Tag Archives: Rigid

Types of Beam Connections

In general, all beam connections are classified as either framed or seated. In the framed type, the beam is connected to the supporting member with fittings (short angles are common) attached to the beam web. With seated connections, the ends of the beam rest on a ledge or seat, in much the same manner as if the beam rested on a wall.

Bolted Framed Connections

When a beam is connected to a support, a column or a girder, with web connection angles, the joint is termed ‘‘framed.’’ Each connection should be designed for the end reaction of the beam, and type, size and strength of the fasteners, and bearing strength of base materials should be taken into account. To speed design, the AISC Manual lists a complete range of suitable connections with capacities depending on these variables. Typical connections for beam or channels ranging in depth from 3 to 30 in are shown in Fig. 7.46.
To provide sufficient stability and stiffness, the length of connection angles should be at least half the clear depth of beam web.
For economy, select the minimum connection adequate for the load. For example, assume an 18-in beam is to be connected. The AISC Manual (ASD) lists three- and four-row connections in addition to the five-row type shown in Fig. 7.46. Total shear capacity ranges from a low of 26.5 kips for 3⁄4-in-diam A307 bolts in a three-row regular connection to a high of 263.0 kips for 1-in-diam A325 bolts in a five-row heavy connection, bearing type. This wide choice does not mean that all types of fasteners should be used on a project, but simply that the tabulated data cover many possibilities, enabling an economical selection. Naturally, one type of fastener should be used throughout, if practical; but shop and field fasteners may be different.
Bearing stresses on beam webs should be checked against allowable stresses (Arts. 7.30.1 and 7.30.2), except for slip-critical connections, in which bearing is not a factor. Sometimes, the shear capacity of the field fasteners in bearing-type connections may be limited by bearing on thin webs, particularly where beams frame into opposite sides of a web. This could occur where beams frame into column or girder webs.

One side of a framed connection usually is shop connected, the other side field connected. The capacity of the connection is the smaller of the capacities of the shop or field group of fasteners.
In the absence of specific instructions in the bidding information, the fabricator should select the most economical connection. Deeper and stiffer connections, if desired by the designer, should be clearly specified.

Bolted Seated Connections

Sizes, capacities, and other data for seated connections for beams, shown in Fig. 7.47, are tabulated in the AISC Manual. Two types are available, stiffened seats (Fig. 7.47a) and unstiffened seats (Fig. 7.47b).
Unstiffened Seats. Capacity is limited by the bending strength of the outstanding horizontal leg of the seat angle. A 4-in leg 1 in thick generally is the practical limit.
In ASD, an angle of A36 steel with these dimensions has a top capacity of 60.5 kips for beams of A36 steel, and 78.4 kips when Fy  50 ksi for the beam steel.
Therefore, for larger end reactions, stiffened seats are recommended.
The actual capacity of an unstiffened connection will be the lesser of the bending strength of the seat angle, the shear resistance of the fasteners in the vertical leg, or the bearing strength of the beam web. (See also Art. 7.22 for web crippling stresses.) Data in the AISC Manual make unnecessary the tedious computations of balancing the seat-angle bending strength and beam-web bearing.
The nominal setback from the support of the beam to be seated is 1⁄2 in. But tables for seated connections assume 3⁄4 in to allow for mill underrun of beam length.

Stiffened Seats. These may be obtained with either one or two stiffener angles, depending on the load to be supported. As a rule, stiffeners with outstanding legs having a width less than 5 in are not connected together; in fact, they may be separated, to line up the angle gage line (recommended centerline of fasteners) with that of the column.
The capacity of a stiffened seat is the lesser of the bearing strength of the fitted angle stiffeners or the shear resistance of the fasteners in the vertical legs. Crippling strength of the beam web usually is not the deciding factor, because of ample seat area. When legs larger than 5 in wide are required, eccentricity should be considered, in accordance with the technique given in Art. 7.32. The center of the beam reaction may be taken at the midpoint of the outstanding leg.
Advantages of Seated Connections. For economical fabrication, the beams merely are punched and are free from shop-fastened details. They pass from the punching machine to the paint shed, after which they are ready for delivery. In erection, the seat provides an immediate support for the beam while the erector aligns the connection hole. The top angle is used to prevent accidental rotation of the beam. For framing into column webs, seated connections allow more erection clearance for entering the trough formed by column flanges than do framed connections.
A framed beam usually is detailed to whim 1⁄16 in of the column web.
This provides about 1⁄8 in total clearance, whereas a seated beam is cut about 1⁄2 in short of the column web, yielding a total clearance of about 1 in. Then, too, each seated connection is wholly independent, whereas for framed beams on opposite sides of a web, there is the problem of aligning the holes common to each connection.
Frequently, the angles for framed connections are shop attached to columns.
Sometimes, one angle may be shipped loose to permit erection. This detail, however, cannot be used for connecting to column webs, because the column flanges may obstruct entering or tightening of bolts. In this case, a seated connection has a distinct advantage.

