Fächer

Fächer

Mehr

Suche

Knowunity Pro

AI Knowunity

Schnittgrößen leicht berechnen: Rechner und Aufgaben mit Lösungen

Öffnen

21

0

user profile picture

Markus Leo Nussbaum

29.11.2021

Mathe

Einführung Statik (Tragwerkslehre)

Fächer

/

Mathe

/

Geometrie

/

Trapeze

Schnittgrößen leicht berechnen: Rechner und Aufgaben mit Lösungen

A comprehensive guide to calculating internal forces and structural analysis, focusing on Schnittgrößen berechnen methods and force distribution in construction elements. This technical resource covers:

  • Fundamental principles of statics and strength of materials
  • Detailed analysis of Streckenlast berechnen Beispiel with practical applications
  • Methods for calculating support reactions and internal forces
  • Step-by-step procedures for determining shear forces and bending moments
  • Integration of Querkraft berechnen Formel in structural analysis
...

29.11.2021

1005

EINFUHRUNG STATIO STATIK UND FESTIGKEITSLEHRE BELASTUNGEN IN DEN UNTERGRUND LEITEN BAUWERK IN VIELE EINZELNE TEILE ZERLEGEN TAV J 2 F (1 15

Öffnen

Beam Analysis Example

This page presents a practical example of Schnittgrößen berechnen for a simply supported reinforced concrete beam. The beam is subjected to a uniformly distributed load, demonstrating how to calculate support reactions and internal forces.

Example: A 10-meter long reinforced concrete beam is supported by a roller support at one end and a pin support at the other, with a uniformly distributed load of 10 kN/m.

The page illustrates the process of determining support reactions and introduces the concept of a Schnittgrößen Tabelle (internal force table) to organize calculations. It emphasizes the importance of understanding the distribution of shear forces and bending moments along the beam's length.

Highlight: The analysis demonstrates that the structure is statically determinate, meaning support reactions can be calculated using equilibrium equations alone.

The example serves as a foundation for more complex structural analysis problems, introducing students to the fundamental principles of beam mechanics and the calculation of internal forces.

EINFUHRUNG STATIO STATIK UND FESTIGKEITSLEHRE BELASTUNGEN IN DEN UNTERGRUND LEITEN BAUWERK IN VIELE EINZELNE TEILE ZERLEGEN TAV J 2 F (1 15

Öffnen

Calculation of Support Reactions

This page delves into the detailed calculation of support reactions for the beam example introduced earlier. It demonstrates the application of equilibrium equations to determine the vertical and horizontal reactions at the supports.

Vocabulary: Gleichgewicht (equilibrium) refers to the state where the sum of all forces and moments acting on a structure is zero.

The calculation process involves:

  1. Summing moments about one support to find the vertical reaction at the other
  2. Using vertical force equilibrium to determine the second vertical reaction
  3. Applying horizontal force equilibrium to find the horizontal reaction

Example: The vertical reaction at support A is calculated as 50 kN, while the horizontal reaction is 5 kN.

The page emphasizes the importance of verifying results by checking that all equilibrium equations are satisfied. This step is crucial for ensuring the accuracy of subsequent internal force calculations.

Highlight: Understanding support reactions is fundamental to Schnittgrößen berechnen, as these reactions form the basis for determining internal forces throughout the structure.

EINFUHRUNG STATIO STATIK UND FESTIGKEITSLEHRE BELASTUNGEN IN DEN UNTERGRUND LEITEN BAUWERK IN VIELE EINZELNE TEILE ZERLEGEN TAV J 2 F (1 15

Öffnen

Internal Force Calculation

This page focuses on the calculation of internal forces (Schnittgrößen) at various points along the beam. It introduces the concept of "cutting" the beam at specific locations to determine shear forces, bending moments, and axial forces.

Definition: The Querkraft berechnen Formel (shear force calculation formula) is derived from the equilibrium of vertical forces at a cut section.

The page demonstrates how to calculate internal forces at two key locations:

  1. At the left support (x = 0)
  2. At the midspan (x = 5 m)

Example: At x = 5 m, the bending moment is calculated as 125 kNm, while the shear force is 0 kN.

The calculations illustrate the use of equilibrium equations to determine internal forces, providing a step-by-step approach that can be applied to various beam problems. This method is essential for creating accurate shear force and bending moment diagrams.

