<?xml version="1.0"?>
<ownershipDocument>

    <schemaVersion>X0306</schemaVersion>

    <documentType>4</documentType>

    <periodOfReport>2018-03-06</periodOfReport>

    <notSubjectToSection16>1</notSubjectToSection16>

    <issuer>
        <issuerCik>0000091142</issuerCik>
        <issuerName>SMITH A O CORP</issuerName>
        <issuerTradingSymbol>AOS</issuerTradingSymbol>
    </issuer>

    <reportingOwner>
        <reportingOwnerId>
            <rptOwnerCik>0001464950</rptOwnerCik>
            <rptOwnerName>BROUWER WILFRIDUS M</rptOwnerName>
        </reportingOwnerId>
        <reportingOwnerAddress>
            <rptOwnerStreet1>A. O. SMITH HOLDINGS (BARBADOS) SRL</rptOwnerStreet1>
            <rptOwnerStreet2>8/F AIA KOWLOON TOWER, 100 HOW MING ST.</rptOwnerStreet2>
            <rptOwnerCity>KOWLOON CITY</rptOwnerCity>
            <rptOwnerState>K3</rptOwnerState>
            <rptOwnerZipCode></rptOwnerZipCode>
            <rptOwnerStateDescription>HONG KONG</rptOwnerStateDescription>
        </reportingOwnerAddress>
        <reportingOwnerRelationship>
            <isDirector>0</isDirector>
            <isOfficer>1</isOfficer>
            <isTenPercentOwner>0</isTenPercentOwner>
            <isOther>0</isOther>
            <officerTitle>President - AOSHBSRL</officerTitle>
        </reportingOwnerRelationship>
    </reportingOwner>

    <nonDerivativeTable>
        <nonDerivativeTransaction>
            <securityTitle>
                <value>Common Stock</value>
            </securityTitle>
            <transactionDate>
                <value>2018-03-06</value>
            </transactionDate>
            <transactionCoding>
                <transactionFormType>4</transactionFormType>
                <transactionCode>S</transactionCode>
                <equitySwapInvolved>0</equitySwapInvolved>
            </transactionCoding>
            <transactionAmounts>
                <transactionShares>
                    <value>30000</value>
                </transactionShares>
                <transactionPricePerShare>
                    <value>63.1125</value>
                    <footnoteId id="F1"/>
                </transactionPricePerShare>
                <transactionAcquiredDisposedCode>
                    <value>D</value>
                </transactionAcquiredDisposedCode>
            </transactionAmounts>
            <postTransactionAmounts>
                <sharesOwnedFollowingTransaction>
                    <value>7630</value>
                    <footnoteId id="F2"/>
                </sharesOwnedFollowingTransaction>
            </postTransactionAmounts>
            <ownershipNature>
                <directOrIndirectOwnership>
                    <value>D</value>
                </directOrIndirectOwnership>
                <natureOfOwnership>
                    <value></value>
                </natureOfOwnership>
            </ownershipNature>
        </nonDerivativeTransaction>
    </nonDerivativeTable>

    <footnotes>
        <footnote id="F1">The price in Column 4 is a weighted average price.  The prices actually received ranged from $62.77 to $63.36.  The reporting person has provided to the issuer, and will provide to any security holder of the issuer, or the SEC staff, upon request, information regarding the number of shares sold at each price within the range for all transactions reported on this Form 4 utilizing an average weighted price.</footnote>
        <footnote id="F2">Mr. Brouwer is a participant in the A. O. Smith Dividend Reinvestment Plan ('Plan&quot;) and receives a quarterly dividend pursuant to the Plan.  The total amount of dividends received was 6 shares of Common Stock.</footnote>
    </footnotes>

    <remarks></remarks>

    <ownerSignature>
        <signatureName>James F. Stern, Attorney-in-Fact for Wilfridius M. Brouwer</signatureName>
        <signatureDate>2018-03-07</signatureDate>
    </ownerSignature>
</ownershipDocument>
