Analysis and design of frames

J. agric. Engng Res. (1967) 12 (3) 222-228

Analysis and Design of Frames J.

R. O'CALLAGHAN*;

J. J.

RYANt;

T. P.

DWANEt

A survey of theories applicable to the analysis of braced and unbraced frames is given. Calculated values obtained by elasticity theory, strain energy theory and torsional resistance theory are compared with experimental results, for four different frame configurations. Existing theoreticalmethodsare shown to be inadequate. A newtheory,whichgives good agreement with experimental results, is presented.

1. Introduction The design of frames for lorries, agricultural trailers and similar vehicles is generally on an empirical basis. Since the frame is stressed mainly in a torsional mode, torsional stiffness and strength are of prime importance. A method of predicting the torsional stiffness of different frame arrangements should lead to a more rational approach in the design process. Work was undertaken to assess the accuracy of various theoretical methods in predicting the torsional stiffness of four types of frame, by comparing theoretical values with experimental results.

2. Testing The frames tested, which all had the same outside dimensions (l7t x 8£ in) were:A: plain rectangular frame B: frame having two equally spaced transverse braces C: frame with single diagonal 45° bracing D: frame with double diagonal 45° bracing. The frames were chosen as scale models of a 12 x 6 ft trailer frame, to one-eighth scale approximately. Scale models were chosen for convenience in testing and because the study was primarily a comparative one between the different theories and bracing systems. The models were fabricated from I X t in mild steel bar. Members forming a corner or angle were bevelled at 45° before assembly, all welds being filed flat. Testing was carried out on an Amsler pendulum-type torsion testing machine, modified to hold the frames. 3. Survey of existing theories 3.1. Elasticity The usual torsional equation, T=

G:()

is valid for closed circular sections only. For other

. GR () may b e used , were h . t he ettective a: . . or " torsiona . 1 sections T = -yR IS po 1ar moment 0 f inertia

resistance" of the section, which can be found theoretically for sections with regular

boundaries.i-" For a rectangular section of dimensions b x d, R=fJbd3 where fJ is a function of ~ given by Fig. 1, and b is the larger of the section dimensions. • Department of Agricultural Engineering, University of Newcastle upon Tyne, formerly Mechanical Engineering Department, University College, Dublin t Mechanical Engineering Department, University College, Dublin

222

223

1. R. O'CALLAGHAN; J. J. RYAN; T. P. DW ANE

o-r '-_'--_'----1_--'_---1._--'

o

2

6

4

8

10

12

b d

Fig. 1. ~ as a/unction

F = T = J = I =

R = a, () =

force torque polar moment of inertia second moment of area torsional resistance angular deflection

o/~

KEY TO SYMBOLS t5, .d = linear displacement b, d, L = length E = Young's modulus G = shear modulus k, K = torsional stiffness n = number of cells in frame

3.1.1.

Plain frame, transverse-braced frame The sides of the frame (Fig. 2, top left) may be considered" as cantilevers in bending with

imposed twisting, leading to the equation 1 5L2 -=

K

+ - - - - - -10- - - - - G(R 1 + R 2) 3L 2 E( /1 + /2) + 3G (R + R ) 1 L 3 L 3 2L 1 2 1

2

... (1)

2

Where there are n cells or compartments, the frame stiffness is K where K is the stiffness of one n cell. 3.1.2.

Double diagonal-braced frame The cross-bracing may be considered.' as two fixed-end beams, giving a torsional stiffness K 24 E /3 L 1 2 ... (2) n (L 12+L 22)i n

where /3 is the second moment of area of the diagonal members. 3.2. Strain energy The strain energy approach uses the fact that the deflection of a structure at the point of application of a load, in the direction of that load, is theoretically equal to the partial derivative of the total strain energy with respect to the load. The following expression can be used for plain frames. 3 K

... (3)

224

ANALYSIS AND DESIGN OF FRAMES

Fig. 2. The frame and fo rce and moment system acting on its sides

where ... (4)

4

For n cells, the stiffness is approximately!i-

n

No treatment for diagonal-braced frames was found.

