.FORMULAS FOR COLUMNS WITH 

SIDE LOADS AND ECCENTRICITY 

November 1950 

No. R1782 

UNITED STATES DEPARTMENT OF AGRICULTURE 

FOREST SERVICE 

FOREST PRODUCTS LABORATORY 

Madison 5, Wisonsin 

In Cooperation with the University of Wisconsin 
FORMULAS FOR COLUMNS WITH SIDE LOADS AND ECCENTRICITY

By

LYMAN W. WOOD, Engineer

Forest Products Laboratory, Forest Service

U. S. Department of Agriculture

---- -

Introduction

The design of columns with side loads and eccentric end loads is an old problem

that his received attention from many authors. General textbooks such as those

by Church or Maurer and Withey, and the more specialized work of Salmon on

columns are among the many publications that have dealt with the subject. The

treatment of short columns, considered as prisms and with stresses due to deflection neglected, is relatively simple, but long columns with deflection

stresses are more complex.

Critical loads on centrally loaded long columns that fail,by bending can be

calculated satisfactorily with the Euler formula. Additional formulas of satisfactory accuracy have been developed for long columns with eccentric end load

or with certain specific combinations of side load with eccentric end load 

These latter formulas, however, require rather cumbersome trial calculations

involving the secant of an angle that can be determined only indirectly; they

have the further disadvantage that a separate formula must be developed for

each condition of side loading.

Recognizing the difficulty in applying the secant formulas to problems of safe

loads, on long columns with side loads and eccentricity, Newlin sought a sim-

 This is an amplification and explanation of an article of the same title by J.

 A. Newlin, formerly Chief, Division of Timber Mechanics, Forest Products

Laboratory, published in Building Standards Monthly, December 1940. The

formulas are also given in “National Design Specification for Stress-grade

Lumber and Its Fastenings," published by the National Lumber Manufacturers

Association. Acknowledgment is made to C. B. Norris of the Forest Products

Laboratory staff for review and confirmation of Newlin’s analysis. First

report of this number published in 1950.

 Maintained at Madison, Wis., in cooperation with the University of Wisconsin.

 Church, Irving P., Mechanics of Engineering, New York, 1914.

 Maurer, Edward R. and Withey, Morton O., Strength of Materials, New York, 1940.

 Salmon, E. H., Columns, London England, 1921.

Report No. 1782 -1-
lified method of design. He succeeded in developing a general formula in which

eocentrioity is simply represented and in which any condition of side loading

can be expressed in terms of the bending stress induced by it. His formula,

while somewhat more approximate than the secant formulas, has as much accuracy

as exists in the present state of knowledge of the properties of wood or of the

conditions under which wood columns are loaded. The formula is not difficult

to apply and is believed to be highly useful in the design of wood columns.

Newlin's original report and published article were rather brief, and a number

of steps in his derivation of the long-column formula were omitted. The derivation of a similar formula for short columns was also omitted. The present

report has been prepared to fill in the omitted material and to illustrate the

application of the formulas by means of suitable examples.

Notation

(All units except ratios are in pounds, inches, or combinations thereof)

A =

C =

E =

F =

I =

Report No.

area of the cross section of a column.

unit failing stress in compression parallel to grain.

Young's modulus of elasticity of the column material.

unit failing stress flexure.

moment of inertia of the cross section of a column around the neutral

axis. As used here, the neutral axis is perpendicular to

direction of eccentricity or side load.

value of the slenderness ratio dividing

columns.

length of a column or, more specifically,

In a rectangular

intermediate from

unsupported length.

the

column,

long

slenderness ratio of a column. As used here, d is measured in the

direction of eccentricity or side loads.

generally, the bending moment on a column from eccentricity or side

load, but is used here also in the more restricted sense of bending moment caused by the portion of side load that is independent

of axial load.

generally, the flexural stress induced in the outer fiber of a column from eccentricity or side load, but used here also in the

more restricted sense of stress in the outer fiber induced by the

portion of side load that is independent

axial load (end load) on a column, whether

tally applied.

direct compressive

column.

section modulus of

in the direction

of a rectangular

1782

stress induced by axial

of axial load.

