-3.4655
-0.0483
3.1676
-3.0000
3.000C
2.9773
-3.0000
-2.9773
5.9773
1.3644
3.0093
-1.3644
-3.0093
1.0000
-1.0000
-0. 924
0.9924
-0.9924
-0.4548
-0.4996
0.4548
0.4996
1.0000
-1.0000
-0.5035
0.5035
Left-Hand Member of Equation
Story No. 15 Story No. 16 Story No. 17 Story No. 18
R15 0A15 B15 R16 I A16 B16 1R7 A17 IA 17 RB17 RI8 A18 I B18
Coefficients of Unknown Slopes and Ratios of Deflection to Story Height
1-0
Co-
efficient
of .0001
0.1414
0.3210
-0.5209
0.0000
-0.1796
0.6624
-0.5209
0.1178
-0.6249
-0.5102
0.0000
0.7427
-0.1147
-0.5102
0.2823
0.0540
-0.2579
0.2282
0.5399
0.1220
0.2802
-0.4542
0.0000
-0.1582
0.5762
-0.4542
0.1028
-0.5507
-0.4507
0 0000
0.6535
-0.0999
-0.4507
0.2407
0.0458
-0.2269
0.1949
0.4677
1
1
1
1
0
0
0
1.000(
--1.0000
-0.5035
0.503
ILLINOIS ENGINEERING EXPERIMENT STATION
TABLE 14.-(Continued).
ELIMINATION OF THE UNKNOWN QUANTITIES IN THE EQUATIONS FOR THE SYMMETRICAL
THREE-SPAN TWENTY-STORY BENT SHOWN IN FIG. 5.
zi
129
130
z
A'
129-130
129-s
z-A'
131
132
133
B'
131-132
132-133
133-B'
134
135
136
134-135
134-136
137
138
C'
D'
137-138
137-C'
C'-D'
139
140
141
E'
139-140
140-141
141-E'
142
143
144
142-143
142-144
145
146
F'
G'
145-146
145-F'
F'-G'
147
148
149
H'
Left-Hand Member of Equation
Story No. 17 Story No. 18 Story No. 19 Story No. 20
R17 0A17 B17 O R18 I 0A18 B18 R19 0A19 B19 R20 A20 0B20
Coefficients of Unknown Slopes and Ratios of Deflection to Story Height
-0.0245
1.5017
1.0092
-1.5262
-1.0337
1.0092
1
1
0
0
0
-0.6935
-1.4268
-4.0214
-3.0000
0.7333
3.3279
-1.0214
-0.4805
-3.2197
-1.0120
-3.0000
2.7392
-2.2077
1.9880
1
1
1
0
0
0.2311
0.2369 0.2387
1.0000 1.0092
5.1755 0.5878
--0.0057
-0.7689
-4.1755
0.0037
0.7438
-4.1373
0.5833
-0.7401
4.8811
-4.7206
--0.2702
-2.2110
-2.3745
1.9408
2.1043
1
1
1
1
0
0
0
-0.2387
-1.0092
0.4214
0.1564
0.9764
0.417C
7.3021
-0.820C
0.5588
-6.884C
-0.2993
-0.2531
-3.4620
-0.0462
3.1632
-0.0238
1.5034
1.0092
-1.5272
-1.0330
1.0095
1
1
1
1
0
0
0
-3.0000
3.0000
2.9727
-3.0000
-2.9727
5.9727
1.3465
3.0042
-1.3465
-3.0042
-0.6938
-1.4277
-4.0214
-3.0000C
0.7339
3.3276
-1.0214
-0.480U
-3.2214
-1.0121
-3.000 C
2.740E
-2.2003
1.9875
1
1
1
0
0
1.0000
-1.0000
-0.9909
0.9909
-0.9909
-0.4485
-0.4980
0.4485
0.4080
0.2311
0.2367
1.0000
5.1755
-0.0056
-0.7689
-4.1755
0.0036
0.7444
-4.1377
0.5833
-0.7407
4.8821
-4.7210C
-0 2703
-2.2099
-2.374Q
1.9396
2.1047
1
1
1
1
0
0
0
1.0000
-1.0000C
-0.9900
0.9900
-0.9900
-0. 4485
--0.4985
0.4485
0.4981
0.231c
0.236i
1.000(
3.1759
-0.005(
-0.7681
-2.1751
0.0031
0.7442
-2.156W
0.5834
-1.0000
-0.5031
0.5031
0.23900
1.0092
0.5878
-0.2390
-1.0092
0.4214
0.1565
0.977C
0.417(l
5.3033
Co-
efficient
of .0001
0.0991
0.2217
-0.3412
0.0000
-0.1226
0.4403
-0.3412
0.0804
-0.4260
-0.3381
0.0000
0.5063
-0.0879
-0.3381
0.1849
0.0398
-0.1701
0.1451
0.3549
0.0747
0.1687
-0.2275
0.0000
-0.0939
0.3022
-0.2275
0.0615
-0.2926
-0.2254
0.0000
0.3541
-0.0672
-0.2254
0.1292
0.0304
-0.1134
0.0988
0.2426
0.0509
0.1153
-0.1137
0.0000
-0.0643
0.1647
-0.1137
0.0421
-0.1594
-0.1127
0.0000
1.0000
-1.0000
-0.5030
0.5030
0.2390
1.0092
0.5878
-0.2390
-1.0092
0.4214
0.1565
0.9769
0.4176
7.3021
-0.8204
0.5594
-6.8846
-0.2993
-0.2532
-3.4632
-0.0461
3.1639
-0.0238
1.5034
1.0092
-1.5272
-1.0330C
1.0092
1
1
1
1
-3.000(
3.000(
2.9721
-3.000(
-2.9721
5.972;
1.3451
3.004(
-1.3451
-3.0041
-0.6931
-1.4271
-4.0214
-3.0001
0.7341
3.3271
-1.0214
-0.4801
-3.221:
-1.012!
-3.0001
I
WILSON-MiANEY-WIND STRESSES
TABLE 14.-(Continued).
ELIMINATION OF THE UNKNOWN QUANTITIES IN THE EQUATIONS FOR THE SYMMETRICAL
THREE-SPAN TWENTY-STORY BENT SHOWN IN FIG. 5.
Left-Hand Member of Equation
Story No. 19 Story No. 20
Rg M 0A19 0B19 R20 0A20 0BSO
S----Co-
efficient
;ZS Coefficients of Unknown Slopes and Ratios of Deflection to Story Height of .0001
147-148 0 2.7411 -0.7406 -0.8205 0.2015
149-149 0 -2.2095 2.9005 0.5595 -0.0467
149-H' 0 1.9878 -2.7397 -4.8857 -0.1127
150 1 -0.2702 -0.2993 0.0735
151 1 -1.3129 -0.2531 0.0211
152 1 -1.3783 -2.4580 -0.0567
150-151 0 1.0427 -0.0462 0.0524
150-152 0 1.1081 2.1587 0.1302
153 1 -0.0443 0.0502
154 1 1.9480 0.1175
153-154 * -1.9923 -0.0673.
155 1.0000 0.0338
ILLINOIS ENGINEERING EXPERIMENT STATION
TABLE 15.
DETERMINATION OF THE CHANGES IN THE SLOPES AND OF THE RATIOS
OF DEFLECTION TO STORY HEIGHT IN THE SYMMETRICAL THREE-
SPAN TWENTY-STORY BENT SHOWN IN FIG. 5.
The equations are taken from Table 14.
