## Kcp

Advanced analysis Analysis predicting directly the stability of a structural system and its component members and not needing separate member capacity checks. ASD Acronym for allowable stress design. Beam columns Structural members whose primary function is to carry axial force and bending moment. Braced frame Frame in which lateral deflection is prevented by braces or shear walls. Column Structural member whose primary function is to carry axial force. CRC Acronym for Column Research Council....

## 1853 Soil Structure Interaction to Improve Earthquake Response

Soil-structure interaction modifies the ground-input motion at the foundation. By taking advantage of soil-structure interaction, structures can be protected during earthquakes. Frank Lloyd Wright's 1915 Imperial Tokyo Hotel survived the 1923 Great Kanto Earthquake because he built his foundation in 70 ft of weak clay. During the earthquake, the ground underneath moved violently while his hotel (effectively isolated by this plastic material) remained relatively immobile. Similarly, a common...

## 20826Structural Steel Design Requirements Guide [36

20.8.2.6.1 Limiting Width-to-Thickness Ratios For capacity-protected components, width-thickness ratios of compression elements shall not exceed the limiting value 1r as specified in Table 20.13. For ductile components, width-thickness ratios shall not exceed the 1p as specified in Table 20.13. Welds located in the expected inelastic region of ductile components are preferably complete penetration welds. Partial penetration groove welds are not recommended in these regions. If the fillet welds...

## 15291

AS 93.54 cm2 v 0.0089 cm year Exposure time 25 years FIGURE 36.23 Optimized concrete T-girder designed for a 25-year service life and a minimum acceptable reliability index of 3.0 while sustaining a deterioration rate of 0.0089 cm year (0.0035 in. year) on corroding reinforcing steel (beam dimensions are in centimeters). (Frangopol et al. 1997a. Reprinted with permission from the American Society of Civil Engineers.) 0.0045 in. year) were considered for the steel reinforcement once corrosion...

## Cable Supported Bridges

26-1 Features of Cable-Supported Bridges Historical Sketch of Modern Cable-Supported Bridges Steel Wires Basic Feature of Wire Cable Types of Bridge Cables End Fittings of Cable Corrosion Protection Vibration Control Use of Composite Materials Structural System Analysis of Suspension Bridges Design Procedure Stress Ribbon Bridge Erection 26.4 Cable-Stayed Bridges 26-22 Manabu Ito Layout of Structural System Configuration and Design of University of Tokyo Structural Components Erection of...

## 1

By setting time difference t 0, Equation 3.114b becomes where a is called variance of X or mean square. The square root of the mean square is aX, known as the root-mean-square (RMS). For a zero mean stochastic process E(X(t)) 0, RMS represents the magnitude of the response. Let the design value be where l represents ith d.o.f. and m is the coefficient related to the probability of the safety. By the sum of the modal contribution, the mean square value of the lth d.o.f. can be obtained as where...

## Asd

When _ ,, OCP ObCmxMx ObCmyMy OcP Pn > 0.15, -C- + x + bCmy y < 1.0 P fc -n < 015, fCPn + fbMnx' fbMn P fcPn > 0.15, -P- + + < 1.0 (6.86) OCP ObMx ObMy c ' b b < 1.0 (6.87) Note Mx and My are the required moments with respect to the centroidal axes of the effective section determined for the required axial strength alone, Mnx and Mny are the nominal flexural strengths about the centroidal axes, Mx and My are the required flexural strengths with respect to the centroidal axes of the...

## Reliability Based Structural Design

12-1 12.2 Available Structural Design Concepts 12-2 12.3 Introduction of the Reliability-Based Structural Design 12-3 12.4 Fundamental Concept of Reliability-Based Structural 12-4 12.5 Reliability-Based Structural Design Using FORM 12-12 12.6 Reliability Evaluation with Nonnormal Correlated Random 12-15 12.7 Reliability Evaluation Using Simulation 12-18 Steps in Simulation Variance Reduction Techniques Simulation in Structural Design 12.8 Future Directions in Reliability-Based Structural 12-23...

## 146 Current Usage

The first bridge that uses HPS70W steel in design and construction was the Snyder South Bridge on State Rt. 79 in Dodge County, Nebraska (Figure 14.2). It is a 150-ft (45.7-m) long simple span bridge that provides a 35.5 ft (10.8 m) clear roadway on a concrete deck supported by five 4'6 (137 cm) deep welded plate girders. The bridge was opened to traffic in October 1997. Since then, the number of bridges designed and constructed using some HPS components is on the rise. As of 2003, some 35...

## Y

Of inflection in the longitudinal element occur midway between the battens, the shear factor av is obtained as I p2EI ah a2 V1 + (KL) 12Eb Ib + 24 IfJ where Eb is the modulus of elasticity of materials for batten plates, Ib is the moment inertia of all battens in one panel in the buckling plane, and If is the moment inertia of one side of the main components taken about the centroid axis of the flange in the buckling plane.

## 174 Distribution of Seismicity

This section discusses and characterizes the nature and distribution of seismicity. It is evident from Figure 17.1 that some parts of the globe experience more and larger earthquakes than others. The two major regions of seismicity are the circum-Pacific Ring of Fire and the TransAlpide belt, extending from the western Mediterranean through the Middle East and the northern Indian subcontinent to Indonesia. The Pacific plate is created at its South Pacific extensional boundary its motion is...

## 07

A Flexure for factory-produced precast prestressed concrete members, f 1.0. For posttensioned cast-in-place concrete members, f 0.95 by AASHTO. b Shear and Torsion Reduction factor for prestressed members, f 0.90 by AASHTO. PHOTO 8.5 Walt Disney World Monorail, Orlando, Florida a series of hollow prestressed concrete 100-ft box girders individually posttensioned to provide a six-span continuous structure. Design by ABAM Engineers and owned by Walt Disney World Company (courtesy E.G. Nawy)....

