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| ´´Treppen auf Platten Beispiel 1 - Treppenhauses in einem mehrgeschossigen Wohnhaus Treppenlauf biegesteif Podest angeschlossen´´
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|
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| = Aufgabenstellung =
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| Bild
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|
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|
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| Für den gegebenen Grundriss eines Treppenhauses in einem mehrgeschossigen Wohnhaus mit sechs Wohnungen soll eine gegenläufige Treppe Entworfen und Bemessen werden. Baustoffe, Umgebungsbedingungen und die Bauteilabmessungen wurden vorab festgelegt. Der Bauherr verzichtet auf ein besonderes maß des Schallschutzes. Als Treppenlauf Belag wurde ein Natursteinplattenbelag oberhalb und unterhalb wird die Treppe mit einem Gipsputz versehen. gewählt
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| Gewählt wurde eine Ausführung mit Arbeitsfuge
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| Die Podestplatten werden im Rahmen des Beispiels als einachsig gespannte platte betrachtet. Die Berechnung dient als Vergleich zu der Berechnung mit dem MB Modul
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|
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| == Vorgaben ==
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| :{|
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| |-
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| | Geschosshöhe <math> h </math>: || 2,72m
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| |-
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| | Plattenstärke <math> h_{L} </math>: || 20 cm
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| |-
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| | Natursteinplattenstärke <math> N_{s} </math>: || 6,0 cm
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| |-
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| | Gipsputz <math> G_{s} </math>: || 1,5 cm
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| |-
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| | Treppenform: || Gegenläufige Treppe
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| |-
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| | Expositionsklasse: || XC1 - trocken, ständig nass
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| |-
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| | Betonfestigkeitsklasse: || C25/30
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| |-
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| | Betonstahl: || B500
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| |}
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|
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| = Anforderungen an Treppenkonstruktionen =
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|
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| ==geometrische Bestimmung==
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|
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| zu Entwerfen ist eine Treppe für ein Treppenhauses in einem mehrgeschossigen Wohnhaus mit sechs Wohnungen
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| da die Treppe zu einem nicht zu ebener Erde liegende Geschoss führt spricht man von einer Baurechtlich notwendige Treppe
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| {| class="wikitable"
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| |+style="text-align:left;"|Grenzmaße <ref Name = "DIN18065" group="N" >DIN 18065:2015-03 Gebäudetreppen - Begriffe, Messregeln, Hauptmaße</ref>
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| |rowspan="3"|
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| |1
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| |2
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| |5
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| |6
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| |7
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| |-
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| !rowspan="2" style="background: #eaecf0;"|Gebäudeart
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| !rowspan="2" style="background: #eaecf0;"|Treppenart
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| !rowspan="2" style="background: #eaecf0;"|minimale nutzbare Laufbreite (b) [cm]
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| !colspan="2" style="background: #eaecf0;"|Steigung (s) [cm]
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| !colspan="2" style="background: #eaecf0;"|Auftritt (a) [cm]
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| |-
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| !style="background: #eaecf0;"|min.
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| !style="background: #eaecf0;"|max.
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| !style="background: #eaecf0;"|min.
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| !style="background: #eaecf0;"|max.
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| |-
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| |1
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| |rowspan="2" style="background: #FFFF40"|Gebäude im Allgemeinen (Fertigmaße im Endzustand)
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| |style="background: #FFFF40"|Baurechtlich notwendige Treppe
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| |style="background:#FFFF40"|100
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| |style="background:#FFFF40"|14
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| |style="background:#FFFF40"|19
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| |style="background:#FFFF40"|26
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| |style="background:#FFFF40"|37
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| |-
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| |2
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| |style="background: #eaecf0;"|Baurechtlich nicht notwendige (zusätzliche)
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| |50
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| |14
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| |21
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| |21
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| |37
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| |-
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| |3
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| |rowspan="2" style="background: #eaecf0;"|Wohngebäude mit bis zu zwei Wohnungen und innerhalb von Wohnungen
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| |style="background: #eaecf0;"|Baurechtlich notwendige Treppe
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| |80
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| |14
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| |20
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| |23
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| |37
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| |-
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| |4
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| |style="background: #eaecf0;"|Baurechtlich nicht notwendige (zusätzliche)
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| |50
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| |14
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| |21
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| |21
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| |37
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| |}
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| ===minimale nutzbare Laufbreite <math>b</math>===
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|
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| <br />
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| ::<math> b \le \underline{100cm} </math>.
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| <br />
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| ===Steigung <math>s</math>===
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|
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| um die Geschosshöhe von 272 cm zu überbrücken, wurde eine gegenläufige Treppe mit jeweils 8 Steigungen pro Treppenlauf gewählt
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| ::{|
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| |<math> s = \frac{h}{\text{Anzahl Steigungen}}= \frac{272cm}{16} = \underline{17cm} </math>
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| |-
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| |<math> 14cm \le \underline{s = 17} \le 19cm</math>
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| |}
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| === Auftritt <math>a</math>===
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|
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| ::{|
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| | <math> a - s \approx 12cm </math> || <math> | +s </math>
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| |-
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| | <math> a \approx s + 12cm </math>|| <math>| mit: s= 17cm </math>
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| |-
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| | <math> a \approx 17cm + 12cm </math>||
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| |-
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| | <math> a \approx 29cm </math>||
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| |-
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| | <math> 26cm \le \underline{a=29cm} \le 37cm</math> ||
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| |}
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| ===Überprüfung Schrittmaß===
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|
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| ::{|
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| | <math> 59cm \le 2 \cdot s + a \le 65cm </math> ||<math>| mit: s= 17cm </math>
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| |-
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| | ||<math>| mit:a=29cm </math>
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| |-
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| | <math> 59cm \le 2 \cdot 17cm + 29 \le 65cm </math>||
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| |-
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| | <math> 59cm \le \underline{63cm} \le 65cm </math>||
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| |}
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| ===Treppenaugebreite <math> b^{'} </math>===
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|
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| Es wird ein Treppenauge von 25 cm gewählt
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| ::<math>20cm \le b^{'} =25 \le 30cm </math>
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| * Steigungswinkel
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|
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| ::{|
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| | <math> \alpha = tan^{-1} (\frac{s}{a}) </math>||<math>| mit: s= 17cm </math>
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| |-
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| | ||<math>| mit:a=29cm </math>
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| |-