Highlight: Understanding the distribution of internal forces along a beam is crucial for identifying critical sections that require careful design consideration.

EINFUHRUNG STATIO STATIK UND FESTIGKEITSLEHRE BELASTUNGEN IN DEN UNTERGRUND LEITEN BAUWERK IN VIELE EINZELNE TEILE ZERLEGEN TAV J 2 F (1 15

Öffnen

Bending Moment Diagram Construction

This page provides a detailed guide on constructing a bending moment diagram for the beam example. It introduces a graphical method for visualizing the distribution of bending moments along the beam's length.

Vocabulary: Parabel (parabola) refers to the shape of the bending moment diagram for a uniformly loaded beam.

The construction process involves several steps:

  1. Calculating the maximum bending moment at midspan
  2. Determining the "stich" (rise) of the parabola
  3. Drawing tangent lines at the supports
  4. Sketching the parabolic curve

Example: The maximum bending moment (Mo) is calculated as 125 kNm at the beam's midpoint.

The page emphasizes the importance of accurately representing the bending moment distribution, as this information is crucial for structural design and identifying critical sections of the beam.

Highlight: The bending moment diagram provides valuable insights into the beam's behavior under load, helping engineers optimize the design for strength and efficiency.

EINFUHRUNG STATIO STATIK UND FESTIGKEITSLEHRE BELASTUNGEN IN DEN UNTERGRUND LEITEN BAUWERK IN VIELE EINZELNE TEILE ZERLEGEN TAV J 2 F (1 15

Öffnen

Complex Beam Analysis Example

This page presents a more complex beam analysis problem, demonstrating the application of Schnittgrößen berechnen techniques to a beam with multiple loads and support conditions.

Example: A beam with a span of 7 m is subjected to a concentrated force, an inclined force, and a uniformly distributed load.

The problem is broken down into several steps:

  1. Calculating support reactions
  2. Analyzing internal forces at multiple sections along the beam
  3. Determining the maximum bending moment location

Vocabulary: Streckenlast q berechnen refers to the process of calculating the effects of a uniformly distributed load on a beam.

The page provides detailed calculations for each step, illustrating how to handle more complex loading scenarios and support conditions. It emphasizes the importance of carefully considering the direction and magnitude of all forces acting on the structure.

Highlight: This example demonstrates how to apply the principles of structural analysis to more realistic and challenging scenarios encountered in engineering practice.

EINFUHRUNG STATIO STATIK UND FESTIGKEITSLEHRE BELASTUNGEN IN DEN UNTERGRUND LEITEN BAUWERK IN VIELE EINZELNE TEILE ZERLEGEN TAV J 2 F (1 15

Öffnen

Internal Force Analysis for Complex Beam

This page continues the analysis of the complex beam example, focusing on calculating internal forces at various sections along the beam's length. It demonstrates how to handle changes in loading conditions and geometry when determining shear forces, bending moments, and axial forces.

Example: At section 3, the bending moment is calculated as 30.76 kNm, while the shear force is -4.424 kN.

The analysis involves:

  1. Dividing the beam into multiple sections based on loading changes
  2. Applying equilibrium equations at each section
  3. Calculating internal forces considering the cumulative effects of loads

Vocabulary: Streckenlast berechnen Beispiel (distributed load calculation example) is demonstrated through the analysis of the uniformly loaded section of the beam.

The page emphasizes the importance of considering the direction and sign conventions of forces and moments when performing calculations. It also illustrates how axial forces can arise in beams subjected to inclined loads.

Highlight: Understanding how to analyze complex loading scenarios is crucial for engineers dealing with real-world structural design problems.

EINFUHRUNG STATIO STATIK UND FESTIGKEITSLEHRE BELASTUNGEN IN DEN UNTERGRUND LEITEN BAUWERK IN VIELE EINZELNE TEILE ZERLEGEN TAV J 2 F (1 15

Öffnen

Maximum Bending Moment and Force Diagrams

This final page focuses on determining the location and magnitude of the maximum bending moment in the complex beam example. It also presents the shear force, bending moment, and axial force diagrams for the entire beam.

Example: The maximum bending moment is calculated as 36.87 kNm, occurring at x = 4.74 m from the left support.

