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ADDIS ABABA UNIVERSITY
FACULTY OF TECHNOLOGY
CIVIL ENGINEERING DEPARTMENT
CENG-2501 Theory of Structures I
Course Outline
1. Stability & Determinacy of Structures 1.1 Introduction 1.2 Stability of Structures, 1.3 Determinacy of Structures 2. Loads on Structures 2.1 Dead Load 2.2 Live Load 2.3 Environmental Loads (Wind loads, earthquake forces, …) 2.4 Load Combinations 3. Influence Lines (IL)for Determinate Structures 3.1 IL for Beams (IL for shear forces and bending moment) 3.2 IL for paneled girders 3.3 IL for trusses 4. Deflection of Determinate Structures 4.1 Direct Integration Method 4.2 Moment-Area Method 4.3 Conjugate-Beam Method 4.4 Method of Virtual Work 4.5 Graphical Multiplication 4.6 Castiglione’s Theorem 4.7 Maxwell Beti law of reciprocal deflection 5. Consistent Deformation Method 5.1 Indeterminate beams 5.2 Trusses by Consistent Deformation Method
References:
- Nigussie Tebeje (Prof.), Statically Indeterminate Structural Analysis, 1984;
- Harry H. West and L.F. Geshwindner, Fundamentals of Structural Analysis 2nd ed. 2002,
- Popov, E.P., Mechanics of Materials
- EBCS-1, 1995 (Ethiopian Building Code Standards, part 1 - Loadings)
- EBCS-8, 1995 (Ethiopian Building Code Standards, part 8- Design of structures for Earthquake Resistance)
- Lecture note by Abrham Gebre
CHAPTER 1
1. Stability & Determinacy of Structures
1.1 Introduction
A structure refers to a system of connected parts used to support loads. The fundamental purpose of a structure is to transmit loads from the point of application to the point of support and, through the foundations to the ground.
Before going into the analysis of any structure, it is necessary to identify its statical type (classification), i.e., whether it is determinate or indeterminate, stable or unstable. An unstable arrangement of supports and structural members should be avoided.
All structures are subjected to loads from their functions and to other unavoidable loads. Establishment of the loads that act on a structure is one of the most difficult and yet important steps in the design process.
In this chapter; Criteria for statical classification will be established and different structures will be checked for stability and determinacy.
1.2 Stability of Structures
To ensure the equilibrium of a structure or its members, it is not only necessary to satisfy the equations of equilibrium, but the members must also be properly held or, constrained by their supports. In structural analysis a structure is said to be stable when it can support any possible system of applied loads.
Stability can be divided into two as external and internal.
A structure in which there are insufficient numbers of reactions to prevent motion from taking place is called an unstable structure. This is external instability.
What matters is not only the number of support reactions but also their arrangement. Structures for which the numbers of reaction components are greater than or equal to the number of available equilibrium equations but that are unstable due to arrangement of these reaction components are said to be geometrically unstable.
When the reaction elements are three or more like supports that are either parallel or concurrent, they are not sufficient to maintain static equilibrium.
For the case of parallel reactions, they will offer no resistance to horizontal motion, thus making the arrangement unstable. The point of intersection of the concurrent reactions becomes an instantaneous center of rotation and the system is instantaneously unstable.
On the other hand, however, indeterminate structures are stiffer and in the case of over loads indeterminate structures can provide an advantage of redistribution of loads within the structure.
The indeterminacy of a structure can be external (with respect to reactions) or internal (with respect to member forces). The question of identifying external or internal indeterminacy is largely of academic interest. What is of primary importance is the total degree of indeterminacy, Nevertheless, determining external and internal indeterminacy is desirable as a method to evaluate the total degree of indeterminacy.
A structure is internally indeterminate when it is not possible to determine all internal forces by using the equations of static equilibrium. For the great majority of structures, the question of whether or not they are indeterminate can be decided by inspection. For certain structures this is not so, and for these types rules have to be established. The internal indeterminacy of trusses will be first considered, and then that of continuous frames
1.4 Criteria for Stability and Determinacy of Structures-Trusses, Beams and Frames
Internal stability of structures and determining which conditions exist in a given case need experience, especially for trusses. In some cases the structure is different from what our mathematical criteria tell us. Therefore, stability of trusses is most easily settled by inspection.