centrally or eccentriload (end Load) on a

the cross section of a column related to flexure

of eccentricity or side loads. Section modulus

column is

-2-
W = critical Euler load on a pin-ended column.

b = width of a rectangular column.

c = allowable unit working stress in compression parallel to gr

column of the length under consideration with centrally a

axial load and no side load; this may be a short-column, 

diate-column, or long-column stress.

d = depth of a rectangular column) measured in the direction of 

tricity or side load.

d = additional deflection at midlength of a long column due to 

load.

e = eccentricity, the distance from the center of gravity of th

section to the center of gravity of the applied load. Ec

city in this analysis is considered as being parallel to 

the sides of a rectangular column.

= increments of deflection at midlength of a column 

Salmon’s analysis.

f = allowable unit working stress in flexure, as in a simple be

transverse loading only.

= flexural stress corresponding to the total deflection at mi

of a long column when all eccentric and side loads are on

= increments of flexural stress corresponding to incr

deflection

 = fictitious flexural stress due to eccentricity of axial loa

to replace the original deflection of the column in Salmo

analysis.

m,n = exponents in the general form of the equation for strength 

combined loading.

w = critical Euler unit stress on a pin-ended column.

= total deflection at midlength of a column in Salmon’s analy

all loads are on.

z = ratio of flexural to direct compressive stress; that is, 

divided by P/A more specifically, a ratio of flexural st

direct compressive stress when both result from the same 

so that the ratio remains constant while

Analysis of Combined Stresses

the load varies.

Textbooks on strength of materials show how the combined stress on a 

short column due to axial and flexural loads is represented by the su

P/A. In this expression, M is bending moment, which may be induced by 

acting side loads, by eccentricity of the axial load, or by both. For 

of design in some structural materials, the sum M/S + P/A is not permi

exceed a safe stress value, assumed to be the same in bending as in co

Where strength in bending is unequal to that

with wood, there is a problem of determining

Report No. 1782 -3-

in compression, as is the 

what is the ultimate stre
value for various combinations of load. It is recognized that the strength

value under combined stress lies generally somewhere between the two values

for separated stresses and is influenced by the relative proportions of each.

The relationship which describes the condition for failure under combined

stress may be of the interaction type.

(1)

where M/S and P/A are stresses applied in flexure and compression and F and

C are the corresponding ultimate strength values under separate loading. Since

F and C are known values, either M/S or P/A can be calculated by assuming a

value for the other. Then the sum M/S + P/A becomes the strength under the

assumed combination of load.

Newlin and Trayer made tests of clear Sitka spruce under combined loading

which showed the relation of the strength to the relative proportions of bending and axial stress. The curve of maximum stress from figure 6 of their report, as obtained from these test results, is shown as a solid line in figure 1.

The most simple equation of type (1) is with exponents m and n each equal to

unity; this relation gives combined strength values in clear Sitka spruce indicated by the lower broken-line curve in figure 1. A much closer agreement

with the test results is obtained if m = 2, giving the upper

of figure 1.

broken-line curve

From consideration of figure 1, it is apparent that strength

stress is estimated very closely from the relationship

under combined

(2)

This relationship, however, is somewhat complex to handle as a formula, especially when M/S is broken up into two or three components representing side

loads and eccentricity of axial load. Newlin’s formulas: for short columns

indicate that he used the form

(3)

which is simpler of application. This relationship (3) has also had extensive

use by other designers dealing with the problem of combined loads. Figure 1

shows that it gives rather conservative estimates of strength, but not far in

error, with the lower ratios of bending to total stress such as are most likely

to occur in short columns.

6

-Newlin, J. A. and Trayer, G. W., Stresses In Wood Members Subjected to Combined Colume and Beam Action, Forest Products Laboratory Report No. 1311,

1941.