No. of Right-Hand
Equa- Left-Hand Member of Equation Member of
tion Equation
The first line in each group is the algebraic form of the equation. Coefficient of
The successive lines are the numerical values of the terms. .0001
155 0BW .0338
153 A20 - .443B20 = .0502
.0443 x .00000338 = .0015
0A20 = .0517
150 R0 - .2702 A20 -.2993 B2= .0735
.2702 x .00000517 = .0140
.2993 x .00000338 = .0101
R2o = .0976
147 OB9I- .4806 R20 + .0037 OA20 + .1565 OB2o = .0421
.4806 x .00000976 = 0469 .0890
.0037 x.,00000517 = .0002
.1565 x .00000338 = 0053 -.0055
0B19 = .0835
145 0A19 - .0238 0B11 - .6938 R20 + .2313 0A20 = .0509
.0238 x .00000835 = .0020
.6938 x .00000976 = .0676 .1205
.2313 x .00000517 = -.0120
9A19 = .1085
142 R,9 - .2703 OA19 - .2993 B19 =.1292
.2703 x .00001085 = .0294
.2993 x .00000835 = .0250
R19 = .1836
139 OB1- .4806 R1 + .0037 0At + .15660B19 = .0615
.4806 x .00001836 = .0881 .1496
.0037 x .00001085 = .0004
.1565 x .00000835 = .0131 -.0135
0B18 = .1361
137 A18 - .0238 B18 - .6938 R1 + .2311 OA19 = .0748
.0238 x .00001361 = .0032
.6938 x .00001836 = .1272 .2052
.2311 x .00001085 = -.0251
0A18 = 1801
134 R18 - 2702 06A1 - .2994 0B18 = .1849
.2702 x .00001801 = .0486
.2994 x .00001361 0407
R18 = .2742
WILSON-M'ANEY-WIND STRESSES 67
TABLE 15.-(Continued).
DETERMINATION OF THE CHANGES IN THE SLOPES AND OF THE RATIOS
OF DEFLECTION TO STORY HEIGHT IN THE SYMMETRICAL THREE-
SPAN TWENTY-STORY BENT SHOWN IN FIG. 5.
No. of Right-Hand
Equa- Left-Hand Member of Equation Member of
tion Equation
The first line in each group is the algebraic form of the equation. Coefficient of
The successive lines are the numerical values of the terms. .0001
131
0B17 - .4805R18 + .0038 0A18 + .1564 0B18 = .0804
.4805 x .00002742 = .1319 .2123
.0038 x .00001801 = .0007
.1564 x .00001361 = .0213 -.0220
OB17 .1903
129 0A17 - .0246 0B7 - .6935R1, + .2311 0A18 .0991
.0246 x .00001903 = .0047
.6935 x .00002742 = .1905 2943
.2311 x .00001801 = -.0416
0A17 .2527
126 R17 - 2703 OA17 - .2979 0B17 .2407
.2703 x .00002527 = .0682
.2979 x .00001903 = .0566
R17 - .3655
123 B16 - .4810R17 + .0061 0A17 + .1542 07 = .1028
.4810 x .00003655 = .1758 .2786
.0061 x .00002527 = .0015
.1542 x .00001903 = .0294 -.0309
0Bs1 = .2477
121 0A,16 - .0387 OB16 - .6774R17 + .2259 0A17 .1220
.0387 x .00002477 = .0096
.6774 x .00003655 = .2473 .3789
.2259 x .00002527 = -.0570
0A16 .3219
118 R16 - .2714 0A16 - .3035 0B16 .2823
.2714 x .00003219 = .0873
.3035 x .00002477 = .052
R16 = 4448
115 OB15 - .5132R16 + .0062 0A6 + .1647 B16 = .1178
.5132 x .00004448 = .2280 .3458
.0062 x .00003219 = .0020
.1647 x .00002477 = .0408 -.0428
0B15 = .3030
113 0A15 - .0389 0B15 - .7091R1 + .2364 A1 = .1414
.0389 x .00003030 = .0118
.7091 x .00004448 = .3180 .4712
.2364 x .00003219 = -.0762
OA15 = .3950
110 R15 - .2718 0A1 - .3006 01 = .3361
.2718 x .00003950 = .1072
.3006 x .00003030 = .0912
R, = .5345
107 li - .5109R15 + .0120 0A15 + .1581 OB15 .1444
.5109 x .00005345 = .2730 .4174
.0120 x .00003950 = .0047
.1581 x .00003030 = .0480 - 0527
B1 = 3647
ILLINOIS ENGINEERING EXPERIMENT STATION
TABLE 15.-(Continued).
DETERMINATION OF THE CHANGES IN THE SLOPES AND OF THE RATIOS
OF DEFLECTION TO STORY HEIGHT IN THE SYMMETRICAL THREE-
SPAN TWENTY-STORY BENT SHOWN IN FIG. 5.
No. of Right-Hand
Equs- Left-Hand Member of Equation Member of
tion Equation
The first line in each group is the algebraic form of the equation. Coefficient of
The successive lines are the numerical values of the terms. .0001
105
1 A14 - .0775 OB14 - .6624 R, + .2210 OA15 = .1627
.0775 x .00003647 = .0283
.6624 x .00005345 = .3541 .5451
.2210 x .00003950 = -.08t3
0A14 = A4578
102 R,. - .2790 OA14 - .3096 0BH = .3412
.2790 x .00004578 = .1278
.3096 x .00003647 = .1160
R1 = .5850
99 6B13 - .5928R1 + .0124 A11 + .1853 B1 = .1487
.5928 x .00005850 = .3470 .4957
.0124 x .00004578 = .0057
.1853 x .00003647 = .0676 -.0733
0B13 = .4224
97 A13 - .0695 B13 - .750R14 + 2520 0A1R = .1658
.0695 x .00004224 = .0294
.7560 x .00005850 = .4420 .6372
.2520 x .00004578 = -.1154
OA13 = .5218
94 R1, - .2789 0A1, - .3092 0B,3 - .3864
.2789 x .00005218 = .1458
.3092 x .00004224 = .1308
Rl I = .6630
91 OB - .5920 R13 + .0138 OA13 + .1835 OB13 = .1715
.5920 x .00006630 = .3921 .5636
.0138 x .00005218 = .0072
.1835 x .00004224 = .0775 -.0847
0 12 .4789
89 0A12 - .0766 0B12 - 7472 R13 + .2490 0A13 .1816
.0766 x .00004789 = .0366
.7472 x .00006630 = .4960 .7142
.2490 x .00005218 = -.1300
0AlA = .5842
86 R - .2750 0A1, - .3160 -0,, = .4216
.2750 x .00005842 = .1603
.3160 x .00004789 = .1511
RI, = .7330
83 0otI - .6150 R12 + .0139 0A12 + .1912 02 = .1831
.6150 x .00007330 = .4512 .6343
.0139 x .00005842 = .0081
.1912 x .00004789 = .0916 -.0997
0B11 = .5346
81 0A11 - .0770 ODBI - .7560 R12 + .2520 OA12 .2018
.0770 x .00005346 = .0411
.7560 x .00007330 = .5550 .7979
.2520 x .00005842 = -.1471
0Al1 = .6508
78 R1, - .2748 011 - .3147 B11 = .4669
.2746 x .00006508 = .1790
.3147 x .00005346 = .1681
R11= .8140
WILSON-1MANEY-WIND STRESSES
TABLE 15.-(Continued).
DETERMINATION OF THE CHANGES IN THE SLOPES AND OF THE RATIOS
OF DEFLECTION TO STORY HEIGHT IN THE SYMMETRICAL THREE-
SPAN TWENTY-STORY BENT SHOWN IN FIG. 5.
No. of
Equa-
tion
75
73
70
67
65
62
59
57
54
51
Left-Hand Member of Equation
The first line in each group is the algebraic form of the equation.
The successive lines are the numerical values of the terms.
0Bo - .6061 R, + .0179 0A11 + .1841 O1 =
.6061 x .00008140
.0179 x .00006508 =
.1841 x .00005346
0B10 =
0A10 - .1082 B10 - 7278 R + .2426 0All
.1082 x .00006018 =
.7278 x .00008140 =
.2426 x .00006508
0A10
R10 - .2601 0A10 - .3423 1o =
.2601 x .00007051 =
.3423 x .00006018
RI0 =
0B9 - .6991 Rio + .0181 0A10 + .2151 0B10 =
.6991 x .00008432 =
.0181 x .00007051
.2151 x .00006018
B9 =
09 - .1038 0,, - .7655 R, + .2551 0A10
.1038 x .00006526
.7655 x .00008432 =
.2551 x .00007051
0A =
R9 - .2602 0A9 - .3372 0B=
.2602 x .00007390 =
.3372 x .00006526
R=
0BS - .6908 R, + .0231 0.A + .2072 0B=
.6908 x .00008841 =
.0231 x .00007390 =
.2072 x .00006526
0BS =
0A8 - .1382 0B - .7000R9 + .2334 04A=
.1382 x .00006586
.7000 x .00008841
.2334 x .00007390
R8 - .2738 0OA - .3118 0BS
.2738 x .00007446
.3118 x .00006586
0B7 - .5814 R, + .0112 0A. + .1826 0B8
.5814 x .00007867
.0112 x .00007446
.1826 x .00006586
0B7 :
Right-Hand
Member of
Equation
Coefficient of
.0001
.2178
.4940 .7118
.0116
.0984 -.1100
.6018
2055
6651
.5925 .8631
-.1580
.7051
.4540
.1833
.2059
.8432
.2056
.5890 .7946
.0127
.1293 -.1420
.6526
2053
.0677
.6460 .9190
-.1800
.7390
.4719
.1922
.2200
.8s41
.2008
.6100 .8108
.0171
.1351 -.1522
6586
.2079
.0910
.6182 .9171
-.1725
.7446
.3774
.2038
.2055
.7867
.1562
.4570 .6132
.0083
.1203 -.1286
.4846
ILLINOIS ENGINEERING LEXPERIMENT STATION
TABLE 15.-(Continued).