Welded Framed Connections

The AISC Manual tabulates sizes and capacities of angle connections for beams for three conditions: all welded, both legs (Fig. 7.48); web leg shop welded, outstanding leg for hole-type fastener; and web leg for hole-type fastener installed in shop, outstanding leg field welded. Tables are based on E70 electrodes. Thus, the connections made with A36 steel are suitable for beams of both carbon and highstrength structural steels.
Eccentricity of load with respect to the weld patterns causes stresses in the welds that must be considered in addition to direct shear. Assumed forces, eccentricities, and induced stresses are shown in Fig. 7.48b. Stresses are computed as in the example in Art. 7.34, based on vector analysis that characterizes elastic design. The capacity of welds A or B that is smaller will govern design.
If ultimate strength (plastic design) of such connections is considered, many of the tabulated ‘‘elastic’’ capacities are more conservative than necessary. Although AISC deemed it prudent to retain the ‘‘elastic’’ values for the weld patterns, recognition was given to research results on plastic behavior by reducing the minimum beam-web thickness required when welds A are on opposite sides of the web. As a result, welded framed connections are now applicable to a larger range of rolled beams than strict elastic design would permit.

Shear stresses in the supporting web for welds B should also be investigated, particularly when beams frame on opposite side of the web.

Welded-Seat Connections

Also tabulated in the AISC Manual, welded-seat connections (Fig. 7.49) are the welded counterparts of bolted-seat connections for beams (Art. 7.35.2). As for welded frame connections (Art. (7.35.3), the load capacities for seats, taking into account for eccentricity of loading on welds, are computed by ‘‘elastic’’ vector analysis. Assumptions and the stresses involved are shown in Fig. 7.49c.
In ASD, an unstiffened seat angle of A36 steel has a maximum capacity of 60.5 kips for supporting beams of A36 steel, and 78.4 kips for steel with Fy  50 ksi (Fig. 7.49a). For heavier loads, a stiffened seat (Fig. 7.49b) should be used.
Stiffened seats may be a beam stub, a tee section, or two plates welded together to form a tee. Thickness of the stiffener (vertical element) depends on the strengths of beam and seat materials. For a seat of A36 steel, stiffener thickness should be at least that of the supported beam web when the web is A36 steel, and 1.4 times thicker for web steel with Fy  50 ksi.

When stiffened seats are on line on opposite sides of a supporting web of A36 steel, the weld size made with E70 electrodes should not exceed one-half the web thickness, and for web steel with Fy  50 ksi, two-thirds the web thickness.
Although top or side lug angles will hold the beam in place in erection, it often is advisable to use temporary erection bolts to attach the bottom beam flange to the seat. Usually, such bolts may remain after the beam flange is welded to the seat.

End-Plate Connections

The art of welding makes feasible connections that were not possible with oldertype fasteners, e.g., end-plate connections (Fig. 7.50).

Of the several variations, only the flexible type (Fig. 7.50c) has been ‘‘standardized’’ with tabulated data in the AISC Manual. Flexibility is assured by making the end plate 1⁄4 in thick wherever possible (never more than 3⁄8 in). Such connections in tests exhibit rotations similar to those for framed connections.
The weld connecting the end plate to the beam web is designed for shear. There is no eccentricity. Weld size and capacity are limited by the shear strength of the beam web adjoining the weld. Effective length of weld is reduced by twice the weld size to allow for possible deficiencies at the ends.
As can be observed, this type of connection requires accurate cutting of the beam to length. Also the end plates must be squarely positioned so as to compensate for mill and shop tolerances.
The end plate connection is easily adapted for resisting beam moments (Fig. 7.50b, c and d). One deterrent, however, to its use for tall buildings where column flanges are massive and end plates thick is that the rigidity of the parts may prevent drawing the surfaces into tight contact. Consequently, it may not be easy to make such connections accommodate normal mill and shop tolerances.

Special Connections

In some structural frameworks, there may be connections in which a standard type (Arts. 7.35.1 to 7.35.5) cannot be used. Beam centers may be offset from column centers, or intersection angles may differ from 90, for example.
For some skewed connections the departure from the perpendicular may be taken care of by slightly bending the framing angles. When the practical limit for bent angles is exceeded, bent plates may be used (Fig. 7.51a).
Special one-sided angle connections, as shown in Fig. 7.51b, are generally acceptable for light beams. When such connections are used, the eccentricity of the fastener group in the outstanding leg should be taken into account. Length l may be reduced to the effective eccentricity (Art. 7.32).
Spandrel and similar beams lined up with a column flange may be conveniently connected to it with a plate (Fig. 7.51c and d). The fasteners joining the plate to the beam web should be capable of resisting the moment for the full lever arm l for the connection in Fig. 7.51c. For beams on both sides of the column with equal reactions, the moments balance out. But the case of live load on one beam only must be considered. And bear in mind the necessity of supporting the beam reaction as near as possible to the column center to relieve the column of bending stresses.

