Nonlinear bar statics

The non-linear bar statics is a statics that can be applied to bar-shaped components and is particularly important for realistic calculations of slender compression members , columns and beams made of reinforced or prestressed concrete . It takes into account non-linear internal forces - distortion relationships (physical non-linearity); this also includes the consideration of linear internal forces-distortion relationships as a special case.

If necessary, the equilibrium is calculated according to the second order theory , taking into account the effects of structural deformations (geometric non-linearity).

Mathematical formulation

The internal forces-distortion relationships are calculated using non-linear or linear-elastic stress-strain relationships . The cross-section must remain level. To the expansion plane ${\displaystyle \sigma =f(\varepsilon )}$

${\displaystyle \varepsilon (y,z)=\varepsilon _{0}+y\cdot \mathrm {d} \varepsilon /\mathrm {d} y+z\cdot \mathrm {d} \varepsilon /\mathrm {d} z}$

are the internal internal forces, the stress resultants

${\displaystyle R=[N,M_{y},M_{z}]=\int [1,z,-y]\cdot \sigma (y,z)\ \mathrm {d} A,}$

preferably by numerical integration of the voltages

${\displaystyle \sigma (y,z)=f[\varepsilon (y,z)]}$

calculated (voltage integration).

In composite cross-sections, for example in reinforced or prestressed concrete cross-sections, instead of the strains that are determined by the strain plane, stress-dependent strains are to be formed as follows in order to use them to determine the stresses from stress-strain lines. ${\displaystyle \varepsilon (y,z)}$ ${\displaystyle \varepsilon _{\sigma }(y,z)}$ ${\displaystyle \sigma (\varepsilon _{\sigma })}$

With the thermal expansion results: ${\displaystyle \varepsilon _{th}(y,z)}$

• for reinforcing steel :

${\displaystyle \varepsilon _{\sigma s}(y,z)=\varepsilon (y,z)-\varepsilon _{th}(y,z)}$

• for prestressing steel with the pre-expansion : ${\displaystyle \varepsilon _{p}}$

${\displaystyle \varepsilon _{\sigma p}(y,z)=\varepsilon (y,z)-\varepsilon _{th}(y,z)+\varepsilon _{p}}$.

The stress-dependent concrete strain is used to take into account creep (creep coefficient ) and shrinkage (shrinkage strain ): ${\displaystyle \varphi }$ ${\displaystyle \varepsilon _{cs}}$

${\displaystyle \varepsilon _{\sigma c}(y,z)=[\varepsilon (y,z)-\varepsilon _{th}(y,z)-\varepsilon _{cs}]/(1+\varphi )}$.

To an outside influence

${\displaystyle S=[N,\ M_{y},\ M_{z}]}$

The corresponding expansion plane must meet the condition . For this purpose, a non-linear system of equations has to be solved iteratively for the three unknowns, for the strain in the zero point of the - coordinate system of the cross-section and for the two curvatures and . The Broyden method is very suitable for this (see below). ${\displaystyle R=S}$ ${\displaystyle \varepsilon _{0}}$ ${\displaystyle y}$ ${\displaystyle z}$ ${\displaystyle 1/r_{y}=\mathrm {d} \varepsilon /\mathrm {d} y}$ ${\displaystyle 1/r_{z}=\mathrm {d} \varepsilon /\mathrm {d} z}$

For the linear elastic stress-strain relationship and the - - the principal axis system provides the voltage integration: ${\displaystyle \sigma (\varepsilon )=E\cdot \varepsilon }$ ${\displaystyle y}$ ${\displaystyle z}$

${\displaystyle R=[N,M_{y},M_{z}]=[\varepsilon _{0}\cdot EA,EI_{y}/r_{y},EI_{z}/r_{z}]}$.

Only for this special case the individual elements of the distortion vector or the strain state can immediately without iteration with the cross-section values , and are calculated: ${\displaystyle D}$ ${\displaystyle A,\ I_{y}}$ ${\displaystyle I_{z}}$

${\displaystyle D=[\varepsilon _{0},1/r_{y},1/r_{z}]=[N/EA,M_{y}/EI_{y},M_{z}/EI_{z}].}$

Practical importance in construction

The consideration of non-linear internal forces-distortion relationships is essential for calculating the realistic load-bearing behavior (component deformations, load-bearing capacities ) of structures, especially those made of reinforced or prestressed concrete. Cracks lead to greater component deformation and reduced component stiffness .

• Because of the reduction in component stiffness, it is a safety imperative to verify the load-bearing capacity of slender compression members and columns using non-linear methods according to the second order theory.
• The effects of hindered or impressed deformations ( constraint ) are more accurately determined with non-linear methods because of the reduction in component stiffness, which enables more economical designs.
• For the average size determination in statically indeterminate bearing beams linear methods are sufficient. It depends on the ratio of the deformations of positively curved field areas to negatively curved support areas (relative deformations). This ratio changes only insignificantly as a result of the crack formation.
• The reinforcement must be known for stress integration . This is a major disadvantage for the application of non-linear methods. In contrast, this disadvantage does not exist for the realistic calculation of existing structures with known reinforcement.
• Because of the numerical complexity, the application of non-linear methods requires suitable matrix-oriented mathematics programs or corresponding application programs.