Report No. 1782 -4-
In the case of long columns, Newlin simplified his safe-loading formu

assuming that the ultimate strength under combined load is equal to t

strength. This is explained as follows. Long columns are those in t

in which the Euler formula applies. The Euler formula is based on el

rather than strength and can be applied only if P/A iS less than th

tional-limit stress in compression. This stress is usually assumed to 

thirds of the compressive strength of wood, or 2C/3 by the notation 

port. The shortest column in the Euler class will reach its maximum 

this stress. The unit flexural strength, being a property of the mate

unaffected by the length of the column. When stress due to an axial 

long column equals 2C/3, and if it is assumed that the maximum stress 

under combined load is equal to the flexural strength, the available 

stress is F - 2C/3, and the ratio of bending stress to total stress be

This is a minimum value for that ratio in long

Sitka spruce (fig. 1), the minimum ratio is

columns. In the case o

It follows that the long-column formula leads to ratios within the ran

to 1.00, as indicated by the heavy horizontal line at the top of figur

responding ratios based on safe rather than ultimate stresses and appl

other species range from about 0.55 to 1.OO, as indicated on figure 1

be seen that, in the range from 0.55 to 1.00, the maximum stress actua

veloped in test is not much short of the flexural strength.

If the relation expressed in equation (2) were used to estimate maximu

in a long column, a formula of considerable complexity would result. 

(3) would give

mates of load.

columns equals

a-usable formula, but would result in overly conservati

The assumption that strength under combined loading of 

flexural strength has therefore been chosen.

Short Columns

Wood columns with slenderness ratios (ratio of unsupported length to 

dimension of cross section) of 11 or less receive both compression an

stresses under eccentric axial load, but it is not necessary to assume 

additional stress due to deflection. The combined stress, derived as 

by well-known principles of mechanics is expressed by the quantity P

In this expression M is bending moment, which may be induced by eccent

an axial load, by direct-acting side loadsj or by both. For maximum s

the relationship of equation (3) is used with safe stresses instead of 

strength values.

 Newlin, J. A. and Gahagan, J. M., Tests of Large Timber Columns and 

tion of the Forest Products Laboratory Column Formula. U. S. Dept. 

Tech. Bull. No. 167, 1930.

Report No. 1782 -5-
(4)

Equation (4) is directly applicable where there are side loads and concentric

axial load. If the end load is proportional to the side load (as in upper

chord members of roof trusses, with bending stress induced by roof loads acting

through purlins, and with axial loads proportional to the same roof loads), M/S

can be replaced by z P/A, and equation (4) becomes

(5)

The bending stress M/S may be induced by eccentricity of the axial load instead

of by side load. In that case, the external moment M is expressed by the quantity, Pe . The section modulus S may be replaced by ( d measured in the

direction of eccentricity). Then

(6)

But in a rectangular column, A = bd and I = Substitution of these values

in (6) gives

Substitution of expression (7) in equation (4) gives

for the condition of

Where both side load

represents the total

may be considered as

(7)

(8)

eccentric axial load and no side load.

and eccentric axial load are present the term M/S in (4)

of bending stress from both. Furthermore, the side load

composed of two portions,

(M/S) and the other proportional to axial load

stress is thus expressed by

one independent of axial load

(z P/A). The total bending

and equation (4) becomes

where M/S has the more restricted

of side load which is independent

Report No. 1782

(9)

meaning of bending stress due to that portion

of axial load.

-6-
Equation

rived by

to axial

(9) is a general equation from which any of the others is read

dropping certain terms. For example, if all side load is prop

load, and axial load is eccentrically applied, the term M/S is 

If all side load is proportional to axial load, and axial load is centr

applied, both M/S and P/A are dropped, and equation (5) results. 

is eccentricity but no side load, M/S and z P/A are dropped, and equati

results. Other loading conditions can be similarly represented by suit

lection of terms. Where there is neither side load nor eccentricity, a

terms in the numerator of the first fraction of (9) disappear, leaving 

It is to be noted that the term P/A in equation (9) is developed f

sideration of a rectangular cross section. Equation (9) is therefore a

in this form only to columns of square-rectangular cross section.

Any of the foregoing equations can be solved, either for P/A or for M/S

sired to facilitate any particular design problem. Solution for P/A or 

will be easier if any redundant terms are first dropped. Newlin publis

these equations in a form in which they are solved for P/A 

Long Columns

In long columns of the Euler class, the problem of eccentric and side l

made more complex by the addition of stress induced by the curvature of 

column itself. The secant formulas are applicable but rather difficult 

in problems of determination of safe load. Furthermore, it is desirable 

press all conditions of side loading in terms of their resulting moment

stresses, thus making one general formula applicable to all loading con

Newlin succeeded in doing this.

