Debt Instrument, Covenant, Leverage Ratio, Maximum

No definition available.

Name:	b_DebtInstrumentCovenantLeverageRatioMaximum
Namespace Prefix:	b_
Data Type:	xbrli:pureItemType
Balance Type:	na
Period Type:	instant

- Definition

Number of Banks Transacted With for Interest Rate Swap Agreements

+ References

No definition available.

+ Details

Name:	b_NumberOfBanksTransactedWithForInterestRateSwapAgreements
Namespace Prefix:	b_
Data Type:	xbrli:integerItemType
Balance Type:	na
Period Type:	instant

- Definition

Number of Interest Rate Swaps

+ References

No definition available.

+ Details

Name:	b_NumberOfInterestRateSwaps
Namespace Prefix:	b_
Data Type:	xbrli:integerItemType
Balance Type:	na
Period Type:	instant

- Definition

Amount of hedged item in hedging relationship. Excludes hedged asset or liability.

+ References

No definition available.

+ Details

Name:	us-gaap_DerivativeAmountOfHedgedItem
Namespace Prefix:	us-gaap_
Data Type:	xbrli:monetaryItemType
Balance Type:	na
Period Type:	instant

- Definition

Fixed interest rate related to the interest rate derivative.

+ References

No definition available.

+ Details

Name:	us-gaap_DerivativeFixedInterestRate
Namespace Prefix:	us-gaap_
Data Type:	dtr-types:percentItemType
Balance Type:	na
Period Type:	instant

- Definition

Amount of realized and unrealized gain (loss) of derivative instruments not designated or qualifying as hedging instruments.

+ References

Reference 1: http://fasb.org/us-gaap/role/ref/legacyRef
-Name Accounting Standards Codification
-Topic 815
-SubTopic 10
-Section 50
-Paragraph 4C
-Subparagraph (e)
-Publisher FASB
-URI https://asc.fasb.org//1943274/2147480434/815-10-50-4C

Reference 2: http://www.xbrl.org/2009/role/commonPracticeRef
-Name Accounting Standards Codification
-Section 50
-Paragraph 4CC
-SubTopic 10
-Topic 815
-Publisher FASB
-URI https://asc.fasb.org//1943274/2147480434/815-10-50-4CC

+ Details

Name:	us-gaap_DerivativeInstrumentsNotDesignatedAsHedgingInstrumentsGainLossNet
Namespace Prefix:	us-gaap_
Data Type:	xbrli:monetaryItemType
Balance Type:	credit
Period Type:	duration

- Definition

Line items represent financial concepts included in a table. These concepts are used to disclose reportable information associated with domain members defined in one or many axes to the table.

+ References

Reference 1: http://www.xbrl.org/2003/role/disclosureRef
-Topic 815
-SubTopic 40
-Name Accounting Standards Codification
-Section 50
-Paragraph 5
-Subparagraph (f)
-Publisher FASB
-URI https://asc.fasb.org//1943274/2147480237/815-40-50-5

+ Details

Name:	us-gaap_DerivativeLineItems
Namespace Prefix:	us-gaap_
Data Type:	xbrli:stringItemType
Balance Type:	na
Period Type:	duration

- Definition

Period the derivative contract is outstanding, in 'PnYnMnDTnHnMnS' format, for example, 'P1Y5M13D' represents the reported fact of one year, five months, and thirteen days.

+ References

No definition available.

+ Details

Name:	us-gaap_DerivativeTermOfContract
Namespace Prefix:	us-gaap_
Data Type:	xbrli:durationItemType
Balance Type:	na
Period Type:	duration

- Definition

Fair value portion of other assets.

+ References

No definition available.

+ Details

Name:	us-gaap_OtherAssetsFairValueDisclosure
Namespace Prefix:	us-gaap_
Data Type:	xbrli:monetaryItemType
Balance Type:	debit
Period Type:	instant

- Definition

Amount of cash inflow (outflow) from financing activities classified as other.

+ References

Reference 1: http://www.xbrl.org/2009/role/commonPracticeRef
-Topic 230
-SubTopic 10
-Name Accounting Standards Codification
-Section 45
-Paragraph 14
-Publisher FASB
-URI https://asc.fasb.org//1943274/2147482740/230-10-45-14

Reference 2: http://www.xbrl.org/2009/role/commonPracticeRef
-Name Accounting Standards Codification
-Section 45
-Paragraph 15
-SubTopic 10
-Topic 230
-Publisher FASB
-URI https://asc.fasb.org//1943274/2147482740/230-10-45-15

+ Details

Name:	us-gaap_ProceedsFromPaymentsForOtherFinancingActivities
Namespace Prefix:	us-gaap_
Data Type:	xbrli:monetaryItemType
Balance Type:	debit
Period Type:	duration

