Analysis Basics Linear Analysis

Bending

The total moment of inertia of a layered element is made up by the sum of each layer’s moment of inertia and the Steiner-part of the individual layers. Here an eccentricity of the centroid’s position could be created due to stiffer layers on one side of the element, e.g. sandwich-element with different top-layer thicknesses. The eccentricity is established automatically and is considered for simple plate bending, it also leads the correct length deformation of the elements. This ef- fect also becomes apparent for an eccentric connection with a homogeneous element.

The input of orthotropic materials is not allowed, due to the occurrence of various eccentricities in various directions. This is blocked by the program and leads to an error message.

Linear normal-stresses in the layers are generated by the strains in the layers. They are calculated as usual by the stress-strain matrix D of the material in a layer:

δσ= D_{· δϵ (2.39)}

where the matrix D can also be orthotropic.

The linear total stress is made up of all the stress components including the allowed factors out of the ASE-GRP input:

σ_ = FACP · FCREEP2 · σ_,PLC+

+ FACS · FCREEP1 · D· (δϵ+ ϵ,LOAD) + σ (2.40)

σ_{,PLC Primary stress (initial stress)}

FACP Factor of the primary stress record ASE-GRP

FACS Factor of the stiffness record ASE-GRP

FCREEP1 Stiffness reduction creep and shrinkage without PLC FCREEP1 = 1 / (1+PHI) with PHI from record ASE-GRP

FCREEP2 Reduction of the primary stress through creep and shrinkage, by taking over the primary load case with ro and dphi from record GRP

FCREEP2 = 1 - dphi/(1 + ro·dphi))

(dphi = creep increment of a creep step component)

ε_{,LOAD Load expansion (temperature or expansion load)} σ_{ Pre-stress (record GRP-PREX)}

The GRPfactor FACL is generally multiplied to the primary load-case as a value of 1.0. It generates the expansion loads from the primary stresses σ_{,PLC. If the}

stresses of the PLC together with the loads of the PLC are multiplied with the factors FACL=1.0 and FACP=1.0, then the system will remain in equilibrium and no additional expansions or displacements are created. The FACL expansion loads are then in equilibrium with the external loads:

Figure 2.18: Load equilibrium when taking over the primary load case without any

new loads

The nodal load resulting from FACL and the element stress is generated be- cause the element wants expand due to the primary compressive stress.

The internal forces and moments are calculated by integrating the stresses in the layers, over the element thickness of each layer.

Shear

Initially the shear stiffnesses of the individual layers are summed up for the stiff- ness determination.

The following equation is used to calculate the shear stress from the shear force q.

τ_= q_{· S}

_{· b} (2.41)

For homogenous material definitions, in the linear domain, this would result in a parable-shaped shear stress distribution over the height of the element, with the maximum value of τ_m = 1.5 · q/h_{. For sandwich elements, with thick (strong)}

top-layers, it would mean that a nearly constant shear stress is present in the middle of the element; given by τ_m= 1.0 · q/h_{(h=element thickness).}

Non-linear Analysis STEEL

Example see ase12_buckling_slab.dat

For a non-linear analysis, the calculation of the new linear stresses is initially made by assuming a linear material behaviour for every layer xi. The following applies when proceeding with the primary load case:

σ_ = σ_,PLC+ D_{· dϵ (2.42)}

and

τ_= τ_,PLF+ dτ_{ (2.43)}

(simplified)

The total stress σ_{ is therefore not just put together by the total strain multiplied}

with the stiffness, instead it might be that the non-linear eigen-stresses of the individual layers of σ_{,PLC have to be considered. For the consistent treatment}

of the problem, including the correct generation of the loading- and unloading curves of the layer model, it is of importance that not only the internal forces and moments are stored in the database, but also all the stress in all the layers and all the Gauss-points. This information is needed for the next load case asσ_,PLC.

From these initial linear stresses a new linear comparison stress is calculated: For QUAD elements the following applies:

σ_ =rσ2

 + σy2− σ· σy+ 3τ2y+ 3τ2+ 3τ2y (2.44)

If the so calculated linear comparison stress σ_{, is above the allowed stress}

(by considering the hardening, which is calculated by summing up the plastic strains, by entering a trilinear stress-strain curve); then first of all the linear com- ponent is established (Break-through point through the plastic area). Then the remaining strain incrementδdε_{ with the elastoplastic material matrix D-P is ap-}

plied incrementally, with the consideration of possible hardening. The non-linear relaxation lies on the surface of the plastic area. The number of plastic incre- ments of the strain increment can be changed in the input CTRL MSTE. The non-linear material behaviour is according to the elasto-plastic plastic-law, de- scribed in TALPA, which is according to van MISE and includes hardening. For more information on this topic you are referred to ZIENKIEWICZ [21].

