# OPTIMISATION OF REINFORCEMENT OF RC FRAMED STRUCTURES

Size: px
Start display at page:

## Transcription

1 Engineering MECHANICS, Vol. 17, 2010, No. 5/6, p OPTIMISATION OF REINFORCEMENT OF RC FRAMED STRUCTURES Petr Štěpánek, Ivana Laníková* This paper presents the entire formulation of longitudinal reinforcement minimisation in a concrete structure of known sections and shape under loading by normal force and bending moment. Constraint conditions are given by the conditions of structure reliability in accordance with the relevant codes for ultimate strength and applicability of the sections specified by a designer. Linearization of the non-linear formulation is described, and possibilities of applying linear programming algorithms are discussed. The functioning of the process described is demonstrated on a plane frame structure design. Keywords : optimisation, reinforcement, frame 1. Introduction The methods for optimising the design of linear elastic structures from the standpoint of minimal cost, shape, or volume of materials are highly developed. The methods used are based on linear methods of mathematical programming and they have been mostly applied to steel structures. Recently, methods of optimisation design have also been developed in the non-linear field [1 4]. These methods are based on the geometrical non-linear behaviour of a structure and on the influence of possible geometrical imperfections. Unlike in steel structures (especially slender structures), physical non-linear behaviour has an important function in concrete structure design. The influence of geometric nonlinearity on the design optimisation of reinforcement is less important, as the minimum slenderness of a pressured member is defined in most codes. Thus, we should formulate reinforcement optimisation design not as minimising the critical load in relation to the parts of a pressured structure, but as the search for minimum reinforcement for the recommended values of load. The method of non-linear mathematical programming can be used to solve this problem [5]. Reinforcement optimisation design of an RC frame structure for constraint conditions based on load-carrying capacity according to Baldur s method of inscribed hyper-spheres is described in [6]. Design of reinforcement for use in columns loaded by eccentric normal force by means of a genetic algorithm is described in [7]. This paper describes an application of the non-linear method of mathematical programming for RC structure reinforcement design with reference to minimising its volume in compliance with the criteria of load-carrying capacity and applicability. We present the complete mathematical formulation of the non-linear problem of optimised reinforcement design for concrete structures. This problem is linearized with the incremental method. *prof.rndr.ing.p.štěpánek, CSc., Ing. I. Laníková, Ph.D., Institute of Concrete and Masonry Structures, Faculty of Civil Engineering, Brno University of Technology, Veveří 331/95, Brno, Czech Republic

2 286 Štěpánek P. et al.: Optimisation of Reinforcement of RC Framed Structures The simplified method for calculating creep effects is used in order to eliminate the solution of that time-dependent task. The response of a structure under constant load with varying reinforcement is solved by a deformation variant of a FEM that takes physical and geometrical non-linearity into account. The set-up programme enables simultaneous solution of several loading states. 2. Calculation model Thetaskisdefinedasfollows a) the target function reaches the extreme {f({a s })} =extreme, while keeping the restrictive conditions implicit in the form of b) the equalities {h({a s })} = {0 1 }, (1a) (1b) c) the inequalities {g({a s })} {0 2 }, (1c) where {A s } = {A s1,...,a snt } T are the design variables; {f({a s })} = {f 1 ({A s }),..., f t ({A s })} T is the vector of the target functions; f i ({A s }) is the i-th target function; {h({a s })} is the vector of the restrictive conditions in the form of equations; {g({a s })} is the vector of the restrictive conditions in the form of inequalities, and {0 1 } and {0 2 } are zero vectors of the relevant type Target function The optimisation problem of the concrete structure may be, e.g. by [9], expressed by the target function : {f({a s })} = f(min E tot, min C tot, max S tot ), (2) where E tot is the gross environmental impact, C tot is the gross cost and S tot is the gross social-cultural quality. The multi-criterion problem is possible to convert to the mono-criterion one by the weight implementation among the individual criteria but it is evident that at the different weights given to the individual criteria, obtained optimums can differ. Furthermore, it is not guaranteed that at the searching for optimum by the different method of multi-criterion optimisation (selection strategy), the given method will always converge. Longitudinal reinforcement optimisation in a plane frame structure is mathematically defined as ne ke f({a s })= = minimal, (3) e=1 l e i=1 where f({a s }) is the target function expressing the overall volume of reinforcement in an RC framed structure; l e is the length of finite element e; A si is the area of reinforcement in the i-th layer of the finite element e; ne is the number of finite elements of which the model of the structure is composed; ke is the number of reinforcing layers in finite element e. A e si

3 Engineering MECHANICS 287 For practical reasons, e.g. to make it possible either to reinforce several members identically or to reduce the number of the optimised quantities, a vector of the reinforcing types A s = {A s1,a s2,...,a snt } T is introduced by means of which the reinforcement in each layer of each finite member can be defined; nt is the number of reinforcing types in the structure. It is generally valid that A sk A sl for k l; the components of vector A s are optimised variables. When we denote {A e s} vector of the reinforcing areas of element e, type (1,ke), its components can be determined from the relation {A e s} =[B e ] {A s }, (4) where [B e ] is the matrix comprising 0 and 1; its elements are defined when a task is assigned. The target function (3) of the optimisation is formulated as ne f({a s })= l e {i e } T [B e ] {A s }, (5) e=1 where {i e } is a vector of type (1,ke) comprising only Constraint conditions For an established vector of the design variables, the conditions of equilibrium of the solved structure with the application of discretion of the solved task via the FEM can be expressed in the form (1b), where {h({a s })} =[K({A s })] {Δ} {F }. (6) [K({A s })] is the global stiffness matrix of the solved structure; {Δ} is a vector of the nodal parameter of deformation and {F } is the loading vector of a structure. Restrictive conditions in the form of inequalities (1c) are possible to specify g di ({A s }) 0 for i =1, 2,...,nd, (7a) g sj ({A s }) 0 for j =1, 2,...,nc, (7b) {A su } {A s } {A sl } {0}, (7c) where the function g di ({A s }) expresses the constraint condition for the structural deformations; nd is the overall number of deforming constraint conditions; g sj ({A s }) is the function expressing the condition of the reinforcement design reliability from the viewpoint of the cross-section load-bearing capacity j (the force constraint condition); nc is the number of sections in which the force condition is controlled. The conditions (7c) guarantee the minimum reinforcement required by a designer. {A sl } = {A sl,1,...,a sl,nt } T or {A su } = {A su,1,...,a su,nt } T is the vector of the minimum/maximum sectional area that restricts the vector components of the reinforcing type Deformation constraint conditions The codes for concrete structure design or for the technology to be installed in a construction generally require compliance with the conditions for deformations for specified