DETERMINATION OF THE CHANGES IN THE SLOPES AND OF THE RATIOS
OF DEFLECTION TO STORY HEIGHT IN THE SYMMETRICAL THREE-
SPAN TWENTY-STORY BENT SHOWN IN FIG. 5.
No. of Right-Hand
Equa- Left-Hand Member of Equation Member of
tion Equation
The first line in each group is the algebraic form of the equation. Coefficient of
The successive lines are the numerical values of the terms. .0001
49
A7 - .0640 0B7 - .7422 R8 + .2474 0A8 = .1729
.0640 x .00004846 = .0310
.7422 x .00007867 = .5845 .7884
.2474 x .00007446 = -.1840
A7= .6044
46 R7 - .2711 9A7 - .3090 0B7= .4011
.2711 x .00006044 = .1639
.3090 x .00004846 = .1497
R7 = .7147
43 OB6 - .5489 R7 + .0090 0A7 + .1739 9B7 = .1706
.5489 x .00007147 = .3920 .5626
.0090 x .00006044 = .0054
.1739 x .00004846 = .0842 -.0896
OB6 = .4730
41 OA6 - .0540 OB6 - .7192R7 + .2397 OA7 = .1946
.0540 x .00004730 = .0256
.7192 x .00007147 = .5140 .7342
.2397 x .00006044 = -.1448
0A6 = .5894
38 R6 - .2648 0A6 - .3000 OB, = .4216
.2648 x .00005894 = .1561
.3000 x .00004730 = .1420
R6 = .7197
35 0B5 - .4496R6 + .0013 OA6 + .1487 OBO = .1486
.4496 x .00007197 = .3240 .4726
.0013 x .00005894 = .0008
.1487 x .00004730 = .0704 -.0712
OB5 = .4014
33 8A6 - .0092 .B5 - .6676 R + .2225 OA6 = .1879
.0092 x .00004014 = .0037
.6676 x .00007197 = .4802 .6718
.2225 x .00005894 = -.1310
O -5 = .5408
30 R5 - .2638 A5 - .2981 B5 .4473
.2638 x .00005408 = .1424
.2981 x .00004014 = .1196
R5 = .7093
27 0B4 - .4348 R, + .0048 OA, + -1401 0B5 = .1550
.4348 x .00007093 = .3080 .4630
.0048 x .00005408 = .0026
.1401 x .00004014 = .0563 -.0589
0B4 = .4041
25 01 , - .0355 01 - .6371 R5 + .2123 OA = .2000
.0355 x .00004041 = .0143
.6371 x .00007093 = .4515 .6658
.2123 x .00005408 = -.1150
A4 = .5508
WILSON-MANEY-WIND STRESSES
TABLE 15.-(Continued).
DETERMINATION OF THE CHANGES IN THE SLOPES AND OF THE RATIOS
OF DEFLECTION TO STORY HEIGHT IN THE SYMMETRICAL THREE-
SPAN TWENTY-STORY BENT SHOWN IN FIG. 5.
No. of Right-Hand
Equa- Left-Hand Member of Equation Member of
tion. Equation
The first line in each group is the algebraic form of the equation. Coefficient of
The successive lines are the numerical values of the terms. .0001
22 R4 + .3642 0t4 - .3206 0B, - .4044R5 + .1348 5 = .5512
.3206 x .00004041 = .1295
.4044 x .00007093 = .2862 .96C9
.3642 x .00005508 = .2008
.1348 x .00005408 = .0729 -.2737
R4 = .6932
19 OB3 - .4696 R4 + .0030 0At + .1535 0B4 = .1540
.4696 x .00006932 = .3257 .4797
.0030 x .00005508 = .0017
.1535 x .00004041 = .0620 -.0637
OB3 = .4160
17 OA - .0196 013 - .6881 R4 + .2294 0A4 = .1917
.0196 x .00004160 = .0082
.6881 x .00006932 = .4770 .6769
.2294 x .00005508 = -.1263
0A3 = .5506
14 R3 - .2689 0A3 - .2980 0B3 = .4564
.2689 x .00005506 = .1481
.2980 x .00004160 = .1240
R3 = .7285
11 0O, - .4511R, + .0027 0A3 + .1476 B = .1768
.4511 x .0000-285 = .3285 .5053
.0027 x .00005506 = .0015
.1476 x .00004160 = .0614 -.0629
0B2 = .4424
9 0A2 - .0191 0B2 - .6857 R + .2286 0A3 = 2151
.0191 x .00004424 = .0085
.6857 x .00007285 = .4990 .7226
.2286 x .00005506 = -.1259
0A2 = .5967
6 R2 - .2627 0A2 - .2918 OB2 = .5021
.2627 x .00005967 = .1565
.2918 x .00004424 = .1291
R = .7877
3 0 - .3612 R - .0088 OA2 + .1292 B2 .1483
.3612 x .00007877 = .2845
.0088 x .00005967 = .0053 .4381
.1292 x .00004424 = -.0572
OR1= .3809
1 OA1 + .0680 9B, - .6494R2 + .2164 OA2 .2667
.6494 x .00007877 = .5110 .7777
.0680 x .00003809 = .0259
.2164 x .00005967 = .1291 -.1550
0A1 = .6227
A R1 - .2495 A1 - .2495 0B, = .5667
.2495 x .00006227 = .1553
.2495 x .00003809 R .0951
R1 -- .8171
ILLINOIS ENGINEERING EXPERIMENT STATION
TABLE 16.
VALUES OF R AND 0 FOR THE SYMMETRICAL THREE-SPAN TWENTY-
STORY BENT SHOWN IN FIG. 5, AND THE FUNCTIONS OF THESE
VALUES THAT OCCUR IN THE EQUATIONS USED TO DETERMINE THE
MOMENTS IN THE COLUMNS AND GIRDERS.