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| | <math> \alpha = tan^{-1} (\frac{17cm}{29cm}) </math> ||
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| |-
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| | <math> \alpha \approx 30,38° </math> ||
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| |}
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|
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| ==Schallschutz==
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| Da der Bauherr keine privatrechtlichen Anforderungen gestellt hat. wird bei der Planung lediglich das Mindestschallniveau nach DIN 4109 einzuhalten. Für die Konstruktion wird in dieser Treppe das typische beispiel für die Mindestanforderungen des Schallschutzes
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|
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| Treppenlauf und Treppenpodest werden biegesteif miteinander verbunden. Die Treppenpodeste werden hierbei mit schwimmendem Estrich ausgestattet.Desweiteren wird der Lauf schalltechnisch mit einer Fugenplatte von den Wänden entkoppelt.
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|
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| ==Brandschutz==
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| es wird keine besondere Anforderungen an den Brandschutz gestellt daher reicht die Normale Betondeckung aus dem EC2.
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|
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|
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| = Lösung für den Treppenlauf =
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|
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| ==Einwirkungen==
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| ===Teilsicherheiten ===
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|
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| ::{|
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| |<math> \gamma_\mathrm{Q} =1,50 </math> ||
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| |-
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| |<math> \gamma_\mathrm{G} =1,35 </math> ||
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| |}
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| === Ständige===
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| Die ständigen Lasten werden auf den Grundriss bezogen.
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|
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| :{|
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| | <math> g_{d}= g_{k} \cdot \gamma_\mathrm{G}</math> ||
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| |}
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| <br />
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| ::{|
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| |<math> g_{k} = g^{*}_{k} + g^{**}_{k} </math> ||
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| |}
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| <br />
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| :::{|
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| |<math> g^{**}_{k} = \frac{s \cdot \gamma_{2}}{2} </math>|| <math>| mit: s= 17cm </math>
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| |-
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| | || <math>| mit: \gamma_{2} = 24 \frac{kN}{m^{3}} </math>
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| |-
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| | <math> g^{**}_{k} = \frac{0,17m\cdot 24 \frac{kN}{m^{3}}}{2} </math>||
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| |-
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| | <math> g^{**}_{k} = 2.04 \frac{kN}{m^{2}} </math>||
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| |}
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| <br />
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| :::{|
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| | <math> g^{*}_{k} = \frac{ h \cdot \gamma_{1} + \gamma_{G_{s}=1,5 cm} + N_{s} \cdot \gamma_{Naturstein} }{cos(\alpha)} </math>||<math>| mit: h= 20cm </math>
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| |-
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| | || <math>| mit: \gamma_{1}= 25 \frac{kN}{m^{3}} </math>
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| |-
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| | || <math>| mit: \gamma_{G_{s}=1,5 cm} = 0,18 \frac{kN}{m^{2}} </math>
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| |-
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| | || <math>| mit: \gamma_{Naturstein} = 0,3 \frac{\frac{kN}{m^{2}}}{cm} </math>
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| |-
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| | || <math>| mit: N_{s} = 6 cm </math>
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| |-
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| | || <math>| mit: \alpha = 30,38° </math>
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| |-
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| |<math> g^{*}_{k} = \frac{ 0,20m \cdot 25 \frac{kN}{m^{3}} + 0,18 \frac{kN}{m^{2}}+ 6 cm \cdot 0,3 \frac{\frac{kN}{m^{2}}}{cm} }{cos(30,38°)} </math>||
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| |-
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| |<math> g^{*}_{k} \approx 8,09 \frac{kN}{m^{2}} </math>||
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| |}
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| <br />
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| ::{|
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| |<math> g_{k} = g^{*}_{k} + g^{**}_{k} </math> ||<math>| mit: g^{*}_{k} = 8,09 \frac{kN}{m^{2}} </math>
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| |-
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| | || <math>| mit: g^{**}_{k} = 2.04 \frac{kN}{m^{2}} </math>
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| |-
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| |<math> g_{k} = 8,09 \frac{kN}{m^{2}} + 2.04 \frac{kN}{m^{2}} </math> ||
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| |-
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| |<math> g_{k} = 10,13 \frac{kN}{m^{2}} </math> ||
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| |}
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| <br />
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| :{|
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| | <math> g_{d}= g_{k} \cdot \gamma_\mathrm{G}</math> || <math>| mit: \gamma_\mathrm{G} = 1,5 </math>
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| |-
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| | || <math>| mit:g_{k} = 10,13 \frac{kN}{m^{2}} </math>
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| |-
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| | <math> g_{d}= 10,13 \frac{kN}{m^{2}} \cdot 1,5 </math> ||
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| |-
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| | <math> g_{d}=15,2 \frac{kN}{m^{2}} </math> ||
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| |}
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|
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| === Veränderliche===
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|
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| {| class="wikitable"
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| |+style="text-align:left;"|Lotrechte Nutzlasten für Treppen <ref Name = "HandbuchEC1" group="F">Handbuch Eurocode 1 Einwirkungen – Band 1 Grundlagen, Nutz- und Eigenlasten, Brandeinwirkungen, Schnee-, Wind-, Temperaturlasten Ausgabedatum: 06.2012 </ref>
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| |rowspan="2"|
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| |colspan="2"|1
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| |2
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| |3
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| |4
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| |5
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| |-
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| !colspan="2"|Kategorie
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| !Nutzung
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| !Beispiele
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| !<math> q_{k} [ \frac{kN}{m^{2}}] </math>
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| !<math> Q_{k} [kN] </math>
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| |-
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| |19
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| |rowspan="3" style="background:#FFFF40"|T
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| |style="background:#FFFF40"|T1
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| |rowspan="3" style="background:#FFFF40"|Treppen und Treppenpodeste
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| |style="background:#FFFF40"|Treppen und Treppenpodeste in Wohngebäuden, Bürogebäuden und von Arztpraxen ohne schweres Gerät
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| |style="background:#FFFF40"|3,0
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| |2,0
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| |-
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| |20
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| |T2
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| |alle Treppen und Treppenpodeste, die nicht in TI oder T3 eingeordnet werden können
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| |5,0
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| |2,0
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| |-
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| |21
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| |T3
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| |Zugänge und Treppen von Tribünen ohne feste Sitzplätze, die als Fluchtwege dienen
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| |7,5
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| |3,0
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| |}
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|
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| ::{|
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| |<math> \underline{ q_{k} = 3,0 \frac{kN}{m^{2}} } </math>||
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| |}
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| <br />
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|
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| :{|
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| |<math> q_{d} =q_{k} \cdot \gamma_\mathrm{ Q } </math>|| <math>| mit: q_{k} = 3,0 \frac{kN}{m^{2}} </math>
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| |-
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| | || <math>| mit: \gamma_\mathrm{ Q } = 1,5 </math>
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| |-
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| |<math> q_{d} = 3,0 \frac{kN}{m^{2}} \cdot 1,5 </math>||
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| |-
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| |<math> q_{d} = 4,5 \frac{kN}{m^{2}} </math>||
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| |}
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|
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| ===Gesamtlasten===
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|
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| :{|
