Tag Archives: Stress

Creep (Dimensional stability)

Creep is defined as the gradual increase in strain with time for a constant applied stress after accounting for other time-dependent deformations not associated with stress, namely, shrinkage or swelling and thermal strain. Hence, creep is reckoned from the elastic strain at loading, which depends upon the rate of application of stress (see Section 3.2) so that the time taken to apply the load should be quoted. Also, since the secant modulus of elasticity increases with time, the elastic strain decreases so that creep should be taken as the strain in excess of the elastic strain at the time in question. However, the change in elastic strain is usually small and creep is reckoned from the elastic strain on first application of load. Like other engineering materials, concrete can suffer a time-dependent failure, which is known as creep rupture or static fatigue. Figure 3.7 represents the general strain-time history of such a material. Initially, there is a high rate of primary creep and then a steady rate of secondary creep before failure occurs after the tertiary creep stage, as characterised by the rapid development of strain. In the case of concrete, the stress needs to exceed approximately 0.6 to 0.8 of the short-term strength for creep rupture to occur in either compression or tension. Concrete is often described as a brittle material since it readily cracks under small strains, but in the case of creep rupture it can develop large strains prior to failure, providing an advantage of early warning in the case of sudden and catastrophic failure. Figure 3.8 illustrates the pattern of stress and strain for varying rate of loading and level of sustained stress leading to creep rupture; actual experimental results for compression were obtained by Rusch (1960) and for tension by Domone (1974). Very high rate of loading produces a near linear stress-strain curve with a higher strength than in the usual standard test (Fig. 3.8(a)), but decreasing the rate of loading or increasing the test duration produces non-linear curves due to creep and microcracking with a lower strength (creep rupture). The creep rupture envelope tends to a constant limit of between 0.6 and 0.8 of the usual short-term strength, depending on the type of concrete and mode of loading. With sustained stresses below approximately 0.6 of short-term strength, creep rupture is avoided and time-dependent strain due to primary and some secondary creep takes place for several years. Depending on the type of mix and other factors, creep of dry-stored concrete can be between two and nine times the elastic strain at loading, with around 70% occurring after one year under load (Brooks, 2005).

As stated earlier, the definition of creep is the increase in strain for a constant sustained stress and is determined from concrete specimens after deducting any drying shrinkage (measured on a separate specimen) and the initial elastic strain. Figure 3.9(a) shows the components of strain involved. In the case of sealed concrete, which represents mass or large-volume concrete where little or no moisture is lost, only basic creep occurs, but when the concrete is allowed to dry additional drying creep occurs even though drying shrinkage has been deducted from the measured strain; the sum of basic and drying creep is sometimes called total creep. Creep is a partly reversible phenomenon. When the load is removed, there is an immediate, almost full, elastic recovery of the initial elastic strain, followed by a gradual decrease of strain called the creep recovery (Fig. 3.9(b)). The recovery quickly reaches a maximum and is only small, e.g. 1±25% of the 30-year creep (Brooks, 2005). Consequently, creep is mostly irreversible in nature.

Like shrinkage, creep can be expressed in units of microstrain (10ÿ6), but because of the dependency on stress, specific creep (Cs) is often used with units of 10ÿ6 per MPa. Other terms are creep coefficient (@) . The creep coefficient is defined as the ratio of creep to the elastic strain on loading and for a unit stress:

 

The effects of creep can be realised from another viewpoint. When a concrete specimen is loaded and then prevented from deforming, then creep will manifest itself as a gradual decrease or relaxation of stress with time, as illustrated in Fig. 3.10. Relaxation is of interest in connection with loss of prestress in prestressed and post-tensioned concrete and cracking processes (see Section 3.3). Apparatus for the determination of creep of any type of concrete is recommended in ASTM C512 (1987) and for prefabricated aerated or lightweight concrete by BS EN 1355 (1997). Alternative methods are described in Neville et al. (1983) and Newman and Choo (2003). There have been many theories proposed to explain the creep of hardened cement paste and concrete, which are too numerous to include within the scope of this chapter. Most agree that creep at normal stresses is caused by the internal movement of water adsorbed or held within the C-S-H, since concrete from which all the evaporable water has been removed exhibits little or no creep unless high temperatures are involved. Movement of water to the outside environment is required for drying creep to occur. Although basic creep is associated with sealed or mass concrete, it is thought that internal movement of water occurs because all the pores do not remain full of water. The dependency of basic creep on strength is indirect evidence that empty or part-empty pores are a main factor. Creep also occurs at high temperatures when non-evaporable water is removed and this is thought to be due to movement of interlayer or zeolitic water, viscous flow or sliding between gel particles (Illston et al., 1979; Mindess and Young, 1981; Neville et al., 1983; Bazant, 1988; Brooks, 2001). The source of creep of concrete is the hydrated cement paste and not normal weight aggregate of good quality. Some light aggregates may exhibit creep. However, like shrinkage, the role of aggregate is an important factor in creep due to the restraining influence on the cement paste through its stiffness or modulus of elasticity and volume concentration. The relationship between modulus of elasticity of aggregate and relative creep of concrete is shown in Fig. 3.11. For a constant water/cement ratio, the volume of cement paste or total aggregate in concrete has a significant effect on creep (Neville and Brooks, 2002). However, in practice, concretes with similar workability normally have  similar cement paste contents and so the differences in creep are not large.

