<PAGE>   1
 
================================================================================
 
                       SECURITIES AND EXCHANGE COMMISSION
                             Washington, D.C. 20549
 
                                   FORM 10-Q
 
         [X]   QUARTERLY REPORT PURSUANT TO SECTION 13 OR 15(D) OF THE
                        SECURITIES EXCHANGE ACT OF 1934
 
                FOR THE QUARTERLY PERIOD ENDED DECEMBER 31, 1997
                                       OR
          [ ]   TRANSITION REPORT PURSUANT TO SECTION 13 OR 15D OF THE
                        SECURITIES EXCHANGE ACT OF 1934
             FROM THE TRANSITION PERIOD FROM             TO
 
                         COMMISSION FILE NUMBER 1-7521
 
                       FRIEDMAN INDUSTRIES, INCORPORATED
             (Exact name of registrant as specified in its charter)
 
<TABLE>
<S>                                            <C>
                    TEXAS                                             74-1504405
       (State or other jurisdiction of                     (I.R.S. Employer Identification
        incorporation or organization)                                 Number)
</TABLE>
 
                 4001 HOMESTEAD ROAD, HOUSTON, TEXAS 77028-5585
                (Address of principal executive office zip code)
 
       Registrant's telephone number, including area code (713) 672-9433
 
--------------------------------------------------------------------------------
Former name, former address and former fiscal year, of changed since last report
 
     Indicate by check mark whether the registrant (1) has filed all reports
required to be filed by Section 13 or 15(d) of the Securities Exchange Act of
1934 during the preceding 12 months, and (2) has been subject to such filing
requirements for the past 90 days.
 
                               Yes  X     No ___
 
     At December 31, 1997, the number of shares outstanding of the issuer's only
class of stock was 6,488,987 shares of Common Stock.
 
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<PAGE>   2
 
                        PART I -- FINANCIAL INFORMATION
 
                       FRIEDMAN INDUSTRIES, INCORPORATED
                    CONSOLIDATED BALANCE SHEETS -- UNAUDITED
 
                                     ASSETS
 
<TABLE>
<CAPTION>
                                                              DECEMBER 31,     MARCH 31,
                                                                  1997           1997
                                                              ------------    -----------
<S>                                                           <C>             <C>
CURRENT ASSETS
  Cash and cash equivalents.................................  $  3,193,609    $   168,245
  Accounts receivable, less allowance for doubtful accounts
     ($7,276 at December 31, 1997 and March 31, 1997,
     respectively)..........................................    12,025,211     11,902,925
  Inventories -- Note B.....................................    21,555,566     21,203,665
  Prepaid expenses and other current assets.................       296,350         82,325
                                                              ------------    -----------
          Total Current Assets..............................    37,070,736     33,357,160
PROPERTY, PLANT AND EQUIPMENT
  Land......................................................       198,021        198,021
  Buildings and improvements................................     2,732,072      2,695,913
  Machinery and equipment...................................    13,440,527     11,724,974
  Less allowance for depreciation...........................   (10,309,629)    (9,909,444)
                                                              ------------    -----------
                                                                 6,060,991      4,709,464
OTHER ASSETS
  Cash value of officers' life insurance....................        74,744         50,567
                                                              ------------    -----------
                                                              $ 43,206,471    $38,117,191
                                                              ============    ===========
 