4 288 Štěpánek P. et al.: Optimisation of Reinforcement of RC Framed Structures combinations of loads at various time stages of a structural action and in various sections. The i-th condition of deformation is expressed as g di ({A s })=w i (F i,t i1,t is ) w iu or (8a) g di ({A s })= w i (F i,t i1,t is )+w iu, (8b) where w i (F i,t i1,t is ) is the structure deformation at time t is in node (section) s from load F i, whose action begun at time t i1 ; w iu is the specified limit value of deformation. The expression (8a) is valid for positive half-axis and (8b) for negative deformation value. In order not to consider the problem as time-dependent the influence of creep on deformation w i can be simplified by a transforming equation (8). In accordance with the regulations for concrete structure design, it is considered to be valid that w i (F i,t i1,t is )=w ip (F i,t i1 )+Δw(F i,t i1,t is ), (9) where w ip = w ip (F i,t i1 ) is the initial deformation at node s at time t i1 generated by the load F i,δw(f i,t i1,t is ) is the deformation generated by creep from load F i acting in the time interval (t i1,t is ). The influence of shrinkage in condition (8) can generally be neglected. In accordance with the regulations for concrete structure design the value of deflection Δw(F i,t i1,t is )causedbycreepcanbeexpressedas Δw(F i,t i1,t is )=w ip (F i,t i1 ) ϕ(t i1,t is ), (10) where ϕ(t i1,t is ) is the creep coefficient given by the relevant code e.g. [1 2]. By inserting (9), (10) into (8a) or (8b), taking into account consideration (7a), the condition of deformation can be adapted as a new condition in the form w iu g di ({A s })=w ip (F i,t i1 ) 1+ϕ(t i1,t is ), w iu g di ({A s })= w ip (F i,t i1 )+ 1+ϕ(t i1,t is ). (11a) (11b) to The initial deformation w ip in a cross-section of local coordinate x i is calculated according w ip (F i,t i1 )={w e (x)} T {Δ e } = {w e (x)} T [L e ] {Δ}, (12) where {w e (x)} is a vector of the type (ndp, 1); {Δ e } = {Δ e ({A s })} (or {Δ} = {Δ({A s })}) is a vector of the nodal deformation parameters of element e (or the structure) deformation loaded by F i, when it is solved with reinforcement defined by vector {A s };[L e ]isatransforming matrix comprising 0 and 1 of the type (ndp, nk); {Δ e } is of the type (ndp, 1), {Δ} is of the type (nk, 1), and ndp (or nk) is the number of nodal parameters of the deformation in element e (or in the whole structure) Force constraint conditions The formulation of force constraint conditions is based on the assumption of RC section behaviour in agreement with the relevant code. Most of the recommendations accept a nonlinear model for a section loaded by normal force and bending moment. The model is based on the following preconditions see Fig. 1 :

5 Engineering MECHANICS 289 Perfect bond between concrete and reinforcement. Linear dependence of the strains along the height of the cross-section. The stress in the individual materials (steel, concrete) is determined from the stressstrain diagrams defined in the relevant code. The cross-section is considered to be reliably designed for the given load when the strains of extremely loaded fibres of individual materials are lower than the limit values defined by the design recommendation under fulfilment of these preconditions. Generally, fulfilment of all the conditions is required a) for concrete fibres under pressure ε(z bc ) ε bd, (13a) b) for reinforcement in a layer i ε sc ε(z si ) ε st, (13b) where ε bd is the limit value of the strains for pressured concrete; ε sc (or ε st ) is the limit value of the strains for pressured/tensioned reinforcement; z bc is z-th coordinate of the extremely stressed fibres of the concrete; z si is z-th coordinate of the i-th reinforcing layer. Fig.1: Shape of a cross-section and precondition for calculation of a) a acting loading, b) the course of the strains generated by load N, M; stress-strain diagram of concrete σ c = σ c(ε); stress-strain diagram of reinforcement σ s = σ s(ε) In the relations (13) and in Fig. 1 the sign convention is considered to be in accordance with the theory of elasticity (tension > 0, pressure < 0). Some Codes (e.g., Eurocode EC 2, Czech Code , German DIN 1045) do not require evaluation of tensile reinforcement as the equation ε bd ε sc is valid.

6 290 Štěpánek P. et al.: Optimisation of Reinforcement of RC Framed Structures The strain ε e (x, z) of the fibres of the coordinates z in the cross-section x of finite element e can be expressed as the linear combination of the nodal parameters of the deformation of element e ndp ε e (x, z) = n k (x, z)δ e,k, (14) k=1 where n k (x, z) is a coefficient that can be derived in dependence on coordinates x, z and on the option of approximate functions when formulating the basic relationship for the finite element e, Δ e,k is the k-th nodal parameter describing the deformation of finite element e. Equation (14) can be written in matrix form as ε e (x, z) ={N e (x, z)} T {Δ} = {N e (x, z)} T [L e ] {Δ}, (15) where {N e (x, z)} = {n 1,...,n ndp } T. Hence, the conditions of reliability (13) are {N e (x, z)} T [L e ] {Δ} ε bd 0, z = z bc, (16a) ε st {N e (x, z)} T [L e ] {Δ} ε sc, z = z st and z = z sc, (16b) where z st, z sc and C gb. is the distance between extremely tensioned or compressed reinforcement 3. Solution of optimisation design The optimisation of reinforcement design into an RC structure is determined by a) the target function (1a), which is the non-linear function of the vector elements of reinforcement {A s }, b) the constraint conditions that express (i) the conditions of equilibrium when a structure is solved by the FEM (6), (ii) the conditions of deformation (7a) reducing the displacement of a structure that can be written as (11), (iii) the reliability of the cross-sections of the designed structure expressed in view of the code recommendation used that can be written as (7b) or (16), (iv) the recommended limits of the reinforcement areas (7c). The constraint conditions expressed by Eqs. (6), (7a) and (7b), respectively, are a nonlinear function of the vector elements {A s }, and the conditions (7c) subject to the elements {A s } are linear. Linearization of the non-linear constraint conditions in terms of using an incremental method is the next step in the theoretical solution. The basic relations of the linear optimisation after linearization and after the introduction of additional variables and auxiliary variables are expressed as a) target function nt { } T f {f({a k s })} T {da s } + da si {da s } + α {e u } T {u} = minimal, (17) A si i=1