For example, in normal weight concretes having aggregate/cement ratios of 9, 6 and 4.5, and corresponding water/cement ratios of 0.75, 0.55 and 0.40, the cement  paste contents are 24, 27 and 29%, respectively. On the other hand, when the cement paste content is constant, an increase in water/cement ratio increases creep as shown in Fig. 3.12.

Since creep is related to the water/cement ratio it can be expected to be related to strength. Indeed, it has been found that over a wide range of mixes made with similar materials, creep is approximately inversely proportional to the strength of the concrete at the age of application of load. Moreover, for stresses less than approximately 0.6 of the short-term strength, creep is proportional to the applied stress, so that combining the dual influences leads to the stress/ strength ratio rule, namely, that creep is approximately proportional to the stress/ strength ratio (Neville et al., 1983). Consequently, since the strength increases with age, creep can be expected to decrease as the age at loading increases (Fig. 3.13).

Experimental work on saturated concrete specimens exposed to the atmosphere at different relative humidities shows that the drier the atmosphere the higher the creep. Figure 3.14 demonstrates that trend, the 100% curve approximating to that of basic creep so that the additional creep for 75 and 50%relative humidities is drying creep. If the concrete is allowed to dry out prior to application of load then creep is much less. However, it is ill-advised to allow concrete to dry out prematurely in order to reduce creep because of the risk of inadequate curing and cracking due to restrained or differential drying shrinkage (see Section 3.3.6). Creep of concrete is affected by size of member in a similar manner to drying shrinkage (see Fig. 3.6). Expressing size as the volume/exposed surface area ratio (V/S) or effective thickness (ˆ 2V=S), which represents the average drying path length, total creep decreases as the volume/surface ratio or size increases (Fig. 3.15). Of course, it is the drying creep component that is influenced since basic creep is unaffected by size. This can be seen when the size of the member is large (mass concrete) when no moisture transfer to the environment takes place and basic creep occurs. It is also apparent from Fig. 3.15 that the influence of shape of concrete member is a secondary factor in creep and can be neglected. The influence of temperature on creep is complex and not fully understood, as it depends upon the time when the temperature of the concrete rises relative to the time of application of load (Neville et al., 1983). Figure 3.16 demonstrates different experimental results for concrete stored at elevated temperature in water (basic creep). Compared with normal-temperature creep, heating just prior to loading accelerates creep, and heating just after loading produces an additional component termed transitional thermal creep. In the case of drying concrete (total creep), elevated temperature causes increases in creep in a similar manner, but when the evaporable water has been removed (between approximately 80 and 120 ëC) there is a decrease in creep before increasing again (Neville et al., 1983). At very high temperatures, such as in fire, very high creep occurs, termed transient thermal strain (Khoury et al., 1985). The type of cement will affect creep if the strength changes at the time of application of load. When the stress/strength ratio at the age of loading is the same, most Portland cements lead to approximately the same creep, but the strength development under load is a factor, e.g. creep will be lower for a greater strength development (Neville and Brooks, 2002). The latter influence is apparent when certain mineral admixtures, such as fly ash and ground granulated blast-furnace slag, are used as partial replacements for Portland cement. The resulting slower pozzolanic reaction often leads to a later strength development than in the case of Portland cement. Figure 3.17 shows the general trends for fly ash, blast-furnace slag, silica fume and meta- kaolin, which are mainly based on the analysis of results of previously published data (Brooks, 2000). That analysis accounted for any changes in aggregate content and water/cementitious materials ratio on creep by considering the relative stress/strength ratio so that the effect of just the mineral admixture could be assessed. Although there is a large variation, it can be seen that fly ash and slag can lead to significant reductions in creep. In the case of the finer mineral admixtures: silica fume and metakaolin, there are larger reductions in creep for replacement levels of up to 15%, but then for silica fume creep increases as the replacement level increases. Generally, the addition of chemical admixtures, plasticisers and super- plasticisers, to make flowing concrete causes a 20% reduction in creep at a constant stress/strength ratio, although the effects do vary widely (Brooks, 2000). When the same admixtures are used as workability aids or water reducers, creep will be less due to the lower water content. As well as reducing shrinkage (see Section 3.3), the use of a shrinkage-reducing admixture appears to reduce drying creep, although experimental verification is limited. For design purposes, estimation of elastic deformation, creep and drying shrinkage are considered together in Codes of Practice. From only a knowledge of strength, mix composition and physical conditions, BS 8110: Part 2 (1985) gives creep and shrinkage after 6 months and 30 years, depending upon the relative humidity and size of member. ACI 209 (1992) and CEB-FIP (1999) methods express creep and shrinkage as functions of time and allow for all the main influencing factors that have been discussed earlier. Alternative models are available by Bazant and Baweja (1995) and Gardner and Lockman (2001). Creep and drying shrinkage estimates by all methods are not particularly accurate (+- (30% at best) mainly because they fail to account for the type of aggregate (Brooks, 2005). For more accurate estimates and for high performance concretes containing several admixtures, short-term tests are recommended. The  test duration should be of at least 28 days using small laboratory specimens made from the actual concrete mix and then measured creep and shrinkage-time data extrapolated to obtain long-term values, which are then adjusted according to the required member size and the average relative humidity of the storage conditions (Brooks and Al-Quarra, 1999). All methods of prediction give an estimate of the creep function (equation 3.6) and drying shrinkage, the total strain, …t; to†, at age t when determined from age to being given by:

The importance of creep in structural concrete lies mainly in the fact that, in the long term, it can be several times the elastic deformation when first loaded.
Consequently, the designer has to assess creep in order to comply with the serviceability requirement of deflection in particular. There are other effects of creep, most of which are detrimental, such as loss of prestress in prestressed concrete and differential movements in tall buildings, but creep can be beneficial when relieving stress induced by restraint of deformations.