The page demonstrates:

  1. Using the method of sections to find the maximum bending moment
  2. Constructing shear force, bending moment, and axial force diagrams
  3. Interpreting the diagrams to understand the beam's structural behavior

Vocabulary: Schnittgrößen Rechner online tools can be used to verify hand calculations and quickly generate internal force diagrams.

The force diagrams provide a visual representation of how internal forces vary along the beam's length, helping engineers identify critical sections for design and analysis.

Highlight: The ability to accurately determine and visualize internal forces is fundamental to effective structural design and analysis.

The page concludes with a summary of key points, reinforcing the importance of thorough structural analysis in engineering practice.

EINFUHRUNG STATIO STATIK UND FESTIGKEITSLEHRE BELASTUNGEN IN DEN UNTERGRUND LEITEN BAUWERK IN VIELE EINZELNE TEILE ZERLEGEN TAV J 2 F (1 15

Öffnen

Force Diagram Construction

The final page presents the complete force diagrams showing the distribution of internal forces.

Definition: Force diagrams provide visual representation of internal force distribution along the structure.

Highlight: Integration of all calculated values into comprehensive force diagrams for practical application.

Nichts passendes dabei? Erkunde andere Fachbereiche.

Knowunity ist die #1 unter den Bildungs-Apps in fünf europäischen Ländern

Knowunity wurde bei Apple als "Featured Story" ausgezeichnet und hat die App-Store-Charts in der Kategorie Bildung in Deutschland, Italien, Polen, der Schweiz und dem Vereinigten Königreich regelmäßig angeführt. Werde noch heute Mitglied bei Knowunity und hilf Millionen von Schüler:innen auf der ganzen Welt.

Ranked #1 Education App

Laden im

Google Play

Laden im

App Store

Knowunity ist die #1 unter den Bildungs-Apps in fünf europäischen Ländern

4.9+

Durchschnittliche App-Bewertung

17 M

Schüler:innen lieben Knowunity

#1

In Bildungs-App-Charts in 17 Ländern

950 K+

Schüler:innen haben Lernzettel hochgeladen

Immer noch nicht überzeugt? Schau dir an, was andere Schüler:innen sagen...

iOS User

Ich liebe diese App so sehr, ich benutze sie auch täglich. Ich empfehle Knowunity jedem!! Ich bin damit von einer 4 auf eine 1 gekommen :D

Philipp, iOS User

Die App ist sehr einfach und gut gestaltet. Bis jetzt habe ich immer alles gefunden, was ich gesucht habe :D

Lena, iOS Userin

Ich liebe diese App ❤️, ich benutze sie eigentlich immer, wenn ich lerne.

Fächer

/

Mathe

/

Geometrie

/

Trapeze

Schnittgrößen leicht berechnen: Rechner und Aufgaben mit Lösungen

user profile picture

Markus Leo Nussbaum

@markusnussibaum

·

244 Follower

Follow

A comprehensive guide to calculating internal forces and structural analysis, focusing on Schnittgrößen berechnen methods and force distribution in construction elements. This technical resource covers:

  • Fundamental principles of statics and strength of materials
  • Detailed analysis of Streckenlast berechnen Beispiel with practical applications
  • Methods for calculating support reactions and internal forces
  • Step-by-step procedures for determining shear forces and bending moments
  • Integration of Querkraft berechnen Formel in structural analysis
...

29.11.2021

1005

 

11/10

 

Mathe

21

EINFUHRUNG STATIO STATIK UND FESTIGKEITSLEHRE BELASTUNGEN IN DEN UNTERGRUND LEITEN BAUWERK IN VIELE EINZELNE TEILE ZERLEGEN TAV J 2 F (1 15

Melde dich an, um den Inhalt freizuschalten. Es ist kostenlos!

Zugriff auf alle Dokumente

Verbessere deine Noten

Werde Teil der Community

Mit der Anmeldung akzeptierst du die Nutzungsbedingungen und die Datenschutzrichtlinie

Beam Analysis Example

This page presents a practical example of Schnittgrößen berechnen for a simply supported reinforced concrete beam. The beam is subjected to a uniformly distributed load, demonstrating how to calculate support reactions and internal forces.

Example: A 10-meter long reinforced concrete beam is supported by a roller support at one end and a pin support at the other, with a uniformly distributed load of 10 kN/m.