1.4.1 Beams
A beam is a structural element that is capable of withstanding load primarily by resisting bending. The bending force induced into the material of the beam as a result of the external loads, own weight and external reactions to these loads is called a bending moment. Beams generally carry vertical gravitational forces but can also be used to carry horizontal loads (i.e., loads due to an earthquake or wind).
- Stability depends on external supports
- Determinacy relates on the number of available and conditional equations. o r (^) a < r; structure is statically unstable o r (^) a = r; structure is statically determinate o r (^) a > r; structure is statically indeterminate
where r (^) a is the available number of reaction components r is the minimum number of reaction components required for stability, usually 3+n n is the number of special/ conditional equation
Remark: r = 3 is not a sufficient conditions for stability
1.4.2 Trusses A simple truss can be made by combining three bars to form a triangle. Stability depends partly on external supports and partly on the arrangement of members or bars. Three reaction components are required for external stability and determinacy of a plane truss without condition equations.
1.4.2.1 External classification The external statical classification of the structure depends on the total number of reaction components, r (^) a and their arrangement. Therefore, the following criteria hold true: o r (^) a < r; structure is statically unstable externally o r (^) a = r; structure is statically determinate externally o r (^) a > r; structure is statically indeterminate externally
where r (^) a is the available number of reaction components r is the minimum number of reaction components required for external stability, usually 3+n n is the number of special/ conditional equation
The condition for ra ≥ r is necessary but not sufficient conditions for statical classification because the arrangement of the reaction components may render the truss unstable.
1.4.2.2 Internal classification For internal classification, in addition to the above definition for r; let m be the total number of bars and j the total number of joints. Then 2j = m + r The above equation can be rewritten as: m = 2j – r In this form, m is the number of members required to form an internally statically determinate truss that connects j joints and has r reaction components required for external stability. If ma is the actual number of bar forces in the truss, then the following criteria hold true for internal classification
o ma < m; truss is statically unstable internally o ma > m; truss is statically determinate internally o ma > m; truss is statically indeterminate internally Consider the trusses shown below. The truss shown in fig (a) is stable where as the truss shown in fig (b) is unstable since the geometric arrangement of the members is not maintained.
Fig (a) Fig (b)
1.4.3 Frames
Frames are composed of continuous members and rigidly connected joints, The degree of indeterminacy (DI) is determined as the difference of the total number of unknown reaction components and the number of static equilibrium equations available. Note that the frame with the hinge has a fourth condition equation, since the bending moment at the hinge must be zero. Stability depends partly on external supports and partly on moment resisting joints.
1.4.3.1 External classification The external statical classification of the structure depends on the total number of reaction components, r (^) a and their arrangement. Therefore, the following criteria hold true: o r (^) a < r; structure is statically unstable externally o r (^) a = r; structure is statically determinate externally o r (^) a > r; structure is statically indeterminate externally
where r (^) a is the available number of reaction components r is the minimum number of reaction components required for external stability, usually 3+n n is the number of special/ conditional equation r (^) a ≥ r is necessary but not sufficient conditions for statical classification because the arrangement of the reaction components may render the frame unstable.
1.4.3.2 Internal classification (3 + r) < (3j + n); Let ma =the actual number of members r = the minimum number of independent reaction components required for external stability j= the total number joints n= number of special/condition equations Therefore, 3 ma + r =the number of unknowns 3j+n=the number of available equations
Then the following criteria hold true for internal classification of frames o (3ma + r) < (3j + n); structure is statically unstable o (3ma + r) = (3j + n); structure is statically determinate o (3ma + r) > (3j + n); structure is statically indeterminate
Overall classification The criterion already established for both trusses and frames hold also for investigation of overall effect. To determine the overall classification of a frame, in the above expressions replace r by r (^) a.