 

Stresses and deflections in service

A composite beam is usually designed first for ultimate limit states. Its behaviour in service must then be checked. For a simply-supported beam, the most critical serviceability limit state is usually excessive deflection, which can govern the design where unpropped construction us used. Floor structures subjected to dynamic loading (e.g. as in a gymnasium) are also susceptible to excessive vibration (Section 3.11.3.2).
Cracking of concrete is a problem only in fully-encased beams. Which are rarely used, and in hogging regions of continuous beams (Section 4.2.5).
Some codes of practice limit stresses in service; but excessive stress is not itself a limit state. It may however invalidate a method of analysis (e.g.linear0elastic theory) that would otherwise be suitable for checking compliance with a serviceability criterion. No stress limits are specified in Eurocode 4; Part 1.1. The policy is to use elastic analysis, allowing for shear lag and creep; and to modify the results, where necessary, to allow for yielding of steel and, where partial shear connection is used, for excessive slip.
If yielding of structural steel occurs in service, in a typical composite beam for a building, it will be in the bottom flange, near midspan. The likelihood of this depends on the ratio between the characteristic variable and permanent loads, given by

where My is the bending moment at which yield of steel first occurs. For sagging bending, it is typically between 1.25 and 1.35 for propped construction, but can rise to 1.45 or above, for unpropped construction.
Deflections are usually checked for the rare combination of actions, given in equation (1.8). So for a beam designed for distributed loads gk and qk only, the ratio of design bending moments (ultimate/serviceability) is

The values given above show that this is unlikely for propped construction, but might occur for unpropped construction when a γ for structural steel is assumed to be 1.05 or less, rather than 1.10 as recommended in Eurocode 4.
Where the bending resistance of a composite section is governed by local buckling, as in c Class 3 section, elastic section analysis is used for ultimate limit states, and then stresses and/or deflections in service are less likely to influence design.
As shown below, elastic analysis of a composite section is more complex than plastic analysis, because account has to be taken of the method of construction and of the effects of creep. The following three types of loading then have to be considered separately:
load carried by the steel beam, short-term load carried by the composite beam long-term load carried by the composite beam

3.7.1 Elastic analysis of composite sections in sagging bending

It is assumed first that full shear connection is provided, so that the effect of slip can be neglected.
All other assumptions are as for the elastic analysis of reinforced concrete sections by the method of transformed sections. The algebra is different because the flexural rigidity of the steel section alone is so much greater than that of reinforcing bars.
For generality, the steel section is assumed to be asymmetrical (Fig.3.25) with cross-sectional area Aa, second moment of area Ia, and centre of area distance zs below the top surface of the concrete slab, which is of uniform overall thickness ht and effective width beff.
The modular ratio for short-term loading is

n= Ea / Ec

where the subscripts a and c refer to structural steel and concrete, respectively. For long-term loading, a value n/3 is a good approximation. For simplicity, a single value n/2 is sometimes used for both types of loading. From here onwards, the symbol n is used for whatever modular ration is appropriate, so it is defined by

n= Ea / E’c

where E’c
E is the relevant effective modulus for the concrete.
It is usual to neglect reinforcement in compression, concrete in tension, and also concrete between  the ribs of profiled sheeting, even when the sheeting spans longitudinally. The condition for the neutral-axis depth x to be less than he is

In global analyses, it is sometimes convenient to use values of I based on the uncracked composite section. The values of x and I are then given by equations (3.88) and (3.89) above, whether x exceeds hc or not. In sagging bending, the difference between the ‘cracked’ and ‘uncracked’ values of I is usually small.
Stresses due to a sagging bending moment M are normally calculated in concrete only at level I in Fig.3.25, and in steel at levels 3 and 4. these stresses are, with tensile stress positive:

Effect of slip on stresses and deflections

Full-interaction and no-interaction elastic analyses are given in Section 2.2 for a composite beam made from two elements of equal size and stiffness. Its cross-section (Fig.2.2(b))can be considered as the transformed section for the steel and concrete beam in Fig .2.16 Partial-interaction analysis of this beam (Appendix A)illustrates well the effect of connector flexibility on interface slip and hence onstresses and deflections, even though the cross-section is not that would be used in practice. The numerical values, chosen to be typical of a composite beam, are given in section A.2.Substituon in (2.27)gives the relation between sand x for a beam of depth 0.6m and span 10m as. 10^4 s =1.05x − 0.007 sinh(1.36x). The maximum slip occurs at the ends of the span, where x= ± 5.m From equation(2.28) it is ± 0.45mm. The results obtained in Sections 22.1 and 22.2are also applicable to this beam. From equation (2.6) the maximum slip if there were no shear connection would be ± 8.1mm.Thus the shear connectors reduce end slip substantially, but do not eliminate it .The variations of slip strain and slip along the span for no interaction and parial interaction are shown in Fig 2.3 The connector modulus k was taken as 150 kN / mm(Appendix A).The maximum load per connetor is k timews the maximum slip ,so the maximum slip so the partial-interaction theory gives this load as 67 kN, which is sufficiently far below the ultimate strength of 100 kN per connector for the assumption of a linear Load –slip relationship to be reasonable. Longitudinal strains at midspan given by full-interaction and partial-interaction and partial –interaction theories are shown in Fig 2.17 The increase in extreme-fibre strain due to slip , 28 ×10−6 , is much less stress in the concrete is increasede,104×10−6 .The maximum compressive stress in the concrete is increased by slip form 12.2 to 12.8 N/mm2,a change of 5%.This higher stress is 43% of the cube strength, so the assumption of elastic behaviour is reasonable