## 1812 Structural Damage

Every day, regions of high seismicity experience many small earthquakes. However, structural damage does not usually occur until the magnitude approaches 5.0. Most structural damage during earthquakes is caused by the failure of the surrounding soil or from strong shaking. Damage also results from surface ruptures, the failure of nearby lifelines, or the collapse of more vulnerable structures. We consider these effects secondary because they are not always present during an earthquake. However,...

## Further Reading

The following publications provide additional sources of information for semi-rigid frame structures Chan, S.L. and Chui, P.P.T., Static and Cyclic Analysis of Semi-Rigid Steel Frames, Elsevier Science, London, 2000. Chen, W.F., Goto, Y., and Liew, J.Y.R., Stability Design of Semi-rigid Frames, John Wiley & Sons, New York, 1996. Chen, W.F. and Lui, E.M., Stability Design of Steel Frames, CRC Press, Boca Raton, FL, 1991. Faella, C., Piluso, V., and Rizzano, G., Structural Steel Semirigid...

## Masonry Structures

9.2 Masonry in the United Fundamentals of Masonry in the United States Modern Masonry Construction in the United States Historical Structural Masonry in the United States 9.3 Fundamental Basis for Design of Masonry in the United Design Approaches for Modern U.S. Masonry 9.4 Masonry Design Codes Used in the United States 9-10 Introduction to Masonry Design Codes in the United States Masonry Design Provisions of Modern Model Codes in the United States Seismic Design Provisions for Masonry in the...

## 3521 Single DegreeofFreedom Structural Systems

Consider the lateral motion of the basic SDOF model, shown in Figure 35.2, consisting of a mass m, supported by springs with total linear elastic stiffness k, and a damper with linear viscosity c. This SDOF system is then subjected to an external disturbance characterized by f(t). The excited model responds with a lateral displacement x( t) relative to the ground that satisfies the equation of motion in which a superposed dot represents differentiation with respect to time. For a specified...

## Jt

FIGURE 32.15 Cross-section distortion. Attach when brace is attached to flange FIGURE 32.16 Web stiffener details for torsional beam bracing. Attach when brace is attached to flange FIGURE 32.16 Web stiffener details for torsional beam bracing. bs stiffener width for one-sided stiffeners, which is twice the individual stiffener width for pairs of stiffeners bT represents the torsional stiffness of the brace member itself. For flexible connections, Equation 32.2 should be used. bsec is the...

## P

FIGURE 2.120 Beam-column with end springs. FIGURE 2.120 Beam-column with end springs. Solving Equations 2.317 and 2.318 simultaneously for MA and MB gives In writing Equations 2.319 and 2.320, the equality sjj sii has been used. Note that as RkA and RkB approach infinity, Equations 2.319 and 2.320 reduce to Equations 2.308 and 2.309, respectively. 2.12.4.4 Member with Transverse Loading For members subjected to transverse loading, the slope deflection Equations 2.308 and 2.309 can be modified...

## D E F

600 ft s< vs < 1200 ft s vs < 600 ft s Soils requiring site-specific evaluations 1. Soils vulnerable to potential failure or collapse under seismic loading such as liquefiable soils, quick and highly sensitive clays, collapsible weakly cemented soils 2. Peats and or highly organic clays (H > 10 ft of peat and or highly organic clay where H thickness of soil) 3. Very high plasticity clays (H> 25 ft 8 m with PI > 75) 4. Very thick soft medium stiff clays (H > 120 ft 36 m ) Note vs,...

## S11 S12 S13

The three roots of Equation 1.12, herein denoted as ffP1, ffP2, and cP3, are the principal stresses acting on the three orthogonal planes. The components of a unit vector that defines the principal plane (i.e., n1Pi, n2Pi, n3Pi) corresponding to a specific principal stress sPi (with i 1, 2, 3) can be evaluated using any two of the following equations 1P> (S11 Spi ) + 2P S12 + 3P S13 0 1P> S12 + 2Pi (s22 Spi ) + 3pi S23 0 1Pi S13 + 2Pi S23 + 3Pi (s33 SPi ) 0 The unit vector calculated for...

## 2

Which can be derived with unit impulse. For practical structural engineering problems, the damping factors are usually small (p < 15 ). p* and p* may be replaced by p and p, respectively. Thus, Equation 3.24a becomes p f e-pp(t -A) F (A) sin p(t - A) dA K J 0 When the structure is subjected to ground acceleration xg, the motion equation is The displacement response is similar to Equation 3.24b in which F(D) is replaced by Mxg. To find the maximum displacement over the entire or partial record...

## Info

A more rational approach is to use the difference between the rate of return on a risk-free investment and the inflation rate as the appropriate discount rate, which is about 2 to 3 . Since these rates are well publicized over time, the discount rate could easily become a random variable based on years of past experience. Using this approach, Figure 36.19 shows a histogram reflecting the discount rate in the United States from 1962 to 2002, which reflects the difference between the...

## Example

A simply supported beam of span L 9.144 m is loaded by a uniformly distributed load w in kN m and a concentrated load P (in kN) applied at the midspan. The maximum deflection of the beam at the midspan can be calculated as where E is the Young's modulus and I is the moment of inertia of the cross-section of the beam. A beam with EI 182,262 kN m2 is selected to carry the load. Suppose w is a normal random variable with a mean of 35.03 kN m and a standard deviation of 5.25 kN m and P is a...