3.3. Torsional resistance The torsional resistance method" consists of finding the sum of the torsional resistances of the members of the braced frame and using it directly in the torque equation, ignoring bending and coupling effects. If there are m longitudinal members of length L 2 in a frame having individual torsional resistances of Rh R 2 ••••• •• R m, then the angular deflection due to a torque T 1 about a longitudinal axis is

Similarly for n transverse members, deflection due to T 2 is T 2 L)

T2 L1

Bn = G (R 1+R 2 + . . . R n) = GR"

225

J. R. O'CALLAGHAN; 1. J. RYAN; T. P. DW ANE

But T 1=F1 L 1 and T 2=F2 L 2 (Fig. 2, left), assuming the force Fto have components F1 and F 2 , i.e. F=F1+F2 • 2L1 2L1 Also L = Om, L- = On where L1 = deflection of one corner. 2

1

R"]

R' :.F=2L1G [ L L 2 + L 2 L 1 2 2 1 Therefore the torsional stiffness

T

[R' + L; R"]

FL 1 Om = 2L1/L 2 = G L 2

The total twist of frame = 2 Om

(R' + R") L

G ~ :. torsional stiffness = 2

... (5)

1

The method cannot be used for more complex cases. 4. Proposed theory 4.1. Analysis of force systems Fig. 2 shows the deflected shape of the frame (bottom) and the force and moment system acting on two sides (right). Side L 1 can be treated as two cantilevers of length ~1 acted on by force F1 at the free end and by bending moment T 2 , which is equal to the torsional moment acting on side L 2 • The free end of the cantilever deflects by an amount 0 relative to its support. A vertical plane through the root of the cantilever in the unloaded state will not remain vertical but will rotate through an angle a due to the vertical displacements of the sides L 1• The total deflection of the cantilever end is, therefore, (01

+ Ll

a1)= L! , and the slope at the end is (a 1

+slope of cantilever relative to its support).

4.1.1. Plain and transverse-braced frames L 1 The bending moment acting at distance x from the free end of the cantilever 2 is (Fl x- T 2)

d 2y Ell dx 2 =F1 x - T 2 dy Ell dx

=

Fl x2 -2- -T2 x+A

... (6)

dy i, T 2 t; F1 L 12 dx=0 forx= 2" :.A=-2---83

2

. () F1 x T2 x (F1 L 12 T 2 L 1 ) B .. Ell Y = -6- - -2-- -8- - -2- x+ L1 F1 L 13 T 2L12 y=O at x=2" :.B= ~ - -8-

... (7)

226

ANALYSIS AND DESIGN OF FRAMES

... (8)

and 8 2=angle of twist of side L 2

[~~]

= a1 +

x=o ... (9)

2 • 2 . . by 8 2 reIative ' to t he T 2 = 2 GR L 8 SInce each h aIf 0 f sride L 2 twists From the torque equation, 2

centre section.

Torsional stiffness T 2 () -

(k 1+k 2) ALI 4A/L 1

... (10)

For a frame with transverse bracing (B), the torsional stiffness is .!..- times that derived by n Eqn (10) approximately. More accurately, the effect of each intermediate transverse member can be divided equally between adjacent cells. The stiffness of a frame with n cells is given by

~[ K

1+(n-l)K2 ]

where K1 and K 2 are the torsional stiffnesses of cells with transverse members of torsional resistances R1 and

~ • "2 and second moments of area II and 2' respectively.

R1

4.1.2. Diagonal-bracedframes In the case of the double diagonal-braced frame, where the diagonals are at 90° approximately, twisting of the diagonals can be ignored, and each diagonal of length La can be treated as two

cantilevers of length ~a. In this case, the free ends are deflected in the same direction, so the root section does not change its position under load.

227

J. R. a'CALLAGHAN; J. J. RYAN; T. P. DWANE

24 st,

Fa

:. LJ = J;T=k a is the stiffness of the diagonals, and the total stiffness is k+k a, where k is the stiffness of the outline frame. For 45° bracing

La=V2L 1

u,

:. k a=8'49 D

1

and torsional stiffness is

(k+ kna)L412

... (11)

For single diagonal bracing (C), the frame will have a stiffness at the braced corner according to Eqn (11) and at the unbraced corner according to Eqn (1). The mean torsional stiffness for the test frame is '" (12) which is a reasonable approximation. The situation is complicated in this case since the geometrical longitudinal centre line does not coincide with the torsional axis.