The ratio of the partial-interaction curaturv to the full-interaction curvature is 690/610, or 1.13, Integration due to slip, is also about 13%. The effects of slip on deflection are found in pratice to be less than is implied by this example, because here a rather low value of conncctor modulus has been used ,and the effect of bond has been neglected.
The longitudinal compressive force in the concrete at midspan is proportional to the mean compressive strain. From Fig 2.17 , this is 305×10-6for full interaction and 293×10-6for partial interaction ,a reduction of 4%

The influence of slip on the flexural behaviour of the member may be summarized as follows ,The bending moment at midspan ,WL^2/8. Can be considered to be the sum of a ‘concrete’ moment Mc, a ‘steel’ moment Ms and a ‘composite’ moment Fdc(Fig.A.1):

In the full-interaction analysis c Fd contributes 75%.of the total moment ,and c M and a M 12.5% each. The Parial-interaction analysis shows that slip reduces the contribution from Fdc to 72% of the total, so that the contributions from Mc and Ma rise to 14%,corresponding to an increase in curvature of (14-12.5) ,or about 13%. The interface shear force per unit length Vx is given by equation (2.12)for full interaction and by equations (A.1)and (2.28)for partial inter action The expressions for Vx over a half span are plotted in Fig 2.18,and show that in the elastic range ,the distribution of loading on the connectors is similar to that given by full-interaction theory. The reasons for using uniform rather than ‘triangular’ spacing of connectors are discussed in section 3.6.

Uplift Shear Connection

In the preceding examplem the stress normal to the interface AOB (Fig.2.2) was everywhere compressive and equal to w/2b except at the ends of the beam .The stress would have been tensile if the load w had been applied to the lower member. Such loading is unlikely, except when traveling cranes are suspended from the steelwork of a composite floor above: but there are other situations in which stresses tending to cause uplift can occur at the interface. These arise from complex effects such as the torsional stiffness of reinforced concrete slabs forming flanges of composite beams, the triaxial stresses in the vicinity of shear connectors and, in box-girder bridges, the torsional stiffness of the steel box.
Tension across the interface can also occur in beams of non-uniform, Section or with partially completed flanges. Two members without shear connection, as shown in Fig. 2.5,provide a simple example. AB. Is sup-ported on CD and carries distributed loading. It can easily be shown by elastic theory that if the flexural rigidity of AB exceeds about one-tenth of that of CD, then the whole of the load on AB is transferred to CD at points A and B, with separation of the beams between these points. If AB was connected to CD, there would be uplift forces at midspan.

Almost all connectors used in practice are therefore so shaped that they provide resistance to uplift as well as to slip. Uplift forces are so much less than shear forces that it is not normally necessary to calculate or estimate them for design purposes, provided that connectors with some uplift resistance are used.

No shear connection

We assume first that there is no shear connection or friction on the interface AB. The upper beam
cannot deflect more than the lower one, so each carries load w/2 per unit length as if it were an
isolated beam of second moment of area bh3/12, and the vertical compressive stress across the
interface is w/2b. The midspan bending moment in each beam is wL2/16. By elementary beam
theory, the stress distribution at midspan is as in Fig. 2.2.(c), and the maximum bending stress in
each component, σ , is given by

There is an equal and opposite strain in the top fibre of the lower beam, so that the difference between the strains in these adjacent fibres, known as the slip strain, is x 2ε ,

It is easy to show by experiment with two or more flexible wooden laths or rulers that under load, the end faces of the two-component beam have the shape shown in Fig,2.3(a).The slip at the interface, s, is zero at x = 0 is the only one where plane sections remain plane. The slip strain, defined above, is not the same as slip. In the same way that strain is rate of change of displacement,
slip strain is the rate of change of slip along the beam. Thus from(2.4),

The constant of integration is zero, since s = 0 when x = 0, so that (2.6) gives the distribution of slip along the beam.
Results(2.5)and(2.6)for the beam studied in Section 2.7 are plotted in fig.2.3.This shows that at midspan, slip strain is a maximum and slip is zero, and at the ends of the beam, slip is a maximum and slip strain is zero. From (2.6), the maximum slip (when x = L / 2 )is wL3 / 4Ebh2 .Some idea of the magnitude of this slip is given by relating it to the maximum deflection of the two beams. From (2.3), the ratio of slip to deflection is3.2h / L , The ratio L / 2h for a beam is typically about 20,so that the end slip is less than a tenth of the deflection. We conclude that shear
connection must be very stiff if it is to be effective.

Bearing Stresses

These may occur in a wood structural member parallel to the grain (end bearing), perpendicular to the grain, or at an angle to the grain.

Bearing Parallel to Grain

The bearing stress parallel to grain ƒg should be computed for the net bearing area.
This stress may not exceed the design value for bearing parallel to grain Fg multiplied by load duration factor CD and temperature factor Ct (Art. 10.5). The adjusted design value applies to end-to-end bearing of compression members if they have adequate lateral support and their end cuts are accurately squared and parallel to each other.
When ƒg exceeds 75% of the adjusted design value, the member should bear on a metal plate, strap, or other durable, rigid, homogeneous material with adequate strength. In such cases, when a rigid insert is required, it should be a steel plate with a thickness of 20 ga or more or the equivalent thereof, and it should be inserted with a snug fit between abutting ends.