7 Engineering MECHANICS 291 b) constraint conditions given by (i) the conditions of equilibrium of a structure (for j =1, 2,...,nk) nt i=1 [ ] K da si {Δ k } +[K] A si nt i=1 where {0} is zero vector of type (1,nk), (ii) the conditions of deformation (1b) for i =1, 2,...,nd { } Δ k da si + {u} = {0}, (18) A si {w e (x i )} T [L e ] {Δ k } + { we (x i )} T {da s } w iu 1+ϕ + w i = 0 or (19a) {w e (x i )} T [L e ] {Δ k } { we (x i )} T {da s } + w iu 1+ϕ + w i = 0 (19b) (iii) the reliability of the cross-sections in accordance with the concrete design recommendation for i =1, 2,...,nc the extremely pressured concrete layer {N e (z i,z bc } T [L e ] {Δ k } { Ne (z i,z bc )} T {da s } + ε bd + c i =0, (20a) the extremely pressured layer of reinforcement (if the relations for the given design recommendation have physical justification) {N e (z i,z sc } T [L e ] {Δ k } { Ne (z i,z sc )} T {da s } + ε sc + s ci =0, (20b) the extremely tensioned layer of reinforcement {N e (z i,z st } T [L e ] {Δ k } + { Ne (z i,z st )} T {da s } ε st + s ti =0, (20c) (iv) the recommended limits of the reinforcement areas {A sl } {A k s } {da s} + {a l } = {0}, (21a) {A k s } {A su} + {da s } + {a u } = {0}, (21b) c) additional conditions of non-negativeness for the auxiliary variables {u} {0}, (22a) the additional variables {w} {0}, {c} {0}, {s t } {0}, {s c } {0}, {a l } {0}, {a u } {0}. (22b) In equations (17) to (21), {A k s } expresses the vector of the reinforcement areas in the k-th iterative step, {da s } is the vector of the increment of the reinforcement areas by means of which we define the vector of areas in the (k+1)-th step {A k+1 s } = {A k s } + {da s}. (23)

9 Engineering MECHANICS 293 Fig.2: Geometry of the frame structure, division into finite elements Fig.3: Loading scheme Load notation intensity γ f (ČSN) γ F (EN) dead gk b 50 kn/m gk col 10 kn/m G k 200 kn G c k 50 kn live q b k 50 kn/m Tab.1: Loads and loading factors Fig.4: Geometry of the cross-sections a) column, b) primary beam The calculations were performed in accordance with two standards : ČSN [1] and EN [2]. The concrete used was class B20 (C16/20) and the steel was (B400). In the calculation according to the ČSN standard a stress-strain diagram of steel was considered as bilinear with a horizontal plastic line and with calculating values of ultimate tensile strain ε sd =10 0 / 00 of reinforcement. The ultimate compressive strain of concrete is ε bd =2.5 0 / 00 (bilinear stress-strain diagram); concrete under tension did not act. Both stress-strain diagrams corresponded to the principles of the ČSN standard. The calculation according to

10 294 Štěpánek P. et al.: Optimisation of Reinforcement of RC Framed Structures the EN principles was performed with a bilinear stress-strain diagram for strengthened steel that corresponds to ductile class A. But the ultimate tensile strain of reinforcement was restricted to the value ε ud =10 0 / 00 from a possible / 00. The stress-strain diagram of concrete was designed as bilinear with an ultimate tensile strain ε cu3 =3.5 0 / 00. In the calculations we considered both physical non-linearity (according to the described stress-strain diagrams of materials) and geometrical non-linearity under the premise of small deformations and torsional displacement. Variant A : a comparative calculation according to both standards under the presumption that in compressed fibres of concrete and in tensioned fibres of reinforcement the values of ultimate strains will not be exceeded, i.e. the requirement of reliability fulfilment from the viewpoint of ULS. These values were verified in 27 cross-sections of the structure. They are defined on the top/end of the members so the area of reinforcement of each reinforcement type was restricted by this condition from below see Fig. 5a. In the middle of the primary beam the value of ultimate deflection w lim was 20 mm (nodes 6 and 16) and on the end of the cantilever in node 19 it was 7.5 mm. However, these conditions were not of use in the calculation. Fig.5: Reinforcement types and cross-sections a) variant A and B, b) variant C1, c) variant C2 In the variant A-ČSN the ultimate strain of reinforcement (or the values close to ultimate strain) was reached in the cross-sections 1 ( / 00), 11 ( / 00), 13 ( / 00), 14 ( / 00), 16 ( / 00), 18 ( / 00), 21 ( / 00) and27( / 00). The ultimate strain of concrete was reached in the cross-sections 1 ( / 00), 13 ( / 00), 14 ( / 00) and27( / 00), namely at the loading state LS1. In the cross-sections 1, 13, 14 and 27 the optimal failure values were reached (or values close this failure). In the variant A-EN the ultimate strain of reinforcement in the loading state LS1(or the values close to ultimate strain) was reached in the cross-sections 1 ( / 00), 11 ( / 00), 13 ( / 00), 14 ( / 00), 16 ( / 00) and18( / 00), and the ultimate strain of con-

11 Engineering MECHANICS 295 crete in the cross-sections 1 ( / 00), 13 ( / 00), 14 ( / 00), 16 ( / 00) and 27 ( / 00). The loading state LS2 was decisive in the cross-sections 19 ( / 00), 21 ( / 00) and23( / 00). The resulting areas of reinforcement of different reinforcement types are compared in Table 2. Reinforcement type A s,max A s,min A-ČSN A-EN B-ČSN B-EN columns upper primary beam upper reinforcement lower primary beam upper primary beam lower reinforcement lower primary beam Volume of reinforcement [10 2 m 3 ] Tab.2: Areas of the reinforced layers for individual variants of the solution Variant B : the calculation was identical to variant A, but the conditions of ultimate serviceability were stricter. This means that the values of ultimate deflections w lim were considered so as to restrict the deflections of nodes calculated in variant A see Table 3, i.e. w lim (6) = 6 mm, w lim (16) = 4 mm and w lim (9) = 2 mm. The resulting deflections from individual loading states are presented in Table 3. In a column denoted MAX-ČSN and MAX-EN the deflections in nodes during the reinforcing of the structure with reinforcement correspond to the maximal allowable areas of the reinforcement types. In these calculations the ultimate strains of fibres, concrete and reinforcement were not reached in any cross- sections. The restriction of deflection (ULS) was decisive for reinforcement. The ultimate deflection in node 16 was reached in the loading state LS2, and in nodes 6 and 9 in the loading state LS3 (the values are marked by bolts). The loading state LS1 was not limiting from the point of view of the deflection. The areas of reinforcement type are presented in Table 2. The restriction of deflection represented an increase in reinforcement volume in the variant B-ČSNof82%andinthevariantB-ENof63%.