For Column A* For Column B
Sz I
1 8171 .6227 .3809 2.4413 1.2454 .7618 1.6263 1.3845 1.1427 1.8186 1.1959 2.0604 1.6795 1
2 .7877 .5967 .4424 2.3631 1.1934 .8848 1.6358 1.4815 1.3372 .5210 .5470 1.1589 1.0974 2
3 .7285 .5506 .4160 2.1855 1.1012 8320 1.5172 1.3826 1.2480 .4415 .4870 .8847 .9111 3
4 .6932 .5508 .4041 2.0798 1.1016 .8082 1.5057 1.3590 1.2123 .4276 .4274 .8435 .8554 4
5 .7093 .540 .4014 2.1279 1.0816 .8028 1.4830 1.34361.2042 .4855 .4955 .9183 .9210 5
6 .7197 .5894 .4730 2.1591 1.1788 .9460 1.6518 1.5354 1.4190 .4881 .4395 .8833 .8117 6
7 .7147 .6044 .4846 2.1441 1.2088 .9692 1.6934 1.5736 1.4538 .3609 .3459 .7135 .7019 7
8 .7867 .7446 .6586 2.3601 1.4892 1.3172 2.1478 2.0618 1.9758 .4067 .2665 .7323 .5583 8
9 .8841 .7390 .6526 2.6523 1.47801.3052 2.1306 2.0442 1.9578 .4241 .4297 .6825 .6885 9
10 .8432 .7051 .6018 2.5296 1 4102 1.2036 2.0120 1.9087 1.8054 .3465 .3804 .6226 .6734 10
11 .8140 .6508 .5346 2.4420 1.3016 1.0692 18362 1 7200 1.6038 .3810 .4353 .7038 .7710 11
12 .7330 .5842 .4789 2.1990 1.1684 .9578 1.6473 1.54201.4367 .3132 .3798 .6509 .7066 12
13 .6630 .5218 .4224 1. 1.0436 .8448 1.4660 1.3666 1.2682 .2988 .3612 .6088 .6653 13
14 .5850 .4578 .3647 1.7550 -.9156 .7294 1.2803 1.18721.0941 .2536 .3176 .5455 .6032 14
15 .5345 .3950 .3030 1.6035 .7900 .6060 1.0930 1.0010 .9090 .2929 .3557 .5711 .6328 15
16 .4448 .3219 .2477 1.3344 6438 .4954 .8915 .8173 .7431 .2225 .2956 .4807 .5360 16
17 .3655 .2527 .1903 1.0965 .5054 .3806 .6957 .6333 .5709 .2000 .2692 .4108 .4682 17
18 .2742 .1801 .1361 .8226 .3602 .2722 .4963 .4523 .4083 .1371 .2097 .3059 .3601 18
19 .1836 .1085 .0335 .5503 .2170 .1670 .3005 .2755 .2505 .0821 .1537 .1951 .2477 19
20 .0976 .0517 .0338 .2923 .1034 .0676 .1372 .1193 .1014 .0231 .0809 .0920 .1417 20
* N represents the number of the story in question, and (N-1) the story below.
WILSON-MANEY-WIND STRESSES
TABLE 17.
VALUES OF K FOR THE COLUMNS AND THE GIRDERS OF THE SYMMETRI-
CAL THREE-SPAN TWENTY-STORY BENT SHOWN IN FIG. 5, AND THE
FUNCTIONS OF THESE VALUES THAT OCCUR IN THE EQUATIONS
USED TO DETERMINE THE MOMENTS IN THE COLUMNS AND GIRDERS.
K for Girder .0001 x 2EK for
at Top of K for Girder at Top of .0001 x 2EK for
Story in Story in
S Bay Bay Column Column Bay Bay Column Column
a b A B a b A B
1 30.5 37.3 25.8 25.8 177000 216200 149600 149600
2 21.4 29.2 35.6 35.6 124100 169400 206300 206300
3 21.4 26.2 35.4 35,5 124100 152000 205 500 205800
4 19.5 26.2 35.4 35.5 113050 152000 205 500 205 800
5 19.5 23.8 29.4 30.4 113050 138050 170500 176300
6 14.1 17.2 29.4 30.4 81750 99700 170500 176300
7 12.8 15.7 28.7 30.0 74250 91100 166 500 174000
8 7.7 9.4 28.7 30.0 44600 54500 166500 174000
9 7.7 9.4 21.1 26.1 44 600 54 500 122 400 151300
10 7.7 9.4 21.1 26.1 44600 54500 122400 151300
11 7.7 9.4 18.8 19.9 44600 54500 109000 115400
12 7.7 9.4 18.8 19.9 44600 54500 109000 115400
13 7.7 9.4 18.3 18.6 44600 54500 106100 107900
14 7.7 9.4 18.3 18.6 44 600 54 500 106100 107 900
15 7.7 9.4 14.3 14.6 44600 54500 82900 84600
16 7.7 9.4 14 3 14.6 44600 54500 82900 84600
17 7.7 9.4 13.1 13.2 44600 54500 76000 76 500
18 7.7 9.4 13.1 13.2 44600 54500 76000 76500
19 7.7 9.4 13.1 13.2 44600 54500 76000 76500
20 7.7 9.4 13.1 13.2 44600 54500 76000 76500
ILLINOIS ENGINEERING EXPERIMENT STATION
TABLE 18.
MOMENTS AT THE ENDS OF THE COLUMNS AND GIRDERS OF THE SYM-
METRICAL THREE-SPAN TWENTY-STORY BENT SHOWN IN FIG. 5.
Moments are expressed in inch-pounds.
Moment in Girder Moment in
Moment in Column A* Moment in Column B at Top of Story a op
in Bay a of Story in
2 113 000 107 500 226 500 239 000 203 000 184 000 226 000
3 100 300 90 700 187 500 182 000 187 100 171 700 189 600
4 87 800 87 800 176 100 173 500 170 500 153 600 184 400
S + II + g I£
5 84 500 82 800 162 300 162 000 107 700 152 000 106 000
7 57 600 60 000 122 000 124 000 125 00 117 000 132 500
9 52 550 51 900 104 200 103 300 95 000 91 100 100 800
1 17846 5800 272 000 102 000 308 200 2879 800 245 000 2478 500
2 11347 500 107500 22689 500 23900 81203 9000 184000 226 000
3 100 4300 9034 1700 187 500 182000 187100 178 700 189 3600
4 87 3800 8731 7800 761 800 17365 600 17065 300 15360 800 184400
5 84300 82800 1623 100 58 62000 167700 52 800 159 6000
6 75000 83200 143300 155700 135000 125600 141500
716 24 450 18 450 45 400 4000 1 00 125900 11736 4000 132 500
8 44400 67700 97200 127500 95800 92000 107800
9 520 4550 15 41900 10435 8200 1031 4300 95 000 91128 200 106 1800
18 15 00 42400 102000 94200 89600 85000 08500
19 11 47500 4150 18 900 14 930 1381200 81900 76612 250 13 6500
12 41400 34100 81500 7100 73 400 685 32600 78 300
13 38300 1700 the71800 65600number 65 300 60 800 69100
14 33700 26900 65100 58900 5700 52800 59600
15 29420 24300 53600 48350 48700 49600 49600
16 24450 18450 45400 40600 39700 36400 40500
17 20450 15400 35800 31400 31000 28200 31100
18 15920 10430 27600 23 400 22100 20150 22300
19 11680 6 240 18900 14930 13380 12250 13650
20 6150 1750 10830 7040 6110 5320 5530
*N represents the number of the story in question, and (N-i) the story below.
WILSON-MlANEY-WIND STRESSES
TABLE 19.
DIRECT STRESSES IN THE COLUMNS, AND THE SHEARS IN THE COLUMNS
AND GIRDERS OF THE SYMMETRICAL THREE-SPAN TWENTY-STORY
BENT SHOWN IN FIG. 5.
All quantities are in pounds.
Shear in Shear in Girder at Direct Stress in
Top of Story in
Column Column Bay Bay Column Column
A B a b A B
1 1 709 2 115 2 020 2 290 14 464 4 587
2 1 149 2 421 1 467 2 095 12 444 4 317
3 1 136 2 200 1 360 1 755 10 977 3 689
4 1044 2 080 1228 1 709 9 617 3 294
5 995 1 931 1210 1 536 8 389 2 813
6 940 1 780 986 1 310 7 179 2 489
7 816 1 709 920 1 228 6 193 2 163
8 779 1 560 712 996 5 273 1855
9 725 1 440 705 988 4 561 1 571
10 617 1363 661 912 3 856 1 288
11 618 1 180 600 810 3 195 1037
12 524 1 088 538 725 2 595 827
13 486 954 478 640 2 057 640
14 421 861 416 552 1 579 478
15 373 710 354 459 1 163 342
16 298 597 288 375 809 237
17 249 466 221 288 521 150
18 182 354 160 206 300 83
19 124 235 97 126 140 37
20 55 124 43 51 43 8
ILLINOIS ENGINEERING EXPERIMENT STATION
TABLE 20.
CHECK ON THE NUMERICAL VALUES OF THE
MOMENTS AT THE ENDS OF THE COLUMNS AND
GIRDERS OF THE SYMMETRICAL THREE-SPAN
TWENTY-STORY BENT SHOWN IN FIG. 5.
Quantities are in inch-pounds.
