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**STRUCTURAL DYNAMICS IN BULDING CODES**

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**BUILDING CODES : ANALYSES**

STATIC ANALYSIS Structures be designed to resist specified static lateral forces related to the properties of the structure and seismicity of the region. Formulas based on an estimated natural period of vibration are specified for base shear and distribution of lateral forces over the height of the building. Static analysis provides the design forces including shears and overturning moments for various stories. DYNAMIC ANALYSES RESPONSE SPECTRUM ANALYSIS RESPONSE HISTORY ANALYSIS .

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**International Building Code - USA Base Shear**

Vb = csw where Cs = Ce & Ce= IC R Cs corresponding to R = 1 is called the elastic seismic coefficient W = total dead load and applicable portions of other loads R = 1.0 I = 1.0, 1.25 or 1.5 C depends on the location of structure and the site classes defined in the code accounting for local soil effects on ground motion. C is also related to pseudo-acceleration design spectrum values at short periods and and at T = 1 second.

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**International Building Code - USA LATERAL FORCES**

Fj = Vb wjhkj ∑ni=1wihik Where K is a coefficient related to the vibration period .

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**International Building Code - USA Story Forces**

The design values of story shears are determined by static analysis of the structure subjected to the lateral forces; the effects of gravity and other loads should be included. Similarly determined overturning moments are multiplied by a reduction factor J defined as follows: J = 1.0 for top 10 stories; between 1.0 and 0.8 for the next 10 stories from the top; varying linearly with height; 0.8 for remaining stories.

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**National Building Code of Canada Base Shear**

Vb = csw where Cs = Ce U & Ce = vSIF R U= 0.6 Calibration Factor zonal velocity v = 0 to 0.4 Seismic importance factor I = 1.5, 1.3, 1.0 Foundation factor F = 1.0, 1.3, 1.5, or 2.0 Seismic response factor S varies with fundamental natural vibration period of the building. Canada is divided in 7 velocity and acceleration related seismic zones

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**National Building Code of Canada LATERAL FORCE**

Fj = (Vb-Ft) wjhj ∑ni=1wihi with the exception that force at the top floor is increased by an additional force , the top force, Ft .

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**National Building Code of Canada STORY FORCES**

The design value of story shears are determined by static analysis of the structure subjected to these lateral forces. Similarly determined overturning moments are multiplied by reduction factors J and Ji at the base and at the i th floor level.

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**EuroCode 8 Base Shear Vb = csw = A/g {(Tb / TI)-1/3}**

where Cs = Ce / q’ Ce = A/g = A/g {(Tb / TI)-1/3} q’ = 1+(T1 / Tb) (q-1) = q Seismic behavior factor q varies between 1.5 and 8 depending on various factors including structural materials and structural system.

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**EuroCode 8 LATERAL FORCES**

Fj = Vb wj Φj1 ∑ni=1wi ΦJ1 where Φj1 is the displacement of the jth floor in the fundamental mode of vibration. The code permits linear approximation of the this mode which becomes: Fj = Vb wjhj ∑ni=1wihi

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EuroCode 8 STORY FORCES The design values of story shears, story overturning moments, and element forces are determined by static analysis of the building subjected to these lateral forces; the computed moments are not multiplied by a reduction factor.

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**Fundamental Vibration Period**

Period formulae used in IBC, NBCC and others codes are derived out of Rayleigh’s method using the shape function given by the static deflection, Ui due to a set of lateral forces Fi at the floor levels.

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**Elastic seismic coefficient**

Elastic seismic coefficient Ce is related to the pseudo – acceleration spectrum for linearly elastic systems. The Ce and A/g as specified in codes are not identical. The ratio of Cc A/g is plotted as a function of period and it exceeds unity for most periods.

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CONCLUSION There can be major design deficiencies, if the building code is applied to structures whose dynamic properties differ significantly from these of ordinary buildings. Building codes should not be applied to special structures, such as high-rise buildings, dams, nuclear power plants, offshore oil- drilling platforms, long spane bridges etc.

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**Requirement of RC Design**

Sufficiently stiff against lateral displacement. Strength to resist inertial forces imposed by the ground motion. Detailing be adequate for response in nonlinear range under displacement reversals.

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**Design Process Pre-dimensioning Analysis. Review. Detailing.**

Production of structural drawings. Final Review.

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**Requirement for structural Response**

Stiffness Stiffness defines the dynamic characteristics of the structure as in fundamental mode and vibration modes. Global and individual members stiffness affects other aspects of the response including non participating structural elements behavior, nonstructural elements damage, and global stability of the structure. Contd

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STRENGTH The structure as a whole, its elements and cross sections within the elements must have appropriate strength to resist the gravity effects along with the forces associated with the response to the inertial effects caused by the earthquake ground motion.

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Toughness The term toughness describes the ability of the reinforced concrete structure to sustain excursions in the non linear ranges of response without critical decrease of strength.