When spandrels and girts are offset from the column, a Z-type connection (Fig. 7.51e) may be used. The eccentricity for beam-web fasteners should be taken as l1, for column-flange fasteners as l2, and for fasteners joining the two connection angles as l3 when l3 exceeds 21⁄2 in; smaller values of l3 may be considered negligible.

Simple, Rigid, and Semirigid Connections

Moment connections are capable of transferring the forces in beam flanges to the column. This moment transfer, when specified, must be provided for in addition to and usually independent of the shear connection needed to support the beam reaction.

Framed, seated, and end-plate connections (Arts. 7.35.1 to 7.35.5) are examples of shear connections. Those in Fig. 7.17 (p. 7.32), are moment connections.
In Fig. 7.17a to g, flange stresses are developed independently of the shear connections, whereas in h and i, the forces are combined and the entire connection resolved as a unit.
Moment connections may be classified according to their design function: those resisting moment due to lateral forces on the structure, and those needed to develop continuity, with or without resistance to lateral forces.
The connections generally are designed for the computed bending moment, which often is less than the beam’s capacity to resist moment. A maximum connection is obtained, however, when the beam flange is developed for its maximum allowable stress.
The ability of a connection to resist moment depends on the elastic behavior of the parts. For example, the light lug angle shown connected to the top flange of the beam in Fig. 7.52b is not designed for moment and accordingly affords negligible resistance to rotation. In contrast, full rigidity is expected of the direct welded flange-to-column connection in Fig. 7.52a. The degree of fixity, therefore, is an important factor in design of moment connections.
Fixity of End Connections. Specifications recognize three types of end connections:
simple, rigid, and semirigid. The type designated simple (unrestrained) is  intended to support beams and girders for shear only and leave the ends free to rotate under load. The type designated rigid (known also as rigid-frame, continuous, restrained frame) aims at not only carrying the shear but also providing sufficient rigidity to hold virtually unchanged the original angles between members connected.

Semirigid, as the name implies, assumes that the connections of beams and girders possess a dependable and known moment capacity intermediate in degree between the simple and rigid types. Figure 7.54 illustrates these three types together with the uniform-load moments obtained with each type.

Although no definite relative rigidities have been established, it is generally conceded that the simple or flexible type could vary from zero to 15% (some researchers recommend 20%) end restraint and that the rigid type could vary from 90 to 100%. The semirigid types lie between 15 and 90%, the precise value assumed in the design being largely dependent on experimental analysis. These percentages of rigidity represent the ratio of the moment developed by the connection, with no column rotation, to the moment developed by a fully rigid connection under the same conditions, multiplied by 100.
Framed and seated connections offer little or no restraint. In addition, several other arrangements come within the scope of simple-type connections, although they appear to offer greater resistance to end rotations. For example, in Fig. 7.52a, a top plate may be used instead of an angle for lateral support, the plate being so designed that plastic deformation may occur in the narrow unwelded portion. Naturally, the plate offers greater resistance to beam rotation than a light angle, but it can provide sufficient flexibility that the connection can be classified as a simple type. Plate and welds at both ends are proportional for about 25% of the beam moment capacity. The plate is shaped so that the metal across the least width is at yield stress when the stresses in the wide portion, in the butt welds, and in the fillet welds are at allowable working values. The unwelded length is then made from 20 to 50% greater than the least width to assure ductile yielding. This detail can also be developed as an effective moment-type connection.
Another flexible type is the direct web connection in Fig. 7.52b. Figured for shear loads only, the welds are located on the lower part of the web, where the rotational effect of the beam under load is the least. This is a likely condition when the beam rests on erection seats and the axis of rotation centers about the seat rather then about the neutral axis.
Tests indicate that considerable flexibility also can be obtained with a property proportioned welded top-plate detail as shown in Fig. 7.52c without narrowing it as in Fig. 7.52a. This detail is usually confined to wind-braced simple-beam designs.
The top plate is designed for the wind moment on the joint, at the increased stresses permitted for wind loads.
The problem of superimposing wind bracing on what is otherwise a clear-cut simple beam with flexible connections is a complex one. Some compromise is usually effected between theory and actual design practice. Two alternatives usually are permitted by building codes:
1. Connections designed to resist assumed wind moments should be adequate to resist the moments induced by the gravity loading and the wind loading, at specified increased unit stresses.
2. Connections designed to resist assumed wind moments should be so designed that larger moments, induced by gravity loading under the actual condition of restraint, will be relieved by deformation of the connection material.
Obviously, these options envisage some nonelastic, but self-limiting, deformation of the structural-steel parts. Innumerable wind-braced buildings of riveted, bolted, or welded construction have been designed on this assumption of plastic behavior and have proved satisfactory in service.