12 296 Štěpánek P. et al.: Optimisation of Reinforcement of RC Framed Structures Deflection in nodes [mm] MAX-ČSN A-ČSN B-ČSN MAX-EN A-EN B-EN LS LS LS Volume of reinforcement [10 2 m 3 ] Tab.3: Deflections in the nodes, maximal volume of reinforcement Variants C1 and C2 : the calculation was identical to variant A, but the reinforcement types were designed so that they went through more elements (Fig. 5b and 5c). It means that in these members the same reinforcement (area) was required. The resulting areas of the reinforcement types and the reinforcement volumes are summarised in Table 4. The reinforcement types were designed to correspond as much as possible to the real reinforcement of the structure. When the areas of reinforcement are changed the rigidity of the members are changed too and with this comes the redistribution of the internal forces in the structure. The internal forces were compared from optimisation calculations according to the EN standard, variant Reinforcement type (variant A) A-ČSN C1-ČSN C2-ČSN A-EN C1-EN C2-EN columns upper primary beam upper reinforcement lower primary beam upper primary beam lower reinforcement lower primary beam Volume of reinforcement [10 2 m 3 ] Tab.4: Comparison of the resulting areas of reinforcement during submission of other reinforcement types

13 Engineering MECHANICS 297 A-EN and variant B-EN with the calculation of the internal forces in the structure reinforced with the maximal allowable reinforcing areas (variant MAX-EN). (This calculation was also performed under the consideration of the physically non-linear behaviour of materials Variant MAX-EN A-EN B-EN LS section M[kNm] N[kN] M[kNm] N[kN] M[kNm] N[kN] LS LS LS Tab.5: Internal forces in the important cross-sections Fig.6: Influence of the change in frame structure rigidity on the internal forces (M)

### Design of reinforced concrete sections according to EN and EN

Design of reinforced concrete sections according to EN 1992-1-1 and EN 1992-2 Validation Examples Brno, 21.10.2010 IDEA RS s.r.o. South Moravian Innovation Centre, U Vodarny 2a, 616 00 BRNO tel.: +420-511

### Standardisation of UHPC in Germany

Standardisation of UHPC in Germany Part II: Development of Design Rules, University of Siegen Prof. Dr.-Ing. Ekkehard Fehling, University of Kassel 1 Overvie Introduction: Work of the Task Group Design

### 9.5 Compression Members

9.5 Compression Members This section covers the following topics. Introduction Analysis Development of Interaction Diagram Effect of Prestressing Force 9.5.1 Introduction Prestressing is meaningful when

### Reliability analysis of slender reinforced concrete column using probabilistic SBRA method

Proceedings of the third international conference ISBN 978-80-7395-096-5 "Reliability, safety and diagnostics of transport structures and means 2008" University of Pardubice, Czech Republic, 25-26 September

### A METHOD OF LOAD INCREMENTS FOR THE DETERMINATION OF SECOND-ORDER LIMIT LOAD AND COLLAPSE SAFETY OF REINFORCED CONCRETE FRAMED STRUCTURES

A METHOD OF LOAD INCREMENTS FOR THE DETERMINATION OF SECOND-ORDER LIMIT LOAD AND COLLAPSE SAFETY OF REINFORCED CONCRETE FRAMED STRUCTURES Konuralp Girgin (Ph.D. Thesis, Institute of Science and Technology,

### Influence of residual stresses in the structural behavior of. tubular columns and arches. Nuno Rocha Cima Gomes

October 2014 Influence of residual stresses in the structural behavior of Abstract tubular columns and arches Nuno Rocha Cima Gomes Instituto Superior Técnico, Universidade de Lisboa, Portugal Contact:

### Bridge deck modelling and design process for bridges

EU-Russia Regulatory Dialogue Construction Sector Subgroup 1 Bridge deck modelling and design process for bridges Application to a composite twin-girder bridge according to Eurocode 4 Laurence Davaine

### SERVICEABILITY LIMIT STATE DESIGN

CHAPTER 11 SERVICEABILITY LIMIT STATE DESIGN Article 49. Cracking Limit State 49.1 General considerations In the case of verifications relating to Cracking Limit State, the effects of actions comprise

### CE5510 Advanced Structural Concrete Design - Design & Detailing of Openings in RC Flexural Members-

CE5510 Advanced Structural Concrete Design - Design & Detailing Openings in RC Flexural Members- Assoc Pr Tan Kiang Hwee Department Civil Engineering National In this lecture DEPARTMENT OF CIVIL ENGINEERING

### Design of AAC wall panel according to EN 12602

Design of wall panel according to EN 160 Example 3: Wall panel with wind load 1.1 Issue Design of a wall panel at an industrial building Materials with a compressive strength 3,5, density class 500, welded

### Comparison of Structural Models for Seismic Analysis of Multi-Storey Frame Buildings

Comparison of Structural Models for Seismic Analysis of Multi-Storey Frame Buildings Dj. Ladjinovic, A. Raseta, A. Radujkovic & R. Folic University of Novi Sad, Faculty of Technical Sciences, Novi Sad,

### Finite Element Modelling with Plastic Hinges

01/02/2016 Marco Donà Finite Element Modelling with Plastic Hinges 1 Plastic hinge approach A plastic hinge represents a concentrated post-yield behaviour in one or more degrees of freedom. Hinges only

### Redistribution of force concentrations in reinforced concrete cantilever slab using 3D non-linear FE analyses

y x m y m y Linear elastic isotropic Linear elastic orthotropic Plastic Redistribution of force concentrations in reinforced concrete cantilever slab using 3D non-linear FE analyses x Master of Science

### Seismic Pushover Analysis Using AASHTO Guide Specifications for LRFD Seismic Bridge Design

Seismic Pushover Analysis Using AASHTO Guide Specifications for LRFD Seismic Bridge Design Elmer E. Marx, Alaska Department of Transportation and Public Facilities Michael Keever, California Department

### STRUCTURAL ANALYSIS CHAPTER 2. Introduction

CHAPTER 2 STRUCTURAL ANALYSIS Introduction The primary purpose of structural analysis is to establish the distribution of internal forces and moments over the whole part of a structure and to identify

### Fire Analysis of Reinforced Concrete Beams with 2-D Plane Stress Concrete Model

Research Journal of Applied Sciences, Engineering and Technology 5(2): 398-44, 213 ISSN: 24-7459; E-ISSN: 24-7467 Maxwell Scientific Organization, 213 Submitted: April 29, 212 Accepted: May 23, 212 Published:

### Civil Engineering Design (1) Design of Reinforced Concrete Columns 2006/7

Civil Engineering Design (1) Design of Reinforced Concrete Columns 2006/7 Dr. Colin Caprani, Chartered Engineer 1 Contents 1. Introduction... 3 1.1 Background... 3 1.2 Failure Modes... 5 1.3 Design Aspects...