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**Seismic Design Categories**

Category A: Ordinary moment resisting frames. Category B. Ordinary moment resisting frames. Flexural members have two continuous longitudinal bars at top & bottom Columns having slenderness ratio of 5 or less Shear design must be made for a factored shear twice that obtained from analysis.

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Category C. Intermediate moment frames. Chapter 21 of ACI 318 implemented. Shear walls designed like a normal wall. Category D, E and F. Special moment frames Special reinforced concrete walls.

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**Earth quake Design Ground Motion Earth quake Design ground Motion**

Maximum Considered Earthquake and Design Ground Motion For most regions, the minimum considered earthquake ground motion is defined with a uniform likelihood of excudance of 2% in 50 years (approximate return period of 2500 years). In regions of high seismicity, it is considered more appropriate to determine directly maximum considered earthquake ground motion base on the characteristic earthquakes of these defined faults multiplied by 1.5.

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Site Classification Where Vs = average shear wave velocity. N = average standard penetration resistance. Nch = average standard penetration resistance for cohesiveless soils. Su = average un-drianed shear strength in cohesive soil.

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All ordinates of this site specific response spectrum must be greater or aqual to 80% of the spectural value of the response spectra obtained from the umpped values of Ss and Si, as shown on previous slide. Use Groups. As per SEI/ASCE 7-02.

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**Required Seismic Design Category**

The structure must be assigned to the most severe seismic design category obtained from.

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**Reinforced concrete lateral Force – Resisting Structural System**

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Bearing Wall. Any concrete or masonry wall that supports more than 200 lbs/ft of vertical loads in addition to its own weight. Braced Frame. An essentially vertical bent, or its equivalent of the concentric or eccentric type that is provided in a bearing walls, building frame or dual system to resist seismic forces . Moment frame. A frame in which members and joints are capable of resisting forces by flexure as well as along the axis of the members. Contd

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Shear Wall. A wall bearing or non bearing designed to resist lateral seismic forces acting on the face of the wall. Space Frame. A structural system composed of inter connected members. Other than bearing walls, which are capable of supporting vertical loads and, when designed for such an application, are capable of providing resistance to seismic forces.

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**The approximate fundamental building period Ta is seconds is obtained**

Ta = C1 hxn

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**The over turning moment at any storey MX is obtained from**

MX = ∑n Fi (hi – hx) i=x

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**Reinforced Brick Masonry**

Allowable stress design provisions for reinforced masonry address failure in combined flexural and axial compression and in shear. Stresses in masonry and reinforcement are computed using a cracked transformed section. Allowable tensile stresses in deformed reinforcement are the specified field strength divided by a safety factor of 2.5. Allowable flexural compressive stresses are one third the specified compressive strength of masonry.

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**Shear stresses are computed elastically, assuming a uniform distribution of shear stress.**

If allowable stresses are exceeded, all shear must be resisted by shear reinforcement and shear stresses in masonry must not exceed a second, higher set of allowable values.

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**Seismic Design Provisions for Masonry in IBC**

General. The three basic characteristics to determine the building’s “Seismic design category” are Building geographic location Building function Underlying soil characteristics Categories A to F Determination of Seismic Design Forces. Forces are based on Structure Location Underlying soil type Degree of structural redundancy System expected in elastic deformation capacity

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**Seismic related Restriction on Materials**

In seismic Design categories A through C, no additional seismic related restrictions apply beyond those related to design in general. In seismic design Categories D & E, type N mortar and masonry cement are prohibited because of their relatively low tensile bond strength. Seismic Related Restrictions on Design Methods Seismic Design Category A. Strength design, allowable stress design or empirical design can be used.

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**Seismic Design Category B and C**

Seismic Design Category B and C elements that are part of lateral force resisting system can be designed by strength design or allowable stress design. Non-contributing elements may be designed by empirical design. Seismic Design Category D, E and F. Elements that are part of lateral force resisting system must be designed by either strength design or allowable stress design. No empirical design be used.

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**Seismic Related Requirement for Connectors.**

Seismic Design Category A and B. No mechanical connections are required between masonry walls and roofs or floors. Seismic Design Category C, D E and F. Connectors are required to accommodate story drift. Seismic Related Requirements for Locations and Minimum Percentage of Reinforcement Seismic Design Categories A and B. No restriction . Seismic Design Category C. In Seismic Design Categories A and B. No requirement.

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In Seismic Design category C, masonry partition walls must have reinforcement meeting requirements for minimum percentage and maximum spacing. Masonry walls must have reinforcement with an area of at least 0.2 sq in at corners. In seismic design category D, masonry walls that are part of lateral force-resisting system must have uniformly distributed reinforcement in the horizontal and vertical directions with a minimum percentage of in each direction and a minimum summation of (both directions). Maximum spacing in either direction is 48 in.