5. Results The results of the torque/deflection tests for the four frames tested are presented in Fig. 3. The slope of each characteristic represents the stiffness of the corresponding frame. Values calculated using the various theories are presented in Table I, together with experimental values. TABLE I Torsional stiffness, x 10· lbf • in/rad

Theory ------

Frame

A

Elasticity - - - - - Strain energy 1 2

B

1·15 1·17

C D

3·12

-

9·75 13-6 47·6 85·5

Torsional resistance

10·6 13-9

Experimental

9·07 12·1 50·9

-

-

9·18 13-1 42·0 86·1

,

6. Discussion For the simple frame, the torsional resistance method gave the closest approximation to the actual case. This result is surprising in view of the simplicity of the approach, and considering that bending of the members was ignored. The assumption that bending is negligible appears to be valid for the particular dimensions chosen but if the ratio

~: were much less than

0'5, or

if the second moment of area (I) was comparable to the torsional resistance (R), the assumption would not hold. The proposed theory offered a less accurate value in this case (Table I, column 2), being 6 % in error, but was superior in all other cases, the increased stiffness being due to the

228

ANALYSIS AND DESIGN OF FRAMES 600 500

-...

:e

tV ::J

~

~

400 300 200 100

0

234

5

6

7

8

9

Angular deflection, deg

Fig. 3. Angular deflection as a function of torque

consideration of bending. The strain energy method also gave a reasonable estimate for the plain frame. The transverse-braced frame (B) stiffness was also predictable with a fair degree of accuracy by the three methods, the proposed theory offering the more accurate figure, when the transverse members are considered to be divided between adjacent cells. The torsional resistance method was the least accurate, being 7·6 % in error. Elasticity and strain-energy theory tended to overestimate the stiffness in this case also, while the torsional resistance method gave a low value. In the case of the single-diagonal frame (C), the theory proposed, while being approximate, gives a value 18 % in error, compared with 21 % for the torsional resistance approach, the only other method yielding a solution. No solution by the strain-energy method is available for the two more complex cases of single and double diagonal bracing. The proposed theory gave satisfactory agreement for the double-diagonal braced frame (D). The calculated value includes the stiffness of the outline frame, which cannot be neglected without serious error. The values for three frames given by other theories based on the elasticity method are given in Table I, column 1. The double-diagonal bracing gave an increase in stiffness of 830 % over the plain frame for a 94 %increase in material, corresponding figures for the single diagonal frame were 357 %increase in stiffness for 47 %more material, and for the cross-braced frame, 47 %increase in stiffness for 33 %more material. Compared with the cross-braced system, the double diagonal frame yielded a 558 % stiffness gain for a 46 % increase in material, the corresponding figures for the singlediagonal frame being 220 and 10·3 %.

7. Conclusions (a) Elasticity, strain energy and torsional resistance theories can be used to predict the stiffness of plain and transverse braced frames with a tolerable degree of accuracy. The proposed elasticity theory is less cumbersome than the strain-energy theory and more rigorous than the torsional resistance theory. (b) The proposed elasticity method gives reasonable accuracy in the case of single and double diagonal bracing, where other theories are inadequate. (c) A much more economic use of material is possible using diagonal rather than cross-bracing. REFERENCES I

2 3

4 5

Faupel, J. H. Engineering design. J. Wiley & Sons Inc., N.Y., 1964 Seely, F. B.; Smith, A. M. Advancedmechanics of materials. J. Wiley & Sons Inc., N.Y., 1932 Cooke, C. J. Torsional stiffness. Auto. des. Engng, 1963, 2 (14) 91 Torsional stiffness. Auto. des. Engng, 1962, 1 (2) 53 Blodgett, O. W. Design of'weldments. J. F. Lincoln Arc Weld. Found., Cleveland, Ohio, 1963