The page illustrates the process of determining support reactions and introduces the concept of a Schnittgrößen Tabelle (internal force table) to organize calculations. It emphasizes the importance of understanding the distribution of shear forces and bending moments along the beam's length.

Highlight: The analysis demonstrates that the structure is statically determinate, meaning support reactions can be calculated using equilibrium equations alone.

The example serves as a foundation for more complex structural analysis problems, introducing students to the fundamental principles of beam mechanics and the calculation of internal forces.

EINFUHRUNG STATIO STATIK UND FESTIGKEITSLEHRE BELASTUNGEN IN DEN UNTERGRUND LEITEN BAUWERK IN VIELE EINZELNE TEILE ZERLEGEN TAV J 2 F (1 15

Melde dich an, um den Inhalt freizuschalten. Es ist kostenlos!

Zugriff auf alle Dokumente

Verbessere deine Noten

Werde Teil der Community

Mit der Anmeldung akzeptierst du die Nutzungsbedingungen und die Datenschutzrichtlinie

Calculation of Support Reactions

This page delves into the detailed calculation of support reactions for the beam example introduced earlier. It demonstrates the application of equilibrium equations to determine the vertical and horizontal reactions at the supports.

Vocabulary: Gleichgewicht (equilibrium) refers to the state where the sum of all forces and moments acting on a structure is zero.

The calculation process involves:

  1. Summing moments about one support to find the vertical reaction at the other
  2. Using vertical force equilibrium to determine the second vertical reaction
  3. Applying horizontal force equilibrium to find the horizontal reaction

Example: The vertical reaction at support A is calculated as 50 kN, while the horizontal reaction is 5 kN.

The page emphasizes the importance of verifying results by checking that all equilibrium equations are satisfied. This step is crucial for ensuring the accuracy of subsequent internal force calculations.

Highlight: Understanding support reactions is fundamental to Schnittgrößen berechnen, as these reactions form the basis for determining internal forces throughout the structure.

EINFUHRUNG STATIO STATIK UND FESTIGKEITSLEHRE BELASTUNGEN IN DEN UNTERGRUND LEITEN BAUWERK IN VIELE EINZELNE TEILE ZERLEGEN TAV J 2 F (1 15

Melde dich an, um den Inhalt freizuschalten. Es ist kostenlos!

Zugriff auf alle Dokumente

Verbessere deine Noten

Werde Teil der Community

Mit der Anmeldung akzeptierst du die Nutzungsbedingungen und die Datenschutzrichtlinie

Internal Force Calculation

This page focuses on the calculation of internal forces (Schnittgrößen) at various points along the beam. It introduces the concept of "cutting" the beam at specific locations to determine shear forces, bending moments, and axial forces.

Definition: The Querkraft berechnen Formel (shear force calculation formula) is derived from the equilibrium of vertical forces at a cut section.

The page demonstrates how to calculate internal forces at two key locations:

  1. At the left support (x = 0)
  2. At the midspan (x = 5 m)

Example: At x = 5 m, the bending moment is calculated as 125 kNm, while the shear force is 0 kN.

The calculations illustrate the use of equilibrium equations to determine internal forces, providing a step-by-step approach that can be applied to various beam problems. This method is essential for creating accurate shear force and bending moment diagrams.

Highlight: Understanding the distribution of internal forces along a beam is crucial for identifying critical sections that require careful design consideration.

EINFUHRUNG STATIO STATIK UND FESTIGKEITSLEHRE BELASTUNGEN IN DEN UNTERGRUND LEITEN BAUWERK IN VIELE EINZELNE TEILE ZERLEGEN TAV J 2 F (1 15

Melde dich an, um den Inhalt freizuschalten. Es ist kostenlos!

Zugriff auf alle Dokumente

Verbessere deine Noten

Werde Teil der Community

Mit der Anmeldung akzeptierst du die Nutzungsbedingungen und die Datenschutzrichtlinie

Bending Moment Diagram Construction

This page provides a detailed guide on constructing a bending moment diagram for the beam example. It introduces a graphical method for visualizing the distribution of bending moments along the beam's length.

Vocabulary: Parabel (parabola) refers to the shape of the bending moment diagram for a uniformly loaded beam.

The construction process involves several steps:

  1. Calculating the maximum bending moment at midspan
  2. Determining the "stich" (rise) of the parabola
  3. Drawing tangent lines at the supports
  4. Sketching the parabolic curve

Example: The maximum bending moment (Mo) is calculated as 125 kNm at the beam's midpoint.