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| | <math> f_{d}=g_{d}+q_{d} </math> || <math>| mit: q_{d} = 4,5 \frac{kN}{m^{2}} </math>
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| |-
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| | || <math>| mit: g_{d}= 15,2 \frac{kN}{m^{2}} </math>
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| |-
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| | <math> f_{d}=15,2 \frac{kN}{m^{2}} + 4,5 \frac{kN}{m^{2}} </math> ||
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| |-
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| | <math> f_{d}=19,7 \frac{kN}{m^{2}} </math> ||
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| |}
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|
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| ==Statisches System==
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| :{|
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| | <math> l_{L} = 8 \cdot a </math> || <math>| mit: a=29cm </math>
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| |-
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| | <math> l_{L} = 8 \cdot 29cm </math> ||
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| |-
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| | <math> l_{L} = 2,32 m </math> ||
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| |}
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| [[Bild]]
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|
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| ==Schnittgrößen==
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| ===maximales Feldmoment===
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| :{|
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| |<math> M_{Ed,F} =f_{d} \cdot \frac{l_{L}^{2}}{8}</math>|| <math>| mit: f_{d}=19,7 \frac{kN}{m^{2}} </math>
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| |-
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| | || <math>| mit: l_{L} = 2,32 m </math>
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| |-
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| |<math> M_{Ed,F} =19,7 \frac{kN}{m^{2}} \cdot \frac{2,32 m^{2}}{8}</math>||
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| |-
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| |<math> M_{Ed,F} =13.25 \frac{kNm}{m} </math>||
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| |}
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|
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|
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|
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| ===Stützmoment===
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| :{|
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| |<math> M_{Ed,S} =-f_{d} \cdot \frac{l_{L}^{2}}{16}</math>|| <math>| mit: f_{d}=19,7 \frac{kN}{m^{2}} </math>
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| |-
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| | || <math>| mit: l_{L} = 2,32 m </math>
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| |-
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| |<math> M_{Ed,S} =-19,7 \frac{kN}{m^{2}} \cdot \frac{2,32 m^{2}}{16}</math>||
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| |-
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| |<math> M_{Ed,S} =6.63 \frac{kNm}{m} </math>||
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| |}
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|
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| ===Auflagekraft===
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| :{|
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| | <math> C_{Ed} =f_{d} \cdot \frac{l_{L}}{2} </math> || <math>| mit: f_{d}=19,7 \frac{kN}{m^{2}} </math>
| |
| |-
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| | || <math>| mit: l_{L} = 2,32 m </math>
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| |-
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| | <math> C_{Ed} =19,7 \frac{kN}{m^{2}} \cdot \frac{ 2,32 m }{2} </math> ||
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| |-
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| | <math> C_{Ed} =22.86 \frac{kN}{m} </math> ||
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| |}
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|
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| === Maximale Normalkraft===
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|
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| :{|
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| | <math> extr n = \pm C_{Ed} \cdot sin(\alpha ) </math> || <math>| mit: \alpha = 30,38° </math>
| |
| |-
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| | || <math>| mit: C_{Ed} = 22.86 \frac{kN}{m} </math>
| |
| |-
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| | <math> extr n = \pm 22.86 \frac{kN}{m} \cdot sin( 30,38° ) </math> ||
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| |-
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| | <math> extr n = \pm 11,56 \frac{kN}{m} </math> ||
| |
| |}
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|
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| == Bemessung im Grenzzustand der Tragfähigkeit==
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| ===Materialparameter===
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|
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| ::{|
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| | <math> f_{cd} = \frac{ \alpha_{cc} \cdot f_{ck} }{ \gamma_{C} } </math>||<math>| mit: \gamma_{C} = 1.5 </math>
| |
| |-
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| | || <math>| mit: \alpha_{cc} = 0.85 </math>
| |
| |-
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| | || <math>| mit: f_{ck} = 25 \frac{kN}{cm^{2}} </math>
| |
| |-
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| | <math> f_{cd} = \frac{ 0.85 \cdot 25 \frac{kN}{cm^{2}}}{ 1.5 } </math> ||
| |
| |-
| |
| | <math> f_{cd} = 14,2 \frac{kN}{cm^{2}} </math> ||
| |
| |}
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| <br />
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|
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| <br />
| |
|
| |
|
| |
| ::{|
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| | <math> f_{yd} = \frac{ f_{yk}}{\gamma_{s}} </math>||<math>| mit: f_{yk} = 500 \frac{N}{mm^{2}} </math>
| |
| |-
| |
| | || <math>| mit: \gamma_{s} = 1.15 </math>
| |
| |-
| |
| | <math> f_{yd} = \frac{ 50 \frac{kN}{cm^{2}}}{1,15} </math>||
| |
| |-
| |
| | <math> f_{yd} = 43,5 \frac{kN}{cm^{2}} </math>||
| |
| |}
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|
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|
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| ===Biegebemessung===
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|
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| ====Feldbereich des Lauf´s====
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| =====Vorbemessung=====
| |
| ::{|
| |
| | <math> z_{est} = 0,75 \cdot h </math>||<math>| mit: h = h_{L} = 20 cm </math>
| |
| |-
| |
| | <math> z_{est} = 0,75 \cdot 20 cm </math>||
| |
| |-
| |
| | <math> z_{est} = 15 cm </math>||
| |
| |}
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| <br />
| |
|
| |
| <br />
| |
| ::{|
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| | <math> M_{Ed,est} = M_{Ed} - N_{Ed} \cdot z_{s1,est} </math>||<math>| mit: z_{s1,est} = 15 cm </math>
| |
| |-
| |
| | ||<math>| mit: N_{Ed} = 0 </math>
| |
| |-
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| | ||<math>| mit: M_{Ed} = M_{Ed,F} =13.25 \frac{kNm}{m} </math>
| |
| |-
| |
| | <math> M_{Ed,est} = 13.25 \frac{kNm}{m} - 0 kN \cdot 0,15 m </math>||
| |
| |-
| |
| | <math> M_{Ed,est} = 13.25 \frac{kNm}{m} </math>||
| |
| |}
| |
| <br />
| |
|
| |
| <br />
| |
| ::{|
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| | <math> A_{s,est} = \frac{M_{Ed,est}}{z_{s1,est} \cdot f_{yd} } + \frac{N_{Ed}}{f_{yd}} </math>||<math>| mit: z_{s1,est} = 15 cm </math>
| |
| |-
| |
| | ||<math>| mit: f_{yd} = 43,5 \frac{kN}{cm^{2}} </math>
| |
| |-
| |
| | ||<math>| mit: N_{Ed} = 0 </math>
| |
| |-
| |
| | ||<math>| mit: M_{Ed,est} = 13.25 \frac{kNm}{m} </math>
| |
| |-
| |
| | <math> A_{s,est} = \frac{1325 \frac{kNcm}{m}}{15 cm \cdot 43,5 \frac{kN}{cm^{2}} } + \frac{0}{43,55 \frac{kN}{cm^{2}}} </math>||
| |
| |-
| |
| | <math> A_{s,est} \approx 2,03 \frac{kNcm}{m} </math>||
| |
| |}
| |
|
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| gewählt: R257 ø7/15cm, <math>{{a}_{s}}= 2,57 \frac{cm^{2}}{m} </math>
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|
| |
|
| |
| =====Querschnittsgeometrie=====
| |
|
| |
| :<math>c_{v}=\mathrm{max}\begin{cases}
| |
| C_{nom,dur} \\
| |
| C_{nom,b,Bü} \\
| |
| C_{nom,b,L}
| |
| \end{cases}</math><br /><br />
| |
|
| |
| <br />
| |
| ::{|
| |
| | <math> C_{nom,dur} = C_{min,dur} + \Delta C_{dev} </math>|| <math>| mit: C_{min,dur} = 10 mm </math> für XC1
| |
| |-
| |
| | || <math>| mit: \Delta C_{dev} = 10 mm </math>
| |
| |-
| |
| | <math> C_{nom,dur} = 10 mm + 10 mm </math>||
| |
| |-
| |
| | <math> C_{nom,dur} = 20 mm </math>||
| |
| |}
| |
| <br />
| |
|
| |
| <br />
| |
| ::{|
| |
| | <math> C_{nom,b,Bü} = C_{min,b,Bü} + \Delta C_{dev} </math>|| <math>| mit: C_{min,b,Bü} = 0 mm </math>
| |
| |-
| |
| | ||<math>| mit: \Delta C_{dev} = 10 mm </math>
| |
| |-
| |
| | <math> C_{nom,b,Bü} = 0 mm + 10 mm </math>||
| |
| |-
| |
| | <math> C_{nom,b,Bü} = 10 mm </math>||
| |
| |}
| |
|
| |
|
| |
| <br />
| |
| ::{|
| |
| | <math> C_{nom,b,L} = C_{min,b,L} - \varnothing bue + \Delta C_{dev} </math>|| <math>| mit: C_{min,b,L} = 7 mm </math>
| |
| |-
| |
| | ||<math>| mit: \Delta C_{dev} = 10 mm </math>
| |
| |-
| |
| | ||<math>| mit: \varnothing bue = 0 mm </math>
| |
| |-
| |
| | <math> C_{nom,b,L} = 7 mm - 0 mm + \Delta 10 mm </math>||
| |
| |-
| |
| | <math> C_{nom,b,L} = 17 mm </math>||
| |
| |}
| |
|
| |
| :<math>c_{v}=\mathrm{max}\begin{cases}
| |
| 20 mm \\
| |
| 10 mm \\
| |
| 17 mm
| |
| \end{cases}</math><br /><br />
| |
|
| |
| ::{|
| |
| | <math> d_{1} = c_{v} + \varnothing bue + \frac{\varnothing L}{2}</math>|| <math>| mit: c_{v} = 20 mm </math>
| |
| |-
| |
| | || <math>| mit: \varnothing bue = 0 mm </math>
| |
| |-
| |
| | || <math>| mit: \varnothing L = 7 mm </math>
| |
| |-
| |
| | <math> d_{1} = 20 mm + 0 mm + \frac{7 mm}{2} </math>||
| |
| |-
| |
| | <math> d_{1} = 23,5 mm</math>||
| |
| |}
| |
|
| |
| ::{|
| |
| | <math> d = h_{L} - d_{1} </math>|| <math>| mit: d_{1} = 27 mm </math>
| |
| |-
| |
| | || <math>| mit: h_{L} = 200 mm </math>
| |
| |-
| |
| | <math> d = 200 mm - 23,5 mm </math>||
| |
| |-
| |
| | <math> d = 176,5 mm \approx 17,6 cm </math>||
| |
| |}
| |
|
| |
|
| |
|
| |
| =====Bemessung mit dem ω-Verfahren=====
| |
|
| |
| ::{|
| |
| |<math> \mu_{Eds} = \frac{M_{Eds}}{b\cdot d^{2} \cdot f_{cd}} </math>|| <math>| mit: d = 17,6 cm </math>
| |
| |-
| |
| | || <math>| mit: b = 100 cm </math>
| |
| |-
| |
| | || <math>| mit: f_{cd} = 1,42 \frac{kN}{cm^{2}}</math>
| |
| |-
| |
| | || <math>| mit: M_{Ed} = M_{Ed,F} =1325 \frac{kNcm}{m} </math>
| |
| |-
| |
| |<math> \mu_{Eds} = \frac{ 1325 \frac{kNcm}{m} }{100 cm \cdot (17,6 cm)^{2} \cdot 1,42 \frac{kN}{cm^{2}} } </math>||
| |
| |-
| |
| |<math> \mu_{Eds} = 0,03012 </math>||
| |
| |}
| |
| <br />
| |
|
| |
| <br />
| |
|
| |
| ::{|
| |
| |<math> \omega = \omega_{1} + \frac{ \omega_{2} - \omega_{1} } { \mu_{Eds,2} - \mu_{Eds,1}} \cdot ( \mu_{Eds} - \mu_{Eds,1} ) </math>||
| |
| <math>| mit: \omega_{1} = 0,03012 </math>
| |
| |-
| |
| | || <math>| mit: \omega_{2} = 0,0410 </math>
| |
| |-