In developing a general formula for the condition of side and eccentric 

loads, use is made of a relationship developed by Salmon~ for columns b

sinusoidal curvature. While it is recognized that the most common side 

ings do not produce sinusoidal curvature, Salmon showed that with small 

of curvature, the deflection is practically the same whether the elasti

is circular, parabolic, or sinusoidal. The error introduced by assumin

soidal curvature is probably of lesser magnitude than approximations in 

present state of knowledge on the strength propetiies of wood or on the 

ditions under which wood columns are loaded.

Salmon considered a column originally bent to a sinusoidal curvature wi

flection at midlength (fig. 2). Under the action of an axial load P 

Report No. 1782 -7-
fleotion increases to a value The original curvature of the column is represented by the equation

(10)

For this condition, Salmon arrived at the relationship

(11)

where P is the axial load on the oolumn and W is the critical Euler load (notation not the same as that used by Salmon), By algebraic transformation, equation (11) becomes

(12)

One property of a sinusoidal elastic curve is that deflection is proportional

to the stress causing it. Taking, for example, the equation for sinusoidal

curve (10), there follows by differentiation

(13)

Since this

moments in

is an elastic curve, the general equation of external and internal

bending may be applied:

Substituting from equation (13)

(14)

(15)

From the general expression for bending stress equal to M/S,

stress (16)

Equation (15) shows the proportionality of moment, and equation (16) shows the

proportionality of stress to deflection.

Report No. 1782 -8-
Newlin used Salmon's originally bent column (fig. 2) to represent an eccentritally loaded long column with axial load P and an eccentricity corresponding

to the deflection If side loads are applied to such a column, additional

deflections may result. If curvature remains sinusoidal, the summation of deflections will hold the same relationship to as

holds in equation (12). Then

(17)

It has already been shown in equation (16) that, with sinusoidal curvature, deflection is proportional to stress. Since each of the curves in figure 2 is

assumed to be sinusoidal, all of the deflections 

have the same factor of proportionality to the

the deflection ratio

is thus equal to a stress ratio

The load ratio P/W can also be replaced by the corresponding stress ratio

where w is the critical Euler stress. Equation (17) then becomes

Now let be replaced by a fictitious stress which, if it had been present

bending stresses that cause them

(18)

in Salmon's column, would have caused the deflection The fictitious stress

 which could have caused deflection can be evaluated in terms of the critical Euler load by substituting in equation (16) for a sinusoidal

elastic curve

(19)

It is now necessary to determine what eccentricity e on a straight column corresponds to the deflection in Salmon’s originally bent column. The two

Report No. 1782 -9-
conditions are compared in figure 3. Consider Salmon's column (fig. 3A) with

an original deflection and bent under a small load P to an additional deflection δ with sinusoidal curvature. From equation (12)

By algebraic traneformation

which, when P/W is small, becomes approximately

(20) 

(21) 

(22)

If the column were originally straight with the same load P applied with an

eccentricity e such that the same deflection δ resulted (fig. 3B), there is

obtained from the secant formula for eccentric load

Since in the Euler formula

Substituting (24) in (23) gives

Tables of integrals express the secant of an angle in terms of the series

(23)

(24)

(25)

(26)

With P/W small, terms beyond the first two of (26) are negligible, and (25) becoms very nearly

(27)

 Pierce, B. O., A Short Table of Integrals, third edition revised, New York,

1929.

Report No. 1782 -10-
Equating (27) with (22) gives

Since is very nearly 

Newlin estimated the relationship of e to from a different basis, 

rived at the same values as in (29), pointing out that could be rep

with results that would check quite closely with the secant formul

Substitution of (29) in (19) gives

Substitution of expression (24) for the critical

column in (30) gives

Euler load on a pin-e

In a rectangular column, A = bd and S = ( d measured in the directi

eccentricity of the load), from which

Now let in equation (18) be replaced by M/S , a stress induced by a 

that is independent of the axial load. Let be replaced by z P/A , a 

induced by a side load that is proportional to the axial load. Note th

M/S or z P/A may be entirely missing in actual loading conditions.