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In Seismic Design Categories E and F, stack bonded masonry partition walls have minimum horizontal reinforcement requirements. Analysis Approaches for Modern U.S. Masonry Analysis of masonry structures for lateral loads, along or in combination with gravity loads, must address the following issues. Analytical approaches Elastic vs. inelastic behavior Selection of earthquake input Two dimensional vs. three dimensional behavior Contd

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Modeling of materials Modeling of gravity loads Modeling of structural elements Flexural working Soil foundation Flexibility Floor diaphragm flexibility

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**Overall Analytical Approach**

Hand type approaches usually emphasize the plan distribution of shear forces in wall elements. Hand methods are not sufficiently accurate for computing wall movements, critical design movements can be overestimated by factors as high as 3. Elastic vs Inelastic Behavior Flexural yielding or shear degradation of significant portions of a masonry structure in anticipated, inelastic analysis should be considered.

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In many cases, masonry structures can be expected to respond in the cracked elastic regime, even under extreme lateral loads. Selection of Earthquake Input. Because structural response in generally expected to be linear elastic, linear elastic response spectra are sufficient.

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**Two Dimensional vs three Dimensional Analysis of Linear Elastic Structures**

In two dimensional analysis, a building is modified as an assemblage of parallel plan as frames, free to displace laterally in their own planes only subject to the requirement of lateral displacements compatibility between all frames at each floor level. In the “Pseudo three dimensional” approach, a building is modeled as an assemblage of planar framers, each of which is free to displace parallel and perpendicular to its own place. The frames exhibit lateral displacement compatibility at each floor level.

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**Modeling of Gravity Loads**

Gravity loads should be based on self weight plus an estimate of the probable live load. A uniform distribution of man should be assumed over each floor except exterior walls. Modeling of Material Properties Material properties should be estimated based on test results. A poisson's ratio of 0.35 can be used for masonry. Modeling of Structural Elements Masonry wall buildings are normally modeled using beams and panels with occasional columns.

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**Flexural Cracking of Walls**

Flexural Cracking Criterion. The cracking movement for a wall should be determined by multiplying the modulus of rapture of the wall under in plane flexure, by the section modulus of the wall. Consequences of Flexural Cracking of walls. Flexural cracking reduces the wall’s stiffness from that of the un-cracked transformed section so that of the cracked transformed section.

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**Soil Foundation Flexibility.**

Regardless of how the building’s foundation in modeled, the building’s periods of vibration significantly increase, and lateral force levels can change significantly. If the building’s foundation is considered flexible the resulting increase in support flexibility at the basis of wall elements causes their base movement to decrease substantially. In –Plane Floor Diaphragm Flexibility Structures in general an often modeled using special purpose analysis programs that assume that floor diaphragms are rigid in their own planes.

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Many masonry wall structures have floor slabs with features that could increase the affects of in-plane floor flexibility. Small openings in critical sections of the floor slab. Rectangular floor plans with large aspect ratios in plan. Variations of in-plane rigidity with in slab. Explicit Inelastic Design and Analysis of Masonry Structures Subjected to Extreme Lateral loads. If in elastic response of a masonry structure is anticipated, a general design and analysis approach involving the following steps in proposed.

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Select a stable collapse mechanism for the wall, with reasonable inelastic deformation demand in hinging regions. Using general plans section theory to describe the flexural behavior of reinforced masonry elements, provide sufficient flexural capacity and flexural ductility in hinging regions. Using a capacity design philosophy, provide wall elements with sufficient shear capacity to resist the shear consistent with the development of intended collapse mechanism.

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Using reinforcing details from current strength design provisions detail the wall reinforcement to develops the necessary strength and inelastic deformation capacity. Inelastic Finite Element Analysis of Masonry Structure In the absence of experimental data, finite element analysis in the most viable method to quantify the ductility and post peak behavior of masonry structures

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The load – deformation relation of a masonry components obtained from a finite element analysis can be used to calibrate structural component models which can in turn be used for the push over analysis or dynamic analysis of large structural systems.

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**Structural Dynamics in Binary Codes**

CANADA IBC Euro Code Beam Shear Vb=csw Where Cs=Ce U R Ce = עSIF Where U=0.6 ע= 0 to 0.4 i= 1.3 or 1.5 S=fundamental natural vibration period Where Cs=Ce Ce= IC W= total dead load R = 1 I = 1.0, 1.25 or 1.5 CS= seismic coefficient Ce= Elastic seismic coefficient θ’ Cc= A/g A/g {1+0.5r[1-(Tc /TI)]} θ’ = { θ 1+(T1/Tb)(θ -1)} Where θ varies from 1 to 4 Lateral Forces Fj=(Vb-Ft) wjhj ∑Ni=1wihi Fj=Vb wjhj ∑Ni=1wihik Where K= coefficient related to the vibration period T1 Fj= Vb wj Φj1 ∑Ni=1wi ΦJ1 Code Allows Linear approx. Fj=Vb wjhj Storey J=Reduction Factor for over turning moments J=Reduction Factor for over turning moments 1 to 0.8 Computed over tuning moments are not reduced

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Lecture 2 January 19, 2006.

Lecture 2 January 19, 2006.

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