| |
| | || <math>| mit: \mu_{Eds} = 0,0312 </math>
| |
| |-
| |
| | || <math>| mit: \mu_{Eds,1} = 0,03 </math>
| |
| |-
| |
| | || <math>| mit: \mu_{Eds,2} = 0,04 </math>
| |
| |-
| |
| |<math> \omega = 0,0306 + \frac{ 0,0410 - 0,0306 } { 0,04 - 0,03} \cdot ( 0,03012 - 0,03 ) </math>||
| |
| |-
| |
| |<math> \omega = 0,0307 </math>||
| |
| |}
| |
| <br />
| |
|
| |
| <br />
| |
| ::{|
| |
| |<math> a_{s,1} = \frac{1}{ \sigma_{sd}} \cdot ( \omega \cdot b \cdot d \cdot f_{cd} + N_{Ed} ) </math>|| <math>| mit: \omega = 0,0307 </math>
| |
| |-
| |
| | || <math>| mit: f_{cd} = 1,42 \frac{kN}{cm^{2}} </math>
| |
| |-
| |
| | || <math>| mit: d = 17,3 cm </math>#
| |
| |-
| |
| | || <math>| mit: b = 100 cm </math>
| |
| |-
| |
| | || <math>| mit: N_{Ed} = 0 kN </math>
| |
| |-
| |
| |<math> a_{s,1} = \frac{1}{ 43,5 \frac{kN}{cm^{2}}} \cdot ( 0,0307 \cdot 100 cm \cdot 17,3 cm \cdot 1,42 \frac{kN}{cm^{2}} + 0 kN ) </math>||
| |
| |-
| |
| |<math> a_{s,1} = 1,73 \frac{cm^{2}}{m} </math>||
| |
| |}
| |
|
| |
| gewählt:R257 ø7/15cm, <math>{{a}_{sw}}= 2,57 \frac{cm^{2}}{m} </math>
| |
|
| |
| ====Bereich der Arbeitsfuge Kopfpunkt====
| |
|
| |
| =====Vorbemessung=====
| |
| ::{|
| |
| | <math> z_{est} = 0,75 \cdot h </math>||<math>| mit: h = h_{L} = 20 cm </math>
| |
| |-
| |
| | <math> z_{est} = 0,75 \cdot 20 cm </math>||
| |
| |-
| |
| | <math> z_{est} = 15 cm </math>||
| |
| |}
| |
| <br />
| |
|
| |
| <br />
| |
| ::{|
| |
| | <math> M_{Ed,est} = M_{Ed} - N_{Ed} \cdot z_{s1,est} </math>||<math>| mit: z_{s1,est} = 15 cm </math>
| |
| |-
| |
| | ||<math>| mit: N_{Ed} = extr n = - 11,56 \frac{kN}{m}</math>
| |
| |-
| |
| | ||<math>| mit: M_{Ed} = M_{Ed,S} =6.63 \frac{kNm}{m} </math>
| |
| |-
| |
| | <math> M_{Ed,est} = 6.63 \frac{kNm}{m} - (- 11,56 \frac{kN}{m}) \cdot 0,15 m </math>||
| |
| |-
| |
| | <math> M_{Ed,est} = 8.36 \frac{kNm}{m}</math>||
| |
| |}
| |
| <br />
| |
|
| |
| <br />
| |
| ::{|
| |
| | <math> a_{s,est} = \frac{M_{Ed,est}}{z_{s1,est} \cdot f_{yd} } + \frac{N_{Ed}}{f_{yd}} </math>||<math>| mit: z_{s1,est} = 15 cm </math>
| |
| |-
| |
| | ||<math>| mit: f_{yd} = 43,5 \frac{kN}{cm^{2}} </math>
| |
| |-
| |
| | ||<math>| mit: N_{Ed} = extr n = - 11,56 \frac{kN}{m} </math>
| |
| |-
| |
| | ||<math>| mit: M_{Ed,est} = 6.63 \frac{kNm}{m} </math>
| |
| |-
| |
| | <math> a_{s,est} = \frac{8.36 \frac{kNm}{m}}{15 cm \cdot 43,5 \frac{kN}{cm^{2}} } + \frac{- 11,56 \frac{kN}{m}}{43,5 \frac{kN}{cm^{2}}} </math>||
| |
| |-
| |
| | <math> a_{s,est} \approx 1,02 \frac{kNcm}{m} </math>||
| |
| |}
| |
|
| |
| gewählt: R188 ø6/15cm, <math>{{a}_{s}}= 1,88 \frac{cm^{2}}{m} </math>
| |
|
| |
|
| |
| =====Querschnittsgeometrie=====
| |
|
| |
| :<math>c_{v}=\mathrm{max}\begin{cases}
| |
| C_{nom,dur} \\
| |
| C_{nom,b,Bü} \\
| |
| C_{nom,b,L}
| |
| \end{cases}</math><br /><br />
| |
|
| |
| <br />
| |
| ::{|
| |
| | <math> C_{nom,dur} = C_{min,dur} + \Delta C_{dev} </math>|| <math>| mit: C_{min,dur} = 10 mm </math> für XC1
| |
| |-
| |
| | || <math>| mit: \Delta C_{dev} = 10 mm </math>
| |
| |-
| |
| | <math> C_{nom,dur} = 10 mm + 10 mm </math>||
| |
| |-
| |
| | <math> C_{nom,dur} = 20 mm </math>||
| |
| |}
| |
| <br />
| |
|
| |
| <br />
| |
| ::{|
| |
| | <math> C_{nom,b,Bü} = C_{min,b,Bü} + \Delta C_{dev} </math>|| <math>| mit: C_{min,b,Bü} = 0 mm </math>
| |
| |-
| |
| | ||<math>| mit: \Delta C_{dev} = 10 mm </math>
| |
| |-
| |
| | <math> C_{nom,b,Bü} = 0 mm + 10 mm </math>||
| |
| |-
| |
| | <math> C_{nom,b,Bü} = 10 mm </math>||
| |
| |}
| |
|
| |
|
| |
| <br />
| |
| ::{|
| |
| | <math> C_{nom,b,L} = C_{min,b,L} - \varnothing bue + \Delta C_{dev} </math>|| <math>| mit: C_{min,b,L} = 7 mm </math>
| |
| |-
| |
| | ||<math>| mit: \Delta C_{dev} = 10 mm </math>
| |
| |-
| |
| | ||<math>| mit: \varnothing bue = 0 mm </math>
| |
| |-
| |
| | <math> C_{nom,b,L} = 6 mm - 0 mm + \Delta 10 mm </math>||
| |
| |-
| |
| | <math> C_{nom,b,L} = 16 mm </math>||
| |
| |}
| |
|
| |
| :<math>c_{v}=\mathrm{max}\begin{cases}
| |
| 20 mm \\
| |
| 10 mm \\
| |
| 16 mm
| |
| \end{cases}</math><br /><br />
| |
|
| |
| ::{|
| |
| | <math> d_{1} = c_{v} + \varnothing bue + \frac{\varnothing L}{2}</math>|| <math>| mit: c_{v} = 20 mm </math>
| |
| |-
| |
| | || <math>| mit: \varnothing bue = 0 mm </math>
| |
| |-
| |
| | || <math>| mit: \varnothing L = 6 mm </math>
| |
| |-
| |
| | <math> d_{1} = 20 mm + 0 mm + \frac{6 mm}{2} </math>||
| |
| |-
| |
| | <math> d_{1} = 23,0 mm</math>||
| |
| |}
| |
|
| |
| ::{|
| |
| | <math> d = h_{L} - d_{1} </math>|| <math>| mit: d_{1} = 23 mm </math>
| |
| |-
| |
| | || <math>| mit: h_{L} = 200 mm </math>
| |
| |-
| |
| | <math> d = 200 mm - 23,0 mm </math>||
| |
| |-
| |
| | <math> d = 177 mm = 17,7 cm </math>||
| |
| |}
| |
|
| |
| =====Bemessung mit dem ω-Verfahren=====
| |
|
| |
|
| |
| ::{|
| |
| |<math> z_{s} = d - \frac{h_{L} }{2} </math>|| <math>| mit: d = 17,7 cm </math>
| |
| |-
| |
| | || <math>| mit: h_{L} = 20 cm </math>
| |
| |-
| |
| |<math> z_{s} = 17,7 cm - \frac{20 cm}{2} </math>||
| |
| |-
| |
| |<math> z_{s} = 7,7 cm </math>||
| |
| |}
| |
|
| |
| ::{|
| |
| |<math> M_{Eds} = M_{Ed,S} - extr n \cdot z_{s} </math>|| <math>| mit: z_{s} = 7,7 cm </math>
| |
| |-
| |
| | || <math>| extr n = (-11,56) \frac{kN}{m} </math>
| |
| |-
| |
| | ||<math>| mit: M_{Ed,S} =663 \frac{kNcm}{m} </math>
| |
| |-
| |
| |<math> M_{Eds} = 663 \frac{kNcm}{m} - (-11,56) \frac{kN}{m} \cdot 7,7 cm </math>||
| |
| |-
| |
| |<math> M_{Eds} = 752,01 \frac{kNcm}{m} </math>||
| |
| |}
| |
|
| |
|
| |
| ::{|
| |
| |<math> \mu_{Eds} = \frac{M_{Eds}}{b\cdot d^{2} \cdot f_{cd}} </math>|| <math>| mit: d = 17,7 cm </math>
| |
| |-
| |
| | || <math>| mit: b = 100 cm </math>
| |
| |-
| |
| | || <math>| mit: f_{cd} = 1,42 \frac{kN}{cm^{2}}</math>
| |
| |-
| |
| | || <math>| mit: M_{Eds} = 752,01 \frac{kNcm}{m} </math>
| |
| |-
| |
| |<math> \mu_{Eds} = \frac{752,01 \frac{kNcm}{m}}{100 cm \cdot (17,7 cm)^{2} \cdot 1,42 \frac{kN}{cm^{2}} } </math>||
| |
| |-
| |
| |<math> \mu_{Eds} = 0,0169 </math>||
| |
| |}
| |
| <br />
| |
|
| |
| <br />
| |
|
| |
| ::{|
| |
| |<math> \omega = \omega_{1} + \frac{ \omega_{2} - \omega_{1} } { \mu_{Eds,2} - \mu_{Eds,1}} \cdot ( \mu_{Eds} - \mu_{Eds,1} ) </math>||
| |
| <math>| mit: \omega_{1} = 0,0101 </math>
| |
| |-
| |
| | || <math>| mit: \omega_{2} = 0,0203 </math>
| |
| |-
| |
| | || <math>| mit: \mu_{Eds} = 0,0169 </math>
| |
| |-
| |
| | || <math>| mit: \mu_{Eds,1} = 0,01 </math>
| |
| |-
| |
| | || <math>| mit: \mu_{Eds,2} = 0,02 </math>
| |
| |-
| |
| |<math> \omega = 0,0101 + \frac{ 0,0203 - 0,0101 } { 0,02 - 0,01 } \cdot ( 0,0169 - 0,01 ) </math>||
| |
| |-
| |
| |<math> \omega = 0,01714 </math>||
| |
| |}
| |
| <br />
| |
|
| |
| <br />
| |
| ::{|
| |
| |<math> a_{s,1} = \frac{1}{ \sigma_{sd}} \cdot ( \omega \cdot b \cdot d \cdot f_{cd} + N_{Ed} ) </math>|| <math>| mit: \omega = 0,01714 </math>
| |
| |-
| |
| | || <math>| mit: f_{cd} = 1,42 \frac{kN}{cm^{2}} </math>
| |
| |-
| |
| | || <math>| mit: d = 17,7 cm </math>
| |
| |-
| |
| | || <math>| mit: b = 100 cm </math>
| |
| |-
| |
| | || <math>| mit: N_{Ed} = extr n = (-11,56) \frac{kN}{m} </math>
| |
| |-
| |
| |<math> a_{s,1} = \frac{1}{ 43,5 \frac{kN}{cm^{2}}} \cdot ( 0,0151 \cdot 100 cm \cdot 17,7 cm \cdot 1,42 \frac{kN}{cm^{2}} + (-11,56) \frac{kN}{m} ) </math>||
| |
| |-
| |
| |<math> a_{s,1} = 0,61 \frac{cm^{2}}{m} </math>||
| |
| |}
| |
|
| |
| gewählt: R188 ø6/15cm, <math>{{a}_{s}}= 1,88 \frac{cm^{2}}{m} </math>
| |
|
| |
| ====Bereich der Arbeitsfuge Fußpunkt====
| |
| =====Vorbemessung=====
| |
| ::{|
| |
| | <math> z_{est} = 0,75 \cdot h </math>||<math>| mit: h = h_{L} = 20 cm </math>
| |
| |-
| |
| | <math> z_{est} = 0,75 \cdot 20 cm </math>||
| |
| |-
| |
| | <math> z_{est} = 15 cm </math>||
| |
| |}
| |
| <br />
| |
|
| |
| <br />
| |
| ::{|
| |
| | <math> M_{Ed,est} = M_{Ed} - N_{Ed} \cdot z_{s1,est} </math>||<math>| mit: z_{s1,est} = 15 cm </math>
| |
| |-
| |
| | ||<math>| mit: N_{Ed} = extr n = 11,56 \frac{kN}{m}</math>
| |
| |-
| |
| | ||<math>| mit: M_{Ed} = M_{Ed,S} =6.63 \frac{kNm}{m} </math>
| |
| |-
| |
| | <math> M_{Ed,est} = 6.63 \frac{kNm}{m} - 11,56 \frac{kN}{m} \cdot 0,15 m </math>||
| |
| |-
| |
| | <math> M_{Ed,est} = 4.9 \frac{kNm}{m}</math>||
| |
| |}
| |
| <br />
| |
|
| |
| <br />
| |
| ::{|
| |
| | <math> a_{s,est} = \frac{M_{Ed,est}}{z_{s1,est} \cdot f_{yd} } + \frac{N_{Ed}}{f_{yd}} </math>||<math>| mit: z_{s1,est} = 15 cm </math>
| |
| |-
| |
| | ||<math>| mit: f_{yd} = 43,5 \frac{kN}{cm^{2}} </math>
| |
| |-
| |
| | ||<math>| mit: N_{Ed} = extr n = 11,56 \frac{kN}{m} </math>
| |
| |-
| |
| | ||<math>| mit: M_{Ed,est} = 4.9 \frac{kNm}{m} </math>
| |
| |-
| |
| | <math> a_{s,est} = \frac{490 \frac{kNcm}{m}}{15 cm \cdot 43,5 \frac{kN}{cm^{2}} } + \frac{11,56 \frac{kN}{m}}{43,5 \frac{kN}{cm^{2}}} </math>||
| |
| |-
| |
| | <math> a_{s,est} \approx 1,02 \frac{kNcm}{m} </math>||
| |
| |}
| |
|
| |
| gewählt: R188 ø6/15cm, <math>{{a}_{s}}= 1,88 \frac{cm^{2}}{m} </math>
| |
|
| |
| =====Querschnittsgeometrie=====
| |
|
| |
| :<math>c_{v}=\mathrm{max}\begin{cases}
| |
| C_{nom,dur} \\
| |
| C_{nom,b,Bü} \\
| |
| C_{nom,b,L}
| |
| \end{cases}</math><br /><br />
| |
|
| |
| <br />
| |
| ::{|
| |
| | <math> C_{nom,dur} = C_{min,dur} + \Delta C_{dev} </math>|| <math>| mit: C_{min,dur} = 10 mm </math> für XC1
| |
| |-
| |
| | || <math>| mit: \Delta C_{dev} = 10 mm </math>
| |
| |-
| |
| | <math> C_{nom,dur} = 10 mm + 10 mm </math>||
| |
| |-
| |
| | <math> C_{nom,dur} = 20 mm </math>||
| |
| |}
| |
| <br />
| |
|
| |
| <br />
| |
| ::{|
| |
| | <math> C_{nom,b,Bü} = C_{min,b,Bü} + \Delta C_{dev} </math>|| <math>| mit: C_{min,b,Bü} = 0 mm </math>
| |
| |-
| |
| | ||<math>| mit: \Delta C_{dev} = 10 mm </math>
| |
| |-
| |
| | <math> C_{nom,b,Bü} = 0 mm + 10 mm </math>||
| |
| |-
| |
| | <math> C_{nom,b,Bü} = 10 mm </math>||
| |
| |}
| |
|
| |
|
| |
| <br />
| |
| ::{|
| |
| | <math> C_{nom,b,L} = C_{min,b,L} - \varnothing bue + \Delta C_{dev} </math>|| <math>| mit: C_{min,b,L} = 7 mm </math>
| |
| |-
| |
| | ||<math>| mit: \Delta C_{dev} = 10 mm </math>
| |
| |-
| |
| | ||<math>| mit: \varnothing bue = 0 mm </math>
| |
| |-
| |
| | <math> C_{nom,b,L} = 6 mm - 0 mm + \Delta 10 mm </math>||
| |
| |-
| |
| | <math> C_{nom,b,L} = 16 mm </math>||
| |
| |}
| |
|
| |
| :<math>c_{v}=\mathrm{max}\begin{cases}
| |
| 20 mm \\
| |
| 10 mm \\
| |
| 16 mm
| |
| \end{cases}</math><br /><br />
| |
|
| |
| ::{|
| |
| | <math> d_{1} = c_{v} + \varnothing bue + \frac{\varnothing L}{2}</math>|| <math>| mit: c_{v} = 20 mm </math>
| |
| |-
| |
| | || <math>| mit: \varnothing bue = 0 mm </math>
| |
| |-
| |
| | || <math>| mit: \varnothing L = 6 mm </math>
| |
| |-
| |
| | <math> d_{1} = 20 mm + 0 mm + \frac{6 mm}{2} </math>||
| |
| |-
| |
| | <math> d_{1} = 23,0 mm</math>||
| |
| |}
| |
|
| |
| ::{|
| |
| | <math> d = h_{L} - d_{1} </math>|| <math>| mit: d_{1} = 23 mm </math>
| |
| |-
| |
| | || <math>| mit: h_{L} = 200 mm </math>
| |
| |-
| |
| | <math> d = 200 mm - 23,0 mm </math>||
| |
| |-
| |
| | <math> d = 177 mm = 17,7 cm </math>||
| |
| |}
| |
|
| |
| =====Bemessung mit dem ω-Verfahren=====
| |
|
| |
|
| |
| ::{|
| |
| |<math> z_{s} = d - \frac{h_{L} }{2} </math>|| <math>| mit: d = 17,7 cm </math>
| |
| |-
| |
| | || <math>| mit: h_{L} = 20 cm </math>
| |
| |-
| |
| |<math> z_{s} = 17,7 cm - \frac{20 cm}{2} </math>||
| |
| |-
| |
| |<math> z_{s} = 7,7 cm </math>||
| |
| |}
| |
|
| |
| ::{|
| |
| |<math> M_{Eds} = M_{Ed,S} - extr n \cdot z_{s} </math>|| <math>| mit: z_{s} = 7,7 cm </math>
| |
| |-
| |
| | || <math>| extr n = 11,56 \frac{kN}{m} </math>
| |
| |-
| |
| | ||<math>| mit: M_{Ed,S} =663 \frac{kNcm}{m} </math>
| |
| |-
| |
| |<math> M_{Eds} = 663 \frac{kNcm}{m} - 11,56 \frac{kN}{m} \cdot 7,7 cm </math>||
| |