                          LIABILITIES AND STOCKHOLDERS' EQUITY
 
CURRENT LIABILITIES
  Trade accounts payable and accrued expenses...............  $ 10,499,610    $ 8,112,096
  Current portion of long-term debt.........................       800,000        800,000
  Dividends payable.........................................       486,141        369,715
  Contribution to profit-sharing plan.......................       187,497        242,000
  Federal income taxes payable..............................            --        256,434
  Employee compensation and related expenses................       373,188        392,427
                                                              ------------    -----------
          Total Current Liabilities.........................    12,346,436     10,172,672
LONG-TERM DEBT, less current portion........................     5,733,333      4,600,000
PROVISION FOR NONPENSION RETIREMENT BENEFITS................       113,000        113,000
DEFERRED INCOME TAXES.......................................       404,560        449,560
STOCKHOLDERS' EQUITY
Common stock:
  Par value $1 per share:
     Authorized 10,000,000 shares; Issued and outstanding
       shares -- 6,488,987 at December 31, 1997 and
       6,161,994 at March 31, 1997..........................     6,488,987      6,161,994
  Additional paid-in capital................................    23,678,173     22,377,246
  Retained earnings.........................................    (5,558,018)    (5,757,281)
                                                              ------------    -----------
          Total Stockholders' Equity........................    24,609,142     22,781,959
                                                              ------------    -----------
                                                              $ 43,206,471    $38,117,191
                                                              ============    ===========
</TABLE>
 
                                        1
<PAGE>   3
 
                       FRIEDMAN INDUSTRIES, INCORPORATED
 
                CONSOLIDATED STATEMENT OF EARNINGS -- UNAUDITED
 
<TABLE>
<CAPTION>
                                             THREE MONTHS ENDED           NINE MONTHS ENDED
                                                 DECEMBER 31                 DECEMBER 31
                                          -------------------------   --------------------------
                                             1997          1996           1997          1996
                                          -----------   -----------   ------------   -----------
<S>                                       <C>           <C>           <C>            <C>
Net sales...............................  $34,300,676   $28,468,809   $109,562,478   $86,707,042
Costs and expenses:
  Costs of goods sold...................   31,533,234    26,169,987    100,664,324    79,255,098
  General, selling and administrative
     costs..............................    1,151,863       986,574      3,757,117     3,071,552
  Interest..............................      116,799       125,588        344,533       389,416
                                          -----------   -----------   ------------   -----------
                                           32,801,896    27,282,149    104,765,974    82,716,066
Interest and other income...............      (30,233)      (12,293)       (57,503)      (69,800)
                                          -----------   -----------   ------------   -----------
Earnings before federal income taxes....    1,529,013     1,198,953      4,854,007     4,060,776
Provision (benefit) for federal income
  taxes:
  Current...............................      534,865       420,144      1,695,364     1,418,163
  Deferred..............................      (15,000)      (12,500)       (45,000)      (37,500)
                                          -----------   -----------   ------------   -----------
                                              519,865       407,644      1,650,364     1,380,663
                                          -----------   -----------   ------------   -----------
Net earnings............................  $ 1,009,148   $   791,309   $  3,203,643   $ 2,680,113
                                          ===========   ===========   ============   ===========
 
Weighted average shares outstanding
  Basic.................................    6,483,359     6,436,583      6,476,114     6,436,583
                                          ===========   ===========   ============   ===========
  Diluted...............................    6,625,480     6,436,583      6,609,678     6,436,583
                                          ===========   ===========   ============   ===========
 
Earnings per share -- Note C
  Basic.................................  $      0.16   $      0.12   $       0.49   $      0.42
                                          ===========   ===========   ============   ===========
  Diluted...............................  $      0.15   $      0.12   $       0.48   $      0.42
                                          ===========   ===========   ============   ===========
Cash dividends per common share.........  $     0.075   $      0.06   $       0.22   $      0.16
                                          ===========   ===========   ============   ===========
</TABLE>
 