- Details

Name:	us-gaap_DerivativeInstrumentRiskAxis=us-gaap_ForeignExchangeContractMember
Namespace Prefix:
Data Type:	na
Balance Type:
Period Type:

- Details

Name:	us-gaap_IncomeStatementLocationAxis=us-gaap_OtherNonoperatingIncomeExpenseMember
Namespace Prefix:
Data Type:	na
Balance Type:
Period Type:

- Details

Name:	us-gaap_DerivativeInstrumentRiskAxis=b_Amended2021SwapMember
Namespace Prefix:
Data Type:	na
Balance Type:
Period Type:

- Details

Name:	us-gaap_DerivativeInstrumentRiskAxis=b_EuriborSwapMember
Namespace Prefix:
Data Type:	na
Balance Type:
Period Type:

- Details

Name:	us-gaap_DerivativeInstrumentRiskAxis=b_A2023SwapMember
Namespace Prefix:
Data Type:	na
Balance Type:
Period Type:

- Details

Name:	us-gaap_HedgingDesignationAxis=us-gaap_DesignatedAsHedgingInstrumentMember
Namespace Prefix:
Data Type:	na
Balance Type:
Period Type:

- Details

Name:	us-gaap_DerivativeInstrumentRiskAxis=us-gaap_InterestRateContractMember
Namespace Prefix:
Data Type:	na
Balance Type:
Period Type:

- Details

Name:	b_DerivativeExpirationPeriodAxis=b_DerivativeExpirationPeriodOneMember
Namespace Prefix:
Data Type:	na
Balance Type:
Period Type:

- Details

Name:	b_DerivativeExpirationPeriodAxis=b_DerivativeExpirationPeriodTwoMember
Namespace Prefix:
Data Type:	na
Balance Type:
Period Type:

- Details

Name:	b_DerivativeExpirationPeriodAxis=b_DerivativeExpirationPeriodThreeMember
Namespace Prefix:
Data Type:	na
Balance Type:
Period Type:

- Details

Name:	b_DerivativeExpirationPeriodAxis=b_DerivativeExpirationPeriodFourMember
Namespace Prefix:
Data Type:	na
Balance Type:
Period Type:

- Details

Name:	b_DerivativeExpirationPeriodAxis=b_DerivativeExpirationPeriodFiveMember
Namespace Prefix:
Data Type:	na
Balance Type:
Period Type:

Derivatives - Narrative (Details) € in Thousands, $ in Thousands	3 Months Ended
	Mar. 31, 2024 USD ($)	Mar. 31, 2023 USD ($)	Dec. 31, 2023 USD ($)	Sep. 12, 2023 USD ($)	Sep. 12, 2023 EUR (€)	Jul. 19, 2023 EUR (€) Bank interestRate	Apr. 30, 2022	Mar. 24, 2021 USD ($)	Apr. 28, 2017 Bank
Derivative [Line Items]
Net cash payments from settlement	$ (1,556)	$ (2,492)
Foreign Exchange Contract
Derivative [Line Items]
Net cash payments from settlement	11,213	(1,353)
Foreign Exchange Contract \| Other Nonoperating Income (Expense)
Derivative [Line Items]
Amount of gain (loss) recognized in income on derivative	(11,903)	$ 1,599
Amended 2021 Swap
Derivative [Line Items]
Asset fair value	(6,183)		$ (5,976)
Euribor Swap
Derivative [Line Items]
Asset fair value	(3,545)		(5,485)
2023 Swap
Derivative [Line Items]
Asset fair value	$ (9,352)		$ (19,984)
Derivatives designated as hedging instruments \| Interest Rate Contract
Derivative [Line Items]
Number of banks transacted with for interest rate swap agreements (in banks) \| Bank						1			1
Derivative amount of hedge					€ 600,000	€ 150,000		$ 100,000
Leverage ratio, maximum				0.04321	0.04321	0.03257
Number of interest rate swap \| interestRate						6
Fixed interest rate							1.075%	1.17%
Term of contract	2 years
Derivatives designated as hedging instruments \| Interest Rate Contract \| Derivative Expiration Period One
Derivative [Line Items]
Derivative amount of hedge				$ 50,000		€ 50,000
Derivatives designated as hedging instruments \| Interest Rate Contract \| Derivative Expiration Period Two
Derivative [Line Items]
Derivative amount of hedge				100		€ 100,000
Derivatives designated as hedging instruments \| Interest Rate Contract \| Derivative Expiration Period Three
Derivative [Line Items]
Derivative amount of hedge				200,000
Derivatives designated as hedging instruments \| Interest Rate Contract \| Derivative Expiration Period Four
Derivative [Line Items]
Derivative amount of hedge				50,000
Derivatives designated as hedging instruments \| Interest Rate Contract \| Derivative Expiration Period Five
Derivative [Line Items]
Derivative amount of hedge				$ 200,000