| |-
| |
| |<math> M_{Eds} = 573.988 \frac{kNcm}{m} </math>||
| |
| |}
| |
|
| |
|
| |
| ::{|
| |
| |<math> \mu_{Eds} = \frac{M_{Eds}}{b\cdot d^{2} \cdot f_{cd}} </math>|| <math>| mit: d = 17,7 cm </math>
| |
| |-
| |
| | || <math>| mit: b = 100 cm </math>
| |
| |-
| |
| | || <math>| mit: f_{cd} = 1,42 \frac{kN}{cm^{2}}</math>
| |
| |-
| |
| | || <math>| mit: M_{Eds} = 573.99 \frac{kNcm}{m} </math>
| |
| |-
| |
| |<math> \mu_{Eds} = \frac{ 573.99 \frac{kNcm}{m}}{100 cm \cdot (17,7 cm)^{2} \cdot 1,42 \frac{kN}{cm^{2}} } </math>||
| |
| |-
| |
| |<math> \mu_{Eds} = 0,0129 </math>||
| |
| |}
| |
| <br />
| |
|
| |
| <br />
| |
|
| |
| ::{|
| |
| |<math> \omega = \omega_{1} + \frac{ \omega_{2} - \omega_{1} } { \mu_{Eds,2} - \mu_{Eds,1}} \cdot ( \mu_{Eds} - \mu_{Eds,1} ) </math>||
| |
| <math>| mit: \omega_{1} = 0,0101 </math>
| |
| |-
| |
| | || <math>| mit: \omega_{2} = 0,0203 </math>
| |
| |-
| |
| | || <math>| mit: \mu_{Eds} = 0,0129 </math>
| |
| |-
| |
| | || <math>| mit: \mu_{Eds,1} = 0,01 </math>
| |
| |-
| |
| | || <math>| mit: \mu_{Eds,2} = 0,02 </math>
| |
| |-
| |
| |<math> \omega = 0,0101 + \frac{ 0,0203 - 0,0101 } { 0,02 - 0,01 } \cdot ( 0,0169 - 0,01 ) </math>||
| |
| |-
| |
| |<math> \omega = 0,01306 </math>||
| |
| |}
| |
| <br />
| |
|
| |
| <br />
| |
| ::{|
| |
| |<math> a_{s,1} = \frac{1}{ \sigma_{sd}} \cdot ( \omega \cdot b \cdot d \cdot f_{cd} + N_{Ed} ) </math>|| <math>| mit: \omega = 0,01306 </math>
| |
| |-
| |
| | || <math>| mit: f_{cd} = 1,42 \frac{kN}{cm^{2}} </math>
| |
| |-
| |
| | || <math>| mit: d = 17,7 cm </math>
| |
| |-
| |
| | || <math>| mit: b = 100 cm </math>
| |
| |-
| |
| | || <math>| mit: N_{Ed} = extr n = 11,56 \frac{kN}{m} </math>
| |
| |-
| |
| |<math> a_{s,1} = \frac{1}{ 43,5 \frac{kN}{cm^{2}}} \cdot ( 0,0151 \cdot 100 cm \cdot 17,7 cm \cdot 1,42 \frac{kN}{cm^{2}} + 11,56 \frac{kN}{m} ) </math>||
| |
| |-
| |
| |<math> a_{s,1} = 1,02 \frac{cm^{2}}{m} </math>||
| |
| |}
| |
|
| |
| gewählt: R188 ø6/15cm, <math>{{a}_{s}}= 1,88 \frac{cm^{2}}{m} </math>
| |
|
| |
| ===Querkraftbemessung===
| |
|
| |
| ====Bauteile ohne Querkraftbewehrung====
| |
|
| |
| ::{|
| |
| |<math> V_{Ed} = C_{Ed} \cdot cos( \alpha ) </math>||<math>| mit: C_{Ed} = 22.86 \frac{kN}{m} </math>
| |
| |-
| |
| |||<math>| mit: \alpha = 30,38° </math>
| |
| |-
| |
| |<math> V_{Ed} = 22.86 \frac{kN}{m} \cdot cos( 30,38° ) </math>||
| |
| |-
| |
| |<math> V_{Ed} = 19.72 \frac{kN}{m} </math>||
| |
| |}
| |
|
| |
|
| |
|
| |
| ::{|
| |
| |<math> C_{Rdc} = \frac{0,15}{\gamma_{c}} </math>||<math>| mit: \gamma_{c} = 1,5</math>
| |
| |-
| |
| |<math> C_{Rdc} = \frac{0,15}{1,5} </math>||
| |
| |-
| |
| |<math> C_{Rdc} = 0,1 </math>||
| |
| |}
| |
|
| |
|
| |
|
| |
|
| |
|
| |
| ::{|
| |
| |<math> k=1+\sqrt{\frac{200}{d}} </math>|| <math>\begin{cases}
| |
| \ge 1,0 \\
| |
| \le 2,0
| |
| \end{cases}</math>
| |
| |-
| |
| | || <math>| mit: d = 177 mm </math>
| |
| |-
| |
| |<math>k=1+\sqrt{\frac{200}{177}}</math> ||
| |
| |-
| |
| |<math>k=2,06</math> || <math>\begin{cases}
| |
| \ge 1,0 \\
| |
| \ge 2,0
| |
| \end{cases}</math>
| |
| |-
| |
| |<math>k=2</math> ||
| |
| |}
| |
|
| |
|
| |
|
| |
|
| |
|
| |
|
| |
| ::{|
| |
| |<math> \rho_{l} = \frac{A_{sl}}{b_{w} \cdot d} </math>||<math> \le 0,02 </math>
| |
| |-
| |
| | || <math>| mit: A_{sl} = 1,88 \frac{cm^{2}}{m} </math>
| |
| |-
| |
| | || <math>| mit: b_{w} = 100 cm </math>
| |
| |-
| |
| | || <math>| mit: d = 17,7 cm</math>
| |
| |-
| |
| |<math> \rho_{l} = \frac{1,88 \frac{cm^{2}}{m} }{100 cm \cdot 17,7 cm }</math>||
| |
| |-
| |
| |<math> \rho_{l} =1,06 \cdot 10^{-3} </math>||<math> \le 0,02 </math>
| |
| |}
| |
|
| |
|
| |
|
| |
| ::{|
| |
| |<math> A_{c} = A - A_{s} </math>|| <math>| mit: A = b_{w} \cdot h_{L} = b_{w} \cdot h_{L} =1000 mm \cdot 200 mm = 200000 </math>
| |
| |-
| |
| | || <math>| mit: A_{s, R188, R257} = 188 \frac{mm^{2}}{m} + 257 \frac{mm^{2}}{m} = 445 \frac{mm^{2}}{m} </math>
| |
| |-
| |
| |<math> A_{c} = 200000 mm^{2} - 445 mm^{2} </math>||
| |
| |-
| |
| |<math> A_{c} = 199555 mm^{2} </math>||
| |
| |}
| |
|
| |
|
| |
|
| |
| ::{|
| |
| |<math> \sigma_{cp}=\frac{ N_{Ed} }{ A_{c} } </math> || <math>| mit: N_{Ed} = extr n = \pm 11560 \frac{N}{m} </math>
| |
| |-
| |
| | || <math>| mit: A_{c} = 199555 mm^{2}</math>
| |
| |-
| |
| |<math> \sigma_{cp}=\frac{ 11560 \frac{N}{m} }{ 199555 mm^{2} } </math> ||
| |
| |-
| |
| |<math> \sigma_{cp} \approx 0,06 \frac{N}{m \cdot mm^{2}} </math> ||
| |
| |}
| |
|
| |
|
| |
|
| |
|
| |
| ::{|
| |
| |<math> V_{Rd,c}= \left[ C_{Rdc} \cdot k \cdot \left( 100 \cdot \rho_{l} \cdot f_{ck} \right)^{\frac{1}{3}} + 0,12 \cdot \sigma_{cp} \right] \cdot b_{w} \cdot d </math> || <math>| mit: \sigma_{cp} = 0,06 \frac{N}{ mm^{2}} </math>
| |
| |-
| |
| | ||<math>| mit: \rho_{l} =1,06 \cdot 10^{-3} </math>
| |
| |-
| |
| | ||<math>| mit: k=2 </math>
| |
| |-
| |
| | ||<math>| mit: C_{Rdc} = 0,1 </math>
| |
| |-
| |
| | ||<math>| mit: f_{ck} = 25 \frac{kN}{cm^{2}} </math>
| |
| |-
| |
| | ||<math>| mit: b_{w} = 1000 mm </math>
| |
| |-
| |
| | ||<math>| mit: d = 177 mm </math>
| |
| |-
| |
| |<math> V_{Rd,c}= \left[ 0,1 \cdot 2 \cdot \left( 100 \cdot 1,06 \cdot 10^{-3} \cdot 25 \frac{N}{mm^{2}} \right)^{\frac{1}{3}} + 0,12 \cdot 0,06 \frac{N}{mm^{2}} \right] \cdot 1000 mm \cdot 177 mm </math> ||
| |
| |-
| |
| |<math> V_{Rd,c}= 50261 N</math> ||<math>\ge V_{Ed}</math>
| |
| |-
| |
| |<math> V_{Rd,c}= 50,26 kN </math> ||<math>\ge 19.72 kN </math>
| |
| |}
| |
|
| |
| Weitere Bemessung Hinsichtlich der Querkraft für dieses Bauteil nicht erforderlich da der Nachweis erfüllt ist .
| |
|
| |
| == Bemessung im Grenzzustand der Gebrauchstauglichkeit==
| |
| === Begrenzung der Biegeschlankheit des Treppenlauf===
| |
|
| |
| :{|
| |
| |<math> \frac{l}{d} \leq K \left[ 11 + 1,5 \cdot \sqrt{ f_{ck}} \cdot \frac{\rho_{0} }{\rho} + 3,2 \cdot \sqrt{ f_{ck}} \cdot (\frac{\rho_{0} }{\rho} -1 )^{3/2} \right] \leq \left( \frac{l}{d} \right)_{max} </math>|| ||| für: <math> \rho \le \rho_{0} </math>
| |
| |-
| |
| |<math> \frac{l}{d} \leq K \left[ 11 + 1,5 \sqrt{ f_{ck}} \cdot \frac{\rho_{0} }{\rho - \rho_{'}} + \frac{1}{12} \cdot \sqrt{ f_{ck}} \cdot (\frac{ \rho_{'}}{\rho_{0}})^{1/2} \right] \leq \left( \frac{l}{d} \right)_{max} </math>|| |||für: <math> \rho > \rho_{0} </math>
| |
| |}
| |
|
| |
|
| |
| ::{|
| |
| |<math> \rho_{0} = 10^{-3} \cdot \sqrt{ f_{ck}} </math>||<math>| mit: f_{ck} = 25 \frac{N}{mm^{2}} </math>
| |
| |-
| |
| |<math> \rho_{0} = 10^{-3} \cdot \sqrt{25 \frac{N}{mm^{2}}} </math>||
| |
| |-
| |
| |<math> \rho_{0} = 5 \cdot 10^{-3} </math>||
| |
| |}
| |
|
| |
|
| |
| ::{|
| |
| |<math> \rho = \frac{A_{s1}}{b \cdot d} </math>||<math>| mit: A_{s1} = 2,57 \frac{cm^{2}}{m} </math>
| |
| |-
| |
| | ||<math>| mit: d = 17,3 cm </math>
| |
| |-
| |
| | ||<math>| mit: b = 100 cm </math>
| |
| |-
| |
| |<math> \rho = \frac{2,57 \frac{cm^{2}}{m} }{100 cm \cdot 17,3 cm} </math>||
| |
| |-
| |
| |<math> \rho = 1,49 \cdot 10^{-3} </math>|| <math>\le \rho_{0} </math>
| |
| |}
| |
|
| |
|
| |
| :{|
| |
| |<math> \frac{l}{d} \leq K \left[ 11 + 1,5 \cdot \sqrt{ f_{ck}} \cdot \frac{\rho_{0} }{\rho} + 3,2 \cdot \sqrt{ f_{ck}} \cdot (\frac{\rho_{0} }{\rho} -1 )^{3/2} \right] \leq \left( \frac{l}{d} \right)_{max} </math>||
| |
| |}
| |
|
| |
|
| |
| :{|
| |
| |<math> \left( \frac{l}{d} \right)_{vorh} = \frac{l}{d} </math>||<math>| mit: d = 0,173 m </math>
| |
| |-
| |
| | ||<math>|mit: l = \sqrt{ (8 \cdot 0,29m)^{2} + (8 \cdot 0,17m)^{2} } = 2,69 m </math>
| |
| |-
| |
| |<math> \left( \frac{l}{d} \right)_{vorh} = \frac{2,69 m}{0,173 m} </math>||
| |
| |-
| |
| |<math> \left( \frac{l}{d} \right)_{vorh} = 15,55 </math>||
| |
| |}
| |
|
| |
|
| |
| :{|
| |
| |<math> \frac{l}{d}_{zul} = K \left[ 11 + 1,5 \cdot \sqrt{ f_{ck}} \cdot \frac{\rho_{0} }{\rho} + 3,2 \cdot \sqrt{ f_{ck}} \cdot (\frac{\rho_{0} }{\rho} -1 )^{3/2} \right] </math>||<math>| mit: K_{Innenfeld} = 1,5 </math>
| |
| |-
| |
| | ||<math>| mit: f_{ck} = 25 \frac{N}{mm^{2}} </math>
| |
| |-
| |
| | ||<math>| mit: \rho = 1,49 \cdot 10^{-3} </math>
| |
| |-
| |
| | ||<math>| mit: \rho_{0} = 5 \cdot 10^{-3} </math>
| |
| |-
| |
| |<math> \frac{l}{d}_{zul} = 1,5 \left[ 11 + 1,5 \cdot \sqrt{ 25 \frac{N}{mm^{2}}} \cdot \frac{5 \cdot 10^{-3} }{1,49 \cdot 10^{-3}} + 3,2 \cdot \sqrt{ 25 \frac{N}{mm^{2}}} \cdot (\frac{5 \cdot 10^{-3}}{1,49 \cdot 10^{-3}} -1 )^{3/2} \right] </math>||
| |
| |-
| |
| |<math> \frac{l}{d}_{zul} = 155.64 </math>||<math> \ge \left( \frac{l}{d} \right)_{vorh} </math>
| |
| |}
| |
|
| |
|
| |
|
| |
|
| |
|
| |
|
| |
| :<math>\left( \frac{l}{d} \right)_{max} \le \begin{cases}
| |
| K \cdot 35 \\
| |
| K^{2} \cdot \frac{150}{l}
| |
| \end{cases}</math><br /><br />
| |
|
| |
|
| |
|
| |
| :{|
| |
| |<math> \left( \frac{l}{d} \right)_{max} K \cdot 35 </math>||<math>| mit: K_{Innenfeld} = 1,5 </math>
| |
| |-
| |
| |<math> \left( \frac{l}{d} \right)_{max} = 1,5 \cdot 35 </math>||
| |
| |-
| |
| |<math> \left( \frac{l}{d} \right)_{max} = 52,5 </math>||<math> \ge \left( \frac{l}{d} \right)_{vorh} </math>
| |
| |}
| |
|
| |
|
| |
|
| |
|
| |
|
| |
| :{|
| |
| |<math> \left( \frac{l}{d} \right)_{max} = K^{2} \cdot \frac{150}{l} </math>||<math>| mit: K_{Innenfeld} = 1,5 </math>
| |
| |-
| |
| | ||<math>| mit: l = \sqrt{ (8 \cdot 0,29m)^{2} + (8 \cdot 0,17m)^{2} } = 2,69 m </math>
| |
| |-
| |
| |<math> \left( \frac{l}{d} \right)_{max} = 1,5^{2} \cdot \frac{150}{2,69 m} </math>||
| |
| |-
| |
| |<math> \left( \frac{l}{d} \right)_{max} = 125, 46</math>||<math> \ge \left( \frac{l}{d} \right)_{vorh} </math>
| |
| |}
| |
|
| |
| = Lösung des Zwischenpodest=
| |
|
| |
| ==Berechnung und Bemessung des Zwischenpodest==
| |
| Das Zwischenpodest wird in diesem beispiel als Dreiseitig gelagert und somit zweiseitig gespannt betrachtet
| |
|
| |
| ==Einwirkungen==
| |
| ===Teilsicherheiten ===
| |
|
| |
| ::{|
| |
| |<math> \gamma_\mathrm{Q} =1,50 </math> ||
| |
| |-
| |
| |<math> \gamma_\mathrm{G} =1,35 </math> ||
| |
| |}
| |
| === Ständige===
| |
|
| |
|
| |
| :{|
| |
| | <math> g_{d}= g_{k} \cdot \gamma_\mathrm{G}</math> ||
| |
| |}
| |
|
| |
|
| |
| ::{|
| |
| | <math> g_{k} = h \cdot \gamma_{1} + \gamma_{G_{s}=1,5 cm} + N_{s} \cdot \gamma_{Naturstein} + \gamma_{Estrich} \cdot d_{Estrich} </math>|| <math>| mit: h_{P}= 20cm </math>
| |
| |-
| |
| | || <math>| mit: \gamma_{1}= 25 \frac{kN}{m^{3}} </math>
| |
| |-
| |
| | || <math>| mit: \gamma_{G_{s}=1,5 cm} = 0,18 \frac{kN}{m^{2}} </math>
| |
| |-
| |
| | || <math>| mit: \gamma_{Naturstein} = 0,3 \frac{\frac{kN}{m^{2}}}{cm} </math>
| |
| |-
| |
| | || <math>| mit: N_{s} = 3 cm </math>
| |
| |-
| |
| | || <math>| mit: \gamma_{Estrich} = 22 \frac{\frac{kN}{m^{2}}}{cm} </math>
| |
| |-
| |
| | || <math>| mit: d_{Estrich} = 4 cm </math>
| |
| |-
| |
| | || <math>| mit: \gamma_{Trittschalldämmung} = 0,01 \frac{\frac{kN}{m^{2}}}{cm} </math>
| |
| |-
| |
| | || <math>| mit: d_{Trittschalldämmung} = 4 cm </math>
| |
| |-
| |
| |<math> g_{k} = 0,20 m \cdot 25 \frac{kN}{m^{3}} + 0,18 \frac{kN}{m^{2}} + 3 cm \cdot 0,3 \frac{\frac{kN}{m^{2}}}{cm} + 0,01 \frac{\frac{kN}{m^{2}}}{cm} \cdot 4 cm + 0,22 \frac{\frac{kN}{m^{2}}}{cm} \cdot 4 cm </math>||
| |
| |-
| |
| |<math> g_{k} = 7 \frac{kN}{m^{2}} </math>||
| |
| |}
| |
|
| |
| :{|
| |
| | <math> g_{d}= g_{k} \cdot \gamma_\mathrm{G}</math> || <math>| mit: \gamma_\mathrm{G} = 1,5 </math>
| |
| |-
| |
| | || <math>| mit:g_{k} = 7 \frac{kN}{m^{2}} </math>
| |
| |-
| |
| | <math> g_{d}= 7 \frac{kN}{m^{2}} \cdot 1,5 </math> ||
| |
| |-
| |
| | <math> g_{d}=10,5 \frac{kN}{m^{2}} </math> ||
| |
| |}
| |
|
| |
| === Veränderliche===
| |
|
| |
| {| class="wikitable"
| |