Note. The number of conditional equation introduced by a hinge joint is equal to the number of members at the joint minus one.
Examples
Table 2. Material Unit wt. (kN/m^3 ) Solid Brick 22 Hollow Concrete Block (HCB) 14- Trachyte (Masonry) 26 Concrete (with reinforcement) 25 Steel 77 Zigba (podacargus Gracillior) 6 PVC floor covering 16
2.2 Live loads In a general sense, live loads are considered to include all loads on the structure that are not classified as dead loads. However, it has become common to narrow the definition of live loads to include only loads that are produced through the construction, use, or occupancy of the structure and not to include environmental.
These loads are dynamic in character in that they vary both in magnitude and position. Live loads where the dynamic nature has significance because of the rapidity with which change in position occurs are called moving loads, whereas live loads in which change occurs over an extended period of time, or where there is the potential for change whether exercised or not, are referred to as movable loads. Moving loads include vehicular loads on bridges or crane loads in industrial build- ings. Another type of live load is a variable load or a time dependent load-that is, one whose magnitude changes with time, such as a load induced through the operation of machinery.
2.2.1 Occupancy live loads Occupancy live loads for buildings are usually specified in terms of the minimum values that must be used for design purposes. Some representative values are given in Table 2.2. (Refer EBCS-1, 1995 Ethiopian Building Code Standards, part 1 - Loadings)
Table 2. Category Uniform Load (kN/m^2 ) Private dwelling 1. Bed rooms, class rooms,... 2. Offices, Café,... 3. Assembly halls, Cinema,... 4.
2.2.2 Traffic Loads for Bridges Bridges must be designed to support the vehicular loads associated with their functional use and minimum loads are mandated for designed purposes. In the case of highway bridges, these loads are specified in Bridge Design Manuals. The approach is to specify the weights and spacing of axles and wheels for a design truck, a design tandem, and the design lane load. These loadings provide for a set of concentrated loads (which represent a truck type loading) and a uniform load (which simulates a line of vehicles).
2.2.3 Impact loads Loads that are applied over a very short period of time have a greater effect on the structure than would occur if the same loads were applied statically. The manner in which a load varies with time and the time over which the full load is placed on the structure will determine the factor by which the static response should be increased to obtain the dynamic response.
For building occupancy loads, the minimum design loads normally include adequate allowance for ordinary impact conditions. However, provisions must be made in the structural design for uses and loads that involve unusual vibrations and impact forces. One situation in which an impact effect (IM is defined as the dynamic load allowance) is applied for moving vehicular loads on a highway bridge.
2.3 Environmental loads Structures experience numerous loading conditions as a result of the environment in which they exist. These are Snow and Ice Loads, Roof loads, Wind loads and Earthquake Loads.
2.3.1 Snow and Ice Loads The procedure for establishing the static snow loads on a building is normally based on ground snow loads and an appropriate ground-to-roof conversion.
The distribution of snow on a roof is complex, and many different approaches are used. Factors considered in calculating snow and ice loads are location, exposure factor, thermal factor, the effects of unloaded portions of roof, unbalanced or nonuniform loads on various roof configurations, drifting, sliding snow, and extra loads induced by rain on snow.
Snow loads are not normally considered in bridge design because they are usually small when compared with other loadings on the structure. However, ice loads can be appreciable on bridge structures. The icing not only creates loads on the structure but also increases the member sizes, which, in turn, increases the magnitude of the wind induced loads.
2.3.2 Rain loads Roof loads that result from the accumulation of rainwater on flat roofs can be a serious problem. This condition is produced by the ponding that occurs when the water accumulates faster than it runs off, either because of the intensity of the rainfall or because of the inadequacy or blockage of the drainage system. The real danger is that as ponding occurs the roof deflects into a dished configuration, which can accommodate more water, and thus greater loads result.
The best way to prevent the problem is to provide a modest slope to the roof (0.25 in. per ft or 2cm. per m or more) and to design an adequate drainage system. In addition to the primary drainage, there should be a secondary system to preclude the accumulation of standing water above a certain level.