Fully rigid, bolted beam end connections are not often used because of the awkward, bulky details, which, if not interfering with architectural clearances, are often so costly to design and fabricate as to negate the economy gained by using smaller beam sections. In appearance, they resemble the type shown in Fig. 7.17 for wind bracing; they are developed for the full moment-resisting capacity of the beam.
Much easier to accomplish and more efficient are welded rigid connections (Fig. 7.53). They may be connected simply by butt welding the beam flanges to the columns—the ‘‘direct’’ connection shown in Fig. 7.53a and b. Others may prefer the ‘‘indirect’’ method, with top plates, because this detail permits ordinary mill tolerance for beam length. Welding of plates to stiffen the column flanges, when necessary, is also relatively simple.
In lieu of the erection seat angle in Fig. 7.53b, a patented, forged hook-and-eye device, known as Saxe erection units, may be used. The eye, or seat, is shop welded to the column, and the hook, or clip, is shop welded to the underside of the beam bottom flange. For deep beams, a similar unit may be located on the top flange to prevent accidental turning over of the beams. Saxe units are capable of supporting normal erection loads and deadweight of members; but their contribution to the strength of the connection is ignored in computing resistance to shear.

A comparison of fixities intermediate between full rigidity and zero restrain in Fig. 7.54 reveals an optimum condition attainable with 75% rigidity; end and centerspan moments are equal, each being WL/ 16, or one-half the simple-beam moment.
The saving in weight of beam is quite apparent.
Perhaps the deterrent to a broader usage of semirigid connections has been the proviso contained in specifications: ‘‘permitted only upon evidence that the connections to be used are capable of resisting definite moments without overstress of the fasteners.’’ As a safeguard, the proportioning of the beam joined by such connections is predicated upon no greater degree of end restraint than the minimum known to be effected by the connection. Suggested practice, based on research with welded connections, is to design the end connections for 75% rigidity but to provide a beam sized for the moment that would result from 50% restraint; i.e., WL/ 12.
(‘‘Report of Tests of Welded Top Plate and Seat Building Connections,’’ The Welding Journal, Research Supplement 146S–165S, 1944.) The type of welded connection in Fig. 7.52c when designed for the intended rigidity, is generally acceptable.
End-plate connections (Fig. 7.50) are another means of achieving negligible, partial, and full restraint.

Continuous Beams and Frames

Fixed-end beams, continuous beams, continuous trusses, and rigid frames are statically indeterminate. The equations of equilibrium are not sufficient for the deter mination of all the unknown forces and moments. Additional equations based on a knowledge of the deformation of the member are required.

Hence, while the bending moments in a simply supported beam are determined only by the loads and the span, bending moments in a statically indeterminate member are also a function of the geometry, cross-sectional dimensions, and modulus of elasticity.

Sign Convention

For computation of end moments in continuous beams and frames, the following sign convention is most convenient: A moment acting at an end of a member or at a joint is positive if it tends to rotate the joint clockwise, negative if it tends to rotate the joint counterclockwise.
Similarly, the angular rotation at the end of a member is positive if in a clockwise direction, negative if counterclockwise. Thus, a positive end moment produces a positive end rotation in a simple beam.
For ease in visualizing the shape of the elastic curve under the action of loads and end moments, bending-moment diagrams should be plotted on the tension side of each member. Hence, if an end moment is represented by a curved arrow, the arrow will point in the direction in which the moment is to be plotted.

Carry-Over Moments

When a member of a continuous beam or frame is loaded, bending moments are induced at the ends of the member as well as between the ends. The magnitude of the end moments depends on the magnitude and location of the loads, the geometry of the member, and the amount of restraint offered to end rotation of the member by other members connected to it. Because of the restraint, end moments are induced in the connecting members, in addition to end moments that may be induced by loads on those spans.
If the far end of a connecting member is restrained by support conditions against rotation, a resisting moment is induced at that end. That moment is called a carryover moment. The ratio of the carry-over moment to the other end moment is called carry-over factor. It is constant for the member, independent of the magnitude and direction of the moments to be carried over. Every beam has two carry-over factors, one directed toward each end.
As pointed out in Art. 5.10.6, analysis of a continuous span can be simplified by treating it as a simple beam subjected to applied end moments. Thus, it is convenient to express the equations for carry-over factors in terms of the end rotations of simple beams: Convert a continuous member LR to a simple beam with the same span L. Apply a unit moment to one end (Fig. 5.60). The end rotation at the support where the moment is applied is , and at the far end, the rotation is B. By the dummy-load method (Art. 5.10.4), if x is measured from the B end,