Ground Support 2013 Y. Potvin and B. Brady (eds) 2013 Australian Centre for Geomechanics, Perth, ISBN 978-0-9806154-7-0 https://papers.acg.uwa.edu.au/p/1304_31_horyl/ Loading capacity of yielding connections

### Structural Steelwork Eurocodes Development of A Trans-national Approach

Structural Steelwork Eurocodes Development of A Trans-national Approach Course: Eurocode Module 7 : Worked Examples Lecture 0 : Simple braced frame Contents: 1. Simple Braced Frame 1.1 Characteristic Loads

### Reinforced Concrete Structures

Reinforced Concrete Structures MIM 232E Dr. Haluk Sesigür I.T.U. Faculty of Architecture Structural and Earthquake Engineering WG Ultimate Strength Theory Design of Singly Reinforced Rectangular Beams

### Sabah Shawkat Cabinet of Structural Engineering Walls carrying vertical loads should be designed as columns. Basically walls are designed in

Sabah Shawkat Cabinet of Structural Engineering 17 3.6 Shear walls Walls carrying vertical loads should be designed as columns. Basically walls are designed in the same manner as columns, but there are

### Pre-stressed concrete = Pre-compression concrete Pre-compression stresses is applied at the place when tensile stress occur Concrete weak in tension

Pre-stressed concrete = Pre-compression concrete Pre-compression stresses is applied at the place when tensile stress occur Concrete weak in tension but strong in compression Steel tendon is first stressed

### NUMERICAL EVALUATION OF THE ROTATIONAL CAPACITY OF STEEL BEAMS AT ELEVATED TEMPERATURES

8 th GRACM International Congress on Computational Mechanics Volos, 12 July 15 July 2015 NUMERICAL EVALUATION OF THE ROTATIONAL CAPACITY OF STEEL BEAMS AT ELEVATED TEMPERATURES Savvas Akritidis, Daphne

### CHAPTER 4. ANALYSIS AND DESIGN OF COLUMNS

4.1. INTRODUCTION CHAPTER 4. ANALYSIS AND DESIGN OF COLUMNS A column is a vertical structural member transmitting axial compression loads with or without moments. The cross sectional dimensions of a column

### - Rectangular Beam Design -

Semester 1 2016/2017 - Rectangular Beam Design - Department of Structures and Material Engineering Faculty of Civil and Environmental Engineering University Tun Hussein Onn Malaysia Introduction The purposes

### Displacement-based methods EDCE: Civil and Environmental Engineering CIVIL Advanced Earthquake Engineering

Displacement-based methods EDCE: Civil and Environmental Engineering CIVIL 706 - Advanced Earthquake Engineering EDCE-EPFL-ENAC-SGC 2016-1- Content! Link to force-based methods! Assumptions! Reinforced

### SHEAR RESISTANCE BETWEEN CONCRETE-CONCRETE SURFACES

, DOI: 10.2478/sjce-2013-0018 M. KOVAČOVIC SHEAR RESISTANCE BETWEEN CONCRETE-CONCRETE SURFACES Marek KOVAČOVIC Email: marek.kovacovic@gmail.com Research field: composite structures Address: Department

### Flexure: Behavior and Nominal Strength of Beam Sections

4 5000 4000 (increased d ) (increased f (increased A s or f y ) c or b) Flexure: Behavior and Nominal Strength of Beam Sections Moment (kip-in.) 3000 2000 1000 0 0 (basic) (A s 0.5A s ) 0.0005 0.001 0.0015

### CONSULTING Engineering Calculation Sheet. Job Title Member Design - Reinforced Concrete Column BS8110

E N G I N E E R S Consulting Engineers jxxx 1 Job Title Member Design - Reinforced Concrete Column Effects From Structural Analysis Axial force, N (tension-ve and comp +ve) (ensure >= 0) 8000kN OK Major

### Lecture-08 Gravity Load Analysis of RC Structures

Lecture-08 Gravity Load Analysis of RC Structures By: Prof Dr. Qaisar Ali Civil Engineering Department UET Peshawar www.drqaisarali.com 1 Contents Analysis Approaches Point of Inflection Method Equivalent

### ε t increases from the compressioncontrolled Figure 9.15: Adjusted interaction diagram

CHAPTER NINE COLUMNS 4 b. The modified axial strength in compression is reduced to account for accidental eccentricity. The magnitude of axial force evaluated in step (a) is multiplied by 0.80 in case

### EUROCODE EN SEISMIC DESIGN OF BRIDGES

Brussels, 18-20 February 2008 Dissemination of information workshop 1 EUROCODE EN1998-2 SEISMIC DESIGN OF BRIDGES Basil Kolias Basic Requirements Brussels, 18-20 February 2008 Dissemination of information

### Serviceability Limit States

Serviceability Limit States www.eurocode2.info 1 Outline Crack control and limitations Crack width calculations Crack width calculation example Crack width calculation problem Restraint cracking Deflection

### SERVICEABILITY OF BEAMS AND ONE-WAY SLABS

CHAPTER REINFORCED CONCRETE Reinforced Concrete Design A Fundamental Approach - Fifth Edition Fifth Edition SERVICEABILITY OF BEAMS AND ONE-WAY SLABS A. J. Clark School of Engineering Department of Civil

### Available online at ScienceDirect. Procedia Engineering 172 (2017 )

Available online at www.sciencedirect.com ScienceDirect Procedia Engineering 172 (2017 ) 1093 1101 Modern Building Materials, Structures and Techniques, MBMST 2016 Iterative Methods of Beam-Structure Analysis

### Validation of the Advanced Calculation Model SAFIR Through DIN EN Procedure

Validation of the Advanced Calculation Model SAFIR Through DIN EN 1991-1-2 Procedure Raul Zaharia Department of Steel Structures and Structural Mechanics The Politehnica University of Timisoara, Ioan Curea

### 3. Stability of built-up members in compression

3. Stability of built-up members in compression 3.1 Definitions Build-up members, made out by coupling two or more simple profiles for obtaining stronger and stiffer section are very common in steel structures,

### STEEL JOINTS - COMPONENT METHOD APPLICATION

Bulletin of the Transilvania University of Braşov Vol. 5 (54) - 2012 Series 1: Special Issue No. 1 STEEL JOINTS - COPONENT ETHOD APPLICATION D. RADU 1 Abstract: As long as the rotation joint stiffness

### THE EC3 CLASSIFICATION OF JOINTS AND ALTERNATIVE PROPOSALS

EUROSTEEL 2002, Coimbra, 19-20 September 2002, p.987-996 THE EC3 CLASSIFICATION OF JOINTS AND ALTERNATIVE PROPOSALS Fernando C. T. Gomes 1 ABSTRACT The Eurocode 3 proposes a classification of beam-to-column

### Annex - R C Design Formulae and Data

The design formulae and data provided in this Annex are for education, training and assessment purposes only. They are based on the Hong Kong Code of Practice for Structural Use of Concrete 2013 (HKCP-2013).