The following diagram results from uniaxial stress:

Figure 2.19: Uniaxial stress

In the case of combined stress, which is made up of normal stress (N/ A_{± M/ )}

and shear force stress, it is assumed that on reaching the elasticity limit (plas- tic area) the shear stress (from the shear force) remains constant and can not be increased any further through hardening. The thus established shear force stress is then basically substituted as a constant component into the calculation of the comparison stress. It has started to plasticising. This would then lead to the following: e.g. in plate bending; the shear stresses in the plastified plate edge would not increase anymore, however in the middle of the plate they would still get bigger, this in turn would cause a deviation from the parable-shaped shear stress distribution over the plate thickness, which would in turn cause a concentration of the shear stresses in the middle of the plate.

Non-linear Analysis CONCRETE

Examples see ase.dat\...\nonlinear_quad\ a1_introduction_example.datThe following literature was consulted on the concrete material law: STEMPNIEWSKI AND EIBL [14], FEENSTRA AND DE BORST [6], SCHIESSEL [12]

Following current assessments and explanations are mentioned here addi- tionally: XX [20], ZILCH AND ROGGE [22], BELLMANN AND RÖTZER [3], XX

[19], SCHNEIDER [13]

The material behaviour of reinforced concrete can be described by the following properties:

• Non-linear stress-strain curve in tension and compressive zone • Contribution of the concrete between cracks (tension stiffening) • Non-linear material behaviour of the steel inserts

• Simplified check of the plate’s shear stress Usual procedure:

The element is subdivided into NLAY layers. The stresses sigma-x, sigma-y and tau-xy and the principal stresses sigma-I and sigma-II are calculated for every layer’s boundary. For each principal stress direction a stress-strain curve is generated, which results from the principal stress relation in the respective direction. The thus established non-linear stresses are then integrated over all the layers to find the internal forces. After this all the forces of the reinforcement including the tension-stiffening-effect are added. Finally an independent check is made for the plate’s shear stresses.

The following is a list of the concrete parameters taken from record CONC: CONC-FC = calculation value of the concrete stiffness

CONC-FCT = average tension stiffness for tension stiffening

CONC-FCTK = lower fractile of the tension stiffness for bare concrete CONC-GC = GC compression fracture energy

CONC-GF = GF tension fracture energy CONC-MUEC = friction value in the crack splice Further inputs in ASE:

LC - BET2 = load duration coefficient (beta2)

CTRL- NLAY = number of layers to be calculated >_=6,

default = 10 Analysis on Serviceability Stress Level

Using the 1.0-times serviceability loads the maximum desired stress is input for this serviceability state at the material. The deformation and crack width to be expected is in this case mostly interesting. The input of the concrete tensile strength of the (pure) concrete layer is particularly important. This value is input in AQUA in CONC...FCTK and it can be modified subsequently temporarily in ASE withCTRLCONC V3+V4. The serviceability stress-strain curve without any additional material safeties is requested then in ASE (NSTRKSV SL = default). The selection of a realistic concrete tensile strength fctk (pure strength without reinforcement) is here very important. If fctk or CTRLCONC V4 is not input, the plate remains in uncracked state I. It can be therefore reasonable to decrease the value e.g. onto 60 % in order to consider a crack predamage from construc- tion stage (hydration heat). On the other hand realistic deflections are resulted often only with a high initial value for fctk.

Analysis with gamma-times Loads

If using gamma-times loads the corresponding material stress-strain curve has to be selected in record NSTR in ASE. There are two possibilities that are also well shown in the beam example aseaqb_1_column_cracked.dat:

• Analysis according to "non-linear method": Here an averaged material safety of 1.3 is used. The material strengths are modified for this purpose. They are available AQUA and can be requested in ASE withNSTRKSV CALD. • Analysis in ultimate limit stateNSTRKSV ULD

In both cases the pure concrete alone must include any tensile strengths. CTRL

CONC V4 0.0 or 0.01 must be input!

The increase of the steel stress due to the concrete action between the cracks may be brought into approach (default for fct or CTRLCONC V3).