For a perfect check the corresponding quantities in the two columns of each
group of columns should be identical.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
S w
2 035 000
1 370 000
1 123 000
1 053 000
983 000
913 000
725 000
673 000
622 000
570 000
518000
466 000
414 000
363 000
311 000
259 000
207 500
155 300
103 600
51 800
2020 000
1 372 000
1 121 000
1 050 400
983 200
914 400
727 200
673 600
624 000
570 200
518400
464 200
414 800
360 200
311 400
259 800
206 200
154 800
103 500
51 540
203 700
187 100
170 500
167 700
135 000
125 900
95 800
95 000
89 600
81 900
73 400
65 300
57 000
48700
39700
31000
22 100
13 380
6110
287800
203 000
188100
170 600
167 700
135 000
125 300
96 300
94 950
88 000
81 600
72 100
65 200
58 000
47 870
39850
30 880
22 160
13 430
6 150
490 000
408 500
338 100
318000
267 300
249 500
200 500
198 400
183 200
164 100
147 100
130 700
113 450
94 00
76 800
59 00
42 30
25 940
108300
JO0830
492 500
410 000
361 300
338 000
318 000
267 100
249 500
199 800
197 900
183 500
164 100
146 900
129 900
112 400
94 200
76 O0W
59 300
42 450
25 00
10 850
10 850
WILSON-MIANEY-WIN STRESSES I
TABLE 21.
ELIMINATION OF THE UNKNOWN QUANTITIES IN THE EQUATIONS
USED TO DETERMINE THE SLOPES AND THE DEFLECTIONS IN
THE BOTTOM STORY OF THE SYMMETRICAL THREE-SPAN
TWENTY-STORY BENT SHOWN IN FIG. 5, BY A MODIFICATION
OF THE SLOPE-DEFLECTION METHOD.
Left-Hand Member of Equation
Story No. 1 Story No. 2 No. 3
R 0A1 I O 2 0A OA2 B2 R3
Coefficients of Unknown Slopes and Ratios of Deflection to Story Height
-206.8
- 77.4
- 77.4
1
1
0
0
0I
51.6
30.5
295.7
71.2
35.6
-0,2495
-0.3941
-3.8204
0.1445
3.4264
0.0680
-1.7300
1.0000
1.7980
-2.7300
1.0000
1
1
1
1
0
0
0
-106.8
-106.8
-284.8
-106.8
-106.8
1.3798
1.3798
-1.3798
-0.6494
-4.0000
-3.0000
-0.6494
4.0000
-1.0000
-0.3612
-1.4652
-1.0000
-3.0000
1.1041
-0.4652
2.0000
1
1
1
0
0
35.6
71.2
220.2
21.4
141.6
-0.4599
0.4599
-0.4599
0.2164
0.2322
1.000C
6.1856
-0.0158
-0.7678
-5.1856
-0.0088
0.2812
-5.185C
0.6012
-0.2900
5.4668
-5.7868
-0.2627
-11.7518
-2.8934
11.4891
-8.8554
1
1
1
0
0
35.6
71.2
21.4
308.1
142.0
-0.4599
0,4599
-0,2322
1.0000
0.6012
0.2322
-1.2322
0.3988
0.1292
0.4513
0.3988
8.6547
-0.3222
0.0525
8.2559
-0.2918
-0.1128
-4.1279
-0.1790
4.0151
-0.0156
-0.454C
1.003C
0.4384
-1.457(
1
1
0
-106.2
-106.5
-283.6
-2.9834
2.9834
2.9334
-2.9915
-2.9834
5.9749
6.4127
2.9875
-6 4127
3.4252
-0.5600
-0.3862
-2.0010
-0.1738
1.6148
-0.3961
-1.109C
0.7131
1
Co-
efficient
of .0001
-117.2
0.0
0.0
- 78.9
0.0
0.0
- 64.5
0.5667
0.0000
0.0000
0.5667
0.0000
0.2667
0.0000
-1.1082
0.0000
0.2667
1.1082
-1.1082
0.1483
-0.4059
-1.1082
0.0000
0.5543
0.7023
-1.1082
0.5021
-1.5095
-0.5541
2.0116
-0.9554
0.1750
0.1079
-0.4555
0.0671
0.5634
0.1530
-0.3865
0.5395
0.7560
51.6
183 8
30.5
71.2
35.6
-0.2495
-2.3 47
-0.3941
2.1252
-1.9807
1
1
1
1
0
0
0
ILLINOIS ENGINEERING EXPERIMENT STATION
TABLE 22.
DETERMINATION OF THE CHANGES IN THE SLOPES AND THE RATIO
OF THE DEFLECTION TO STORY HEIGHT IN THE BOTTOM STORY OF
THE SYMMETRICAL THREE-SPAN TWENTY-STORY BENT SHOWN IN
FIG. 5, BY A MODIFICATION OF THE SLOPE-DEFLECTION METHOD.
No. of Right-Hand
Equa- Left-Hand Member of Equation Member of
tion Equation
The first line in each group is the algebraic form of the equation. Coefficient of
The successive lines are the numerical values of the terms. .0001
13
13 R = .7560
11 6B2 - .3961R = .1530
.3961 x .00007560 = .2995
tB2 = .4525
9 OA2 - .0156 B2 - .5600 R3 = .1750
.0156 x .00004525 = .0071
.5600 x .00007560 = .4240
0A2= .6061
6 R, - .2627 0A - .2918 0B2 .5021
.2627 x .00006061 = .1593
.2918 x .00004525 .1320
R2 .7934
3 B1,- .3612 R2 - .0088 0A.+ .1292 B2 .1483
.3612 x .00007934 = .2865
.0088 x .00006061 = .0053 .4401
.1292 x .00004525 = -.0585
-01 = .3816
1 .A1 + .0680 0B1 - .6494 R2 + .2164 02 = .2667
.6494 x .00007934 = .5150 .7817
.0680 x .00003816 = .0260
.2164 x .00006061 = .1313 -.1573
0A1 .6244
A R1 - .2495 0A1 - .2495 0B1 = .5667
.2495 x .00006244 = .1556
.2495 x .00003816 = .0951
RI = .8174
WILSON-MANEY--WIND STRESSES
0
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t0.0 00N N00 ON. 0000 CON. N.' 000000000000 000000001000
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ILLINOIS ENGINEERING EXPERIMENT STATION
Z
2q
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00
RH
H0
000
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u
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C OQO -'00 000 00t 0 000o 000 000n C-C~ O 00 00 00 CON
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CO 00~~C 00 000 IC =CA 000 00 CO 0 00 0
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uotopa-aoi co oo oo ?o oo co O so ®o ob co q
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WILSON-MANEY-WIND STRESSES
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ILLINOIS ENGINEERING FXPERIMENT STATION
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WILSON-MANEY-WIND STRESSES
TABLE 27.
EFFECT OF THE PROPORTIONS OF A BENT UPON THE ACCURACY OF METHOD I.
ALL GIRDER SECTIONS ARE THE SAME.
All stories of a bent are identical, and the shears on all stories are equal. All column sections are equal, all girder sections
are equal, and the column sections are equal to the girder sections.
For each bent the upper line is the moment in per cent of W x h, and the lower line is the moment in per cent of the moment
determined by the slope-deflection method.
4
7
Proportions of
the Bent
Ratio of
Story
Height
to Width
of Bay a
2
0.5
Ratio of
Width of
Bay a to
Width of
Bay b
2
2
2
Moment at Top
and Bottom of
Column A
Slope-
tion
Method
9.09
100.0
10.00
100.0
10.87
roo.o
8.09
100.0
9.38
00oo.o0
10.53
100.0
7.32
10oo0.0
8.93
0oo.0
10.33
Ioo.o
Method
I
9.62
10o5.9
7.50
75.0
5.00
46.o
9.62
119.00
7.50
79.9
5.00
47.5
9.62
131.3
7.50
83.9
5.00
48.4
Moment at Top
and Bottom of
Column B
tion Method
Method I
15.91 15.38
1oo.o 96.5
15.00 17.50
ioo.o 116.5
14.13 20.00
1oo.o 141.5
16.91 15.38
100.0 90.9
15.62 17.50
100.0 111.9
14.47 20.00
1oo.o 138.0
17.67 15.38
1oo.o 87.0
16.06 17.50
oo.o o108.7
14.68 20.00
zoo.o 136.1
Moment at Right
End of Girder a
ec-
tion Method
Method I
18.20 19.24
100.0 10o5.9
20.00 15.00
1oo.o 75.0
21.70 10.00
1oo.o 46.1
16.18 19.24
100.0 119.3
18.75 15.00
1oo.o 8o.o
21.04 10.00
1oo.o 47.5
14.66 19.24
100.0 131.0
17.84 15.00
oo. o 83.9
20.61 10.00
100oo.o 48.5
Moment at Left
End of Girder a
Slope-
Deflec-
tion
Method
13.67
100.0
16.66
roo0.