Substituting the equivalents for in equation (18) 

Equation (33) expresses the condition for breaking load, since w is the 

critical or breaking stress. The same relation holds for safe or desi

by substituting c , the Euler safe working stress, for w . Equation (33

becomes

Report No. 1782 -11-
The available value for the bending stress when all loads are on is now limited by the safe bending stress value f. In the extreme fiber on the concave

side of the column) and P/A are additive, and their sum cannot be permitted

to exceed f (as previously noted under "Analysis of Combined Stresses)"

analysis leading to equation (11) was not applied, so for Salmon's relationship to hold, or its equivalent should be deducted from the sum

 + P/A. Then, for the condition of safe loading, the sum 

cannot exceed f . To solve for the maximum safe load using the assumption developed under "Analysis of Combined Stresses,”

By transposition, and equation (34) then becomes

(35)

Algebraic transformation of equation (35) by clearing fractions and collecting.

terms gives a more simple form

(36)

the general equation for the maximum safe value of any combination of side loads

and eccentricity on long columns.

Equation (36) is in a general form from which equations for a number of special

cases can be readily derived by dropping the inapplicable terms. For example,

if all side load is proportional to axial load, and axial load is eccentrically

applied, the term M/S is dropped. If all side load is proportional to axial

load, and axial load is centrally applied, both M/S and P/A are dropped.

Where there is eccentricity but no side load, both M/S and z (P/A) are dropped.

Other loading conditions can be similarly represented by suitable selection

of terms. Where there is neither side load nor eccentricity, all three terms

in the numerator of the first fraction of (36) disappear, leaving only

P/A =

c

l or P/A = c . These are the same processes as in equation (9) for short

columns; it is to be remembered, however, that c in equation (36) is the safe

long-column stress, while c in equation (9) is the safe short-column stress.

Because of the method of its derivation equation (36), like equation (9), is

applicable only to columns of square or rectangular cross section.

Report No. 1782 -12-
Equation (36) or any of the specialized equations derived from it can be 

for P/A or for M/Sc , if desired to facilitate my particular design probl

is to be noted that solution for P/A results in most cases in a quadrati

tionj for which there are two roots of the form

The root using the minus sign before the radical is used, since the root 

the plus sign would give an impossible result. Newlin published these e

in a form in which they are solved for P/A. 

Equation (36) is developed from the assumption that maximum moment and m

deflection occur at midlength of the column. This is true for eccentric 

load and for side load applied symmetrically along the length, but is no

with large side loads unsymmetrically placed along the length of the colu

Recognizing this point, Newlin made the following recommendations in reg

unsymmetrically applied side load.

1. For a single concentrated side load, the stress under the load can be 

regardless of the position of the side load with reference to the length 

column. Flexural stress from side load iS maximum at this point.

2. The stress to use with a system of side loads is the maximum stress 

the system. Where the system of side loads is such that maximum moment 

side load is present near the end of the column, some slight error on th

of overload will occur.

Columns of Intermediate Length

Newlin chose a lower limiting value of 20 for the slenderness ratio i

long-column formula (36). This limit was chosen as being approximately 

lower limit for values of K (slenderness ratio separating intermediate- 

long-column groups) in the most common species and grades for structural 

In many species and grades, the K value exceeds 20, and formula (36) may 

used for intermediate columns coming under the Forest Products Laboratory

fourth-power parabolic formula~ instead of the Euler formula. Where this 

the case, the value of c in equation (36) is determined from the paraboli

stead of the Euler formula, but use of equation (36) is otherwise the sam

For columns with slenderness ratios between 11 and 20, the safe load

under side load and eccentricity may be determined by straight-line inter

tion between equation (9) for = 11 and equation (36) for = 20. Load

thus determined may be somewhat in error on the conservative side.

Report No. 1782 -13-
Examples

Assume a species and grade of wood having the following properties:

E = 1,600,000 pounds per square inch

f = 1,600 pounds per square inch

c = 1,200 pounds per square inch (for short column)

Values of c for intermediate or long columns are obtained with the Wood Handbook formulas.2

Assume a 6- by 8-inch (nominal dimensions) column in various lengths and under

various conditions of loading. The side loads and eccentricity are assumed as

acting to bend the column in the 8-inch direction. The actual cross-sectional

dimensions are 5-1/2 by 7-1/2 inches.