Terrain Category : The terrain category attempts to take into account the effect of the land coverage, and is given below. The terrain type is classified into 4 groups as follows:
Category I: Lakes with at least 5 km fetch upwind and smooth flat country without obstacles. Category II: Farmland with boundary hedges, occasional small farm structure, houses or trees Category III: Suburban or industrial areas and permanent forests. Category IV: Urban areas in which at least 15% of the surface is covered with buildings and their average height.
Topography Coefficient : The topography coefficient Ct accounts for the increase in mean wind speed over isolated hills and escarpments and mountainous regions. It is defined by: Ct =1 for Φ<0. Ct =1+2SΦ for 0.05≤Φ<0. Ct = 1+0.6S for Φ>0. Where: S is a factor to be obtained by interpolation from the value of s=1.0 at the crest of a hill or escarpment and the value of S=0 at the boundary on the topography affected zone, Φ is the upwind slope in the wind direction.
Factor S : Cliffs and escarpments
Factor S: Hills and ridges
Pressure Coefficient : The shape factor takes into account the effect of shape of structure on the pressure distribution.
The external pressure coefficients cpe for buildings and individual parts of building depend on the size of the loaded area A. They are given for loaded area A of 1m2 and 10m2 in the relevant tables for the appropriate building configuration as cpe.1 and cpe.10, respectively. For areas between 1m2 and 10m2, values are obtained by linear interpolation. That is: Cpe =c (^) pe. 1 for A≤1m^2 Cpe = c (^) pe. 1 +( cpe. 10 - cpe. 1 )log 10 A for 1m^2 <A<10m^2 Cpe = c (^) pe. 10 for A≥10m^2 The values of pressure coefficient are applicable to buildings.
Values of external pressure coefficients for different cases are given in Table A.1 to Table A.5 of EBCS-1, 1995.
2.3.4 Earthquake Loads A common dynamic loading that structures must resist is that associated with earthquake motions. Here, loads are not applied to the structure in the normal fashion. Instead, the base of the structure is subjected to a sudden movement. Since the upper portion of the structure resists motion because of its inertia, a deformation is induced in the structure. This deformation, in turn, induces a horizontal vibration that causes horizontal shear forces throughout the structure.
It results from the acceleration of the supporting earth. Movement of the ground during EQ in the direction parallel to the ground surface has the most damaging effect on structures. The resulting earthquake loads are dependent on the nature of the ground movement and the inertia response characteristics of the structure. The computation of lateral loads due to EQ and load distribution to various levels of a building frame as of EBCS-8, 1995 (Ethiopian Building Code Standards part 8- Design of structures for Earthquake Resistance) is presented below. F (^) b = Sd ( T 1 ). W
S d ( T 1 )= αβ γ
Where: Fb = Total lateral load on the structure (seismic base shear) S (^) d ( T 1 ) =Ordinate of the design spectrum T 1 = Fundamental period
W = Seismic dead load DL (+25% for storage and warehouses)
α = α 0 I , ( α is the ratio of the design bedrock acceleration to the acceleration
of gravity, g) I = importance of a structure
T
S (^) ( β is the design response factor)
γ = accounts the ductility level (behavior factor)
Fig. Seismic Hazard Map of Ethiopia
ii) Importance of a structure (I) I = 1. 4 → hospitals, fire stations, power plants = 1. 2 →schools, halls, … = 1. 0 → ordinary buildings = 0. 8 → buildings of minor importance
iii) Site coefficient (S) For different soil types, the value of S is given:- Subsoil A − 1. 0 →rock, stiff deposits of sand, gravel or over consolidated clay B − 1. 2 → deep deposits of medium dense sand, gravel or medium stiff clays, .. C − 1. 5 → loose cohesion less soil deposits with or without soft cohesive layers,
iv) The behavior factor ( )γ
It depends on the structural system:
- for concrete structures γ≤^0.^70
- for steel structures, it ranges from 0.17 to 1.