19.54
1oo.o
13.23
oo.o0
16.66
roo0.0
19.74
oo.o0
12.93
100.0
16.66
100.0
19.76
10oo.
Method
I
19.24
140.7
15.0
90.0
10.00
51.2
19.24
145.3
15.00
90.0
10.00
50.6
19.24
149.o0
15.00
00.00
10.0
50.6
Moment at End
of Girder b
tion
Method
18.20
100.0
13.33
100.0
8.70
10oo0.0
20.60
00oo.o0
14.60
100.0
9.20
100.0
22.40
00oo.o0
15.45
100.0
9.60
100.0
Method
I
11.52
63.2
20.00
15o.o
30.00
345.0
11.52
55.9
20.00
137.0
30.00
326.o
11.52
51.3
20.00
r29.5
30.00
312.0
1
ILLINOIS ENGINEERING EXPERIMENT STATION
TABLE 28.
EFFECT OF THE PROPORTIONS OF A BENT UPON THE ACCURACY OF METHOD I.
ALL GIRDERS PROPORTIONAL TO THE BENDING MOMENTS.
All stories of a bent are identical and the shears on all stories are equal. All column sections are equal, the moments of
inertia of column A and girder a are equal, and the ratio of the moment of inertia of girder a to the moment of inertia of
girder b equals the ratio of the bending moment in girder a to the bending moment in girder b, as determined by method I.
For each bent the upper line is the moment in per cent of W x h, and the lower line is the moment in per cent of the moment
as determined by the slope-deflection method.
7
Proportions of
the Bent
Ratio of
Story
Height
to Width
of Bay a
2
1
0.5
0.5
tatio of
Width of
atoy a to
Width of
Bay b
2
2
2
.1
0.5
Moment at Top
and Bottom of
Column A
tion
Methed
9.75
100.0
9.60
100.0
9.45
0oo.0
9.05
100 .0
8.85
Z100.0
8.60
100.0
8.55
100.0
8.30
8.00
00o.0
Method
I
9.62
98.8
7.50
78.1
5.00
52.9
9.62
io6.z2
7.50
84.8
5.00O
58.o
9.62
112.5
7.50
90.3
5.00
62.5
Moment at Top
and Bottom of
Column B
tion
Method
15.25
roo.o0
15.40
roo.o
15.55
roo.0
15.95
roo.o
16.15
oo00.0
16.40
roo.0
16.45
OO.O0
16.70
100.0
17.00
oo00.0
Method
I
15.38
100.5
17.50
113.6
20.00
129.0
15.38
96.2
17.50
1o8.3
20.00
122.0
15.38
93.2
17.50
o105.o
20.00
r17.5
Moment at Right
End of Girder a
Slope-
tion
Method
19.50
100.0
19.2
100.0
18.9
100.0
18.10
100.0
17.70
zOO.0
17.20
100.0
17.10
100.0
16.60
100.0
16.00
100.0
Method
I
19.24
98.8
15.00
78.1
10.00
52.9
19.24
ro6.2
15.00
84.8
10.00
58.1
19.24
112.5
15.00
90.4
10.00
62.5
Moment at Left
End of Girder a
tion
Method
16.05
15.60
1OO.O
15.10
100.0
15.70
100.0
15.20
100.0
14.7
100.O
15.80
100.O
15.20
100.0
14.60
100.0
Method
I
19.24
120.0
15.00
96. z
10.00
66.3
19.24
122.5
15.00
98.8
10.00
68.o
19.24
122.0
15.00
98.8
10.00
68.5
Moment at End
of Girder b
Slope-
tion
Method
14.15
roo.o0
15.20
roo.0
16.00
oo.o0
16.20
1oo.o
17.10
100.0
18.1
oo00.0
17.1
100.0
18.2
19.4
100.0
Method
I
11.52
79.6
20.00
131.5
30.00
187.5
11.52
92.6
20.00
I17.0
30.00
165.8
11.52
67.3
20.00
110.0
30.00
154.8
WILSON-MANEY-WIND STRESSES
TABLE 29.
EFFECT OF THE PROPORTIONS OF A BENT UPON THE ACCURACY OF
METHOD II.
All stories of a bent are identical and the shears on all stories are equal. All column sections are equal, the momenta of
inertia of column A and girder a are equal, and the ratio of the moment of inertia of girder a to the moment of inertia of
girder b equals the ratio of the bending moment in girder a to the bending moment in girder b, as determined by method II.
For each bent the upper line is the moment in per cent of W x h, and the lower line is the moment in per cent of the moment
as determined by the slope-deflection method.
4
7
Proportions of
the Bent
Ratio of
Story
Height
to Width
of Bay a
2
2
2
1
0.5
Ratio of
Width of
Bay a to
Width of
Bay b
2
1
0.5
2
2
Moment at Top
and Bottom of
Column A
tion
Method
9.65
100.0
10.55
100.0
11.25
100.0
8.82
zoo.o
10.10
100.0
11.00
100o.o
8.30
100.0
9.80
100.0
10.85
100.0
Method
II
12.50
129.5
12.50
Ii8.5
12.50
111.0
12.50
141.8
12.50
123.8
12.50
113.5
12.50
150.6
12.50
127.5
12.50
115.2
Moment at Top
and Bottom of
Column B
tion Method
Method II
15.35 12.50
1oo.o 8.s5
14.45 12.50
1oo.o 86.5
13.75 12.50
100.0 91.0
16.18 12.50
1oo.o 77.3
14.90 12.50
Zoo.o 83.9
14.00 12.50
1oo.o 89.3
16.70 12.50
100.0 74.9
15.20 12.50
100.0 82.3
14.15 12.50
1oo.o 88.4
Moment at Right
End of Girder a
tion
Method
19.30
O10.0
21.10
zOO. 0
22.50
I00.0
17.64
oo0.0
20.20
100.0
22.00
100.0
16.60
100.0
19.60
100.O
21.70
100.0
Method
II
25.00
129.5
25.00
118.5
25.00
11i.1
25.00
141.8
25.00
124.0
25.00
113.7
25.00
150.7
25.00
127.o
25.00
1I5.0
Moment at Left
End of Girder a
Slope-
Deflec-
tion Method
Method II
15.45 8.33
1oo.o 53.9
18.30 8.33
joo.o 45.6
20.95 8.33
1oo.o 39.8
15.26 8.33
1oo.o 54.6
18.45 8.33
oo00. 45.20
21.20 8.33
1oo.o 39.3
15.20 8.33
1oo.o 54.9
18.55 8.33
0oo.o 44.9
21.40 8.33
1oo.o 38.9
Moment at End
of Girder b
tion Method
Method II
15.25 16.66
100.0 109.3
10.60 16.66
100.0 157.1
6.55 16.66
100.0 255.0
17.10 16.66
1oo.o 97.5
11.35 16.66
1oo.0o 46.8
6.80 16.66
1oo.o 245.0
18.20 16.66
joo.o or.6
11.85 16.66
1oo.o 140.5
6.9 16.66
1oo.o 241.5
ILLINOIS ENGINEERING EXPERIMENT STATION
TABLE 30.
EFFECT OF THE PROPORTIONS OF A BENT UPON THE ACCURACY
OF METHOD III.
All stories of a bent are identical and the shears on all stories are equal. All column sections are equal, the moments of
inertia of column A and girder a are equal, and the ratio of the moment of inertia of girder a to the moment of inertia of
girder b equals the ratio of the bending moment in girder a to the bending moment in girder b, as determined by method III.
For each bent the upper line is the moment in per cent of W x h, and the lower line is the moment in per cent of the moment
as determined by the slope-deflection method.