Carry-Over Factors. The preceding equations can be used to determine carryover factors for any magnitude of end restraint. The carry-over factors toward fixed ends, however, are of special importance.
The bending-moment diagram for a continuous span LR that is not loaded except for a moment M applied at end L is shown in Fig. 5.61a. For determination of the carry-over factor CR toward R, that end is assumed fixed (no rotation can occur there). The carry-over moment to R then is CRM. The moment diagram in Fig. 5.61a can be resolved into two components: a simple beam with M applied at L (Fig. 5.61b) and a simple beam with CRM applied at R (Fig. 5.61c). As indicated in Fig. 5.61d, M causes an angle change at R of . As shown in Fig. 5.61e, CR M induces an angle change at R of CRMR.

With the use of Eqs. (5.107) and (5.111), the stiffness of a beam with constant moment of inertia is given by

This equation indicates that a prismatic beam hinged at only one end has threefourths the stiffness, or resistance to end rotation, of a beam fixed at both ends.

Fixed-End Moments

A beam so restrained at its ends that no rotation is produced there by the loads is called a fixed-end beam, and the end moments are called fixed-end moments. Fixedend moments may be expressed as the product of a coefficient and WL, where W is the total load on the span L. The coefficient is independent of the properties of other members of the structure. Thus, any member can be isolated from the rest of the structure and its fixed-end moments computed.
Assume, for example, that the fixed-end moments for the loaded beam in Fig. 5.63a are to be determined. Let MF be the moment at the left end L and MF the L R
moment at the right end R of the beam. Based on the condition that no rotation is permitted at either end and that the reactions at the supports are in equilibrium with the applied loads, two equations can be written for the end moments in terms of the simple-beam end rotations, L at L and R, at R for the specific loading.
Let KL be the fixed-end stiffness at L and KR the fixed-end stiffness at R, as given by Eqs. (5.112) and (5.113). Then, by resolution of the moment diagram into simple-beam components, as indicated in Fig. 5.63ƒ to h, and application of the superposition principle (Art. 5.10.6), the fixed-end moments are found to be

Deflection of Supports. Fixed-end moments for loaded beams when one support is displaced vertically with respect to the other support may be computed with the use of Eqs. (5.116) to (5.121) and the principle of superposition: Compute the fixedend moments induced by the deflection of the beam when not loaded and add them to the fixed-end moments for the loaded condition with immovable supports.
The fixed-end moments for the unloaded condition can be determined directly from Eqs. (5.116) and (5.117). Consider beam LR in Fig. 5.64, with span L and support R deflected a distance d vertically below its original position. If the beam were simply supported, the angle change caused by the displacement of R would be very nearly d/L. Hence, to obtain the fixed-end moments for the deflected conditions, set @L =@R = d/L and substitute these simple-beam end rotations in Eqs. (5.116) and (5.117):

where MF is the fixed-end moment at the left support and MF at the right support. L R
As an example of the use of the curves, find the fixed-end moments in a prismatic beam of 20-ft span carrying a triangular loading of 100 kips, similar to the loading shown in Case 4, Fig. 5.70, distributed over the entire span, with the maximum intensity at the right support.

Slope-Deflection Equations

In Arts. 5.11.2 and 5.11.4, moments and displacements in a member of a continuous beam or frame are obtained by addition of their simple-beam components. Similarly, moments and displacements can be determined by superposition of fixed-end-beam components. This method, for example, can be used to derive relationships between end moments and end rotations of a beam known as slope-deflection equations.
These equations can be used to compute end moments in continuous beams.
Consider a member LR of a continuous beam or frame (Fig. 5.72). LR may have a moment of inertia that varies along its length. The support R is displaced vertically

The slope-deflection equations can be used to determine end moments and rotations of the spans of continuous beams by writing compatibility and equilibrium equations for the conditions at each support. For example, the sum of the moments at each support must be zero. Also, because of continuity, the member must rotate through the same angle on both sides of every support. Hence, ML for one span, given by Eq. (5.133) or (5.135), must be equal to MR for the adjoining span, given by Eq. (5.134) or (5.136), and the end rotation  at that support must be the same on both sides of the equation. One such equation with the end rotations at the supports as the unknowns can be written for each support. With the end rotations determined by simultaneous solution of the equations, the end moments can be computed from the slope-deflection equations and the continuous beam can now be treated as statically determinate.
See also Arts. 5.11.9 and 5.13.2.
(C. H. Norris et al., ‘‘Elementary Structural Analysis,’’ 4th ed., McGraw-Hill Book Company, New York.)