## U

Which represents static stiffness coefficients. 3.4.3 Dynamic System Matrix and Eigensolutions Let a rigid frame have joint rotations Xy , sideways Xs , and concentrated masses M (associated with sideways such as floor masses), then the dynamic system matrix without externally applied force maybe expressed as 1, pp. 175-186

## 0330

Note Uncorrected PGA is to be used only when an estimate of PGA is required. Corrected PGA is to be used when an estimate of PGA compatible with PSA is required. Source Adapted from Campbell, K.W. and Bozorgnia, Y. (2003) Updated Near-Source Ground Motion (Attenuation) Relations for the Horizontal and Vertical Components of Peak Ground Acceleration and Acceleration Response Spectra, Bull. Seismol. Soc. Am.

## Alz

Kc 4 (h tw), 0.35 < kc < 0.763 bf compression flange width tf compression flange thickness Lb lateral unbraced length of the girder rT P (tfb3 12 + hctW 72) (bftf + hctw 6) hc twice the distance from the neutral axis to the inside face of the compression flange less Fyf yield stress of compression flange (ksi) Cb Bending coefficient (see Section 4.5) Fcr must be calculated for both flange local buckling and lateral torsional buckling. The smaller value of Fcr is used in Equation 4.93. The...

## 153424 Cement Aggregate Bond

There is now accumulated evidence that the microstructure of the interfacial zone between the cement paste and the aggregate is not so typical of the bulk paste. It is characterized by a more porous structure and oriented CH crystals and as a result is more prone to cracking, particularly under stress concentration that may occur at the interface. In the presence of MS, the interfacial zone has a denser, more uniform structure that does not appear to differ significantly from the bulk paste....

## 1 V

FIGURE 26.11 Examples of stiffening girder of modern suspension bridge (a) Severn Bridge (b) Tsing Ma Bridge and (c) Akashi Kaikyo Bridge. FIGURE 26.11 Examples of stiffening girder of modern suspension bridge (a) Severn Bridge (b) Tsing Ma Bridge and (c) Akashi Kaikyo Bridge. bridges, adoption of big fairings on both sides of the cross-section and longitudinal openings on upper and lower decks features the Tsing Ma Bridge in Hong Kong (Figure 26.11b). Even if the stiffening girder is simply...

## 1563Modulus of Elasticity

To appreciate the differences in code expressions for the modulus of elasticity of concrete, it is important to realize that both ACI 318-02 and EC2-02 use a secant modulus. The definition of the secant modulus, however, is different in the two codes and the ACI 318-02 bases the modulus on the specified compressive strength, whereas EC2-02 bases the modulus on the average compressive strength. ACI 318-02 gives an expression for the secant modulus of elasticity defined as the slope of a line...

## Example 3619

A highway bridge is expected to have a service life of 45 years. The concrete bridge deck is not expected to last that long and will potentially have to be replaced several times during the life of the bridge. The reinforcing in the bridge deck is expected to corrode as chlorides from the deicing salts penetrate the deck and reach a critical concentration at the reinforcing steel. The tensile forces resulting from the corrosion expansion cause the bridge deck to spall. Based on the...

## Xpo

Limit, the tangent modulus Et becomes progressively smaller than the initial modulus of elasticity. For cold-formed steel design, the shear modulus is taken as G 11,300 ksi (77.9 GPa or 0.794 x 106 kg cm2) according to the AISI Specification. According to the AISI Specification, the ratio Fu Fy for the steels used for structural framing members should not be less than 1.08, and the total elongation should not be less than 10 for a 2-in. (50.8 mm) gage length. If these requirements cannot be...

## A

Diagrams can be broken down into single triangles and rectangles. Beams supporting uniform loads or uniformly varying loads may be handled by integration. Properties of some of the shapes of M EI diagrams that designers usually come across are given in Figure 2.11. It should be understood that the slopes and deflections that are obtained using the moment area theorems are with respect to tangents to the elastic curve at the points being considered. The theorems do not directly give the slope or...

## 0 370

The problems discussed so far have involved concentrated forces and moments applied to nodes only. But real structures are subjected to distributed or concentrated loading between nodes, as shown in Figure 2.79. Loading may range from a few concentrated loads to an infinite variety of uniform or nonuniformly distributed loads. The solution method of matrix analysis must be modified to account for such load cases. One way to treat such loads in the matrix analysis is to insert artificial nodes,...

## 0656

And the length of the half-buckle wavelength is given by A reasonable approximation based on test data obtained by Bank et al. (1994b, 1995) for common pultruded WF beams is to assume that the critical local buckling stress for the compression flange is equal to the average of the simply supported and the fixed conditions given in Equations 16.17 and 16.19. For compression flanges where both longitudinal edges are restrained, such as in box beams or hat sections, the flange is assumed to be...

## 414 Plastic Design

Plastic analysis and design is permitted only for steels with yield stress not exceeding 65 ksi. The reason for this is that steels with high yield stress lack the ductility required for inelastic deformation at hinge locations. Without adequate inelastic deformation, moment redistribution, which is an important characteristic for PD, cannot take place. In PD, the predominant limit state is the formation of plastic hinges. Failure occurs when sufficient plastic hinges have formed for a collapse...

## V3

Where au is the ultimate (or tensile) strength of the material. For materials that exhibit brittle behavior in tension, but ductile behavior in confined compression (e.g., concrete, rocks, and soils), the Rankine criterion is sometimes combined with the Tresca or von Mises criterion to describe the failure behavior of the materials. If used in this context, the criterion is referred to as the Tresca or von Mises criterion with a tension cut-off. 2. Mohr-Coulomb criterion. This criterion is...

## Zpa

1 Structural Fundamentals Eric M. 2 Structural Analysis J. Y. Richard Liew and N. E. Shanmugam 2-1 3 Structural Dynamics Franklin Y. 4 Steel Structures Eric M. 5 Steel Frame Design Using Advanced Analysis S. E. Kim and Wai-Fah 6 Cold-Formed Steel Structures Wei-Wen Yu 6-1 7 Reinforced Concrete Structures Austin Pan 7-1 8 Prestressed Concrete Edward G. 9 Masonry Structures Richard E. 10 Timber Structures J. Daniel 11 Aluminum Structures Maurice L. 12 Reliability-Based Structural Design Achintya...