Moment at Top
and Bottom of
Column B
Method
III
12.50
84.1
12.50
85.2
12.50
87.3
12.50
8o.p
12.50
82.3
12.50
84.9
12.50
78.8
12.50
80.5
12.50
83.70
Moment at Right
End of Girder a
tion
Method
20.24
oo0.0
20.60
roo.o
21.34
oo .0
19.10
1O0.0
19.60
roo.o
20.52
roo. 0
18.30
roo.o
18.92
too.0
20.10
o00.o
Method
III
25.00
123.5
25.00
121.5
25.00
117.3
25.00
131.0
25.00
127.6
25.00
Z22.0
25.00
136.8
25.00
132.1
25.00
124.5
Moment
End of G
tion
Method
17.06
100.0
17.60
100.0
18.79
100.0
17.00
10oo0.0
17.60
100.0
18.98
100.0
17.10
100.0
17.63
100.0
18.95
100.0
at Left Moment at End
Girder a of Girder b
Method tion Method
III Method III
13.48 12.70 11.52
70.0 ioo.o 9o.8
5.0 11.80 20.00
28.4 oo.o 170o.o
-5.0 9.87 30.00
..... . 100.0 304.5
13.48 13.90 11.52
79.3 1oo.o 83.0
5.0 12.80 20.00
28.4 100.0 156.3
-5.0 10.50 30.00
....... oo00.0 286.
13.48 14.60 11.52
78.8 1oo.o 79.0
5.0 13.45 20.00
28.4 Zoo.o 148.8
-5.0 10.95 30.00
...... 00oo.o0 274.5
Proportions of
the Bent
Ratio of
Story
Height
to Width
of Bay a
2
2
2
1
0.5
4
Moment at Top
and Bottom of
Column A
tion Method
Method III
10.12 12.50
1oo.o 123.5
10.30 12.50
100.0 121.5
10.67 12.50
oo.o0 z17.4
9.55 12.50
100.0 131.0
9.80 12.50
oo.o0 127.7
10.26 12.50
100.0 122.0
9.15 12.50
roo.o 136.9
9.46 12.50
100.0 132.2
10.05 12.50
roo.o 124.3
Ratio of
Width of
Bay a to
Width of
Bay b
2
1
0.5
2
2
1
0.5
Slope-
tion
Method
14.88
j00.0
14.70
joo.0
14.33
100.0
15.45
1oo.0
15.20
100.0
14.74
100.0
15.85
100.0
15.54
100.0
14.95
r10 n
7
WILSON-M.ANEY--WIND STRESSES
TABLE 31.
EFFECT OF THE PROPORTIONS OF A BENT UPON THE ACCURACY
OF METHOD IV.
All stories of a bent are identical and the shears on all stories are equal. All column sections are equal, the moments of
inertia of column A and girder a are equal, and the ratio of the moment of inertia of girder a to the moment of inertia of
girder b equals the ratio of the bending moment in girder a to the bending moment in girder b, as determined by method IV.
For each bent the upper line is the moment in per cent of W x h, and the lower line is the moment in per cent of the moment
as determined by the slope-deflection method.
4
Proportions of
the Bent
Ratio of
Story
Height
to Width
of Bay a
2
1
1
0.5
0.5
0.5
Ratio of
Width of
Bay a to
Width of
Bay b
2
2
1
0.5
2
1
0.5
Moment at Top
and Bottom of
Column A
Slope-
Deflec-
tion
Method
9.09
100.0
10.0
100.0
10.89
roo.o
8.09
100.0
9.38
100.0
10.53
100.0
7.32
100.0
8.93
100.0
10.33
JOo.O
Method
IV
8.33
91.8
8.33
83.3
8.33
76.8
8.33
103.2
8.33
88.8
8.33
79.2
8.33
113.9
8.33
93.3
8.33
80.7
Moment at Top
and Bottom of
Column B
Slope-
Deflec-
tion
Method
15.91
100.0
15.00
100.0
14.13
100.0oo
16.91
100.0
15.62
I00.0
14.47
100.0
17.68
16.07
100.0
14.67
100.0
Method
IV
16.66
104.5
16.66
1I1. 1
16.66
117.8
16.66
98.3
16.66
io6.4
16.66
I15.o
16.66
94.2
16.66
103.7
16.66
113.5
Moment at Right
End of Girder a
Slope-
Deflec-
tion Method
Method IV
18.18 16.66
1oo.o 91.7
20.00 16.66
100.0 83.3
21.74 16.66
oo0.o 76.6
16.18 16.66
100.0 103.0
18.76 16.66
1oo.o 88.9
21.06 16.66
1oo.o 79.1
14.64 16.66
1oo.o 113.8
17.86 16.66
100oo.o 93.3
20. 6 16.66
1oo.o 8o.6
Moment at Left
End of Girder a
Slope-
Deflec-
tion
Method
13.62
zoo.0
16.66
100.0
19.56
100.0
13.22
100.0
16.64
100.0
19.74
100.0
12.96
100.0
16.69
100.0
19.74
100.0
Method
IV
16.66
122 . I
16.66
oo .0
16.66
85.20
16.66
126.o
16.66
100.0
16.66
84.4
16.66
128.7
16.66
100.0
16.66
84.4
Moment at End
of Girder b
Slope-
Deflec-
tion
Method
18.20
100.0
13.33
100.0
8.70
100.0
20.60
100.0
14.60
100.0
9.20
100.0
22.40
100.0
15.45
o100.0
9.60
100.0
Method
IV
16.66
91.5
16.66
125.0
16.66
191.5
16.66
80.9
16.66
114.0
16.66
18i.o
16.66
74.4
16.66
o07.8
16.66
173.5
1
ILLINOIS ENGINEERING EXPERIMENT STATION
TABLE 32.
LOG OF THE TEST OF CELLULOID MODEL No. 4.
Horizontal Deflection
at
I Center of Girder
at
Top of Story
in Inches
Mid-
dle
Span
.04
.08
.11
.08
.16
.24
.14
.29
.42
.18
.36
.55
Left-
Hand
Span
.04
.08
.11
.08
.16
.24
.14
.28
.42
.18
.37
.55
Change in Slope Measured on an Arc Having a 19-inch Radius, in Inches.
Force at Top of Model
Acting Toward the Right
0 at Top of 0 at Top of
Column A Column B
Right- Left- Right- Left-
Hand Hand Hand Hand
Side of Side of Side of Side of
Model Model Model Model
.20 .20 .09 .10
.40 .40 .20 .21
.61 .59 .31 .33
.21 .21 .10 .10
.43 .43 .21 .20
.66 .66 .34 .33
.18 .17 .11 .12
.37 .35 .26 .26
.59 .54 .39 .40
.13 .10 ,06 .07
.25 .19 .13 .15
.40 .30 .20 .22
Force at Top of Model
Acting Toward the Left
0 at Top of 9 at Top of
Column A Column A
Right- Left- Right- Left-
Hand Hand Hand Hand
Side of Side of Side of Side of
Model Model Model Model
.19 .18 .09 .10
.40 .40 .19 .23
.61 .63 .30 .38
.19 .19 .11 .10
.41 .41 .24 .22
.63 .65 .37 .37
.17 .16 .13 .14
.36 .35 .26 .29
.56 .55 .40 .42
.13 .09 .06 .07
.26 .20 .13 .15
.39 .27 .18 .25
4
T .
I
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*Bulletin No. 45. The Strength of Oxyacetylene Welds in Steel, by Herbert L. Whitte-
more. 1911. Thirty-five cents.
*Bulletin No. 46. The Spontaneous Combustion of Coal, by S. W. Parr and F. W.
Kressmann. 1911. Forty-five cents.
*Bulletin No. 47. Magnetic Properties of Heusler Alloys, by Edward B. Stephenson.
1911. Twenty-five cents.
*Bulletin No. 48. Resistance to Flow Through Locomotive Water Columns, by Arthur
N. Talbot and Melvin L. Enger. 1911. Forty cents.
*Bulletin No. 49. Tests of Nickel-Steel Riveted Joints, by Arthur N. Talbot and Her-
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*Bulletin No. 50o. Tests of a Suction Gas Producer, by C. M. Garland and A. P. Kratz.
1912. Fifty cents.
*Bulletin No. 5r. Street Lighting, by J. M. Bryant and H. G. Hake. 1912. Thirty-
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*Bulletin No. 52. An Investigation of the Strength of Rolled Zinc, by Herbert F.