Bearing Perpendicular to Grain

This is equivalent to compression perpendicular to grain. The compressive stress should not exceed the design value perpendicular to grain multiplied by applicable adjustment factors, including the bearing area factor (Art. 10.5.11). In the calculation of bearing area at the end of a beam, an allowance need not be made for the fact that, as the beam bends, it creates a pressure on the innr edge of the bearing that is greater than at the end of the beam.

Bearing at an Angle to Grain

The design value Fg for bearing parallel to grain and the design value for bearing perpendicular to grain Fc differ considerably. When load is applied at an angle 

Air-Stabilized Structures

A true membrane is able to withstand tension but is completely unable to resist bending. Although it is highly efficient structurally, like a shell, a membrane must be much thinner than a shell and therefore can be made of a very lightweight material, such as fabric, with considerable reduction in dead load compared with other types of construction. Such a thin material, however, would buckle if subjected to compression. Consequently, a true membrane, when loaded, deflects and assumes a shape that enables it to develop tensile stresses that resist the loads.
Membranes may be used for the roof of a building or as a complete exterior enclosure. One way to utilize a membrane for these purposes is to hang it with initial tension between appropriate supports. For example, a tent may be formed by supporting fabric atop one or more tall posts and anchoring the outer edges of the stretched fabric to the ground. As another example, a dish-shaped roof may be constructed by stretching a membrane and anchoring it to the inner surface of a ring girder. In both examples, loads induce only tensile stresses in the membrane.
The stresses may be computed from the laws of equilibrium, because a membrane is statically determinate.
Another way to utilize a membrane as an enclosure or roof is to pretension the membrane to enable it to carry compressive loads. For the purpose, forces may be applied, and retained as long as needed, around the edges or over the surface of the membrane to induce tensile stresses that are larger than the larger compressive stresses that loads will impose. As a result, compression due to loads will only reduce the prestress and the membrane will always be subjected only to tensile stresses.

Pneumatic Construction

A common method of pretensioning a membrane enclosure is to pressurize the interior with air. Sufficient pressure is applied to counteract dead loads, so that the membrane actually floats in space. Slight additional pressurization is also used to offset wind and other anticipated loads. Made of lightweight materials, a membrane thus can span large distances economically. This type of construction, however, has the disadvantage that energy is continuously required for operation of air compressors to maintain interior air at a higher pressure than that outdoors.
Pressure differentials used in practice are not large. They often range between 0.02 and 0.04 psi (3 and 5 psf). Air must be continually supplied, because of leakage. While there may be some leakage of air through the membrane, more important sources of air loss are the entrances and exits to the structure. Air locks and revolving doors, however, can reduce these losses.
An air-stabilized enclosure, in effect is a membrane bag held in place by small pressure differentials applied by environmental energy. Such a structure is analogous to a soap film. The shape of a bubble is determined by surface-tension forces.
The membrane is stressed equally in all directions at every point. Consequently, the film forms shapes with minimum surface area, frequently spherical. Because of the stress distribution, any shape that can be obtained with soap films is feasible for an air-stabilized enclosure. Figure 5.105c shows a configuration formed by a conglomeration of bubbles as an illustration of a shape that can be adopted for an air-stabilized structure.
In practice, shapes of air-stabilized structures often resemble those used for thinshell enclosures. For example, spherical domes (Fig. 5.105a) are frequently con structed with a membrane. Also, membranes are sometimes shaped as semi-circular cylinders with quarter-sphere ends (Fig. 5.105b).

Air-stabilized enclosures may be classified as air-inflated, air-supported, or hybrid structures, depending on the type of support.

Air-inflated enclosures are completely supported by pressurized air entrapped within membranes. There are two main types, inflated-rib structures and inflated dual-wall structures.
In inflated-rib construction, the membrane enclosure is supported by a framework of air-pressurized tubes, which serve much like arch ribs in thin-shell construction (Art. 5.15.1). The principle of their action is demonstrated by a water hose. A flexible hose, when empty, collapses under its own weight on short spans or under loads normal to its length; but it stiffens when filled with water. The water pressure tensions the hose walls and enables them to withstand compressive stresses.
In inflated dual-walled construction, pressurized air is trapped between two concentric membranes (Fig. 5.106). The shape of the inner membrane is maintained by suspending it from the outer one. Because of the large volume of air compressed between the membranes, this type of construction can span longer distances than can inflated-rib structures.
Because of the variation of air pressure with changes in temperature, provision must be made for adjustment of the pressure of the compressed air in air-inflated structures. Air must be vented to relieve excessive pressures, to prevent overtensioning of the membranes. Also, air must be added to compensate for pressure drops, to prevent collapse.
Air-supported enclosures consist of a single membrane supported by the difference between internal air pressure and external atmospheric pressure (Fig. 5.107).
The pressure differential deflects the membrane outward, inducing tensile stresses in it, thus enabling it to withstand compressive forces. To resist the uplift, the construction must be securely anchored to the ground. Also, the membrane must be completely sealed around its perimeter to prevent air leakage.
Hybrid structures consist of one of the preceding types of pneumatic construction augmented by light metal framing, such as cables. The framing may be merely a safety measure to support the membrane if pressure should be lost or a means of shaping the membrane when it is stretched. Under normal conditions, air pressure against the membrane reduces the load on the framing from heavy wind and snow loads.

