M+

x

z

0,5 m 0,5 m

F = 10 kN

45°

Aufgabe 4

A

B

H.

SCHOLZ-O1FV1a-10.01.06

FAX

FAZF = 10 kN

FB

FBX

FBZ

45°

0,5 m 0,5 m

Freischneiden:

A

B

M+

x

z

H.

SCHOLZ-O1FV1a-10.01.06

Gleichgewichtsbedingungen

I. Σ FX = 0 = FAX - FBX

II. Σ FZ = 0 = FAZ – F + FBZ

III. Σ M(A) = 0 = 10kN * 0,5m – FBZ * 1m

[FBX kann kein Moment erzeugen]

FAX

FAZF = 10 kN

FB

FBX

FBZ

45°

0,5 m

0,5 m

B

A

M+

x

z

H.

SCHOLZ-O1FV1a-10.01.06

NR für FB

sin α = a / c ; cos α = b / c ; sin 45° = cos 45° = √2 / 2 = 0,7071

a = b = √2 / 2 * c

→ FBX = FBZ = √2 / 2 *FB

FAX

FAZF = 10 kN

FB

FBX

FBZ

45°

0,5 m

0,5 m

A

B

I. Σ FX = 0 = FAX - FBX

II. Σ FZ = 0 = FAZ – F + FBZ

III. Σ M(A) = 0 = 10kN * 0,5m – FBZ * 1m

H.

SCHOLZ-O1FV1a-10.01.06

I. FAX = FBX

II. FAZ = 10 kN - √2 / 2 * FB

FAX = √2 / 2 * FB

III. 10kN * 0,5m = √2 / 2 * FB * 1m

5kNm = √2 / 2 * m * FB

5kN = √2 / 2 * FB

FB = 5 * 2 / √2 kN = 10 / √2 kN = 7,071 kN

FAX

FAZF = 10 kN

FB

FBX

FBZ

45°

0,5 m

0,5 m

B

AI. Σ FX = 0 = FAX - FBX

II. Σ FZ = 0 = FAZ – F + FBZ

III. Σ M(A) = 0 = 10kN * 0,5m – FBZ * 1m

H.

SCHOLZ-O1FV1a-10.01.06

FAZ = 10 kN - 5 kN

FAZ = 5 kN

III. → I. FAX = (√2 / 2) * (10 / √2) kN

FAX = 5 kN

I. Σ FX = 0 = FAX - FBX

II. Σ FZ = 0 = FAZ – F + FBZ

III. Σ M(A) = 0 = 10kN * 0,5m – FBZ * 1m

FAX

FAZF = 10 kN

FB

FBX

FBZ

45°

0,5 m

0,5 m

A

B

H.

SCHOLZ-O1FV1a-10.01.06

III. → II. FAZ = 10 kN - √2 / 2 * FB

FAX = 5 kN

FAZ = 5 kNF = 10 kN

FB = 7,071 kN

FBX

FBZ

45°

0,5 m 0,5 m

A

B

M+

x

z

H.

SCHOLZ-O1FV1a-10.01.06

I. FAX = FBX

II. FAZ = 10 kN - √2 / 2 * FB

FAX = √2 / 2 * FB

III. 10kN * 0,5m = √2 / 2 FB * 1m

5kNm = √2 / 2 * m * FB

5kN = √2 / 2 * FB

FB = 5 * 2 / √2 kN = 10 / √2 kN = 7,071 kN

I. Σ FX = 0 = FAX - FBX

II. Σ FZ = 0 = FAZ – F + FBZ

III. Σ M(A) = 0 = 10kN * 0,5m – FBZ * 1m

III. → II. FAZ = 10 kN – 5 kN

FAZ = 5 kN

III. → I. FAX = (√2 / 2) * (10 / √2) kN

FAX = 5 kN

NR für FB

sin α = a / ccos α = b / csin 45° = cos 45° = √2 / 2 = 0,7071a = b = √2 / 2 * c

→ FBX = FBZ = √2 / 2 * FB

M+

x

z

Zusammenstellung:

FAZ = 5 kNF = 10 kN

FB = 7,071 kN

FBX

FBZ

45°

0,5 m 0,5 m

A B

FAX = 5 kN

H.

SCHOLZ-O1FV1a-10.01.06