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"Bulletin No. 53. Inductance of Coils, by Morgan Brooks and H. M. Turner. 1912.
Forty cents.
*Bulletin No. 54. Mechanical Stresses in Transmission Lines, by A. Guell. 1912.
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"Bulletin No. 58. A New Analysis of the Cylinder Performance of Reciprocating En-
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"Bulletin No. 59. The Effect of Cold Weather Upon Train Resistance and Tonnage
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*Bulletin No. 60o. The Coking of Coal at Low Temperatures, with a Preliminary Study
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"Bulletin No. 61. Characteristics and Limitations of the Series Transformer, by A. R.
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Bulletin No. 62. The Electron Theory of Magnetism, by Elmer H. Williams. 1918.
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"Bulletin No. 63. Entropy-Temperature and Transmission Diagrams for Air, by C. R.
Richards. 1913. Twenty-five cents.
*Bulletin No. 64. Tests of Reinforced Concrete Buildings Under Load, by Arthur N.
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*Bulletin No. 65. The Stsam Consumption of Locomotive Engines from the Indicator
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Bulletin No. 66. The Properties of Saturated and Superheated Ammonia Vapor, by G.
A. Goodenough and William Earl Mosher. 1918. Fifty cents.
Bulletin No. 67. Reinforced Concrete Wall Footings and Column Footings, by Arthur
N. Talbot 1918. Fifty cents.
*Bulletin No. 68. Strength of I-Beams in Flexure, by Herbert F. Moore. 1918. Twenty
cents.
*Bulletin No. 69. Coal Washing in Illinois, by F. C. Lincoln. 1913. Fifty cents.
Bulletin No. 70. The Mortar-Making Qualities of Illinois Sands, by C. C. Wiley. 1913.
Twenty cents.
"Bulletin No. yr7. Tests of Bond between Concrete and Steel, by Duff A. Abrams. 1914.
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*Bulletin No. 7p. Magnetic and Other Properties of Electrolytic Iron Melted in
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*Bulletin No. 73. Acoustics of Auditoriums, by F R. Watson. 1914. Twenty cents.
*Bulletin No. 74. The Tractive Resistance of a 28-Ton Electric Car, by Harold H.
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Bulletin No. 75. Thermal Properties of Steam, by G. A. Goodenough. 1914. Thirty.
live cents.
*Bulletin No. 76. The Analysis of Coal with Phenol as a Solvent, by S. W. Parr and
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*Bulletin No. 77. The Effect of Boron upon the Magnetic and Other Properties of
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*Bulletin No. 78. A Study of Boiler Losses, by A. P. Kratz. 1915. Thirty-five cents.
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*Bulletin No. 79. The Coking of Coal at Low Temperatures, with Special Reference to
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*A limited number of copies of those bulletins which are starred are available for free
distribution.
A'
'A'.
THETUNIVERSITY OF ILLINOIS
THE STATE UNIVERSITY
Urbana
EDMUND J, JAMES, Ph. D., LL. D., President
The University includes the following dep rtments:
The Graduate School
The College of Liberal Aris and Sciences (Ancient and Modern Lan
Rguages and Literatures; History, Economics and Accountancy.
Plitical Science, Socioogy; Philosophy, Psychology, Education;
Mathematics; Astronomy; Geology; Phyics; Chemistry; Botany,
Zoology, Entomology; Physiology; Art and. Design; Ceramics)
The College of Enginieering (Architecture; Architectural, Civil, Elec-
trical, Mechanical, Miniag, Municipal and Sanitary, and Railway
Engingecring)
The College of Agriculture (Agronomy; Animal Husbandry; Dairy
Husbandry; Hiortilture and Landscape: Gardening; Veterinary
Science; Agricultural Extension; Teachers' Course; Household
Science)
The College of Law (three years" course
The School of Education
The Courses in Business (General Business: Banking; Accountancy;
Railway Administration; Insurance; Secretarial;; Commercial
#Teacherg')
The Course in Journalism
The Course in Chemistry and Chemical Engineering
The Course in Ceramics and Ceramic Engineering
The School of Railway Engineering and Administratioii
The School-of Music (four years' course)
The School of Library Sieence (two years' course)
The College of Medicine (in Chicago)
The College of Dentistry (in Chicago)
The School of Pharmat y (in Chicago; Ph. G., and Ph. C. courses)
The Sumnmer Session (eight weeks)
Experiment Stations: U. S. Agricultural Experinient Station- En-
, gineering Experimenit Station; State Laboratory of Natdral His-
tory; State Extomologist's Office; Biological Experiment Station
on-Illinois River; State Water Survey; State* (eological Survey;
1Mine, Rescue .Station
The iibrary collections contain (May1, -1914) 24,0 volunies, in-
ciding ihe library ofý the State Laboratory of Natur Histo
the, Quine Medical Libr4ry and the library of the School
Pharmacy.
-#or caqtadks ond informatiots address. ,
TR£ RGISTRAR,'
'VA
ThE UNWERSITY OF ILUNOIS
TUE STATh UNWERSITY
Urbana
EpMPm 3", J~s, ~'h. D., LL. P., ~resi~x~t
The U11Ai~rslty ~ticludea "t1~e iol1~win~' dcp~rtn~iita~
't~he Qra4i~ata School
Th~ CaUe~e of' LThClAa~ Arts 3and S~er~cee (A~ujent ~nd M~dern tail
guag~s a~d Li~erat~utes- Uistoty, Ec&n~4cs and A~cou~It~t~y'.
.42o11~ical S~cjen~e, Soci~f~gy~ Phllo&9phy? Payebotogy, ~d~ication;
AMat'be~natic5 Aetroxioi~iy; ~ebIogy; Phy$cs; Chemistry; B~tany1
'Z9Q~Ogy, ~donioiogy~ Physiology; Art a~ad flesi~n4 Ceraniicu)
The College of E±xgIneerin~ (4rehite~tu~e; A~chlt~tutal, Civil, Elec-
trival, Meclrnnic4 Mining, Mun~idpaLaiid Sanitary,, a~id Railway
E~gin.e~ering)
The College of Agrici4ti*~e (Agrt~m1uy; Animal Th1~ban4ry'~ Daity
I'lusbandty; floi'tie~tlture' and tan4sca~ie Gartlening~ Vetetinary'
Scj~nee$ Agricultural Extenslomi; Teachers' Course;'4i~uselidld.
Scien~e~ 'A
The College of Law (tlree yea~ conrs4
The 8'~hooI of ~4ucatiou
The Coi~rsea h~ Rizsines~ (General litisinessi ranking; Ac~,o~zitat~cy;
RaU~ay Adiniiiistration; InsuraA~ice; Secretarial; Commercial
~'eecher~') N
lii. Cour,. In Jot~r*4ismhfld C k*h BngI~i
The Course hi Chemi~try ee~ng
The 'Conr5e in C~razui~a an4 Cer~mIc~nghieezi±x% . A,
The Schoci, ~f r?~aflw~y ~)ngineerisig 'and. A~i~tiitmti0i
Th~ S Ii oh-of ?~nskA (1ourjyear~' coerseY 'N
~he Sdhool ~ i.ibreiy' $oi~i*c.~ (t~. yeaii' course) . -
a.,w &b '~ ~ r~.,.or, A 'A
~h Ca1Leg~ o~ Dn~is~ (in C?&i~ago)
~ School otNPIia~tta~ n'~hic~go, 'Ph. G~ gn~l Ph. C~ c~nr~ep~Y
'.t'ile PUkWIeXA S~4oa (eight weeks)
~~$~iae*~t St~*k~ ~,;A8. A~rlcIiltu41 E~erh~Iknt S*atio~ Eu-
'~hiee~ruig Experiment. S*ition; ~tahe La'b6r~q1y of Nah~raf MIs-
'llI~r~4~ Cf(ive1; State 4~14~~ 5OOvob~rt~kes, liP
cb~din~ Abe ~i6~rary~of~ 'the -$tMe {~'ratoi)~ ~d N~atIiral Histb~
tli~ Q~lRe l4edical Libr~ ~n~1 tb~ lib'razj, A~I the $ch~4u1~i
Pharm4~4 2 .N.
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