Membrane Stresses

Air-supported structures are generally spherical or cylindrical because of the supporting uniform pressure.
When a spherical membrane with radius R, in, its subjected to a uniform radial internal pressure, p, psi, the internal unit tensile force, lb / in, in any direction, is given by

T=pR/2

In a cylindrical membrane, the internal unit tensile force, lb / in, in the circumferential direction is given by

T=pR

where R  radius, in, of the cylinder. The longitudinal membrane stress depends on the conditions at the cylinder ends. For example, with immovable end enclosures, the longitudinal stress would be small. If, however the end enclosure is flexible, a tension about half that given by Eq. (5.234) would be imposed on the membrane in the longitudinal direction.
Unit stress in the membrane can be computed by dividing the unit force by the thickness, in, of the membrane.

Thin-Shell Structures

A structural membrane or shell is a curved surface structure. Usually, it is capable of transmitting loads in more than two directions to supports. It is highly efficient structurally when it is so shaped, proportioned, and supported that it transmits the loads without bending or twisting.
A membrane or a shell is defined by its middle surface, halfway between its extrados, or outer surface and intrados, or inner surface. Thus, depending on the geometry of the middle surface, it might be a type of dome, barrel arch, cone, or hyperbolic paraboloid. Its thickness is the distance, normal to the middle surface, between extrados and intrados.

Thin-Shell Analysis

A thin shell is a shell with a thickness relatively small compared with its other dimensions. But it should not be so thin that deformations would be large compared with the thickness.
The shell should also satisfy the following conditions: Shearing stresses normal to the middle surface are negligible. Points on a normal to the middle surface before it is deformed lie on a straight line after deformation. And this line is normal to the deformed middle surface.
Calculation of the stresses in a thin shell generally is carried out in two major steps, both usually involving the solution of differential equations. In the first, bending and torsion are neglected (membrane theory, Art. 5.15.2). In the second step, corrections are made to the previous solution by superimposing the bending and shear stresses that are necessary to satisfy boundary conditions (bending theory, Art. 5.15.3).

Ribbed Shells. For long-span construction, thin shells often are stiffened at intervals by ribs. Usually, the construction is such that the shells transmit some of the load imposed on them to the ribs, which then perform structurally as more than just stiffeners. Stress and strain distributions in shells and ribs consequently are complicated by the interaction between shells and ribs. The shells restrain the ribs, and the ribs restrain the shells. Hence, ribbed shells usually are analyzed by approximate methods based on reasonable assumptions.
For example, for a cylindrical shell with circumferential ribs, the ribs act like arches. For an approximate analysis, the ribbed shell therefore may be assumed to be composed of a set of arched ribs with the thin shell between the ribs acting in the circumferential direction as flanges of the arches. In the longitudinal direction, it may be assumed that the shell transfers load to the ribs in flexure. Designers may adjust the results of a computation based on such assumptions to correct for a variety of conditions, such as the effects of free edges of the shell, long distances between ribs, relative flexibility of ribs and shell, and characteristics of the structural materials.

Membrane Theory for Thin Shells

Thin shells usually are designed so that normal shears, bending moments, and torsion are very small, except in relatively small portions of the shells. In the membrane theory, these stresses are ignored.
Despite the neglected stresses, the remaining stresses ae in equilibrium, except possibly at boundaries, supports, and discontinuities. At any interior point, the number of equilibrium conditions equals the number of unknowns. Thus, in the membrane theory, a thin shell is statically determinate.
The membrane theory does not hold for concentrated loads normal to the middle surface, except possibly at a peak or valley. The theory does not apply where boundary conditions are incompatible with equilibrium. And it is in exact where there is geometric incompatibility at the boundaries. The last is a common condition, but the error is very small if the shell is not very flat. Usually, disturbances of membrane equilibrium due to incompatibility with deformations at boundaries, supports, or discontinuities are appreciable only in a narrow region about each source of disturbance. Much larger disturbances result from incompatibility with equilibrium conditions.
To secure the high structural efficiency of a thin shell, select a shape, proportions, and supports for the specific design conditions that come as close as possible to satisfying the membrane theory. Keep the thickness constant; if it must change, use a gradual taper. Avoid concentrated and abruptly changing loads. Change curvature gradually. Keep discontinuities to a minimum. Provide reactions that are tangent to the middle surface. At boundaries, ensure, to the extent possible, compatibility of shell deformations with deformations of adjoining members, or at least keep restraints to a minimum. Make certain that reactions along boundaries are equal in magnitude and direction to the shell forces there.
Means usually adopted to satisfy these requirements at boundaries and supports are illustrated in Fig. 5.97. In Fig. 5.97a, the slope of the support and provision for movement normal to the middle surface ensure a reaction tangent to the middle  surface. In Fig. 5.97b, a stiff rib, or ring girder, resists unbalanced shears and transmits normal forces to columns below. The enlarged view of the ring girder in Fig. 5.97c shows gradual thickening of the shell to reduce the abruptness of the change in section. The stiffening ring at the lantern in Fig. 5.97d, extending around the opening at the crown, projects above the middle surface, for compatibility of strains, and connects through a transition curve with the shell; often, the rim need merely be thickened when the edge is upturned, and the ring can be omitted. In Fig. 5.97e, the boundary of the shell is a stiffened edge. In Fig. 5.97f, a scalloped  shell provides gradual tapering for transmitting the loads to the supports, at the same time providing access to the shell enclosure. And in Fig. 5.97g, a column is flared widely at the top to support a thin shell at an interior point.