- for timber structures it varies from 0.30 to 1.
2.4 Load Combinations
Ultimate Design Load
The ultimate design load acting on a member will be the summation of the relevant characteristic load combinations multiplied by their respective partial safety factors. Thus, the ultimate design load for the combination of dead and imposed loads will be expressed as follows:
Partial Safety Factors for Load In practice the applied load may be greater than the characteristic load for any of the following reasons: a. Calculation errors b. Constructional inaccuracies c. Unforeseen increases in load.
To allow for these the respective characteristic loads are multiplied by a partial safety factor γf to give the ultimate design load appropriate to the limit state being considered. That is,
Ultimate design load = γf x characteristic load
Load combinations depend on the design philosophy adopted.
Load Combinations for Ultimate Limit States (ULS) o Permanent action (Gk ) and only one variable action (Qki ) Fd = 1.3Gk + 1.6Qki
o Permanent action (Gk ) and two or more variable actions Fd = l.3Gk + 1.35ΣQki
o Permanent action, variable action and accidental (seismic) action Fd = GK + Q (^) ki +AEd = 0.75 (l.3Gk + 1.6 Qk ) + AEd
Load Combinations for Serviceability Limit States (SLS) o Permanent action (Gk ) and only one variable action (Qk1 ) Fd = Gk + Qki
o Permanent action (Gk ) and two or more variable actions Fd = Gk + 0.9ΣQki
The final design of a structure must be consistent with the most critical combination of loads that the structure is to support. However, some judgment is necessary in selecting loading conditions that can reasonably be combined. Obviously, the maximum effects of all loading conditions should not be combined because it is unlikely that they will all occur simultaneously.
Figure 3.1b - Influence line for the support reaction at A
ii. Influence Line for Reaction at C, RC
From figure 3.1a above, MA = 0 RC (L)=1(x) R (^) c = X/L At X=0, RC =0 and at X=L, RC =
Figure 3.1c - Influence line for the reaction at support C
iii. Influence Line for Shear at B The influence line for the shear at point B can be found by developing equations for the shear at the section using statics. This can be accomplished as follows:
Figure 3.1d Figure 3.1e
a) if the load moves from A to B, consider figure 3.1d Fy = 0 VB =RA- But, RA = (L-x)/L Therefore the shear force at B, VB becomes VB =-1 +(L-x)/L = -X/L At X=0, VB =0 and at X=Ll , VB = -Ll /L
Fig 3.1f Influence line for the Shear at point B.
b) if the load moves from B to C, figure 3.1e Fy = 0 VB = 1- Rc But, Rc = X/L Up on substitution, the shear force at B, VB becomes: V (^) B =( L-X)/L At X=Ll , VB =( L-L 1 )/L= L (^) r/L and at X=L, V (^) B = 0
iv. Influence line for Moment at B The influence line for the moment at point B can be found by using statics to develop equations for the moment at the point of interest, due to a unit load acting at any location on the structure.
a) if the load moves from A to B, consider figure 3.1d MB = 0 M (^) B +1(Ll -X)-RA(Ll ) = 0 B r r LL r BM = RL = X at x = 0 BMB = 0
at L
L L x = L l BMB =^ l r L
L L m = l r
Fig 3.1g Influence line for Bending moment at point B.
b) if the load moves from B to C, figure 3.1e MB = 0 RC (Lr) -1(X-Ll )-MB = 0 l (^) L L l BM R L x B r t = =^ −
at
L
L L
x = l l , BMB = r^ l
at x = L BMB = 0
- Determination of a stress at a paint due to different leads using influence curve i) Series of concentrated loads say P 1 and P 2 as shown.
The product of the load P 1 and the ordinate of the influence line at the position of the respective load shall give the magnitude of the stress at the section due to the induced load
In the particular case SFa = S 1 P 1 + S 2 P 2 Where S 1 (^) and S 2 represent the ordinates of the IL with appropriate sign for SFa , at the respective positions of the loads P 1 and P 2.