Moment Distribution

The frame in Fig. 5.74 consists of four prismatic members rigidly connected together
at O at fixed at ends A, B, C, and D. If an external moment U is applied at  O, the sum of the end moments in each member at O must be equal to U. Furthermore, all members must rotate at O through the same angle , since they are assumed to be rigidly connected there. Hence, by the definition of fixed-end stiffness, the proportion of U induced in the end of each member at O is equal to the ratio of the stiffness of that member to the sum of the stiffnesses of all the members at the joint (Art. 5.11.3).

can be taken when a member has a hinged end to reduce the work of distributing moments. This is done by using the true stiffness of a member instead of the fixedend stiffness. (For a prismatic beam with one end hinged, the stiffness is threefourth the fixed-end stiffness; for a beam with variable I, it is equal to the fixedend stiffness times 1 - CLCR, where CL and CR are the carry-over factors for the beam.) Naturally, the carry-over factor toward the hinge is zero.
When a joint is neither fixed nor pinned but is restrained by elastic members connected there, moments can be distributed by a series of converging approximations.
All joints are locked against rotation. As a result, the loads will create fixed-end moments at the ends of every member. At each joint, a moment equal to the algebraic sum of the fixed-end moments there is required to hold it fixed. Then, one joint is unlocked at a time by applying a moment equal but opposite in sign to the moment that was needed to prevent rotation. The unlocking moment must be distributed to the members at the joint in proportion to their fixed-end stiffnesses and the distributed moments carried over to the far ends.
After all joints have been released at least once, it generally will be necessary to repeat the process—sometimes several times—before the corrections to the fixed  end moments become negligible. To reduce the number of cycles, the unlocking of joints should start with those having the greatest unbalanced moments.

Suppose the end moments are to be found for the prismatic continuous beam ABCD in Fig. 5.75. The I /L values for all spans are equal; therefore, the relative fixed-end stiffness for all members is unity. However, since A is a hinged end, the computation can be shortened by using the actual relative stiffness, which is 3⁄4.
Relative stiffnesses for all members are shown in the circle on each member. The distribution factors are shown in boxes at each joint.
The computation starts with determination of fixed-end moments for each member (Art. 5.11.4). These are assumed to have been found and are given on the first line in Fig. 5.75. The greatest unbalanced moment is found from inspection to be at hinged end A; so this joint is unlocked first. Since there are no other members at the joint, the full unlocking moment of +400 is distributed to AB at A and onehalf of this is carried over to B. The unbalance at B now is +400 - 480 plus the carry-over of +200 from A, or a total of -120. Hence, a moment of 120 must be applied and distributed to the members at B by multiplying by the distribution factors in the corresponding boxes.
The net moment at B could be found now by adding the entries for each member at the joint. However, it generally is more convenient to delay the summation until the last cycle of distribution has been completed.
The moment distributed to BA need not be carried over to A, because the carryover factor toward the hinged end is zero. However, half the moment distributed to BC is carried over to C.
Similarly, joint C is unlocked and half the distributed moments carried over to B and D, respectively. Joint D should not be unlocked, since it actually is a fixed end. Thus, the first cycle of moment distribution has been completed.
The second cycle is carried out in the same manner. Joint B is released, and the distributed moment in BC is carried over to C. Finally, C is unlocked, to complete the cycle. Adding the entries for the end of each member yields the final moments.

Maximum Moments in Continuous Frames

In design of continuous frames, one objective is to find the maximum end moments and interior moments produced by the worst combination of loading. For maximum moment at the end of a beam, live load should be placed on that beam and on the  beam adjoining the end for which the moment is to be computed. Spans adjoining these two should be assumed to be carrying only dead load.

For maximum midspan moments, the beam under consideration should be fully loaded, but adjoining spans should be assumed to be carrying only dead load.

For maximum midspan moments, the beam under consideration should be fully loaded, but adjoining spans should be assumed to be carrying only dead load.
The work involved in distributing moments due to dead and live loads in continuous frames in buildings can be greatly simplified by isolating each floor. The tops of the upper columns and the bottoms of the lower columns can be assumed fixed. Furthermore, the computations can be condensed considerably by following the procedure recommended in ‘‘Continuity in Concrete Building Frames.’’
EB033D, Portland Cement Association, Skokie, IL 60077, and indicated in Fig. 5.74.
Figure 5.74 presents the complete calculation for maximum end and midspan moments in four floor beams AB, BC, CD, and DE. Building columns are assumed to be fixed at the story above and below. None of the beam or column sections is known to begin with; so as a start, all members will be assumed to have a fixedend stiffness of unity, as indicated on the first line of the calculation.
On the second line, the distribution factors for each end of the beams are shown, calculated from the stiffnesses (Arts. 5.11.3 and 5.11.4). Column stiffnesses are not shown, because column moments will not be computed until moment distribution to the beams has been completed. Then the sum of the column moments at each joint may be easily computed, since they are the moments needed to make the sum of the end moments at the joint equal to zero. The sum of the column moments at each joint can then be distributed to each column there in proportion to its stiffness.
In this example, each column will get one-half the sum of the column moments.
Fixed-end moments at each beam end for dead load are shown on the third line, just above the heavy line, and fixed-end moments for live plus dead load on the fourth line. Corresponding midspan moments for the fixed-end condition also are shown on the fourth line and, like the end moments, will be corrected to yield actual midspan moments.