| |+style="text-align:left;"|Lotrechte Nutzlasten für Treppen <ref Name = "HandbuchEC1" group="F">Handbuch Eurocode 1 Einwirkungen – Band 1 Grundlagen, Nutz- und Eigenlasten, Brandeinwirkungen, Schnee-, Wind-, Temperaturlasten Ausgabedatum: 06.2012 </ref>
| |
| |rowspan="2"|
| |
| |colspan="2"|1
| |
| |2
| |
| |3
| |
| |4
| |
| |5
| |
| |-
| |
| !colspan="2"|Kategorie
| |
| !Nutzung
| |
| !Beispiele
| |
| !<math> q_{k} [ \frac{kN}{m^{2}}] </math>
| |
| !<math> Q_{k} [kN] </math>
| |
| |-
| |
| |19
| |
| |rowspan="3" style="background:#FFFF40"|T
| |
| |style="background:#FFFF40"|T1
| |
| |rowspan="3" style="background:#FFFF40"|Treppen und Treppenpodeste
| |
| |style="background:#FFFF40"|Treppen und Treppenpodeste in Wohngebäuden, Bürogebäuden und von Arztpraxen ohne schweres Gerät
| |
| |style="background:#FFFF40"|3,0
| |
| |2,0
| |
| |-
| |
| |20
| |
| |T2
| |
| |alle Treppen und Treppenpodeste, die nicht in TI oder T3 eingeordnet werden können
| |
| |5,0
| |
| |2,0
| |
| |-
| |
| |21
| |
| |T3
| |
| |Zugänge und Treppen von Tribünen ohne feste Sitzplätze, die als Fluchtwege dienen
| |
| |7,5
| |
| |3,0
| |
| |}
| |
|
| |
| ::{|
| |
| |<math> \underline{ q_{k} = 3,0 \frac{kN}{m^{2}} } </math>||
| |
| |}
| |
| <br />
| |
|
| |
| :{|
| |
| |<math> q_{d} =q_{k} \cdot \gamma_\mathrm{ Q } </math>|| <math>| mit: q_{k} = 3,0 \frac{kN}{m^{2}} </math>
| |
| |-
| |
| | || <math>| mit: \gamma_\mathrm{ Q } = 1,5 </math>
| |
| |-
| |
| |<math> q_{d} = 3,0 \frac{kN}{m^{2}} \cdot 1,5 </math>||
| |
| |-
| |
| |<math> q_{d} = 4,5 \frac{kN}{m^{2}} </math>||
| |
| |}
| |
|
| |
| ===Gesamt Einwirkungen===
| |
|
| |
| :{|
| |
| | <math> F_{d}=g_{d}+q_{d} </math> || <math>| mit: q_{d} = 4,5 \frac{kN}{m^{2}} </math>
| |
| |-
| |
| | || <math>| mit: g_{d} = 10,5 \frac{kN}{m^{2}} </math>
| |
| |-
| |
| | <math> F_{d}= 10,5 \frac{kN}{m^{2}} + 4,5 \frac{kN}{m^{2}} </math> ||
| |
| |-
| |
| | <math> F_{d}=15,0 \frac{kN}{m^{2}} </math> ||
| |
| |}
| |
|
| |
|
| |
|
| |
|
| |
| :{|
| |
| | <math> F_{0} = C_{Ed} </math> || <math>| mit: C_{Ed} =22.86 \frac{kN}{m} </math>
| |
| |-
| |
| | <math> F_{0} = 22.86 \frac{kN}{m} </math> ||
| |
| |}
| |
|
| |
|
| |
|
| |
|
| |
| :{|
| |
| | <math> m_{0} = M_{Ed,S} </math> || <math>| mit: M_{Ed,S} =6.63 \frac{kNm}{m} </math>
| |
| |-
| |
| | <math> m_{0} = 6.63 \frac{kNm}{m} </math> ||
| |
| |}
| |
|
| |
|
| |
| ==Statisches System==
| |
| Es wurde sich im Rahmen dieses Beispiels für eine Zwischenpodestplatte mit dreiseitig frei drehbar gelagerten Rändern entschieden.
| |
|
| |
| :{|
| |
| |<math>a_{1}=\mathrm{min}\begin{cases}
| |
| \frac{h_{P}}{2} \\
| |
| \frac{t}{2}
| |
| \end{cases}</math>||
| |
| |-
| |
| | ||<math>| mit: h_{P} = 20 cm</math>
| |
| |-
| |
| | ||<math>| mit: t_{1} = 36,5 cm</math>
| |
| |-
| |
| |<math>a_{1}=\mathrm{min}\begin{cases}
| |
| \frac{20 cm}{2} \\
| |
| \frac{36,5 cm}{2}
| |
| \end{cases}</math>||
| |
| |-
| |
| |<math>a_{1}=\mathrm{min}\begin{cases}
| |
| 10 cm \\
| |
| 18,25 cm
| |
| \end{cases}</math>||
| |
| |-
| |
| |<math>a_{1} = 10 cm </math>||
| |
| |}
| |
|
| |
|
| |
|
| |
|
| |
| :{|
| |
| |<math>a_{2}=\mathrm{min}\begin{cases}
| |
| \frac{h_{P}}{2} \\
| |
| \frac{t}{2}
| |
| \end{cases}</math>||
| |
| |-
| |
| | ||<math>| mit: h_{P} = 20 cm</math>
| |
| |-
| |
| | ||<math>| mit: t_{2} = 24 cm</math>
| |
| |-
| |
| |<math>a_{2}=\mathrm{min}\begin{cases}
| |
| \frac{20 cm}{2} \\
| |
| \frac{24 cm}{2}
| |
| \end{cases}</math>||
| |
| |-
| |
| |<math>a_{2}=\mathrm{min}\begin{cases}
| |
| 10 cm \\
| |
| 12 cm
| |
| \end{cases}</math>||
| |
| |-
| |
| |<math>a_{2} = 10 cm </math>||
| |
| |}
| |
|
| |
|
| |
|
| |
| :{|
| |
| | <math> b_{P} = 2 \cdot b + b^{'} + a_{2} + a_{2} </math> || <math>| mit: b = 1,0 m </math>
| |
| |-
| |
| | ||<math>| mit: a_{1} = 0,1 m </math>
| |
| |-
| |
| | ||<math>| mit: a_{2} = 0,1 m </math>
| |
| |-
| |
| | ||<math>| mit: b^{'} = 0,25 m </math>
| |
| |-
| |
| | <math> b_{P} = 2 \cdot 1,0 m + 0,25 m + 0,1 m + 0,1 m </math> ||
| |
| |-
| |
| | <math> b_{P} = 2,45 m </math> ||
| |
| |}
| |
|
| |
|
| |
|
| |
|
| |
| :{|
| |
| |<math>a_{1}=\mathrm{min}\begin{cases}
| |
| \frac{h_{P}}{2} \\
| |
| \frac{t}{2}
| |
| \end{cases}</math>||
| |
| |-
| |
| | ||<math>| mit: h_{P} = 20 cm</math>
| |
| |-
| |
| | ||<math>| mit: t_{1} = 12,5 cm</math>
| |
| |-
| |
| |<math>a_{1}=\mathrm{min}\begin{cases}
| |
| \frac{20 cm}{2} \\
| |
| \frac{12,5 cm}{2}
| |
| \end{cases}</math>||
| |
| |-
| |
| |<math>a_{1}=\mathrm{min}\begin{cases}
| |
| 10 cm \\
| |
| \approx 6 cm
| |
| \end{cases}</math>||
| |
| |-
| |
| |<math>a_{1} = 6 cm </math>||
| |
| |}
| |
|
| |
|
| |
|
| |
|
| |
| :{|
| |
| | <math> t_{P} = ln + a_{1} + a_{2} </math> || <math>| mit: b = 1,0 m </math>
| |
| |-
| |
| | ||<math>| mit: a_{1} = 0,1 m </math>
| |
| |-
| |
| | ||<math>| mit: a_{2} = 0 m </math>
| |
| |-
| |
| | <math> t_{P} = 1,0 m + 0,06 m + 0 m </math> ||
| |
| |-
| |
| | <math> t_{P} = 1,06 m </math> ||
| |
| |}
| |
|
| |
| ==Schnittgrößen ==
| |
|
| |
|
| |
| =====Verhältnis Podesttiefe zu Podestbreite feststellen =====
| |
| ::{|
| |
| | <math> \frac {t_{P}}{b_{P} } </math>||<math>| mit: b_{P} = 2,45 m </math>
| |
| |-
| |
| | || <math>| mit: t_{P} = 1,06 m </math>
| |
| |-
| |
| | <math> \frac {t_{P}}{b_{P} } = \frac {1,06 m}{2,45 m} </math> ||
| |
| |-
| |
| | <math> \frac {t_{P}}{b_{P} } \approx 0,4 </math> ||
| |
| |}
| |
|
| |
| ====Ermittnlung der Schnittkräfte über Tabelle ====
| |
|
| |
| {| class="wikitable"
| |
| |+style="text-align:left;"|Tafel zur Schnittgrößen Ermittlung von Podestplatten mit dreiseitig frei drehbar gelagerten Rändern <ref Name = "Köseoglu" group="F">Beton-Kalender, Jahrgang 1980, Band 2, Abschnitt E, Abschnitt Treppen, Köseoglu, S.</ref> <ref Name = "AVAK" group="F">Stahlbetonbau in Beispielen - Teil 2: Bemessung von Flächentragwerken nach EC 2 - Konstruktionspläne für Stahlbetonbauteile, Ralf Avak, René Conchon, Markus Aldejohann 2017 Auflage 5</ref>
| |
| |rowspan="3"|
| |
| |1
| |
| |2
| |
| |3
| |
| |-
| |
| !style="background: #eaecf0;" rowspan="2"|Belastungsvariante
| |
| !style="background: #eaecf0;"|<math> \frac {t_{P}}{b_{P} } </math>
| |
| !style="background: #eaecf0;"|0,4
| |
| |-
| |
| !style="background: #eaecf0;"|
| |
| !style="background: #eaecf0;" |<math> \chi </math>
| |
| |-
| |
| |1
| |
| |rowspan="3" style="background: #eaecf0;"|I
| |
| |style="background: #eaecf0;"|<math> m_{x,m} = \frac{F_{d} \cdot t_{P}^{2}}{\chi} </math>
| |
| |8,04
| |
| |-
| |
| |2
| |
| |style="background: #eaecf0;"|<math> m_{y,m} = \frac{F_{d} \cdot t_{P}^{2}}{\chi} </math>
| |
| |10,5
| |
| |-
| |
| |3
| |
| |style="background: #eaecf0;"|<math> m_{x,r} = \frac{F_{d} \cdot t_{P}^{2}}{\chi} </math>
| |
| |4,41
| |
| |-
| |
| |4
| |
| |rowspan="3" style="background: #eaecf0;"|II
| |
| |style="background: #eaecf0;"|<math> m_{x,m} = \frac{F_{0} \cdot b_{P}}{\chi} </math>
| |
| |10,5
| |
| |-
| |
| |5
| |
| |style="background: #eaecf0;"|<math> m_{x,m} = - \frac{F_{0} \cdot b_{P}}{\chi} </math>
| |
| |91,0
| |
| |-
| |
| |6
| |
| |style="background: #eaecf0;"|<math> m_{x,r} = \frac{F_{0} \cdot b_{P}}{\chi} </math>
| |
| |5,60
| |
| |-
| |
| |7
| |
| |rowspan="3" style="background: #eaecf0;"|III
| |
| |style="background: #eaecf0;"|<math>m_{y,m} = \frac{m_{0}}{\chi} </math>
| |
| |5,70
| |
| |-
| |
| |8
| |
| |style="background: #eaecf0;"|<math> m_{y,m} = - \frac{m_{0}}{\chi} </math>
| |
| |2,20
| |
| |-
| |
| |9
| |
| |style="background: #eaecf0;"|<math> m_{x,r} = \frac{m_{0}}{\chi} </math>
| |
| |2,35
| |
| |-
| |
| |colspan="11" style="text-align:left;"|<math>\chi</math> = Wert in der Tabelle
| |
| * in Belastungsvariante I wird eine Podestplatte betrachtet die ausschließlich durch eine Gleichflächenlast <math>F_{d}</math> belastet ist
| |
| * in Belastungsvariante II wird eine Podestplatte betrachtet die ausschließlich Streckenlast <math>F_{0}</math> am Rand aus der Auflagerkraft des Treppenlaufs belastet ist
| |
| * in Belastungsvariante III wird eine Podestplatte betrachtet die ausschließlich Streckenmoment <math>m_{0}</math> aus der elastischen Einspannung des Treppenlaufs belastet ist
| |
| |}
| |
|
| |
| ======m_{x,m}======
| |
|
| |
| :<math> m_{i} = m_{i,I} + m_{i,II} + m_{i,III} </math>
| |
|
| |
|
| |
| ::{|
| |
| | <math> m_{x,m,I} = \frac{F_{d} \cdot t_{P}^{2}}{\chi} </math>||<math>| mit: \chi = 8,04 </math>
| |
| |-
| |
| | || <math>| mit: t_{P} = 1,06 m </math>
| |
| |-
| |
| | || <math>| mit: F_{d} = 15,0 \frac{kN}{m^{2}} </math> ||
| |
| |-
| |
| | <math> m_{x,m,I} = \frac{ 15,0 \frac{kN}{m^{2}} \cdot (1,06 m)^{2}}{ 8,04} </math> ||
| |
| |-
| |
| | <math> m_{x,m,I} \approx 2,1 \frac{kNm}{m} </math> ||
| |
| |}
| |
|
| |
|
| |
| ::{|
| |
| | <math> m_{x,m,II} = \frac{F_{0} \cdot b_{P}}{\chi} </math>||<math>| mit: \chi = 10,5 </math>
| |
| |-
| |
| | || <math>| mit: b_{P} = 2,45 m </math>
| |
| |-
| |
| | || <math>| mit: F_{d} = 22.86 \frac{kN}{m} </math> ||
| |
| |-
| |
| | <math> m_{x,m,II} = \frac{ 22.86 \frac{kN}{m} \cdot 2,45 m }{10,5} </math> ||
| |
| |-
| |
| | <math> m_{x,m,II} \approx 5,33 \frac{kNm}{m} </math> ||
| |
| |}
| |
|
| |
|
| |
| ::{|
| |
| | <math> m_{x,m,III} = \frac{ m_{0} }{ \chi } </math>||<math>| mit: \chi = 5,70 </math>
| |
| |-
| |
| | || <math>| mit: m_{0} = 6.63 \frac{kNm}{m} </math> ||
| |
| |-
| |
| | <math> m_{x,m,III} = \frac{ 6.63 \frac{kNm}{m} }{5,70} </math> ||
| |
| |-
| |
| | <math> m_{x,m,III} \approx 1,16 \frac{kNm}{m} </math> ||
| |
| |}
| |
|
| |
|
| |
| :{|
| |
| |<math> m_{x,m} = m_{x,m,I} + m_{x,m,II} + m_{x,m,III} </math>||<math>| mit: m_{x,m,I} = 2,1 \frac{kNm}{m} </math>
| |
| |-
| |
| | || <math>| mit: m_{x,m,II} = 5,33 \frac{kNm}{m} </math> ||
| |
| |-
| |
| | || <math>| mit: m_{x,m,III} = 1,16 \frac{kNm}{m} </math> ||
| |
| |-
| |
| |<math> m_{x,m} = 2,1 \frac{kNm}{m} + 5,33 \frac{kNm}{m} + 1,16 \frac{kNm}{m} </math>||
| |
| |-
| |
| |<math> m_{x,m} = 8,59 \frac{kNm}{m} </math>||
| |
| |}
| |
|
| |
| ======m_{y,m}======
| |
|
| |
| :<math> m_{i} = m_{i,I} + m_{i,II} + m_{i,III} </math>
| |
|
| |
| ::{|
| |
| | <math> m_{y,m,I} = \frac{F_{d} \cdot t_{P}^{2}}{\chi} </math>||<math>| mit: \chi = 10,5 </math>
| |
| |-
| |
| | || <math>| mit: t_{P} = 1,06 m </math>
| |
| |-
| |
| | || <math>| mit: F_{d} = 15,0 \frac{kN}{m^{2}} </math> ||
| |
| |-
| |
| | <math> m_{y,m,I} = \frac{ 15,0 \frac{kN}{m^{2}} \cdot (1,06 m)^{2}}{10,5} </math> ||
| |
| |-
| |
| | <math> m_{y,m,I} \approx 1,61 \frac{kNm}{m} </math> ||
| |
| |}
| |
|
| |
|
| |
|
| |
|
| |
| ::{|
| |
| | <math> m_{y,m,II} = - \frac{F_{0} \cdot b_{P}}{\chi} </math>||<math>| mit: \chi = 91,0 </math>
| |
| |-
| |
| | || <math>| mit: b_{P} = 2,45 m </math>
| |
| |-
| |
| | || <math>| mit: F_{d} = 22.86 \frac{kN}{m} </math> ||
| |
| |-
| |
| | <math> m_{y,m,II} = - \frac{ 22.86 \frac{kN}{m} \cdot 2,45 m}{91,0} </math> ||
| |
| |-
| |
| | <math> m_{y,m,II} \approx - 0,62 \frac{kNm}{m} </math> ||
| |
| |}
| |
|
| |
|
| |
|
| |
|
| |
|
| |
| ::{|
| |
| | <math> m_{y,m,III} = - \frac{ m_{0} }{ \chi } </math>||<math>| mit: \chi = 2,20 </math>
| |
| |-
| |
| | || <math>| mit: m_{0} = 6.63 \frac{kNm}{m} </math> ||
| |
| |-
| |
| | <math> m_{y,m,III} = - \frac{ 6.63 \frac{kNm}{m} }{2,20} </math> ||
| |
| |-
| |
| | <math> m_{y,m,III} \approx - 3,01 \frac{kNm}{m} </math> ||
| |
| |}
| |
|
| |
|
| |
|
| |
|
| |
| :{|
| |
| |<math> m_{y,m} = m_{y,m,I} + m_{y,m,II} + m_{y,m,III} </math>||<math>| mit: m_{y,m,I} = 1,61 \frac{kNm}{m} </math>
| |
| |-
| |
| | || <math>| mit: m_{y,m,II} = - 0,62 \frac{kNm}{m} </math> ||
| |
| |-
| |
| | || <math>| mit: m_{y,m,III} = - 3,01 \frac{kNm}{m} </math> ||
| |
| |-
| |
| |<math> m_{y,m} = 1,61 \frac{kNm}{m} - 0,62 \frac{kNm}{m} - 3,01 \frac{kNm}{m} </math>||
| |
| |-
| |
| |<math> m_{y,m} = - 2,02 \frac{kNm}{m} </math>||
| |
| |}
| |
|
| |
| ======m_{x,r}======
| |
| ::{|
| |
| | <math> m_{x,r} = \frac{F_{d} \cdot t_{P}^{2}}{\chi} </math>||<math>| mit: \chi = 4,41 </math>