## N

FIGURE 36.28 Bridge network consisting of 14 highway bridges in the Denver-Boulder corridor of Colorado. (Agkul and Frangopol 2003. Reprinted with permission from the American Society of Civil Engineers.) FIGURE 36.28 Bridge network consisting of 14 highway bridges in the Denver-Boulder corridor of Colorado. (Agkul and Frangopol 2003. Reprinted with permission from the American Society of Civil Engineers.) this study is described in Akg l and Frangopol (2004). A study of networks of bridges...

## 20

Mcr Sb(0. 5lVfffi + fce - fd) Vcw 0 . 3 (l-p7 + 0 . 3fc) bw d + Vp where Mcr for shear analysis is equal to the moment causing flexural cracking at section due to externally applied load Vc_ ,'M+Mb,* M< 1 .0 c 20 Mu w Mu where bw, s, and d are in millimeters and fy is in MPa. 8.7.4 Design of Prestressed Concrete Beams Subjected to Combined Torsion, Shear, and Bending in Accordance with the ACI 318-02 Code In statically indeterminate systems, stiffness assumptions, compatibility of strains at...

## 416 Seismic Design

Special provisions (AISC 2002) apply for the design of steel structures to withstand earthquake loading. To ensure sufficient ductility, more stringent limiting width-thickness ratios than those shown in Table 4.8 for compression elements are required. These seismic limiting width thickness ratios are shown in Table 4.21. TABLE 4.21 Seismic Limiting Width-Thickness Ratios for Compression Elements Flanges of I-shaped rolled, hybrid, or

## 15422Transportation

Transportation of the concrete from the mixer to the site has to be accomplished without any significant change in the slump, w b ratio, air content, consistency, and temperature. Quality control and quality assurance personnel have to be cognizant of prolonged mixing causes slump loss, and hence lower workability. Consequently, adequate job control must be established so as to prevent delays in delivery. Where practical, withholding some of the superplasticizer until the truck arrives at the...

## S12

Following the same procedure for sP2 or, more conveniently, by realizing that the two principal planes are orthogonal to each other, we have (Note that the planes on which the maximum shear stress acts make an angle of 45 with the principal planes, that is, 0s1 0P1 (p 4), 0s2 0P2 (p 4) 0P1 + (p 4).) 1.1.5 Octahedral, Mean, and Deviatoric Stresses Octahedral normal and shear stresses are stresses that act on planes with direction indices satisfying the condition 1 2 2 3 with respect to the three...

## 276 Methods of Analysis 2761 Response Behavior

Thin shells may resist external loading through forces acting parallel to the shell surface, forces acting perpendicular to the shell surface, and moments. While the analysis of such shells may be formulated within the three-dimensional theory of elasticity, there are reduced theories that are two-dimensional and are expressed in terms of force and moment intensities. These intensities are traditionally based on a reference surface, generally the middle surface, and are forces and moments per...

## 0250

In addition to the design requirements discussed above, the AISI North American Specification also includes some provisions for built-up compression members composed of two sections in contact or for compression elements joined to other parts of built-up members by intermittent connections. In connection design, due consideration should be given to shear rupture, tension rupture, and block shear rupture. For details, see the AISI Specification 7 .

## 05

0.0 -I 1 1 1 1 I ' i'1'ii FIGURE 21.20 Original design spectrum, scaled spectrum, and spectra of selected ground motions after scaling. Component demands from each time-history analysis are recorded, and the maximum values from each earthquake are used to estimate plastic rotation demands. Since seven records were used in the simulation, both FEMA-356 and ATC-40 stipulate that mean values of the peak demands are an adequate measure of the seismic demand. Table 21.16 summarizes the peak and...

## 211 Inelastic Analysis 2111 An Overall View

Inelastic analyses can be generalized into two main approaches. The first approach is known as plastic hinge analysis. The analysis assumes that structural elements remain elastic except at critical regions where plastic hinges are allowed to form. The second approach is known as spread of plasticity analysis. This analysis follows explicitly the gradual spread of yielding throughout the structure. Material yielding in the member is modeled by discretization of members into several line...

## U30

Ln is length of clear in long direction, face-to-face of support. 2. Minimum thickness if slabs without drop panels should not be less than 5 in. 3. Minimum thickness of slabs with drop panel should not be less than 4 in. 1. ln is length of clear in long direction, face-to-face of support. 2. Minimum thickness if slabs without drop panels should not be less than 5 in. 3. Minimum thickness of slabs with drop panel should not be less than 4 in. The minimum thickness of two-way slabs is dependent...

## 64 Materials and Mechanical Properties

In the AISI Specification 7 , 15 different steels are presently listed for the design of cold-formed steel members. Table 6.2 lists steel designations, American Society for Testing Materials (ASTM) designations, and yield points, tensile strengths, and elongations for these steels. From a structural standpoint, the most important properties of steel are as follows 1. Yield point or yield strength, Fy 4. Modulus of elasticity, tangent modulus, and shear modulus In addition, formability,...

## 309 Application

What follows is a step-by-step design procedure for simple tubular trusses, applying the charts presented in the foregoing section Step 1. Lay out the truss and calculate member forces using statically determinate pin-end assumptions. Flexibility of the connections results in secondary bending moments being lower than those given by typical rigid-joint computer frame analyses. Step 2. Select members to carry these axial loads, using the appropriate governing design code, for example, AISC....

## 3422 Fatigue Loading

Since fatigue is typically only a serviceability problem, members and connections are designed for fatigue using service loads. Most structures experience what is known as long-life variable-amplitude loading, that is, very large numbers of random-amplitude cycles greater than the number of cycles associated with the CAFL 29 . For example, a structure loaded continuously at an average rate of once per minute (0.016 Hz), would accumulate 10 million cycles in 19 years. With long-life...