Moment-Influence Factors

In certain types of framing, particularly those in which different types of loading conditions must be investigated, it may be convenient to find maximum end moments from a table of moment-influence factors. This table is made up by listing for the end of each member in the structure the moment induced in that end when a moment (for convenience, +1000) is applied to every joint successively. Once this table has been prepared, no additional moment distribution is necessary for computing the end moments due to any loading condition.
For a specific loading pattern, the moment at any beam end MAB may be obtained from the moment-influence table by multiplying the entries under AB for the various  joints by the actual unbalanced moments at those joints divided by 1000, and summing (see also Art. 5.11.9 and Table 5.6).

Procedure for Sidesway

Computations of moments due to sidesway, or drift, in rigid frames is conveniently executed by the following method:
1. Apply forces to the structure to prevent sidesway while the fixed-end moments due to loads are distributed.
2. Compute the moments due to these forces.
3. Combine the moments obtained in Steps 1 and 2 to eliminate the effect of the forces that prevented sidesway.

Suppose the rigid frame in Fig. 5.77 is subjected to a 2000-lb horizontal load acting to the right at the level of beam BC. The first step is to compute the moment-
influence factors (Table 5.6) by applying moments of 1000 at joints B and C,  assuming sidesway prevented.

Since there are no intermediate loads on the beams and columns, the only fixed-end moments that need be considered
are those in the columns resulting from lateral deflection of the frame caused by the horizontal load. This deflection, however is not known initially.
So assume an arbitrary deflection, which produces a fixed-end moment of -1000M at the top of column CD. M is an unknown constant to be determined fr  the columns and hence are equal in AB to -1000M x 6⁄2 = -3000M. The columnom the fact that the sum of the shears in the deflected columns must be equal to the 2000-lb load. The same deflection also produces a moment of -1000M at the bottom of CD [see Eq. (5.126)].

From the geometry of the structure, furthermore, note that the deflection of B  relative to A is equal to the deflection of C relative to D. Then, according to Eq. (5.126) the fixed-end moments in the columns are proportional to the stiffnesses of  fixed-end moments are entered in the first line of Table 5.7, which is called a moment-collection table.

from which M = 2.30. This value is substituted in the sidesway total in Table 5.7 to yield the sidesway moments for the 4000-lb load. The addition of these moments to the totals for no sidesway yields the final moments.
This procedure enables one-story bents with straight beams to be analyzed with the necessity of solving only one equation with one unknown regardless of the number of bays. If the frame is several stories high, the procedure can be applied to each story. Since an arbitrary horizontal deflection is introduced at each floor or roof level, there are as many unknowns and equations as there are stories.
The procedure is more difficult to apply to bents with curved or polygonal members between the columns. The effect of the change in the horizontal projection of the curved or polygonal portion of the bent must be included in the calculations.
In many cases, it may be easier to analyze the bent as a curved beam (arch).
(A. Kleinlogel, ‘‘Rigid Frame Formulas,’’ Frederick Ungar Publishing Co., New York.)