| |
| |-
| |
| | || <math>| mit: t_{P} = 1,06 m </math>
| |
| |-
| |
| | || <math>| mit: F_{d} = 15,0 \frac{kN}{m^{2}} </math> ||
| |
| |-
| |
| | <math> m_{x,r} = \frac{ 15,0 \frac{kN}{m^{2}} \cdot (1,06 m)^{2}}{4,41} </math> ||
| |
| |-
| |
| | <math> m_{x,r} \approx 3,82 \frac{kNm}{m} </math> ||
| |
| |}
| |
|
| |
|
| |
|
| |
|
| |
| ::{|
| |
| | <math> m_{x,r,II} = \frac{F_{0} \cdot b_{P}}{\chi} </math>||<math>| mit: \chi = 5,60 </math>
| |
| |-
| |
| | || <math>| mit: b_{P} = 2,45 m </math>
| |
| |-
| |
| | || <math>| mit: F_{d} = 22.86 \frac{kN}{m} </math> ||
| |
| |-
| |
| | <math> m_{x,r,II} = \frac{ 22.86 \frac{kN}{m} \cdot 2,45 m}{5,60} </math> ||
| |
| |-
| |
| | <math> m_{x,r,II} \approx 10,0 \frac{kNm}{m} </math> ||
| |
| |}
| |
|
| |
|
| |
|
| |
|
| |
|
| |
| ::{|
| |
| | <math> m_{x,r,III} = \frac{ m_{0} }{ \chi } </math>||<math>| mit: \chi = 2,35 </math>
| |
| |-
| |
| | || <math>| mit: m_{0} = 6.63 \frac{kNm}{m} </math> ||
| |
| |-
| |
| | <math> m_{x,r,III} = \frac{ 6.63 \frac{kNm}{m} }{2,35} </math> ||
| |
| |-
| |
| | <math> m_{x,r,III} \approx 2,82 \frac{kNm}{m} </math> ||
| |
| |}
| |
|
| |
|
| |
|
| |
|
| |
| :{|
| |
| |<math> m_{x,r} = m_{x,r,I} + m_{x,r,II} + m_{x,r,III} </math>||<math>| mit: m_{x,r,I} = 3,82 \frac{kNm}{m} </math>
| |
| |-
| |
| | || <math>| mit: m_{x,r,II} = 10,0 \frac{kNm}{m} </math> ||
| |
| |-
| |
| | || <math>| mit: m_{x,r,III} = 2,82 \frac{kNm}{m} </math> ||
| |
| |-
| |
| |<math> m_{x,r} = 3,82 \frac{kNm}{m} + 10,0 \frac{kNm}{m} + 2,82 \frac{kNm}{m} </math>||
| |
| |-
| |
| |<math> m_{x,r} = 16,64 \frac{kNm}{m} </math>||
| |
| |}
| |
|
| |
| == Bemessung im Grenzzustand der Tragfähigkeit==
| |
| ===Materialparameter===
| |
| ::{|
| |
| | <math> f_{cd} = \frac{ \alpha_{cc} \cdot f_{ck} }{ \gamma_{C} } </math>||<math>| mit: \gamma_{C} = 1.5 </math>
| |
| |-
| |
| | || <math>| mit: \alpha_{cc} = 0.85 </math>
| |
| |-
| |
| | || <math>| mit: f_{ck} = 25 \frac{kN}{cm^{2}} </math>
| |
| |-
| |
| | <math> f_{cd} = \frac{ 0.85 \cdot 25 \frac{kN}{cm^{2}}}{ 1.5 } </math> ||
| |
| |-
| |
| | <math> f_{cd} = 14,2 \frac{kN}{cm^{2}} </math> ||
| |
| |}
| |
| <br />
| |
|
| |
| <br />
| |
|
| |
|
| |
| ::{|
| |
| | <math> f_{yd} = \frac{ f_{yk}}{\gamma_{s}} </math>||<math>| mit: f_{yk} = 500 \frac{N}{mm^{2}} </math>
| |
| |-
| |
| | || <math>| mit: \gamma_{s} = 1.15 </math>
| |
| |-
| |
| | <math> f_{yd} = \frac{ 50 \frac{kN}{cm^{2}}}{1,15} </math>||
| |
| |-
| |
| | <math> f_{yd} = 43,5 \frac{kN}{cm^{2}} </math>||
| |
| |}
| |
|
| |
|
| |
| ===Biegebemessung m_{x,m}===
| |
|
| |
| ====Vorbemessung m_{x,m}====
| |
| ::{|
| |
| | <math> z_{est} = 0,75 \cdot h </math>||<math>| mit: h = h_{p} = 20 cm </math>
| |
| |-
| |
| | <math> z_{est} = 0,75 \cdot 20 cm </math>||
| |
| |-
| |
| | <math> z_{est} = 15 cm </math>||
| |
| |}
| |
| <br />
| |
|
| |
| <br />
| |
| ::{|
| |
| | <math> M_{Ed,est} = M_{Ed} - N_{Ed} \cdot z_{s1,est} </math>||<math>| mit: z_{s1,est} = 15 cm </math>
| |
| |-
| |
| | ||<math>| mit: N_{Ed} = 0 \frac{kN}{m}</math>
| |
| |-
| |
| | ||<math>| mit: M_{Ed} = m_{x,m} = 8,59 \frac{kNm}{m} </math>
| |
| |-
| |
| | <math> M_{Ed,est} = 8,59 \frac{kNm}{m} - 0 \frac{kN}{m} \cdot 0,15 m </math>||
| |
| |-
| |
| | <math> M_{Ed,est} = 8,59 \frac{kNm}{m}</math>||
| |
| |}
| |
| <br />
| |
|
| |
| <br />
| |
| ::{|
| |
| | <math> a_{s,est} = \frac{M_{Ed,est}}{z_{s1,est} \cdot f_{yd} } + \frac{N_{Ed}}{f_{yd}} </math>||<math>| mit: z_{s1,est} = 15 cm </math>
| |
| |-
| |
| | ||<math>| mit: f_{yd} = 43,5 \frac{kN}{cm^{2}} </math>
| |
| |-
| |
| | ||<math>| mit: N_{Ed} = 0 </math>
| |
| |-
| |
| | ||<math>| mit: M_{Ed,est} = 8,59 \frac{kNm}{m} </math>
| |
| |-
| |
| | <math> a_{s,est} = \frac{859 \frac{kNcm}{m}}{15 cm \cdot 43,5 \frac{kN}{cm^{2}} } + \frac{0}{43,5 \frac{kN}{cm^{2}}} </math>||
| |
| |-
| |
| | <math> a_{s,est} \approx 1,32 \frac{kNcm}{m} </math>||
| |
| |}
| |
|
| |
| gewählt: R188 ø6/15cm, <math>{{a}_{s}}= 1,88 \frac{cm^{2}}{m} </math>
| |
|
| |
| ====Querschnittsgeometrie m_{x,m}====
| |
|
| |
| :<math>c_{v}=\mathrm{max}\begin{cases}
| |
| C_{nom,dur} \\
| |
| C_{nom,b,Bü} \\
| |
| C_{nom,b,L}
| |
| \end{cases}</math><br /><br />
| |
|
| |
| <br />
| |
| ::{|
| |
| | <math> C_{nom,dur} = C_{min,dur} + \Delta C_{dev} </math>|| <math>| mit: C_{min,dur} = 10 mm </math> für XC1
| |
| |-
| |
| | || <math>| mit: \Delta C_{dev} = 10 mm </math>
| |
| |-
| |
| | <math> C_{nom,dur} = 10 mm + 10 mm </math>||
| |
| |-
| |
| | <math> C_{nom,dur} = 20 mm </math>||
| |
| |}
| |
| <br />
| |
|
| |
| <br />
| |
| ::{|
| |
| | <math> C_{nom,b,Bü} = C_{min,b,Bü} + \Delta C_{dev} </math>|| <math>| mit: C_{min,b,Bü} = 0 mm </math>
| |
| |-
| |
| | ||<math>| mit: \Delta C_{dev} = 10 mm </math>
| |
| |-
| |
| | <math> C_{nom,b,Bü} = 0 mm + 10 mm </math>||
| |
| |-
| |
| | <math> C_{nom,b,Bü} = 10 mm </math>||
| |
| |}
| |
|
| |
|
| |
| <br />
| |
| ::{|
| |
| | <math> C_{nom,b,L} = C_{min,b,L} - \varnothing bue + \Delta C_{dev} </math>|| <math>| mit: C_{min,b,L} = 7 mm </math>
| |
| |-
| |
| | ||<math>| mit: \Delta C_{dev} = 10 mm </math>
| |
| |-
| |
| | ||<math>| mit: \varnothing bue = 0 mm </math>
| |
| |-
| |
| | <math> C_{nom,b,L} = 6 mm - 0 mm + \Delta 10 mm </math>||
| |
| |-
| |
| | <math> C_{nom,b,L} = 16 mm </math>||
| |
| |}
| |
|
| |
| :<math>c_{v}=\mathrm{max}\begin{cases}
| |
| 20 mm \\
| |
| 10 mm \\
| |
| 16 mm
| |
| \end{cases}</math><br /><br />
| |
|
| |
| ::{|
| |
| | <math> d_{1} = c_{v} + \varnothing bue + \frac{\varnothing L}{2}</math>|| <math>| mit: c_{v} = 20 mm </math>
| |
| |-
| |
| | || <math>| mit: \varnothing bue = 0 mm </math>
| |
| |-
| |
| | || <math>| mit: \varnothing L = 6 mm </math>
| |
| |-
| |
| | <math> d_{1} = 20 mm + 0 mm + \frac{6 mm}{2} </math>||
| |
| |-
| |
| | <math> d_{1} = 23,0 mm</math>||
| |
| |}
| |
|
| |
| ::{|
| |
| | <math> d = h_{L} - d_{1} </math>|| <math>| mit: d_{1} = 23 mm </math>
| |
| |-
| |
| | || <math>| mit: h_{L} = 200 mm </math>
| |
| |-
| |
| | <math> d = 200 mm - 23,0 mm </math>||
| |
| |-
| |
| | <math> d = 177 mm = 17,7 cm </math>||
| |
| |}
| |
|
| |
| ====Bemessung mit dem ω-Verfahren m_{x,m}====
| |
|
| |
|
| |
| ::{|
| |
| |<math> z_{s} = d - \frac{h_{L} }{2} </math>|| <math>| mit: d = 17,7 cm </math>
| |
| |-
| |
| | || <math>| mit: h_{L} = 20 cm </math>
| |
| |-
| |
| |<math> z_{s} = 17,7 cm - \frac{20 cm}{2} </math>||
| |
| |-
| |
| |<math> z_{s} = 7,7 cm </math>||
| |
| |}
| |
|
| |
| ::{|
| |
| |<math> M_{Eds} = M_{Ed,S} - extr n \cdot z_{s} </math>|| <math>| mit: z_{s} = 7,7 cm </math>
| |
| |-
| |
| | || <math>| extr n = 0 </math>
| |
| |-
| |
| | ||<math>| mit: M_{Ed,S} = m_{x,m} = 859 \frac{kNcm}{m} </math>
| |
| |-
| |
| |<math> M_{Eds} = 859 \frac{kNcm}{m} - 0 \frac{kN}{m} \cdot 7,7 cm </math>||
| |
| |-
| |
| |<math> M_{Eds} = 859 \frac{kNcm}{m} </math>||
| |
| |}
| |
|
| |
|
| |
| ::{|
| |
| |<math> \mu_{Eds} = \frac{M_{Eds}}{b\cdot d^{2} \cdot f_{cd}} </math>|| <math>| mit: d = 17,7 cm </math>
| |
| |-
| |
| | || <math>| mit: b = 100 cm </math>
| |
| |-
| |
| | || <math>| mit: f_{cd} = 1,42 \frac{kN}{cm^{2}}</math>
| |
| |-
| |
| | || <math>| mit: M_{Eds} =m_{x,m} = 859,0 \frac{kNcm}{m} </math>
| |
| |-
| |
| |<math> \mu_{Eds} = \frac{859,0 \frac{kNcm}{m}}{100 cm \cdot (17,7 cm)^{2} \cdot 1,42 \frac{kN}{cm^{2}} } </math>||
| |
| |-
| |
| |<math> \mu_{Eds} = 0,0193 </math>||
| |
| |}
| |
| <br />
| |
|
| |
| <br />
| |
|
| |
| ::{|
| |
| |<math> \omega = \omega_{1} + \frac{ \omega_{2} - \omega_{1} } { \mu_{Eds,2} - \mu_{Eds,1}} \cdot ( \mu_{Eds} - \mu_{Eds,1} ) </math>||
| |
| <math>| mit: \omega_{1} = 0,0101 </math>
| |
| |-
| |
| | || <math>| mit: \omega_{2} = 0,0203 </math>
| |
| |-
| |
| | || <math>| mit: \mu_{Eds} = 0,0193 </math>
| |
| |-
| |
| | || <math>| mit: \mu_{Eds,1} = 0,01 </math>
| |
| |-
| |
| | || <math>| mit: \mu_{Eds,2} = 0,02 </math>
| |
| |-
| |
| |<math> \omega = 0,0101 + \frac{ 0,0203 - 0,0101 } { 0,02 - 0,01 } \cdot ( 0,0193 - 0,01 ) </math>||
| |
| |-
| |
| |<math> \omega = 0,0196 </math>||
| |
| |}
| |
| <br />
| |
|
| |
| <br />
| |
| ::{|
| |
| |<math> a_{s,1} = \frac{1}{ \sigma_{sd}} \cdot ( \omega \cdot b \cdot d \cdot f_{cd} + N_{Ed} ) </math>|| <math>| mit: \omega = 0,0196 </math>
| |
| |-
| |
| | || <math>| mit: f_{cd} = 1,42 \frac{kN}{cm^{2}} </math>
| |
| |-
| |
| | || <math>| mit: d = 17,7 cm </math>
| |
| |-
| |
| | || <math>| mit: b = 100 cm </math>
| |
| |-
| |
| | || <math>| mit: N_{Ed} = 0 </math>
| |
| |-
| |
| |<math> a_{s,1} = \frac{1}{ 43,5 \frac{kN}{cm^{2}}} \cdot ( 0,0196 \cdot 100 cm \cdot 17,7 cm \cdot 1,42 \frac{kN}{cm^{2}} + 0 \frac{kN}{m} ) </math>||
| |
| |-
| |
| |<math> a_{s,1} = 1,13 \frac{cm^{2}}{m} </math>||
| |
| |}
| |
|
| |
| gewählt: R188 ø6/15cm, <math>{{a}_{s}}= 1,88 \frac{cm^{2}}{m} </math>
| |
|
| |
| ===Biegebemessung m_{x,m}===
| |
|
| |
| ====Vorbemessung m_{x,m}====
| |
| ::{|
| |
| | <math> z_{est} = 0,75 \cdot h </math>||<math>| mit: h = h_{p} = 20 cm </math>
| |
| |-
| |
| | <math> z_{est} = 0,75 \cdot 20 cm </math>||
| |
| |-
| |
| | <math> z_{est} = 15 cm </math>||
| |
| |}
| |
| <br />
| |
|
| |
| <br />
| |
| ::{|
| |
| | <math> M_{Ed,est} = M_{Ed} - N_{Ed} \cdot z_{s1,est} </math>||<math>| mit: z_{s1,est} = 15 cm </math>
| |
| |-
| |
| | ||<math>| mit: N_{Ed} = 0 \frac{kN}{m}</math>
| |
| |-
| |
| | ||<math>| mit: M_{Ed} = m_{x,m} = 8,59 \frac{kNm}{m} </math>
| |
| |-
| |
| | <math> M_{Ed,est} = 8,59 \frac{kNm}{m} - 0 \frac{kN}{m} \cdot 0,15 m </math>||
| |
| |-
| |
| | <math> M_{Ed,est} = 8,59 \frac{kNm}{m}</math>||
| |
| |}
| |
| <br />
| |
|
| |
| <br />
| |
| ::{|
| |
| | <math> a_{s,est} = \frac{M_{Ed,est}}{z_{s1,est} \cdot f_{yd} } + \frac{N_{Ed}}{f_{yd}} </math>||<math>| mit: z_{s1,est} = 15 cm </math>
| |
| |-
| |
| | ||<math>| mit: f_{yd} = 43,5 \frac{kN}{cm^{2}} </math>
| |
| |-
| |
| | ||<math>| mit: N_{Ed} = 0 </math>
| |
| |-
| |
| | ||<math>| mit: M_{Ed,est} = 8,59 \frac{kNm}{m} </math>
| |
| |-
| |
| | <math> a_{s,est} = \frac{859 \frac{kNcm}{m}}{15 cm \cdot 43,5 \frac{kN}{cm^{2}} } + \frac{0}{43,5 \frac{kN}{cm^{2}}} </math>||
| |
| |-
| |
| | <math> a_{s,est} \approx 1,32 \frac{kNcm}{m} </math>||
| |
| |}
| |
|
| |
| gewählt: R188 ø6/15cm, <math>{{a}_{s}}= 1,88 \frac{cm^{2}}{m} </math>
| |
|
| |
| ====Querschnittsgeometrie m_{x,m}====
| |
|
| |
| :<math>c_{v}=\mathrm{max}\begin{cases}
| |
| C_{nom,dur} \\
| |
| C_{nom,b,Bü} \\
| |
| C_{nom,b,L}
| |
| \end{cases}</math><br /><br />
| |
|
| |
| <br />
| |
| ::{|
| |
| | <math> C_{nom,dur} = C_{min,dur} + \Delta C_{dev} </math>|| <math>| mit: C_{min,dur} = 10 mm </math> für XC1
| |
| |-
| |
| | || <math>| mit: \Delta C_{dev} = 10 mm </math>
| |
| |-
| |
| | <math> C_{nom,dur} = 10 mm + 10 mm </math>||
| |
| |-
| |
| | <math> C_{nom,dur} = 20 mm </math>||
| |
| |}
| |
| <br />
| |
|
| |
| <br />
| |
| ::{|
| |
| | <math> C_{nom,b,Bü} = C_{min,b,Bü} + \Delta C_{dev} </math>|| <math>| mit: C_{min,b,Bü} = 0 mm </math>
| |
| |-
| |
| | ||<math>| mit: \Delta C_{dev} = 10 mm </math>
| |
| |-
| |
| | <math> C_{nom,b,Bü} = 0 mm + 10 mm </math>||
| |
| |-
| |
| | <math> C_{nom,b,Bü} = 10 mm </math>||
| |
| |}
| |
|
| |
|
| |
| <br />
| |
| ::{|
| |
| | <math> C_{nom,b,L} = C_{min,b,L} - \varnothing bue + \Delta C_{dev} </math>|| <math>| mit: C_{min,b,L} = 7 mm </math>
| |
| |-
| |
| | ||<math>| mit: \Delta C_{dev} = 10 mm </math>
| |
| |-
| |
| | ||<math>| mit: \varnothing bue = 0 mm </math>
| |
| |-
| |
| | <math> C_{nom,b,L} = 6 mm - 0 mm + \Delta 10 mm </math>||
| |
| |-
| |
| | <math> C_{nom,b,L} = 16 mm </math>||
| |
| |}
| |
|
| |
| :<math>c_{v}=\mathrm{max}\begin{cases}
| |
| 20 mm \\
| |
| 10 mm \\
| |
| 16 mm
| |
| \end{cases}</math><br /><br />
| |
|
| |
| ::{|
| |