Member calculations

Classic structural engineering methods apply to linear internal forces-distortion relationships. Matrix methods with computer programs are used to take into account non-linear internal forces-distortion relationships. The best known is the finite element method  (FEM). In FEM calculations with non-linear stress-strain relationships, the convergence behavior of the iterative calculation is influenced by the size of the finite elements, the size of the load increment and the discontinuity of the stress-strain relationship. Appropriate knowledge of suitable structural modeling must be assumed for the user.

For beam structures, the combination of the angle of rotation method with the single member calculation is a useful alternative to FEM calculations. The member-by-member calculation of the individual members of the structure can be carried out using the transfer method or by numerically solving the system of differential equations. Overall, considerably fewer unknowns remain. This significantly improves the convergence of the iterative computation.

Numerical calculation of the system of differential equations

The classical differential equation of the 4th order for the linear-elastic bar under uniaxial bending and without torsion

${\displaystyle EI\ w''''(x)+N\ w''(x)=q(x)}$

is formulated as a system of four first-order differential equations.

The matrix with the four state variables

${\displaystyle Z=[Q,M,\varphi ,w;0\ldots x\ldots l]}$

for all points of the rod, the gradient at the points of the rod becomes ${\displaystyle x}$ ${\displaystyle x}$

${\displaystyle \mathrm {d} Z(x)/\mathrm {d} x=[q(x)+N(x)\cdot 1/r(M(x)),Q(x),1/r(M(x)),\varphi (x)]}$

calculated with the help of numerical methods for the solution of differential equations of the first order (Runge, Heun).

In the case of biaxial bending, there are eight first-order differential equations.

The curvatures are determined for the bending moments with special functions: ${\displaystyle 1/r(M(x))=k(M(x))}$ ${\displaystyle M(x)}$

• for linear internal forces-distortion relationships ${\displaystyle k(M(x))=M(x)/EI}$
• For non-linear internal forces-distortion relationships, the curvature can be determined iteratively using the stress integration method. ${\displaystyle k(M(x))=\mathrm {d} \varepsilon /\mathrm {d} x}$

For members with constant longitudinal force and only uniaxial bending, it is more effective to calculate the curvatures in advance for individual bending moments and to interpolate them from the value pairs and . The value pairs and can be displayed as a moment-warp line. The increasing or decreasing gradient of this - line indicates the load- dependent reduction in flexural rigidity. ${\displaystyle N}$ ${\displaystyle k(M)}$ ${\displaystyle M}$ ${\displaystyle k(M(x))}$ ${\displaystyle M}$ ${\displaystyle k(M)}$ ${\displaystyle M}$ ${\displaystyle k(M)}$ ${\displaystyle M}$ ${\displaystyle k(M)}$ ${\displaystyle \mathrm {d} M/\mathrm {d} k}$ ${\displaystyle M}$ ${\displaystyle k}$

Initial values ​​as the solution of a nonlinear system of equations

For the numerical solution of the differential equations, two unknown initial values must be determined iteratively in the state variables in the case of uniaxial bending , and four in the case of biaxial bending. With the initial values, the numerical calculation of the differential equations must not leave any differences compared to the two known final values in the state variables at the member end. The Broyden method can be used for the iterative calculation of the initial values . ${\displaystyle ZA}$ ${\displaystyle Z=[Q,M,\varphi ,w]}$ ${\displaystyle ZE}$

See also

• Young's modulus of concrete
• Beam theory
• Bending line

literature

• Olaf Ehrigsen: A general calculation method for bars and its application to bars in solid construction. TU Hamburg-Harburg, dissertation, 2003. Göttingen: Cuvillier, 2003, ISBN 3-89873-755-1 .
• Piotr Noakowski and Horst G. Schäfer: Rigidity-oriented statics in reinforced concrete construction. Simply calculate reinforced concrete structures correctly. Berlin, Ernst & Sohn, 2003, ISBN 3433017514 .
• Uwe Pfeiffer: The non-linear calculation of flat frames made of reinforced or prestressed concrete, taking into account the axial expansion caused by the tear. TU Hamburg-Harburg, dissertation, 2004. Göttingen: Cuvillier, 2004, ISBN 3-86537-298-8 .
• Ulrich Quast: Nonlinear statics in reinforced concrete construction. Berlin, Bauwerk Verlag, 2006, ISBN 3-89932-158-8 .
• Ulrich Quast: Non-linear computing. Avak / Goris (ed.), Stahlbetonbau aktuell, Praxishandbuch 2009. Chapter C Statics, Berlin, Bauwerk Verlag, 2009, ISBN 978-3-89932-205-7 .
• Ulrich Quast: Stress-dependent and thermal expansions. Concrete and reinforced concrete construction 104 (2009), Issue 9, 616–618, ISSN  0005-9900 .
• Fei Chen: Numerical simulation of the non-linear load-bearing and damage behavior of reinforced concrete rod structures with monotonic and cyclical loading. Update report VDI Series 4 No. 171, Düsseldorf, VDI-Verlag, 2001, ISBN 3-18-317104-X , ISSN  0178-9511 .

Web links

• The INCA2 programs developed for teaching purposes for calculating cross-sections and Stab2D-NL for calculating flat frames are provided by the solid construction work area at the TU Hamburg-Harburg .
• About FEM in solid construction, Dr.-Ing. Casimir Katz ( Memento from August 28, 2007 in the Internet Archive ) very informative lecture documents in PDF format.