## Structural Design for Fire Safety

37-1 Standard Fire Exposure Natural Fire Exposure 37.3 Introduction to Heat Transfer 37-8 Conduction Convection Radiation Some Simplified Solutions of Heat Transfer Section Factors Thermal Properties of Materials 37.4 Design of Structural Elements at Elevated 37-14 General Mechanical Properties of Steel and Concrete at Elevated Temperatures Design of Steel Elements Composite Composite Steel Concrete Members 37.5 Design for Unprotected Steelwork 37-27 Integration of Structural Load Bearing and...

## 1000

Note b' is the distance measured from toe of flange to centerline of web, h' is the distance between centerline lines of flanges, h'' is the distance from centerline of flange to tip of stem, l1, l2 are the length of the legs of the angle, ti, t2 are the thickness of the legs of the angle, bf is the flange width, tf is the average thickness of flange, tw is the thickness of web, Ic is the moment of inertia of compression flange taken about the axis of the web, It is the moment of inertia of...

## 75Design Criteria

In achieving the design objectives, there are four general design criteria of SAFE that must be satisfied 1. Safety, strength, and stability. Structural systems and member must be designed with sufficient margin of safety against failure. 2. Aesthetics. Aesthetics include such considerations as shape, geometrical proportions, symmetry, surface texture, and articulation. These are especially important for structures of high visibility such as signature buildings and bridges. The structural...

## 63

K K + - 2 M + C (3.101a) Ap AFy + M Q + C R (3.101b) AFy 8 F(t + At) - F(t) (3.102a) Q At x_ (t) + 3 (t) (3.102b) Aty R 3 x (t) + At6 (t) (3.102c) After solving Ax8 from Equation 3.100 as Ax8 -1 AF (3.103) Ax is determined by the following formula The incremental velocity vector, Ax , and displacement vector, Ax , are obtained from Ax At _ (t) + (t) + A Total displacement, velocity, and acceleration vectors are then determined from x( t + At) x(t ) + Ax (3.104) x (t + At) x (t ) + Ax (3.105)...

## 352 Basic Principles and Methods of Analysis

FIGURE 35.3 SDOF structure model with passive damper element. Consider now the addition of a generic passive or active control element into the SDOF model, as indicated in Figure 35.3. The response of the system is now influenced by this additional element. The symbol r in Figure 35.3 represents a generic integrodifferential operator, such that the force corresponding to the control device is written simply as Gx. This permits quite general response characteristics, including displacement-,...

## Example 369

The three-bar indeterminate truss shown in Figure 36.6 (Estes 1997 Estes and Frangopol 2001b) is modeled as a series-parallel system where the failure of any two bars will cause failure of the system. Let the event Bar 2 1'' indicate failure of bar 2 given that bar 1 has already failed, and let Bar 1'' indicate failure of bar 1. The resistances of the three perfectly ductile bars are assumed random variables as follows Rbar1 N 15,1.5 , Rbar2 N 15,1.5 , Rbar3 N 10,1.0 where N 10, 1.0 indicates...

## Fiber Reinforced Polymer Composites

16.1 Analysis and Design of FRP Pultruded Structures 16-1 Introduction Properties of Pultruded Shapes Design Basis for Pultruded Structures Design of Pultruded Structural Members Design of Pultruded Structural Connections 16.2 Analysis and Design of FRP Reinforcements for 16-24 Introduction Properties of FRP Reinforcing Bars Design Basis for FRP Reinforced Concrete Design of Flexural Members with FRP Reinforcing Bars Detailing of FRP Reinforcements 16.3 Analysis and Design of FRP Strengthening...

## 28 Energy Methods

Energy methods are a powerful tool in obtaining numerical solutions of statically indeterminate problems. The basic quantity required is the strain energy, or work stored due to deformations, of the structure. 2.8.1 Strain Energy Due to Uniaxial Stress In an axially loaded bar with constant cross-section, the applied load causes normal stress sy as shown in Figure 2.59. The tensile stress ay increases from 0 to a value sy as the load is gradually applied. The original, unstrained position of...

## 13

28.4.5.2 Analysis of Earthquake Effects All immersed tunnels should be designed for seismic events appropriate to their location 1, Chapter 8 . Seismic events during the construction phase should also be considered. Liquefaction of soils around an immersed tunnel should be avoided, perhaps requiring special measures to be taken. An appropriate level of risk should be agreed on with the client because cost implications may be significant. It may be appropriate to consider three magnitudes of...

## 4

ATC-40 also recommends changing the lateral load pattern to reflect the displaced shape of the building at the end of each phase of the analysis. It is also possible to consider degrading effects in the simplified procedure described above by reinitializing the analysis at the end of each phase and plotting a new capacity curve. Each new curve begins with a degraded system that accounts for the state of elements at the end of each phase. Figure 21.11b provides a conceptual view of three such...

## 2072 Elastic Response Spectrum Analysis ERSA

The ERSA including uniform load method and multimode dynamic analysis method is a linear elastic spectral analysis with the appropriate response spectrum. An adequate number of modes considered to capture a minimum of 90 mass participation shall be used for complex structures. The uniform load method specified in ATC MCEER Guidelines 37 is essentially an ESA method that uses a uniform lateral load distribution to approximate the effect of seismic loads. It may be used for both transverse and...

## 10

FIGURE 36.8 The hazard function for a retaining wall beam over a 20-year period. warning that something is wrong. The entire wall would need to be replaced, control measures would be installed, and an accelerated schedule would result for an estimated cost of 200,000. (b) There is a 50 chance that the wall will fail catastrophically, slide down the hill, but not spill onto the roadway below. The cost associated with a clean-up, immediate repair, and delaying of traffic while the project is...