| | <math> d_{1} = c_{v} + \varnothing bue + \frac{\varnothing L}{2}</math>|| <math>| mit: c_{v} = 20 mm </math>
| |
| |-
| |
| | || <math>| mit: \varnothing bue = 0 mm </math>
| |
| |-
| |
| | || <math>| mit: \varnothing L = 6 mm </math>
| |
| |-
| |
| | <math> d_{1} = 20 mm + 0 mm + \frac{6 mm}{2} </math>||
| |
| |-
| |
| | <math> d_{1} = 23,0 mm</math>||
| |
| |}
| |
|
| |
| ::{|
| |
| | <math> d = h_{L} - d_{1} </math>|| <math>| mit: d_{1} = 23 mm </math>
| |
| |-
| |
| | || <math>| mit: h_{L} = 200 mm </math>
| |
| |-
| |
| | <math> d = 200 mm - 23,0 mm </math>||
| |
| |-
| |
| | <math> d = 177 mm = 17,7 cm </math>||
| |
| |}
| |
|
| |
| ====Bemessung mit dem ω-Verfahren m_{x,m}====
| |
|
| |
|
| |
| ::{|
| |
| |<math> z_{s} = d - \frac{h_{L} }{2} </math>|| <math>| mit: d = 17,7 cm </math>
| |
| |-
| |
| | || <math>| mit: h_{L} = 20 cm </math>
| |
| |-
| |
| |<math> z_{s} = 17,7 cm - \frac{20 cm}{2} </math>||
| |
| |-
| |
| |<math> z_{s} = 7,7 cm </math>||
| |
| |}
| |
|
| |
| ::{|
| |
| |<math> M_{Eds} = M_{Ed,S} - extr n \cdot z_{s} </math>|| <math>| mit: z_{s} = 7,7 cm </math>
| |
| |-
| |
| | || <math>| extr n = 0 </math>
| |
| |-
| |
| | ||<math>| mit: M_{Ed,S} = m_{x,m} = 859 \frac{kNcm}{m} </math>
| |
| |-
| |
| |<math> M_{Eds} = 859 \frac{kNcm}{m} - 0 \frac{kN}{m} \cdot 7,7 cm </math>||
| |
| |-
| |
| |<math> M_{Eds} = 859 \frac{kNcm}{m} </math>||
| |
| |}
| |
|
| |
|
| |
| ::{|
| |
| |<math> \mu_{Eds} = \frac{M_{Eds}}{b\cdot d^{2} \cdot f_{cd}} </math>|| <math>| mit: d = 17,7 cm </math>
| |
| |-
| |
| | || <math>| mit: b = 100 cm </math>
| |
| |-
| |
| | || <math>| mit: f_{cd} = 1,42 \frac{kN}{cm^{2}}</math>
| |
| |-
| |
| | || <math>| mit: M_{Eds} =m_{x,m} = 859,0 \frac{kNcm}{m} </math>
| |
| |-
| |
| |<math> \mu_{Eds} = \frac{859,0 \frac{kNcm}{m}}{100 cm \cdot (17,7 cm)^{2} \cdot 1,42 \frac{kN}{cm^{2}} } </math>||
| |
| |-
| |
| |<math> \mu_{Eds} = 0,0193 </math>||
| |
| |}
| |
| <br />
| |
|
| |
| <br />
| |
|
| |
| ::{|
| |
| |<math> \omega = \omega_{1} + \frac{ \omega_{2} - \omega_{1} } { \mu_{Eds,2} - \mu_{Eds,1}} \cdot ( \mu_{Eds} - \mu_{Eds,1} ) </math>||
| |
| <math>| mit: \omega_{1} = 0,0101 </math>
| |
| |-
| |
| | || <math>| mit: \omega_{2} = 0,0203 </math>
| |
| |-
| |
| | || <math>| mit: \mu_{Eds} = 0,0193 </math>
| |
| |-
| |
| | || <math>| mit: \mu_{Eds,1} = 0,01 </math>
| |
| |-
| |
| | || <math>| mit: \mu_{Eds,2} = 0,02 </math>
| |
| |-
| |
| |<math> \omega = 0,0101 + \frac{ 0,0203 - 0,0101 } { 0,02 - 0,01 } \cdot ( 0,0193 - 0,01 ) </math>||
| |
| |-
| |
| |<math> \omega = 0,0196 </math>||
| |
| |}
| |
| <br />
| |
|
| |
| <br />
| |
| ::{|
| |
| |<math> a_{s,1} = \frac{1}{ \sigma_{sd}} \cdot ( \omega \cdot b \cdot d \cdot f_{cd} + N_{Ed} ) </math>|| <math>| mit: \omega = 0,0196 </math>
| |
| |-
| |
| | || <math>| mit: f_{cd} = 1,42 \frac{kN}{cm^{2}} </math>
| |
| |-
| |
| | || <math>| mit: d = 17,7 cm </math>
| |
| |-
| |
| | || <math>| mit: b = 100 cm </math>
| |
| |-
| |
| | || <math>| mit: N_{Ed} = 0 </math>
| |
| |-
| |
| |<math> a_{s,1} = \frac{1}{ 43,5 \frac{kN}{cm^{2}}} \cdot ( 0,0196 \cdot 100 cm \cdot 17,7 cm \cdot 1,42 \frac{kN}{cm^{2}} + 0 \frac{kN}{m} ) </math>||
| |
| |-
| |
| |<math> a_{s,1} = 1,13 \frac{cm^{2}}{m} </math>||
| |
| |}
| |
|
| |
| gewählt: R188 ø6/15cm, <math>{{a}_{s}}= 1,88 \frac{cm^{2}}{m} </math>
| |
|
| |
|
| |
| ===Biegebemessung m_{x,r}===
| |
|
| |
| ====Vorbemessung m_{x,r}====
| |
| ::{|
| |
| | <math> z_{est} = 0,75 \cdot h </math>||<math>| mit: h = h_{p} = 20 cm </math>
| |
| |-
| |
| | <math> z_{est} = 0,75 \cdot 20 cm </math>||
| |
| |-
| |
| | <math> z_{est} = 15 cm </math>||
| |
| |}
| |
| <br />
| |
|
| |
| <br />
| |
| ::{|
| |
| | <math> M_{Ed,est} = M_{Ed} - N_{Ed} \cdot z_{s1,est} </math>||<math>| mit: z_{s1,est} = 15 cm </math>
| |
| |-
| |
| | ||<math>| mit: N_{Ed} = 0 \frac{kN}{m}</math>
| |
| |-
| |
| | ||<math>| mit: M_{Ed} = m_{x,m} = 16,64 \frac{kNm}{m} </math>
| |
| |-
| |
| | <math> M_{Ed,est} = 16,64\frac{kNm}{m} - 0 \frac{kN}{m} \cdot 0,15 m </math>||
| |
| |-
| |
| | <math> M_{Ed,est} = 16,64 \frac{kNm}{m}</math>||
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| |}
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| <br />
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|
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| <br />
| |
| ::{|
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| | <math> a_{s,est} = \frac{M_{Ed,est}}{z_{s1,est} \cdot f_{yd} } + \frac{N_{Ed}}{f_{yd}} </math>||<math>| mit: z_{s1,est} = 15 cm </math>
| |
| |-
| |
| | ||<math>| mit: f_{yd} = 43,5 \frac{kN}{cm^{2}} </math>
| |
| |-
| |
| | ||<math>| mit: N_{Ed} = 0 </math>
| |
| |-
| |
| | ||<math>| mit: M_{Ed,est} = 16,64 \frac{kNm}{m} </math>
| |
| |-
| |
| | <math> a_{s,est} = \frac{1664 \frac{kNcm}{m}}{15 cm \cdot 43,5 \frac{kN}{cm^{2}} } + \frac{0}{43,5 \frac{kN}{cm^{2}}} </math>||
| |
| |-
| |
| | <math> a_{s,est} \approx 2,55 \frac{kNcm}{m} </math>||
| |
| |}
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|
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| gewählt: ø 8/15cm, <math>{{a}_{s}}= 3,35 \frac{cm^{2}}{m} </math>
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|
| |
| ====Querschnittsgeometrie m_{x,r}====
| |
|
| |
| :<math>c_{v}=\mathrm{max}\begin{cases}
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| C_{nom,dur} \\
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| C_{nom,b,Bü} \\
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| C_{nom,b,L}
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| \end{cases}</math><br /><br />
| |
|
| |
| <br />
| |
| ::{|
| |
| | <math> C_{nom,dur} = C_{min,dur} + \Delta C_{dev} </math>|| <math>| mit: C_{min,dur} = 10 mm </math> für XC1
| |
| |-
| |
| | || <math>| mit: \Delta C_{dev} = 10 mm </math>
| |
| |-
| |
| | <math> C_{nom,dur} = 10 mm + 10 mm </math>||
| |
| |-
| |
| | <math> C_{nom,dur} = 20 mm </math>||
| |
| |}
| |
| <br />
| |
|
| |
| <br />
| |
| ::{|
| |
| | <math> C_{nom,b,Bü} = C_{min,b,Bü} + \Delta C_{dev} </math>|| <math>| mit: C_{min,b,Bü} = 0 mm </math>
| |
| |-
| |
| | ||<math>| mit: \Delta C_{dev} = 10 mm </math>
| |
| |-
| |
| | <math> C_{nom,b,Bü} = 0 mm + 10 mm </math>||
| |
| |-
| |
| | <math> C_{nom,b,Bü} = 10 mm </math>||
| |
| |}
| |
|
| |
|
| |
| <br />
| |
| ::{|
| |
| | <math> C_{nom,b,L} = C_{min,b,L} - \varnothing bue + \Delta C_{dev} </math>|| <math>| mit: C_{min,b,L} = 7 mm </math>
| |
| |-
| |
| | ||<math>| mit: \Delta C_{dev} = 10 mm </math>
| |
| |-
| |
| | ||<math>| mit: \varnothing bue = 0 mm </math>
| |
| |-
| |
| | <math> C_{nom,b,L} = 8 mm - 0 mm + \Delta 10 mm </math>||
| |
| |-
| |
| | <math> C_{nom,b,L} = 18 mm </math>||
| |
| |}
| |
|
| |
| :<math>c_{v}=\mathrm{max}\begin{cases}
| |
| 20 mm \\
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| 10 mm \\
| |
| 16 mm
| |
| \end{cases}</math><br /><br />
| |
|
| |
| ::{|
| |
| | <math> d_{1} = c_{v} + \varnothing bue + \frac{\varnothing L}{2}</math>|| <math>| mit: c_{v} = 20 mm </math>
| |
| |-
| |
| | || <math>| mit: \varnothing bue = 0 mm </math>
| |
| |-
| |
| | || <math>| mit: \varnothing L = 8 mm </math>
| |
| |-
| |
| | <math> d_{1} = 20 mm + 0 mm + \frac{8 mm}{2} </math>||
| |
| |-
| |
| | <math> d_{1} = 24,0 mm</math>||
| |
| |}
| |
|
| |
| ::{|
| |
| | <math> d = h_{L} - d_{1} </math>|| <math>| mit: d_{1} = 24 mm </math>
| |
| |-
| |
| | || <math>| mit: h_{L} = 200 mm </math>
| |
| |-
| |
| | <math> d = 200 mm - 24,0 mm </math>||
| |
| |-
| |
| | <math> d = 176 mm = 17,6 cm </math>||
| |
| |}
| |
|
| |
| ====Bemessung mit dem ω-Verfahren m_{x,r}====
| |
|
| |
|
| |
| ::{|
| |
| |<math> z_{s} = d - \frac{h_{L} }{2} </math>|| <math>| mit: d = 17,6 cm </math>
| |
| |-
| |
| | || <math>| mit: h_{L} = 20 cm </math>
| |
| |-
| |
| |<math> z_{s} = 17,6 cm - \frac{20 cm}{2} </math>||
| |
| |-
| |
| |<math> z_{s} = 7,6 cm </math>||
| |
| |}
| |
|
| |
| ::{|
| |
| |<math> M_{Eds} = M_{Ed} - extr n \cdot z_{s} </math>|| <math>| mit: z_{s} = 7,7 cm </math>
| |
| |-
| |
| | || <math>| extr n = 0 </math>
| |
| |-
| |
| | ||<math>| mit: M_{Ed,S} = m_{x,m} = 1664 \frac{kNcm}{m} </math>
| |
| |-
| |
| |<math> M_{Eds} = 1664 \frac{kNcm}{m} - 0 \frac{kN}{m} \cdot 7,7 cm </math>||
| |
| |-
| |
| |<math> M_{Eds} = 1664 \frac{kNcm}{m} </math>||
| |
| |}
| |
|
| |
|
| |
| ::{|
| |
| |<math> \mu_{Eds} = \frac{M_{Eds}}{b\cdot d^{2} \cdot f_{cd}} </math>|| <math>| mit: d = 17,7 cm </math>
| |
| |-
| |
| | || <math>| mit: b = 100 cm </math>
| |
| |-
| |
| | || <math>| mit: f_{cd} = 1,42 \frac{kN}{cm^{2}}</math>
| |
| |-
| |
| | || <math>| mit: M_{Eds} = 1664 \frac{kNcm}{m} </math>
| |
| |-
| |
| |<math> \mu_{Eds} = \frac{1664 \frac{kNcm}{m}}{100 cm \cdot (17,7 cm)^{2} \cdot 1,42 \frac{kN}{cm^{2}} } </math>||
| |
| |-
| |
| |<math> \mu_{Eds} = 0,0378 </math>||
| |
| |}
| |
| <br />
| |
|
| |
| <br />
| |
|
| |
| ::{|
| |
| |<math> \omega = \omega_{1} + \frac{ \omega_{2} - \omega_{1} } { \mu_{Eds,2} - \mu_{Eds,1}} \cdot ( \mu_{Eds} - \mu_{Eds,1} ) </math>||
| |
| <math>| mit: \omega_{1} = 0,0306 </math>
| |
| |-
| |
| | || <math>| mit: \omega_{2} = 0,0410 </math>
| |
| |-
| |
| | || <math>| mit: \mu_{Eds} = 0,0378 </math>
| |
| |-
| |
| | || <math>| mit: \mu_{Eds,1} = 0,03 </math>
| |
| |-
| |
| | || <math>| mit: \mu_{Eds,2} = 0,04 </math>
| |
| |-
| |
| |<math> \omega = 0,0306 + \frac{ 0,0410 - 0,0306 } { 0,04 - 0,03 } \cdot ( 0,0378 - 0,03 ) </math>||
| |
| |-
| |
| |<math> \omega = 0,0387 </math>||
| |
| |}
| |
| <br />
| |
|
| |
| <br />
| |
| ::{|
| |
| |<math> a_{sx,r} = \frac{1}{ \sigma_{sd}} \cdot ( \omega \cdot b \cdot d \cdot f_{cd} + N_{Ed} ) </math>|| <math>| mit: \omega = 0,0306 </math>
| |
| |-
| |
| | || <math>| mit: f_{cd} = 1,42 \frac{kN}{cm^{2}} </math>
| |
| |-
| |
| | || <math>| mit: d = 17,6 cm </math>
| |
| |-
| |
| | || <math>| mit: b = 100 cm </math>
| |
| |-
| |
| | || <math>| mit: N_{Ed} = 0 </math>
| |
| |-
| |
| |<math> a_{sx,r} = \frac{1}{ 43,5 \frac{kN}{cm^{2}}} \cdot ( 0,0387 \cdot 100 cm \cdot 17,6 cm \cdot 1,42 \frac{kN}{cm^{2}} + 0 \frac{kN}{m} ) </math>||
| |
| |-
| |
| |<math> a_{sx,r} = 2,20 \frac{cm^{2}}{m} </math>||
| |
| |}
| |
|
| |
| gewählt: ø8 / 20cm, <math>{{a}_{s}}= 2,51 \frac{cm^{2}}{m} </math>
| |
|
| |
|
| |
| <br />
| |
| ::{|
| |
| |<math> a_{sy,r} = 0,2 \cdot a_{sx,r} </math>|| <math>| mit: a_{sx,r} = 2,20 \frac{cm^{2}}{m} </math>
| |
| |-
| |
| |<math> a_{sy,r} = 0,2 \cdot 2,20 \frac{cm^{2}}{m} </math>||
| |
| |-
| |
| |<math> a_{sy,r} = 0,44 \frac{cm^{2}}{m} </math>||
| |
| |}
| |
|
| |
| gewählt: ø6 / 25cm, <math>a_{sy,r}= 1,13 \frac{cm^{2}}{m} </math>
| |
|
| |
| =Berechnung und Bemessung des Zwischenpodest Hauptpodest=
| |
| Das Hauptpodest wird als zweiseitig gelagert also einachsig gespannt
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| ==''Quellen''==
| |
| :''Normen''
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| <references group="N" /><br />
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| <br />
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| :''Fachliteratur''
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| <references group="F" /><br />
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| <br />
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| :''Links''
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| <references group="L" />
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| <br />
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| {{Seiteninfo
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| |Quality-flag = [[File:quality-flag-white.gif|right|70px]]
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| |Status = Seite in Bearbeitung}}
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| [[Kategorie:Grundlagen/Begriffe-Stahlbetonbau]]-Stahlbetonbau]]
| |