### Engineeringmanuals. Part2

Engineeringmanuals Part2 Engineering manuals for GEO5 programs Part 2 Chapter 1-12, refer to Engineering Manual Part 1 Chapter 13. Pile Foundations Introduction... 2 Chapter 14. Analysis of vertical load-bearing

### Chapter. Materials. 1.1 Notations Used in This Chapter

Chapter 1 Materials 1.1 Notations Used in This Chapter A Area of concrete cross-section C s Constant depending on the type of curing C t Creep coefficient (C t = ε sp /ε i ) C u Ultimate creep coefficient

### Parametric analysis and torsion design charts for axially restrained RC beams

Structural Engineering and Mechanics, Vol. 55, No. 1 (2015) 1-27 DOI: http://dx.doi.org/10.12989/sem.2015.55.1.001 1 Parametric analysis and torsion design charts for axially restrained RC beams Luís F.A.

### A Simplified Method for the Design of Steel Beam-to-column Connections

P P Periodica Polytechnica Architecture A Simplified Method for the Design of Steel Beam-to-column Connections 48() pp. 79-86 017 https://doi.org/10.3311/ppar.11089 Creative Commons Attribution b Imola

### Bending and Shear in Beams

Bending and Shear in Beams Lecture 3 5 th October 017 Contents Lecture 3 What reinforcement is needed to resist M Ed? Bending/ Flexure Section analysis, singly and doubly reinforced Tension reinforcement,

### NUMERICAL SIMULATIONS OF CORNERS IN RC FRAMES USING STRUT-AND-TIE METHOD AND CDP MODEL

Numerical simulations of corners in RC frames using Strut-and-Tie Method and CDP model XIII International Conference on Computational Plasticity. Fundamentals and Applications COMPLAS XIII E. Oñate, D.R.J.

### CAPACITY DESIGN FOR TALL BUILDINGS WITH MIXED SYSTEM

13 th World Conference on Earthquake Engineering Vancouver, B.C., Canada August 1-6, 24 Paper No. 2367 CAPACITY DESIGN FOR TALL BUILDINGS WITH MIXED SYSTEM M.UMA MAHESHWARI 1 and A.R.SANTHAKUMAR 2 SUMMARY

### Structural Steelwork Eurocodes Development of A Trans-national Approach

Structural Steelwork Eurocodes Development of A Trans-national Approach Course: Eurocode 3 Module 7 : Worked Examples Lecture 20 : Simple braced frame Contents: 1. Simple Braced Frame 1.1 Characteristic

### Structural Steelwork Eurocodes Development of A Trans-national Approach

Structural Steelwork Eurocodes Development of A Trans-national Approach Course: Eurocode Module 7 : Worked Examples Lecture 22 : Design of an unbraced sway frame with rigid joints Summary: NOTE This example

### VORONOI APPLIED ELEMENT METHOD FOR STRUCTURAL ANALYSIS: THEORY AND APPLICATION FOR LINEAR AND NON-LINEAR MATERIALS

The 4 th World Conference on Earthquake Engineering October -7, 008, Beijing, China VORONOI APPLIED ELEMENT METHOD FOR STRUCTURAL ANALYSIS: THEORY AND APPLICATION FOR LINEAR AND NON-LINEAR MATERIALS K.

### Design of a Multi-Storied RC Building

Design of a Multi-Storied RC Building 16 14 14 3 C 1 B 1 C 2 B 2 C 3 B 3 C 4 13 B 15 (S 1 ) B 16 (S 2 ) B 17 (S 3 ) B 18 7 B 4 B 5 B 6 B 7 C 5 C 6 C 7 C 8 C 9 7 B 20 B 22 14 B 19 (S 4 ) C 10 C 11 B 23

### Simplified procedures for calculation of instantaneous and long-term deflections of reinforced concrete beams

Simplified procedures for calculation of instantaneous and long-term deflections of reinforced concrete beams José Milton de Araújo 1 Department of Materials and Construction, University of Rio Grande

### A HIGHER-ORDER BEAM THEORY FOR COMPOSITE BOX BEAMS

A HIGHER-ORDER BEAM THEORY FOR COMPOSITE BOX BEAMS A. Kroker, W. Becker TU Darmstadt, Department of Mechanical Engineering, Chair of Structural Mechanics Hochschulstr. 1, D-64289 Darmstadt, Germany kroker@mechanik.tu-darmstadt.de,

### MECHANICS OF MATERIALS Sample Problem 4.2

Sample Problem 4. SOLUTON: Based on the cross section geometry, calculate the location of the section centroid and moment of inertia. ya ( + Y Ad ) A A cast-iron machine part is acted upon by a kn-m couple.

### APPROXIMATE DESIGN OF SLENDER BI-AXIALLY LOADED RC COLUMNS

ГОДИШНИК НА УНИВЕРСИТЕТА ПО АРХИТЕКТУРА, СТРОИТЕЛСТВО И ГЕОДЕЗИЯ СОФИЯ Първа научно-приложна конференция с международно участие СТОМАНОБЕТОННИ И ЗИДАНИ КОНСТРУКЦИИ ТЕОРИЯ И ПРАКТИКА 22 23 октомври 2015

### Fundamentals of Structural Design Part of Steel Structures

Fundamentals of Structural Design Part of Steel Structures Civil Engineering for Bachelors 133FSTD Teacher: Zdeněk Sokol Office number: B619 1 Syllabus of lectures 1. Introduction, history of steel structures,

### CALCULATION OF A SHEET PILE WALL RELIABILITY INDEX IN ULTIMATE AND SERVICEABILITY LIMIT STATES

Studia Geotechnica et Mechanica, Vol. XXXII, No. 2, 2010 CALCULATION OF A SHEET PILE WALL RELIABILITY INDEX IN ULTIMATE AND SERVICEABILITY LIMIT STATES JERZY BAUER Institute of Mining, Wrocław University