## 05s1

The total, lateral base shear force given by Equations 19.7-19.9 must be distributed vertically for application to the various mass or diaphragm levels of the structure. For a structure with n levels, the force at diaphragm level x is given by the equation hx and hi, respectively, are the heights of levels x and i above the structure's base. These formula are based on the assumption that the structure is responding in its first mode, in pure sinusoidal motion, and that the mode shape is linear....

## Example 324

Nodal Column Braces as Effective Continuous Braces Consider a column of length 3Lb with two equally spaced nodal braces giving an unbraced length of Lb. Estimate the critical load of an ideally-braced column by approximating the nodal braces as an equivalent continuous brace. Ideal nodal brace stiffness required to buckle between brace points (2 Pcr 3Pcr Equivalent continuous stiffness using Equation 32.13 Lean-on bracing systems provide a means for some members in a structural system to rely...

## 73 Material Properties

With respect to structural design, the most important property of concrete that must be specified by the structural designer is the compressive strength. The typical compressive strength specified, fc, is one between 3000 and 8000 psi. For steel reinforcement, Grade 60 (American Society for Testing and Materials ASTM A615), with specified yield strength fy 60,000 psi, has become the industry standard in the United States and is widely available (see Photo 7.2). Material properties of concrete...

## 79 Moment Redistribution

The moment values of a continuous beam obtained from structural analysis may be adjusted or redistributed according to guidelines set by ACI 8.4. Negative moment can be adjusted down or up, but not more than 1000et or 20 (see Notation section for et). After the negative moments are adjusted in a span, the positive moment must also be adjusted to maintain the statical equilibrium of the span (see Section 7.13.12). Redistribution of moment is permitted to account for the ductile behavior of...

## 1524 Chemical Admixtures Superplasticizer

Chemical admixtures are important and necessary components for the preparation of HPC. The concrete properties, both in fresh and hardened states, can be significantly modified or improved by chemical admixtures. In some countries, 70 to 80 of concrete contains one or more admixtures (88 in Canada, 85 in Australia, and 71 in the United States). It is thus important for civil engineers to be familiar with commonly used admixtures. Presently, many types of chemical admixtures are being used in...

## 08

FIGURE 15.10 Strain hardening phenomenon of fiber-reinforced concrete. reaches its tensile strength. The multiple cracking process eventually leads to homogenization of matrix cracks, that is, the end of multiple cracking at which the minimum crack spacing, determined by the interfacial bond properties, is reached. In stage 4, no further matrix crack is expected and the additional load is only sustained by the fibers. To study the multiple cracking and associated interfacial bond behavior,...

## 1553Flexural Strength

Flexural strength is the strength of concrete under bending. In designing concrete pavements, flexural strength is often used because it better simulates what happens when a slab is loaded by a truck or other vehicle. Most general-use concrete has a flexural strength between 3.5 and 5 MPa. In general, a relationship exists between the compressive and flexural strengths of concrete. Concrete, which has a higher compressive strength, will have a correspondingly higher flexural strength. This...

## 26 Shells

2.6.1 Stress Resultants in the Shell Element A thin shell is defined as a shell with a thickness relatively small compared with its other dimensions. The primary difference between a shell and a plate is that the former has a curvature in the unstressed state, whereas the latter is assumed to be initially flat. The presence of initial curvature is of little consequence as far as flexural behavior is concerned. The membrane behavior, however, is affected significantly by the curvature. Membrane...

## M

FIGURE 12.1 Reliability-based design concept. (Adopted from Probability, Reliability and Statistical Methods in Engineering Design, by Haldar and Mahadevan, 2000, with permission from John Wiley & Sons, Inc.) The concept is shown in Figure 12.1. In Figure 12.1, the nominal values of resistance and load effect, denoted as Rn and Sn, respectively, and the corresponding PDFs of R and S are shown. The overlapped (dashed area) between the two PDFs provides a qualitative measure of the probability...

## 3 Hh

FIGURE 5.37 Configuration of the unbraced eight-story frame. The analysis library consists of three elements a plane frame, a plane truss, and a connection. The connection is represented by a zero-length rotational spring element with a user-specified nonlinear moment-rotation curve. Loading is allowed only at nodal points. Geometric and material nonlinearities can be accounted for by using an iterative load-increment scheme. Zero-length plastic hinges are lumped at the element ends. 5.5.4.4...

## 2413 Basic Concepts

The space frame can be formed on either a flat or a curved surface. The earliest form of space frame structure is single-layer grid. By adding intermediate grids and including rigid connection to the joist and girder framing system, the single-layer grid is formed. The major characteristic of grid construction is the omnidirectional spread of the load as opposed to the linear transfer of the load in an ordinary framing system. Since such load transfer is mainly by bending, for larger spans the...

## 208 597

Fn - (0 6581c )Fy - (0 6580 438 ) 40 - 36 914 ksi 3. Effective area, Ae. Because the given square tube is composed of four stiffened elements, the effective width of stiffened elements subjected to uniform compression can be computed from Equations 6.6 through 6.9 by using k 4.0 1.052 4(70 . 619) 36. 914 29,500 1. 314 Since 1 > 0.673, from Equation 6.7 p (1 0 . 22 1) (1 0 .22 1. 314) 1. 314 0 .634 Ae 3.273 4(7.415 4.701)(0.105) 2 .133 in. 2 4. Nominal and allowable loads. Using Equation 6.75,...

## Ohb

Vs is the average shear wave velocity for top 30 m of a site. N is the average standard penetration test blow count for top 30 m of a site. Nch is the average standard penetration test blow count for cohesionless layers of top 30 m of a site. su is the average undrained shear strength of cohesive layers for top 30 m of a site. Seismic force effects from different vibration modes shall be combined. For modal response closely spaced in frequency and ground motion in one direction, the complete...