                                        2
<PAGE>   4
 
                       FRIEDMAN INDUSTRIES, INCORPORATED
 
               CONSOLIDATED STATEMENTS OF CASH FLOWS -- UNAUDITED
 
<TABLE>
<CAPTION>
                                                                     NINE MONTHS
                                                                  ENDED DECEMBER 31
                                                              --------------------------
                                                                 1997           1996
                                                              -----------    -----------
<S>                                                           <C>            <C>
OPERATING ACTIVITIES
  Net earnings..............................................  $ 3,203,643    $ 2,680,113
  Adjustments to reconcile net earnings to cash provided by
     operating activities:
     Depreciation...........................................      506,678        475,028
     Provision for deferred taxes...........................      (45,000)       (37,500)
  Decrease (increase) in operating assets:
     Accounts receivable....................................     (122,286)     1,296,073
     Inventories............................................     (351,901)    (3,246,164)
     Other..................................................     (214,025)       (44,055)
  Increase (decrease) in operating liabilities:
     Accounts payable and accrued expenses..................    2,387,514      1,172,787
     Contribution to profit sharing plan....................      (54,503)       (47,000)
     Employee compensation and related expenses.............      (19,239)        (5,954)
     Federal income taxes...................................     (256,434)       (34,449)
                                                              -----------    -----------
  NET CASH PROVIDED (USED) BY OPERATING ACTIVITIES..........    5,034,447      2,208,879
INVESTING ACTIVITIES
  Purchase of property, plant and equipment.................   (1,858,205)       (84,466)
  Decrease (increase) in cash value of officers' life
     insurance..............................................      (24,177)       (26,338)
                                                              -----------    -----------
  NET CASH PROVIDED (USED) IN INVESTING ACTIVITIES..........   (1,882,382)      (110,804)
FINANCING ACTIVITIES
  Cash dividends paid.......................................   (1,310,801)      (906,534)
  Principal payments on long-term debt......................    2,733,333             --
  Proceeds from borrowings of long-term debt................   (1,600,000)      (600,000)
  Exercise of stock options.................................       50,767         45,297
                                                              -----------    -----------
  NET CASH PROVIDED (USED) IN FINANCING ACTIVITIES..........     (126,701)    (1,461,237)
  INCREASE (DECREASE) IN CASH AND CASH EQUIVALENTS..........    3,025,364        636,838
  Cash and cash equivalents at beginning of period..........      168,245        595,216
                                                              -----------    -----------
  CASH AND CASH EQUIVALENTS AT END OF PERIOD................  $ 3,193,609    $ 1,232,054
                                                              ===========    ===========
</TABLE>
 
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<PAGE>   5
 
                       FRIEDMAN INDUSTRIES, INCORPORATED
 
                     NOTES TO QUARTERLY REPORT -- UNAUDITED
                      THREE MONTHS ENDED DECEMBER 31, 1997
 
NOTE A -- BASIS OF PRESENTATION
 
     The accompanying unaudited condensed, consolidated financial statements
have been prepared in accordance with the instructions to Form 10-Q and do not
include all of the information and footnotes required by generally accepted
accounting principles for complete financial statements. In the opinion of
management, all adjustments (consisting of normal recurring accruals) considered
necessary for a fair presentation have been included. For further information
refer to the financial statements and footnotes included in the Company's annual
report on Form 10-K for the year ended March 31, 1997.
 
NOTE B -- INVENTORIES
 
     Coil inventory consists primarily of raw materials. Tubular inventory is
comprised of both raw materials and finished goods.
 
NOTE C -- EARNINGS PER SHARE
 
     In February 1997, the Financial Accounting Standards Board issued Statement
of Financial Accounting Standards No. 128, Earnings Per Share, which is required
to be adopted for financial statements issued for periods ending after December
31, 1997. The statement replaces primary and fully diluted earnings per share
with basic and diluted earnings per share. The Company implemented this new
standard for the quarter and year to date ended December 31, 1997. This new
standard did not have a significant effect on earnings per share. The difference
between weighted average shares outstanding used for basic and diluted earnings
per share is the effect of stock options. Applicable per share amounts have been
adjusted to give effect to stock dividends.
 