### MEZNÍ STAVY TRVANLIVOSTI MODELOVÁNÍ A ROZMĚR ČASU

MEZNÍ STAVY TRVANLIVOSTI MODELOVÁNÍ A ROZMĚR ČASU LIMIT STATES FOR DURABILITY DESIGN MODELLING AND THE TIME FORMAT Dita Matesová 1, Florentina Pernica 2, Břetislav Teplý 3 Abstract Durability limit states

### Experimental Study and Numerical Simulation on Steel Plate Girders With Deep Section

6 th International Conference on Advances in Experimental Structural Engineering 11 th International Workshop on Advanced Smart Materials and Smart Structures Technology August 1-2, 2015, University of

Chapter 5 CENTRIC TENSION OR COMPRESSION ( AXIAL LOADING ) 5.1 DEFINITION A construction member is subjected to centric (axial) tension or compression if in any cross section the single distinct stress

### Flexural properties of polymers

A2 _EN BUDAPEST UNIVERSITY OF TECHNOLOGY AND ECONOMICS FACULTY OF MECHANICAL ENGINEERING DEPARTMENT OF POLYMER ENGINEERING Flexural properties of polymers BENDING TEST OF CHECK THE VALIDITY OF NOTE ON

### DESIGN OF STAIRCASE. Dr. Izni Syahrizal bin Ibrahim. Faculty of Civil Engineering Universiti Teknologi Malaysia

DESIGN OF STAIRCASE Dr. Izni Syahrizal bin Ibrahim Faculty of Civil Engineering Universiti Teknologi Malaysia Email: iznisyahrizal@utm.my Introduction T N T G N G R h Flight Span, L Landing T = Thread

### Practical Design to Eurocode 2

Practical Design to Eurocode 2 The webinar will start at 12.30 (Any questions beforehand? use Questions on the GoTo Control Panel) Course Outline Lecture Date Speaker Title 1 21 Sep Jenny Burridge Introduction,

### Basis of Design, a case study building

Basis of Design, a case study building Luís Simões da Silva Department of Civil Engineering University of Coimbra Contents Definitions and basis of design Global analysis Structural modeling Structural

### D : SOLID MECHANICS. Q. 1 Q. 9 carry one mark each. Q.1 Find the force (in kn) in the member BH of the truss shown.

D : SOLID MECHANICS Q. 1 Q. 9 carry one mark each. Q.1 Find the force (in kn) in the member BH of the truss shown. Q.2 Consider the forces of magnitude F acting on the sides of the regular hexagon having

9th International Conference on Fracture Mechanics of Concrete and Concrete Structures FraMCoS-9 Chenjie Yu, P.C.J. Hoogenboom and J.G. Rots DOI 10.21012/FC9.288 ALGORITHM FOR NON-PROPORTIONAL LOADING

### Entrance exam Master Course

- 1 - Guidelines for completion of test: On each page, fill in your name and your application code Each question has four answers while only one answer is correct. o Marked correct answer means 4 points

### INFLUENCE OF FLANGE STIFFNESS ON DUCTILITY BEHAVIOUR OF PLATE GIRDER

International Journal of Civil Structural 6 Environmental And Infrastructure Engineering Research Vol.1, Issue.1 (2011) 1-15 TJPRC Pvt. Ltd.,. INFLUENCE OF FLANGE STIFFNESS ON DUCTILITY BEHAVIOUR OF PLATE

### Slenderness Effects for Concrete Columns in Sway Frame - Moment Magnification Method (CSA A )

Slenderness Effects for Concrete Columns in Sway Frame - Moment Magnification Method (CSA A23.3-94) Slender Concrete Column Design in Sway Frame Buildings Evaluate slenderness effect for columns in a

### Design issues of thermal induced effects and temperature dependent material properties in Abaqus

Materials Characterisation VII 343 Design issues of thermal induced effects and temperature dependent material properties in Abaqus I. Both 1, F. Wald 1 & R. Zaharia 2 1 Department of Steel and Timber

### Reinforced concrete structures II. 4.5 Column Design

4.5 Column Design A non-sway column AB of 300*450 cross-section resists at ultimate limit state, an axial load of 700 KN and end moment of 90 KNM and 0 KNM in the X direction,60 KNM and 27 KNM in the Y

### AXIAL BUCKLING RESISTANCE OF PARTIALLY ENCASED COLUMNS

Proceedings of the 5th International Conference on Integrity-Reliability-Failure, Porto/Portugal 24-28 July 2016 Editors J.F. Silva Gomes and S.A. Meguid Publ. INEGI/FEUP (2016) PAPER REF: 6357 AXIAL BUCKLING

### Slender Structures Load carrying principles

Slender Structures Load carrying principles Basic cases: Extension, Shear, Torsion, Cable Bending (Euler) v017-1 Hans Welleman 1 Content (preliminary schedule) Basic cases Extension, shear, torsion, cable

### Final Report. Predicting of the stiffness of cracked reinforced concrete structure. Author: Yongzhen Li

Predicting of the stiffness of cracked reinforced concrete structure Final Report Predicting of the Stiffness of Cracked Reinforced Concrete Structures Author: 1531344 Delft University of Technology Faculty

### Sensitivity and Reliability Analysis of Nonlinear Frame Structures

Sensitivity and Reliability Analysis of Nonlinear Frame Structures Michael H. Scott Associate Professor School of Civil and Construction Engineering Applied Mathematics and Computation Seminar April 8,

### Plastic design of continuous beams

Budapest University of Technology and Economics Department of Mechanics, Materials and Structures English courses Reinforced Concrete Structures Code: BMEEPSTK601 Lecture no. 4: Plastic design of continuous

### PLATE GIRDERS II. Load. Web plate Welds A Longitudinal elevation. Fig. 1 A typical Plate Girder

16 PLATE GIRDERS II 1.0 INTRODUCTION This chapter describes the current practice for the design of plate girders adopting meaningful simplifications of the equations derived in the chapter on Plate Girders

### SIMPLIFIED METHOD FOR PREDICTING DEFORMATIONS OF RC FRAMES DURING FIRE EXPOSURE

SIMPLIFIED METHOD FOR PREDICTING DEFORMATIONS OF RC FRAMES DURING FIRE EXPOSURE M.A. Youssef a, S.F. El-Fitiany a a Western University, Faculty of Engineering, London, Ontario, Canada Abstract Structural

### Estimation of the Residual Stiffness of Fire-Damaged Concrete Members

Copyright 2011 Tech Science Press CMC, vol.22, no.3, pp.261-273, 2011 Estimation of the Residual Stiffness of Fire-Damaged Concrete Members J.M. Zhu 1, X.C. Wang 1, D. Wei 2, Y.H. Liu 2 and B.Y. Xu 2 Abstract:

### Experimental and numerical studies of the effect of high temperature to the steel structure

Experimental and numerical studies of the effect of high temperature to the steel structure LENKA LAUSOVÁ IVETA SKOTNICOVÁ IVAN KOLOŠ MARTIN KREJSA Faculty of Civil Engineering VŠB-Technical University

### Strengthening of columns with FRP

with FRP Professor Dr. Björn Täljsten Luleå University of Technology Sto Scandinavia AB 9/12/2013 Agenda Case study Restrained transverse expansion (confinement) Circular and rectangular cross sections

### CHAPTER 5. T a = 0.03 (180) 0.75 = 1.47 sec 5.12 Steel moment frame. h n = = 260 ft. T a = (260) 0.80 = 2.39 sec. Question No.