## 4487

Since the above computed stress is close to the assumed value, it is alright. 3. Check the effectiveness of the web. Use the AISI Specification to check the effectiveness of the web element. From Figure 6.23, f1 50(4.1945 5.513) 38.04 ksi (compression) f2 50(5.2205 5.513) 47.35 ksi (tension) C l 2 f1 1.245. k 4 + 2(1 + C)3 + 2(1 + C) 4 + 2(2.245)3 + 2(2.245) 31.12 h t 9.415 0.105 89.67 < 200 OK. FCT v 3Ll2V9.67 297500 be h 9.415 in. Since ho 10.0 in., bo 15.0 in., ho bo 0.67 < 4, use...

## 0 1000 2000 3000 4000 5000 6000

FIGURE 15.19 Comparison of stress-strain curves in compression of HPC and normal concrete. 15.5.2.1 Direct Tensile Strength The direct tensile test is difficult to conduct. Due to the difficulty in testing, only limited and often conflicting data are available. For normal-strength concrete, it is often assumed that the direct tensile strength of concrete is about 10 of its compressive strength. For HPC, the ratio of tensile strength to compressive strength is greatly reduced to even 5 . Among...

## 0040

A quantitative approach for determining weldability of steel is to calculate its carbon equivalent value. One definition of the carbon equivalent value Ceq is (manganese + silicon) (copper + nickel) Ceq Carbon H-- 6--1-- - H (chromium + molybdenum + vanadium + columbium) A steel is considered weldable if Ceq < 0.50 for steel in which the carbon content does not exceed 0.12 and if Ceq < 0.45 for steel in which the carbon content exceeds 0.12 . Equation 4.1 indicates that the presence of...

## 411 Connections

Connections are structural elements used for joining different members of a framework. Connections can be classified according to The type of connecting medium used. Bolted connections, welded connections, bolted-welded connections, riveted connections. The type of internal forces the connections are expected to transmit. Shear (semirigid, simple) connections, moment (rigid) connections. The type of structural elements that made up the connections. Single plate angle connections, double web...

## 821

Spans AC and BC FIGURE 2.10 Example continuous beam. where M is the bending moment at the point and EI is the flexural rigidity of the beam section. Since the deflection is small, 1 R is approximately taken as and Equation 2.13 may be rewritten as In Equation 2.14, y is the deflection of the beam at distance x measured from the origin of the coordinate. The change in slope in a distance dx can be expressed as M dx EI and hence the slope in a beam is obtained as Equation 2.15 may be stated as...

## 1565 Minimum Reinforcement for Flexure

The primary role of minimum reinforcement is to provide adequate reserve of strength after cracking. ACI 318-02 and NZS 3101-95 code expressions for the minimum reinforcement ratio for rectangular sections are 3 f 200 As min f c bwd > 2 bwd in.2 (psi and in.) As,min bwd > bwd mm2 (MPa and mm) 4f fy where bw is the web width and d is the effective depth. For a statically determinate T-beam, with the flange in tension, ACI 318-02 requires that bw be taken as the smaller of 2bw or the flange...

## Ft A md

+691 .2 745 .7 -55 psi (C), no tension at transfer, OK. 376,110 ( 15 .0 x 18 .84 2,490,638 1 +-- +- - -2501.3 + 664.2 - -1837 psi (C) < fci - 2,250 psi, OK . Analysis of stresses at service load. From Equation 8.25a Pe _ 13 x 0 .153 x 154,980 308,255 lb (1,371 kN) Msd + Ml - (100 + 1,100)(65)2 x 12 8 - 7,605,000 in . lb Total moment MT MD + MSD + ML 2,490,638 + 7,605,000 10,095,638 in. lb (1,141 kN m) p _ 308,25 15 . 0 x 21.1 10,095,638 +566.5 - 3022 . 6 -2,456 psi (C) > f -2,250 psi...

## 005

FIGURE 22.58 Strouhal number for a rectangular section. 22.5.3.2 Forces Due to Turbulent Flow If the wind is turbulent, then the velocity of the wind along the wind direction is described as follows where J is the mean wind and u(t) is the turbulent component along the wind direction. The time dependent drag force per unit height is obtained from Equation 22.18 by replacing J by U(t). As the ratio u(t) U is small, the time-dependent drag force can be expressed as where JD and fD are the mean...

## 075

For the seismic design of ductile reinforced concrete members. These more conservative limits were chosen because of the concern over the ductility of high-strength concrete members and the lack of reversed cyclic loading test results at the time the standards were written. In the comparison of the different maximum strengths, it is important to realize that the codes determine the compressive strengths with a number of different standard test specimens (Table 15.11).

## 322Types of Bracing

Bracing used in structural systems generally serve two primary functions. They resist secondary loads on structures (e.g., wind bracing) and increase the strength of individual members by resisting deformation in the weakest direction 1 . For the latter case, structural bracing forces higher modes of deformation by providing resistance to lateral and or rotational displacement. This is achieved through axial, shear, and or flexural deformations of the bracing member. Diaphragms, for instance,...

## Deu

Note a indicates site-specific evaluation required. Note a indicates site-specific evaluation required. FIGURE 19.11 Maximum considered earthquake response spectrum. FIGURE 19.11 Maximum considered earthquake response spectrum. spectra based on mapped values are used, the actual design values are taken as two thirds of the MCE values. The resulting design parameters are, respectively, labeled SDs and SDi and the design spectrum is identical to the MCE spectrum, except that the ordinates are...

## Ei

Ma e- (s sj)0a (2.328) 2.12.5 Second-Order Elastic Analysis There are two methods for incorporating second-order effects, the stability function approach and the geometric stiffness (or finite element) approach. The stability function approach is based on the governing differential equations of the problem as described in Section 2.12.4, whereas the stiffness approach is based on an assumed cubic polynomial variation of the transverse displacement along the element length. Therefore, the...