Even when the conditions for geometric compatibility are not satisfactory, the membrane theory is a useful approximation. Furthermore, it yields a particular solution to the differential equations of the bending theory.
(D. P. Billington, ‘‘Thin Shell Concrete Structures,’’ 2d ed., and S. Timoshenko and S. Woinowsky-Krieger, ‘‘Theory of Plates and Shells,’’ McGraw-Hill Book Company, New York: V. S. Kelkar and R. T. Sewell, ‘‘Fundamentals of the Analysis and Design of Shell Structures,’’ Prentice-Hall, Englewood Cliffs, N.J.)

Bending Theory for Thin Shells

When equilibrium conditions are not satisfied or incompatible deformations exist at boundaries, bending and torsion stresses arise in the shell. Sometimes, the design of the shell and its supports can be modified to reduce or eliminate these stresses (Art. 5.15.2). When the design cannot eliminate them, provisions must be made for the shell to resist them.

But even for the simplest types of shells and loading, the stresses are difficult to compute. In bending theory, a thin shell is statically indeterminate; deformation conditions must supplement equilibrium conditions in setting up differential equations for determining the unknown forces and moments. Solution of the resulting equations may be tedious and time-consuming, if indeed solution if possible.
In practice, therefore, shell design relies heavily on the designer’s experience and judgment. The designer should consider the type of shell, material of which it is made, and support and boundary conditions, and then decide whether to apply a bending theory in full, use an approximate bending theory, or make a rough estimate of the effects of bending and torsion. (Note that where the effects of a disturbance are large, these change the normal forces and shears computed by the membrane theory.) For concrete domes, for example, the usual procedure is to use as support a deep, thick girder or a heavily reinforced or prestressed tension ring, and the shell is gradually thickened in the vicinity of this support (Fig. 5.97c).
Circular barrel arches, with ratio of radius to distance between supporting arch ribs less than 0.25 may be designed as beams with curved cross section. Secondary stresses, however, must be taken into account. These include stresses due to volume change of rib and shell, rib shortening, unequal settlement of footings, and temperature differentials between surfaces.
Bending theory for cylinders and domes is given in W. Flu¨gge, ‘‘Stresses in Shells,’’ Springer-Verlag, New York; D. P. Billington, ‘‘Thin Shell Concrete Structures,’’ 2d ed., and S. Timoshenko and S. Woinowsky-Krieger, ‘‘Theory of Plates and Shells,’’ McGraw-Hill Book Company, New York; ‘‘Design of Cylindrical Concrete Shell Roofs,’’ Manual of Practice No. 31, American Society of Civil Engineers.

Stresses in Thin Shells

The results of the membrane and bending theories are expressed in terms of unit forces and unit moments, acting per unit of length over the thickness of the shell. To compute the unit stresses from these forces and moments, usual practice is to assume normal forces and shears to be uniformly distributed over the shell thickness and bending stresses to be linearly distributed.
Then, normal stresses can be computed from equations of the form

Folded Plates

A folded-plate structure consists of a series of thin planar elements, or flat plates, connected to one another along their edges. Usually used on long spans, especially for roofs, folded plates derive their economy from the girder action of the plates and the mutual support they give one another.
Longitudinally, the plates may be continuous over their supports. Transversely, there may be several plates in each bay (Fig. 5.98). At the edges, or folds, they may be capable of transmitting both moment and shear or only shear.
A folded-plate structure has a two-way action in transmitting loads to its supports.
Transversely, the elements act as slabs spanning between plates on either side. The plates then act as girders in carrying the load from the slabs longitudinally to supports, which must be capable of resisting both horizontal and vertical forces.
If the plates are hinged along their edges, the design of the structure is relatively simple. Some simplification also is possible if the plates, though having integral edges, are steeply sloped or if the span is sufficiently long with respect to other dimensions that beam theory applies. But there are no criteria for determining when such simplification is possible with acceptable accuracy. In general, a reasonably accurate analysis of folded-plate stresses is advisable.
Several good methods are available (D. Yitzhaki, ‘‘The Design of Prismatic and Cylindrical Shell Roofs,’’ North Holland Publishing Company, Amsterdam; ‘‘Phase  I Report on Folded-plate Construction,’’ Proceedings Paper 3741, Journal of the Structural Division, American Society of Civil Engineers, December 1963; and A. L. Parme and J. A. Sbarounis, ‘‘Direct Solution of Folded Plate Concrete Roofs,’’ EB021D, Portland Cement Association, Skokie, Ill.). They all take into account the effects of plate deflections on the slabs and usually make the following assumptions:
The material is elastic, isotropic, and homogeneous. The longitudinal distribution of all loads on all plates is the same. The plates carry loads transversely only by bending normal to their planes and longitudinally only by bending within their planes. Longitudinal stresses vary linearly over the depth of each plate. Supporting members, such as diaphragms, frames, and beams, are infinitely stiff in their own planes and completely flexible normal to their own planes. Plates have no torsional stiffness normal to their own planes. Displacements due to forces other than bending moments are negligible.