A non-linear analysis for the ultimate limit state is particularly necessary for ad- ditional effects from second-order theory. Such an analysis with temporarily switched-off tensile strength of the pure concrete causes however often big de- formations and bad convergences.

forces and moments with average values of the material strengths (analysis in serviceability limit state) and a definitive design of the redistributed internal forces and moments with an average load safety coefficient (e.g. 1.45)

Futher explanations see example a2_nonlinear_slab.dat

Non-linear Stress-Strain Curve in the Compressive Zone

The maximum concrete compression stiffness beta-ic, found in the compressive zone, is deduced from the principal stress relation. Beta-ic can either be read from the Kupfer curve, or it can be calculated by the respective equations [ 1] , pg. 260.

Figure 2.20: Biaxial failure curve according to Kupfer-Hilsdorf-Ruesch

With this maximum value beta-ic an uniaxial stress-strain line can be generated according to the concrete stress-strain curve for every of both principal stress directions.

An increase value higher than 1.0 is only allowed for calculations in service- ability limit state. For calculations with gamma-times loads (ultimate limit state) this increase is deactivated in the default, because it is mostly desired that the maximum stress increases about the basic value of the concrete compressive strength beta-ic - see CTRLCONC V2. A reduction of the permissible compres- sive stresses is always considered for lateral tension.

the stress is reduced parallelly. The calculation is repeated again with the pos- sible modified principle stress ratio.

Tensile zone

In the tensile zone of concrete, the maximum value beta-z, is always taken as the lower fractile of the concrete stiffness fctk. The length of the descending curve results from the tension crack energy GF of the processing zone. Typical values lie between 0.10 and 0.25 Nmm/mm2_{. The program restricts the length} of the descending curve to 5·epslin - see CTRLCONC VAL.

If a stress-strain curve for concrete is already defined in the tensile zone in AQUA, then this one is used instead of the here described program- internal curve! Thus it is possible to calculated steel fibre concrete ->

steel_fibre_concrete.dat

Figure 2.21: Uniaxial stress-strain curve for the tensile zone

The element is seen as cracked as soon as the tensile-strain crosses the linear limit value of epslin. Any further strain is stored as plastic tensile-strain and is taken into account for reloading after an element has been unloaded (hystere- sis). Due to the possibility of excessive tensile stiffness perpendicular to the first crack, the program has to store two plastic tensile-strains at each point (first crack and second crack).

It could be that a crack has already emerged when a primary load case is taken over. In this case the fixed crack direction of the primary load case is used for the calculation of the stresses. For this calculation the strains in the direction of the crack and perpendicular to it are used. When a possible shear stress is present at the crack it is lowered by a simple friction consideration (Crack-toothing input with AQUA-CONC-MUEC). This would prevent the occurrence of a second crack perpendicular to the first crack. For biaxial coated material, without the primary load case, two cracks are always perpendicular to each other.

Reinforcement

The program takes the defined reinforcement as the default reinforcement. The non-linear analysis is then performed for the default reinforcement. An automatic increase in lacking structural safety does not take place! It is therefore the users responsibility to check the certainty of the convergence of the analysis! Possible residual forces of the non-linear iteration have to be checked. Since these resid- ual forces are stored as support forces they can be checked with the program WinGRAF, this is done by generating a plot of the support forces. During a plate analysis residual forces are also generated in the plate’s plane (normal forces), this is because the program needs to find equilibrium of the normal stresses. The reinforcement parameters and a given minimum reinforcement is taken from BEMESS-PARA or from the corresponding SSD design parameter di- alog. REIQ is used to import a reinforcement from a previously generat- ed BEMESS-analysis. The recommended method is used in the example

a2_nonlinear_slab.dat . An analysis can also be made with non-reinforced concrete, when no reinforcement is defined. Further information on the program ASE can be found in the chapter ’Definition of Reinforcement’ as well as the latest TEDDY-Help .

The consideration of the tension stiffening is done generally with a modification of the steel stress-strain curve described in [ 2] page 269. Since ASE 11.76-21 the consideration in serviceability limit state (NSTR SL/SLD) occurs according to the method of SchieSSl (DAfStB Heft 400) or EC 2, because more realistic deformation values result here. For the ultimate limit state and the non-linear determination of the internal forces and moments (NSTRUL/ULD or CAL/CALD) the consideration of the tension stiffening is done according to the simplified method of the modified steel strains according to DAfStB Heft 525. For a better clarity the in each case used method in ASE is output again at the non-linear properties of the plane elements.