CHAPTER 5 Question Brief Explanation No. 5.1 From Fig. IBC 1613.5(3) and (4) enlarged region 1 (ASCE 7 Fig. -3 and -4) S S = 1.5g, and S 1 = 0.6g. The g term is already factored in the equations, thus

### MODELLING NON-LINEAR BEHAVIOUR OF STEEL FIBRE REINFORCED CONCRETE

6th RILEM Symposium on Fibre-Reinforced Concretes (FRC) - BEFIB - September, Varenna, Italy MODELLING NON-LINEAR BEHAVIOUR OF STEEL FIBRE REINFORCED CONCRETE W. A. Elsaigh, J. M. Robberts and E.P. Kearsley

### CHAPTER 6: ULTIMATE LIMIT STATE

CHAPTER 6: ULTIMATE LIMIT STATE 6.1 GENERAL It shall be in accordance with JSCE Standard Specification (Design), 6.1. The collapse mechanism in statically indeterminate structures shall not be considered.

### Multi Linear Elastic and Plastic Link in SAP2000

26/01/2016 Marco Donà Multi Linear Elastic and Plastic Link in SAP2000 1 General principles Link object connects two joints, i and j, separated by length L, such that specialized structural behaviour may

### Structural behaviour of traditional mortise-and-tenon timber joints

Structural behaviour of traditional mortise-and-tenon timber joints Artur O. Feio 1, Paulo B. Lourenço 2 and José S. Machado 3 1 CCR Construtora S.A., Portugal University Lusíada, Portugal 2 University

### A NEW SIMPLIFIED AND EFFICIENT TECHNIQUE FOR FRACTURE BEHAVIOR ANALYSIS OF CONCRETE STRUCTURES

Fracture Mechanics of Concrete Structures Proceedings FRAMCOS-3 AEDFCATO Publishers, D-79104 Freiburg, Germany A NEW SMPLFED AND EFFCENT TECHNQUE FOR FRACTURE BEHAVOR ANALYSS OF CONCRETE STRUCTURES K.

### SENSITIVITY ANALYSIS OF LATERAL BUCKLING STABILITY PROBLEMS OF HOT-ROLLED STEEL BEAMS

2005/2 PAGES 9 14 RECEIVED 18.10.2004 ACCEPTED 18.4.2005 Z. KALA, J. KALA SENSITIVITY ANALYSIS OF LATERAL BUCKLING STABILITY PROBLEMS OF HOT-ROLLED STEEL BEAMS ABSTRACT Doc. Ing. Zdeněk KALA, PhD. Brno

### Earthquake-resistant design of indeterminate reinforced-concrete slender column elements

Engineering Structures 29 (2007) 163 175 www.elsevier.com/locate/engstruct Earthquake-resistant design of indeterminate reinforced-concrete slender column elements Gerasimos M. Kotsovos a, Christos Zeris

### STATIC NONLINEAR ANALYSIS. Advanced Earthquake Engineering CIVIL-706. Instructor: Lorenzo DIANA, PhD

STATIC NONLINEAR ANALYSIS Advanced Earthquake Engineering CIVIL-706 Instructor: Lorenzo DIANA, PhD 1 By the end of today s course You will be able to answer: What are NSA advantages over other structural

### International Journal of Scientific & Engineering Research Volume 9, Issue 3, March-2018 ISSN

347 Pushover Analysis for Reinforced Concrete Portal Frames Gehad Abo El-Fotoh, Amin Saleh Aly and Mohamed Mohamed Hussein Attabi Abstract This research provides a new methodology and a computer program

### DEFORMATION CAPACITY OF OLDER RC SHEAR WALLS: EXPERIMENTAL ASSESSMENT AND COMPARISON WITH EUROCODE 8 - PART 3 PROVISIONS

DEFORMATION CAPACITY OF OLDER RC SHEAR WALLS: EXPERIMENTAL ASSESSMENT AND COMPARISON WITH EUROCODE 8 - PART 3 PROVISIONS Konstantinos CHRISTIDIS 1, Emmanouil VOUGIOUKAS 2 and Konstantinos TREZOS 3 ABSTRACT

### TIME-DEPENDENT ANALYSIS OF PARTIALLY PRESTRESSED CONTINUOUS COMPOSITE BEAMS

2nd Int. PhD Symposium in Civil Engineering 998 Budapest IME-DEPENDEN ANAYSIS OF PARIAY PRESRESSED CONINUOUS COMPOSIE BEAMS M. Sar and Assoc. Prof. J. apos 2 Slova echnical University, Faculty of Civil

### 9-3. Structural response

9-3. Structural response in fire František Wald Czech Technical University in Prague Objectives of the lecture The mechanical load in the fire design Response of the structure exposed to fire Levels of

### Chapter Objectives. Design a beam to resist both bendingand shear loads

Chapter Objectives Design a beam to resist both bendingand shear loads A Bridge Deck under Bending Action Castellated Beams Post-tensioned Concrete Beam Lateral Distortion of a Beam Due to Lateral Load

### CHAPTER 14 BUCKLING ANALYSIS OF 1D AND 2D STRUCTURES

CHAPTER 14 BUCKLING ANALYSIS OF 1D AND 2D STRUCTURES 14.1 GENERAL REMARKS In structures where dominant loading is usually static, the most common cause of the collapse is a buckling failure. Buckling may

### Project data Project name Project number Author Description Date 26/04/2017 Design code AISC dome anchor. Material.

Project data Project name Project number Author Description Date 26/04/2017 Design code AISC 360-10 Material Steel A36, A529, Gr. 50 Concrete 4000 psi dome anchor Connection Name Description Analysis Design