The design of the column in these examples provides only for flexure in the 8-

inch direction. The column is assumed to be stable or supported so that there

is no deflection in the 6-inch direction; if simultaneous deflection in both

directions is possible, the analysis becomes more complex. For the purpose of

determining limits of application of the formulas in the examples, the slenderness ratio is based on 7-1/2 rather than 5-1/2 inches.

Long Column with Side Load and Concentric Axial Load

Let the column be 20 feet long, so that = 32. Asumme a side load of 75

pounds per foot of length, uniformly distributed. Determine the safe axial

load centrally applied.

Equation (36) is utilized by dropping the terms representing eccentricity and

side load proportional to end load, so that it becomes

Since M/S is known, the equation is solved for P/A giving

(37)

 Forest Products Laboratory Wood Handbook. U. S. Dept. of Agr., revised 1955)

p. 216.

Report No. 1782 -14-
The safe load is 5.5 x 7.5 x 167 = 6,900 pounds.

Eccentric Axial Load on Long Column

Let the column be 20 feet long, and the eccentricity be 2-1/2 inches, wi

side load.

Equation (36) takes the form

which, solved for P/A gives

f = 1,600 p.s.i. and c = 428 p.s.i. as before

The safe load is 5.5 x 7.5 x 240 = 9,900 pounds.

If the length of the column is 12-1/2 feet, so that = 20, the solution 

same except for the value of c . Since K = 23.4, which is greater than 2

fourth-power parabolic formula is used, and c = 986 pounds per square in

Report No. 1782 -15-
 Substitution of this value in equation (38) gives P/A = 335 pounds per square

inch. The safe load is 5.5 x 7.5 x 335 = 13,800 pounds.

Eccentric Axial Load on Short Column

Let the column be 7 feet long, so that = 11. The eccentricity is 2-1/2 inches

and there is no side load. Determine the safe load.

When solved for P/A, this becomes

(39)

The safe load is 5.5 x 7.5 x 480 = 19,800 pounds.

Eccentric Axial Load on Intermediate Column

If the column is 10 feet long, so that = 16, the safe load is obtained by

straight-line interpolation between the values for = 11 and = 20.

19,800 - 5/9 (19,800 - 13,800), = 16,500 pounds.

Axial Load Known

Where P/A has a known value and it is desired to determine the permissible side

load, equations (9) or (36) or any modifications thereof can be solved for M/S .

Since M/S appears only in the first power in either equation, solution for it

offers no special problems. Having determined M/S , the permissible side load

for the assumed conditions can be found.

Report No. 1782 -16-
Eccentricity and Two Side Loads

A 4- by 16-inch (nominal) member is used with width vertical in the upp

of a roof truss. It supports a number if uniformly distributed purli

with purlin assumed to give full lateral support so that the member q

as a short column. The roof system is designed so that the flexural 

in the member from the roof loads through the purlins in one-half th

stress in the member from the truss reactions. There is a additon a 

trated load suspended near the center of length of the member, causing 

ural stress of 200 pounds per square inch. The design at the panel point

truss is such that the axial load is centered at a point 1.55 inches a

center of width of member. Physical properties of the material 

same as in preceding examples. Determine the safe axial load, a

actual dimensions of 3-5/8 by 15-1/2 inches.

In this case, equation (9) in its complete form is used. When solved f

this becomes

z P/A = = 262 pounds per square inch flexural stress from purlin loa

Safe axial load is 525 x 3-5/8 x 15-1/2 = 29,500 pounds.

Column Formula Summarized

The general formulas developed by Newlin are

for columns with slenderness ratios of 11 or less, and

for columns with slenderness ratios of 20 or more, where

Report No. 1782 -17-
P/A =

M/S =

direct compressive stress induced by axial load.

flexural stress induced by side loads that are independent of end

load.

c = allowable unit working stress in compression parallel to grain for

a column of the slenderness ratio under consideration with centrally applied axial load and no side load.

f = allowable unit working stress in flexure that is permitted where

flexural stress only exists.

e = eccentricity.

d = depth of column, measured in the direction of side loads or

eccentricity.

z = ratio of flexural to direct compressive stress when both result

from the same loading, so that the ratio remains constant while

the load varies.