Please note, that the serviceability analysis (NSTR KSV SL) should be done usually according to Heft 400 also for DIN 1045-1 and respectively acc. EN 1992-1-1, because it leads to a better agreement with the test result according our experiences.

Figure 2.22: Simplified method of the tensile stiffening acc. to Heft 525 (Bild H

8-4)

As the pure concrete layers also work in tension, the following working method is used:

• In a first step the strains in the steel layers in reinforcement direction are determined. These strains are equal to the mean steel strainsε_{sm according}

to SCHIESSEL[12].

• Using the tensile working law the two majoring strains I and II are determined based on the actual tensile strength amd the process zone length LZ (see below):

I: average strain when cracking starts

II: average strain for finished crack development = at the end of the decreas- ing part of the tensile work law

• The streel stress is now calculated as follows:

– In interval 0-I the steel stress is linear, concrete works linear.

– In interval I-II the additional strain due to tension stiffening is interpolated linear. Concrete descends linear.

– After II the full effect of tension stiffening is applied, concrete stress is 0. Reaching the steel yielding point a trilinear part follows,e.g. the steel working law is used.

example EC=27700 GF=0.3 fctk=3.71:

Process zone length LZ = GF*EC/FCTK/FCTK = 0.3*27.7/3.71/3.71 = 0.605 m

In SOFiSTiK this length is limited to 0.400 m, because otherwise an FCTK of e.g. 0.5 N/mm2 would result in a very long and unrealistic length. In

ˇ

TFinite Elemente im Stahlbeton - Betonkalender 1993/I Stempniewski ˇT a value between 200-600 mm is recommended.

• With this process zone length LZ and GF a crack opening delta= 2 * GF / FCTK can be calculated. With eps=delta/LZ the length of the descending part will be DEPSX= 0.404 promille (relative to LZ). This value is then limited to 5*length of the increasing part = 5*0.134 - not controlling here. This strain DEPSX is used in ASE for the plot of the stress strain curve ˝U that means for an element with the element gauss point size LZ.

• In the real analysis now this strain DEPSX= 0.404 promille is scaled to the actual element gauss point size.

e.g. element area = 0.05m*0.05m = 0.0025m2 = per gauss point 0.000625m2 -> _{element gauss point size L_Gauss = square-}

root(0.000625m2) = 0.025m.

– For an actual element with L_Gauss > _{LZ, DEPSX_GAUSS is calculat-}

ed to DEPSX_GAUSS = DEPSX*LZ/L_Gauss (descending part is short- ened).

– For an actual element with L_Gauss < _{LZ, DEPSX_GAUSS = DEPSX.}

That means that the descending part will not be elongated!

So for an element size of 0.05m*0.05m a descending part of DEP- SX_GAUSS = DEPSX = 0.404 promille is taken into account.

• For the new design codes (and without the input of CTRL CONC V5 400) the crack width is then calculated according to DIN 1045.1 11.2.4 or according to the Eurocode equation.

The average force of the steel insert is calculated by multiplying the steel stress for the crack cross section in the cracked condition (state II) σ_{s with}

the reinforced concrete area. This value is added to the concreteŠs internal forces and moments.

The crack widths are first calculated in the direction of the reinforcement! If the crack direction is not perpendicular to the reinforcement, the crack distance and the crack width are modified according to EN 1992-I-I 7.3.4(4). For non-reinforced elements it is only possible to calculate one crack direc- tion, but the crack width can not be established. The the average strain is plotted as crack width.

The coefficient describing the connection properties is to be defined in AQUA- STEE. The factor for the influence of the load period is input in ASE-LC.

For ultimate limit state the calculation is done according to Heft 525, if DIN 1045- 1, DIN FB 102 or EN 1992-1-1 is set.

Shear force

The shear stresses for the concrete law are not calculated for each layer, as is the case for the plastic yield criteria of STEEL, instead a simple shear limitation of the shear force is set with an assumed shear stress in the cracked condition (state II) of

τ= q/z = q/(0.8 · h) _(2.45)

where h represents average of all the reinforcement layers.

If the linear calculated shear stress τ _{rises over the input value} τ_{02, then the}

shear force is reduced accordingly and the element undergoes plastic shear deformation. The value τ_{02 is input with ASE-CTRL FRIC in N/mm}2 _{and the}

default value is set to 2.4 N/mm2_{. With TAU2...V2 a descending part with a final} strength can be defined.

In document ase_1_sofistik analysis (Page 45-58)