Regardless of the method selected, the computations are rather involved; so it is wise to carry out the work by computer or, when done manually, in a wellorganized table. The Yitzhaki method offers some advantages over others in that the calculations can be tabulated, it is relatively simple, it requires the solution of no more simultaneous equations than one for each edge for simply supported plates, it is flexible, and it can easily be generalized to cover a variety of conditions.
Yitzhaki Method. Based on the assumptions and general procedure given above, the Yitzhaki method deals with the slab and plate systems that comprise a foldedplate structure in two ways. In the first, a unit width of slab is considered continuous over supports immovable in the direction of the load (Fig. 5.99b). The strip usually is taken where the longitudinal plate stresses are a maximum. Second, the slab reactions are taken as loads on the plates, which now are assumed to be hinged along the edged (Fig. 5.99c). Thus, the slab reactions cause angle changes in the plates at each fold. Continuity is restored by applying to the plates an unknown moment at each edge. The moments can be determined from the fact that at each edge the sum of the angle changes due to the loads and to the unknown moments must equal zero.
The angle changes due to the unknown moments have two components. One is the angle change at each slab end, now hinged to an adjoining slab, in the transverse strip of unit width. The second is the angle change due to deflection of the plates.
The method assumes that the angle change at each fold varies in the same way longitudinally as the angle changes along the other folds.
For example, for the folded-plate structure in Fig. 5.99a, the steps in analysis are as follows:

(Figure 5.99d shows the resolution of forces at edge n; n  1 is similar.) Equation (5.179) does not apply for the case of a vertical reaction on a vertical plate, for R/k is the horizontal component of the reaction.
Step 4. Calculate the midspan (maximum) bending moment in each plate. In this example, each plate is a simple beam and M  PL^2 / 8, where L is the span in feet.
Step 5. Determine the free-edge longitudinal stresses at midspan. In each plate, these can be computed from

where ƒ is the stress in psi, M the moment in ft-lb from Step 4, A  plate crosssectional area and tension is taken as positive, compression as negative.
Step 6. Apply a shear to adjoining edges to equalize the stresses there. Compute the adjusted stresses by converging approximations, similar to moment distribution.
To do this, distribute the unbalanced stress at each edge in proportion to the reciprocals of the areas of the plates, and use a carry-over factor of 1⁄2 to distribute the tress to a far edge. Edge 0, being a free edge, requires no distribution of the stress there. Edge 3, because of symmetry, may be treated the same, and distribution need be carried out only in the left half of the structure.
Step 7. Compute the midspan edge deflections. In general, the vertical component S can be computed from

 

Stresses in Arches

An arch is a curved beam, the radius of curvature of which is very large relative to the depth of the section. It differs from a straight beam in that: (1) loads induce both bending and direct compressive stresses in an arch; (2) arch reactions have horizontal components even though loads are all vertical; and (3) deflections have horizontal as well as vertical components (see also Arts. 5.6.1 to 5.6.4). Names of arch parts are given in Fig. 5.93.

The necessity of resisting the horizontal components of the reactions is an important consideration in arch design. Sometimes these forces are taken by the tie rods between the supports, sometimes by heavy abutments or buttresses.
Arches may be built with fixed ends, as can straight beams, or with hinges at the supports. They may also be built with a hinge at the crown.

Three-Hinged Arches

An arch with a hinge at the crown as well as at both supports (Fig. 5.94) is statically determinate. There are four unknowns—two horizontal and two vertical components of the reactions—but four equations based on the laws of equilibrium are available: (1) The sum of the horizontal forces must be zero. (2) The sum of the moments about the left support must be zero. (3) The sum of the moments about the right support must be zero. (4) The bending moment at the crown hinge must be zero (not to be confused with the sum of the moments about the crown, which also must be equal to zero but which would not lead to an independent equation for the solution of the reactions).

Stresses and reactions in threehinged arches can be determined graphically by taking advantage of the fact that the bending moment at the crown hinge is zero. For example, in Fig. 5.94a, a concentrated load P is applied to segment AB of the arch. Then, since the bending moment at B must be zero, the line of action of the reaction at C must pass through the crown hinge. It intersects the line of action of P at X.
The line of action of the reaction at A must also pass through X. Since P is equal to the sum of the reactions, and since the directions of the reactions have thus been determined, the magnitude of the reactions can be measured from a parallelogram of forces (Fig. 5.94b). When the reactions have been found, the stresses can be computed from the laws of statics (see Art. 5.14.3) or, in the case of a trussed arch, determined graphically.

Two-Hinged Arches

When an arch has hinges at the supports only (Fig. 5.95), it is statically indeterminate, and some knowledge of its deformations is required to determine the reactions.
One procedure is to assume that one of the supports is on rollers. This makes the arch statically determinate. The reactions and the horizontal movement of the support are computed for this condition (Fig. 5.95b). Then, the magnitude of the horizontal force required to return the movable support to its original position is calculated (Fig. 5.95c). The reactions for the two-hinged arch are finally found by superimposing the first set of reactions on the second (Fig. 5.95d).
For example, if x is the horizontal movement of the support due to the loads, and if x is the horizontal movement of the support due to a unit horizontal force applied to the support, then

where M  moment at any section resulting from loads
N  normal thrust on cross section
A  cross-sectional area of arch
y  ordinate of section measured from A as origin, when B is on rollers
I  moment of inertia of section
E  modulus of elasticity
ds  differential length along axis of arch
dx  differential length along horizontal

where   the angle the tangent to the axis at the section makes with the horizontal.
Unless the thrust is very large and would be responsible for large strains in the direction of the arch axis, the second term on the right-hand side of Eq. (5.169) can usually be ignored.
In most cases, integration is impracticable. The integrals generally must be evaluated by approximate methods. The arch axis is divided into a convenient number of sections and the functions under the integral sign evaluated for each section. The sum is approximately equal to the integral. Thus, for the usual two-hinged arch,

(S. Timoshenko and D. H. Young, ‘‘Theory of Structures,’’ McGraw-Hill Book Company, New York; S. F. Borg and J. J. Gennaro, ‘‘Modern Structural Analysis,’’ Van Nostrand Reinhold Company, Inc., New York.)

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.