The page emphasizes the importance of accurately representing the bending moment distribution, as this information is crucial for structural design and identifying critical sections of the beam.

Highlight: The bending moment diagram provides valuable insights into the beam's behavior under load, helping engineers optimize the design for strength and efficiency.

EINFUHRUNG STATIO STATIK UND FESTIGKEITSLEHRE BELASTUNGEN IN DEN UNTERGRUND LEITEN BAUWERK IN VIELE EINZELNE TEILE ZERLEGEN TAV J 2 F (1 15

Melde dich an, um den Inhalt freizuschalten. Es ist kostenlos!

Zugriff auf alle Dokumente

Verbessere deine Noten

Werde Teil der Community

Mit der Anmeldung akzeptierst du die Nutzungsbedingungen und die Datenschutzrichtlinie

Complex Beam Analysis Example

This page presents a more complex beam analysis problem, demonstrating the application of Schnittgrößen berechnen techniques to a beam with multiple loads and support conditions.

Example: A beam with a span of 7 m is subjected to a concentrated force, an inclined force, and a uniformly distributed load.

The problem is broken down into several steps:

  1. Calculating support reactions
  2. Analyzing internal forces at multiple sections along the beam
  3. Determining the maximum bending moment location

Vocabulary: Streckenlast q berechnen refers to the process of calculating the effects of a uniformly distributed load on a beam.

The page provides detailed calculations for each step, illustrating how to handle more complex loading scenarios and support conditions. It emphasizes the importance of carefully considering the direction and magnitude of all forces acting on the structure.

Highlight: This example demonstrates how to apply the principles of structural analysis to more realistic and challenging scenarios encountered in engineering practice.

EINFUHRUNG STATIO STATIK UND FESTIGKEITSLEHRE BELASTUNGEN IN DEN UNTERGRUND LEITEN BAUWERK IN VIELE EINZELNE TEILE ZERLEGEN TAV J 2 F (1 15

Melde dich an, um den Inhalt freizuschalten. Es ist kostenlos!

Zugriff auf alle Dokumente

Verbessere deine Noten

Werde Teil der Community

Mit der Anmeldung akzeptierst du die Nutzungsbedingungen und die Datenschutzrichtlinie

Internal Force Analysis for Complex Beam

This page continues the analysis of the complex beam example, focusing on calculating internal forces at various sections along the beam's length. It demonstrates how to handle changes in loading conditions and geometry when determining shear forces, bending moments, and axial forces.

Example: At section 3, the bending moment is calculated as 30.76 kNm, while the shear force is -4.424 kN.

The analysis involves:

  1. Dividing the beam into multiple sections based on loading changes
  2. Applying equilibrium equations at each section
  3. Calculating internal forces considering the cumulative effects of loads

Vocabulary: Streckenlast berechnen Beispiel (distributed load calculation example) is demonstrated through the analysis of the uniformly loaded section of the beam.

The page emphasizes the importance of considering the direction and sign conventions of forces and moments when performing calculations. It also illustrates how axial forces can arise in beams subjected to inclined loads.

Highlight: Understanding how to analyze complex loading scenarios is crucial for engineers dealing with real-world structural design problems.

EINFUHRUNG STATIO STATIK UND FESTIGKEITSLEHRE BELASTUNGEN IN DEN UNTERGRUND LEITEN BAUWERK IN VIELE EINZELNE TEILE ZERLEGEN TAV J 2 F (1 15

Melde dich an, um den Inhalt freizuschalten. Es ist kostenlos!

Zugriff auf alle Dokumente

Verbessere deine Noten

Werde Teil der Community

Mit der Anmeldung akzeptierst du die Nutzungsbedingungen und die Datenschutzrichtlinie

Maximum Bending Moment and Force Diagrams

This final page focuses on determining the location and magnitude of the maximum bending moment in the complex beam example. It also presents the shear force, bending moment, and axial force diagrams for the entire beam.

Example: The maximum bending moment is calculated as 36.87 kNm, occurring at x = 4.74 m from the left support.