Stresses for columns with slenderness ratios between 11 and 20 are determined

by straight-line interpolation between formula (9) for a slenderness ratio of

11 and formula (36) for a slenderness ratio of 20.

These formulas may be simplified for some conditions of loading by dropping

out certain terms; for example, if there are side loads and a concentrically

applied end load, e becomes zero, and the first term in the numerator of equations (9) and (36) disappears. The formulas can be solved for P/A or for M/S

where this will facilitate their use.

These formulae are applicable only to columns of square or rectangular cross

section.

Where side loads are such that maximum deflection axial maximum flexural stress

do not occur at midlength of the column, it is generally satisfactory to consider M/S as the maximum flexural stress due to the load or loads, regardless

of its position in the length of the column. When the point of this maximum

stress is near the end of the column, a slight error on the side of overload

will occur.

Report No. 1782 -18-
RATIO OF BENDING STRESS TO TOTAL COMB

Figure 1. --Strength values of clear Sitka spruce u

stress, as determined by test and calcualted by 
Figure 2 .--Load on column with sinusoidal curvature.
Figure 3.--Straight column with eccentricity equivalent to Salmon’s

originally bent column, A , Salmon's originally bent column; B ,

 straight column with eccentric loading.
SIDE LOADS AND ECCENTRICITIES ACTING IN TWO DIRECTIONS ON COLUMNS

Supplement

FORMULAS FOR COLUMNS WITH SIDE

to

LOADS AND ECCENTRICITY

By

C. B. NORRIS, Engineer

and

LYMAN W. WOOD, Engineer

October 1951 

---- -

This supplement presents formulas for columns with stresses and deflections resulting from side loads or eccentricities in more than one direction. The concept is thus broadened from that of the main body of this report, which deals

with side loads and eccentricities in the same plane and assumes that deflection

of the column takes place only in that plane.

A solution of this problem for short columns is not difficult, since it is

necessary only to sum up the stresses without taking into account the deflections. Consider equation (9), using the same notation for stresses and ecoentricities in one direction and an equivalent notation with prime superscripts

for the counterparts in a direction at right angles. Equation (9) then becomes

( 9

1

)

Note in the above that P is the full compressive load, causing bending stresses

from eccentricity in both directions in addition to the direct compression.

Equation (91) defines the condition for maximum allowable stress in the extreme

fiber, which in this case is on a corner rather than along one side of the

column.

The problem in long columns is somewhat more complex, since the deflections

must be considered, but an analysis along lines similar to that for short columns seems possible.

Report No. 1782 -19-
Equation (17) has for its counterpart

representing deflections at right angles to those of equation (17). No

that W

1

 the critical Euler load, differs from W except for columns of 

cross section, where the two are the same.

Substitution of stresses for deflections gives a counterpart to equation 

Substitution of equivalent stresses gives a counterpart to equation (33

Equations (33) and (331) are solved for and respectively, the 

corresponding to the total deflections when all eccentric and side loads 

This gives

and

Let be a total stress corresponding to the total of all deflections 

directions. For safe loadings, use c and c

1

 instead of w and w

1

 (in squ

columns, c = c

1

 ). Then

Report No. 1782 -20-
(37)

By a reasoning analogous to that leading to equation (35), the safe stress f

must be equal to

or by transposition

(38)

Elimination of between equations (37) and (38) gives

a counterpart of equation (35).

Algebraic transformation of equation (39) by clearing fractions and collecting

terms gives the form

(40)

Equation (40) defines the condition for safe loading. It is somewhat complex,

but in its general form, any combination of side loads and eccentricities can

be represented by it. It may be simplified for some conditions; for example,

z and z

1

 may be equal or one or both may be zero; or in square columus,

c = c

1

 and d = d

1

 .

Equation (40) is a cubic equation in P/A so that, with other,quantities known,

three possible values for P/A are indicated. Of the three, the least value that

that is not imaginary should be used in design. Solution for

 or is more simple and yields only one value.

Report No. 1782 -21- .5-27