                                        4
<PAGE>   6
 
                       FRIEDMAN INDUSTRIES, INCORPORATED
 
                    MANAGEMENT'S DISCUSSION AND ANALYSIS OF
                 FINANCIAL CONDITION AND RESULTS OF OPERATIONS
 
 NINE MONTHS ENDED DECEMBER 31, 1997 COMPARED TO NINE MONTHS ENDED DECEMBER 31,
                                      1996
 
     During the nine months ended December 31, 1997, sales, costs of goods sold
and gross profit increased by $22,855,436, $21,409,226 and $1,446,210,
respectively, from the comparable amounts recorded during the nine month period
ended December 31, 1996. The increases in sales and costs of goods sold were
related to both coil and tubular operations which produced a combined increase
in tons sold of approximately 26%. The increase in gross profit was primarily
related to tubular operations which reflected both increased sales and margins
earned on sales. Market conditions for tubular goods remained strong during the
1997 period. Gross profit as a percentage of sales declined from 8.6% in the
1996 period to 8.1% in the 1997 period. This decline in margins was related to
the coil operations which were adversely affected by stiff competition within
the hot-rolled coil industry. The Company increased sales of coil products at
the expense of margins during the 1997 period.
 
     General, selling and administrative costs during the 1997 period increased
$685,565 from the amount recorded during the 1996 period. This increase was
primarily associated with variable expenses associated with volume and/or
earnings, to an increase in bad debt expense and to sales expenses related to
additional sales employees.
 
     Interest expense declined $44,883 due primarily to reductions in term debt
during the 1997 period.
 
     Interest and other income declined $12,297. This decline was primarily
related to a transaction involving a gain on the sale of assets in September
1996.
 
     Federal income taxes increased $269,701 period to period due to increased
earnings before taxes. Tax rates were the same for both periods.
 
THREE MONTHS ENDED DECEMBER 31, 1997 COMPARED TO THREE MONTHS ENDED DECEMBER 31,
                                      1996
 
     During the three months ended December 31, 1997, sales, costs of goods sold
and gross profit increased by $5,831,867, $5,363,247 and $468,620, respectively,
from the comparable amounts recorded during the three months ended December 31,
1996. These increases were primarily related to an increase in tons sold of
approximately 20%. Gross profit as a percentage of sales was approximately the
same for both quarters.
 
     General, selling and administrative costs during the 1997 quarter increased
$165,289 from the amount recorded during the 1996 quarter. This increase was
primarily related to increases in selling costs associated with an increase in
sales employees and to increases in variable expenses associated with volume
and/or earnings.
 
     Interest and other income increased $17,940 during the 1997 quarter. This
increase primarily resulted from improved invested cash positions during this
quarter.
 
     Federal income taxes increased $112,221 as a result of the increase in
earnings before taxes. Tax rates were the same for both quarters.
 
FINANCIAL POSITION, LIQUIDITY AND CAPITAL RESOURCES
 
     The Company remained in a strong, liquid position at December 31, 1997.
Current ratios were 3.0 and 3.3 at December 31, 1997 and March 31, 1997,
respectively. Working capital was $24,724,300 at December 31, 1997 and
$23,184,488 at March 31, 1997.
 
     The Company has a credit arrangement with a bank which provides for a
revolving line of credit facility (the "revolving facility") and a term credit
facility (the "term facility"). Pursuant to the revolving facility which expires
April 1, 2000, the Company may borrow up to $8 million at an interest rate no
greater than the
 
                                        5
<PAGE>   7
 
FINANCIAL POSITION, LIQUIDITY AND CAPITAL RESOURCES (CONTINUED)
bank's prime rate. At December 31, 1997, the Company had borrowings outstanding
under the revolving facility of $4 million. The term facility includes
borrowings of $1.2 million from the previous term note and also provides for
additional advances up to $3.5 million, all of which convert to a term loan on
December 31, 1998. The amount outstanding under the term facility bears interest
at a stated rate of LIBOR plus 1.25% and requires quarterly principal payments
of $200,000 plus accrued interest through March 1, 2003. In July 1997, the
Company entered into a swap transaction with the bank pursuant to which it
exchanged the term facility's LIBOR-based interest rate obligation for a fixed
interest rate obligation of 8% to remain in effect for the entire term of the
term facility. As of December 31, 1997, the principal amount of indebtedness
outstanding under the term facility was $2,533,333.
 