The page demonstrates:

  1. Using the method of sections to find the maximum bending moment
  2. Constructing shear force, bending moment, and axial force diagrams
  3. Interpreting the diagrams to understand the beam's structural behavior

Vocabulary: Schnittgrößen Rechner online tools can be used to verify hand calculations and quickly generate internal force diagrams.

The force diagrams provide a visual representation of how internal forces vary along the beam's length, helping engineers identify critical sections for design and analysis.

Highlight: The ability to accurately determine and visualize internal forces is fundamental to effective structural design and analysis.

The page concludes with a summary of key points, reinforcing the importance of thorough structural analysis in engineering practice.

EINFUHRUNG STATIO STATIK UND FESTIGKEITSLEHRE BELASTUNGEN IN DEN UNTERGRUND LEITEN BAUWERK IN VIELE EINZELNE TEILE ZERLEGEN TAV J 2 F (1 15

Melde dich an, um den Inhalt freizuschalten. Es ist kostenlos!

Zugriff auf alle Dokumente

Verbessere deine Noten

Werde Teil der Community

Mit der Anmeldung akzeptierst du die Nutzungsbedingungen und die Datenschutzrichtlinie

Force Diagram Construction

The final page presents the complete force diagrams showing the distribution of internal forces.

Definition: Force diagrams provide visual representation of internal force distribution along the structure.

Highlight: Integration of all calculated values into comprehensive force diagrams for practical application.

EINFUHRUNG STATIO STATIK UND FESTIGKEITSLEHRE BELASTUNGEN IN DEN UNTERGRUND LEITEN BAUWERK IN VIELE EINZELNE TEILE ZERLEGEN TAV J 2 F (1 15

Melde dich an, um den Inhalt freizuschalten. Es ist kostenlos!

Zugriff auf alle Dokumente

Verbessere deine Noten

Werde Teil der Community

Mit der Anmeldung akzeptierst du die Nutzungsbedingungen und die Datenschutzrichtlinie

Introduction to Statics and Strength of Materials

This page introduces the fundamental concepts of structural analysis, focusing on how forces are transmitted through a structure to its foundation. It emphasizes the importance of understanding internal forces, or Schnittgrößen, in structural design.

Definition: Schnittgrößen (internal forces) are the forces and moments that act within a structural member, such as beams or columns.

The page outlines the process of analyzing a structure by breaking it down into individual components and examining the forces acting on each part. This approach is crucial for designing structures in steel, wood, and reinforced concrete.

Highlight: The analysis of internal forces is essential for determining the appropriate material and cross-section for structural elements.

The document introduces three key aspects of structural analysis:

  1. External forces and support reactions
  2. Internal forces (Schnittgrößen)
  3. Material properties and cross-section design

Vocabulary: Auflagerkräfte (support reactions) are the forces exerted by the supports on the structure to maintain equilibrium.

Understanding these concepts allows engineers to effectively design structures that can safely transfer loads to the ground while meeting strength and serviceability requirements.

Nichts passendes dabei? Erkunde andere Fachbereiche.

Knowunity ist die #1 unter den Bildungs-Apps in fünf europäischen Ländern

Knowunity wurde bei Apple als "Featured Story" ausgezeichnet und hat die App-Store-Charts in der Kategorie Bildung in Deutschland, Italien, Polen, der Schweiz und dem Vereinigten Königreich regelmäßig angeführt. Werde noch heute Mitglied bei Knowunity und hilf Millionen von Schüler:innen auf der ganzen Welt.

Ranked #1 Education App

Laden im

Google Play

Laden im

App Store

Knowunity ist die #1 unter den Bildungs-Apps in fünf europäischen Ländern

4.9+

Durchschnittliche App-Bewertung

17 M

Schüler:innen lieben Knowunity

#1

In Bildungs-App-Charts in 17 Ländern

950 K+

Schüler:innen haben Lernzettel hochgeladen

Immer noch nicht überzeugt? Schau dir an, was andere Schüler:innen sagen...

iOS User

Ich liebe diese App so sehr, ich benutze sie auch täglich. Ich empfehle Knowunity jedem!! Ich bin damit von einer 4 auf eine 1 gekommen :D

Philipp, iOS User

Die App ist sehr einfach und gut gestaltet. Bis jetzt habe ich immer alles gefunden, was ich gesucht habe :D

Lena, iOS Userin

Ich liebe diese App ❤️, ich benutze sie eigentlich immer, wenn ich lerne.