## 2762 Linear Elastic Analysis

The linear static analysis method is based on the classical bending theory of thin shells. While this theory has been formulated for many years, solutions for doubly curved shells have not been readily achievable until the development of computer-based numerical methods, most notably the finite element method. The outputs of such an analysis are the stress resultants and couples, defined in Figure 27.15, over the entire shell surface and the accompanying displacements. The analysis is based on...

## S

FIGURE 25.45 Nodal joints of truss bridge (JASBC 1985). FIGURE 25.45 Nodal joints of truss bridge (JASBC 1985). often make use of the Howe truss since the connections of the diagonals in wood prefer compression. A K-truss, so named since the web members form a K, is most economical in large bridges because the short member lengths reduce the risk of buckling. 25.7.3 Structural Analysis and Secondary Stress The truss is a framed structure of bars, theoretically assumed to be connected by hinges,...

## C

(2 293a) (2 .293b) (2 293c) (2 .293d) Upon substitution of these constants into Equation 2.291, the deflection function can be written as sin kx H--T- - kx---- - H-- The maximum moment for this beam-column occurs at the fixed ends and is equal to Mmax Ely I x 0 Ely I x L Since wL2 12 is the maximum first-order moment at the fixed ends, the term in the brackets represents the theoretical moment amplification factor due to the P-d effect. For beam-columns with other transverse loading and...

## 15362Property Improvements Compared with Plain Concrete

Quasibrittle materials such as concrete have two major deficiencies, that is, a rather low tensile strength and a rather low energy consumption capacity or toughness. One way to overcome these weaknesses is to incorporate high-strength small-diameter fibers. The effect of the incorporation of the fiber can be largely attributed to the contribution of the bond between the fiber and matrix. The role of the interface can be viewed in two different aspects. First, in conventional applications of...

## 68

FIGURE 32.11 Beam with inflection point. FIGURE 32.12 Lateral buckling of cantilever and simple beams (a) max moment at max twist and (b) zero moment at max twist. FIGURE 32.12 Lateral buckling of cantilever and simple beams (a) max moment at max twist and (b) zero moment at max twist.

## 410 Plate Girders

They are used as flexural members to carry extremely large lateral loads. A flexural member is considered a plate girder if the width-thickness ratio of the web, hc tw, exceeds 760 v Fb (Fb is the allowable flexural stress) according to ASD, or 1r (see Table 4.8) according to LRFD. Because of the large web slenderness, plate girders are often designed with transverse stiffeners to reinforce the web and to allow for postbuckling (shear) strength (i.e., tension...

## 1515 Future Development

Significant evolutions relating to HPC have been taking place in the past two decades. These evolutions have been facilitated considerably by increased knowledge of the atomic and molecular structure, studies of long-term durability, development of more powerful instruments and monitoring techniques, and the need for stronger, higher-performance materials suitable for larger structures, long span, and more ductility. The last two decades can be described as the decades of HPC. Despite these...

## Passive Energy Dissipation and Active Control

Department of Civil Engineering, State University of New York, Buffalo, NY Department of Civil Engineering, State University of New York, Buffalo, NY 35.2 Basic Principles and Methods of Single-Degree-of-Freedom Structural Systems Multi-Degree-of-Freedom Structural Systems Energy Formulations Energy-Based Design 35.3 Recent Development and Passive Energy Dissipation Active Control 35.4 Code 35.5 Concluding References

## 74 Design Objectives

For reinforced concrete structures, the design objectives of the structural engineer typically consist of the following 1. To configure a workable and economical structural system. This involves the selection of the appropriate structural types and laying out the locations and arrangement of structural elements such as columns and beams. 2. To select structural dimensions, depth and width, of individual members, and the concrete cover. 3. To determine the required reinforcement, both...

## 171

Note Normal-weight concrete, Grade 60 reinforcement. a 1.0, b 1.0, c 1.5 in., and Ktr 0. Note Normal-weight concrete, Grade 60 reinforcement. a 1.0, b 1.0, c 1.5 in., and Ktr 0. 7.17.2 Compression Development Lengths For bars under compression, such as in columns, yielding is assured if the development length meets the largest value of (0.02f yf)db, (0.0003fy)db, and 8 in. Compression development lengths ldc are given in Table 7.14. Compression development length may be reduced by the factor...

## Qcr L6ER2

The critical load for overall buckling may also be expressed as the following formula for comparison where tm is the effective in-plane thickness, tb is the effective bending thickness, and the values of k are 0.377 16 0.365 17 0.247 15 0.294 18 Discrete Analysis is a more powerful tool to study the whole process of instability for space frames. As shown in Figure 24.27, a structure may lose its stability when it has reached a limit point,'' where

## 1728

The time of initiation of deterioration of reliability is equal to the time for corrosion to start (i.e., 10 years). The deterioration rate of the reliability index is given by a -- -- - 0.079 per year The condition index deterioration rate, in absolute value, is The effect of concrete cover replacement is analyzed at different points in time. To ensure a reliability index above the minimum target during the entire lifetime, maintenance should be applied when the condition index reaches the...

## H

FIGURE 2.115 Basic differential equation of a beam-column. FIGURE 2.115 Basic differential equation of a beam-column. Considering equilibrium of forces 1. Horizontal equilibrium S + ds - S + w ds 0 (2.282) dM ( dS (ds ( dH , fds M + ds - M - ( S + + cos 6(y j + (H + ds + Hj sin 6(yj 0 (2.283) Since (dS ds)ds and (dH ds)ds are negligibly small compared to S and H, the above equilibrium equations can be reduced to dM - S cos 6 + Hsin 6 0 (2.284c) For small deflections and neglecting shear...