                                        6
<PAGE>   8
 
                       FRIEDMAN INDUSTRIES, INCORPORATED
 
                        QUARTER ENDED DECEMBER 31, 1997
 
                          PART II -- OTHER INFORMATION
 
ITEM 1. LEGAL PROCEEDINGS
 
       Not applicable
 
ITEM 2. CHANGES IN SECURITIES
 
     a). Not applicable
 
     b). Not applicable
 
     c). Not applicable
 
     d). Not applicable
 
ITEM 3. DEFAULTS UPON SENIOR SECURITIES
 
     a). Not applicable
 
     b). Not applicable
 
ITEM 4. SUBMISSION OF MATTERS TO A VOTE OF SECURITY HOLDERS
 
       None
 
ITEM 5. OTHER INFORMATION
 
       Not applicable
 
ITEM 6. EXHIBITS AND REPORTS ON FORM 8-K
 
     a). Exhibits
         27 -- Financial Data Schedule
 
     b). Reports on Form 8-K
         None
 
                                        7
<PAGE>   9
 
                                   SIGNATURES
 
     Pursuant to the requirements of the Securities Exchange Act of 1934, the
registrant has duly caused this report to be signed on its behalf by the
undersigned thereunto duly authorized.
 
                                            FRIEDMAN INDUSTRIES, INCORPORATED
 
Date November 13, 1997                      By        /s/ BEN HARPER
                                             -----------------------------------
                                                   Ben Harper, Senior Vice
                                                     President -- Finance
                                                 (Chief Accounting Officer)
 
Date November 13, 1997                      By     /s/ HAROLD FRIEDMAN
                                             -----------------------------------
                                               Harold Friedman, Vice Chairman
<PAGE>   10


                   Independent Accountants' Review Report

Board of Directors
Friedman Industries, Incorporated

We have reviewed the accompanying condensed consolidated balance sheet of
Friedman Industries, Incorporated, as of December 31, 1997, the related
consolidated statements of earnings for the three and nine month periods ended
December 31, 1997 and 1996 and the consolidated statements of cash flows for the
nine month periods ended December 31, 1997 and 1996. These financial statements
are the responsibility of the Company's management.

We conducted our reviews in accordance with standards established by the
American Institute of Certified Public Accountants. A review of interim
financial information consists principally of applying analytical procedures to
financial data and making inquiries of persons responsible for financial and
accounting matters. It is substantially less in scope than an audit conducted in
accordance with generally accepted auditing standards, which will be performed
for the full year with the objective of expressing an opinion regarding the
financial statements taken as a whole. Accordingly, we do not express such an
opinion.

Based on our reviews, we are not aware of any material modifications that should
be made to the accompanying condensed consolidated financial statements referred
to above for them to be in conformity with generally accepted accounting
principles.

We have previously audited, in accordance with generally accepted auditing
standards, the consolidated balance sheet of Friedman Industries, Incorporated,
as of March 31, 1997, and the related consolidated statements of earnings,
stockholders' equity and cash flows for the year then ended (not presented
herein) and in our report dated May 30, 1997, we expressed an unqualified
opinion on those consolidated financial statements. In our opinion, the
information set forth in the accompanying condensed consolidated balance sheet
as of March 31, 1997, is fairly stated, in all material respects, in relation to
the consolidated balance sheet from which it has been derived.



                           /s/ Ernst & Young LLP



Houston, Texas
February 13, 1997


<PAGE>   11
 
                               INDEX TO EXHIBITS
 
<TABLE>
<CAPTION>
        EXHIBIT
          NO.                                      EXHIBIT
        -------                                    -------
<C>                      <S>                                                          <C>
          27             -- Financial Data Schedule
</TABLE>