Rapid Approximate Analysis of Multistory Frames

Exact analysis of multistory rigid frames subjected to lateral forces, such as those from wind or earthquakes, involves lengthy calculations, and they are timeconsuming and expensive, even when performed with computers. Hence, approximate methods of analysis are an alternative, at least for preliminary designs and, for some structures, for final designs.
It is noteworthy that for some buildings even the ‘‘exact’’ methods, such as those described in Arts. 5.11.8 and 5.11.9, are not exact. Usually, static horizontal loads are assumed for design purposes, but actually the forces exerted by wind and earthquakes are dynamic. In addition, these forces generally are uncertain in intensity, direction, and duration. Earthquake forces, usually assumed as a percentage of the mass of the building above each level, act at the base of the structure, not at each floor level as is assumed in design, and accelerations at each level vary nearly linearly with distance above the base. Also, at the beginning of a design, the sizes of the members are not known. Consequently, the exact resistance to lateral deformation cannot be calculated. Furthermore, floors, walls, and partitions help resist the lateral forces in a very uncertain way. See Art. 5.12 for a method of calculating the distribution of loads to rigid-frame bents.
Portal Method. Since an exact analysis is impossible, most designers prefer a wind-analysis method based on reasonable assumptions and requiring a minimum of calculations. One such method is the so-called ‘‘portal method.’’
It is based on the assumptions that points of inflection (zero bending moment) occur at the midpoints of all members and that exterior columns take half as much shear as do interior columns. These assumptions enable all moments and shears throughout the building frame to be computed by the laws of equilibrium.
Consider, for example, the roof level (Fig. 5.78a) of a tall building. A wind load of 600 lb is assumed to act along the top line of girders. To apply the portal method, we cut the building along a section through the inflection points of the top-story columns, which are assumed to be at the column midpoints, 6 ft down from the top of the building. We need now consider only the portion of the structure above this section.
Since the exterior columns take only half as much shear as do the interior columns, they each receive 100 lb, and the two interior columns, 200 lb. The moments at the tops of the columns equal these shears times the distance to the inflection point. The wall end of the end girder carries a moment equal to the moment in the column. (At the floor level below, as indicated in Fig. 5.78b, that end of the end girder carries a moment equal to the sum of the column moments.) Since the inflection point is at the midpoint of the girder, the moment at the inner end of the girder must the same as at the outer end. The moment in the adjoining girder can be found by subtracting this moment from the column moment, because the sum of the moments at the joint must be zero. (At the floor level below, as shown in Fig. 5.78b, the moment in the interior girder is found by subtracting the moment in the exterior girder from the sum of the column moments.)
Girder shears then can be computed by dividing girder moments by the half span. When these shears have been found, column loads can be easily computed from the fact that the sum of the vertical loads must be zero, by taking a section around each joint through column and girder inflection points. As a check, it should be noted that the column loads produce a moment that must be equal to the moments of the wind loads above the section for which the column loads were computed.
For the roof level (Fig. 5.78a), for example, -50 x 24 + 100  x 48 = 600 x 6.

Cantilever Method. Another wind-analysis procedure that is sometimes employed is the cantilever method. Basic assumptions here are that inflection points are at the midpoints of all members and that direct stresses in the columns vary as the distances of the columns from the center of gravity of the bent. The assumptions are sufficient to enable shears and moments in the frame to be determined from the laws of equilibrium.
For multistory buildings with height-to-width ratio of 4 or more, the Spurr modification
is recommended (‘‘Welded Tier Buildings,’’ U.S. Steel Corp.). In this
method, the moments of inertia of the girders at each level are made proportional to the girder shears.
The results obtained from the cantilever method generally will be different from those obtained by the portal method. In general, neither solution is correct, but the answers provide a reasonable estimate of the resistance to be provided against lateral deformation. (See also Transactions of the ASCE, Vol. 105, pp. 1713–1739, 1940.)

Beams Stressed into the Plastic Range

When an elastic material, such as structural steel, is loaded in tension with a gradually increasing load, stresses are proportional to strains up to the proportional limit (near the yield point). If the material, like steel, also is ductile, then it continues to carry load beyond the yield point, though strains increase rapidly with little increase in load (Fig. 5.79a).

Similarly, a beam made of a ductile material continues to carry more load after the stresses in the outer surfaces reach the yield point. However, the stresses will no longer vary with distance from the neutral axis, so the flexure formula [Eq. (5.54)] no longer holds. However, if simplifying assumptions are made, approximating the stress-strain relationship beyond the elastic limit, the load-carrying capacity of the beam can be computed with satisfactory accuracy.

Modulus of rupture is defined as the stress computed from the flexure formula for the maximum bending moment a beam sustains at failure. This is not a true stress but it is sometimes used to compare the strength of beams.
For a ductile material, the idealized stress-strain relationship in Fig. 5.79b may be assumed. Stress is proportional to strain until the yield-point stress ƒy is reached, after which strain increases at
a constant stress.
For a beam of this material, the following assumptions will also be made:
1. Plane sections remain plane, strains thus being proportional to distance from the neutral axis.
2. Properties of the material in tension are the same as those in compression.
3. Its fibers behave the same in flexure as in tension.
4. Deformations remain small.

Strain distribution across the cross section of a rectangular beam, based on these assumptions, is shown in Fig. 5.80a. At the yield point, the unit strain is y and the curvature y, as indicated in (1). In (2), the strain has increased several times, but the section still remains plane. Finally, at failure, (3), the strains are very large and nearly constant across upper and lower halves of the section.
Corresponding stress distributions are shown in Fig. 5.80b. At the yield point, (1), stresses vary linearly and the maximum if ƒy . With increase in load, more and more fibers reach the yield point, and the stress distribution becomes nearly constant, as indicated in (2). Finally, at failure, (3), the stresses are constant across the top and bottom parts of the section and equal to the yield-point stress.
The resisting moment at failure for a rectangular beam can be computed from the stress diagram for stage 3. If b is the width of the member and d its depth, then the ultimate moment for a rectangular beam is

Since the resisting moment at stage 1 is My  ƒybd2 / 6, the beam carries 50% more moment before failure than when the yield-point stress is first reached at the outer surfaces.